Boundary Terms, Variational Principles and Higher ... · Boundary Terms, Variational Principles and Higher Derivative Modi ed Gravity Ethan Dyer1 and Kurt Hinterbichler2 Institute

Boundary Terms Variational Principles andHigher Derivative Modified Gravity

Ethan Dyer1 and Kurt Hinterbichler2

Institute for Strings Cosmology and Astroparticle Physics

and Department of Physics

Columbia University New York NY 10027 USA

We discuss the criteria that must be satisfied by a well-posed variationalprinciple We clarify the role of Gibbons-Hawking-York type boundary termsin the actions of higher derivative models of gravity such as F (R) gravity andargue that the correct boundary terms are the naive ones obtained thoughthe correspondence with scalar-tensor theory despite the fact that variationsof normal derivatives of the metric must be fixed on the boundary We showin the case of F (R) gravity that these boundary terms reproduce the correctADM energy in the hamiltonian formalism and the correct entropy for blackholes in the semi-classical approximation

1esd2107columbiaedu2kurthphyscolumbiaedu

1 Introduction and outline

The Einstein-Hilbert action for general relativity (GR) is

SEH simintd4xradicminusgR (11)

When varying this action one finds surface contributions that must vanish ifthe action is to be stationary The surface contributions contain the metricvariation δgmicroν and variations of the derivatives of the metric δ(partσgmicroν) Settingδgmicroν = 0 on the boundary is not sufficient to kill all the surface contributionsFixing both the metric and the derivatives of the metric on the boundaryis uncomfortable however it may not be initially obvious why this is so orthat there exists a unique and correct prescription for dealing with it

Gibbons Hawking and York proposed adding the trace of the extrinsiccurvature of the boundary K to the action [1 2] With this modificationthe action takes the form

S = SEH + SGHY simintd4xradicminusgR + 2

∮d3xradic|h|K (12)

h is the determinant of the induced metric on the boundary This modifica-tion is appealing because the variation of the Gibbons-Hawking-York (GHY)boundary term cancels the terms involving δ(partσgmicroν) and so setting δgmicroν = 0becomes sufficient to make the action stationary This has been widely ac-cepted as the correct modification to the action

This modification raises some questions Is the modification unique Isit necessarily incorrect to require δ(partσgmicroν) = 0 If this is incorrect why is itappropriate to require the fixing of all components of gmicroν when the gravitonhas only two degrees of freedom More generally what criteria determinewhich quantities to fix on the boundary Should it be related to the numberof degrees of freedom in the theory

Despite these questions the GHY term is desirable as it possesses a num-ber of other key features The term is required to ensure the path integralhas the correct composition properties [1] When passing to the hamiltonianformalism it is necessary to include the GHY term in order to reproduce thecorrect Arnowitt-Deser-Misner (ADM) energy [3] When calculating black

hole entropy using the euclidean semiclassical approach the entire contribu-tion comes from the GHY term [44] These considerations underscore thenecessity of a complete understanding of boundary terms

Higher derivative theories of gravity have attracted attention recentlymostly as modifications to GR that have the potential to explain cosmicacceleration without vacuum energy A simple example of such a modificationis F (R) gravity where the action becomes some arbitrary function F of theRicci scalar (for reviews see [4 5])

S simintd4xradicminusgF (R) (13)

It is well known that this theory is dynamically equivalent to a scalar-tensortheory [25 26 27 28 30] By extending this equivalence to the GHY termwe find that the F (R) action is left with a boundary termint

d4xradic|g|F (R) + 2

∮d3xradic|h|F prime(R)K (14)

This boundary term must be present if the correspondence to scalar-tensortheory is to hold at the boundary

This term has been arrived at before both directly and indirectly and inseveral different contexts [15 16 17 18 19] There has been some confusionas this boundary term does not allow δ(partσgmicroν) to remain arbitrary on theboundary Various ways around this have been attempted for example whenrestricting to maximally symmetric backgrounds a boundary term for F (R)theory can be found that allows δ(partσgmicroν) to remain arbitrary on the boundary[18]

F (R) theory is a higher derivative theory however and the derivatives ofthe metric encode true degrees of freedom δ(partσgmicroν) should not remain arbi-trary on the boundary Instead δ(partσgmicroν) must be subject to the constraintthat the variation of the four-dimensional Ricci scalar be held fixed on theboundary This corresponds to holding the scalar field fixed in the equivalentscalar-tensor theory We will show that the above boundary term reproducesthe expected ADM energy upon passing to the hamiltonian formalism andthe expected entropy for black holes

In what follows we will elaborate upon the above in detail

An outline of this paper is as follows In section 2 we go over the criteriathat are necessary to have a well-posed variational principle In section 3we examine several toy examples of lagrangians in classical mechanics thatshare some of the features of the more complicated GR case We discusswhat adding total derivatives can do to the variational principle and whathappens in higher derivative theories We also discuss the complications dueto constraints and gauge invariance and we work out the case of electro-magnetism In section 4 we review the GHY term its variation and how itrenders the GR action well-posed In section 5 we discuss higher derivativemodified gravity theories scalar-tensor theories and their equivalence usingF (R) theory as the prime example We find the boundary terms for F (R)theory using this equivalence In section 6 we derive the hamiltonian formu-lation of scalar-tensor theory and F (R) theory keeping all boundary termsand show how the boundary term is essential for obtaining the ADM energyIn section 7 we calculate the entropy of a Schwartzschild black hole in F (R)theory using the euclidean semi-classical approach and compare the resultto the Wald entropy formula We conclude in section 8 and some formulaetheorems and technicalities are relegated to the appendices

Conventions We use the conventions of Carroll [20] These include us-ing the mostly plus metric signature (minus+++ ) and the following defi-nitions of the Riemann and Ricci tensors Rκ

λmicroν = partmicroΓκνλminuspartνΓκmicroλ + ΓκmicroηΓηνλminus

ΓκνηΓηmicroλ Rλν = Rmicro

λmicroν The weight for anti-symmetrization and symmetriza-

tion is eg A[microν] = 12

(Amicroν minus Aνmicro) For spacelike hypersurfaces the normalvector will always be inward pointing For timelike hypersurfaces it will beoutward pointing The dimension of spacetime is n Further conventions andnotation for foliations of spacetime are laid out in appendices A and B

2 What makes a variational principle well-

A traditional approach to field theory is to integrate by parts at will ignor-ing boundary contributions The reasoning is that if there are no physicalboundaries in the space under consideration or if they are so far away thattheir effects can be expected not to interfere with the system under studythen they can be ignored

If one is interested only in the local equations of motion of a theory thisis a valid approach In this case one uses the action only as a formal devicefor arriving at the equations of motion Denoting the fields collectively byφi one writes down the lagrangian which is a local density

L ([φ] x) (21)

Here [φ] stands for dependence on φi and any of its higher derivatives [φ] equivφi partmicroφ

i partmicropartνφi and x are the coordinates on spacetime One then defines

the equations of motion to be the Euler-Lagrange equations

δELLδφi

= 0 (22)

whereδEL

δφi=

partφiminus partmicro

part(partmicroφi)+ partmicropartν

part(partmicropartνφi)minus middot middot middot

is the Euler-Lagrange derivative and the symmetric derivative is defined by

part(partmicro1 partmicrokφi)partν1 partνkφ

j = δijδmicro1

(ν1middot middot middot δmicrokνk) (23)

(The symmetric derivative is to avoid over-counting multiple derivativeswhich are not independent)

All the relevant local theory ie equations of motion symmetries con-served currents gauge symmetries Noether identities etc can be developedin terms of the lagrangian density alone without ever writing an integral andwithout consideration of boundary contributions For example Noetherrsquostheorem can be stated as an infinitesimal transformation δφi ([φ] x) is asymmetry if the lagrangian changes by a total derivative

δL = partmicrokmicro (24)

from which one can show directly that the following current is conserved onshell

jmicro = minuskmicro +nsumr=1

partmicro1 partmicrorminus1δφi

nsuml=r

(minus1)lminusrpartmicror partmicrolminus1

partSLpart(partmicropartmicro1 partmicrolminus1

(This reduces to jmicro = δφi partLpart(partmicroφi)

minus kmicro in the usual case where the lagrangian

depends at most on first derivatives of the fields)1

Without specifying boundary conditions the Euler-Lagrange equations(22) typically have many solutions (if there are no solutions the lagrangianis said to be inconsistent) To select out the true solution boundary condi-tions must be set There should be some class of boundary conditions thatrender the system well-posed A class of boundary conditions is well-posedif given any choice of boundary conditions from this class there exists aunique solution to the system compatible with that choice The equations ofmotion are typically hyperbolic so the class will generally involve all possi-ble choices of initial conditions and velocities for the fields and all possiblechoices of spatial boundary conditions at all times There may be severaldifferent classes of well-posed data but each class will involve specifying thesame number of boundary data This number is the number of degrees offreedom in the theory

The choice of spatial boundary conditions generally corresponds to achoice of ldquovacuum staterdquo for the theory For example in GR if spatial bound-ary conditions (fall-off behavior for the fields in this case) are chosen so thatthe spacetime is asymptotically flat we find ourselves in the asymptoticallyflat vacuum of the theory A choice of initial condition generally correspondsto a choice of state within the vacuum For example both Minkowski spaceand the Schwartzschild black hole have the same spatial boundary behaviorso they are both in the asymptotically flat vacuum of GR but they havedifferent initial data and so they represent different states Thus from thepoint of view that the action is just a formal device to arrive at the localequations of motion information about the possible vacua of the theory andthe space of states in each vacuum is not encoded directly in the actiononly indirectly through the boundary conditions required of the equations ofmotion

Ideally the action should do more We want to say that the true field

1If one wishes still keeping with this line of thought the action integral can be intro-duced as a formal device

S =intdnx L ([φ] x) (26)

The integration region need not be specified and the equations of motion are obtained bysetting δS = 0 and integrating by parts ignoring all boundary contributions

Figure 1 Region over which a field theory is defined

configuration among all field configurations with given boundary data ex-tremizes the action Consider a spacetime region V such as that in figure1 The boundary of this region consists of a timelike part B (which may beat infinity) which we will refer to as the spatial boundary and two space-like edges Σ1 and Σ2 which we will refer to as the endpoints We considerconfigurations of the fields φi in V that have some given boundary valuesφi(B) as well as initial and final values φi(Σ1) φi(Σ2) (possibly involvingderivatives of the fields or other combinations) chosen from a class which iswell-posed meaning that for each choice in the class there exists a uniquesolution to the equations of motion2

The action is a functional of field configurations in V

S = S[φ(V)] (27)

in the form of an integral over over V of the lagrangian L ([φ] x)

S[φ(V)] =

intVdnx L ([φ] x) (28)

2Well-posedness of hyperbolic equations usually requires initial data such as the valuesof φi and its time derivatives at Σ1 rather than endpoint data such as the value of φi atΣ1 and Σ2 The action naturally wants endpoint data however As we will elaborate onin the next section endpoint data is usually just as well posed as initial data with theexception of ldquounfortunate choicesrdquo which are measure zero in the space of all choices

The criteria for a well-posed action should be the following If we restrictthe action to those field configurations consistent with our choice of boundaryand initial data

intVdnx L ([φ] x)

∣∣∣∣φi consistent with data φi(B)φi(Σ1)φi(Σ2)

the unique solution to the Euler-Lagrange equations should be the only ex-tremum of the action It is important that the space of paths over which wevary be all paths that have the given boundary data

Of course a necessary condition for the action to be stationary is thatthe field satisfy the Euler-Lagrange equations

δELLδφi

= 0 (210)

For the action to be truly stationary any boundary contributions arisingfrom the variation must vanish No conditions other than those implicitin the class of well-posed data may be used in checking whether boundarycontributions vanish

There will be many possible choices of the boundary data φi(B) (corre-sponding to different choices of vacuum) and endpoint data φi(Σ1) φi(Σ2)(corresponding to different choices of the state) within a well-posed classEach such choice leads to a different set of paths to consider in the actionand a well-posed action will give a unique stationary field configuration foreach The action when evaluated on the classical path φic is thus a welldefined functional on the class of possible boundary data

S[φ(B) φ(Σ1) φ(Σ2)]|φ=φc=

intVdnx L ([φ] x)

∣∣∣∣φ=φc

The number of physical degrees of freedom in the theory is the number offree choices of φi(partV) (modulo gauge invariance see later) ie the size ofthe class of well-posed boundary data

These criteria are essential to setting up a Hamilton-Jacobi theory wherethe stationary value of the action as a function of endpoint data gives thecanonical transformation that solves the equations of motion [22] These

criteria are also essential to setting up a quantum theory based on the pathintegral [1]

We will now turn to some examples We will look at some actions whichare well-posed in the sense we have just described and some which are notWe will see that adding boundary terms or equivalently total derivativesto the action affects whether the action can be well-posed Second classconstraints and gauge freedom complicate the issue and we will have moreto say about these as well

3 Some Toy Examples

We start with some simple examples from point particle mechanics These areone-dimensional field theories with fields qi(t) that depend on one parameterthe time t The issue of spatial boundary data does not arise in this caseit is only the endpoint data qi(t1) and qi(t2) that we need to worry about3

The number of degrees of freedom in the theory is half the number of freechoices that may be made in specifying endpoint data

31 Standard classical mechanics

The best example of a well-defined variational principle is an ordinary uncon-strained lagrangian system for n variables qi(t) i = 1 n without higherderivatives

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

The action is a function of the endpoint data qi1 qi2 and the endpoint times

t1 t2 The set of paths over which we vary is all qi(t) satisfying qi(t1) = qi1

3 In this sense one dimensional theories do not possess different vacua only a space ofstates given by the various values the endpoint data are allowed to take

and qi(t2) = qi2 Varying the action we have

int t2

[partL

partqiminus d

partqi

]δqi +

partqiδqi∣∣∣∣t2t1

The boundary variation vanishes since the qi are fixed there so the nec-essary and sufficient condition for the action to be stationary is that theEuler-Lagrange equations of motion hold

partqiminus d

partqi= 0 (34)

or upon expanding out the time derivative

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

Because of (32) we can solve algebraically for the highest derivative qi interms of the lower derivatives and we have a well formed initial value prob-lem given a choice from the class of data qi(t1) and qi(t1) we will have aunique solution to the equations of motion

However the action principle is not well suited to the initial value classIt is suited to the class where we choose endpoint data ie where we fix qi

on the two endpoints but do not specify the qi anywhere For a second orderdifferential equation such as (35) the endpoint problem will be well posedfor all but a few ldquounfortunaterdquo choices of the endpoint data (borrowing thelanguage of [23]) An example of such an unfortunate choice is the followingTake the action of a simple harmonic oscillator in one dimension with unitmass and angular frequency ω S =

int t2t1dt 1

2(q2 minus ω2q2) The equation of

motion is q + ω2q = 0 If the time interval is chosen to be a half integermultiple of the period of the oscillator t2 minus t1 = nπ

ω n = 1 2 and we

choose q1 = q2 = 0 then there will be an infinite number of distinct solutionsto the equations of motion compatible with the endpoint data This reflectsthe fact that the period of oscillations is independent of the amplitude Itis not hard to think of other such examples of ldquounfortunaterdquo choices Theproblem is that the equations of motion want initial conditions but the actionprinciple wants endpoint conditions and the result is a failure of the actionprinciple to be well defined for all choices

These unfortunate choices will make up a set of measure zero among allpossible choices for the endpoint data This is because finding degeneracies inthe initial value formulation is equivalent to an eigenvalue problem over theinterval t2 minus t1 and the eigenvalues are discrete We will still consider sucha variational principle well-posed With this caveat the action principleabove using the class of endpoint data is well-posed because for almostany choice of endpoint data qi1 q

i2 t1 t2 there is a unique solution to the

equation of motion corresponding to the unique path which extremizes theaction The action evaluated on this path becomes a well defined functionalof qi1 q

i2 t1 t2 almost everywhere and so the number degrees of freedom (ie

the size of the class of data) or the possibility of differentiating the actionwith respect to endpoint data is not affected There are 2n free choicesamong the field endpoint values so the model has n degrees of freedom

Going to the hamiltonian poses no problem We introduce fields pi(t)

pi =partL

partqi (36)

which by virtue of (32) can be inverted to solve for qi = qi(p q) We thenwrite the action

int t2

dt piqi minusH (37)

whereH(p q) =

i minus L)∣∣qi=qi(pq)

The variation with respect to the pirsquos is done without fixing the endpointsso their endpoint values are not specified and the action is not a functionof them The equations of motion for p are (36) which upon pluggingback into the action reproduces the original lagrangian action (the transitionto the hamiltonian is a special case of the fundamental theorem of auxiliaryvariables see appendix D) Note that adding a total derivative d

dtF (q) of any

function of the qi to the original lagrangian does not change the variationalprinciple or the equations of motion but does change the canonical momentain a way that amounts to a canonical transformation

32 Constraints

Consider now a lagrangian that does not satisfy det(

part2Lpartqipartqj

)6= 0 This is

the case for essentially all theories of interest to most high energy physicistswith the exception of scalar field theories on fixed backgrounds The theorythen has constraints and the Dirac constraint algorithm or some equivalentmethod must be applied [24 23]

As an example consider a theory with a free point particle of mass mlabeled by q1 along with a harmonic oscillator of mass M and angular fre-quency ω labeled by q2 coupled through a derivative interaction λq1q2

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

This lagrangian is free of constraints except when λ = plusmnradicMm in which

case there is a single primary constraint and a single secondary constraintwhich taken together are second class It is easy to see why these values ofλ are special because for these values we can factor the action

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

after which a change of variables qprime equivradicmq1 plusmn

radicMq2 shows that qprime behaves

as a free point particle while q2 which has no kinetic term is constrained tobe zero

Varying the action we find the endpoint term(radicmq1 plusmn

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

The action is stationary if we choose the quantityradicmq1plusmn

radicMq2 to be fixed

on the endpoints q2 must be chosen to be zero so choosing this quantity isequivalent to choosing q1

This illustrates what should be the case in general with constrained sys-tems The variations of the unconstrained variables are fixed on the bound-ary and the constrained variables are not fixed Each choice of endpoint

data for the unconstrained variables determines a choice of data for the con-strained variables which then determines a unique solution to the equationsof motion The action is thus a function of the endpoint data for the uncon-strained variables The action should be such that even though all variablesmay appear in the endpoint variation the number that must be fixed tomake the action stationary is fewer equal in number to the number of un-constrained degrees of freedom

33 Gauge invariance

Gauge invariance also complicates the well-posedness of variational princi-ples Consider the following action for two variables q1 and q2

int t2

2(q1 minus q2)2 (312)

Varying with respect to q1 gives the equation of motion q1 = q2 with noendpoint contribution q1 can thus be eliminated as an auxiliary field andplugging this back into the action gives identically zero The general solutionto the equation of motion is to let q2 be a completely arbitrary function oft and then set q1 = q2 The presence of arbitrary functions in the generalsolution to the equations of motion is the hallmark of a gauge theory andindeed this theory is invariant under the gauge transformation δq1 = ε δq2 =ε where ε(t) is an arbitrary function of t

Arbitrary functions in the solution also indicate that the action principleis not well defined as it stands Given endpoint data we can make a gaugetransformation that changes the solution somewhere in the bulk away fromthe endpoints and so the solution for the given endpoint data will not beunique We salvage uniqueness by identifying as equivalent all solutionswhich differ by a gauge transformation For example given any q2(t) wecan without changing the endpoint values q2(t1) q2(t2) bring it to the gaugeq2 = 0 by making a gauge transformation with the gauge parameter ε(t)obtained by solving ε + q2 = 0 subject to the endpoint conditions ε(t1) =ε(t2) = 0 Then given any choice of q2(t1) q2(t2) there is a unique solutioncompatible with q2 = 0

The gauge condition q2 = 0 still allows residual gauge transformationsthose where ε(t) is linear in t These gauge transformations do not vanish at

the endpoints If a gauge transformation does not vanish at the endpointswe must identify endpoint data that differ by such a gauge transformationIn the case above q2(t1) and q2(t2) can be set to any desired value by aresidual gauge transformation so all the q2 endpoint data is to be identifiedinto a single equivalence class The q1 data is constrained so in total thismodel has only a single state and carries no degrees of freedom

The variational principle is still well-posed The action is stationary with-out fixing the constrained variable q1 The unconstrained variable q2 mustbe fixed so the action is a function of its chosen boundary values How-ever gauge invariance of the action ensures that the action takes the samevalue over all the gauge identified endpoint data so it is well defined on theequivalence classes4

34 Field theoryspatial boundaries

As an example of a field theory with gauge invariance we will show how theaction for electromagnetism is well-posed Consider Maxwellrsquos equations fora vector field Amicro to be solved in a region such as that shown in figure 1

Amicro minus partmicro(partνAν) = 0 (313)

They are gauge invariant under Amicro rarr Amicro + partmicroΛ for any function Λ(x)

We wish to know what boundary data are required to make this systemwell-posed Start by fixing Lorentz gauge partmicroA

micro = 0 in the bulk by choosingΛ to satisfy Λ = minuspartmicroAmicro The equations of motion are then equivalent to

Amicro = 0 partmicroAmicro = 0 (314)

These still allow for a residual gauge transformation Amicro rarr Amicro + partmicroΛ whereΛ satisfies Λ = 0 in the bulk Solving the equation Λ = 0 requires

4In general gauge transformations may force the identification of other data besides theendpoint data An example is the lagrangian for a relativistic point particle which containsreparametrization invariance of the world-line In this case it is the time interval t2 minus t1that is identified [48] In the case of general relativity there are gauge transformationsof both kind Spatial diffeomorphisms(generated by the momentum constraints) generateequivalences among the endpoint data whereas the diffeomorphisms generated by thehamiltonian constraint generates equivalences among the boundary data

specifying Λ on the boundary so the residual gauge transformations will betransformations of the data on the boundary which will be used to generateequivalence classes

The wave equation Amicro = 0 requires specifying Amicro on Σ1 Σ2 and B (ex-cept for ldquounfortunaterdquo choices) Differentiating we have (partmicroA

micro) = 0 whichis the wave equation for the quantity partmicroA

micro Thus if we fix boundary condi-tions for Amicro then extend into the bulk using Amicro = 0 we need only checkthat partmicroA

micro = 0 on the boundary as the extension will then automaticallysatisfy partmicroA

micro = 0 in the bulk

For a given choice of Amicro on the boundary partmicroAmicro is entirely specified by

Laplacersquos equation Amicro = 0 thus there will be a single constraint on theallowed choices of Amicro on the boundary to ensure partmicroA

micro = 0 there To see theform of the constraint note that the perpendicular derivatives of the Amicro arenot set as part of the boundary data but are determined by extending thesolution into the bulk For instance we can write

partperpAperp = f(Aperp|partV A∣∣partV) (315)

where f is some function of the components of Amicro on the boundary specifiedby solving Laplacersquos equation Therefore we need only impose the restriction[

partA + f(Aperp A)]partV = 0 (316)

on the boundary data For every choice of boundary data satisfying this re-striction there is a unique solution to Maxwellrsquos equations in Lorentz gauge

We can think of this condition as determining Aperp given A so we obtaina unique solution given any choice of A with Aperp determined through theconstraint To have a well defined variational problem given this choice ofthe free data the action should be stationary with only A fixed on theboundary

We now return to the residual gauge freedom A gauge transformationsuch that Λ = 0 is still permissible These residual gauge transformationsleave the constraint (315) invariant and serve to identity the unconstrainedboundary data into equivalence classes We can select a member of eachclass by fixing the residual gauge For example we can use it to set Aperp = 0We then have two independent functions that must be specified namely A

subject to the single constraint (315) These two functions correspond tothe two polarization states of the photon

Consider now the Maxwell action in this volume

intVd4x minus 1

4FmicroνF

microν Fmicroν equiv partmicroAν minus partνAmicro (317)

Variation gives intVd4x partmicroF

microνδAν minus∮partVd3x nmicroF

microνδAν (318)

nmicro is the outward pointing normal on the timelike surface B and the inwardpointing normal on the spacelike surfaces Σ1 and Σ2 Notice that the ac-tion is stationary when only A is held fixed at the boundaries due to theanti-symmetry of Fmicroν exactly the data required of the equations of motionFurthermore we are to able to identify as equivalent any data that differ bya gauge transformation because the gauge invariance of the action ensuresthat it is well defined on the equivalence classes Thus we have a well-posedaction principle

35 Adding total derivatives

Adding a boundary term such as the Gibbons-Hawking-York term amountsto adding a total derivative to the lagrangian We would like to understandhow adding a total derivative can change the variational problem The bulkequations of motion are always unaffected by adding a total derivative evenone that contains higher derivatives of the fields However the addition of atotal derivative may render the variational problem inconsistent or requiredifferent boundary data in order to remain well-posed

Consider adding a total derivative to the lagrangian L1 = 12q2 of the free

non-relativistic point particle so that the action reads

int t2

dt L2 L2 = minus1

2qq (319)

The lagrangian contains higher derivatives but they appear only as totalderivatives so the equations of motion are still second order This is analo-gous to the Einstein-Hilbert action without the GHY term

Varying the action produces the endpoint contribution

2(qδq minus qδq)|t2t1 (320)

Setting q(t1) and q(t2) and the associated requirement δq = 0 at the end-points is no longer sufficient to kill the surface contribution in the variationFixing q(t1) and q(t2) as well as q(t1) and q(t2) and hence δq = 0 and δq = 0would be sufficient to kill the endpoint term But this is now setting 4 piecesof boundary data for a second order equation of motion so for most choicesof data the equations of motion will have no solution

One might try to say the following In the four dimensional space ofvariables that are fixed parametrized by q(t1) q(t2) q(t1) q(t2) there is sometwo dimensional subspace that is unconstrained The others are fixed by theequations of motion so the parameters of this subspace are the true degreesof freedom of the theory In fact there are many such subspaces ie thatparametrized by q(t1) q(t2) or that parametrized by q(t1) q(t2) etc Theessential point is that the action should be stationary when only variationsof the quantities parametrizing the subspace are held fixed because we mustvary over all paths This is not true of the two subspaces just given If wecould find a subspace for which this were true we could salvage the actionS2 by saying that the degrees of freedom have been mixed up in the qrsquos andqrsquos

For example we might try writing the boundary contribution as

minus 1

)∣∣∣∣t2t1

and fixing the quantity qq

on the boundary If fixing endpoint data for qq

leads to a unique solution for most choices then the endpoint variation van-ishes upon fixing only this quantity and we can say that the degree of free-dom is encoded in the combination qq However this does not work sincethe quantities q

q(t1) and q

q(t2) do not parametrize a subspace over which the

equation of motion is well-posed We can see this by noting that the generalsolution to the equation of motion is q(t) = At + B so fixing q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

which cannot be solved for AB given generic values of C1 C2 t1 t2

In fact there is no way to find such a subspace for the action based onL25 For this reason if we are given the lagrangian L2 we must add the GHY

type term(

12qq)∣∣t2t1

to bring the lagrangian back to L1 which is well-posed

As a general rule of thumb if an action contains higher derivatives whichappear only as total derivatives a boundary term will need to be added

As a final example which cleanly illustrates why the space of competingcurves in the variational principle must be kept as large as possible considerthe lagrangian L3 = 1

q The boundary variation is then qδq + δq|t2t1 We

are fixing δq and we might think we can simply fix q = 0 as well As longas we can fix q to be those values given by solving the equations of motion(ie q = 0) it is still true that there is a two parameter (q(t1) and q(t2))family of endpoint data all of which yield a unique solution to the equation ofmotion We are tempted to think of q as a second-class constrained variableHowever fixing q and hence δq = 0 at the endpoints restricts the class ofpaths over which we are extremizing Taken to the extreme we could alsofix δ

q = 0 at the endpoints and so forth until the entire Taylor series is

fixed and the only curve competing for the extremum is the solution itselfTo avoid arbitrariness we need to keep the space of competing paths as largeas possible by varying over all paths consistent with the specified boundarydata The action based on L3 is not well-posed

36 Higher derivative lagrangians

We now turn to an example with higher derivatives that is analogous to theF (R) gravity case Consider the action for a single variable q(t) that involves

5In general to set up such a two dimensional subspace we must set functions of q or qon the boundary separately or a function F (q q) on both boundaries It is clear that thefirst approach fails to set the above boundary variation to zero We may imagine functionsF (q q) and f such thatfδF = 1

2 (qδq minus qδq) These would have the desired property thatsetting F = C1|t1 and F = C2|t2 parametrize a two dimensional subspace and would implythe vanishing of the boundary variation It is not difficult to show however that any suchF must be of the form F ( qq ) and thus is not a valid choice for the same reason that q

q isnot

an arbitrary function F of the second derivative q which satisfies F primeprime 6= 0

int t2

dt F (q) F primeprime 6= 0 (323)

Variation gives the following

δS1 =

int t2

dt[F primeprimeprime(q)

q 2 + F primeprime(q)

q]δq + [F prime(q)δq minus F primeprime(q)

q δq]|t2t1 (324)

The bulk equation of motion can be solved for the highest derivativeq in

terms of the lower derivatives This is a fourth order equation in standardform and requires four pieces of boundary data to be well-posed Fixing qand q at both endpoints is a valid choice With this the variation at theendpoints vanishes and we have a well defined action principle for two degreesof freedom

S1 = S1[q1 q1 q2 q2 t] (325)

Consider now introducing an auxiliary field λ(t) to try and get rid ofthe higher derivatives (this can always be done a general method is theOstrogradski method [50 51 9])

int t2

dt F (λ) + F prime(λ)(q minus λ) (326)

Varying gives

δS2 =

int t2

dt[F primeprimeprime(λ)λ2 + F primeprime(λ)λ

]δq+F primeprime(λ) [q minus λ] δλ+

[F prime(λ)δq minus F primeprime(λ)λδq

]∣∣∣t2t1

(327)This action is well-posed if q and q are held fixed at the boundary and λis kept arbitrary The equation of motion for λ can be solved for λ = qwhich when plugged back into the action yields S1 and hence S1 and S2 areequivalent by the fundamental theorem of auxiliary variables (see AppendixD)

The action S2 still involves the higher derivatives q but now they appearonly through a total derivative so we integrate by parts

int t2

dt F (λ)minus λF prime(λ)minus F primeprime(λ)qλ+ F prime(λ)q|t2t1 (328)

The integration by parts has generated a boundary contribution We canrender the action first order by subtracting this boundary term that is byadding to S2 the GHY type term minusF prime(λ)q|t2t1

The action and its variation now read

int t2

dt F (λ)minus λF prime(λ)minus F primeprime(λ)qλ (329)

δS3 =

int t2

]δq+F primeprime(λ) [q minus λ] δλminus

[F primeprime(λ)(λδq + qδλ)

]∣∣∣t2t1

(330)The bulk variation is unchanged because we have only added a total deriva-tive but the boundary variation has changed We must modify the vari-ational principle to keep λ and q fixed on the boundary The action nowbecomes a functional S3 = S3[q1 λ1 q2 λ2 t] The degree of freedom q hashas been shifted into λ

Here adding the GHY type term was not strictly necessary to keep thevariational principle well defined as was the case in the previous sectionHere the effect is simply to shift the degrees of freedom into different vari-ables6

We are still free to eliminate the auxiliary variable λ from the equationsof motion Doing so yields the action

int t2

dt F (q)minus d

dt(F prime(q)q) (331)

with variation

δS4 =

int t2

q 2 + F prime(q)

q]δq minus [F primeprime(q)

q δq + F primeprime(q)qδq]|t2t1 (332)

The action is now a functional S4 = S4[q1 q1 q2 q2 t] The degrees of freedomhave been relabeled as q and q

6This may be true of F (R) theory as well that is the boundary term obtained fromthe correspondence to scalar-tensor theory is not necessarily the only one that rendersthe variational principle well-posed It is however the one that must be used if thecorrespondence to scalar-tensor theory is to be maintained at the boundary

The moral of all this is that while adding total derivatives (equivalentlyboundary terms) to the action does not change the bulk equations of motionit can change the variational principle and the corresponding labeling ofdegrees of freedom or it can make it impossible for the variational principleto be well-posed

4 Review of the Gibbons-Hawking-York term

The GHY boundary term is a modification to the Einstein-Hilbert actionwhich makes the action well-posed The modified action may be written as(conventions and definitions for boundary quantities laid out in appendicesA and B)

16πG S = 16πG(SEH + SGHY ) =

intVdnxradicminusgR + 2

∮partVdnminus1x

radic|h|K (41)

Where G is Newtonrsquos constant Upon varying the action we arrive at

16πG δS =

intVdnxradicminusg(Rmicroν minus

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|hαβnmicropartmicroδgαβ

∮partVdnminus1x

radic|h|δK (42)

Here we have used the assumption δgmicroν = 0 on partV which also implies thatthe tangential derivative vanishes on partV hαβpartαδgmicroβ = 0 Noting that

δK = δ(hαβ(partαnβ minus Γmicroαβnmicro)

)= minushαβδΓmicroαβnmicro

2hαβnmicropartmicroδgαβ (43)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

We now have the boundary variation vanishing without any restriction onthe normal derivatives However if this is the only property we desire the

choice of 2K for the boundary term is not unique We are free to add anarbitrary function of the metric normal vector and tangential derivativesF (gmicroν nmicro h

αβpartβ) because the variation of such an addition vanishes withthe assumption δgmicroν = 0 on partV

In fact because of this freedom Einstein unwittingly used the GHYboundary term well before either Gibbons Hawking or York proposed it[53] He used an object H for the lagrangian instead of R

H = gαβ(ΓνmicroαΓmicroνβ minus ΓmicromicroνΓ

ναβ

) (45)

This is sometimes called the gamma-gamma lagrangian for GR and hasthe advantage that it is first order in the metric so there is no need to fixderivatives of the metric at the boundary

H differs from R by a total derivative

H = RminusnablaαAα

where Aα = gmicroνΓαmicroν minus gαmicroΓνmicroν As such it produces the same equations ofmotion ie the Einstein equations upon variation It also possesses all thesame bulk symmetries as the Einstein-Hilbert action namely diffeomorphisminvariance Under a diffeomorphism it does not change like a scalar but itdoes change by a total derivative

We can see that the gamma-gamma action differs from the Einstein-Hilbert plus GHY action by a boundary term of the form F (gmicroν nmicro h

αβpartβ)intVdnxradicminusgH =

intVdnxradicminusg [RminusnablaαA

intVdnxradicminusgRminus

∮partVdnminus1x Aαnα (46)

Aαnα = minus2K + 2hαβpartβnα minus nmicrohνσpartνgσmicro (47)

so this is an example of a choice of boundary term that differs from K by afunction F namely F = 2hαβpartβnα minus nmicrohνσpartνgσmicro

The Einstein-Hilbert plus GHY action requires even fewer variables thanthose of the metric to be fixed on the boundary Only the induced metrichab need be fixed To see this we foliate in ADM variables with timelikehypersurfaces relative to B

16πG SEH =

intVdnxradicminusgR =

intVdnx N

radicminusγ[

(nminus1)RminusKabKab +K2

+ 2nablaα

(rβnablaβr

α minus rαnablaβrβ)] (48)

The total derivative term when reduced to a surface term cancels againstthe GHY term So the action is

16πG(SEH + SGHY ) =

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

(nminus1)RminusKabKab +K2] (49)

(Here we are ignoring total time derivatives and the GHY terms on theendpoints but a similar cancellation will apply there) There are no radialderivatives of the lapse or shift so their variation need not be set to zeroon the boundary Fixing the induced metric on the boundary is sufficient torender the action stationary

7In detail

nαAα = nα(gmicroνΓαmicroν minus gαmicroΓνmicroν)

= (nσgmicroν minus nmicrogνσ)12

[partmicrogνσ + partνgmicroσ minus partσgmicroν ]

= (nσhmicroν minus nmicrohνσ)12

= nαhmicroνΓαmicroν minus

12nαg

microαhνσpartmicrogνσ

= 2nαhmicroνΓαmicroν minus nmicrohνσpartνgσmicro= minus2K + 2hαβpartβnα minus nmicrohνσpartνgσmicro

For most choices of an induced metric on partV the Einstein equationsshould produce a unique solution in V up to diffeomorphisms that vanishat the boundary In this case the Einstein-Hilbert plus GHY action is well-posed The counting in four dimensions goes as follows Of the ten pieces ofboundary data for the ten components of the metric there are 4 constraintsThe six components of the induced metric can be taken as the unconstrainedcomponents These are subject to equivalence under four gauge transfor-mations leaving two independent pieces of data corresponding to the twodegrees of freedom of the graviton

5 Higher derivative gravity and scalar-tensor

theory

Higher derivative theories of gravity have been studied extensively in manydifferent contexts (for a review see [12]) They are invoked to explain cosmicacceleration without the need for dark energy [6 7 8 9 10] and as quantumcorrections to Einstein gravity [13 14]

One can imagine trying to modify gravity by writing a quite generalfunction of arbitrary curvature invariants of any order in metric derivativesint

dnxradicminusgF (RRmicroνR

microν RmicroνλσRmicroνλσ nablamicroRnablamicroRRmicroνR

microλR

λν )

(51)Typically these are true higher derivative lagrangians ie second and higherderivatives of the metric appear in a way that cannot be removed by addingtotal derivatives to the action so the equations of motion are at least fourthorder The amount of gauge symmetry is typically unchanged from generalrelativity ie diffeomorphism invariance remains the only gauge symmetryThis means that the theories either have more degrees of freedom than generalrelativity the presence of second class constraintsauxiliary fields or both

Such models are not as diverse as it might seem since they are essentiallyall equivalent to various multiple scalar-tensor theories [25 26 27 28 2930 31] The transition to scalar-tensor theory amounts to an eliminationof auxiliary fields represented by some of the higher derivatives of the met-ric They are replaced by scalar fields that more efficiently encapsulate the

physical degrees of freedom

We can get GHY terms for such theories by exploiting this equivalenceThe equivalent scalar-tensor theory is typically a Brans-Dicke like theoryin Jordan frame with a potential for the scalar fields minimally coupledto matter The theory can be brought to Einstein frame by a conformaltransformation The boundary term in Einstein frame is just the GHY termso we can find the boundary term for the original higher derivative theoryby taking the Einstein frame GHY term backwards through the conformaltransformation and through the scalar-tensor equivalence The term foundin this way must be the correct one if the equivalence between the higherderivative theory and scalar-tensor theory is to hold for the boundary terms

The GHY terms obtained in this way are not generally sufficient to killthe boundary variation of the action when only δgmicroν = 0 This is simply areflection of the fact that we are dealing with a higher-order theory Someof the boundary values that the action depends upon involve derivatives ofthe metric in a fashion exactly analogous to the fourth order toy example insection 36 The only metric theories where δgmicroν = 0 should be sufficient arethose with the special property that the equations of motion are still secondorder despite the appearance of higher order terms in the action namelythe Lovelock lagrangians [33] Indeed such GHY terms can be found forLovelock theory [34 35 36]

In what follows we analyze in detail the simplest case namely the casewhere the lagrangian is allowed to be an arbitrary function of the Ricci scalarF (R) but does not contain any other curvature invariants The extensionto more complicated cases follows easily when the scalar-tensor equivalenceis known

51 F (R) gravity

F (R) theory is one of the most widely studied modifications of gravity [5 4]It has the ability to explain cosmic acceleration without dark energy [6 7 8 9]and to evade local solar system constraints [8 11]

The action for F (R) gravity is

intdnxradicminusgF (R) (52)

We would typically add matter which is minimally coupled to the metric butit plays no essential role in the boundary terms so we will omit the matter inwhat follows The Euler-Lagrange variation gives equations of motion whichare fourth order in the metric

The equivalence to scalar-tensor theory is seen by introducing a scalarfield φ

intdnxradicminusg [F (φ) + F prime(φ)(Rminus φ)] (53)

The equation of motion for the scalar is

F primeprime(φ) (Rminus φ) = 0 (54)

which provided F primeprime 6= 0 implies R = φ Note that the scalar has massdimension 2 Plugging this back into the action using the fundamentaltheorem of auxiliary fields recovers the original F (R) action so the two areclassically equivalent This is ω = 0 Brans-Dicke theory with a scalar fieldF prime(φ) and a potential

In the GR limit F (R)rarr R we have F primeprime rarr 0 so the transformation breaksdown As the limit is taken the scalar field decouples from the theory [32]

52 Boundary terms for general scalar-tensor theory

In this section we consider a general scalar-tensor action of the form

intVdnxradicminusg(f(φ)Rminus 1

2λ(φ)gmicroνpartmicroφpartνφminus U(φ)

) (55)

We show that this should be supplemented by the boundary term

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

We do this by showing that this reduces to the usual GHY term upon con-formal transformation to the Einstein frame

Once this is done the equations of motion for the metric are obtained bysetting δgmicroν = 0 on the boundary

2f(φ)

(Rmicroν minus

2Rgmicroν

)+ 2f(φ)gmicroν minus 2nablamicronablaνf(φ) = T φmicroν (57)

T φmicroν = λ(φ)

[partmicroφpartνφminus

2gmicroν(partφ)2

]minus gmicroνU(φ) (58)

The equation of motion for the scalar field is obtained by setting δφ = 0 onthe boundary

λ(φ)φ+1

2λprime(φ) (partφ)2 minus U prime(φ) + f prime(φ)R = 0 (59)

We now proceed with the conformal transformation keeping careful trackof all boundary contributions [17] Assuming

f(φ) 6= 0

we can rewrite the action in terms of a conformaly re-scaled metric

gmicroν = [16πGf(φ)]2

nminus2 gmicroν

G gt 0 if f(φ) gt 0

G lt 0 if f(φ) lt 0(510)

G can be chosen to be anything consistent with the sign of f and willbecome the Einstein frame Newtonrsquos constant Re-writing the action is justa matter of using the conformal transformation formulae we have collectedfor convenience in appendix C In particular we see from the second term of(C8) used to rewrite R that there is an integration by parts that will benecessary to bring the scalar kinetic term to its usual form This will generatea surface term which must be combined with the conformal transformationof the GHY term (56)

The result is

intVdnx

radicminusg(

16πGRminus 1

2A(φ)gmicroνpartmicroφpartνφminus V (φ)

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

[16πGf(φ)]nnminus2

The GHY term has the usual form in Einstein frame so working backwardsthe Jordan frame expression must be correct as well Variation is done withδgmicroν = 0 δφ = 0 on the boundary

53 Boundary term for F (R) theory

Adding the boundary term to the scalar-tensor form of the F (R) action wehave

intVdnxradicminusg [F (φ) + F prime(φ)(Rminus φ)] + 2

∮partVdnminus1x

radic|h|F prime(φ)K (514)

The variation of the action with respect to the scalar field now contains theboundary contribution

∮partVdnminus1x

radic|h|F primeprime(φ)Kδφ (515)

which vanishes since we require δφ = 0 on the boundary

Plugging back in we have

intVdnxradicminusgF (R) + 2

∮partVdnminus1x

radic|h|F prime(R)K (516)

This boundary term has been arrived at before in several different con-texts sometimes indirectly [15 16 17 18 19] However there has been someconfusion because this boundary term is not enough to make the action sta-tionary given only δgmicroν = 0 on the boundary Indeed there is in generalno such boundary term with this property [18] We see that this is becauseR now carries the scalar degree of freedom so we must set δR = 0 on theboundary as well

We will now go on to accumulate some evidence that this is indeed thecorrect boundary term We first calculate the energy in the hamiltonianformalism and show that to obtain the correct energy that reduces to theADM energy when F (R) sim R this boundary term must be included Wethen calculate the entropy of Schwartzschild black holes and show that inorder to reproduce the results of the Wald entropy formula the boundaryterm is necessary

6 Hamiltonian formulation and ADM en-

In this section we will develop the hamiltonian formulation of a generalscalar-tensor theory which will encompass the F (R) case We will stay inJordan frame and keep track of all boundary terms Just as in GR thebulk hamiltonian vanishes on shell and the only contribution comes fromthe boundary so it will be essential to include the GHY terms found in theprevious section

We start with the action (55) which we write as S = SG + Sφ + SBwhere

intVdnxradicminusg f(φ)R (61)

intVdnxradicminusg[minus1

2λ(φ)gαβpartαφpartβφminus U(φ)

] (62)

SGHY = 2

intpartVdnminus1x

radic|h| Kf(φ) (63)

We change to ADM variables [52] (see appendices A and B for conven-tions and definitions of the various quantities) The boundary term splitsinto three integrals over the three boundaries The integral over Σ2 gets anadditional minus sign since by convention the normal should be directedinward for spacelike surfaces whereas for Σ2 it is directed outward We willsuppress the argument of f g and λ over the course of the calculation

First look at the combination S = SG + SGHY

SG + SGHY =

intVdnxradicminusg f [(nminus1)RminusK2 +KabKab minus 2nablaα(nβnablaβn

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

dnminus1yradichKf minus 2

∮Σ2

dnminus1yradichKf

Integrating by parts the last term in the bulk integral we find surface con-tributions that exactly cancel the boundary integrals over Σ1 and Σ2 Thesurface contribution over B does not cancel its corresponding boundary in-tegral and we are left with

SG + SGHY =

intVdnxradichN

(nminus1)RminusK2 +KabKab

)+ 2f prime(nβnablaβn

α minus nαK)partαφ]

∮Bdnminus1z

radicγf [K minus rαnβnablaβn

α] (65)

We can simplify the integrand of the boundary piece

K minus rαnβnablaβnα = K + nαnβnablaβrα

= gαβnablaβrα + nαnβnablaβrα

= (σABeαAeβB minus n

αnβ)nablaβrα + nαnβnablaβrα

= σABeαAeβBnablaβrα

= k (66)

The bulk terms multiplying f prime can be further simplified8

nβnablaβnα = hακnβ (partβnκ minus partκnβ) (67)

Putting all this together we have

SG + SGHY =

intVdnx

radichN

)+ 2f prime

(hακnβ(partβnκ minus partκnβ)minus nαK

)partαφ

∮Bdnminus1z

radicσNfk (68)

We now specialize to the (t ya) coordinate system in which nα = minusNδ0α

eaα = δaα The term hακnβpartβnκ vanishes We are left with

SG + SGHY =

intVdnxradich[Nf

((nminus1)RminusK2 +KabKab

)+ 2f prime(habpartaNpartbφminusKφ+KNapartaφ)

∮Bdnminus1z

radicσNfk (69)

8In detail

nβnablaβnα = gακnβnablaβnκ= (hακ minus nαnκ)nβnablaβnκ= hακnβnablaβnκ= hακnβpartβnκ minus hακnβnλΓλβκ

= hακnβpartβnκ minus12hακnβnσ(partβgκσ + partκgβσ minus partσgβκ)

= hακnβpartβnκ minus12hακnβnσpartκgβσ

= hακnβpartβnκ +12hακnβnσpartκ(nβnσ)

= hακnβpartβnκ minus hακnβpartκnβ

The scalar action in ADM variables is

intVdnx

radichλ

[φ(φminus 2Napartaφ

)minusN2habpartaφpartbφ+ (Napartaφ)2

]minusNradichU

Define now

S primeG equivintVdnxradich(Nf [(nminus1)RminusK2 +KabKab] + 2f primehabpartaNpartbφ

)(611)

S primeφ equiv Sφ minus 2

intVdnxradichf primeK(φminusNapartaφ) (612)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

S = S primeG + S primeφ + S primeB (614)

The action is now in a form amenable to transition to the hamiltoniannamely it is in the form of a time integral over a lagrangian L (which isitself a space integral plus boundary parts) containing no time derivativeshigher than first and no boundary contributions at the time endpoints

int t2

dt L[hab hab NN

a φ φ] (615)

Note that this would not be the case were it not for the GHY term onthe surfaces Σ1 and Σ2 It now remains to transition to the hamiltonianformulation This has been done without keeping surface terms by [37 38]and at the level of the equations of motion by [39] We start by finding thecanonical momentum conjugate to φ

pφ =δL

radichλ

(φminusNapartaφ

)minus 2radichf primeK (616)

To find the canonical momenta conjugate to hab we vary with respect

to Kab then use the relation Kab = 12N

(hab minusnablaaNb minusnablabNa

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

Nhab(φminusN cpartcφ)

Equations (616) and (617) are invertible for φ and Kab (which is essentially

hab) in terms of pφ pab (and φ hab N Na)

φ =Nradich

[(nminus 2)fpφ minus 2f primep

2(nminus 1)f prime2 + (nminus 2)fλ

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

where p = habpab The canonical momenta conjugate to N and Na bothvanish just as in GR

Notice that the map from velocities (φ hab) to momenta (pφ pab) is non-singular even when the scalar kinetic term vanishes λ rarr 0 as in the caseof theories equivalent to F (R) This corresponds to the fact that the scalaris still dynamical in this limit by virtue of its non-minimal coupling Thecase f rarr const 6= 0 is also well behaved provided λ 6= 0 as the scalar hasdynamics stemming from the kinetic term The case f rarr const and λ rarr 0is indeed singular because the scalar field then loses its dynamics

We now start the calculation of the hamiltonian Starting with the scalarfield part we must express everything in terms of the fields and momentaby eliminating all time derivatives

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

2λhabpartaφpartbφ+ U

(pφradichpartaφ

)] (619)

Here we must treat Kab as the function of pab hab φ and pφ given by (618)Naively the above appears singular in the limit λ rarr 0 (the F (R) case)however once we re-express Kab in terms of the momenta the 1λ termscancel and the limit is smooth

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

dnminus1y pab(2NKab + 2nablaaNb)minusradich(Nf [(nminus1)RminusK2 +KabKab]

+2f primehabpartaNpartbφ) (620)

Integrate by parts to pull all the derivatives off of N and Na being sure tokeep the boundary contributions

H primeG =

intΣt

dnminus1yradich

pabradichKab minusNf

)+ 2Nnablaa(f primenablaaφ)minus 2Nanablab

(pabradich

dnminus2θradicσ ra

pabradichminusNf primepartaφ

) (621)

The boundary term in the lagrangian contributes

H primeB = minusLprimeB = minus2

dnminus2θradicσNfk

Combining these terms yields the full hamiltonian

H = H primeG +H primeφ +H primeB

intΣt

dnminus1yradich[Nf

(minus(nminus1)RminusK2 +KabKab

(pabradichKab + 2nablaa(f primenablaaφ) +

h+ f prime

pφradich

(pφradichpartaφminus 2nablab

pabradich

dnminus2θradicσ

pabradichminusN(fk + raf primepartaφ)

] (622)

This hamiltonian is like that for GR in the sense that the equations ofmotion for N and Na are Lagrange multipliers which cause the bulk termsto vanish We have not bothered to write out Kab using (618) because itbecause it does not contain N or Na

The hamiltonian evaluated on solutions reduces to the boundary part

Hsolution = 2

dnminus2θradicσraNb

pabradichminusN(fk + raf primepartaφ) (623)

The ADM energy E is given by choosing the lapse to vanish and the shiftto be unity

E = minus2

dnminus2θradicσ [fk + f primerapartaφ] (624)

Alternatively we could also have obtained this expression by finding thehamiltonian in Einstein frame and then performing a conformal transforma-tion

As an example consider the Schwartzschild solution in four dimensions

ds2 = minusf(r)dt2 +1

f(r)dr2 + r2dω2 φ = φ0 (625)

where f(r) = 1 minus 2GMr

GM is a constant and φ0 is a constant This willbe a solution to the scalar-tensor theory equations of motion (57) (58)and (59) provided that Minkowski space is a solution and φ0 is set to thevacuum value of φ in the Minkowski solution The ADM energy for thissolution is9

E = 16πGMf(φ0) (626)

which is what one expects given that f(φ0) is playing the role of the effectivegravitational constant f(φ0) = 1

16πGeff

F (R) theory is the special case where f(φ) = F prime(φ) U(φ) = F prime(φ)φ minusF (φ) λ(φ) = 0 In order to hamiltonize F (R) theory it must first be broughtto first order form (this is essentially the content of the Ostrogradski methodfor hamiltonizing higher order systems [50 51 9]) Passing to the scalar-tensor description is the simplest way to do this so we have also found the

9Recall that we must measure the energy relative to the energy of Minkowski spacewhich is the vacuum solution The energy of each is individually divergent as r rarrinfin butthe difference is finite and yields the above expression

ADM energy for F (R) theory The boundary term must be passed in thesame way and so plays the same essential role

7 Black hole entropy

When calculating the entropy of a black hole in the euclidean semi-classicalapproximation it is essential to have the correct GHY term [44] In factthis term is responsible for the entire contribution to the euclidean actionIn this section we will calculate the entropy for a Schwartzschild black holein F (R) theory using our boundary term and compare this to the entropygiven by the Wald entropy formula

The Wald entropy formula allows one to calculate the entropy of a blackhole in any diffeomorphism invariant metric theory of gravity It involvesan integral over the bifurcation two sphere of the horizon of the black hole[40 41] The formula does not rely on having a well-posed action principleie it depends only on the lagrangian density and hence the GHY termsare not needed in order to apply it For a Schwartzschild black hole in F (R)theory in four dimensions the Wald formula gives the result [43]

SBH =A

416πF prime(R0) (71)

where A is the area of the horizon and R0 is the (constant) backgroundcurvature of the spacetime the black hole sits in Wald has shown that hisentropy formula gives the same value as the euclidean semi-classical approachfor theories which satisfy several conditions in addition to those needed bythe formula itself [42] One of these additional conditions is that there bea variational principle for the action where only the metric is held fixed onthe boundary so this does not cover the general F (R) case Neverthelesswe will see that our surface term still gives the entropy in agreement withWaldrsquos formula

The partition function in the semiclassical limit is

Z[β] = eminusSE (72)

where β = 1T

is the inverse temperature and SE is the euclidean action of thedominant classical field configuration where the time variable is identifiedwith period β

In Einstein gravity in four dimensions we have the Schwartzschild solu-tion

f(r)dr2 + r2dω2 (73)

where f(r) = 1minus 2GMr

and M is the ADM mass

The temperature can be determined by finding the corresponding solutionto the euclidean action which amounts to taking t rarr iτ and then findingthe period of τ required to eliminate the conical singularity at the horizonThe resulting metric is then

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

The temperature of a black hole is model-independent since no use of theEinstein equations was made other than the fact that Schwartzschild is asolution Thus in any theory of modified gravity where Schwartzschild is asolution the black hole will have the same temperature

We first review the euclidean action calculation in GR since the extensionto F (R) is then trivial The euclidean action for GR is

16πGSE = minusintVd4x

radic|g|Rminus 2

∮partVd3x

radic|h|(K minusK0) (75)

The action for a general theory must be zeroed on the action for the back-ground one is expanding about ie the vacuum state The true finite actionis thus S minus S0 where S0 is the action of the background In our case thebackground is flat space The bulk contribution to S0 vanishes and all thatremains is K0 the extrinsic curvature of the boundary as measured in flatspace The term K0 is often called the boundary counterterm (misappropri-ated from the quantum field theory jargon) and may in general have to takea more complicated form [49])

Since the Ricci curvature of Schwartzschild vanishes the bulk term doesnot contribute to the classical action The entire contribution comes fromthe boundary term Taking the boundary to be a sphere of radius r aboutthe origin we have ∮

partVd3x

radic|h|K = 4πβ(2r minus 3GM) (76)

For the background we periodically identify the time to the period β red-

shifted by the factor(1minus 2GM

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

Taking the difference and the limit r rarrinfin yields

SE =β2

16πG (78)

The free energy entropy and energy are then

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

where A = 4π(2GM)2 is the area of the horizon

We now extend the calculation to F (R) theory The bulk vacuum equa-tions of motion are

F prime(R)Rmicroν minus1

2F (R)gmicroν + gmicroνnabla2F prime(R)minusnablamicronablaνF

prime(R) = 0 (712)

which are fourth order in the metric as expected AdSdS Schwartzschildhas the property Rmicroν = R

4gmicroν with R = R0 a constant Using this ansatz

the equation of motion reduces to

F prime(R0)R0 minus 2F (R0) = 0 (713)

which is an algebraic equation for R0 This is the same equation one wouldobtain seeking constant curvature solutions so for every constant curvaturebackground we also have a Schwartzschild black hole that asymptoticallyapproaches this background

The euclidean action including the GHY term is

SE + S0 = minusintVd4x

radic|g|F (R)minus 2

∮partVd3xradic|h|F prime(R)K (714)

Here S0 is the action for the background Since the Schwartzschild solutionhas the same constant curvature as the background the bulk contributionvanishes and the entire contribution again comes from the boundary term

SE = minus2F prime(R0)

∮partVd3xradic|h| (K minusK0) (715)

where R0 is the scalar curvature of the backgroundblack hole

If we assume for simplicity that F (R) is such that there is a flat spacesolution then the calculation proceeds just as in the GR case with the result

SE = β2F prime(R0) (716)

The thermodynamic quantities are then

F = βF prime(R0) E = 16πGF prime(R0)M S =A

416πF prime(R0) (717)

These formulae make good sense from the point of view of the scalar-tensor theory The scalar field is justR the Ricci curvature so the Schwartzschildsolutions we are considering have constant scalar field everywhere This is inaccord with no hair theorems which forbid non-trivial scalar profiles aroundblack holes in space with zero or positive curvature [45 46 47] The valueof the effective Newtonrsquos constant is set by the constant value of the scalarfield

16πGeff

= F prime(R0) (718)

so the entropy is simply one quarter the area of the horizon in units of theeffective Planck length The energy is the ADM energy we found in theprevious section

An interesting consequence of this formula is that higher curvature termsmake no correction to the entropy From (713) we see that if R = 0 is to bea solution then F (0) = 0 so F can be taylor expanded around the originstarting with the Einstein-Hilbert term F (R) = F prime(0)R + 1

2F primeprime(0)R2 + middot middot middot

All the higher power corrections to the action do not affect the entropy Inparticular the entropy of a black hole in pure Rn gravity for n ge 2 vanishes

8 Conclusions

Having correct boundary terms is essential to the consistency of any theoryOne can generally get away without them but there are instances where theyare vitally important We have argued that consistent GHY terms for higherderivative modified gravity theories can be obtained by using any scalar-tensor equivalence the theory may posses The boundary terms obtainedin this way while not necessarily unique do give a well-posed variationalproblem and give the expected ADM energy and black hole entropy eventhough derivatives of the metric may have to be fixed on the boundary

What we have given is by no means a complete analysis of boundaryterms however For a general lagrangian of any kind of field not necessarilya modified gravity theory it is far from clear whether there always exists aboundary term that renders the variational principle well-posed Further-more even if such a term can be found it is not evident what freedom isallowed in choosing the boundary term or what physical significance thisfreedom entails

Acknowledgements

The authors are grateful to Allan Blaer Adam Brown Norman ChristSolomon Endlich Brian Greene Dan Kabat Alberto Nicolis and ErickWeinberg for discussions and to their anonymous referee for helpful com-ments KH is supported by DOE grant DE-FG02-92ER40699 and by aColumbia University Initiatives in Science and Engineering grant

A The ADM decomposition

Here we review the ADM hypersurface decomposition of spacetime and layout our conventions in the process The conventions and notation are thoseof [21] In this appendix we will describe a generic foliation of a volumeV by spacelike or timelike hypersurfaces and in the next appendix we willdescribe the specific foliations we use throughout the paper

Put coordinates xmicro on V We foliate the volume V with hypersurfaces Σt

by giving a global time function t(xmicro) and declaring the hypersurfaces to be

its level sets We then set up functions ya(xmicro) a = 1 n minus 1 independentof t(xmicro) and each other to serve as coordinates on the submanifolds Takentogether t and ya form a new coordinate system on V and we have aninvertible transformation from the these coordinates to the old ones

xmicro(ya t) t(xmicro) ya(xmicro) (A1)

The coordinate basis vectors of this new coordinate system are

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

The dual one forms are

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

They satisfy the duality and completeness relations

tmicrotmicro = 1 emicrobeamicro = δab tmicroe a

micro = emicroatmicro = 0 (A4)

tmicrotν + emicroaeaν = δmicroν (A5)

Introduce a bulk metric gmicroν There is now a well defined one-dimensionalnormal subspace at each point of Σt which may be different from the sub-space spanned by tmicro We set up a basis consisting of the forward pointingunit normal vector nmicro along with the emicroa The emicroa are not required to beorthonormal among themselves but are orthogonal to nmicro We have

gmicroνnmicronν = ε gmicroνe

microan

ν = 0 (A6)

Here ε is defined by

1 Σt timelike

minus1 Σt spacelike (A7)

We define the associated dual forms e amicro nmicro at each point

nmicronmicro = 1 emicrobeamicro = δab nmicroe a

micro = emicroanmicro = 0 (A8)

nmicronν + emicroaeaν = δmicroν (A9)

(We use nmicro for the dual one form and reserve nmicro for the form gmicroνnν They

differ by a sign for spacelike hypersurfaces ie nmicro = εnmicro)

A vector field Amicro is called parallel if it admits the decomposition Amicro =Aaemicroa A form Amicro is parallel if it admits the decomposition Amicro = Aae

amicro

Similarly a general tensor is parallel if it admits a similar decomposition forexample a (2 1) tensor is parallel if

Amicroνγ = Aabcemicroaeνbe

cγ (A10)

There is a bijective relation between tensors on the submanifold Σt (really aone parameter family of tensors one on each surface parametrized by t) andparallel tensors in the bulk Given a parallel bulk tensor Amicroνγ it corresponds

to the submanifold tensor Aabc and vice versa

Define the projection tensor

P microν = δmicroν minus nmicronν (A11)

It projects the tangent space of V onto the tangent space of Σt along thesubspace spanned by nmicro It satisfies

P microλP

λν = P micro

ν (A12)

P microνeνa = emicroa P micro

νnν = 0 (A13)

P microνe

amicro = e a

ν P microνnmicro = 0 (A14)

Given any bulk tensor eg Amicroνγ we can make a parallel tensor by pro-jecting it

Amicroνγ = P microρP

ρσλ (A15)

A tensor is parallel if and only if it is equal to its projection

We have the relationemicroae

aν = P micro

ν (A16)

Projecting the metric gives the induced metric h on the hypersurfaces

hmicroν = P ρmicroP

σνgρσ = habe

amicro e

bν hab = emicroae

νbhmicroν = emicroae

νbgmicroν (A17)

We raise and lower bulk indices micro ν with gmicroν and its inverse gmicroν and weraise and lower submanifold indices a b with hab and its inverse hab Inparticular we have

habgmicroνeνb = e a

micro gmicroνnν equiv nmicro = εnmicro (A18)

habgmicroνe b

ν = emicroa gmicroνnν equiv nmicro = εnmicro (A19)

as well asgmicroαP

αν = hmicroν gmicroαP ν

α = hmicroν (A20)

To perform the ADM decomposition we want to write the bulk metric inthe (ya t) coordinate system We start by expanding the coordinate vectortmicro over the new basis

tmicro = Nnmicro +Naemicroa (A21)

The coefficients N and Na are called the lapse and the shift respectivelyWe have

tmicro = partmicrot = ε1

Nnmicro nmicro = εNtmicro = εNpartmicrot (A22)

The coordinate one-forms are

dxmicro = tmicrodt+ emicroadya = Nnmicrodt+ (Nadt+ dya) emicroa (A23)

The metric is

gmicroνdxmicrodxν = εN2dt+ hab (Nadt+ dya)

(N bdt+ dyb

) (A24)

In matrix form this is

(εN2 +NaNa Na

Na hab

) (A25)

The inverse metric is

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

The square root of the norm of the determinant isradic|g| = N

radic|h| (A27)

The normal vector is

N na = minusN

N n0 = εN na = 0 (A28)

The extrinsic curvature is a parallel tensor defined by

Kab equiv emicroaeνbnablamicronν (A29)

It is symmetricKab = Kba (A30)

as can be easily shown by noting that the basis vectors have zero Lie bracketeνbnablaνe

microa = eνanablaνe

microb We also have

Kab = nabla(micronν)emicroaeνb =

2emicroae

νbLngmicroν (A31)

where L is the Lie derivative The trace of the extrinsic curvature is givenby

K = habKab = nablamicronmicro (A32)

In terms of ADM variables we have

Kab =1

) (A33)

We often use the decomposition

R = (nminus1)R + ε(K2 minusKabK

+ 2εnablaα

(nβnablaβn

α minus nαK) (A34)

Here R is the Ricci scalar constructed from gmicroν and (nminus1)R is the Ricci scalarconstructed from the induced metric hab

B Foliation of spacetime

Throughout this paper it is necessary to refer to spacetime boundaries ofspacetime foliations of spacetime and all of the geometrical quantities thatthese induce As such it is necessary to layout some standard terminology todescribe the situation See figure 2

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

Figure 2 Foliation of spacetime

Spacetime will be referred to as V bounded by partV partV has three partsa timelike boundary B a spacelike initial surface Σ1 and a spacelike finalsurface Σ2 We foliate the volume V with hypersurfaces Σt by giving aglobal time function t(x) and declaring the hypersurfaces to be its level sets

The top and bottom surfaces Σ1 and Σ2 are to coincide with Σt1 and Σt2 forsome times t1 and t2 The normal vector to the Σt is denoted nmicro and is futurepointing The coordinates on Σt are ya The lapse and shift relative to thisfoliation are N Na The induced metric is hab and the extrinsic curvature isKab

The boundary B can be thought of as part of a foliation by timelikesurfaces The coordinates on B are zi The radially outward pointing normalvector is rmicro We demand that the surfaces Σt intersect B orthogonally sothat gmicroνr

micronν = 0 on B This implies that rmicro is parallel to Σt on B so thatrmicro = raemicroa there for some ra The lapse and shift relative to this foliationare N N i The induced metric is γij and the extrinsic curvature is Kij

The intersections of Σt with B are denoted St = Σt cap B These form aspacelike foliation of B The coordinates on St are θA the induced metric isσAB and the extrinsic curvature of St as embedded in Σt is kAB

C Conformal transformation formulae

Here we collect some formulae on conformal transformations of the metric

gmicroν = ω2(x)gmicroν ω(x) gt 0 (C1)

gmicroν = ωminus2(x)gmicroν (C2)

The connection transforms as

Γρmicroν = Γρmicroν + Cρmicroν (C3)

Cρmicroν = ωminus1

(δρmicropartνω + δρνpartmicroω minus gmicroνgρσpartσω

) (C4)

The curvature scalar transforms as

R = ωminus2Rminus 2(nminus 1)gαβωminus3nablaαnablaβω minus (nminus 1)(nminus 4)gαβωminus4partαωpartβω (C5)

Some convenient covariant derivative transformations are

nablamicronablaνφ = nablamicronablaνφminus(δαmicroδ

βν + δβmicroδ

αν minus gmicroνgαβ

)ωminus1partαωpartβφ (C6)

φ = ωminus2φ+ (nminus 2)gαβωminus3partαωpartβφ (C7)

The inverse transformations are

R = ω2R + 2(nminus 1)gαβωnablaαnablaβω minus n(nminus 1)gαβpartαωpartβω (C8)

nablamicronablaνφ = nablamicronablaνφ+(δαmicroδ

βν + δβmicroδ

αν minus gmicroν gαβ

φ = ω2φminus (nminus 2)gαβωpartαωpartβφ (C10)

The various 3+1 dimensional quantities transform as follows

hab = ω2hab Na = Na N = ωN (C11)

hab = ωminus2hab Na = ω2Na (C12)

nmicro = ωminus1nmicro nmicro = ωnmicro (C13)

Kab = ωKab + habnmicropartmicroω (C14)

K = ωminus1K + ωminus2(nminus 1)nmicropartmicroω (C15)

D The fundamental theorem of auxiliary vari-

Here we recall some facts about auxiliary fields in classical field theory Formore see for example [54 23]

Suppose the lagrangian depends on two sets of fields φi and χA L =L([φ] [χ] x) (here [ ] stands for dependence on the fields and any of itsspacetime derivatives of arbitrary but finite order) and that the equations ofmotion for χA can be solved in terms of the φi

δELLδχA

= 0rArr χA = χA([φ] x) (D1)

Plugging these relations back into L we get a lagrangian depending only onthe φi which we call L

L([φ] x) equiv L([φ] [χ([φ] x)] x) (D2)

We will christen the following the fundamental theorem of auxiliary vari-ables

bull The equations of motion derived from L and L are equivalent in the φi

sector ie if φi(x) is a solution to δELLδφi

= 0 then it is also a solution

to δELLδφi

= 0 and vice versa if φi(x) is a solution to δELLδφi

= 0 then it

is also a solution to δELLδφi

= 0 with the extension to the χA given by

the χA([φ] x) obtained from solving the χA equations of motion

Proving this is a matter of convincing yourself that you can extremize afunction of several variables either by extremizing with respect to all thevariables or by first extremizing with respect to a few and then with respectto the rest

The other important property that hold when eliminating auxiliary vari-ables is that both the global and gauge symmetry groups of L and L arethe same One does not lose or gain symmetries by eliminating or addingauxiliary variables

Notice that in the fundamental theorem there is no requirement that theauxiliary fields be solved for algebraically However when this is not thecase one must be careful about boundary contributions To illustrate thiswe reprint here a nice example taken from the exercises of chapter one of[23]

Consider the action for two variables q(t) and A(t) with values fixed onthe endpoints

int t2

) (D3)

The equations of motion for q and A are respectively

)= 0 (D4)

Notice that the A equation implies A = cq2 where c is a constant of integra-tion Plugging this back into the action we obtain

int t2

(q2 + c2q2

) (D5)

which gives naively for the q equation of motion q minus c2q = 0 Looking backat the original q equation of motion and plugging in A = cq2 we obtainq + c2q = 0 a contradiction for c2 6= 0

It is often said that auxiliary variables can only be eliminated if theirequations of motion can be solved algebraically and ldquocounterexamplesrdquo likethis are quoted to illustrate this However if we are careful to take into ac-count the endpoints we can resolve the problem Consider again the equation

of motion for A ddt

)= 0 We must solve this subject to the endpoint

conditions A(t1) = A1 A(t2) = A2 Integrating both sides of the solutionA = cq2 we find that the constant c actually depends non-locally on q aswell as on the endpoint data for A

c =A2 minus A1int t2t1q(t)2dt

Plugging this more careful result into the action we obtain a non-localaction for q

int t2

(A2 minus A1int t2t1q(tprime)2dtprime

Varying this carefully with respect to q we recover the correct (non-linearnon-local) equation of motion q + c2q = 0

References

[1] G W Gibbons and S W Hawking ldquoAction Integrals And PartitionFunctions In Quantum Gravityrdquo Phys Rev D 15 2752 (1977)

[2] J W York ldquoRole of conformal three geometry in the dynamics ofgravitationrdquo Phys Rev Lett 28 (1972) 1082

[3] S W Hawking and G T Horowitz ldquoThe Gravitational Hamiltonianaction entropy and surface termsrdquo Class Quant Grav 13 (1996)1487 [arXivgr-qc9501014]

[4] T P Sotiriou and V Faraoni ldquof(R) Theories Of GravityrdquoarXiv08051726 [gr-qc]

[5] S Nojiri and S D Odintsov ldquoIntroduction to modified gravity andgravitational alternative for dark energyrdquo eConf C0602061 06 (2006)[Int J Geom Meth Mod Phys 4 115 (2007)][arXivhep-th0601213]

[6] B Boisseau G Esposito-Farese D Polarski and A A StarobinskyldquoReconstruction of a scalar-tensor theory of gravity in an acceleratinguniverserdquo Phys Rev Lett 85 2236 (2000) [arXivgr-qc0001066]

[7] S M Carroll V Duvvuri M Trodden and M S Turner ldquoIs cosmicspeed-up due to new gravitational physicsrdquo Phys Rev D70 (2004)043528 astro-ph0306438

[8] S Nojiri and S D Odintsov ldquoModified gravity with negative andpositive powers of the curvature Unification of the inflation and of thecosmic accelerationrdquo Phys Rev D 68 123512 (2003)[arXivhep-th0307288]

[9] R P Woodard ldquoAvoiding dark energy with 1R modifications ofgravityrdquo Lect Notes Phys 720 403 (2007) [arXivastro-ph0601672]

[10] S M Carroll et al ldquoThe cosmology of generalized modified gravitymodelsrdquo Phys Rev D71 (2005) 063513 astro-ph0410031

[11] W Hu and I Sawicki ldquoModels of f(R) cosmic acceleration that evadesolar-system testsrdquo Phys Rev D76 (2007) 064004 arXiv07051158[astro-ph]

[12] M Farhoudi ldquoOn higher order gravities their analogy to GR anddimensional dependent version of Duffrsquos trace anomaly relationrdquo GenRel Grav 38 1261 (2006) [arXivphysics0509210]

[13] M Asorey J L Lopez and I L Shapiro ldquoSome remarks on highderivative quantum gravityrdquo Int J Mod Phys A 12 5711 (1997)[arXivhep-th9610006]

[14] M Bojowald and A Skirzewski ldquoQuantum gravity and highercurvature actionsrdquo eConf C0602061 03 (2006) [Int J Geom MethMod Phys 4 25 (2007)] [arXivhep-th0606232]

[15] A Balcerzak and M P Dabrowski ldquoGibbons-Hawking BoundaryTerms and Junction Conditions for Higher-Order Brane GravityModelsrdquo arXiv08040855 [hep-th]

[16] N H Barth ldquoThe Fourth Order Gravitational Action For ManifoldsWith Boundariesrdquo Class Quant Grav 2 497 (1985)

[17] R Casadio and A Gruppuso ldquoOn boundary terms and conformaltransformations in curved spacetimesrdquo Int J Mod Phys D 11 703(2002) [arXivgr-qc0107077]

[18] M S Madsen and J D Barrow ldquoDe Sitter Ground States AndBoundary Terms In Generalized Gravityrdquo Nucl Phys B 323 (1989)242

[19] S Nojiri and S D Odintsov ldquoFinite gravitational action for higherderivative and stringy gravitiesrdquo Phys Rev D 62 064018 (2000)[arXivhep-th9911152]

[20] S M Carroll ldquoSpacetime and geometry An introduction to generalrelativityrdquo San Francisco USA Addison-Wesley (2004)

[21] E Poisson ldquoA Relativistrsquos Toolkit The Mathematics of Black-HoleMechanicsrdquo Cambridge University Press (2004)

[22] H Goldstein ldquoClassical Mechanicsrdquo Addison Wesley 3rd edition(2001)

[23] M Henneaux and C Teitelboim ldquoQuantization of gauge systemsrdquoPrinceton USA Univ Pr (1992)

[24] P A M Dirac ldquoLectures on Quantum Mechanicsrdquo DoverPublications (2001)

[25] P Teyssandier and Ph Tourrenc ldquoThe Cauchy problem for the R+R2

theories of gravity without torsionrdquo J Math Phys 24 2793 (1983)

[26] B Whitt ldquoFourth Order Gravity As General Relativity Plus MatterrdquoPhys Lett B 145 176 (1984)

[27] J D Barrow and S Cotsakis ldquoInflation and the Conformal Structureof Higher Order Gravity Theoriesrdquo Phys Lett B 214 515 (1988)

[28] J D Barrow ldquoThe Premature Collapse Problem in ClosedInflationary Universesrdquo Nucl Phys B 296 697 (1988)

[29] D Wands ldquoExtended gravity theories and the Einstein-Hilbertactionrdquo Class Quant Grav 11 269 (1994) [arXivgr-qc9307034]

[30] T Chiba ldquo1R gravity and scalar-tensor gravityrdquo Phys Lett B 5751 (2003) [arXivastro-ph0307338]

[31] T Chiba ldquoGeneralized gravity and ghostrdquo JCAP 0503 008 (2005)[arXivgr-qc0502070]

[32] G J Olmo ldquoLimit to general relativity in f(R) theories of gravityrdquoPhys Rev D 75 023511 (2007) [arXivgr-qc0612047]

[33] D Lovelock ldquoThe Einstein tensor and its generalizationsrdquo J MathPhys 12 498 (1971)

[34] R C Myers ldquoHigher Derivative Gravity Surface Terms and StringTheoryrdquo Phys Rev D 36 392 (1987)

[35] C Charmousis and R Zegers ldquoMatching conditions for a brane ofarbitrary codimensionrdquo JHEP 0508 075 (2005)[arXivhep-th0502170]

[36] J T Liu and W A Sabra arXiv08071256 [hep-th]

[37] L J Garay and J Garcia-Bellido ldquoJordan-Brans-Dicke QuantumWormholes And Colemanrsquos Mechanismrdquo Nucl Phys B 400 416(1993) [arXivgr-qc9209015]

[38] S Capozziello and R Garattini ldquoThe cosmological constant as aneigenvalue of f(R)-gravity Hamiltonian Class Quant Grav 24 1627(2007) [arXivgr-qc0702075]

[39] M Salgado ldquoThe Cauchy problem of scalar tensor theories of gravityrdquoClass Quant Grav 23 4719 (2006) [arXivgr-qc0509001]

[40] R M Wald ldquoBlack hole entropy is the Noether chargerdquo Phys Rev D48 3427 (1993) [arXivgr-qc9307038]

[41] V Iyer and R M Wald ldquoSome properties of Noether charge and aproposal for dynamical black hole Phys Rev D 50 846 (1994)[arXivgr-qc9403028]

[42] V Iyer and R M Wald ldquoA Comparison of Noether charge andEuclidean methods for computing the Phys Rev D 52 4430 (1995)[arXivgr-qc9503052]

[43] F Briscese and E Elizalde ldquoBlack hole entropy in modified gravitymodelsrdquo Phys Rev D 77 044009 (2008) [arXiv07080432 [hep-th]]

[44] J D Brown and J W York ldquoThe Microcanonical functionalintegral 1 The Gravitational fieldrdquo Phys Rev D 47 1420 (1993)[arXivgr-qc9209014]

[45] J D Bekenstein ldquoNonexistence of baryon number for static blackholesrdquo Phys Rev D 5 1239 (1972)

[46] M Heusler ldquoA Mass bound for spherically symmetric black holespacetimesrdquo Class Quant Grav 12 779 (1995) [arXivgr-qc9411054]

[47] E Winstanley ldquoDressing a black hole with non-minimally coupledscalar field hairrdquo Class Quant Grav 22 2233 (2005)[arXivgr-qc0501096]

[48] C Teitelboim ldquoQuantum Mechanics Of The Gravitational FieldrdquoPhys Rev D 25 3159 (1982)

[49] R B Mann and D Marolf ldquoHolographic renormalization ofasymptotically flat spacetimesrdquo Class Quant Grav 23 2927 (2006)[arXivhep-th0511096]

[50] M Ostrogradski Mem Ac St Petersbourg VI 4 385 (1850)

[51] T Nakamura and S Hamamoto ldquoHigher Derivatives and CanonicalFormalismsrdquo Prog Theor Phys 95 469 (1996)[arXivhep-th9511219]

[52] R L Arnowitt S Deser and C W Misner ldquoThe dynamics of generalrelativityrdquo arXivgr-qc0405109

[53] A Einstein ldquoHamiltonrsquos Principle and the General Theory ofRelativityrdquo Sitzungsber Preuss Akad Wiss Berlin (Math Phys )1916 1111 (1916)

[54] M Henneaux ldquoOn the use of auxiliary fields in classical mechanics andin field theoryrdquo Contemporary Mathematics Proceedings of the 1991Joint Summer Research Conference on Mathematical Aspects ofClassical Field Theory

Introduction and outline

What makes a variational principle well-posed

Some Toy Examples

Standard classical mechanics

Constraints

Gauge invariance

Field theoryspatial boundaries

Adding total derivatives

Higher derivative lagrangians

Review of the Gibbons-Hawking-York term

Higher derivative gravity and scalar-tensor theory

F(R) gravity

Boundary terms for general scalar-tensor theory

Boundary term for F(R) theory

Hamiltonian formulation and ADM energy

Black hole entropy

Conclusions

The ADM decomposition

Foliation of spacetime

Conformal transformation formulae

The fundamental theorem of auxiliary variables

1 Introduction and outline

The Einstein-Hilbert action for general relativity (GR) is

SEH simintd4xradicminusgR (11)

When varying this action one finds surface contributions that must vanish ifthe action is to be stationary The surface contributions contain the metricvariation δgmicroν and variations of the derivatives of the metric δ(partσgmicroν) Settingδgmicroν = 0 on the boundary is not sufficient to kill all the surface contributionsFixing both the metric and the derivatives of the metric on the boundaryis uncomfortable however it may not be initially obvious why this is so orthat there exists a unique and correct prescription for dealing with it

Gibbons Hawking and York proposed adding the trace of the extrinsiccurvature of the boundary K to the action [1 2] With this modificationthe action takes the form

S = SEH + SGHY simintd4xradicminusgR + 2

∮d3xradic|h|K (12)

h is the determinant of the induced metric on the boundary This modifica-tion is appealing because the variation of the Gibbons-Hawking-York (GHY)boundary term cancels the terms involving δ(partσgmicroν) and so setting δgmicroν = 0becomes sufficient to make the action stationary This has been widely ac-cepted as the correct modification to the action

This modification raises some questions Is the modification unique Isit necessarily incorrect to require δ(partσgmicroν) = 0 If this is incorrect why is itappropriate to require the fixing of all components of gmicroν when the gravitonhas only two degrees of freedom More generally what criteria determinewhich quantities to fix on the boundary Should it be related to the numberof degrees of freedom in the theory

Despite these questions the GHY term is desirable as it possesses a num-ber of other key features The term is required to ensure the path integralhas the correct composition properties [1] When passing to the hamiltonianformalism it is necessary to include the GHY term in order to reproduce thecorrect Arnowitt-Deser-Misner (ADM) energy [3] When calculating black

L ([φ] x) (21)

δELLδφi

= 0 (22)

whereδEL

δφi=

j = δijδmicro1

nsuml=r

S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





L ([φ] x) (21)

δELLδφi

= 0 (22)

whereδEL

δφi=

j = δijδmicro1

nsuml=r

S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





L ([φ] x) (21)

δELLδφi

= 0 (22)

whereδEL

δφi=

j = δijδmicro1

nsuml=r

S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





L ([φ] x) (21)

δELLδφi

= 0 (22)

whereδEL

δφi=

j = δijδmicro1

nsuml=r

S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





S = S[φ(V)] (27)

S[φ(V)] =

intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





intVdnx L ([φ] x)

δELLδφi

= 0 (210)

intVdnx L ([φ] x)

∣∣∣∣φ=φc

3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





3 Some Toy Examples

S[qi1 qi2 t1 t2] =

int t2

dt L(qi qi t) (31)

(part2L

partqipartqj

)6= 0 (32)

int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





int t2

[partL

partqiminus d

partqi

]δqi +

partqiminus d

partqi= 0 (34)

part2L

partqipartqjqj =

partqiminus part2L

partqipartqjqj (35)

int t2t1dt 1

ω n = 1 2 and we

pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





pi =partL

partqi (36)

int t2

dt piqi minusH (37)

whereH(p q) =

dtF (q) of any

32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





32 Constraints

part2Lpartqipartqj

)6= 0 This is

int t2

2 minus1

2Mω2q2

2 + λq1q2 (39)

int t2

(radicmq1 plusmn

radicMq2

minus 1

2Mω2q2

2 (310)

radicMq2

)δ(radic

mq1 plusmnradicMq2

)∣∣∣t2t1 (311)

33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





33 Gauge invariance

int t2

2(q1 minus q2)2 (312)

intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





intVd4x minus 1

4FmicroνF

microνδAν (318)

int t2

dt L2 L2 = minus1

2qq (319)

minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





minus 1

)∣∣∣∣t2t1

q(t1) and q

q(t2) at the

endpoints yieldsA

At1 +B= C1

At2 +B= C2 (322)

type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





type term(

12qq)∣∣t2t1

q isnot

int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





int t2

δS1 =

int t2

q δq]|t2t1 (324)

S1 = S1[q1 q1 q2 q2 t] (325)

int t2

Varying gives

δS2 =

int t2

]∣∣∣t2t1

int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





int t2

δS3 =

int t2

]∣∣∣t2t1

int t2

dt F (q)minus d

with variation

δS4 =

int t2

q 2 + F prime(q)

∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





∮partVdnminus1x

radic|h|K (41)

16πG δS =

2Rgmicroν

)δgmicroν minus

∮partVdnminus1x

radic|h|δK (42)

We see that

16πG δS =

2Rgmicroν

)δgmicroν (44)

ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





ναβ

) (45)

16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





16πG SEH =

intVdnx N

radicminusγ[

+ 2nablaα

(rβnablaβr

∮Bdnminus1z

radic|h|K

intVdnx N

radicminusγ[

7In detail

12nαg

theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





theory

microλR

λν )

51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





51 F (R) gravity

) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





) (55)

SGHY = 2

∮partVdnminus1x

radic|h|f(φ)K (56)

2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





2f(φ)

(Rmicroν minus

2Rgmicroν

2gmicroν(partφ)2

λ(φ)φ+1

f(φ) 6= 0

nminus2 gmicroν

G lt 0 if f(φ) lt 0(510)

The result is

intVdnx

radicminusg(

16πGRminus 1

∮partVdnminus1x

radic|h|K

A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





A(φ) =1

λ(φ)

f(φ)+nminus 1

nminus 2

f prime(φ)2

f(φ)2

) (512)

V (φ) =U(φ)

∮partVdnminus1x

] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





] (62)

SGHY = 2

intpartVdnminus1x

SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





SG + SGHY =

α minus nαK)]

∮Bdnminus1z

radicminusγKf + 2

∮Σ1

∮Σ2

dnminus1yradichKf

SG + SGHY =

intVdnxradichN

∮Bdnminus1z

α] (65)

= k (66)

SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





SG + SGHY =

intVdnx

radichN

)+ 2f prime

)partαφ

∮Bdnminus1z

radicσNfk (68)

SG + SGHY =

intVdnxradich[Nf

∮Bdnminus1z

radicσNfk (69)

8In detail

intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





intVdnx

radichλ

]minusNradichU

Define now

)(611)

S primeB equiv 2

∮Bdnminus1z

radicσNfk (613)

int t2

dt L[hab hab NN

a φ φ] (615)

pφ =δL

radichλ

(φminusNapartaφ

)to replace

δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





δKab = 12Nδhab

pab =δL

=radich

[f(Kab minusKhab

)minus f prime

φ =Nradich

]+Napartaφ (618)

Kab =1

pabradichminus hab

prime + 2pfprime2

f+ pλ

H primeφ =

(intΣt

dnminus1y pφφ

)minus Lprimeφ

intΣt

dnminus1yradich

h+ 2f prime

pφradich

+ 2f prime2K

(pφradichpartaφ

)] (619)

Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Now the metric part

H primeG =

(intΣt

dnminus1y pabhab

)minus LprimeG

intΣt

H primeG =

intΣt

dnminus1yradich

(pabradich

) (621)

intΣt

dnminus1yradich[Nf

h+ f prime

pφradich

pabradich

dnminus2θradicσ

] (622)

Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Hsolution = 2

E = minus2

f(r)dr2 + r2dω2 φ = φ0 (625)

E = 16πGMf(φ0) (626)

16πGeff

SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





SBH =A

where β = 1T

f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





f(r)dr2 + r2dω2 (73)

ds2E = f(r)dτ 2 +

f(r)dr2 + r2dω2 τ = τ + β β = 8πGM (74)

radic|g|Rminus 2

∮partVd3x

partVd3x

)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





)12∮

partVd3x

radic|h|K0 = 8πβr

(1minus 2GM

SE =β2

16πG (78)

F = minus 1

βlnZ =

βSE =

16πG (79)

E = F + βpartF

partβ=

8πG= M (710)

S = β2partF

partβ=

16πG=

4G (711)

prime(R) = 0 (712)

416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





416πF prime(R0) (717)

16πGeff

= F prime(R0) (718)

8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





8 Conclusions

Acknowledgements

tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





tmicro =partxmicro

partt emicroa =

partxmicro

partya (A2)

tmicro =partt

partxmicro e a

micro =partya

partxmicro (A3)

microan

ν = 0 (A6)

1 Σt timelike

amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





amicro

cγ (A10)

P microλP

λν = P micro

ν (A12)

νnν = 0 (A13)

P microνe

amicro = e a

ρσλ (A15)

aν = P micro

ν (A16)

σνgρσ = habe

amicro e

bν hab = emicroae

νbgmicroν (A17)

habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





habgmicroνe b

as well asgmicroαP

α = hmicroν (A20)

The metric is

(N bdt+ dyb

) (A24)

(εN2 +NaNa Na

Na hab

) (A25)

gminus1 =

(ε 1N2 minusεNa

minusεNa

N2 hab + εNaNb

) (A26)

radic|h| (A27)

N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





N na = minusN

N n0 = εN na = 0 (A28)

microb We also have

2emicroae

νbLngmicroν (A31)

Kab =1

) (A33)

+ 2εnablaα

(nβnablaβn

emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





emicro

tmicro

nmicro

rmicro

emicroA

nmicro

line of constant ya

) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





) (C4)

βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





βν + δβmicroδ

δELLδχA

to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





to δELLδφi

= 0 then it

int t2

) (D3)

)= 0 (D4)

int t2

(q2 + c2q2

) (D5)

of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





of motion for A ddt

int t2

References

Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Some Toy Examples


Constraints

Gauge invariance






F(R) gravity




Black hole entropy

Conclusions





Documents

Boundary Terms, Variational Principles and Higher ... · Boundary Terms, Variational Principles and Higher Derivative Modi ed Gravity Ethan Dyer1 and Kurt Hinterbichler2 Institute