M.A. Dariescu et al- Thermodynamics of Bosons in an Universe with Global Pathology

8/3/2019 M.A. Dariescu et al- Thermodynamics of Bosons in an Universe with Global Pathology

1/10

Romanian Reports in Physics, Vol. 61, No. 3, P. 417426, 2009

Dedicated to Professor Ioan Gottliebs

80th anniversary

THERMODYNAMICS OF BOSONS IN AN UNIVERSEWITH GLOBAL PATHOLOGY

M. A. DARIESCU, C. D. DARIESCU, A. C. PRGHIE

Faculty of Physics, Al. I. Cuza University

Bd. Carol I, no. 11, 700506 Iai, Romania,E-mail: [email protected]

(Received June 4, 2009)

Abstract. The aim of the present paper is to investigate an exact class of solutions belonging tothe Plebanski-PetrovD [2S 2T](11) type with a G6 = VII0 VIII group of motion, supported by asuitable matter-source. In the spacetime endowed with g44 = cosh

2(z), we are pointing out someunusual (pathological) features and solve the KleinGordon equation for the massless bosons. Finally,within a thermodynamic analysis, we derive the characteristic function and the main thermodynamicquantities.

Key words: globally pathological manifolds, Gordon-type equation, thermodynamic properties.

1. THE GEOMETRY

Studies of supernova explosions suggesting an accelerating rate of theexpanding Universe as well as the largest computer simulation of the evolution ofthe Universe, performed by the Virgo consortium two years ago, have revived talkof Einsteins cosmological constant. Although the existence of a non-zero value forthe cosmological constant has come into play previously, now it has gone hand inhand with the increased accuracy of observational data on the distribution ofclustering that, for the first time, perfectly matched the theoretical results [1]. Onthe other hand, in the past decades, a wide interest has been focused on the globally

pathologic manifolds, and radical changes have occured in understanding gravity,matter fields and spacetime [26]. In this respect, not only intensive studies on thecosmic strings, naked singularities, Bianchi spacetimes, dynamical isotropizationor topological domain walls have been the main targets [79], but also black holesin less than four dimnensions [10], revealed new intriguing features due to theircausal structure singularities, such as the closed timelike curves and/or additionalTaub-Nut pathologies at the metricsingular point.


2/10

M. A. Dariescu, C. D. Dariescu, A. C. Prghie 2418

A brief overview of Universes with VII0 VIII isometries pointed out howthe extreme pathology induced by the cosmic temporal trap led to unexpected behaviours of electric or magnetic static modes, such as the appearance of agravitoelectromagnetic resonance, in some spatially finite regions [11]. Moreover,our analysis has led to similar results as for theBTZblack hole, where the reducedone-dimensional motion of the test particle evolves in a parabolic well periodicallycrossing the two r = 1 horizons. As the problem of time remains verycontroversal, it turns out that it is possible to get cosmic-time traps and temporallyimprisoned geodesics even when the metric contains no singular points.

Our interest being motivated by the recent investigations on the eigenvalue problem of scalar fields in BTZ black holes [12], in the present work are arewriting down the univoc-regular solutions to the Gordon equation, using the same

geometry as in the Ref. [11]. Since one of the most attractive aspects of black holephysics is its thermodynamical properties, in the final section, we are deriving thecharacteristic function and the main thermodynamic quantities, for the massless

bosons evolving in a spacetime described by a metric withg44= cosh2(z).Let us start with the metric

( ) ( )222d d d e d ; , 1,3f zs x x t = = (1)

and, using the dually-related pseudo-orthonormal tetrads

( ) ( ) 44, e ; d , e d ,

f z f z e e t x t = = = =

we arrive at the following essential components of the Riemann, Ricci and Einstein

tensors

( )2

3434 33 44 33 3

1

2R R R R f f = = = = + (2)

( )2

33 3 , , 1, 2.AB ABG f f A B = + =

(3)

As the total energy density, T44, should be zero and any conventional source

possesses a positive energy density ( )44cs

T , one must necessarily use a false vacuum

state described by ( )ab abT = , with > 0, in order to get ( )44 0

csT = .

Considering, for simplicity, the conventional matter as being characterized by apressureless ideal fluid at rest, i.e. ( )dab a bT u u= and ua = a4, it obviously results

= , such that ( ) 4 4d

ab a bT = . To also get G33 = 0 without violating the rest of theGab values, we need an extra source that can be thought of as a global cosmic stringof unitary elongation effort = along the Xa = a3 direction. Hence, the totalenergy-momentum tensor


3/10

3 Thermodynamics of bosons in an Universe with global pathology 419

[ ]4 4 3 3ab a b a b abT = + , with > 0, (4)

describes a combined matter-source made of stuck universal dust, with = , on az-directed global string immersed in a medium of negative energy density andequal positive pressure that floods everything all around. Finally, the conservationlaw : 0

ab

bT = requires a constant and expresses the (false) vacuum-typecontribution as a true -term.

Under these assumptions, the Einstein equations turn to the essential one,

( )2

233 3 ,f f+ = (5)

where ( )1/ 2

0 = , which is satisfied by the general solution which brings themetric (1) to the hyperbolic form

( ) ( ) ( )2 22 4 2d d d d cosh d , , 1, 2BABs x x z z t A B= + = . (6)

As it can be noticed, the above metric is defined on M4 =R2M2and is free of finitesingular points. According to EstabrookEllisMacCallum method [13], byanalyzing the invariant properties characterized by 0A C = = and

1

2N C A = , we conclude that we deal with a

6 3 3 0VII VIIIG E G = = group of motion, acting on the manifold described by (6).

By computing the timelike geodesics,

( ) ( ) ( ) ( ) ( )2 20 01

ln sinh cos sinh cos 1 ,z z z = + +

( )( )

( )1 2

0

tan1arctan , with ,

cosh 2 2t t t

z

= = < < =

(7)

represented in Fig. 1, one points out a genuine temporal imprisonment, since they

cannot be emitted earlier than 1 2t

=

, or extended to the future beyond

2 2t = , getting trapped inbetween these two universal moments.

As a matter of fact, in 5D-dimensional models, such hyperbolic metrics havebeen used for the analysis of test particles trajectories and the so-called trappingsolution of multidimensional Einstein equations, cosh(x5), has been considered forexplaining the matter-confinement [14], in agreement with the anzats of RubakovShaposhnikov [15].


4/10


2. SCALAR FIELD QUANTIZATION

In the spacetime endowed with the metric (6), the Gordon equation,

20

1 0

ik

i kg g m

x xg

=

, (8)

does actually read

( )( )

2 2 2 2202 2 2 2 2

1tanh 0

coshz m

zx y z z t

+ + + =

, (9)

wherex1

=x,x2

=y,x3

=z,x4

= t. Thence, it comes to be written as

( )( )

( )2 2

202 2 2

1tanh 0

coshz m

zz z t

+ + =

,

where2 2

2

2 2AB

ABx y

= = +

. Since the three operators, , , ,

t x y

commute

with each other and also with the main operator

( )( )

2 2 2 2202 2 2 2 2

1 tanh cosh

D z mzz z t x y

= + + +

,

they share a common set of eigenfunctions, respectively for the positivefrequency and planary out-going modes

i i ie e e , 1, 2AA

AA

p x t p x t A = = , so that

the field can be expressed as

( ) ( ) ( ) ( )i

, , e , where , ,A

Ap x t A

Ax z t F z p

= = (10)

the amplitude function F(z) being the regular solution, on { }z = , of thecorresponding differential equation,

( )( )

( )2 2

2 202 2

d dtanh 0

dd cosh

F Fz p m F

zz z

+ + + =

, (11)

with 2 AB A Bp p p = . Dividing by 2and defining

( )2 2 2021

, ,z p m

= = = +

,

it yields the form


5/10


( ) ( ) ( )2 2 2 2 2dd1 1 1 0d d

Fs s s F s s

+ + + + = ,

with respect to the integration variable s = sinh, wheres , forz (,).

This suggests the change of variable ( ) arctan2

s

= , so that it becomes

2 22

2 2

d 0

d sin

FF

+ =

.

Finally, employing the function substitution sinF G = , one gets for G the

well-known Legendre (generalized) equation [16], which basically sets the univoc-regular solution ( )cosmG P = , with m and |m| , , where

( )21 1

14 2

= + = + ,

1/ 2

2 21 1 14 2 2

m m m + = = +

.

As it can be noticed, even for the massless case, 1 p =

|, m is restricted by

the inequality |m| 1, so that, because of |m| , the second quantum number must (compulsory) take only natural values, = 1, 2, 3, . . . Thence, with respect

to the local coordinates ( )3, arctan sinh ,2

Ax x t =

, the positive-

frequency modes of take on the form (inducing the normalization constant N ),

( ) ( )

( )

, , , sin cos

1exp i cos sin ,

2

m

mu x y t P

p x y t

=

+ +

N

(12)

where

1/ 2

2 20

1 1

2 2p m m m

= +

,

[1/ 22

02

10, 2 ), Int , and .

4

mm m

+

(13)


6/10


3. THERMODYNAMIC PROPERTIES OF MASSLESS BOSONS

Because of (0, 2) in the momentum space the one-particle stateswith m = |m|, and m , have been already considered. Therefore, in the

massless case, m0 = 0, it yields1/ 22 1/ 4 0p m = > , meaning that m 1 and

0 < m , where = 1, 2, 3, . . . . Thus, the actual (proper) degeneration of

each energy-level1

2 = +

is g = , coming from ( )1

2 1 12

g = + ,

because m = 0 is forbidden and we also have to account only for half of theeigenfunctions, as the azimuth in the momentum space runs from 0 to 2.

Thence, the one-particle partition function does effectively become1

2

1 1e e .Z g

+

= == =

(14)

Inserting the physic dimensionless parameter = , it can be written as

2

1e eZ

==

yielding

( )

2

2

e.

e 1

Z

= (15)

The corresponding free-energy becomes

1 2 ln ln e 1 2

Z

= F , (16)

and the one-particle entropy is going to read,

( )2 2 22 e 2

ln 1 ,e 1

S e

= =

F(17)

i.e. it comes to the (simple) expression (in terms of = )

( )2 ln e 1 .e 1S

= + (18)

As it can be noticed, at T= 0+, where , it readily becomes

( ) ( )

02 2 lim e 0,T

TS e

T+

= + = =


7/10


and therefore it fulfills the Third Thermodynamical Principle. Concerning its sign,one can take the other limit, at large (positive) values of T, where 0+, andobtains

( )0 2 2 ln ,S + =

which goes to + as runs into 0+. Also, taking the derivative of S, it results

( )2

d e2 0,

d e 1

xS

=

Documents

M.A. Dariescu et al- Thermodynamics of Bosons in an Universe with Global Pathology