Research Article Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity - Hindawi

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013 Article ID 253985 15 pageshttpdxdoiorg1011552013253985

Research ArticleEnergy Conditions in a Generalized Second-OrderScalar-Tensor Gravity

M Sharif and Saira Waheed

Department of Mathematics University of the Punjab Quaid-e-Azam Campus Lahore 54590 Pakistan

Correspondence should be addressed to M Sharif msharifmathpuedupk

Received 27 November 2012 Accepted 27 January 2013

Academic Editor Piero Nicolini

Copyright copy 2013 M Sharif and S Waheed This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The study of energy conditions has many significant applications in general relativistic and cosmological contexts This paperexplores the energy conditions in the framework of the most general scalar-tensor theory with field equations involving second-order derivatives For this purpose we use flat FRW universe model with perfect fluid matter contents By taking power lawansatz for scalar field we discuss the strong weak null and dominant energy conditions in terms of deceleration jerk and snapparameters Some particular cases of this theory like 119896-essence model modified gravity theories and so forth are analyzed with thehelp of the derived energy conditions and the possible constraints on the free parameters of the presented models are determined

1 Introduction

ldquoThe increasing rate of cosmic expansion in current phaserdquois one of the primal facts in modern cosmology that issupported by some sorts of energy with negative pressureas well as hidden characteristics refereed to dark energy(DE) (its existence is affirmed by the recent data of manyastronomical observations) [1ndash4] The investigation of thishidden unusual nature of DE has been carried out in twoways one approach utilizes the modified matter sourcesthat is different models like Chaplygin gas [5] quintessence[6ndash9] 119896-essence [10ndash12] cosmological constant [13 14] andso forth which are introduced in the usual matter contentswithin the gravitational framework of general relativity (GR)while in second approach GR framework is modified by theinclusion of some extradegrees of freedom [15 16] Examplesof some well-known modified gravity theories include 119891(119877)gravity [17 18] scalar-tensor theories like Brans-Dicke (BD)gravity [19 20] Gauss-Bonnet gravity [21 22] 119891(119879) theory[23 24] and119891(119877 119879) gravity [25]Modifiedmatter sources arerather interesting but each faces some difficulties and hencecould not prove to be very promising Modified gravitationaltheories being large-distance modifications of gravity havebrought a fresh insight in modern cosmology Among thesescalar-tensor theories are considered to be admirable efforts

for the investigation ofDE characteristics which are obtainedby adding an extra scalar degree of freedom in Einstein-Hilbert action

Scalar field provides a basis for many standard infla-tionary models leading to an effective candidate of DEIn literature [26ndash28] many inflationary models have beenconstructed like chaotic inflation smalllarge inflation 119896-inflation Dirac-Born-Infeld (DBI) inflation single field 119896-inflation and so forth All of these are some peculiarextensions of the 119896-essence models Although the 119896-essencescalar models are considered to be the general scalar fieldtheories described by the Lagrangian in terms of first-orderscalar field derivatives that is 119871 = 119871(120601 nabla120601) HoweverLagrangian with higher-order scalar field derivatives (119871 =

119871(120601 nabla120601 nablanabla120601) can be taken into account which fixes theequations of motion (obtained by metric and scalar fieldvariations of the Lagrangian density) to second-order [2930] Horndeski [29] was the pioneer to discuss the conceptof most general Lagrangian with single scalar field Recentlythis action is discussed by introducing a covariant Galileanfield with second-order equations of motion [30] Kobayashiet al [31] developed a correspondence between theseLagrangians This theory has fascinated many researchersand much work has been done in this context for example[31ndash35]

2 Advances in High Energy Physics

The energy conditions have many significant theoreticalapplications likeHawking Penrose singularity conjecture thatis based on the strong energy condition [36] while thedominant energy condition is useful in the proof of positivemass theorem [37] Furthermore null energy condition is abasic ingredient in the derivation of second law of black holethermodynamics [38] On the cosmological grounds Visser[39] discussed various cosmological terms like distancemod-ulus look back time deceleration and statefinder parametersin terms of red shift using energy condition constraintsTheseconditions are originally formulated in the context of GR andthen extended to modified theories of gravity Many authorshave explored these energy conditions in the framework ofmodified gravity and found interesting results [40ndash43]

Basically modified gravity theories contain some extra-functions like higher-order derivatives of curvature term orsome functions of Einstein tensor or scalar field and so forthThus it is a point of debate that how one can constrain theadded extra degrees of freedom consistently with the recentobservationsThe energy conditions can be used to put someconstraints on these functions that could be consistent withthose already found in the cosmological arena Recently theseenergy conditions have been discussed in 119891(119879) [44] and119891(119877 119879) [45] theories

In this paper we study the energy condition bounds ina most general scalar-tensor theory The paper is designedin the following layout Next section defines the energyconditions inGR aswell as in a generalmodified gravitationalframework Section 3 provides basic formulation of the mostgeneral scalar-tensor theory In the same section we formu-late the energy conditions in terms of some cosmologicalparameters within such modified framework In Section 4we provide some specific cases of this theory and discussthe corresponding constraints Finally we summarize andpresent some general remarks

2 Energy Conditions

In this section we discuss the energy conditions in GRframework and then express the respective conditions in ageneral modified gravity In GR the energy conditions comefrom a well-known purely geometric relationship known asRaychaudhuri equation [38 46] together with the lineamentof gravitational attractiveness In a spacetime manifold withvector fields 119906120583 and 119896120583 as tangent vectors to timelike andnull-like geodesics of the congruence the temporal variationof expansion for the respective curves is described by theRaychaudhuri equation as

119889120579

119889120591= minus

1

31205792minus 120590120583120584120590

120583120584+ 120596120583120584120596

120583120584minus 119877120583120584119906

120583119906120584

119889120579

119889120591= minus

1

21205792minus 120590120583120584120590

120583120584+ 120596120583120584120596

120583120584minus 119877120583120584119896

120583119896120584

(1)

Here 119877120583120584 120579 120590120583120584 and 120596

120583120584 represent the Ricci tensorexpansion shear and rotation respectively related with thecongruence of timelike or null-like geodesics

The characteristic of the gravity that is attractive leads tothe condition 119889120579119889120591 lt 0 For infinitesimal distortions and

vanishing shear tensor 120596120583120584 = 0 that is zero rotation (forany hypersurface of orthogonal congruence) we ignore thesecond-order terms in Raychaudhuri equation and conse-quently integration leads to 120579 = minus120591119877120583120584119906

120583119906120584= minus120591119877120583120584119896

120583119896120584

It further implies that

119877120583120584119906120583119906120584ge 0 119877120583120584119896

120583119896120584ge 0 (2)

Since GR and its modifications lead to a relationship ofthe matter contents that is energy-momentum tensor interms of Ricci tensor through the field equations thereforethe respective physical conditions on the energy-momentumtensor can be determined as follows

119877120583120584119906120583119906120584= (119879120583120584 minus

119879

2119892120583120584) 119906

120583119906120584ge 0

119877120583120584119896120583119896120584= (119879120583120584 minus

119879

2119892120583120584) 119896

120583119896120584ge 0

(3)

where 119879120583120584 and 119879 are the energy-momentum tensor andits trace respectively For perfect fluid with density 120588 andpressure 119901 defined by

119879120583120584 = (120588 + 119901) 119906120583119906120584 minus 119901119892120583120584 (4)

the strong and null energy conditions respectively aredefined by the inequalities 120588 + 3119901 ge 0 and 120588 + 119901 ge 0while the weak and dominant energy conditions are definedrespectively by 120588 ge 0 and 120588 plusmn 119901 ge 0

Raychaudhuri equation being a geometrical statementworks for all gravitational theories Therefore its interestingfeatures like focussing of geodesic congruences as well as theattractiveness of gravity can be used to derive the energyconstraints in the context of modified gravity In case ofmodified gravity we assume that the total matter contentsof the universe act like perfect fluid and consequently theseconditions can be defined in terms of effective energy densityand pressure (matter sources getmodified andwe replace119879120583120584and 119879 in (3) by 119879eff

120583120584and 119879eff resp) These conditions can be

regarded as an extension of the respective conditions in GRgiven by [43]

NEC 120588eff + 119901eff ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0

WEC 120588eff ge 0 120588eff+ 119901

effge 0

DEC 120588eff ge 0 120588effplusmn 119901

effge 0

(5)

For a detailed discussion we suggest the readers to study arecent paper [47]

TheDE requires negative EoS parameter120596 le minus13 for theexplanation of cosmic expansion Indeed for cosmologicalpurposes we are curious for a source with 120588 ge 0 inthat case all of the energy conditions require 120596 ge minus1

[48] The role of possible DE candidates with 120596 lt minus1

was pointed out by Caldwell who referred to null DECviolating sources as phantom components It is argued thatDE models with 119908 ge minus1 such as the cosmological constant

Advances in High Energy Physics 3

and the quintessence satisfy the NEC but the models with120596 lt minus1 (predicted for instance by the phantom theory)where the kinetic term of the scalar field has a wrong(negative) sign does not satisfy However quintom modelscan also satisfy NEC as they yield the phantom era for avery short period of time [49] Usually the discussions onenergy conditions for cosmological constant are available inliterature by introducing it in some other type of matter likeelectromagnetic field [50] The cosmological constant willtrivially satisfy all these energy conditions except SEC

3 Energy Conditions in the Most GeneralScalar-Tensor Gravity

The most general scalar-tensor theory in 4 dimensions isgiven by the action [31ndash35]

119878 = int1198894119909radicminus119892

times [119870 (120601119883) minus 1198663 (120601 119883) ◻120601 + 1198664 (120601119883) 119877

+ 1198664119883 (◻120601)2minus (nabla120583nabla120584120601) (nabla

120583nabla120584120601)

+ 1198665 (120601 119883)119866120583120584 (nabla120583nabla120584120601) minus

1

61198665119883

times (◻120601)3minus 3 (◻120601) (nabla120583nabla120584120601) (nabla

120583nabla120584120601)

+2 (nabla120583nabla120572120601) (nabla

120572nabla120573120601) (nabla

120573nabla120583120601) + 119871119898]

(6)

where 120601 is the scalar field 119892 is the determinant of themetric tensor 119871119898 denotes the matter part of the Lagrangianand 119883 represents the kinetic energy term defined by 119883 =

minus(12)120597120583120601120597120583120601 Moreover 119866120583120584 is the Einstein tensor 119877 is the

Ricci scalar nabla120583 is the covariant derivative operator and ◻ =nabla120583nabla120583 is the dersquoAlembertian operator The functions 119870(120601119883)

and 119866119894(120601 119883) 119894 = 3 4 5 are all arbitrary functions and 119866119894119883 =120597119866119894120597119883 In this action the term 1198663(120601 119883)◻120601 is the Galileanterm 1198664(120601 119883)119877 can yield the Einstein-Hilbert term and1198665(120601 119883) leads to the interaction with Gauss-Bonnet termThis indicates that it covers not only several DE proposalslike 119896-essence 119891(119877) gravity BD theory and Galilean gravitymodels but it also contains 4-dimensional Dvali Gabadadzeand Poratti (DGP) model (modified) the field coupling withGauss-Bonnet term and the field derivative coupling withEinstein tensor as its particular cases

By varying the action (6)with respect to themetric tensorthe gravitational field equation can be written as

119866120583120584 =1

1198664

Θeff120583120584=1

1198664

[119879119898

120583120584+ 119879120601

120583120584] (7)

where Θeff120583120584

is the modified energy-momentum tensor 119879119898120583120584

isthe source of usual matter field that can be described by theperfect fluid while 119879120601

120583120584provides the matter source due to

scalar field and hence yields the source of DE defined in theAppendixThe scalarwave equation for suchmodified gravityhas been described in the literature [32ndash35]

By inverting (7) the Ricci tensor can be expressed interms of effective energy-momentum tensor and its trace asfollows

119877120583120584 = 119879eff120583120584minus1

2119892120583120584119879

eff (8)

where the effective energy-momentum tensor 119879eff120583120584

and itstrace 119879eff are

119879eff120583120584= 119879120583120584 +

1

2119870119883nabla120583120601nabla120584120601 minus

1

21198663119883◻120601nabla120584120601nabla120583120601

minus nabla(1205831198663nabla120584)120601 +1

21198664119883 times 119877nabla120583120601nabla120584120601 +

1

21198664119883119883

times [(◻120601)2minus (nabla120572nabla120573120601)

2

] nabla120583120601nabla120584120601

+ 1198664119883◻120601 times nabla120583nabla120584120601 minus 1198664119883nabla120582nabla120583120601nabla120582nabla120584120601

minus 2nabla1205821198664119883nabla120582nabla(120583120601nabla120584)120601 + nabla1205821198664119883 times nabla

120582120601nabla120583nabla120584120601

minus 2 [1198664119883119877120582(120583nabla120584)120601nabla120582120601 minus nabla(1205831198664119883nabla120584)120601◻120601]

minus 1198664119883 times 119877120583120572120584120573nabla120572120601nabla120573120601 + 1198664120601nabla120583nabla120584120601

+ 1198664120601120601nabla120583120601nabla120584120601 minus 21198664119883120601nabla120582120601 times nabla120582nabla(120583120601nabla120584)120601

+ 1198664119883119883nabla120572120601nabla120572nabla120583120601nabla

120573120601nabla120573nabla120584120601

minus 1198665119883119877120572120573nabla120572120601 times nabla120573nabla(120583120601nabla120584)120601

+ 1198665119883119877120572(120583nabla120584)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573120601

times nabla120583nabla120584120601 +1

21198665119883119877120583120572120584120573nabla

120572120601nabla120573120601◻120601

minus 1198665119883119877120572120582120573(120583nabla120584)120601nabla120582120601nabla120572nabla120573120601

minus 1198665119883119877120572120582120573(120583nabla120584)nabla120582120601nabla120572120601nabla120573120601

+1

2nabla(120583 [1198665119883nabla

120572120601] nabla120572nabla120584)120601◻120601 minus

1

2

times nabla(120583 [1198665120601nabla120584)120601] ◻120601 + nabla120582 [1198665120601nabla(120583120601]nabla120584)nabla120582120601

minus1

2[nabla120582 (1198665120601nabla

120582120601) minus nabla120572 (1198665119883nabla120573120601)nabla

120572nabla120573120601]

times nabla120583nabla120584120601 minus nabla1205721198665nabla120573120601119877120572(120583120584)120573

+ nabla(1205831198665119877120584)120582 times nabla120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601

minus nabla120572 [1198665119883nabla120573120601]nabla120572nabla(120583120601nabla

120573nabla120584)120601 + nabla1205731198665119883

times [◻120601nabla120573nabla(120583120601 minus nabla

120572nabla120573120601nabla120572 times nabla(120583120601]nabla120584)120601

minus1

2nabla120572120601nabla1205721198665119883 [◻120601nabla120583nabla120584120601 minus nabla120573nabla120583120601nabla120573nabla120584120601]


+1

21198665119883 times 119866120572120573nabla

120572nabla120573120601nabla120583120601nabla120584120601 +

1

21198665119883◻120601

times nabla120572nabla120583120601nabla120572nabla120584120601 minus

1

21198665119883(◻120601)

2times nabla120583nabla120584120601

minus1

121198665119883119883 [(◻120601)

3minus 3 (◻120601) (nabla120572nabla120573120601)

2

+ 2(nabla120572nabla120573120601)3

]

times nabla120583120601 times nabla120584120601 minus1

2nabla1205821198665119877120583120584nabla

120582120601

(9)

119879eff= 119870 + nabla1205821198663nabla

120582120601 minus 2 (1198664120601◻120601 minus 21198831198664120601120601)

minus 2 minus 21198664119883120601nabla120572nabla120573120601nabla120572120601nabla120573120601 + 1198664119883119883nabla120572nabla120582120601

timesnabla120573nabla120582120601nabla120572120601nabla120573120601 +

1

21198664119883 [(◻120601)

2minus (nabla120572nabla120573120601)

2

]

+ 2 [1198664119883 times 119877120572120573nabla120572120601nabla120573120601 minus nabla1205821198664119883nabla

120582120601◻120601]

minus1

2119877nabla1205821198665nabla

120582120601

minus 2 minus1

61198665119883 [(◻120601)

3minus 3◻120601(nabla120572nabla120573120601)

2

+2(nabla120572nabla120573120601)3

]

+ nabla1205721198665119877120572120573nabla120573120601 minus

1

2nabla1205721198665120601nabla

120572120601◻120601 +

1

2nabla1205721198665120601

times nabla120573120601nabla120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601 +

1

2nabla1205721198665119883


1

4nabla1205821198665119883nabla120582120601 [(◻120601)

2minus(nabla120572nabla120573120601)

2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883 times 119877120572120582120573120588nabla

120572120601

times nabla120573120601nabla120582120601nabla120588120601 + 119879 +

1

2119870119883(nabla120583120601)

2

minus1

21198663119883◻120601(nabla120583120601)

2

minus nabla1205831198663nabla120583120601 +1

21198664119883119877(nabla120583120601)

2

+1

21198664119883119883 [(◻120601)


2

] (nabla120583120601)2

+ 1198664119883◻120601nabla120583nabla120583120601 minus 1198664119883nabla120582nabla120583120601nabla120582nabla120583120601

minus 2nabla1205821198664119883nabla120582(nabla120601)2+ nabla1205821198664119883 times nabla

120582120601nabla120583nabla120583120601

minus 2 [1198664119883119877120582120583nabla120583120601nabla120582120601 minus nabla1205831198664119883nabla120583120601◻120601]

minus 1198664119883119877120583120572120583120573nabla120572120601nabla120573120601 + 1198664119883nabla120583nabla120583120601 + 1198664120601120601(nabla120583120601)

2

minus 21198664119883120601nabla120582120601 times nabla120582(nabla120583120601)

2

+ 1198664119883119883nabla120572120601nabla120572

times nabla120583120601nabla120573120601nabla120573nabla120583120601 minus 1198665119883119877120572120573nabla

120572120601nabla120573(nabla120583120601)

2

+ 1198665119883119877120572(120583nabla120583)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573

times 120601nabla120583nabla120583120601 +1

21198665119883 times 119877120583120572120583120573nabla

120572120601nabla120573120601◻120601

minus 1198665119883119877120572120582120573120583nabla120583120601nabla120582120601nabla120572nabla120573120601 minus 1198665119883119877120572120582120573120583

times nabla120583nabla120582120601nabla120572120601nabla120573120601 +

1

2nabla(120583 [1198665119883nabla

120572120601] nabla120572nabla120583)120601◻120601

minus1

2nabla(120583 [1198665120601nabla120583)120601] ◻120601 + nabla120582 [1198665120601nabla(120583120601]nabla120583)nabla

120582120601

minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]

times nabla120583nabla120583120601 minus nabla1205721198665nabla120573120601119877120572120583120583120573 + nabla1205831198665119877120583120582nabla

120582120601

minus1

2nabla1205831198665119883 times nabla120583120601 [(◻120601)


2

]

+ nabla1205821198665119877120582120583nabla120583120601 minus nabla120572 [1198665119883nabla120573120601]nabla

120572nabla120583120601 times nabla

120573nabla120583120601

+ nabla1205731198665119883 [◻120601nabla120573nabla(120583120601 minus nabla

120572nabla120573120601nabla120572nabla(120583120601]nabla120583)120601

minus1

2nabla120572120601 times nabla1205721198665119883 [◻120601nabla120583nabla120583120601 minus nabla120573nabla120583120601nabla120573nabla120583120601]

+1

21198665119883119866120572120573nabla

120572nabla120573120601(nabla120583120601)

2

+1

21198665119883◻120601nabla120572nabla120583120601nabla

120572nabla120583120601

minus1

21198665119883(◻120601)


1

121198665119883119883

times [(◻120601)3minus 3 (◻120601) (nabla120572nabla120573120601)

2

+2(nabla120572nabla120573120601)3

] (nabla120583120601)2

minus1

2nabla1205821198665119877nabla

120582120601

+1

2nabla1205721198665119883nabla120573119883nabla

120572nabla120573120601

minus1

4nabla1205721198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883119877120572120582120573120588 (10)

Evaluating the temporal and spatial components of theeffective energy-momentum tensor and its trace definedpreviously and using these values in (3) we can find theenergy conditions for any spacetime

Let us consider the spatially homogeneous isotropic andflat FRW universe model with 119886(119905) as a scale factor describedby the metric

1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (11)

The background fluid is taken as perfect fluid given by (4)with 119906120583 = (1 0 0 0) and the null like vector is taken as119896120583 = (1 119886 0 0) Furthermore we assume that the scalar fieldis a function of time only The Friedmann equations for thegeneralized scalar-tensor theory in terms of effective energydensity and pressure are given by [32ndash35]

31198672=120588eff

1198664

minus (31198672+ 2) =

119901eff

1198664

(12)


where

120588eff=1

2[120588119898+ 2119883119870119883 minus 119870 + 6119883

1206011198671198663119883 minus 21198831198663120601 + 241198672119883

times (1198664119883 + 1198831198664119883119883) minus 121198671198831206011198664119883120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)]

(13)

119901eff=1

2[119901119898+ 119870 minus 2119883 (1198663120601 +

1206011198663119883) minus 1211986721198831198664119883

minus 41198671198664119883 minus 81198831198664119883 minus 81198671198831198664119883119883

+ 2 ( 120601 + 2119867 120601)1198664120601 + 41198831198664120601120601 + 4119883 (120601 minus 2119867 120601)

times 1198664119883120601 minus 2119883 (21198673 120601 + 2119867 120601 + 3119867

2 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 4119867119883( minus 119867119883)1198665119883120601

+2 [2 (119867119883)+ 31198672119883]1198665120601 + 4119867119883

1206011198665120601120601]

(14)

Here 1198664 being an arbitrary function of 120601 and 119883 acts as adynamical gravitational constant and it should be positivefor any gravitational theory Furthermore 120588119898 and 119901119898 aredensity and pressure respectively for ordinary matter Weshall discuss its different forms in the next section Usingthese values in (5) it can be checked that theNECWEC SECand DEC require the following conditions to be satisfied

NEC 120588eff + 119901eff ge 0 997904rArr

1

21198664

[(120588119898+ 119901119898) + 2119883119870119883 + 6119883

1206011198671198663119883 minus 41198831198663120601

+ 1211986721198831198664119883 + 24119867

211988321198664119883119883 minus 20119883119867

1206011198664119883120601

minus 2119867 1206011198664120601 + 61198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 16119867211988321198665119883120601 minus 12119867

21198831198665120601 minus 2119883

1206011198663119883

minus 41198671198664119883 minus 8119883 times 1198664119883 minus 81198671198831198664119883119883 + 21206011198664120601

+ 4119883119866120601120601 + 41198831206011198664119883120601 minus 2119883 (2119867

120601 + 31198672 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 41198671198831198665119883120601

+4 (119867119883)1198665120601 + 4119867119883

1206011198665120601120601] ge 0

WEC 120588eff + 119901eff ge 0 120588effge 0 997904rArr

1

21198664

[120588119898+ 2119883119870119883 minus 119870 + 6119883119867

1206011198663119883 minus 21198831198663120601

+ 241198672119883 times (1198664119883 + 1198831198664119883119883) minus 12119883119867

120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)] ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0 997904rArr

1

21198664

[(120588119898+ 3119901119898) + 2119883119870119883 + 2119870 + 6119883119867

1206011198663119883


21198831198664119883 + 24119867

21198832

times 1198664119883119883 minus 361198831198671206011198664119883120601 + 6119867

1206011198664120601 + 61206011198664120601

minus 21198673119883 1206011198665119883 + 4119867

31198832 1206011198665119883119883 minus 24119867

211988321198665119883120601

+ 121198671198831198665119883120601 + 12 (119867119883)1198665120601

+ 12119867119883 120601119866120601120601 minus 1211986721198832 1206011198665119883119883 minus 12119883119867

1206011198665119883

minus 181198831198672 1206011198665119883 + 12119883

1206011198664119883120601 minus 241198671198831206011198664119883120601

+ 121198831198664120601120601 minus 24119867119883 times 1198664119883119883

minus241198831198664119883 minus 121198671198664119883 minus 3611986721198831198664119883] ge 0

DEC 120588eff ge 0 120588eff+ 119901

effge 0 120588

effminus 119901

effge 0 997904rArr

1

21198664

[(120588119898minus 119901119898) + 2119883119870119883 minus 2119870 + 6119883119867

1206011198663119883 + 21198831206011198663119883

+ 3611986721198831198664119883 + 24119867

21198831198664119883119883 minus 4119883119867

1206011198664119883120601

minus 10119867 1206011198664120601 + 141198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 2411986721198831198665120601 minus 4 (119867119883)

1198665120601 minus 4119867119883

120601119866120601120601

minus 8119867211988321198665119883120601 + 41198671198664119883

+ 81198831198664119883 + 81198671198831198664119883119883 minus 21206011198664120601 minus 41198831198664120601120601

minus 4119883 1206011198664119883120601 + 2119883 (2119867120601 + 3119867

2 120601) 1198665119883

+4119867211988321198665119883119883

120601 minus 41198671198831198665119883120601] ge 0

(15)

In a mechanical framework the terms velocity accel-eration jerk and snap parameters are based on the firstfour time derivatives of position In cosmology the Hubble


deceleration jerk and snap parameters are respectivelydefined as

119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)

In order to have a more precise picture of these conditions(15) we use the relations of time derivatives ofHubble param-eter in terms of cosmological quantities like decelerationsnap and jerk parameters as

= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)

= 1198674(119904 minus 2119895 minus 5119902 minus 3)

(17)

Moreover we assume that the scalar field evolves as a powerof scale factor that is 120601(119905) sim 119886(119905)120573 [51 52] which leads to

120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)

Here 120573 is a nonzero parameter (120573 = 0 yields constant scalarfield 120573 gt 0 yields expanding scalar field and 120573 lt 0

corresponds to contracting scalar field) Clearly 120601 remains asa positive quantity

Introducing these quantities in the energy conditionsgiven by (15) it follows that

1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883

minus 120573311986741198863120573(120573 minus 1 minus 119902)1198663119883 minus 4119867

411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573

times (120573 minus 1 minus 119902)1198664119883119883 + 21198672119886120573(120573 minus 1 minus 119902) 1205731198664120601

+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573

times (120573 minus 1 minus 119902)1198665119883119883 + 2119867612057341198864120573

times (120573 minus 1 minus 119902)1198665119883120601 + 4119867412057321198862120573(120573 minus 1 minus 119902)1198665120601

minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573

times (120573 minus 1 minus 119902)1198664120601 minus 1198676120573311988631205731198665119883

+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573

times (120573 minus 1 minus 119902)1198665119883120601 minus 6119867412057321198862120573(1 + 119902)1198665120601

+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601

minus 3119867812057351198865120573(120573 minus 1 minus 119902) times 1198665119883119883

+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883

+ 6119867412057331198863120573times (120573 minus 1 minus 119902)1198664119883120601

minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601

minus 12119867612057341198864120573times (120573 minus 1 minus 119902)1198664119883119883 + 12119867

412057321198862120573

times (1 + 119902)1198664119883 minus 12119867412057321198862120573(120573 minus 1 minus 119902)1198664119883

minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883

+ 120573211986731198862120573(120573 minus 119902 minus 1) times 1198663119883 + 18120573

2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867

41205732times 1198862120573(120573 minus 119902 minus 1)1198664119883

minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573

times (120573 minus 1 minus 119902)1198664119883119883 minus 21198672119886120573120573 (120573 minus 1 minus 119902)1198664120601


minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883

+ 11986781205735times 1198865120573(120573 minus 1 minus 119902)1198665119883119883

minus2119867612057341198864120573(120573 minus 1 minus 119902)1198665119883120601] ge 0

(19)

These are the most general energy conditions that can yieldthe energy conditions for various DE models like 119896-essenceand modified theories in certain limits In order to satisfythese conditions it must be guaranteed that the function1198664 is a positive quantity However we have discussed earlierthat 1198664 being a gravitational constant would be positive inall cases (if it is not so then we impose this condition andrestrict the free parameters) Clearly these conditions areonly dependent on the Hubble deceleration parameters andarbitrary functions namely 119870 1198663 1198664 and 1198665 Once thesearbitrary functions are specified the energy bounds on theselectedmodels can be determined by using these conditions

In order to have a better understanding of these con-straints we can use either the power law ansataz for thescale factor for example [45] or we can use the estimationof present values of the respective parameters available inliterature In this study we consider the present value of theHubble parameter 1198670 = 0718 the scale factor 1198860 = 1and the deceleration parameter 119902 = minus064 as suggestedby Capozziello et al [53] Since it is well known that theenergy constraints are satisfied for usual matter contentslike perfect fluid therefore we shall focus on validity of theenergy constraints for the scalar field terms only (either wetake vacuum case or assume that the energy conditions forordinary matter hold) It is interesting to mention here thatthe respective energy conditions in GR can be recovered bytaking the arbitrary functions 1198701198663 and 1198665 zero with 1198664 asconstant

4 Energy Conditions in Some Particular Cases

Now we discuss application of the derived conditions tosome particular cases of this theory The violation of energyconditions leads to various interesting results In particularfor a canonical scalar field violation of these conditionsyields instabilities and ghost pathologies It is important todiscuss the violation of these energy conditions in order tocheck the existence of instabilities in Horndeski theory Theprocedure for FRW universe model in most general scalar-tensor theory based on tensor and scalar perturbations isavailable in literature [31] By introducing perturbed metricit has been shown that for the avoidance of ghost and gradientinstabilities the tensor perturbations suggest

F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0

G119879 = 2 [1198664 minus 21198831198664119883 minus 119883(1198671206011198665119883 minus 1198665120601)] gt 0

(20)

while scalar perturbations impose

F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)

where the quantities Σ and Θ are defined in [31] We simplyplug the values in these conditions for the following cases andshow that violation of energy conditions leads to the existenceof ghost instabilities

41 119896-Essence Models in General Relativity The 119896-essencedynamical models of DE play a dominant role in the solutionof various problems in cosmological context [54] The action(6) can be reduced to the action for 119896-essence model in GRframework defined by 119878 = intradicminus119892[119870(120601119883) + (119872

2

1199011198972)119877 +

119871119898]1198894119909 with the following choice of the functions

119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897

is Planck mass The 119896-essence models can beclassified into three forms

(i) 119870(120601119883) = 1198701(119883) (Kinetic case)(ii) 119870(120601119883) = 1198701(119883)119861(120601)(iii) 119870(120601119883) = 1198701(119883) + 119861(120601)

For the choice of arbitrary functions given by (22) the energyconditions (15) take the following forms

NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0

DEC 120588eff + 119901eff ge 0 120588effge 0

1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)

Here 119870(120601119883) is arbitraryIn order to see how the function 119870(120601119883) can be con-

strained by using the previous energy conditions we choosea particular model of 119896-essence [55] as follows

119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)


where 1198621 1198622 119861 and1198720 are arbitrary constants In this caseWEC requires the following conditions

1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897

gt 0 For the interpretation of the previousinequalities we consider the power law ansatz for the scalarfield 120601 sim 119886

120573 120573 = 0 which further yields 119883 sim (1205732119886212057311986722)

Consequently the WEC (25) turns out to be

[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)

It is difficult to find the admissible ranges of all constants1198621 1198622 1198611198720 and 120573 from the previous conditions In orderto find the constraints on these parameters we consider thatthese conditions are satisfied for ordinarymatter that is 120588119898 gt0 and 120588119898 + 119901119898 gt 0 Moreover we take the present valueof Hubble parameter and choose some particular values ofthe constants 1198621 and 1198622 to find the ranges of 1198611198720 and 120573consistent with the WEC It turns out from the graphs thatwe can take the parameter 120573 as follows 120573 gt 14 0 lt 120573 lt 14and 120573 lt 0 while 119861 and can be positive or negative For theconsistency of condition (27) we restrict the parameter1198720 as0 lt 1198720 lt 1 From the condition (26) it can be observed thatenergy conditions are satisfied only when we take 120573 gt 14with arbitrary 119861 and 120573 lt 14 with 119861 gt 10 only Other choicesof these parameters lead to violation of WEC Figures 1(a)and 1(b) show that WEC is satisfied with these fixed inputparameters by taking 120573 gt 14 0 lt 1198720 lt 1 and 0 lt 119861 lt 1

42 Brans-Dicke Theory Brans-Dicke gravity with actionintradicminus119892[(119872119901119897119883120596120601) minus 119881(120601) + (12)119872119901119897119877120601 + 119871119898]119889

4119909 can be

defined by the following choice of functions

119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)

Here 120596 is the BD parameter and 119881 is the field potentialThe action for general scalar-tensor gravity can be obtainedby taking 119865(120601) instead of 120601 in 1198664 In this case the energyconditions take the form

NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)

A suitable choice of the field potential has always been amatter of debate in BD gravity We use power laws forthe scalar field and potential as 120601 = 1206010119886

120573 and 119881 =

1198810120601119898 respectively where 119898 and 120573 are nonzero parameters

Consequently the WEC restricts the parameters as

1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)

For accelerated expanding universe the observed rangeof BD parameter is minus2 lt 120596 lt minus32 [56] Clearly both ofthese conditions are independent of the parameter 119898 whichshows that these conditions are valid for both the positiveand inverse power law potentials however these conditionsdepend on the present value of the field potential Figure 2(a)shows that the condition 120588eff ge 0 leads to 120573 lt 0 whichfurther yields positive present value of the BD field potential1198810 Moreover the second condition will be satisfied if we take120573 lt 0with arbitrary 120596 and 120573 gt 0with 120596 gt 0 only Figure 2(b)indicates that 120588eff +119901eff ge 0 is satisfied for a particular choiceof 120573 lt 0 and minus2 lt 120596 lt minus15 Clearly the energy condition120588eff+ 119901

effge 0 is violated for 120573 gt 0 with 120596 lt 0 It is easy

to check that this choice of parameters (120573 = 2 120596 = minus18)


76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0

Figure 1 Plot (a) shows 120588eff + 119901effge 0 versus 120573 and119872

0 Plot (b) represents 120588eff ge 0 versus parameters 120573 and 119861 with119872

0= 02 In both cases

we take the present values of cosmological parameters with 1198621= 1 119862

2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)

Figure 2 Plots (a) and (b) showWEC versus parameters 120596 and 120573 for1198670= 0718 and 119902

0= minus064

also leads to the violation of condition imposed by scalarperturbations given by the expression

21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)

and consequently yields the ghost instabilities for the modelHowever the conditions imposed by tensor perturbation aretrivially satisfied

43 119891(119877) Gravity The action for 119891(119877) gravity described by119878 = intradicminus119892[(119872

2

1199011198972)119891(119877) + 119871119898]119889

4119909 can be obtained from the

action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)

Here 119891(119877) is an arbitrary function of the Rici scalar Usingthese values in energy conditions (15) we obtain

NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)

where prime denotes the derivative with respect to 119877 Let usconsider the example of logarithmic model in 119891(119877) gravitydefined by [57]

119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)

In this case the WEC (120588eff ge 0) leads to

minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)

times [(log (minus612057211986720(1 minus 1199020)))

119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1

minus 1] minus 361198674

0

times (1198950 minus 1199020 minus 2)

times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)

times (log (minus612057211986720(1 minus 1199020)))

119902minus2

] ge 0

(35)

where we have used = minus61198672(1 minus 119902) and the present valuesof respective parameters Notice that we have taken only thecondition 120588eff ge 0 as the other condition involves the presentvalue of snap parameter 119904 which is not correctly estimated inthe literature yet Figure 3 shows that theWEC is satisfied fora suitable range of both the parameters 119902 and 120572

20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0

Figure 3 Plot showsWEC versus free parameters 119902 and 120572 Here wetake 119902

0= minus064119867

0= 0718 and 119895

0= 141

44 Kinetic Gravity Braiding Model Kinetic gravity braidingis defined by the action [58]

119878 = int1198894119909radicminus119892(119870 (120601119883) minus 1198663 (120601 119883) ◻120601 +

1198722

119901119897

2119877 +

1198722

119901119897

2

times [(◻120601)2minus (nabla120583nabla120584120601) (nabla

120583nabla120584120601)] + 119871119898)

(36)

where the functions 119870 and 1198663 are arbitrary (while otherfunctions are taken to be zero in action (6)) A particularchoice of these functions proposed by Dvali and Turner [59]is given by 119870 = minus119883 and 1198663 = 119888119883

119899 where 119888 and 119899 areconstants For this choice of model the respective energyconditions turn out to be

NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0

Figure 4 Plots (a) and (b) showWEC namely 120588eff ge 0 and 120588eff + 119901effge 0 versus parameters 119899 and 120573

By taking the power law evolution of the scalar field WECyields

[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)

Here we have taken the present values of the ordinary densityand pressure to be zero Using the present values of theHubble parameter and the scale factor WEC can be satisfiedonly when both the parameters 119899 and 120573 remain positive asindicated in Figure 4 where we have taken 0 lt 119899 lt 1 and1 lt 120573 lt 2 In this case scalar perturbations lead to thefollowing constraints

1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)

It is easy to check that the energy conditions are violatedfor 06 lt 119899 lt 1 and negative range of 120573 (eg 120573 =

minus10) For this choice of parameters the previous constraintsare also violated and hence the ghost instabilities occurHowever constraints imposed by tensor perturbations aretrivially satisfied as1198722

119901119897120601 gt 0

45 Covariant Galilean Model In the absence of potentialthe covariant Galilean model [59] is defined by the followingchoice of parameters in action (6)

119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)

where 1198882 1198883 1198884 and 1198885 are dimensionless constants while119872 isconstant with dimensions of mass Using these values in (15)it follows that

NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)

By taking the power law ansataz for scalar field andconsequently for kinetic term 119883 the WEC in terms ofpresent values of the involved parameter require the followinginequalities

1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674

0120573 (120573 minus 1 minus 1199020) minus 2119867

4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)

Clearly these conditions are satisfied when both 1198664 andterms inside the brackets are positive Since 1198664 gt 0 requires

120573 gt (7511987261198884)14 therefore a suitable choice of all

these parameters yield the consistency with WEC if param-eter 120573 remains small and positive (120573 lt 78) while1198882 remains negative as shown in Figure 5 Here we havetaken minus5 lt 1198882 lt minus1 or minus50 lt 1198882 lt minus10 and55 lt 120573 lt 8 In this case tensor perturbations sug-gest the following conditions for the avoidance of ghostinstabilities

F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)

Clearly the energy conditions are violated for 120573 gt 8

with minus5 lt 1198882 lt minus1 For this choice of parametersthe previous constraints imposed by tensor perturbationsare also violated and consequently the ghost instabilitiesexist However scalar perturbations lead to very complicatedexpressions so we consider only the conditions imposed bytensor perturbations

5 Summary

The most general scalar-tensor theory being a combina-tion of various DE proposals provides a vast gravitationalframework for the discussion of accelerated expansion of theuniverse The modified theories involve some extradegreesof freedom that are described by the models with someunknown parameters It would be interesting to restrict theseparameters on physical grounds In cosmology this can bedone by making compatibility with local gravity tests In agravitational theory energy conditions can be used as anapproach to restrict these parameters In the present paper weconsider the most general scalar-tensor gravity with the fieldequations involving second-order derivatives Firstly we haveexplored the effective energy-momentum tensor and its traceby inverting the generalized field equations which can beused to find the energy condition bounds for any spacetimemanifold

In order to describe these conditions for specific caseswe consider flat FRW universe model with perfect fluidBy defining the effective energy density and pressure wehave expressed the strong weak null and dominant energyconditions For the sake of convenience we have assumedthat the scalar field evolves as a power of scale factorAlso the derivative terms are removed by expressing theseconditions in terms of cosmological quantities like decel-eration snap and jerk parameters An estimation to thepresent values of these parameters is available in litera-ture [53] that can be used to find the constraints on freeparameters of the model The derived energy conditionsare the most general in nature involving many arbitraryfunctions 119870 1198663 1198664 and 1198665 that correspond to different DEproposals

For the application of these energy conditions we havetaken different choices of the functions1198701198663 1198664 and1198665 andhave deduced the energy conditions for 119896-essence model BDgravity 119891(119877) theory kinetic gravity braiding and covariant


500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0

Figure 5 Plots (a) and (b) showWEC versus parameters 1198882and 120573 with 119888

3= 2 1198884= 4 1198885= 3 and119872 = 2

Galileanmodel In each case there are many free parametersIt is not possible to find their range that would be consistentwith energy conditions as well For this reason we havespecified some of these parameters and restricted the othersThe results can be summarized in the following

(i) The WEC in 119896-essence model is satisfied only if thefree parameters satisfy 0 lt 1198720 lt 1 120573 gt 14 witharbitrary 119861 and 120573 lt 14 with 119861 gt 10

(ii) In the literature [40ndash42] using the equivalencebetween BD and 119891(119877) gravity (120596 = 0) it has beenshown that1198810 should be negative In our case (120596 = 0)theWEC restricts the parameter 120573 to be negative (thescalar field should be of contractive nature) for thepositive present value of BD field potential that is1198810 gt 0 which is physically correct However thecondition 120588eff +119901eff ge 0 is satisfied only when we take120573 lt 0 with arbitrary 120596 and 120573 gt 0 with 120596 gt 0 Thusit can be concluded that expanding scalar field withnegative range of BD parameter allowed for cosmicexpansion is inconsistent with the WEC

(iii) In 119891(119877) gravity many authors have used the energyconditions to find the constraints on the models like119891(119877) = 120572119877

119899 or 119877 + 120572119877119899 [40ndash42] In the present

case we have found the constraints on the logarithmic119891(119877) model It is seen that the restrictions on freeparameters 120572 lt 0 and 119902 gt 0 are consistent withWEC

(iv) In kinetic gravity braiding the WEC is satisfied forthe presented model only when 120573 gt 0 (that showsexpanding scalar field) and 119899 gt 0

(v) In covariant Galilean model of DE the WEC issatisfied when free parameters of the model satisfy120573 gt (75119872

61198884)14 and 1198882 lt 0

All these results are also shown through graphs Further wehave determined the conditions for the avoidance of ghostinstabilities using the constraints based on the scalar andtensor perturbations proposed by Kobayashi et al [31] It

is concluded that the violation of these energy conditionsleads to the occurrence of ghost and gradient instabilities inthe above-mentioned cases of the most general second-orderscalar-tensor theory It would be interesting to investigate theconstraints on other DE models like exponential model of119891(119877) gravity other forms of potentials for BD gravity and soforth by making them consistent with the energy conditions

Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2

] times nabla120583120601nabla120584120601

+ 1198664119883◻120601nabla120583nabla120584120601 minus 1198664119883nabla120582nabla120583120601nabla120582nabla120584120601 minus 2nabla1205821198664119883

times nabla120582nabla(120583120601nabla120584)120601 + nabla1205821198664119883nabla

120582120601nabla120583nabla120584120601

minus 119892120583120584 (1198664120601◻120601 minus 21198831198664120601120601) minus 119892120583120584

times [ minus 21198664119883120601nabla120572nabla120573120601nabla120572120601nabla120573120601 + 1198664119883119883nabla120572nabla120582

times120601nabla120573nabla120582120601nabla120572120601nabla120573120601 +

1

21198664119883 [(◻120601)


2

]]

minus 2 [1198664119883119877120582(120583nabla120584)120601nabla120582120601 minus nabla(1205831198664119883nabla120584) times 120601◻120601]

+ 119892120583120584 [1198664119883119877120572120573nabla120572120601nabla120573120601 minus nabla1205821198664119883nabla

120582120601◻120601]

minus 1198664119883119877120583120572120573120584120573 times nabla120572120601nabla120573120601 + 1198664120601nabla120583nabla120584120601

+ 1198664120601120601nabla120583120601nabla120584120601

minus 21198664119883120601nabla120582120601nabla120582nabla(120583120601nabla120584)120601 + 1198664119883119883nabla


120573

times 120601nabla120573nabla120584120601 minus 1198665119883119877120572120573nabla120572120601nabla120573nabla(120583120601nabla120584)120601 + 1198665119883


times 119877120572(120583nabla120584)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573120601nabla120583nabla120584120601

times1

21198665119883119877120583120584120572120573nabla

120572120601 times nabla120573120601◻120601 minus 1198665119883119877120572120582120573(120583nabla120584)120601nabla

120582120601

times nabla120572nabla120573120601 minus 1198665119883119877120572120582120573(120583nabla120584)nabla

120582120601nabla120572120601 times nabla120573120601 +

1

2

times nabla(120583 [1198665119883nabla120572120601] nabla120572nabla120584)120601◻120601 minus

1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601

+ nabla120582 [1198665120601nabla(120583120601]nabla120584)nabla120582120601 minus

1

2

times [nabla120582 (1198665120601nabla120582120601) minus nabla120572 (1198665119883nabla120573120601)nabla

120572nabla120573120601] times nabla120583nabla120584120601

minus nabla1205721198665nabla120573120601119877120572(120583120584)120573 + nabla(1205831198665119866120584)120582nabla

120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883

times [◻120601nabla120573nabla120583120601 minus nabla

120572nabla120573120601nabla120572nabla(120583120601] nabla120584)120601 minus

1

2nabla120572120601

times nabla1205721198665119883 [◻120601nabla120583nabla120584120601 minus nabla120573nabla120583120601nabla120573nabla120584120601] +1

21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]

times nabla120583120601nabla120584120601 minus1

2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]

times nabla1205721198665119877120572120573nabla120573120601 minus

1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573

timesnabla120572120601nabla120573120601◻120601 minus

1

21198665119883119877120572120582120573120588

(A1)

References

[1] S Perlmutter G Aldering and M Della Valle ldquoDiscovery of asupernova explosion at half the age of theUniverserdquoNature vol391 pp 51ndash54 1998

[2] A G Riess A V Filippenko and P Challis ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[3] M Tegmark M A Strauss M R Blanton et al ldquoCosmologicalparameters from SDSS andWMAPrdquo Physical Review D vol 69Article ID 103501 26 pages 2004

[4] E Komatsu J Dunkley M R Nolta et al ldquoFive-year wilkinsonmicrowave anisotropy probe observations cosmological inter-pretationrdquoTheAstrophysical Journal vol 180 no 2 p 330 2009

[5] M C Bento O Bertolami and A A Sen ldquoGeneralizedChaplygin gas accelerated expansion and dark-energy-matterunificationrdquo Physical Review D vol 66 Article ID 043507 5pages 2002

[6] Y Fujii ldquoOrigin of the gravitational constant andparticlemassesin a scale-invariant scalar-tensor theoryrdquo Physical ReviewD vol26 no 10 pp 2580ndash2588 1982

[7] B Ratra and P J E Peebles ldquoCosmological consequences of arolling homogeneous scalar fieldrdquo Physical ReviewD vol 37 no12 pp 3406ndash3427 1988

[8] T Chiba N Sugiyama and T Nakamura ldquoCosmology with x-matterrdquo Monthly Notices of the Royal Astronomical Society vol289 p L5 1997

[9] R R Caldwell R Dave and P J Steinhardt ldquoCosmologicalimprint of an energy component with general equation of staterdquoPhysical Review Letters vol 80 no 8 pp 1582ndash1585 1998

[10] C Armendariz-Picon T Damour and V Mukhanov ldquo119896-inflationrdquo Physics Letters B vol 458 no 2-3 pp 209ndash218 1999

[11] T Chiba T Okabe and M Yamaguchi ldquoKinetically drivenquintessencerdquo Physical Review D vol 62 Article ID 023511 8pages 2000

[12] L P Chimento ldquoExtended tachyon field Chaplygin gas andsolvable 119896-essence cosmologiesrdquo Physical Review D vol 69 no12 Article ID 123517 10 pages 2004

[13] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[14] T Padmanabhan ldquoDark energy and gravityrdquo General Relativityand Gravitation vol 40 no 2-3 pp 529ndash564 2008

[15] F S N Lobo ldquoThe dark side of gravity modified theo-ries of gravityrdquo Dark Energy-Current Advances and Ideashttparxivorgabs08071640

[16] E E Flanagan ldquoHigher-order gravity theories and scalar-tensortheoriesrdquo Classical and Quantum Gravity vol 21 no 2 pp 417ndash426 2004

[17] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1pp 115ndash145 2007

[18] K Karami and M S Khaledian ldquoReconstructing f(R)modifiedgravity from ordinary and entropy-corrected versions of theholographic and new agegraphic dark energy modelsrdquo Journalof High Energy Physics vol 2011 article 086 2011

[19] C H Brans and R H Dicke ldquoMachrsquos principle and a relativistictheory of gravitationrdquo Physical Review vol 124 pp 925ndash9351961

[20] P A M Dirac ldquoA new basisfor cosmologyrdquo Proceedings of theRoyal Society A vol 165 pp 199ndash208 1938

[21] M Mohseni ldquoNon-geodesic motion in 119891(119866) gravity with non-minimal couplingrdquo Physics Letters B vol 682 no 1 pp 89ndash922009


[22] A D Felice and S Tsujikawa ldquoSolar system constraints on f (G)gravity modelsrdquo Physical Review D vol 80 Article ID 06351615 pages 2009

[23] E V Linder ldquoEinsteinrsquos other gravity and the acceleration of theuniverserdquo Physical Review D vol 81 Article ID 127301 3 pages2010

[24] R Myrzakulov ldquoAccelerating universe from F(T) gravityrdquo TheEuropean Physical Journal C vol 71 article 1752 2011

[25] T Harko et al ldquof(RT) gravityrdquo Physical Review D vol 84Article ID 024020 11 pages 2011

[26] A D Linde ldquoA new inflationary universe scenario a possiblesolution of the horizon flatness homogeneity isotropy andprimordial monopole problemsrdquo Physics Letters B vol 108 no6 pp 389ndash393 1982

[27] A D Linde ldquoChaotic inflationrdquo Physics Letters B vol 129 no3-4 pp 177ndash181 1983

[28] J Garriga and V F Mukhanov ldquoPerturbations in 119896-inflationrdquoPhysics Letters B vol 458 no 2-3 pp 219ndash225 1999

[29] GWHorndeski ldquoSecond-order scalar-tensor field equations ina four-dimensional spacerdquo International Journal of TheoreticalPhysics vol 10 no 6 pp 363ndash384 1974

[30] C Deffayet et al ldquoFrom k-essence to generalized GalileonsrdquoPhysical Review D vol 84 Article ID 064039 15 pages 2011

[31] T Kobayashi M Yamaguchi and J Yokoyama ldquoGeneralized g-inflationrdquo Progress of Theoretical Physics vol 126 no 3 pp 511ndash529 2011

[32] AD Felice and S Tsujikawa ldquoInflationary non-Gaussianities inthe most general second-order scalar-tensor theoriesrdquo PhysicalReview D vol 84 Article ID 083504 15 pages 2011

[33] A D Felice and S Tsujikawa ldquoConditions for the cosmologicalviability of the most general scalar-tensor theories and theirapplications to extended Galilean dark energy modelsrdquo Journalof Cosmology and Astroparticle Physics vol 007 p 1202 2012

[34] X Gao and D A Steer ldquoInflation and primordial non-Gaussianities of generalized Galileansrdquo Journal of Cosmologyand Astroparticle Physics vol 019 p 12 2011

[35] RKimura T Kobayashi andKYamamoto ldquoVainshtein screen-ing in a cosmological background in the most general second-order scalar-tensor theoryrdquo Physical Review D vol 85 ArticleID 024023 11 pages 2012

[36] S W Hawking and G F R Ellis The Large Scale Structure ofSpace-Time Cambridge University Press London UK 1973

[37] R Schoen and S T Yau ldquoProof of the positive mass theoremIIrdquo Communications in Mathematical Physics vol 79 no 2 pp231ndash260 1981

[38] S Carroll Spacetime andGeometrymdashAn Introduction toGeneralRelativity Addison Wesley San Francisco Calif USA 2004

[39] M Visser ldquoGravitational vacuum polarization IV Energyconditions in the Unruh vacuumrdquo Physical Review D vol 56no 2 pp 936ndash952 1997

[40] J Santos J S Alcaniz M J Reboucas and F C CarvalholdquoEnergy conditions in 119891(119877) gravityrdquo Physical Review D vol 76no 8 Article ID 083513 6 pages 2007

[41] J Santos M J Reboucas and J S Alcaniz ldquoEnergy conditionsconstraints on a class of f (r)-gravityrdquo International Journal ofModern Physics D vol 19 p 1315 2010

[42] Y Y Zhao Y BWu J B Lu et al ldquoModified f(G) gravitymodelswith curvature-matter couplingrdquoTheEuropean Physical JournalC vol 72 article 1924 2012

[43] N M Garcia T Harko F S N Lobo et al ldquoEnergy conditionsin modified Gauss-Bonnet gravityrdquo Physical Review D vol 83Article ID 104032 8 pages 2011

[44] D Liu and M J Reboucas ldquoEnergy conditions bounds on f(T)gravityrdquo Physical Review D vol 86 Article ID 083515 8 pages2012

[45] M Sharif and M Zubair ldquoEnergy conditions constraints andstability of power law solutions in f(RT) gravityrdquo Journal of thePhysical Society of Japan vol 82 p 014002 2013

[46] S Kar and S Sengupta ldquoThe raychaudhuri equations a brietreviewrdquo Pramana vol 69 no 1 pp 49ndash76 2007

[47] F G Alvarenga M J S Houndjo A V Monwanou et alldquoTesting some f(RT) gravity models from energy conditionsrdquoJournal of Modern Physcis vol 7 130 pages 2013

[48] S M Carroll M Hoffman and M Trodden ldquoCan the darkenergy equation-of-state parameter w be less than -1rdquo PhysicalReview D vol 68 Article ID 023509 11 pages 2003

[49] T Qiu Y Cai and X Zhang ldquoNull energy condition and darkenergymodelsrdquoModern Physics Letters A vol 23 no 32 p 27872008

[50] O Goldoni and M F A da Silva ldquoEnergy Conditions forElectromagnetic Field in presence of Cosmological ConstantrdquoProceedings of Science

[51] N Banerjee and D Pavon ldquoHolographic dark energy in Brans-Dicke theoryrdquo Physics Letters B vol 647 no 5-6 pp 477ndash4812007

[52] A Sheykhi ldquoMagnetic strings in 119891(119877) gravityrdquo General Relativ-ity and Gravitation vol 44 no 9 pp 2271ndash2281 2012

[53] S Capozziello et al ldquoCosmography in f(T) gravityrdquo PhysicalReview D vol 84 Article ID 043527 11 pages 2011

[54] R Myrzakulov ldquoFermionic K-essencerdquo httparxivorgabs10114337v5

[55] Y Arefva N V Bulatov L V Joukovskaya and S Yu VernovldquoNull energy condition violation and classical stability in theBianchi I metricrdquo Physical Review D vol 80 no 8 Article ID083532 13 pages 2009

[56] M R Setare and M Jamil ldquoHolographic dark energy in BransndashDicke cosmology with chameleon scalar fieldrdquo Physics Letters Bvol 690 no 1 pp 1ndash4 2010

[57] Z Girones ldquoCosmological data analysis of f(R) gravity modelsrdquoJournal of Cosmology and Astroparticle Physics vol 11 p 0042010

[58] C Deffayet et al ldquoImperfect dark energy from kinetic gravitybraidingrdquo Journal of Cosmology and Astroparticle Physics vol10 p 026 2010

[59] G Dvali andM S Turner ldquoDark energy as amodification of thefriedmann equationrdquo httparxivorgabsastro-ph0301510

Submit your manuscripts athttpwwwhindawicom

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The energy conditions have many significant theoreticalapplications likeHawking Penrose singularity conjecture thatis based on the strong energy condition [36] while thedominant energy condition is useful in the proof of positivemass theorem [37] Furthermore null energy condition is abasic ingredient in the derivation of second law of black holethermodynamics [38] On the cosmological grounds Visser[39] discussed various cosmological terms like distancemod-ulus look back time deceleration and statefinder parametersin terms of red shift using energy condition constraintsTheseconditions are originally formulated in the context of GR andthen extended to modified theories of gravity Many authorshave explored these energy conditions in the framework ofmodified gravity and found interesting results [40ndash43]

Basically modified gravity theories contain some extra-functions like higher-order derivatives of curvature term orsome functions of Einstein tensor or scalar field and so forthThus it is a point of debate that how one can constrain theadded extra degrees of freedom consistently with the recentobservationsThe energy conditions can be used to put someconstraints on these functions that could be consistent withthose already found in the cosmological arena Recently theseenergy conditions have been discussed in 119891(119879) [44] and119891(119877 119879) [45] theories

In this paper we study the energy condition bounds ina most general scalar-tensor theory The paper is designedin the following layout Next section defines the energyconditions inGR aswell as in a generalmodified gravitationalframework Section 3 provides basic formulation of the mostgeneral scalar-tensor theory In the same section we formu-late the energy conditions in terms of some cosmologicalparameters within such modified framework In Section 4we provide some specific cases of this theory and discussthe corresponding constraints Finally we summarize andpresent some general remarks

2 Energy Conditions

In this section we discuss the energy conditions in GRframework and then express the respective conditions in ageneral modified gravity In GR the energy conditions comefrom a well-known purely geometric relationship known asRaychaudhuri equation [38 46] together with the lineamentof gravitational attractiveness In a spacetime manifold withvector fields 119906120583 and 119896120583 as tangent vectors to timelike andnull-like geodesics of the congruence the temporal variationof expansion for the respective curves is described by theRaychaudhuri equation as

119889120579

119889120591= minus

1

31205792minus 120590120583120584120590

120583120584+ 120596120583120584120596

120583120584minus 119877120583120584119906

120583119906120584

119889120579

119889120591= minus

1

21205792minus 120590120583120584120590

120583120584+ 120596120583120584120596

120583120584minus 119877120583120584119896

120583119896120584

(1)

Here 119877120583120584 120579 120590120583120584 and 120596

120583120584 represent the Ricci tensorexpansion shear and rotation respectively related with thecongruence of timelike or null-like geodesics

The characteristic of the gravity that is attractive leads tothe condition 119889120579119889120591 lt 0 For infinitesimal distortions and

vanishing shear tensor 120596120583120584 = 0 that is zero rotation (forany hypersurface of orthogonal congruence) we ignore thesecond-order terms in Raychaudhuri equation and conse-quently integration leads to 120579 = minus120591119877120583120584119906

120583119906120584= minus120591119877120583120584119896

120583119896120584

It further implies that

119877120583120584119906120583119906120584ge 0 119877120583120584119896

120583119896120584ge 0 (2)

Since GR and its modifications lead to a relationship ofthe matter contents that is energy-momentum tensor interms of Ricci tensor through the field equations thereforethe respective physical conditions on the energy-momentumtensor can be determined as follows

119877120583120584119906120583119906120584= (119879120583120584 minus

119879

2119892120583120584) 119906

120583119906120584ge 0

119877120583120584119896120583119896120584= (119879120583120584 minus

119879

2119892120583120584) 119896

120583119896120584ge 0

(3)

where 119879120583120584 and 119879 are the energy-momentum tensor andits trace respectively For perfect fluid with density 120588 andpressure 119901 defined by

119879120583120584 = (120588 + 119901) 119906120583119906120584 minus 119901119892120583120584 (4)

the strong and null energy conditions respectively aredefined by the inequalities 120588 + 3119901 ge 0 and 120588 + 119901 ge 0while the weak and dominant energy conditions are definedrespectively by 120588 ge 0 and 120588 plusmn 119901 ge 0

Raychaudhuri equation being a geometrical statementworks for all gravitational theories Therefore its interestingfeatures like focussing of geodesic congruences as well as theattractiveness of gravity can be used to derive the energyconstraints in the context of modified gravity In case ofmodified gravity we assume that the total matter contentsof the universe act like perfect fluid and consequently theseconditions can be defined in terms of effective energy densityand pressure (matter sources getmodified andwe replace119879120583120584and 119879 in (3) by 119879eff

120583120584and 119879eff resp) These conditions can be

regarded as an extension of the respective conditions in GRgiven by [43]

NEC 120588eff + 119901eff ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0

WEC 120588eff ge 0 120588eff+ 119901

effge 0

DEC 120588eff ge 0 120588effplusmn 119901

effge 0

(5)

For a detailed discussion we suggest the readers to study arecent paper [47]

TheDE requires negative EoS parameter120596 le minus13 for theexplanation of cosmic expansion Indeed for cosmologicalpurposes we are curious for a source with 120588 ge 0 inthat case all of the energy conditions require 120596 ge minus1

[48] The role of possible DE candidates with 120596 lt minus1

was pointed out by Caldwell who referred to null DECviolating sources as phantom components It is argued thatDE models with 119908 ge minus1 such as the cosmological constant





119878 = int1198894119909radicminus119892

times [119870 (120601119883) minus 1198663 (120601 119883) ◻120601 + 1198664 (120601119883) 119877


120583nabla120584120601)


1

61198665119883


120583nabla120584120601)


120572nabla120573120601) (nabla

120573nabla120583120601) + 119871119898]

(6)






119866120583120584 =1

1198664

Θeff120583120584=1

1198664

[119879119898

120583120584+ 119879120601

120583120584] (7)







119877120583120584 = 119879eff120583120584minus1

2119892120583120584119879

eff (8)



119879eff120583120584= 119879120583120584 +

1


1

21198663119883◻120601nabla120584120601nabla120583120601



1

21198664119883119883


2

] nabla120583120601nabla120584120601



120582120601nabla120583nabla120584120601





120573120601nabla120573nabla120584120601


+ 1198665119883119877120572(120583nabla120584)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573120601


21198665119883119877120583120572120584120573nabla

120572120601nabla120573120601◻120601



+1

2nabla(120583 [1198665119883nabla


1

2


minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]



1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



minus1



+1

21198665119883 times 119866120572120573nabla


1

21198665119883◻120601


1

21198665119883(◻120601)


minus1

121198665119883119883 [(◻120601)


2

+ 2(nabla120572nabla120573120601)3

]


2nabla1205821198665119877120583120584nabla

120582120601

(9)

119879eff= 119870 + nabla1205821198663nabla

120582120601 minus 2 (1198664120601◻120601 minus 21198831198664120601120601)



1

21198664119883 [(◻120601)


2

]


120582120601◻120601]

minus1

2119877nabla1205821198665nabla

120582120601

minus 2 minus1

61198665119883 [(◻120601)


2

+2(nabla120572nabla120573120601)3

]


1

2nabla1205721198665120601nabla

120572120601◻120601 +

1

2nabla1205721198665120601


1

2nabla1205721198665119883nabla

120572119883◻120601 +

1

2nabla1205721198665119883


1

4nabla1205821198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883 times 119877120572120582120573120588nabla

120572120601


1

2119870119883(nabla120583120601)

2

minus1

21198663119883◻120601(nabla120583120601)

2


21198664119883119877(nabla120583120601)

2

+1

21198664119883119883 [(◻120601)


2

] (nabla120583120601)2



120582120601nabla120583nabla120583120601



2


2

+ 1198664119883119883nabla120572120601nabla120572


120572120601nabla120573(nabla120583120601)

2

+ 1198665119883119877120572(120583nabla120583)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573


21198665119883 times 119877120583120572120583120573nabla

120572120601nabla120573120601◻120601



1

2nabla(120583 [1198665119883nabla

120572120601] nabla120572nabla120583)120601◻120601

minus1


120582120601

minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]


120582120601

minus1



2

]



120573nabla120583120601



minus1


+1

21198665119883119866120572120573nabla

120572nabla120573120601(nabla120583120601)

2

+1


120572nabla120583120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+2(nabla120572nabla120573120601)3

] (nabla120583120601)2

minus1

2nabla1205821198665119877nabla

120582120601

+1


120572nabla120573120601

minus1

4nabla1205721198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883119877120572120582120573120588 (10)



1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (11)


31198672=120588eff

1198664

minus (31198672+ 2) =

119901eff

1198664

(12)


where

120588eff=1

2[120588119898+ 2119883119870119883 minus 119870 + 6119883

1206011198671198663119883 minus 21198831198663120601 + 241198672119883

times (1198664119883 + 1198831198664119883119883) minus 121198671198831206011198664119883120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)]

(13)

119901eff=1

2[119901119898+ 119870 minus 2119883 (1198663120601 +

1206011198663119883) minus 1211986721198831198664119883


+ 2 ( 120601 + 2119867 120601)1198664120601 + 41198831198664120601120601 + 4119883 (120601 minus 2119867 120601)

times 1198664119883120601 minus 2119883 (21198673 120601 + 2119867 120601 + 3119867

2 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 4119867119883( minus 119867119883)1198665119883120601

+2 [2 (119867119883)+ 31198672119883]1198665120601 + 4119867119883

1206011198665120601120601]

(14)



1

21198664

[(120588119898+ 119901119898) + 2119883119870119883 + 6119883

1206011198671198663119883 minus 41198831198663120601

+ 1211986721198831198664119883 + 24119867

211988321198664119883119883 minus 20119883119867

1206011198664119883120601

minus 2119867 1206011198664120601 + 61198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 16119867211988321198665119883120601 minus 12119867

21198831198665120601 minus 2119883

1206011198663119883


+ 4119883119866120601120601 + 41198831206011198664119883120601 minus 2119883 (2119867

120601 + 31198672 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 41198671198831198665119883120601

+4 (119867119883)1198665120601 + 4119867119883

1206011198665120601120601] ge 0


1

21198664

[120588119898+ 2119883119870119883 minus 119870 + 6119883119867

1206011198663119883 minus 21198831198663120601

+ 241198672119883 times (1198664119883 + 1198831198664119883119883) minus 12119883119867

120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)] ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0 997904rArr

1

21198664

[(120588119898+ 3119901119898) + 2119883119870119883 + 2119870 + 6119883119867

1206011198663119883


21198831198664119883 + 24119867

21198832

times 1198664119883119883 minus 361198831198671206011198664119883120601 + 6119867

1206011198664120601 + 61206011198664120601

minus 21198673119883 1206011198665119883 + 4119867

31198832 1206011198665119883119883 minus 24119867

211988321198665119883120601

+ 121198671198831198665119883120601 + 12 (119867119883)1198665120601

+ 12119867119883 120601119866120601120601 minus 1211986721198832 1206011198665119883119883 minus 12119883119867

1206011198665119883

minus 181198831198672 1206011198665119883 + 12119883

1206011198664119883120601 minus 241198671198831206011198664119883120601

+ 121198831198664120601120601 minus 24119867119883 times 1198664119883119883


DEC 120588eff ge 0 120588eff+ 119901

effge 0 120588

effminus 119901

effge 0 997904rArr

1

21198664

[(120588119898minus 119901119898) + 2119883119870119883 minus 2119870 + 6119883119867

1206011198663119883 + 21198831206011198663119883

+ 3611986721198831198664119883 + 24119867

21198831198664119883119883 minus 4119883119867

1206011198664119883120601

minus 10119867 1206011198664120601 + 141198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 2411986721198831198665120601 minus 4 (119867119883)

1198665120601 minus 4119867119883

120601119866120601120601

minus 8119867211988321198665119883120601 + 41198671198664119883

+ 81198831198664119883 + 81198671198831198664119883119883 minus 21206011198664120601 minus 41198831198664120601120601

minus 4119883 1206011198664119883120601 + 2119883 (2119867120601 + 3119867

2 120601) 1198665119883

+4119867211988321198665119883119883

120601 minus 41198671198831198665119883120601] ge 0

(15)




119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)


= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)


(17)


120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)




1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883


411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573


+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573



minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573


+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573


+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601


+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883


minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601


412057321198862120573


minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883


2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867


minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573



minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119878 = int1198894119909radicminus119892

times [119870 (120601119883) minus 1198663 (120601 119883) ◻120601 + 1198664 (120601119883) 119877


120583nabla120584120601)


1

61198665119883


120583nabla120584120601)


120572nabla120573120601) (nabla

120573nabla120583120601) + 119871119898]

(6)






119866120583120584 =1

1198664

Θeff120583120584=1

1198664

[119879119898

120583120584+ 119879120601

120583120584] (7)







119877120583120584 = 119879eff120583120584minus1

2119892120583120584119879

eff (8)



119879eff120583120584= 119879120583120584 +

1


1

21198663119883◻120601nabla120584120601nabla120583120601



1

21198664119883119883


2

] nabla120583120601nabla120584120601



120582120601nabla120583nabla120584120601





120573120601nabla120573nabla120584120601


+ 1198665119883119877120572(120583nabla120584)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573120601


21198665119883119877120583120572120584120573nabla

120572120601nabla120573120601◻120601



+1

2nabla(120583 [1198665119883nabla


1

2


minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]



1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



minus1



+1

21198665119883 times 119866120572120573nabla


1

21198665119883◻120601


1

21198665119883(◻120601)


minus1

121198665119883119883 [(◻120601)


2

+ 2(nabla120572nabla120573120601)3

]


2nabla1205821198665119877120583120584nabla

120582120601

(9)

119879eff= 119870 + nabla1205821198663nabla

120582120601 minus 2 (1198664120601◻120601 minus 21198831198664120601120601)



1

21198664119883 [(◻120601)


2

]


120582120601◻120601]

minus1

2119877nabla1205821198665nabla

120582120601

minus 2 minus1

61198665119883 [(◻120601)


2

+2(nabla120572nabla120573120601)3

]


1

2nabla1205721198665120601nabla

120572120601◻120601 +

1

2nabla1205721198665120601


1

2nabla1205721198665119883nabla

120572119883◻120601 +

1

2nabla1205721198665119883


1

4nabla1205821198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883 times 119877120572120582120573120588nabla

120572120601


1

2119870119883(nabla120583120601)

2

minus1

21198663119883◻120601(nabla120583120601)

2


21198664119883119877(nabla120583120601)

2

+1

21198664119883119883 [(◻120601)


2

] (nabla120583120601)2



120582120601nabla120583nabla120583120601



2


2

+ 1198664119883119883nabla120572120601nabla120572


120572120601nabla120573(nabla120583120601)

2

+ 1198665119883119877120572(120583nabla120583)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573


21198665119883 times 119877120583120572120583120573nabla

120572120601nabla120573120601◻120601



1

2nabla(120583 [1198665119883nabla

120572120601] nabla120572nabla120583)120601◻120601

minus1


120582120601

minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]


120582120601

minus1



2

]



120573nabla120583120601



minus1


+1

21198665119883119866120572120573nabla

120572nabla120573120601(nabla120583120601)

2

+1


120572nabla120583120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+2(nabla120572nabla120573120601)3

] (nabla120583120601)2

minus1

2nabla1205821198665119877nabla

120582120601

+1


120572nabla120573120601

minus1

4nabla1205721198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883119877120572120582120573120588 (10)



1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (11)


31198672=120588eff

1198664

minus (31198672+ 2) =

119901eff

1198664

(12)


where

120588eff=1

2[120588119898+ 2119883119870119883 minus 119870 + 6119883

1206011198671198663119883 minus 21198831198663120601 + 241198672119883

times (1198664119883 + 1198831198664119883119883) minus 121198671198831206011198664119883120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)]

(13)

119901eff=1

2[119901119898+ 119870 minus 2119883 (1198663120601 +

1206011198663119883) minus 1211986721198831198664119883


+ 2 ( 120601 + 2119867 120601)1198664120601 + 41198831198664120601120601 + 4119883 (120601 minus 2119867 120601)

times 1198664119883120601 minus 2119883 (21198673 120601 + 2119867 120601 + 3119867

2 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 4119867119883( minus 119867119883)1198665119883120601

+2 [2 (119867119883)+ 31198672119883]1198665120601 + 4119867119883

1206011198665120601120601]

(14)



1

21198664

[(120588119898+ 119901119898) + 2119883119870119883 + 6119883

1206011198671198663119883 minus 41198831198663120601

+ 1211986721198831198664119883 + 24119867

211988321198664119883119883 minus 20119883119867

1206011198664119883120601

minus 2119867 1206011198664120601 + 61198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 16119867211988321198665119883120601 minus 12119867

21198831198665120601 minus 2119883

1206011198663119883


+ 4119883119866120601120601 + 41198831206011198664119883120601 minus 2119883 (2119867

120601 + 31198672 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 41198671198831198665119883120601

+4 (119867119883)1198665120601 + 4119867119883

1206011198665120601120601] ge 0


1

21198664

[120588119898+ 2119883119870119883 minus 119870 + 6119883119867

1206011198663119883 minus 21198831198663120601

+ 241198672119883 times (1198664119883 + 1198831198664119883119883) minus 12119883119867

120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)] ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0 997904rArr

1

21198664

[(120588119898+ 3119901119898) + 2119883119870119883 + 2119870 + 6119883119867

1206011198663119883


21198831198664119883 + 24119867

21198832

times 1198664119883119883 minus 361198831198671206011198664119883120601 + 6119867

1206011198664120601 + 61206011198664120601

minus 21198673119883 1206011198665119883 + 4119867

31198832 1206011198665119883119883 minus 24119867

211988321198665119883120601

+ 121198671198831198665119883120601 + 12 (119867119883)1198665120601

+ 12119867119883 120601119866120601120601 minus 1211986721198832 1206011198665119883119883 minus 12119883119867

1206011198665119883

minus 181198831198672 1206011198665119883 + 12119883

1206011198664119883120601 minus 241198671198831206011198664119883120601

+ 121198831198664120601120601 minus 24119867119883 times 1198664119883119883


DEC 120588eff ge 0 120588eff+ 119901

effge 0 120588

effminus 119901

effge 0 997904rArr

1

21198664

[(120588119898minus 119901119898) + 2119883119870119883 minus 2119870 + 6119883119867

1206011198663119883 + 21198831206011198663119883

+ 3611986721198831198664119883 + 24119867

21198831198664119883119883 minus 4119883119867

1206011198664119883120601

minus 10119867 1206011198664120601 + 141198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 2411986721198831198665120601 minus 4 (119867119883)

1198665120601 minus 4119867119883

120601119866120601120601

minus 8119867211988321198665119883120601 + 41198671198664119883

+ 81198831198664119883 + 81198671198831198664119883119883 minus 21206011198664120601 minus 41198831198664120601120601

minus 4119883 1206011198664119883120601 + 2119883 (2119867120601 + 3119867

2 120601) 1198665119883

+4119867211988321198665119883119883

120601 minus 41198671198831198665119883120601] ge 0

(15)




119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)


= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)


(17)


120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)




1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883


411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573


+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573



minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573


+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573


+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601


+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883


minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601


412057321198862120573


minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883


2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867


minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573



minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




+1

21198665119883 times 119866120572120573nabla


1

21198665119883◻120601


1

21198665119883(◻120601)


minus1

121198665119883119883 [(◻120601)


2

+ 2(nabla120572nabla120573120601)3

]


2nabla1205821198665119877120583120584nabla

120582120601

(9)

119879eff= 119870 + nabla1205821198663nabla

120582120601 minus 2 (1198664120601◻120601 minus 21198831198664120601120601)



1

21198664119883 [(◻120601)


2

]


120582120601◻120601]

minus1

2119877nabla1205821198665nabla

120582120601

minus 2 minus1

61198665119883 [(◻120601)


2

+2(nabla120572nabla120573120601)3

]


1

2nabla1205721198665120601nabla

120572120601◻120601 +

1

2nabla1205721198665120601


1

2nabla1205721198665119883nabla

120572119883◻120601 +

1

2nabla1205721198665119883


1

4nabla1205821198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883 times 119877120572120582120573120588nabla

120572120601


1

2119870119883(nabla120583120601)

2

minus1

21198663119883◻120601(nabla120583120601)

2


21198664119883119877(nabla120583120601)

2

+1

21198664119883119883 [(◻120601)


2

] (nabla120583120601)2



120582120601nabla120583nabla120583120601



2


2

+ 1198664119883119883nabla120572120601nabla120572


120572120601nabla120573(nabla120583120601)

2

+ 1198665119883119877120572(120583nabla120583)120601nabla120572120601◻120601 +

1

21198665119883119877120572120573nabla

120572120601nabla120573


21198665119883 times 119877120583120572120583120573nabla

120572120601nabla120573120601◻120601



1

2nabla(120583 [1198665119883nabla

120572120601] nabla120572nabla120583)120601◻120601

minus1


120582120601

minus1

2[nabla120582 (1198665120601nabla


120572nabla120573120601]


120582120601

minus1



2

]



120573nabla120583120601



minus1


+1

21198665119883119866120572120573nabla

120572nabla120573120601(nabla120583120601)

2

+1


120572nabla120583120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+2(nabla120572nabla120573120601)3

] (nabla120583120601)2

minus1

2nabla1205821198665119877nabla

120582120601

+1


120572nabla120573120601

minus1

4nabla1205721198665119883nabla120582120601 [(◻120601)


2

]

+1

21198665119883119877120572120573nabla

120572120601nabla120573120601◻120601 minus

1

21198665119883119877120572120582120573120588 (10)



1198891199042= minus119889119905

2+ 1198862(119905) (119889119909

2+ 1198891199102+ 1198891199112) (11)


31198672=120588eff

1198664

minus (31198672+ 2) =

119901eff

1198664

(12)


where

120588eff=1

2[120588119898+ 2119883119870119883 minus 119870 + 6119883

1206011198671198663119883 minus 21198831198663120601 + 241198672119883

times (1198664119883 + 1198831198664119883119883) minus 121198671198831206011198664119883120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)]

(13)

119901eff=1

2[119901119898+ 119870 minus 2119883 (1198663120601 +

1206011198663119883) minus 1211986721198831198664119883


+ 2 ( 120601 + 2119867 120601)1198664120601 + 41198831198664120601120601 + 4119883 (120601 minus 2119867 120601)

times 1198664119883120601 minus 2119883 (21198673 120601 + 2119867 120601 + 3119867

2 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 4119867119883( minus 119867119883)1198665119883120601

+2 [2 (119867119883)+ 31198672119883]1198665120601 + 4119867119883

1206011198665120601120601]

(14)



1

21198664

[(120588119898+ 119901119898) + 2119883119870119883 + 6119883

1206011198671198663119883 minus 41198831198663120601

+ 1211986721198831198664119883 + 24119867

211988321198664119883119883 minus 20119883119867

1206011198664119883120601

minus 2119867 1206011198664120601 + 61198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 16119867211988321198665119883120601 minus 12119867

21198831198665120601 minus 2119883

1206011198663119883


+ 4119883119866120601120601 + 41198831206011198664119883120601 minus 2119883 (2119867

120601 + 31198672 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 41198671198831198665119883120601

+4 (119867119883)1198665120601 + 4119867119883

1206011198665120601120601] ge 0


1

21198664

[120588119898+ 2119883119870119883 minus 119870 + 6119883119867

1206011198663119883 minus 21198831198663120601

+ 241198672119883 times (1198664119883 + 1198831198664119883119883) minus 12119883119867

120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)] ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0 997904rArr

1

21198664

[(120588119898+ 3119901119898) + 2119883119870119883 + 2119870 + 6119883119867

1206011198663119883


21198831198664119883 + 24119867

21198832

times 1198664119883119883 minus 361198831198671206011198664119883120601 + 6119867

1206011198664120601 + 61206011198664120601

minus 21198673119883 1206011198665119883 + 4119867

31198832 1206011198665119883119883 minus 24119867

211988321198665119883120601

+ 121198671198831198665119883120601 + 12 (119867119883)1198665120601

+ 12119867119883 120601119866120601120601 minus 1211986721198832 1206011198665119883119883 minus 12119883119867

1206011198665119883

minus 181198831198672 1206011198665119883 + 12119883

1206011198664119883120601 minus 241198671198831206011198664119883120601

+ 121198831198664120601120601 minus 24119867119883 times 1198664119883119883


DEC 120588eff ge 0 120588eff+ 119901

effge 0 120588

effminus 119901

effge 0 997904rArr

1

21198664

[(120588119898minus 119901119898) + 2119883119870119883 minus 2119870 + 6119883119867

1206011198663119883 + 21198831206011198663119883

+ 3611986721198831198664119883 + 24119867

21198831198664119883119883 minus 4119883119867

1206011198664119883120601

minus 10119867 1206011198664120601 + 141198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 2411986721198831198665120601 minus 4 (119867119883)

1198665120601 minus 4119867119883

120601119866120601120601

minus 8119867211988321198665119883120601 + 41198671198664119883

+ 81198831198664119883 + 81198671198831198664119883119883 minus 21206011198664120601 minus 41198831198664120601120601

minus 4119883 1206011198664119883120601 + 2119883 (2119867120601 + 3119867

2 120601) 1198665119883

+4119867211988321198665119883119883

120601 minus 41198671198831198665119883120601] ge 0

(15)




119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)


= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)


(17)


120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)




1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883


411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573


+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573



minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573


+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573


+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601


+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883


minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601


412057321198862120573


minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883


2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867


minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573



minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




where

120588eff=1

2[120588119898+ 2119883119870119883 minus 119870 + 6119883

1206011198671198663119883 minus 21198831198663120601 + 241198672119883

times (1198664119883 + 1198831198664119883119883) minus 121198671198831206011198664119883120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)]

(13)

119901eff=1

2[119901119898+ 119870 minus 2119883 (1198663120601 +

1206011198663119883) minus 1211986721198831198664119883


+ 2 ( 120601 + 2119867 120601)1198664120601 + 41198831198664120601120601 + 4119883 (120601 minus 2119867 120601)

times 1198664119883120601 minus 2119883 (21198673 120601 + 2119867 120601 + 3119867

2 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 4119867119883( minus 119867119883)1198665119883120601

+2 [2 (119867119883)+ 31198672119883]1198665120601 + 4119867119883

1206011198665120601120601]

(14)



1

21198664

[(120588119898+ 119901119898) + 2119883119870119883 + 6119883

1206011198671198663119883 minus 41198831198663120601

+ 1211986721198831198664119883 + 24119867

211988321198664119883119883 minus 20119883119867

1206011198664119883120601

minus 2119867 1206011198664120601 + 61198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 16119867211988321198665119883120601 minus 12119867

21198831198665120601 minus 2119883

1206011198663119883


+ 4119883119866120601120601 + 41198831206011198664119883120601 minus 2119883 (2119867

120601 + 31198672 120601) 1198665119883

minus 411986721198832 1206011198665119883119883 + 41198671198831198665119883120601

+4 (119867119883)1198665120601 + 4119867119883

1206011198665120601120601] ge 0


1

21198664

[120588119898+ 2119883119870119883 minus 119870 + 6119883119867

1206011198663119883 minus 21198831198663120601

+ 241198672119883 times (1198664119883 + 1198831198664119883119883) minus 12119883119867

120601

minus 6119867 1206011198664120601 + 21198673119883 120601 (51198665119883 + 21198831198665119883119883)

minus61198672119883(31198665120601 + 21198831198665119883120601)] ge 0

SEC 120588eff + 119901eff ge 0 120588eff+ 3119901

effge 0 997904rArr

1

21198664

[(120588119898+ 3119901119898) + 2119883119870119883 + 2119870 + 6119883119867

1206011198663119883


21198831198664119883 + 24119867

21198832

times 1198664119883119883 minus 361198831198671206011198664119883120601 + 6119867

1206011198664120601 + 61206011198664120601

minus 21198673119883 1206011198665119883 + 4119867

31198832 1206011198665119883119883 minus 24119867

211988321198665119883120601

+ 121198671198831198665119883120601 + 12 (119867119883)1198665120601

+ 12119867119883 120601119866120601120601 minus 1211986721198832 1206011198665119883119883 minus 12119883119867

1206011198665119883

minus 181198831198672 1206011198665119883 + 12119883

1206011198664119883120601 minus 241198671198831206011198664119883120601

+ 121198831198664120601120601 minus 24119867119883 times 1198664119883119883


DEC 120588eff ge 0 120588eff+ 119901

effge 0 120588

effminus 119901

effge 0 997904rArr

1

21198664

[(120588119898minus 119901119898) + 2119883119870119883 minus 2119870 + 6119883119867

1206011198663119883 + 21198831206011198663119883

+ 3611986721198831198664119883 + 24119867

21198831198664119883119883 minus 4119883119867

1206011198664119883120601

minus 10119867 1206011198664120601 + 141198673119883 1206011198665119883 + 4119867

31198832 120601 times 1198665119883119883

minus 2411986721198831198665120601 minus 4 (119867119883)

1198665120601 minus 4119867119883

120601119866120601120601

minus 8119867211988321198665119883120601 + 41198671198664119883

+ 81198831198664119883 + 81198671198831198664119883119883 minus 21206011198664120601 minus 41198831198664120601120601

minus 4119883 1206011198664119883120601 + 2119883 (2119867120601 + 3119867

2 120601) 1198665119883

+4119867211988321198665119883119883

120601 minus 41198671198831198665119883120601] ge 0

(15)




119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)


= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)


(17)


120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)




1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883


411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573


+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573



minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573


+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573


+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601


+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883


minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601


412057321198862120573


minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883


2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867


minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573



minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119867 =119886

119886 119902 = minus

1

1198672

119886

119886 119895 =

1

1198673

119886

119886

119904 =1

1198674

119886

119886

(16)


= minus 1198672(1 + 119902) = 119867

3(119895 + 3119902 + 2)


(17)


120601 sim 120573119867119886120573 120601 sim 120573119886

120573( + 120573119867

2) = 120573119886

1205731198672(120573 minus 1 minus 119902)

119883 sim120573211986721198862120573

2 sim 120573

211986731198862120573(120573 minus 1 minus 119902)

(18)




1

1198664

[119870119883 minus 21198663120601 + 611986721198664119883 minus 6119867

21198665120601 + 21198664120601120601]

times 120573211986721198862120573+[31198671198663119883 minus 101198671198664119883120601+3119867

31198665119883+21198671198665120601120601]

times 120573311986731198863120573+ (6119867

21198664119883119883 minus 4119867

21198665119883120601)

times 120573411986741198864120573minus 211986721205731198861205731198664120601 + 119867

8120573511988651205731198665119883119883


411988621205731205732(120573 minus 1 minus 119902)

times 1198664119883 + 4119867411988621205731205732(1 + 119902)1198664119883 minus 4119867

612057341198864120573


+ 2120573311986741198863120573(120573 minus 1 minus 119902)1198664119883120601 + 2119867

6(1 + 119902) 120573

311988631205731198665119883


812057351198865120573



minus 21198674(1 + 119902) 120573

211988621205731198665120601 + (120588

119898+ 119901119898) ge 0

1

1198664

[120588119898+ 3119901119898+ 2119870 + 3120573

3119886312057311986741198663119883 minus 4120573

21198672

times 11988621205731198663120601 minus 3120573

311988631205731198674(120573 minus 1 minus 119902)1198663119883

minus 61198674120573211988621205731198664119883 + 3119867

6120573411988641205731198664119883119883

minus 18119867412057331198863120573119866119883120601 + 6119867

2120573 times 1198861205731198664120601 + 6119867

2120573119886120573


+ 21198678120573511988651205731198665119883119883 minus 3119867

6120573411988641205731198665119883120601 + 6119867

612057341198864120573


+ 12119867412057321198862120573(120573 minus 1 minus 119902)1198665120601 + 6119867

4120573311988631205731198665120601120601


+ 119867612057331198863120573(1 + 119902)1198665119883 minus 9119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883


minus 121198674120573311988631205731198664119883120601 + 6119867

2120573211988621205731198664120601120601


412057321198862120573


minus181198674120573211988621205731198664119883 + 120573

211988621205731198672119870119883] ge 0

1

1198664

[119867212057321198862120573119870119883 minus 119870+3119867

4120573311988631205731198663119883 minus 119867

2120573211988621205731198663120601

+ 121198674120573211988621205731198664119883 + 6119867

6120573411988641205731198664119883119883

minus 6119867412057331198863120573119866119883120601 minus 6119867

21205731198861205731198664120601

+ 51198676120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883 minus 9119867

4

times120573211988621205731198665120601 minus 3119867

6120573411988641205731198664119883120601 + 120588

119898] ge 0

1

1198664

[120588119898minus 119901119898+ 120573211986721198862120573119870119883 minus 2119870 + 3119867

4120573311988631205731198663119883


2119867411988621205731198664119883

+ 121198674120573211988621205731198664119883119883 minus 2119867

4120573311988631205731198664119883120601 minus 10119867

2

times 1205731198861205731198664120601 + 7119867

6120573311988631205731198665119883 + 119867

8120573511988651205731198665119883119883

minus 121198674120573211988621205731198665120601 minus 4119867

41205732times 11988621205731198665120601

+ 2119867412057321198862120573(1 + 119902)1198665120601 minus 2119867

4120573311988631205731198665120601120601

minus 4119867412057321198862120573119866119883120601 + 4 times 119867


minus 4119867412057321198862120573(1 + 119902)1198664119883 + 4119867

612057341198864120573



minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus 21198672120573211988621205731198664120601120601 minus 2119867

412057331198863120573(120573 minus 119902 minus 1)1198664119883120601

minus 2119867612057331198863120573(1+119902)1198665119883+3119867

612057331198863120573(120573 minus 1 minus 119902)1198665119883



(19)





F119879 = 2 [1198664 minus 119883(1206011198665119883 + 1198665120601)] gt 0


(20)


F119878 =1

119886

119889

119889119905(119886

ΘG2

119879) minusF119879 gt 0

G119878 =Σ

Θ2G2

119879+ 3G119879 gt 0

(21)



2

1199011198972)119877 +


119870 = 119870 (120601119883) 1198664 =1198722

119901119897

2 1198663 = 1198665 = 0

(22)

where 1198722119901119897




NEC 1

1198722119901119897

[2119883119870119883 + 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 minus 119870 + 120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[2119883119870119883 + 2119870 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[2119883119870119883 minus 2119870 + 120588119898minus 119901119898] ge 0

(23)



119870(120601119883) =1

2[1198622 minus 1198621 minus 2119861120601

2] +

1

2(1198621 + 1198622)119883

+1

21198724

0(119883 minus 1)

2

(24)



1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1

1198722119901119897

[1

2119883 (1198621 + 1198622) + 2119883119872

4

0(119883 minus 1)

minus1

21198724

0(119883 minus 1)

2minus1

2(1198622 minus 1198621 minus 2119861120601

2) + 120588119898] ge 0

1

1198722119901119897

[119883 (1198621 + 1198622) + 21198724

0119883(119883 minus 1) + 120588

119898+ 119901119898] ge 0

(25)

where 1198722119901119897




[1

2

120573211988621205731198672

2(1198621 + 1198622) + 2

120573211988621205731198672

21198724

0(120573211988621205731198672

2minus 1)

minus1

21198724

0(120573211988621205731198672

2minus 1)

2

minus1

2(1198622minus1198621minus2119861120601

2)+120588119898] ge 0

(26)

[120573211988621205731198672

2(1198621 + 1198622) + 2119872

4

0

120573211988621205731198672

2

times(120573211988621205731198672

2minus 1) + 120588

119898+ 119901119898] ge 0

(27)



4119909 can be


119870 =119872119901119897119883120596

120601minus 119881 (120601) 1198663 = 1198665 = 0

1198664 =1

2119872119901119897120601

(28)


NEC 1

119872119901119897

[

1206012

1206012119872119901119897120596 minus

119867 120601

120601119872119901119897 +

120601

120601119872119901119897 +

120588119898+ 119901119898

120601] ge 0

WEC 120588eff + 119901eff ge 0

1

119872119901119897

[120596119872119901119897

1206012

21206012+119881 (120601)

120601minus3119867 120601

120601119872119901119897 +

120588119898

120601] ge 0

SEC 120588eff + 119901eff ge 0

1

119872119901119897

[2

1206012

1206012119872119901119897120596 minus

2119881 (120601)

120601+3119867 120601

120601119872119901119897

+3

120601

120601119872119901119897 +

120588119898+ 3119901119898

120601] ge 0


1

119872119901119897

[2119881 (120601)

120601minus 5

119867 120601

120601119872119901119897

minus

120601

120601119872119901119897 +

120588119898minus 119901119898

120601] ge 0

(29)


120573 and 119881 =



1198810 ge 31198672

0120573 minus

120596

212057321198672

0

12057321198672

0120596 minus 119867

2

0120573 + 119867

2

01198861205731205732minus 1198861205731205731198672

0(1 + 1199020) ge 0

(30)





76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




76543

14145

15155

16 0

02

04

06081

119861

120573

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

14145

15155

16 0

02

04

06081

120573

119861

3252

151

120588eff

(b) 120588eff ge 0



0= 02 In both cases


2= 2

080604020minus1

minus08minus06

minus04minus02

0 minus2

minus19

minus18

minus17

minus16

minus15

120573

120596

120588eff

+119901eff

(a) 120588eff + 119901eff ge 0

54321

1214

1618

2

120573

120596

minus2

minus19

minus18

minus17

minus16

minus15

1198810

(b) 120588eff ge 0 (1198810 ge 0)


0= minus064


21198671206012+ 4120601 120601

2119867120601 + 1206012minus21206012(2119867 120601 + 2120601 + 120601)

(2119867120601 + 1206012)2

gt 119886120601

41206012(119883120596120601 minus 3119867

2120601 minus 119867 1206012

(2119867120601 + 1206012)2

) + 3120601 gt 0

(31)



2

1199011198972)119891(119877) + 119871119898]119889


action (6) for

119870 = minus1198722

119901119897

2(119877119891119877 minus 119891) 1198663 = 1198665 = 0

1198664 =119872119901119897

2120601 120601 = 119872119901119897119891119877

(32)


NEC ( minus 119867)11989110158401015840 (119877) + 2119891101584010158401015840 (119877) + 120588119898 + 119901119898 ge 0

WEC 120588eff + 119901eff ge 0

120588119898minus1

2(119891 (119877) minus 119877119891

1015840(119877)) minus 3119867119891

10158401015840(119877) ge 0


SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




SEC 120588eff + 119901eff ge 0

(119891 (119877) minus 1198771198911015840(119877)) + 3 ( + 119867)119891

10158401015840(119877)

+ 32119891101584010158401015840(119877) + 120588

119898+ 3119901119898ge 0


minus (5119867 + ) 11989110158401015840(119877) minus

2119891101584010158401015840(119877)

minus (119891 (119877) minus 1198771198911015840(119877)) + 120588

119898minus 119901119898ge 0

(33)


119891 (119877) = 119877[log (120572119877)]119902 minus 119877 120572 lt 0 119902 gt 0 (34)


minus 61198672

0(1 minus 1199020) [log (minus6120572119867

2

0(1 minus 1199020))]

119902

+ 61198672

0(1 minus 1199020)

+ 61198672

0(1 minus 1199020)


119902

+119902(log (minus612057211986720(1 minus 1199020)))

119902minus1


0


times [119902

611986720(1 minus 1199020)

(log (minus612057211986720(1 minus 1199020))

119902

)

+119902 (119902 minus 1)

611986720(1 minus 1199020)


119902minus2

] ge 0

(35)


20151051

1214

1618

2119902

minus2

minus18

minus16

minus14

minus12

minus1

120572

120588eff

120588eff ge 0


0= minus064119867

0= 0718 and 119895

0= 141



1198722

119901119897

2119877 +

1198722

119901119897

2


120583nabla120584120601)] + 119871119898)

(36)



NEC 1

1198722119901119897

[minus2119883 + 6119899119888119867 120601119883119899minus 2119888119899119883

119899 120601 + 120588119898+ 119901119898] ge 0

WEC 120588eff+119901eff ge 0 1

1198722119901119897

[minus119883+6119899119888119867 120601119883119899+120588119898] ge 0

SEC 120588eff + 119901eff ge 0

1

1198722119901119897

[minus4119883 + 6119899119888119867 120601119883119899minus 6119888119899119883

119899 120601 + 120588119898+ 3119901119898] ge 0


1

1198722119901119897

[6119899119888119867 120601119883119899+ 2119888119899119883

119899 120601 + 120588119898minus 119901119898] ge 0

(37)


5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




5

0

112

1416

182 0

02

04

06081

120573

119899

120588eff

(a) 120588eff ge 0

10501

1214

1618

2 0

02

04

06081

120573

119899

120588eff

+119901eff

(b) 120588eff + 119901eff ge 0



[minus120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

] ge 0

[minus2120573211988621205731198672

2+ 6119899119888119867 120601(

120573211988621205731198672

2)

119899

minus2119888119899(120573211988621205731198672

2)

119899

120601] ge 0

(38)


1198672(2 + 119902) minus

119888119899119867119883119899 120601

119886+ 1198881198992119883119899minus1 120601

+ 119888119899119883119899 120601 gt (119867 minus 119888119899119883

119899 120601)2

(minus119883 + 12119888119899119867119883

119899 120601 + 6119899119888 (119899 minus 1)119867119883119899 120601 minus 3119867

21198722

119901119897

(1198671198722119901119897minus 119888119899119883119899 120601)

2)

times1198722

119901119897gt minus3

(39)



119901119897120601 gt 0


119870 = minus1198882119883 1198663 =1198883

1198723119883 1198664 =

1198722

119901119897

2minus1198884

11987261198832

1198665 =31198885

11987291198832

(40)


NEC 1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus21198831198882 + 61198831198671206011198883

1198723minus 72119867

21198832 1198884

1198726

+ 6011986731198832 1198885

1198729minus 2119883 120601

1198883

1198723+ 24119867119883

1198884

1198726

+ 161198832 1198884

1198726minus 12119883

2times (2119867 120601 + 3119867

2 120601)1198885

1198729

minus2411986721198832 120601

1198885

1198729+ 120588119898+ 119901119898] ge 0

WEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus1198831198882 + 61198831198671206011198883

1198723minus 96119867

21198832 1198884

1198726

+8411986731198832 120601

1198885

1198729+ 120588119898] ge 0


SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




SEC 120588eff + 119901eff ge 0

1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [minus41198831198882 + 61198831198671206011198883

1198723

minus 6119883 1206011198883

1198723+ 24119883119867

2 1198884

1198726minus 48119867

21198832 1198884

1198726

+ 1211986731198832 120601

1198885

1198729minus 180119867

21198832 120601

1198885

1198729minus 72119883

2119867 120601

times1198885

1198729+ 48119883119867

1198884

1198726+ 48119883

21198884

1198726+ 24119867119883

times1198884

1198726+ 72119867

21198832times1198884

1198726+ 120588119898+ 3119901119898] ge 0


1

2 ((11987221199011198972) minus (1198884119872

6)1198832)

times [6119883119867 1206011198883

1198723+2119883 120601

1198883

1198723minus72119867

21198832 1198884

1198726minus48119883119867

2 1198884

1198726

+ 10811986731198832 120601

1198885

1198726minus 24119867119883

1198884

1198726minus 16119883

2 1198884

1198726

+ 121198832(2119867 120601 + 2119867

2 120601)1198885

1198729+ 24119867

21198832 120601

1198885

1198729

+120588119898minus 119901119898] ge 0

(41)


1

(1 minus (119888412057341198674021198726))

times [minus11988821198672

01205732+31198674

011988831205733

1198723minus181198676

012057341198884

1198726

+151198677

01205731198885

1198729minus 12057331198674

0(120573 minus 1 minus 1199020)

1198883

1198723

+ 241198676

01205734(120573 minus 1 minus 1199020)

1198884

1198726minus 16119867

6

01205734times (1 + 119902)

1198884

1198726

minus 31198674

01205734(31198674


4

0120573 (1 + 119902))

1198885

1198729

minus61198678

01205735(120573 minus 1 minus 1199020)

1198885

1198729] ge 0

1

(1 minus (119888412057341198674021198726))

times [minus1198672

012057321198882

2+311988831198674

01205733

1198723minus241198676

012057341198884

1198726+120588119898

0] ge 0

(42)




F119879 = 1198722

119901119897minus 21198832 1198884

1198726minus 12119883

2 1206011198885

1198729gt 0

G119879 = 1198722

119901119897+ 61198832 1198884

1198726minus 12119883

2 1206011198671198885

1198729gt 0

(43)



5 Summary





500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




500

minus50minus100

665

775

8 minus5

minus4

minus3

minus2

minus1

120573

120588eff

+119901eff

119888

(a) 120588eff + 119901eff ge 0

400200

0

665

775

8minus50

minus40

minus30

minus20

minus10

120573

120588eff

119888

(b) 120588eff ge 0


3= 2 1198884= 4 1198885= 3 and119872 = 2









61198884)14 and 1198882 lt 0



Appendix

119879(120601)

120583120584=1

2119870119883nabla120583120601nabla120584120601 +

1

2119870119892120583120584 minus

1

21198663119883◻120601nabla120584120601nabla120583120601


2119892120583120584nabla1205821198663nabla

120582120601 +

1

21198664119883119877nabla120583120601nabla120584120601

+1

21198664119883119883 [(◻120601)


2




120582120601nabla120583nabla120584120601




1

21198664119883 [(◻120601)


2

]]



120582120601◻120601]


+ 1198664120601120601nabla120583120601nabla120584120601



120573




1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1

21198665119883119877120572120573nabla


times1

21198665119883119877120583120584120572120573nabla


120582120601



1

2


1

2nabla(120583 [1198665120601nabla120584)120601] ◻120601


1

2




120582120601 minus

1

2nabla(1205831198665119883nabla120584)120601


2

] + nabla1205821198665119877120582(120583nabla120584)120601


120573nabla120584)120601 + nabla1205731198665119883



1

2nabla120572120601


21198665119883119866120572120573


1


120572nabla120584120601

minus1

21198665119883(◻120601)


1

121198665119883119883


2

+ (nabla120572nabla120573120601)3

]


2nabla1205821198665119866120583120584nabla

120582120601 minus 119892120583120584

times minus1

61198665119883 [(◻120601)


2

+ (nabla120572nabla120573120601)3

]


1

2nabla120572 (1198665120601nabla

120582120601) ◻120601

+1


120572nabla120573120601 minus

1

2nabla1205721198665119883nabla

120572119883◻120601

+1


120572nabla120573120601 minus

1

4nabla1205821198665119883nabla120582120601


2

] +1

21198665119883119877120572120573


1

21198665119883119877120572120582120573120588

(A1)

References


































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Energy Conditions in a Generalized Second-Order Scalar-Tensor Gravity - Hindawi