Thermodynamical Variables
in
Lanczos-Lovelock Gravity
IUCAA-NCRA Graduate School Project ReportSubmitted By
Sumanta Chakraborty 1 2
IUCAA, Post Bag 4, Ganeshkhind,Pune University Campus, Pune 411 007, India
Under Guidance of
Prof. T. Padmanabhan 3
IUCAA, Post Bag 4, Ganeshkhind,Pune University Campus, Pune 411 007, India
Date of Submission
25th July 2014
To
my grandmother
Jharna Chakraborty
Acknowledgements
First and foremost I would like to thank Prof. T. Padmanabhan, whose constant encour-agement, support and breathtaking explanations in these fields of research made this reportpossible. Without Prof. Padmanabhan’s guidance and constant mental support from my par-ents this report would never come into existence. The time when the work presented in thisreport was going on I have lost my grandmother from whom I have learnt the meaning ofstruggle in life. My sincerest respect to her.
Also I would like to thank my classmates, I have been benefited from discussions withthem. I must mention that special thanks are in order to Bibhas Ranjan Majhi, Suprit Singhand Krishnamohan Parattu for fruitful discussions from which I have learnt much. It would becompletely unjustified if I do not thank the Library facility at IUCAA, it is the IUCAA Librarythat prompted me to do research from my childhood. Finally I must thank CSIR, Governmentof India for providing me SPM fellowship.
i
Abstract
The project work consisted of two parts: (a) I have reviewed two papers, one [1] on the ther-modynamical conjugate variables in Einstein-Hilbert action and another [2] on the holographicequipartition and thermodynamics in GR. (b) I have generalized these results to Lanczos-Lovelock gravity by identifying proper conjugate variables.
In the report, the first chapter is a broad introduction to the subject area; in Chap. 2, Ihave presented basic results in Lanczos-Lovelock gravity which are useful for later discussion.In Chap. 3, I have first presented the results of [1] for Einstein-Hilbert action and subsequentlyhave shown the generalization to Lanczos-Lovelock gravity. After describing the conjugatevariables we discuss the thermodynamic properties of these variables in Chap. 4. Again, I havefirst discussed the results in the Einstein-Hilbert action and then have generalized these resultsto Lanczos-Lovelock gravity using the previously introduced conjugate variables. Similarly inChap. 5, I have first reviewed the relation between bulk dynamics and horizon thermodynamicsin Einstein-Hilbert action following Ref. [2] and have subsequently generalized them to Lanczos-Lovelock gravity. Finally I conclude with a discussion on the results. Various results used inthe text are described in the two appendices at the end of the report.
ii
Contents
1 Introduction 1
2 Lanczos-Lovelock Gravity: Overview 32.1 Brief Introduction to Lanczos-Lovelock Gravity . . . . . . . . . . . . . . . . . . 52.2 Noether Current and Entropy of Horizons . . . . . . . . . . . . . . . . . . . . . 7
3 Describing Gravity in Lanczos-Lovelock Theories 103.1 Conjugate Variables in Einstein Gravity . . . . . . . . . . . . . . . . . . . . . . 103.2 Conjugate Variables in Lanczos-Lovelock Theories . . . . . . . . . . . . . . . . . 13
4 Thermodynamics with Conjugate Variables 184.1 Introduction to Gaussian Null Coordinates . . . . . . . . . . . . . . . . . . . . . 184.2 Thermodynamics Related to Einstein-Hilbert Action . . . . . . . . . . . . . . . 19
4.2.1 Static Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.2 Generalization to Arbitrary null surface . . . . . . . . . . . . . . . . . . . 20
4.3 Thermodynamics Related to Lanczos-Lovelock Action . . . . . . . . . . . . . . . 224.3.1 A general Static Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.2 Generalization to Arbitrary Null Surface . . . . . . . . . . . . . . . . . . 24
5 From Bulk Dynamics to Surface Thermodynamics 275.1 Noether Charge and Holographic Equipartition In Einstein-Hilbert Action . . . 275.2 Generalization To Lanczos-Lovelock Gravity . . . . . . . . . . . . . . . . . . . . 31
6 Conclusions 36
A Derivation of Various Identities used in Text 38
B Identities Regarding Lie Variation of P abcd 42
Bibliography 44
iii
Chapter 1
Introduction
Gravity can be described in terms of the underlying geometrical properties of the spacetimeand the field equation for gravity determines dynamics of the spacetime given a source of en-ergy momentum tensor. Though the Einstein-Hilbert action contains second order derivativesof the metric, the field equation in General Relativity has only upto second derivatives of themetric. This prompts further research in order to obtain the most general class of Lagrangiansthat give field equation with only second order derivatives of the metric. This general classof Lagrangians are referred to as Lanczos-Lovelock Lagrangian. The Einstein-Hilbert actionis a special case of this Lagrangian since for spacetime dimension D = 4 Lanczos-LovelockLagrangian becomes the Ricci scalar. Hence the Lanczos-Lovelock Lagrangian turns out tobe the most natural generalization of Einstein-Hilbert action to higher dimensional spacetime[3, 4, 5, 6].
There are several peculiar features in standard General Relativity based on the Einstein-Hilbert action. The most important of them is the curious connection between horizons andthermodynamics. These were first discovered for black holes in the seminal work by Bekenstein,relating black hole entropy to its horizon area [7, 8]. Later it was found that not only entropybut also temperature can be attributed to black holes. Thus with entropy and temperatureblack hole behaves as a thermodynamic system [9, 10, 11, 12]. This connection between grav-ity and thermodynamics gains more significance from the works by Unruh and Davies [13, 14].They showed that the gravity-thermodynamics relationship emerges from the null nature of thehorizons, so that the black hole horizons behave as one way membranes. The above programworks not just for black hole solutions but for any arbitrary null surfaces. These results suggestthat gravity may not be of fundamental origin, there could be something more fundamentalbeneath it. This point of view gets amplified by the following results:
• The Field equations in various theories representing gravity have thermodynamical in-terpretations. This goes well beyond Einstein gravity [15, 16].
• It is possible to obtain Gravitational field equations starting from thermodynamic ex-tremum principles [17, 18] and the action functional for gravity has thermodynamic interpre-tation [19, 20, 21]. Another important aspect is that microscopic degrees of freedom can beprobed via equipartition theorem [22, 23].
1
• The field equations for Einstein gravity can be mapped into Navier-Stokes fluid equationprojected on a null surface by generalizing results on black hole spacetime [24, 25, 26]. Thisresult can be supplemented by the fact that the Euclidean path integral of action for gravityhas partition function like behavior. This can be exploited to get free energy, entropy and otherthermodynamic variables for Lanczos-Lovelock models [27, 28].
Very recently some further results were obtained for General Relativity in Refs. [1, 2] andinteresting features appeared in these studies. In Ref. [1] a new set of variables have been intro-duced, which are fab =
√−ggab and its conjugate momenta. Various results in Einstein-Hilbert
action gets simplified considerably on using these variables, and more importantly variationsof these variables are found to be connected to δT and δS [29, 30, 31, 32, 33, 34]. In Ref. [2]evolution equation for a spacetime has been determined in terms of bulk and surface degreesof freedom.
This raises an interesting question: Can these results be generalized from Einstein-Hilbertto Lanczos-Lovelock models? As we show in this report, most of the results for the Einstein-Hilbert action can be generalized to Lanczos-Lovelock scenario with properly defined conju-gate variables. The thermodynamical results and the dynamical evolution gets generalized instraightforward manner. However, some issues regarding inclusion of matter in the equation ofmotion exists in this present scheme, which needs further study.
The report is organized as follows: In Chap. 2 we will review the basics of Lanczos-Lovelocktheories and its properties, along with the holographic structure and the horizon thermody-namics using Noether current. Then in Chap. 3 we present some results regarding canonicallyconjugate variables in Einstein-Hilbert action for ease comparison and limiting process and ad-dress the issue of conjugate variables in Lanczos-Lovelock theories. In Chap. 4 we present thethermodynamic results. In Sec. 4.1 we present construction of line element around a generalnull surface and the thermodynamic interpretation for Einstein-Hilbert action. The penulti-mate Sec. 4.3 presents thermodynamic quantities for a general static spacetime and then ithas been generalized to arbitrary null surface constructed in the previous section. Finally, weconclude with a discussion of our results. We have also presented all the main calculations usedin the text in two Appendices A and B respectively.
Throughout the report we will use the following conventions: The metric is assumed to havesignature (−,+,+, . . . ,+). All the fundamental constants, G, ~ and c are set to unity. The latinindices a, b, . . . runs over all the spacetime coordinates, while bold faced latin indices A,B, . . .runs over transverse coordinates with greek indices µ, ν, . . . over the spatial coordinates.
2
Chapter 2
Lanczos-Lovelock Gravity: Overview
In this chapter we shall first briefly review the structural aspects of Einstein-Hilbert (EH)action which will be helpful for generalization to Lanczos-Lovelock (LL) gravity. The equationsof motion obtained from varying the action are expected to be of second order. Usually theLagrangian is a scalar, which should contain first derivatives of dynamical variables. But in localinertial frames the first derivatives of the metric are zero, thus the Lagrangian being a scalarremains zero in any other frame and hence in order to circumvent these difficulties the action isconstructed in such a manner that though it contains second derivatives of dynamical variablesthe equations of motion are of second order. In four dimensions this criteria is sufficient touniquely identify the action functional to be EH action which is given by,
16πAEH =
∫Vd4x√−gLEH =
∫Vd4x√−gR (2.1)
where R = gacgbdRabcd, is the Ricci scalar. The above action can be simplified considerably bydefining a quantity, Q bcd
a and the Lagrangian density LEH can be presented as,
LEH ≡ Q bcda Ra
bcd = QabcdR
cdab (2.2)
Q bcda =
1
2
(δcag
bd − δdagbc)
; Qabcd ≡ δabcd ≡
(δac δ
bd − δadδbc
)(2.3)
There exists an interesting result regarding the decomposition of EH action into a bulk term,which is quadratic in derivatives of the metric and a surface term containing all the secondderivatives [35]. As presented in Ref. [5] the bulk part of the action, when varied, leadsto Einstein equation and the surface part is actually related to the horizon entropy. Thisdecomposition can be written in terms of the variable Q bcd
a as,
LEH = Lquad + Lsur
Lquad ≡ 2Q bcda ΓadkΓ
kbc; Lsur ≡
2√−g
∂c(√−gQ bcd
a Γabd)
(2.4)
3
The two parts, quadratic part and the surface part can be further simplified and written in amore compact form as [5]:
Lquad =1
4Mabcijk∂agbc∂igjk (2.5)
Mabcijk = gaigbcgjk − 1
2
(gaigbjgck + gaigcjgbk
)+
1
2
(gakgbjgci + gajgbkgci
)+
1
2
(gakgcjgbi + gajgckgbi
)− 1
2
(gakgbcgij + gajgbcgik + gibgjkgac + gicgjkgab
)(2.6)
√−gLsur = 2∂c
[√−gQ bcd
a Γabd]≡ ∂c
[√−gV c
](2.7)
V c =(gikΓcik − gckΓmkm
)= −1
g∂a(ggab
)(2.8)
In the above expressions the six indexed tensorial object Mabcijk is symmetric in the pairs(b, c) and (j, k). Moreover it is also symmetric under exchange of the triplets (a, b, c) and(i, j, k). Thus the above result suggests that the Lagrangian can be split up into two parts,one containing the quadratic part and the other a surface divergence. Hence we can derivethe equation of motion in terms of the quadratic part alone. For that purpose we vary thequadratic part of the action with respect to the metric leading to,
δ(√−gLquad
)= δ
(√−ggabRab
)− δ
(√−gLsur
)=√−gGabδg
ab +√−ggabδRab − δ
(√−gLsur
)(2.9)
In the above expression we have used the Einstein tensor in the form Gab = Rab − (1/2)gabR.The remaining two terms can be evaluated with the following expression [1, 5] as:
√−ggabδRab = 2∂c
[√−ggbkQcd
akδΓabd
](2.10)
δ(√−gLsur
)= ∂c
(2√−ggbkQcd
akδΓabd +√−gM c
lmδglm)
(2.11)
M clm = 2Qcd
akBbklmΓabd; Bbk
lm =
(δbl δ
km −
1
2gbkglm
)(2.12)
Then substitution of all the above results in Eq. (2.9) leads to the following expression forvariation of the quadratic part of the Lagrangian as:
δAquad =1
16π
∫Vd4xδ
(√−gLquad
)=
1
16π
∫Vd4x√−gGabδg
ab − 1
16π
∫∂Vd3x√hncM
cabδg
ab (2.13)
To arrive at the last expression we have used Gauss’s divergence theorem to convert the di-vergence to a surface term, where the surface is characterized by the normal nc and inducedmetric on the surface, hab. Hence by fixing the metric on the boundary we readily obtain thesurface term to vanish and thus for arbitrary variation the Einstein tensor comes out to be thegeometrical quantity determining the dynamics of gravity.
We also mention one important feature regarding the connection between the bulk andsurface Lagrangian introduced above. In EH action it turns out that the surface term is
4
entirely expressible in terms of the quadratic Lagrangian in D dimension (D > 2) such thatthe following relation is obtained:
Lsur = − 1
{(D/2)− 1}∂c
(gab
∂Lquad∂(∂cgab)
)(2.14)
These type of Lagrangians with the above properties are known as holographic [20].
We must stress the following point that, equations of motion can be determined from thequadratic term alone. As solutions to the equations of motion are obtained these obviouslyincludes the black hole solutions as well. This has no connection with the surface term since allof the above quantities are derived from quadratic term alone. However when the surface termis evaluated on the horizon it produces the entropy associated with the horizon. This couldbe taken as one motivation for the fact that action formulation in gravity has thermodynamicinterpretation.
2.1 Brief Introduction to Lanczos-Lovelock Gravity
The above discussion was restricted to spacetime with gravity described by the EH action.Now we shall concentrate on more general action functionals that can represent gravity athigher dimensional spacetime and has several unique features. All the action functionals weshall consider should contain at most second derivatives of the metric, i.e. it can containcurvature tensor but not its derivatives. Under this circumstances the above results can bereadily generalized to Lanczos-Lovelock theories where we have more than four dimensions andhave more terms than the EH term. In a general D dimensional space-time the action functionaltakes the following form,
A =
∫VdDx√−gL
(gab, Rp
qrs
)(2.15)
where the Lagrangian scalar L is built from the metric and the curvature tensor but doesnot contain any derivatives of curvature tensor. Thus it can be thought of as a functionalin the following quantities
(Rabcd, R
abcd, R
abcd, . . .
)having proper combination with gab or gab
respectively. The most important quantity derived from the Lagrangian is the derivative of theLagrangian with respect to curvature tensor defined as,
P abcd =
(∂L
∂Rabcd
)gij
(2.16)
The above tensor has all the properties of the curvature tensor such that, it is antisymmetricunder the exchange of first two and the last two indices, symmetric under exchange of first twoset and last two set of indices and finally satifies Bianchi identity (for more properties of thistensor see [36]). Note that there is another tensor of importance which is an equivalent of Riccitensor in EH action defined as,
Rab ≡ P aijkRbijk (2.17)
5
Having defined the quantities necessary for our purpose, we now consider variation of the actionfunctional leading to,
δA = δ
∫VdDx√−gL
=
∫VdDx√−gEabδgab +
∫VdDx√−g∇jδv
j (2.18)
where we have the following expression for Eab and the boundary term δva [5]:
Eab ≡1√−g
(∂√−gL∂gab
)Rabcd
− 2∇m∇nPamnb
= Rab −1
2gabL− 2∇m∇nPamnb (2.19)
δvj = 2P ibjd∇bδgdi − 2δgdi∇cPijcd (2.20)
As we have been stressing earlier the field equation should be second order in the metric.However since the quantity P abcd involves second derivative of the metric, the term ∇m∇nPamnbin Eab contain fourth order derivative of the metric. Thus in order to get sensible field equationwe should impose the extra condition on P abcd such that,
∇aPabcd = 0 (2.21)
In addition to the above criteria we need to set the boundary term δvj to zero. This requiressetting both δgab and ∇cδgab to zero, however we can only set δgab = 0. This amounts addingan extra counter term to kill the normal derivative of the variation of metric tensor. With theseconditions the field equation in presence of matter in Lanczos-Lovelock gravity turns out to be:(
Rba −
1
2δbaL
)= 8πT ba (2.22)
Thus the intention to obtain an action functional which would satisfy equation of motion whichis second order in the metric reduces to finding Lagrangian such that Eq. (2.21) is satisfied.Such an action functional satisfying the above criteria obtained is unique and coincides withLanczos-Lovelock Lagrangian in D dimensions given by,
L =∑m
cmLm (2.23)
where, Lm is a Lagrangian which is homogeneous in Rabcd of order m having the followingexpression,
Lm =1
m
∂Lm∂Rabcd
Rabcd =1
mP abcd
(m) Rabcd (2.24)
This can also be written as, Lm = Qabcd(m) Rabcd, which can be used to identify, P abcd
(m) = mQabcd(m) .
From now on we shall work with this mth order Lagrangian only and hence shall skip them index here after. For this mth order Lanczos-Lovelock Lagrangian we have the expressionfor P ab
cd in terms of the curvature tensor as,
P abcd =
∂Lm∂Rcd
ab
= mδaba2b2...ambmcdc2d2...cmdmRc2d2a2b2
. . . Rcmdmambm
(2.25)
6
This relation will be used extensively later. Also note that due to complete antisymmetry in theindices of the determinant tensor we have in a D dimensional space-time the following restriction4m ≤ D otherwise the determinant tensor would vanish identically. In four dimensions thisproperty uniquely fixes EH action for m = 1. In the general action expressed by Eq. (2.15)we can introduce the condition (2.21), by virtue of which we can write the Lagrangian densityalong the similar lines as we have used to describe the EH action such that:
√−gL =
√−gQ bcd
a Rabcd; ∇cQ
bcda = 0 (2.26)
From the above discussion it will be clear that the quantity Q bcda is an arbitrary function of the
Riemann tensor defined through Eq. (2.24). Also for an mth order Lanczos-Lovelock Lagrangianwe have P bcd
a = mQ bcda as mentioned earlier. Then the surface and bulk decomposition in this
case turns out to be,
√−gL = 2
√−gQ bcd
a ΓadkΓkbc +
2√−g
∂c(√−gQ bcd
a Γabd)≡ Lbulk + Lsur (2.27)
The nature of Lanczos-Lovelock models at D = 2m is of quite importance these are knownas critical dimensions for a given Lanczos-Lovelock term. In these situations the variation ofaction functional reduces to a pure surface term [37].
As we have mentioned the holographic principle in EH action such a holographic principleexists in Lanczos-Lovelock gravity as well. In this case as well the surface term is expressiblein terms of the bulk Lagrangian in a straightforward manner such that for mth order Lanczos-Lovelock Lagrangian the following result is obtained [20]:
{(D/2)− 1}L(m)sur = −∂i
[gab
δL(m)bulk
δ (∂igab)+ ∂jgab
∂L(m)bulk
∂ (∂i∂jgab)
](2.28)
where the Euler derivative which is denoted by δ for a dynamical variable φ having the followingexpression:
δK (φ, ∂iφ)
δφ=∂K (φ, ∂iφ)
∂φ− ∂i
{∂K (φ, ∂iφ)
∂ (∂iφ)
}(2.29)
Having discussed the gravity part in Lanczos-Lovelock gravity we now move to discuss thethermodynamic properties in Lanczos-Lovelock gravity.
2.2 Noether Current and Entropy of Horizons
Another important aspect of the Lanczos-Lovelock theories originate from the fact that anygenerally covariant theory must posses diffeomorphism invariance. This implies that under aninfinitesimal coordinate transformation, xa → xa+ξa(x) the theory should lead to conservationof a current which is referred to as Noether current. As we have stressed earlier that variationof the action functional under arbitrary variation of gab leads to the equation of motion termand a surface term from which we can get the Noether current having the following expression[6, 5]:
Ja ≡(2Eabξb + Lξa + δξv
a)
(2.30)
7
where Eab = Rab − (1/2)gabL. In the above expression the last term, δξva represents the
boundary term with metric variation having the form, δgab = ∇aξb +∇bξa. From the propertyof the Noether current ∇aJ
a = 0, we can define an antisymmetric tensor referred to as Noetherpotential by the condition, Ja = ∇bJ
ab. Then from Eq. (2.19) we can substitute for theboundary term leading to an explicit form for both the Noether current and potential. Thesegeneral expressions can be found in Ref. [5]. However we just provide the respective expressionsin connection with Lanczos-Lovelock theories as,
Jab = 2P abcd∇cξd (2.31)
Ja = 2P abcd∇b∇cξd = 2Rabξb + 2P ajik LξΓkaj (2.32)
where Γabc is the metric compatible connection.
The above results concerning Noether current has interesting application in thermodynam-ics. We can define a quantity called Noether charge out of these quantities and this is directlyrelated to the entropy of the horizon [5, 34, 38, 39]. For EH action the Noether potential turnsout to be,
Jab =1
16πG
(∇aξb −∇bξa
)(2.33)
For a timelike Killing vector field if the above expression is evaluated and used to calculatethe Noether charge it turns out to be, κAh/8πG, where κ is the surface gravity and Ah is thehorizon area. Thus these quantities are directly related to the thermodynamic structure ofunderlying spacetime.
The same prescription works in Lanczos-Lovelock gravity as well. Rather than pursuing thegeneral form we will consider the surface term in Rindler metric, which closely approximatesthe metric near a horizon. Hence for the rest of the calculation we can assume the metric neara null surface to have the Rindler form. The surface Lagrangian can be inserted such that thecorresponding action has the following expression in D dimension:
Asur = 2
∫dDx∂c
(√−gQabcdΓabd
)= 2
∫dDx∂c
(√−gQabcd∂bgad
)(2.34)
We need to evaluate this for the metric in the near horizon regime, which is the Rindler metric:
ds2 = −κ2x2dt2 + dx2 + dx2⊥ (2.35)
with x⊥ denoting (D − 2) dimensional transverse sector coordinates. On eliminating the totaldivergence Eq. (2.34) reduces to:
Asur = 2
∫dD−1xnc
(√−gQabcd∂bgad
)= 2
∫dt dD−2x nc
(√−gQabcd∂bgad
)(2.36)
Now the quantity ∂Ssur/∂t behaves as a surface Hamiltonian and has the following expression:
∂Asur∂t
= 2
∫dD−2x nc
(√−gQabcd∂bgad
)(2.37)
8
When evaluated using the Rindler metric Eq. (2.35) the normal turns out to be nc = (0, 1, 0, . . .)and only derivative of the g00 component with respect to x remains non zero. Thus the aboveequation leads to:
∂Asur∂t
= 2
∫dD−2x κx
√σQ0xx0∂x
(−κ2x2
)= 4κ3x2
∫dD−2x
√σQ0x
0xg00gxx
= −4κ
∫dD−2x
√σQ0x
0x =( κ
2π
)(−8π
∫dD−2x
√σQ0x
0x
)=
κ
2πSWald = TSWald (2.38)
Since we are evaluating the surface term in the Rindler metric approximation, which is static,we can integrate this over t only if we have suitable limits for the t integration. This can bedone easily in the Euclidean coordinates obtained by transforming t = −iτ in which case theRindler metric Eq. (2.35) reduces to:
ds2 = κ2x2dτ 2 + dx2 + dx2⊥ (2.39)
From Eq. (2.39) it is evident that κτ = tE behaves as an angular coordinate. Thus as τ variesin the range [0, 2π/κ], the Euclidean time This suggests integrating Eq. (2.38) over the range0 < t < 2π/κ. Then we get [5]:
Asur = −8π
∫HdD−2x⊥
√σQ0x
0x = SWald (2.40)
with σ the metric on the D − 2 dimensional hypersurface. The above result for the surfaceaction can be connected with the Noether current formalism as explained in [5]. This result canalso be related to the change in energy momentum tensor. Integral over δT ab can be representedas the energy flux δE through the horizon. This energy flux will in turn alter the entropy ofthe horizon such that we can interpret (2π/κ)δE as the rate of entropy change.
9
Chapter 3
Describing Gravity inLanczos-Lovelock Theories
3.1 Conjugate Variables in Einstein Gravity
We again return to Einstein gravity to illustrate the fact that we can use a different set ofvariables as canonically conjugate and use them to derive the field equations. This analysis isdone in Ref. [1] and [2]. Under variation the Einstein-Hilbert action has an field equation partand a surface term, which we usually set to zero. The variational principle becomes clearerif we introduce a new dynamical variable fab ≡
√−ggab instead of the metric gab, which is a
tensor density. Then the Einstein-Hilbert action in terms of the new variable fab reads [1, 2],
√−gR =
√−gLquad − ∂c
[fab
∂ (√−gLquad)
∂ (∂cfab)
]=√−gLquad − ∂c
(fabN c
ab
)(3.1)
where we have defined another quantity, representing the conjugate momenta to fab as,
N cab =
∂ (√−gLquad)
∂ (∂cfab)= Qcd
aeΓebd +Qcd
beΓead (3.2)
Then the usefulness of these variables can be grasped from the variation of the Einstein-Hilbertaction. These variables also lead to two terms, the equation of motion and the surface termhaving the following decomposition,
δ(√−gR
)= Rabδf
ab + fabδRab = Rabδfab − ∂c
(fabδN c
ab
)(3.3)
Thus these variables simplify the variation considerably and also the surface term can be elim-inated by setting δNa
bc = 0, which is equivalent to setting variation of momentum to zero atthe end points. Also many expressions will simplify considerably in terms of these variables.Another important aspect is that these variables are directly connected with thermodynamicinterpretation.
10
Next we summarize some important relations and regarding these variables. The most im-portant of these relations are the various connections between Ricci scalar with these variables,
Rab =
(−∂cN c
ab −N cadN
dbc +
1
3N cacN
dbd
)(3.4)
√−gR = −fab∂cN c
ab −1
2N cab∂cf
ab (3.5)
=1
2N cab∂cf
ab − ∂c(fabN c
ab
)(3.6)
= −1
2
[fab∂cN
cab + ∂c
(fabN c
ab
)](3.7)
√−gLquad =
1
2N cab∂cf
ab (3.8)√−gLsur = −∂c
(fabN c
ab
)(3.9)
Having obtained all these relations we now turn to the Hamiltonian formulation and derivationof equation of motion in terms of these variables f ij and Nk
ij. We shall consider a Palatinivariation where we have taken the two conjugate variables, f ij and Nk
ij as independent. Forthat purpose we shall define the Hamiltonian as,
Hg = fab(N cadN
dbc −
1
3N cacN
dbd
)(3.10)
Then the usual Hamilton’s equations of motion obtained from the above Hamiltonian will be:
∂cfab =
∂Hg
∂N cab
= fadN bcd + f bdNa
cd −1
3famNd
dmδbc −
1
3f bmNd
dmδac (3.11)
∂cNcab = −∂Hg
∂fab
= −N cadN
dbc +
1
3N cacN
dbd (3.12)
In order to be consistent the above relations for ∂cfab and ∂cN
cab must be connected up with
those in standard Einstein-Hilbert action, which are ∇cgab = 0 and the Einstein equation itself.The substitution of fab =
√−ggab in Eq. (3.11) leads to:
√−gNa
ba =3
2∂c√−g; =⇒
√−gΓaba = ∂c
√−g (3.13)
Using this result we can evaluate√−g∇cg
ab leading to:
√−g∇cg
ab = ∂cfab − gab∂c
√−g + fmbΓacm + fmaΓbcm (3.14)
using Eq. (3.13) the above equation reduces to:
√−g∇cg
ab = ∂cfab −
(fadN b
cd + f bdNacd −
1
3famNd
dmδbc −
1
3f bmNd
dmδac
)(3.15)
11
which vanishes exactly by virtue of Eq. (3.11) leading to ∇cgab = 0. The other equation (3.12)can be seen to be equivalent to:
Rab = 0 (3.16)
on using Eq. (3.4). Hence both the relations obtained by varying the Lagrangian as presentedin Eqs. (3.11) and (3.12) are equivalent to ∇cgab = 0 and Rab = 0, the source free Einsteinequation.
We have shown earlier in Eq. (2.32) the Noether current is related to the Lie variation ofthe Christoffel Symbols corresponding to the diffeomorphism xa → xa + ξa. In this case weneed to consider how the two quantities fab and N c
ab changes under the diffeomorphism. SinceN cab is a linear combination of the connections, we can easily get the following expression for
the Noether current as [1, 2],√−gJa = 2
√−gRabξb + fpqLξNa
pq (3.17)
Thus we have given some basic identities corresponding to a set of conjugate variables in Ein-stein gravity.
Now we shall try to generalize these ideas to Lanczos-Lovelock theories. In Lanczos-Lovelockgravity as it turns out that the natural generalization of these two variables does not work.However there exists another set of variables for which identical equations of motion are ob-tained. In Lanczos-Lovelock gravity these variables work better rather than those used above.We will first introduce those variables leading to proper equation of motion in the Einstein-Hilbert case before proceeding to Lanczos-Lovelock theory. For this purpose we shall start withthe Einstein-Hilbert Lagrangian written as:
L =√−gR =
√−g1
2
(gbdδca − gbcδda
)Ra
bcd
=√−g(gbdδca − gbcδda
) (∂cΓ
abd − ΓadpΓ
pbc
)≡ U bcd
a
(∂cΓ
abd − ΓadpΓ
pbc
)(3.18)
where we have defined a new variable, U bcda =
√−g(gbdδca − gbcδda
)having all the symmetries of
curvature tensor. Let us see what happens if we treat the variables U bcda and Γabc as independent.
The variation of Γ leads to,
δL |U bcda
= U bcda
(∂cδΓ
abd − ΓpbcδΓ
adp − ΓadpδΓ
pbc
)= U bcd
a ∇cδΓabd (3.19)
In order to get to the final expression we have used the fact that the Lagrangian is a scalardensity and hence can be evaluated in a local inertial frame with U bcd
a and δΓabc as tensors.Then we can rewrite Eq. (3.19) such that:
δL |U bcda
= ∇c
(U bcda δΓabd
)− δΓabd∇cU
bcda (3.20)
The above equation when integrated over the spacetime volume, the first term ∇c
(U bcda δΓabd
)turns out to be total divergence and needs to be evaluated only on the surface and hence doesnot contribute. Thus for arbitrary variations δΓabc we obtain:
∇cUbcda = 0 (3.21)
12
From the definition of U bcda it is evident that Eq. (3.21) implies ∇cgab = 0, standard result
from Palatini variation in Einstein-Hilbert action [5].However we are led to a difficulty in this approach when we vary U bcd
a because the variationof U bcd
a leads to Rabcd = 0, i.e. flat spacetime. It is not possible to get Einstein equation from
arbitrary variations of U bcda since it has four indices, naturally leading to zero curvature tensor,
equivalently flat spacetime. In order to get correct equations of motion we need to restrictvariations. Note that U bcd
a has all the symmetries of the curvature tensor thus it has in fourdimension 20 independent elements. If we vary all of them we are going to obtain the flatspacetime as mentioned earlier. Therefore we will restrict these variations such that only 10of them are independent, which corresponds to the components of some arbitrary symmetricsecond rank tensor Spq . Based on this fact we are led to consider variations of the form:
δU qrsp =
(U qrsm δnp −
1
2U qrsp δnm
)δSmn (3.22)
However the above variation can be slightly generalized by introducing a sixth rank tensorA qrsnpm satisfying the criteria A qrsn
pm Rpqrs = 0. We shall not consider these variations any more
since they have no effect on the equation of motion. With these restricted class of variationswe arrive at:
δL |Γabc
=1
2Ra
bcdδUbcda
=1
2Rp
qrs
(U qrsm δnp −
1
2δnmU
qrsp
)δSmn (3.23)
Then for arbitrary variations of the symmetric tensor Sab , we get the equations of motion:
Rpqrs
(U qrsm δnp −
1
2δnmU
qrsp
)= 0 (3.24)
To prove the equivalence with Einstein equation we note that the following relations:
RpqrsU
qrsm = 2
√−gRp
m; RpqrsU
qrsp = 2
√−gR (3.25)
directly make the equation of motion (3.24) equivalent to, Gab = 0, the source free Einsteinequation. We have not included matter fields to our system about which we shall commentlater. In the next section we will show the validity of the above formalism for Lanczos-Lovelockgravity.
3.2 Conjugate Variables in Lanczos-Lovelock Theories
In Einstein gravity, starting from the equation of motion to all relevant quantities, we can have aunique description using the two conjugate variables, fab =
√−ggab and Na
bc = QcdaeΓ
ebd+Qcd
beΓead.
The most natural choice for the corresponding conjugate variables in Lanczos-Lovelock gravityamounts to setting Qab
cd in Nabc as presented in Eq. (3.2) to a polynomial in curvature tensor
as presented in Eq. (2.25). Thus we will first start with this choice of conjugate variables and
13
shall see whether they satisfy all the requirements.
The most striking feature corresponds to the fact that even in Lanczos-Lovelock theoriesthe decomposition of the Lagarangian into a bulk term and a surface term do exist and haveidentical expression as presented in Eq. (2.4) with the interpretation of Eq. (2.25). Thereare two conditions that the conjugate variables must satisfy, first the surface term should beexpressible as, −∂(qp), secondly the quadratic part of the Lagrangian must be expressibleas, p∂q. Note that q and p are not absolute, since in the Hamiltonian formulation both aregiven equal weightage, we can at ease interchange q and p. In Einstein-Hilbert action theserelations are given by Eq. (3.8) and Eq. (3.9). With these facts in mind we now proceed todetermine whether these two variables, fab which is same as the fab in Einstein-Hilbert actionand Na
bc having the same expression as that of Nabc with proper interpretation of Qab
cd serve as ourconjugate variables. For that purpose we need to evaluate the following combinations, fab∂cN
cab
and Nabc∂af
bc. These are evaluated explicitly in Appendix A [see Eq. (A.5) and Eq. (A.6)], wehere state the final results,
√−gLsur = −∂c
(fpqN c
pq
)(3.26)
√−gLquad = N c
ab∂cfab −
(2√−gQ bqc
p ΓpqbΓmcm − 2
√−ggbmQcq
apΓpqbΓ
acm
)(3.27)
√−gQ qrs
p Rpqrs = −fab∂cN c
ab −(2√−gQ bqc
p ΓpqbΓmcm − 2
√−ggbmQcq
apΓpqbΓ
acm
)(3.28)
Thus even though it leads to the proper surface term it does not do so with other terms as theydid for the Einstein-Hilbert case presented in Eq. (3.8). Hence these variables are not suitable.The same conclusion can also be taken from the Noether current expression as well. In Lanczos-Lovelock theories the Noether current has the expression given by Eq. (B.5), derived withoutany notion of diffeomorphism, using only differential geometry. As in the Einstein-Hilbertscenario where the last term becomes, gabLξN c
ab, here also we should demand the last termto be of the form fpqLξNa
pq. However this does not yield the Noether current. An extra termdepending on Lie variation of the entropy tensor P abcd comes into picture. We have used severalidentities regarding Lie variation of P abcd but none can help to reduce that extra term to zero[though these identities do not help they are quiet interesting and have not been derived earlierthus we present these relations in Appendix. B]. Hence these variables though the most naturalchoice does not fulfill the above mentioned criteria for being conjugate variables.
Now we try to get another set of variables that satisfies all the above conditions. From thestructure of the Noether current it appears as though the proper variables could be:
Γabc =1
2gad (−∂dgbc + ∂bgcd + ∂cgbd) (3.29)
U bcda = 2
√−gQ bcd
a (3.30)
where it is evident that Γ is the standard connection and Q bcda can be identified from Eq. (2.25).
Note that all the original Einstein-Hilbert relations can be written in terms of these two variablesrather than fab and Na
bc as presented in Eqs. (A.7)) and (A.8) matching exactly Eq. (3.8). This
14
assertion can be put to a more strong basis using the following relation,
fpqDNapq = 2
√−ggpqD
(QaepdΓ
deq
)= 2
√−ggpqQae
pdDΓdeq
= 2√−gQ qea
d DΓdqe = U qrap DΓpqr (3.31)
where D is any covariant derivative operator. This is possible only in Einstein-Hilbert actionwhere Qcd
ab involves only the Kronekar delta functions and hence can be taken inside or outsideof any covariant derivative operator. Thus in general Γabc and U pqr
a can also act as the conju-gate variables, in Einstein gravity they become equivalent to fab and N c
ab respectively. Howeverin higher order Lanczos-Lovelock theories Qab
cd becomes dependent on curvature tensor, thusthe above decomposition becomes impossible and we need to work with the two variables aspresented in Eq. (3.29) and Eq. (3.30). In the metric formulation we treat the Lagrangianwith gab and Rabcd as independent variables and variation of gab leads to the equation of mo-tion. While in Palatini we treat both the metric and the connections as independent and theirvariation leads to the relation between them and also the equation of motion. For a generalarbitrary Lagrangian metric and Palatini variation does not coincide [40]. However if the con-dition ∇c (∂L/∂Rabcd) = 0 is satisfied then both the metric and Palatini formulations coincide.This is identical to the condition presented in Eq. (2.21). The appropriate Lagrangian for ourpurpose turns out to be:
L = U bcda (∂cΓ
abd − ΓamdΓ
mbc) (3.32)
which using Eq. (3.30) can be identified to be the Lanczos-Lovelock Lagrangian. Then variationof the above Lagrangian with respect to Γabc leads to,
∇cUbcda = 0 (3.33)
as in the Einstein-Hilbert scenario presented by Eq. (3.19). This condition is the reminiscentof the criteria that in Lanczos-Lovelock gravity ∇cP
abcd = 0.
Next we need to vary the Lagrangian with respect to the other conjugate variable U bcda ,
which leads to trouble just as in GR Since the Lagrangian can equivalently be written asL = U bcd
a Rabcd/2, such that for arbitrary variation of U bcd
a we get Rabcd = 0, i.e. flat spacetime
solution. In order to get the field equation we need to consider only a class of variations aswe did in the Einstein-Hilbert scenario to derive the equations of motion (3.24). Here also weassume that not all the independent components of U bcd
a are contributing to the variation onlya symmetric second rank part has arbitrary variation. This amounts to taking:
δU qrsp =
(U qrsm δnp −
1
2U qrsp δnm
)δSmn (3.34)
where also we can introduce an additional sixth rank tensor as we did after Eq. (3.22). Howeveras the equation of motion is concerned it has no effect and thus will not be considered any more.With these restricted class of variations the Lagrangian variation leads to:
δL |Γabc
=
(mU pqr
a δbs −1
2δbaU
pqrs
)Rs
pqrδSab (3.35)
15
where δSab is variation of an arbitrary symmetric second rank tensor and the factor m comesfrom the fact that we are considering mth order Lanczos-Lovelock Lagrangian. When thevariation is considered arbitrary the equation of motion turns out to be,(
mU pqra δbs −
1
2δbaU
pqrs
)Rs
pqr = 0 (3.36)
To show that the above equation of motion is indeed identical to the equation of motion inLanczos-Lovelock gravity we just use Eq. (3.30) to substitute for U bcd
a leading to:
0 = mQ pqra Rb
pqr −1
2δbaQ
pqrs
= Rab −
1
2δabL (3.37)
which is the Lanczos-Lovelock equation of motion. Thus we observe that these two variablessatisfy all the criteria that conjugate variables should.
The above result is derived for mth order Lanczos-Lovelock Lagrangian and can be easilygeneralized to general Lanczos-Lovelock Lagrangian L =
∑m cmL
(m). Then the above variationof Lanczos-Lovelock Lagrangian leads to the following expression:
δ(√−gL
)|Γa
bc=
∑m
cmδ(√−gL(m)
)|Γa
bc
=
{(∑m
cmmUpqra
)δbs −
1
2δba
(∑m
cmUpqrs
)}Rs
pqrδSab (3.38)
For arbitrary variation of the symmetric tensor Sab the equation of motion can be obtained as:{(∑m
cmmUpqra
)δbs −
1
2δba
(∑m
cmUpqrs
)}Rs
pqr = 0 (3.39)
Note that with the following relations,∑m
cmmUpqra =
∂√−gL
∂Rapqr
=√−gP pqr
a (3.40)∑m
cmUpqrs Rs
pqr =√−gL (3.41)
the above Eq. (3.39) becomes equivalent to,
Rba −
1
2δbaL = 0 (3.42)
which is the Lanczos-Lovelock field equation.Finally the Noether current in terms these variables becomes:
√−gJa = m
(2U pqr
b Rapqrv
b + U cdab LξΓbcd
)= 2
√−gRa
bvb +mU cda
b LξΓbcd (3.43)
16
Note that this is exactly equivalent to Eq. (3.17), the Noether current for the Einstein-Hilbertgravity. Hence from the Lagrangian by variation of respective independent elements we arriveat the condition that needs to be imposed in order to get second order equation of motion andhave also obtained the field equation as well. In spite of all these success there is one point stillunclear, regarding inclusion of matter in this scheme. As the energy momentum tensor comessolely from the variation of the matter Lagrangian with respect to the metric and we have notincluded the metric in our formulation inclusion of matter is not possible. This issue requiresfurther investigation.
17
Chapter 4
Thermodynamics with ConjugateVariables
This chapter we will start by reviewing an earlier work [1] and then extending it to Lanczos-Lovelock theories. In the following we will briefly review the construction of Gaussian NullCoordinate(GNC) system which can be introduced in the vicinity of an arbitrary null surfaceand then shall review the thermodynamics relations obtained from holographically conjugatevariables for Einstein-Hilbert action in this coordinate system. For the geometrical construction,we shall confine ourselves in a four dimensional spacetime, with possibility of generalizing tohigher dimensions in a straightforward manner.
4.1 Introduction to Gaussian Null Coordinates
Let us consider the four dimensional spacetime V 4 = M3 × R, where M3 is a compact threedimensional manifold. We will consider spacetimes to be time orientable with null embeddedhypersurfaces, which are diffeomorphic to M3 with closed null generators. We take N to besuch a null hypersurface with closed null generator [41, 42]. On this null surface N we canintroduce spacelike two surface with coordinates (x1, x2) defined on them. The null geodesicswhich generate the null hypersurface N goes out of this elementary spacelike two surface. Thuswe can use these null generators to define coordinates on the null hypersurface. The intersectionpoint of these null geodesics with the spacelike two surface can be determined by the coordinates(x1, x2), which then evolves along the null geodesic affinely parameterized by u and labels eachpoint on the null hypersurface as (u, x1, x2). The above system of coordinates readily identifythree basis vectors: (a) the tangent to the null geodesics, ` = ∂/∂u, and (b) basis vectors onthe two surface eA = ∂/∂A.
Having fixed the coordinates on the null surface N we now move out of the surface usinganother set of null generators with tangent ka satisfying the following constraints: (a) kak
a = 0,(b) eaAka = 0 and finally `ak
a = −1. Thus these null geodesics are taken to intersect thenull surface at coordinates (u, x1, x2) and then move out with affine parameter r, such thatany point in the neighborhood of the null surface can be characterized by four coordinates(u, r, x1, x2). In this coordinate system the null surface is given by the condition r = 0. Thisdefines a coordinate system {u, r, xA} over the global manifold V 4. This system of coordinates
18
are formed in a manner analogous with Gaussian Normal Coordinate is referred to as GaussianNull Coordinates which will henceforth be referred as GNC for ease of notation.
Having set the full coordinate map near the null surface we now proceed to determine themetric in that region. Note that `a`
a = 0 leads to guu = 0 on the null surface since ` = ∂/∂u.We also note that the basis vectors eaA have to lie on the null surface implying `ae
aA = 0 on
N , which leads to guA = 0. Also the metric on the two surface is given by gAB = eaAebBgab,
which we denote by µAB. We also need the criteria that µAB is positive definite with finitedeterminant ensuring invertibility and non-degeneracy of the two metric. Thus the followingmetric components gets fixed to be:
guu|r=0 = guA|r=0 = 0;
gAB = µAB (4.1)
Let us now proceed to determine the other components of the metric. This time we will use thevector k = −∂/∂r such that from kaka = 0 we get grr = 0 throughout the spacetime manifold.Also from the criteria that the null geodesics are affinely parametrized by r we readily obtain∂rgrα = 0, where α = (u, x1, x2). Again, from the conditions `aka = −1, we readily get gru = 1and from kae
aA = 0 we get grA = 0. From the criteria derived earlier showing ∂rgrα = 0 we
can conclude that the above two metric coefficients are valid everywhere. Thus within theglobal region V 4 we can have smooth functions α and βA such that, α |r=0= (∂guu/∂r) |r=0 andβA |r=0= (∂guA/∂r) |r=0. With these two identifications we have the following expression forthe line element as,
ds2 = gabdxadxb = −2rαdu2 + 2dudr − 2rβAdudx
A + µABdxAdxB (4.2)
where, µAB is the two dimensional metric representing the metric on the null surface. Note thatthe construction presented above is completely general, this can be applied in the neighborhoodof any null hypersurface and, in particular to the event horizon of a black hole.
4.2 Thermodynamics Related to Einstein-Hilbert Action
In this section we will show that the variables fab =√−ggab and N c
ab = QadbeΓecd+Qad
ceΓebd lead tointensify these thermodynamic structures. The variations pδq and qδp obtained from conjugatevariables have direct thermodynamic interpretation associated with them. First we will provethese relations for a general static spacetime before discussing arbitrary null surface constructedabove.
4.2.1 Static Spacetime
We will consider an arbitrary static spacetime with horizons. For this spacetime we have anarbitrary two surface with metric σAB and the line element can be written in the form [44]:
ds2 = −N2dt2 + dn2 + σABdyAdyB (4.3)
In the above line element, n represents spatial direction normal to the (D − 2) dimensionalhypersurface with σAB being transverse metric on the two surface. Let ξ = ∂/∂t be a timelike
19
Killing vector, with Killing horizon located at, N2 → 0. The coordinate system is chosen insuch a way that n = 0 on the Killing horizon. Then, in the near horizon regime, the followingexpansions of N and σAB are valid [44]:
N = κ(xA)n+O(n3)
σAB = [σH(y)]AB +1
2[σ2(y)]AB n
2 +O(n3) (4.4)
In the above expression κ is the local gravity defined as: κ = ∂nN and in the r → 0 limitthe this κ→ κH , satisfying all the standard properties of surface gravity. Also κ/N representsthe normal component of the four acceleration of an observer at fixed (n, xA). Throughout thecalculation we can evaluate quantities on a n = constant surface and then take the n→ 0 limit.The nonzero components of the metric variations are:
δgtt = 2NδN ; δgAB = δσAB (4.5)
We will also require the nonzero components of Nabc which are:
Nntt = −N∂nN ; Nn
nn =∂nN
N+
1
2ΣAB∂nσAB; Nn
AB =1
2∂nσAB (4.6)
Then in the near horizon limit the respective pδq and qδp terms turn out to be:
Nnabδf
ab|H = 2κδ(√σ); fabδNn
ab = 2√σδκ (4.7)
Then integration over the transverse variables and introduction of proper numerical factoryields the variation of the surface term in Einstein-Hilbert action to be :
1
16π
∫d2x⊥N
nabδf
ab =
∫d2x
κ
2πδ
(√σ
4
)=
∫d2xTδs (4.8)
1
16π
∫d2x⊥f
abδNnab =
∫d2x
√σ
4δ( κ
2π
)=
∫d2xsδT (4.9)
where s =√σ/4 is the entropy density of the spacetime. If the zeroth law of black hole ther-
modynamics can be used i.e. the surface gravity κ is independent of the transverse coordinatesthen the above results become TδS and SδT respectively.
4.2.2 Generalization to Arbitrary null surface
In the context of horizon thermodynamics local Rindler frame and Rindler horizons are exten-sively used [43], where local Rindler horizon refers to a patch of null surface in some locallyinertial frame. This immediately leads to temperature and entropy as observed by local Rindlerobservers. However this interpretation can be made more general by working in the near horizonregime of an arbitrary null surface whose construction we have presented in Sec. 4.1. We shalluse the line element in the near horizon limit to be given by Eq. (4.13) with r = 0 representingthe null surface. It will be unrealistic to expect arbitrary variation of the metric representingnear horizon limit of the null surface to have simple thermodynamic interpretation. In orderto obtain suitable thermodynamic interpretation we need to impose some restrictions. These
20
restriction can be of two types: (i) those involving restriction on the background metric itselfvariations being arbitrary or (ii) keeping background metric arbitrary with restricted variations.Among these, restriction on the variation is preferred since we want the background metric torepresent any arbitrary null surface and hence we shall keep it most general. These conditionson metric variation we need turn out to be ∂uδ(µAB) = 0 and also ∂uδ(
õ) = 0. These con-
ditions imply that the variations should not depend on time coordinate u and thus henceforthwill be referred to as stationarity conditions for obvious reasons.
We now proceed to calculate the variation of the surface term and split it into two partssuch that:
δAsur = − 1
16π
∫d3xnc
(fabδN c
ab +N cabδf
ab)≡ −δA1 − δA2 (4.10)
where δA1 contains fabδN cab and δA2 contains fabδN c
ab. Among them fabδN cab has the following
expression near the r = 0 null surface:
fabδN rab = 2
√µδ
(α +
1
4µAC∂uµAC
)+
1
2
√µµAB∂uδµAB (4.11)
It is evident from the above expression that with the stationarity conditions mentioned abovethe following result can be obtained:
δA1 ≡1
16π
∫d3xfabδN r
ab =1
8π
∫d3x√µδα (4.12)
Rather than evaluating the second term N cabf
ab individually we can get it by consideringvariation of the surface term. The surface term in Einstein-Hilbert action evaluated for theabove metric turns out to be [1],
Asur =1
16π
∫d3x√hnrV
r
=1
16π
∫d3x[− 1√µ∂uµ−
√µ(2α + 2r∂rα + 2rβ2 + 2r2βA∂rβ
A)
− 2rα + r2β2
õ
∂rµ−√µr∂Aβ
A − rβA√µ∂Aµ
](4.13)
The integration variables being u and xA we can take the r → 0 limit easily leading to,√hnrV
r = −∂uµ/√µ− 2
√µα. Thus we get,
δAsur =1
16π
∫d3xδ
(−∂uµ√
µ− 2√µα
)(4.14)
Hence again under the stationarity condition the only nonzero term in the above expansionbecomes −2
√µδα− 2αδ
õ. Hence the other term in Eq. (4.10) becomes,
δA2 =1
16π
∫d3xN r
abδfab =
1
8π
∫d3xαδ
õ (4.15)
21
Then we can write Eqs. (4.12) and (4.15) as sδT and Tδs respectively. For that purpose wecan write the following relations as:
∂δA1
∂u=
1
16π
∫d2xfabδN r
ab
=
∫d2x
õ
4δ( α
2π
)=
∫d2x sδT (4.16)
∂δA2
∂u=
1
16π
∫d2xN r
abδfab
=
∫d2x
α
2πδ
(õ
4
)=
∫d2x Tδs (4.17)
where again s =√µ/4 is the entropy density of the spacetime. If the quantities α and
õ are
expanded in a Taylor series in u with leading order term being independent of u. Then averageof the leading order term α0 over the two surface is defined as, α0 =
∫d2xA
√µ(0)α0/A⊥, with
A⊥, the area of the two surface. Thus with these expansions substituting in Eqs. (4.16) and(4.17) we get:
1
16π
∫d3xfabδN r
ab = uSδT (4.18)
1
16π
∫d3xN r
pqδfpq = uTδS (4.19)
where T = α0/2π and S = A⊥/4.
This brings out the connection between these conjugate variables and respective thermody-namic quantities pertaining to the null surface acting as local Rindler horizons. The curiousthing is that the surface term integrated over variables over the horizon leads to the heat con-tent along with the fact that TδS coming from variation of generalized coordinates, while SδTcoming from variation of generalized momenta.
4.3 Thermodynamics Related to Lanczos-Lovelock Ac-
tion
In this section, we shall describe how the variations of conjugate variable in the form pδq andqδp have thermodynamic interpretation when they are integrated over the horizon. We shallfirst illustrate the results for a general static spacetime and then we shall generalize these to thearbitrary null surface constructed in Sec. 4.2. This provides sufficient insight into the behaviorof these conjugate variable in local Rindler horizon. The surface term in Lanczos-LovelockLagrangian has been discussed in Sec. 3, which can be written as,
− ∂c(2√−gQ bdc
a Γabd)≡ −∂c
(√−gV c
)(4.20)
Then under infinitesimal variation the surface term variation can be subdivided into two parts,
δ(√−gV c
)= U bdc
a δΓabd + ΓabdδUbdca (4.21)
22
where one term involves variation of connections, while the other one is quiet complex andinvolves variation of the metric and the entropy tensor both. Hence the variation of the surfaceterm can be written as:
δAsur = − 1
16π
∫dD−1x ncδ
(√−gV c
)= − 1
16π
∫dD−1x ncU
bdca δΓabd −
1
16π
∫dD−1x ncΓ
abdδU
bdca
= −δA1 − δA2 (4.22)
Also the variation of the connection due to infinitesimal change of metric gab → gab + hab is,
δΓpqr =1
2hpa (−∂agqr + ∂qgar + ∂rgaq) +
1
2gpa (−∂ahqr + ∂qhar + ∂rhaq) (4.23)
Having stated the basic facts about surface term variation we shall now calculate the variationexplicitly for different metrices. The curious fact is that the first term in Eq. (4.22) leads toSδT while the second term leads to TδS.
Thus with all the tools in the bag we now proceed for actual calculations, which involvestwo metrices a general static metric and the GNC metric respectively.
4.3.1 A general Static Spacetime
We will consider an arbitrary static spacetime with horizons as presented in Sec. 4.2.1. For theline element presented in Eq. (4.3) using Eq. (A.10) and Eq. (4.4)) we readily obtain that inthe n→ 0 limit and only three connections are non zero. These are respectively: Γntt = N∂nN ,Γtnt = ∂nN/N and ΓABC . Another quantity would be frequently used in the calculation, the(D − 2) dimensional surface element and Wald entropy for Lanczos-Lovelock theories. Thenull surface we are interested in can be defined using the condition, l2 = 0, with la being(approximate) Killing vector. Then if we can introduce another auxiliary null vector ka suchthat, lak
a = −1, then the (D − 2) dimensional surface element turns out to be,
dΣab = dD−2xµab = −dD−2x (lakb − lbka) (4.24)
The Wald entropy expressed in terms of the entropy tensor turns out to be [32, 38]:
S = −1
8
∫ √σdD−2xP abcdµabµcd (4.25)
Also the determinant of the metric on the two surface is σ. Now for the la and ka as definingelements for the (D − 2) dimensional surface we have the entropy for this static spacetime tobe,
S =1
2
∫dD−2x
√σP nt
nt ≡∫dD−2xs (4.26)
where s = (1/2)√σP nt
nt . Next we evaluate the variation U bdna δΓabd. This variation turns out to
be:
U bdna δΓabd |H = 2
√−gQ bdn
a δΓabd= 2N2
√σQ ttn
n δ (∂nN) + 2√σQ ntn
t δ (∂nN)
= 4√σQnt
ntδκ (4.27)
23
Along the similar lines other part of the variation, ΓabdδUbdna can be written in the following
form:ΓabdδU
bdna |H= 4κδ
(√σQnt
nt
)(4.28)
Now from Eq. (4.26) we get the expression for entropy and the temperature can be estimated asκ/2π. With these two identifications we readily arrive at the following results for the variations,
1
16π
∫dD−2x ΓabdδU
bdna =
1
2m
∫dD−2x
κ
2πδ(√
σP ntnt
)=
1
m
∫dD−2xTδs (4.29)
1
16π
∫dD−2x U bdn
a δΓabd =1
2m
∫dD−2x
√σP nt
nt δ( κ
2π
)=
1
m
∫dD−2x sδT (4.30)
In the above expressions s represents the entropy density of the horizon, which reduces to√σ/4
in the Einstein-Hilbert limit. The above results shows that respective two terms in Eq. (4.22)leads to Tδs and sδT , with κ being κ(xA). For κ being independent of transverse coordinateszeroth law of black hole thermodynamics [45, 46] can be applied in order to get TδS and SδTrespectively.
4.3.2 Generalization to Arbitrary Null Surface
Having discussed the thermodynamic interpretation for a general static spacetime, we will nowtry to extend this to an arbitrary null surface. The metric outside the null surface has beenconstructed in Sec. 4.1 and we shall extensively use that metric to evaluate various quantitiesof interest. As in the static situation, here also we shall calculate all the quantities on ar = constant surface and then shall take the limit r → 0 to retrieve the null surface. For thatpurpose we start with normal to the r = constant surface and then take the null limit. Withproper la and ka [47] the entropy turns out to be,
S =1
2
∫dD−2x
√σP ur
ur (4.31)
Let us next calculate the surface term that comes from the r = constant surface. This termhas the following expression,
Asur = − 1
8π
∫dD−2xdu
õQ bdr
a Γabd (4.32)
The only connections that will remain nonzero even in the r → 0 limit are those given byEq. (A.11). With these connections the surface term turns out to be,
Asur = − 1
8π
∫dD−2xdu
√µ[2αQur
ur + 2βAQArur + 2µBDΓABCQ
CrAD −
3
2µAC∂rµABQ
BruC
+1
2
[−∂uµAB
(Q ABrr +Q BAr
r
)+ ∂uµBC
(QCBur +QCuBr
)](4.33)
24
In the Einstein-Hilbert limit only the Qurur term contribute leading to Eq. (4.14). Now using
Eq. (2.25) all these Qabcd terms can be calculated. They all leads to,
QArur = δAruBPQ...urCDRS...R
CDuB R
RSPQ . . . (4.34)
QCrAD = δCrMNuL...
ADurPQ... RurMNR
PQuL + δCruMPQ...
ADurRS... RuruMR
RSPQ . . .
+ δCrMPuKUV ...ADuQLJrW... R
uQMPR
LJuKR
rWUV . . .+ δCrMPuL...
ADuQrK...RuQMPR
rKuL . . .
+ δCruPMN...ADuQrL... R
uQuPR
rLMN (4.35)
QBruC = δBruA...uCrD...R
rDuC + δBruPJK...uCQRrM...R
QRuP R
rMJK (4.36)
In arriving at the above results we have used the fact that the determinant tensor is antisym-metric in any two indices. Thus all the remaining terms in the above expressions are only thecomponents of the curvature tensor with indices depending on coordinates on the null surface.They are all fully characterized by the (D − 2) metric µAB. Now in the null limit we have:
RCQuB = µQPRC
PuB; RuQMP = RMP
uQ ; RrMJK = µMNRr
NJK ; RrDuC = −µADRr
ACu; RuruM = RuM
ur
(4.37)Having derived all the components of the surface term we will now consider the thermodynamiclimit.
As we said earlier thermodynamic interpretations can be given provided some additionalconditions are imposed. As explained in Sec. 4.2.2 we can use two conditions: one on thevariation with metric arbitrary and another on the metric keeping variations arbitrary. Thenatural conditions that one can impose are the stationarity conditions yielding constraint onthe variation but not on the background metric. From the result we had in the Einstein-Hilbertaction we would expect these conditions δ
(µCA∂iµAB
)= 0 and ∂iδµAB = 0. However it turns
out that along with these conditions another constraint must be imposed. This is also on thevariation and which amounts to set δβA = 0. With these three conditions we get,
δAsur = − 1
4π
∫dD−2xdu [δ (
õQur
ur)α +√µQur
urδα] (4.38)
As well as we have the following expression from Eq. (4.22),
δA1 =1
16π
∫dD−1x ncU
bdca δΓabd =
1
4π
∫dD−1x
õQur
urδα (4.39)
This along with Eqs. (4.38) and (4.22) naturally leads to,
δA2 =1
16π
∫dD−1x ncΓ
abdδU
bdca =
1
4π
∫dD−1x αδ (
õQur
ur) (4.40)
Then Eqs. (4.39) and (4.40) can also be interpreted in terms of entropy density s = (m/2)õQur
ur
25
leading to:
∂δA1
∂u=
∫dD−2x
õQur
ur
2δ( α
2π
)=
1
m
∫dD−2x sδT (4.41)
∂δA2
∂u=
∫dD−2x
α
2πδ
(õQur
ur
2
)=
1
m
∫dD−2x Tδs (4.42)
Another interpretation can be given provided we expand both µAB and α in a Taylor seriesaround r = 0 null surface with leading order term being independent of u. This amounts towriting T = α0/2π, where, α0 represents average of α0 over the surface with Qur
ur as a weightingfunction. Then from the above expressions for U bdr
a δΓabd and ΓabdδUbdra we arrive at:
m
16π
∫dD−2xdu U bdr
a δΓabd = uSδT (4.43)
m
16π
∫dD−2xdu ΓabdδU
bdra = uTδS (4.44)
The above results hold for mth order Lanczos-Lovelock Lagrangian, which can be generalizedstraight forwardly to general Lanczos-Lovelock Lagrangian. There the two respective Eqs. (4.45)and (4.46) leads to the following forms:
1
16π
∫dD−2xdu
(∑m
cmmUbdra
)δΓabd = uSδT (4.45)
1
16π
∫dD−2xdu Γabdδ
(∑m
cmmUbdra
)= uTδS (4.46)
Note that the entropy in both the cases is taken as S =∑
m cmS(m), the standard expression.
We should point out that the same result can also be obtained with a constraint on thebackground metric. This constraint is identical to the assumption used in Einstein-Hilbertaction, which is ∂uµAB = 0.
26
Chapter 5
From Bulk Dynamics to SurfaceThermodynamics
As we have stressed earlier there exists a fundamental interrelationship between gravitationaldynamics on the bulk with horizon thermodynamics. As it turns out the conserved chargesconfined to the bulk region being associated with some time evolution vector field has directassociation with the gravitational heat density of the boundary surfaces. This leads to furthurinteresting results [2]: e.g. naturally defined surface and bulk degrees of freedom becomes equalto one another in the static spacetime, also the time evolution of the dynamical quantities canbe related to the difference between the surface and bulk degrees of freedom. In this chapterwe review these results for Einstein-Hilbert action and then generalize these results to Lanczos-Lovelock gravity, where also the same results are obtained.
5.1 Noether Charge and Holographic Equipartition In
Einstein-Hilbert Action
We start with a spacetime foliated by a series of spacelike hypersurfaces each being determinedby the constant value of t, which is taken as a coordinate in this foliated spacetime. The unitnormal to the t = constant surface is, ua = −N∇at = −Nδ0
a. We can also define another usefulvector field: ξa = Nua = −N2δ0
a. Note that for static spacetime the vector field ξa acts as atime-like Killing vector. The next natural thing to define is the acceleration ai = uj∇jui. Notethat:
Diai = Dαa
α = ∇iai − a2 (5.1)
on the t = constant hypersurface, where Di is the surface covariant derivative. Also theacceleration has the following expression:
ai =1
Nhji∇jN =
1
N
(∇iN + uiu
j∇jN)
(5.2)
where we have used the relation hij = δij +uiuj, the projector on the t = constant hypersurface.We will now calculate the Noether current for the previously defined vector ξa. For the
evaluation we shall use a relation between Noether current of two vector fields qa and va such
27
that va = f(x)qa, for arbitrary function f(x). In Ref. [2] it is shown that:
qaJa(vb)− f(x)qaJ
a(qb)
= ∇b
({qaqb − gabq2
}∇af
)(5.3)
The usefulness of this relation can be realised by noting that for qa = ∇aφ for some scalar φthe Noether current vanishes. Thus applying the above result for ua and then for ξa one canarrive at the following simple relation for Noether current of ξa as [2]:
uaJa(ξb)
= 2Dα (Naα) (5.4)
Then we can integrate Eq. (5.4) over the three surface leading to the total Noether chargecontained in the surface. However the above results relies on the fact that 16πG = 1, henceinserting proper numerical factors this leads to (still with G = 1):∫
Vd3x√huaJ
a(ξb)
=
∫V
d3x√h
16π2Dα (Naα) =
∫∂V
√σd2x
8πNrαa
α (5.5)
which holds for any arbitrary region V of the spacetime, with the bounding region beingN (t,x) = constant surface within t = constant hypersurface. This directly helps us to iden-tify the vector rα to be normal to this N (t,x) = constant hypersurface such that, rα =
ε∇αN (∇αN∇αN)−1/2, where the ε factor is +1 chosen such that rα is always the outwardpointing normal. Then εri and acceleration ai are directed along the same direction even forthe most general nonstatic situations. Thus using magnitude for acceleration we can write thenormal rα as: rα = εaα/a, with a representing magnitude of the acceleration. Then we obtain,
Nrαaα = Nε
aαaaα = εNa (5.6)
Choosing the boundary surfaces to be that of N = constant the Tolman shifted Davies-Unruhtemperature takes the following expression Tloc = Na/2π for observers with four velocity ua =−Nδ0
a. Locally free falling observers will observe these observers moving normal to the t =constant hypersurface with an acceleration a and as a consequence the local vacuum will appearas a thermal state of temperature Tloc to these observers. Using all these results Eq. (5.5) canbe written as:
2
∫Vd3x√huaJ
a(ξb)
= ε
∫∂V
√σd2x
2
(Na
2π
)= ε
∫∂V
√σd2x
(1
2Tloc
)(5.7)
The above result can be interpreted as: Twice the Noether charge contained in theN = constantsurface is equal to the equipartition energy of the surface. With the interpretation of
√σ/4 as
entropy density the above result leads to:∫Vd3x√huaJ
a(ξb)
= ε
∫∂V
√σd2x
4Tloc = ε
∫∂Vd2xTs (5.8)
which is the heat density of the bounding surface.
Next we will consider dynamics of gravity in terms of thermodynamic variables using theNoether current formalism. For that we again start with Eq. (5.4) and use Eq. (3.17) leadingto:
uagijLξNa
ij = Dα (2Naα)− 2NRabuaub (5.9)
28
Then we integrate as in the earlier situation over the three dimensional region with boundarysurface being N = constant leading to:∫
Rd3x√huag
ijLξNaij =
∫∂Rd2x√σrα (2Naα)−
∫Rd3x√h2NRabu
aub (5.10)
The dynamics is introduced through Einstein’s equation Rab = 8π (Tab − (1/2)gabT ) = Tab andthen dividing the whole expression by 8π leading to:∫
R
d3x√h
8πuag
ijLξNaij =
∫∂Rd2x√σrα
(Naα
4π
)−∫Rd3x√h2NTabu
aub (5.11)
As explained earlier the normal εri being in the direction of ai leads to the Tolman redshiftedDavies-Unruh temperature as Tloc = Na/2π. Also the energy momentum tensor term can bewritten by noticing that 2NTabu
aub = ρKomar is the Komar energy density. With input fromall these results we arrive at the following form:
1
8π
∫Rd3x√huag
ijLξNaij = ε
∫∂Rd2x√σ
(1
2Tloc
)−∫Rd3x√hρKomar (5.12)
The surface degrees of freedom can be defined as:
Nsur ≡ A =
∫∂R
√σd2x (5.13)
which is always positive. Next we can define an average temperature such that,
Tavg ≡1
A
∫∂R
√σd2xTloc (5.14)
Then the bulk degrees of freedom has the following expression:
Nbulk =ε
(1/2)Tavg
∫d3x√hρKomar (5.15)
Here also the ε is +1 if the total Komar energy within the volume is positive and is −1 if thetotal Komar energy in the volume is negative so as to keep Nbulk always positive. Then Eq.(5.12) can be written in the following manner:
1
8π
∫Rd3x√huag
ijLξNaij =
ε
2Tavg (Nsur −Nbulk) (5.16)
Thus for comoving observers in static spacetime we have the holographic equipartition Nsur =Nbulk. When the difference (Nsur −Nbulk) is nonzero we have departure from holographicequipartition driving the dynamical evolution of the spacetime.
In order to show the meaning of these results and these expressions from a more clearperspective we apply these results to the Friedmann universe. The metric in Friedmann universein the standard form is expressed in the syncronous coordinates. In synchronous frame heobserver at xα = constant is comoving and hence its four velocity can be written as, ua =
29
(−1, 0, 0, 0). This implies that the observer is not accelerating, and thus the local Davies-Unruhtemperature vanishes.
Given this situation Eq. (5.16) reduces to the following form:
1
8π
∫Rd3x√huag
ijLξNaij = − ε
2TavgNbulk −
∫Rd3x√hρKomar (5.17)
where we have used Eq. (5.15). Before specializing to the Friedmann universe we will start bycalculating Eq. (5.17) in an arbitrary synchronous frame in which the general line element is:
ds2 = −dτ 2 + gαβdxαdxβ (5.18)
For the general case as presented in Eq. (5.18) we have the following expression for the Eq. (5.16)which leads to [2]:
√huag
ijLξNaij = 2
√h(KabK
ab − ua∇aK)
= gαβ∂2τgαβ +
1
2∂τg
αβ∂τgαβ
= −16πTabuaub (5.19)
Let us now specialize to Friedmann universe where we have gαβ = a2(t)ηαβ leading to thefollowing expressions:
∂τgαβ = 2aaδαβ (5.20)
∂2τgαβ =
(2a2 + 2aa
)δαβ (5.21)
∂τgαβ = −2
a
a3δαβ (5.22)
Also for the energy momentum tensor we readily obtain: Tabuaub = (1/2) (ρ+ 3p). On substi-
tution of Eqs. (5.20), (5.21) and (5.22) in Eq. (5.19)) we arrive at the following expression forthe time evolution of the scale factor:
a
a= −4π
3(ρ+ 3p) (5.23)
The above equation supplemented by the conservation of energy momentum tensor leads to theFriedmann equations for the scale factor. Thus our results are consistent.
To get a feel for the terms in the static frame we specialize to de-Sitter spacetime whichcan be presented in a static spherically symmetric form with the line element:
ds2 = −(
1− r2
l2
)dt2 +
dr2(1− r2
l2
) + r2(dθ2 + sin2 θdφ2
)(5.24)
In this spacetime observers with xα = constant has the following four velocity and four accel-eration:
ua =
√(1− r2
l2
)(−1, 0, 0, 0) (5.25)
ai =(0,−(r/l2), 0, 0
)(5.26)
30
Thus the acceleration and the normal ri are directed opposite to each other as ri is the outwarddirected normal. Hence in this situation we have ε = −1. The magnitude of the accelerationis:
a =r
l21√(
1− r2
l2
) (5.27)
from Eq. (5.26). Thus the local Davies-Unruh temperature turns out to be:
Tloc =Na
2π=
r
l22π= Tavg (5.28)
Since the spacetime is static ξi becomes a timelike Killing vector. Then the Lie derivative ofthe connection present in Eq. (5.16) vanishes. From Eq. (5.13) the surface degrees of freedomturns out to be:
Nsur ≡ A =
∫∂R
√σd2x = 4πr2 (5.29)
Again the bulk degree of freedom can be obtained from Eq. (5.15) as (notice the ε factor in thedefinition of the bulk degrees of freedom):
Nbulk = 4π8π3ρr3
rl−2(5.30)
Then in de-Sitter spacetime we have 8πρ = (3/l2) we readily observe that:
Nbulk = (8πρ)(l2/3)4πr2 = 4πr2 = Nsur (5.31)
Hence for de-Sitter spacetime in static coordinates we have shown that holographic equipartitionholds i.e. surface and bulk degrees of freedom becomes identical.
5.2 Generalization To Lanczos-Lovelock Gravity
In the previous section we have shown how the departure from holographic equipartition leadsto the dynamics of the spacetime. We have also shown that in static spacetime the surfacedegrees of freedom equals the bulk degrees of freedom. Having completed the description forEinstein-Hilbert action it is now time to move to Lanczos-Lovelock gravity. In Lanczos-Lovelockgravity the Noether current and the Noether potential has the following expression [5]:
Jab (ξi) = 2P abcd∇cξd (5.32)
Ja (ξi) = 2P abcd∇b∇cξd (5.33)
Next we will perform the same calculation i.e. relating the Noether current for a vector qa tothat of another vector f(x)qa = va. Thus from the above equation we obtain:
Jab (vi) = 2P abcd∇c (fqd)
= 2P abcdqd∇cf + 2fP abcd∇cqd (5.34)
31
Then the corresponding Noether current has the following expression:
Ja (vi) = 2P abcd∇b (qd∇cf) + 2P abcd∇b (f∇cqd)
= 2P abcdqd∇b∇cf + 2P abcd∇cf∇bqd + 2P abcd∇bf∇cqd + 2fP abcd∇b∇cqd (5.35)
From the above result we readily arrive at:
Ja (vi)− fJa (qi) = 2P abcdqd∇b∇cf + 2P abcd∇cf∇bqd + 2P abcd∇bf∇cqd
= P abcd∇bAcd + Jab (qi)∇bf (5.36)
where we have defined the antisymmetric tensor Acd as Acd = qd∇cf − qc∇df . Now considerthe following result: qa∇bAcd = ∇b (qaAcd)− Acd∇bqa which leads to:
P abcdqa∇bAcd = ∇b
(P abcdqaAcd
)− 2P abcdqd∇cf∇bqa
= ∇b
(P abcdqaAcd
)− qa∇bfJ
ab (qi) (5.37)
Then Eq. (5.36) can be rewritten in the following manner:
qaJa (fqi)− fqaJa (qi) = Jab (qi)∇bfqa +∇b
(P abcdqaAcd
)− qa∇bfJ
ab (qi)
= ∇b
(2P abcdqaqd∇cf
)(5.38)
It can be easily verified that in the Einstein-Hilbert limit the above equation reduces to Eq. (5.3).Now applying the above equation to ua = −N∇at we arrive at:
uaJa (ui) = 2N∇b
(P abcduaud
∇cN
N2
)(5.39)
In order to proceed we define a new vector field such that:
χa = −2P abcdubud∇cN
N
= −2P abcdubud
(ac +
1
Nucu
j∇jN
)= −2P abcdubacud = −2εaP abβdubrβud (5.40)
Note that in the Einstein-Hilbert limit this vector reduces to the acceleration four vector. Justas in the case of acceleration we have two important properties of the tensor χa:
uaχa = −2aP abβduaubrβud = 0 (5.41)
√σNχαrα
8π= ε− 1
2
(Na
2π
)√σPαbβdrαubrβud
= εTlocs (5.42)
where we have introduced the entropy density of the surface in Lanczos-Lovelock gravity as[5, 38]:
s = −1
8
√σP abcdµabµcd
= −1
2
√σPαbβdrαubrβud (5.43)
32
Though the entropy density seems to have a negative sign, in Einstein-Hilbert limit this becomes√σ/4.
Eq. (5.41) holds due to antisymmetry property of P abcd. We can also have the followingrelation:
Nabχb = χb∇bN + χbubu
j∇jN = χb∇bN (5.44)
Thus Eq. (5.39) can be written in terms of χa as:
uaJa (ui) = N∇b
(χb
N
)= ∇bχ
b − ∇bN
Nχb
= Dαχα (5.45)
The last relation follows from the fact that:
Dαχα = Dbχ
b = ∇bχb − abχb = ∇bχ
b − ∇bN
Nχb (5.46)
Then it is straightforward to get the Noether current for ξa as:
uaJa (ξi) = NuaJa +∇b
(Nχb
)= NDαχ
α +∇b
(Nχb
)= Dα (2Nχα) (5.47)
Here also we have used the following identity:
Dα (Nχα) =(gij + uiuj
)∇i (Nχj)
= ∇i
(Nχi
)+ uiuj∇i (Nχj)
= N∇iχi +Nχiai −Nχj
(ui∇iuj
)= N∇iχ
i (5.48)
Now we can integrate Eq. (5.47) over (d − 1) dimensional volume bounded by t = constantsurface leading to: ∫
V
√hdd−1xuaJa (ξi) =
∫∂V
dd−2x√σ
8πNrαχ
α
= ε
∫∂Vdd−2xTlocs (5.49)
where we have used Eq. (5.42) to arrive at the final equality.Hence the result that Noether charge being the heat content TS holds in Lanczos-Lovelock
gravity as well. We therefore conclude that even for the most general gravitational Lagrangianthe Noether charge measures the enthalpy E − F = TS.
Then we can proceed further using Eq. (2.32) in Eq. (5.47) we obtain the following equation:
2uaPjkai LξΓijk = Dα (2Nχα)− 2NRabu
aub (5.50)
33
for mth order Lanczos-Lovelock Lagrangian we have P jkai = mQ jka
i = (m/√−g)U jka
i . Thusthe conjugate variables introduced in the previous two chapters to describe both gravity andthermodynamics have a direct relation to the Noether current and can determine dynamicalevolution of the spacetime.
Now the field equation in Lanczos-Lovelock gravity leads to: Rab − (1/2)gabL = 8πTab.Being contracted with gab this leads to: L = 8π
[m−(d/2)]T , where d is the space-time dimension.
Hence the field equation can be written as:
Rab = 8πTab = 8π
(Tab −
1
2
1
(d/2)−mgabT
)(5.51)
Then using Eq. (5.51) and integrating Eq. (5.50) over (d−1) dimensional volume we arrive at:∫R
dd−1x√h
8π2uaP
jkai LξΓijk =
∫∂R
dd−2x√σ
4πNχαrα −
∫Rdd−1x
√h2NTabu
aub (5.52)
We can proceed using ∇αN = 2πεTlocrα, where ri is normal to the N = constant hypersurfacewithin t = constant surface. Thus Eq. (5.52) reduces to:∫R
dd−1x√h
8π2uaP
jkai LξΓijk = −2ε
∫∂Rdd−2x
√σPαbβdrαubrβud
(1
2Tloc
)−∫Rdd−1x
√h2NTabu
aub
(5.53)Now the surface degree of freedom is defined as for the Lanczos-Lovelock model as:
Nsur ≡ 4S = −2
∫∂Rdd−2x
√σPαbβdrαubrβud (5.54)
with the following expression for average temperature:
Tavg =
∫∂R d
d−2x√σPαbβdrαubrβudTloc∫
∂R dd−2x√σPαbβdrαubrβud
=1
S
∫dSTloc =
1
Nsur
∫dNsurTloc (5.55)
The bulk degrees of freedom is again related to ρKomar given by 2NTabuaub = ρKomar through:
Nbulk =ε
(1/2)Tavg
∫Rdd−1x
√hρKomar (5.56)
Inserting Eqs. (5.54), (5.55) and (5.56) in Eq. (5.53) we find that the dynamical evolution isdetermined by the following relation:∫
R
dd−1x√h
8π2uaP
jkai LξΓijk = ε
(1
2Tavg
)(Nsur −Nbulk) (5.57)
For a static spacetime in Lanczos-Lovelock gravity the Lie variation of connection vanishesas ξa becomes a timelike Killing vector. Hence in that situation we have Nsur = Nbulk, theholographic equipartition is obeyed even in Lanczos-Lovelock gravity [48]. Also for non staticobservers in static spacetime or in a non-static spacetime itself, departure from holographicequipartition leads to dynamical evolution of the spacetime governed by the Lie derivativeterm on the left hand side of the above equation.
34
The above result was derived for mthe order Lanczos-Lovelock Lagrangian. This can beeasily generalized to a Lanczos-Lovelock Lagrangian of general form. For that the equation ofmotion, Rab − (1/2)gabL = 8πTab on contraction with gab leads to,
∑m cm [m− (D/2)]L(m) =
8πT , which cannot be solved for L in terms of T . Thus in this general situation we have towork with the original expression Eq. (5.50) itself. Then integrating Eq. (5.50) over (d − 1)dimensional hypersurface bounded by N = constant surface leads to the following expression:∫
R
dd−1x√h
8π2uaP
jkai LξΓijk =
∫∂R
dd−2x√σ
4πNχαrα −
∫Rdd−1x
√h2N
(Rab
8π
)uaub (5.58)
Here also we define ri to be normal to N = constant surface within t = constant surface andwith the definition of surface degree of freedom presented in Eq. (5.54) and average temperaturedefined through Eq. (5.55) we arrive at,∫
R
dd−1x√h
8π2uaP
jkai LξΓijk = ε
(1
2Tavg
)Nsur −
∫Rdd−1x
√h2N
(Rab
8π
)uaub (5.59)
In this situation the source of gravity is defined analogously with Komar mass density as:ρ = 2N(Rab/8π)uaun and then the bulk degree of freedom reduces to the following form:
Nbulk =ε
(1/2)Tavg
∫Rdd−1x
√hρ (5.60)
Using this definition of bulk degree of freedom in Eq. (5.59) we arrive at the evolution equationin a general Lanczos-Lovelock gravity:∫
R
dd−1x√h
8π2uaP
jkai LξΓijk = ε
(1
2Tavg
)(Nsur −Nbulk) (5.61)
Thus for static spacetime in Lanczos-Lovelock gravity with most general Lagrangian the holo-graphic equipartition holds as the Lie derivative vanishes identically. While for time depen-dent situation the departure from holographic equipartition determines dynamical evolution ofspacetime.
35
Chapter 6
Conclusions
Principle of equivalence and general covariance makes gravity a geometrical phenomenon. Asin the Einstein-Hilbert action, the field equation for the Lanczos-Lovelock Lagrangian also pos-sesses only second order derivatives of the metric. The Lanczos-Lovelock Lagrangian is the mostnatural generalization of Einstein-Hilbert action to higher dimensional spacetime. Recently inRef. [1] a pair of conjugate variables were introduced with interesting thermodynamic interpre-tation which simplified various expressions in Einstein gravity a lot. Also in Ref. [2] dynamicalevolution equation for the spacetime were determined in the context of General Relativity. Inthis project work we attempt to generalize the results in [1] and [2] from Einstein gravity toLanczos-Lovelock gravity. Below we list the main results that come out of this investigation:
• Simple generalization of the variables fab and N cab used in Ref. [1] do not work. We
have introduced new variables and with these newly defined variables we can obtain the fieldequations in Lanczos-Lovelock theories. The surface term in Lanczos-Lovelock theory can bewritten as −∂(qp) and the quadratic part is written as (1/2)p∂q.
• The most striking feature of these new conjugate variables are that variation of thesevariables are related to δT and δS respectively. Thus these conjugate variables lead to ther-modynamic relations when the background metric is varied with proper restrictions on thevariation. In this sense the action principle encodes the information about horizon thermody-namics, a key result in emergent gravity paradigm that holds in Lanczos-Lovelock theories aswell. The above results hold for any arbitrary null metric.
• The dynamical evolution equation generalizes to Lanczos-Lovelock gravity, with propersurface and bulk degrees of freedom. In Lanczos-Lovelock gravity as well for static sphericallysymmetric spacetime we obtain Nsur = Nbulk, i.e. holographic equipartition holds in Lanczos-Lovelock gravity as well. As in Einstein-Hilbert scenario in Lanczos-Lovelock gravity as well thedifference between surface and bulk degrees of freedom determining departure from holographicequipartition results in dynamical evolution of the spacetime.
• Also the Noether current can be written in terms of the new conjugate variables inLanczos-Lovelock gravity. The Noether charge turns out to be the heat content of the space-time.
36
Though we have addressed many questions we started with, we have also left few questionsunanswered. It is not clear from our analysis that how these conjugate variables play their role inpresence of matter. Also to arrive at the thermodynamic interpretation we have imposed certainrestrictions on the metric variations, their physical origin is not clear. However we have foundout a set of variables that leads to the dynamical evolution of the spacetime to thermodynamicbehavior. Thus we have a clear understanding of how various results in Einstein-Hilbert actiongets generalized to Lanczos-Lovelock gravity.
37
Appendix A
Derivation of Various Identities used inText
We will consider the p∂q and q∂p structure arising from the indentification of fab as coordinateand N c
ab as momentum in LL gravity. For the calculation, the following identity will be usedhere and there:
0 = ∇cQcdab = ∂cQ
cdab + ΓcckQ
kdab
− ΓkcaQcdkb − ΓkcbQ
cdak − ΓdckQ
ckab (A.1)
However Qcdab being antisymmetric in (c,d) while Γcab being symmetric in (a,b) the last term in
the above expansion vanishes. Thus ordinary derivative of the quantity Qcdab has the following
expression,∂cQ
cdab = −ΓcckQ
kdab + ΓkcaQ
cdkb + ΓkcbQ
cdak (A.2)
Note that we can include√−g in the above expression leading to,
∂c(√−gQ bcd
a
)=(√−gQ bcd
p
)Γpac −
(√−gQpcd
a
)Γbcp (A.3)
Thus using Eq. (A.2) we get the following expression:
∂cNcab = ∂c
[QcqbpΓ
paq +Qcq
apΓpbq
]=
(∂cQ
cqbp
)Γpaq +Qcq
bp∂cΓpbq +
(∂cQ
cqap
)Γpbq +Qcq
ap∂cΓpbq
= QcqkpΓ
kcbΓ
paq +Qcq
bkΓkcpΓ
paq −Q
kqbpΓpaqΓ
cck +Qcq
kpΓkcaΓ
pbq
+ QcqakΓ
kcpΓ
pbq −Q
kqapΓ
cckΓ
pbq +Qcq
bp∂cΓpaq +Qcq
ap∂cΓpbq (A.4)
where in order to arrive at the last equality Eq. (A.2) has been used. Now contracting theabove expression with fab we readily obtain,
fab∂cNcab =
√−ggab
[2Qcq
ap∂cΓpbq + 2Qcq
kpΓkcaΓ
pbq + 2Qcq
akΓkcpΓ
pbq − 2Qkq
apΓcckΓ
pbq
]= −2
√−gQ bqc
p
(∂cΓ
pbq + ΓkcpΓ
pbq
)+ 2
√−gQ bkq
p ΓcckΓpbq + 2
√−ggabQcq
kpΓkcaΓ
pbq
= −√−gQ bqc
p Rpbqc + 2
√−gQ bkq
p ΓcckΓpbq + 2
√−ggabQcq
kpΓkcaΓ
pbq (A.5)
38
Note that in the EH limit the last two terms adds up to yield −√−gLquad. Then consider the
other combination which can be expressed as,
N cab∂cf
ab =(QcqapΓ
pqb +Qcq
bpΓpqa
)∂c(√−ggab
)=√−g(QcqapΓ
pqb +Qcq
bpΓpqa
) (∂cg
ab + gabΓpcp)
= 2√−gQcq
apΓpqb∂cg
ab + 2√−gQ bqc
p ΓpqbΓmcm
= 2√−gQ bcq
p ΓlbcΓpql + 2
√−gQ bqc
p ΓpqbΓmcm − 2
√−ggbmQcq
apΓpqbΓ
acm
=√−gLquad + 2
√−gQ bqc
p ΓpqbΓmcm − 2
√−ggbmQcq
apΓpqbΓ
acm (A.6)
In the EH limit the above term leads to√−gLquad. Next we will derive similar relations which
actually behaves as conjugate variables, with the identification, p ≡ 2√−gQ bcd
a and q ≡ Γabc.Then the respective p∂q and q∂p expressions are given in the following results:
2√−gQ bdc
e ∂cΓebd =
√−gQ bdc
e (∂cΓebd − ∂dΓebc)
=√−gQ bdc
e Rebcd − 2
√−gQ bdc
e ΓemcΓmbd
= −√−gQ abc
e Reabc −
√−gLquad (A.7)
and
Γdbe∂c(2√−gQ bec
d
)= 2
√−gΓdbe∂cQ
becd + 2ΓdbeQ
becd ∂c
√−g
= 2√−gΓdbe
(ΓacdQ
beca − ΓbcaQ
aecd − ΓccaQ
bead
)+ 2ΓdbeQ
becd ∂c
√−g
= 2√−gLquad (A.8)
In the text we have used local inertial frame to derive the fact that covariant derivative ofU bcda vanishes. However even without using the local inertial frame condition we obtain,
∂cUbcda = U bcd
p Γpac − U pcda Γbcp, which is also equivalent to Eq. (3.33). Then we have the
following decomposition for the Lagrangian in Lanczos-Lovelock gravity as:
√−gL = U bcd
a
(∂cΓ
abd + ΓackΓ
kbd
)= −∂c
(U bdca Γabd
)+ U bcd
a ΓackΓkbd − Γabd∂cU
bcda
= U bcda ΓamdΓ
mbc − ∂c
(U bdca Γabd
)=√−gLquad +
√−gLsur (A.9)
Hence the two variables U bcda and Γabc leads to identical decomposition of the Lagrangian in
Lanczos-Lovelock gravity as the variables fab and N cab does in Einstein-Hilbert Lagrangian
evident from Eqs. (3.6), (3.8) and (3.9) respectively.The next thing to consider are the connections in the general static metric we are considering.
39
There the connections are given by,
Γnnn = Γntn = ΓnnA = ΓntA = 0
Γntt = N∂nN ; ΓnAB = −1
2∂nσAB
Γtnn = ΓtnA = 0
Γtnt =∂nN
N; Γttt =
∂tN
N
ΓttA =∂AN
N; ΓtAB =
∂tσAB2N2
ΓAnn = ΓAnt = 0
ΓAnB =1
2σAC (∂nσBC) ; ΓAtt = NσAB∂BN
ΓAtB =1
2σAC (∂tσBC) ; ΓABC =
1
2σAD (−∂DσBC + ∂BσCD + ∂CσBD) (A.10)
Also we list below all the connectins in GNC coordinate system that will remain nonzero inthe null surface limit:
Γuuu = α; ΓuuA = βA/2; ΓuAB = −∂rµAB/2Γrur = −α; ΓrrA = −βA/2; ΓrAB = −∂uµAB/2
ΓABC = ΓABC ; ΓABu = µCA∂uµBC/2;
ΓABr = µCA∂rµBC/2; ΓAur = −βA/2 (A.11)
Now we will present curvature tensor components in GNC coordinates, which are relevant forthe calculations in the main text,
RCPuB = ∂uΓ
CPB − ∂BΓCPu + ΓCmuΓ
mPB − ΓCmBΓmPu
= ∂u
(1
2βC∂rµPB + ΓCPB
)− ∂B
(1
2βC∂rβP +
1
2µCA∂uµPA +
1
2µCA
(DP βA − DAβP
))+
(1
2βC∂rβA +
1
2µCB∂uµBA +
1
2µCB
(DAβB − DBβA
))(1
2βA∂rµBP + ΓABP
)−
(1
2βC∂rα−
1
2µCADAα + µAC∂uβA
)(1
2∂rµAB
)−
(1
2µCA∂rβA
)(1
2
{∂uµPB +
(βC βC − α
)∂rµPB
}+
1
2
(DP βB + DBβP
))−
(1
2βC∂rµBA + ΓCAB
)(1
2βA∂rβP +
1
2µCA∂uµPC +
1
2µCA
(DP βC − DC βP
))+
(1
2βC∂rβB +
1
2µCA∂uµBA +
1
2µCA
(DBβA − DAβB
))(1
2∂rβP
)+
(1
2µCA∂rµBA
)(1
2
(βC βC − α
)∂rβP +
1
2DP α−
1
2βB(∂uµPB + DP βB − DBβP
))= ∂uΓ
CPB −
1
2µCM∂B∂uµMP −
1
2∂Bµ
CM∂uµMP +1
2µCM∂uµMAΓAPB
− 1
2∂uµPBΓCru −
1
2µCM∂uµMBΓuPu −
1
2µAC∂uµCP ΓCAB (A.12)
40
RrABu = ∂BΓrAu − ∂uΓrAB + ΓrBmΓmAu − ΓrumΓmBA
= ∂B
[−1
2
(βC βC − α
)∂rβA +
1
2DAα−
1
2βB(∂uµAB + DAβB − DBβA
)]− ∂u
[−1
2
{∂uµAB +
(βC βC − α
)∂rµAB
}+
1
2
(DAβB + DBβA
)]+
([1
2
(∂rβB − βC∂rµCB
)] [−1
2
(βC βC − α
)∂rβA +
1
2DAα−
1
2βB(∂uµAB + DAβB − DBβA
)]+
[−1
2
(βC βC − α
)∂rβA +
1
2DAα−
1
2βB(∂uµAB + DAβB − DBβA
)] [−1
2∂rβA
]+
[−1
2
{∂uµBC +
(βM βM − α
)∂rµBC
}+
1
2
(DC βB + DBβC
)]×
[1
2βC∂rβA +
1
2µCB∂uµBA +
1
2µCB
(DBβA − DAβB
)]− (B ↔ u)
)=
1
2∂2uµAB −
1
4∂uµBCµ
CM∂uµMA +1
2α∂uµAB (A.13)
and
RrPQR = ∂QΓrPR − ∂RΓrPQ + ΓrmQΓmPR − ΓrmRΓmPQ
= −∂Q(
1
2
{∂uµPR +
(βC βC − α
)∂rµPR
}+
1
2
(DP βR + DRβP
))+
(1
2
(βC βC − α
)∂rβQ +
1
2DQα−
1
2βB(∂uµQB + DQβB − DBβQ
))(1
2∂rµPR
)−
(1
2
(∂rβQ − βC∂rµCQ
))(1
2
{∂uµPR +
(βC βC − α
)∂rµPR
}+
1
2
(DP βR + DRβP
))−
(1
2
{∂uµAQ +
(βC βC − α
)∂rµAQ
}+
1
2
(DAβQ + DQβA
))(1
2βA∂rµPR + ΓAPR
)− (Q↔ R)
= −1
2∂Q∂uµPR +
1
2∂R∂uµPQ +
1
4βQ∂uµPR −
1
4βR∂uµPQ
− 1
2∂uµAQΓAPR +
1
2∂uµARΓAPQ (A.14)
41
Appendix B
Identities Regarding Lie Variation ofP abcd
In this section we shall derive some identities related to Lie variation of the entropy tensor,P abcd. First we shall derive the Noether current from identities in differential geometry. Thisdemystifies the connection of conserved Noether current with diffeomorphism invariance of theaction principle. This has already been emphasised in Ref. [2], however only in the contextof Einstein gravity, we shall now derive it for LL models. For that purpose we shall use theantisymmetric part of ∇avb for arbitrary vector vb and identities regarding curvature tensor.We start with the fact that the covariant derivative of any vector field can be decomposedinto a symmetric and an antisymmetric part. However in this case we will consider only theantisymmetric part and define an antisymmetric tensor field as,
Jaj = 2P ajki∇kvi = P ajki (∇kvi −∇ivk) (B.1)
It is evident from the antisymmetry of P abcd that a conserved current exists such that, Ja =∇jJ
aj. Now we have the standard identity,
(∇j∇k −∇k∇j) vi = Ri
cjkvc (B.2)
and,LvΓijk = ∇k∇jv
i −Rikjmv
m (B.3)
Then we can use Eq. (2.17) to get the following result,
Rabvb = P aijkRbijkvb = −P aijk (∇j∇k −∇k∇j) vi
= P aijk∇k∇jvi +(P akij + P ajki
)∇j∇kvi
= P aijk∇k∇jvi + P akij∇j∇kvi +∇j
(P ajki∇kvi
)(B.4)
where in the second line we have used the identity, P a(bcd) = 0. Then from Eq. (B.1) we readilyobtain,
Ja = 2Rabvb − 2P aijk∇k∇jvi − 2P akij∇j∇kvi
= 2Rabvb + 2P ajki ∇k∇jv
i + 2P jaki ∇j∇kv
i
= 2Rabvb + P ajki
(LvΓbkj +Rb
jkmvm)− P jak
i
(LvΓijk +Ri
kjmvm)
= 2Rabvb + 2P jkai LvΓijk (B.5)
42
while arriving at the third line we have used Eq. (B.3) and for the last line we have used thefact that, P ijakRikjm = P akijRikjm = −P kaijRikjm = P kaijRkijm. Thus Eq. (B.5) has beenderived without any reference to the gravitational action only using differential geometry andvarious symmetry properties.
Next we present some identities relating to Lie variation of P abcd. For that purpose we firstconsider Lie variation of the Lagrangian treated as a scalar function of the metric gab and Rabcd
leading to,
LξL (gab, Rijkl) = ξm∇mL (gab, Rijkl) =∂L
∂gabξm∇mgab +
∂L
∂Rijkl
ξm∇mRijkl = P ijklξm∇mRijkl
(B.6)where we have used the fact that covariant derivative of metric tensor vanishes. Then for theLagrangian which is homogeneous function of degree m we get,
LξL = ξm∇m
(1
mP ijklRijkl
)(B.7)
Then using Eq. (B.6) we readily obtain,
Rabcdξm∇mP
abcd = (m− 1)P abcdξm∇mRabcd (B.8)
We also have the following relation,
P ijklLξRijkl = P ijkl (ξm∇mRijkl +Rajkl∇iua +Riakl∇ju
a +Rijal∇kua +Rijka∇lu
a)
= P ijklξm∇mRijkl + 4∇iξmRijklP
mjkl
= P ijklξm∇mRijkl + 4∇iξmRim (B.9)
Again we can also write, mLξL = P ijklLξRijkl +RijklξPijkl. Then we get,
RijklLξP ijkl = mLξL− P ijklLξRijkl
= mLξL− P ijklξm∇mRijkl − 4∇iξmRim
= (m− 1)P abcdξm∇mRabcd − 4∇iξmRim (B.10)
This equation can also be casted in a different form as,
Rabcd
(LξP abcd − ξm∇mP
abcd)
= −4∇iξmRim (B.11)
Now we can rewrite the metric as a function of gab and Rabcd, in which case the Lie variation
leads to,LξL (gij, R
abcd) = P jkl
i ξm∇mRijkl (B.12)
With the Lagrangian as homogeneous function of curvature tensor to mth order leads to,
RabcdLξP bcd
a = (m− 1)P jkli ξm∇mR
ijkl (B.13)
Then we arrive at the following identity,
P abcdLξ (gamRmbcd) = P bcd
a ξm∇mRabcd + 4∇iξmRim (B.14)
43
or,
P bcda LξRa
bcd = P bcda ξm∇mR
abcd + 4∇iξmRim −RamLξgam
= P bcda ξm∇mR
abcd + 2∇iξmRim (B.15)
This leads to the following relation,
Rabcd
(LξP bcd
a − ξm∇mPbcda
)= −2∇iξmRim (B.16)
If we proceed along the same lines we readily obtain another such relation given as,
Rijkl
(LξP kl
ij − ξm∇mPklij
)= 0 (B.17)
These relations illustrate the Lie variation of P abcd when contracted with the curvature tensor.
44
Bibliography
[1] K. Parattu, B.R. Majhi and T. Padmanabhan, Phys. Rev. D 87, 124011 (2013) [arXiv:gr-qc/1303.1535].
[2] T. Padmanabhan, Gen. Rel. Grav 46, 1673 (2014) [arXiv:gr-qc/1312.3253].
[3] C. Lanczos, Z.Phys. 73, 147 (1932);C. Lanczos, Ann. of Math. 39, 842 (1938).
[4] D. Lovelock, J. Math. Phys. 12, 498 (1971).
[5] T. Padmanabhan, Gravitation: Foundation and Frontiers, Cambridge University Press,Cambridge, UK (2010).
[6] T. Padmanabhan and D. Kothawala, Phys. Rept. 531, 115 (2013).
[7] J.D. Bekenstein, Phys. Rev. D 7, 2333 (1973).
[8] J.D. Bekenstein, Phys. Rev. D 9, 3292 (1974).
[9] S. Hawking, Commun. Math. Phys. 43, 199 (1975).
[10] T. Padmanabhan, Phys. Rept. 406, 49 (2005) [arXiv:gr-qc/0311036].
[11] T. Padmanabhan, Rept. Prog. Phys. 73, 046901 (2010) [arXiv:gr-qc/0911.5004].
[12] R.M. Wald, Liv. Rev. Relt. 4, 6 (2001)[arXiv:gr-qc/9912119].
[13] P. C. W. Davies, S. A. Fulling, and W. G. Unruh, Phys. Rev. D 13, 2720 (1976).
[14] W. G. Unruh, Phys. Rev. D 14, 870 (1976).
[15] T. Padmanabhan, Class. Quant. Grav. 19, 5837 (2002) [arXiv:gr-qc/0204019].
[16] D. Kothawala and T. Padmanabhan, Phys. Rev. D 79, 104020 (2009) [arXiv:gr-qc/0904.0215].N. Dadhich, [arXiv:1210.3022].
[17] T. Padmanabhan and A. Paranjape, Phys. Rev. D 75, 064004 (2007) [arXiv:gr-qc/0701003].
[18] T. Padmanabhan, Gen. Relt. Grav. 40, 529 (2008) [arXiv:gr-qc/0705.2533].
[19] T. Padmanabhan, Braz. J. Phys. 35, 362 (2005) [arXiv:gr-qc/0412068].
45
[20] A. Mukhopadhyay and T. Padmanabhan, Phys. Rev. D 74, 124023 (2006) [arXiv:hep-th/0608120].
[21] S. Kolekar and T. Padmanabhan, Phys. Rev. D 82, 024036 (2010) [arXiv:gr-qc/1005.0619].
[22] T. Padmanabhan, Mod. Phys. Lett. A25, 1129 (2010) [arXiv:gr-qc/0912.3165].
[23] T. Padmanabhan, Phys. Rev. D 81, 124040 (2010) [arXiv:gr-qc/1003.5665].
[24] T. Padmanabhan, Phys. Rev. D 83, 044048 (2011) [arXiv:1012.0119].
[25] S. Kolekar and T. Padmanabhan, Phys. Rev. D 85, 024004 (2012) [arXiv:gr-qc/1109.5353].
[26] T. Damour, Surface Effects in Black Hole Physics, Proceedings of the second Marcel Gross-mann Meeting on General Relativity (1982).
[27] S. Kolekar, D. Kothawala and T. Padmanabhan, Phys. Rev. D 85, 064031 (2012) [arXiv:gr-qc/1111.0973].
[28] G. Gibbons and S.W. Hawking, Phys. Rev. D 15, 2752 (1977).
[29] A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (1996).
[30] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998);J.M. Garcia-Islas, Class. Quant. Grav. 25, 245001 (2008).
[31] L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, Phys. Rev. D 34, 373 (1986).
[32] R.M. Wald, Phys. Rev. D 48, R3427 (1993);V. Iyer and R.M. Wald, Phys. Rev. D 50, 846 (1994);R.M. Wald and A. Zoupas, Phys. Rev. D 61, 084027 (2000).
[33] T. Padmanabhan, Gen. Relt. Grav. 44, 2681 (2012) [arXiv:gr-qc/1205.5683].
[34] B.R. Majhi and T. Padmanabhan, Eur. Phys. J C 73, 2651 (2013) [arXiv:gr-qc/1302.1206].
[35] A. Eddington, The Mathematical Theory of Relativity, Cambridge University Press, Cam-bridge, UK (1924);E. Schrodinger, Space-time Structure, Cambridge University Press, Cambridge, UK (1950).
[36] T. Padmanabhan, Phys. Rev. D 84, 124041 (2011) [arXiv:gr-qc/1109.3846].
[37] A. Yale and T. Padmanabhan, Gen. Relt. Grav. 43, 1549 (2011) [arXiv:gr-qc/1008.5154];N. Kiriushcheva and S. Kuzmin, Mod. Phys. Lett. A21 (2006) [arXiv:hep-th/0510260].N. Dadhich, S.G. Ghosh and S. Jhingan, [arXiv:gr-qc/1202.4575]
[38] B.R. Majhi and T. Padmanabhan, Phys. Rev. D 85, 084040 (2012) [arXiv:gr-qc/1111.1809].
[39] B.R. Majhi, Adv. High Energy Phys. 386342 (2013) [arXiv:gr-qc/1210.6736].
[40] Q. Exirifard and M.M. Sheikh-Jabbari, Phys. Lett. B 661, 158 (2008).
[41] V. Moncrief and J. Isenberg, Commun. Math. Phys. 89, 387 (1983).
46
[42] E.M. Morales, On a Second Law of Black Hole Mechanics in a Higher Derivative Theoryof Gravity Ph.D Thesis, Gottingen University.
[43] T. Padmanabhan, AIP Conf. Proc. 1241, 93 (2010) [arXiv:gr-qc/0911.1403].
[44] A. Medved, D. Martin and M. Visser, Class. Quant. Grav. 21 3111 (2004) [arXiv:gr-qc/0402069].
[45] I. Racz and R.M. Wald, Class. Quanta. Grav. 13, 539 (1996) [arXiv:gr-qc/9507055].
[46] B. Carter, Gen. Relt. Grav. 42, 653 (2010).
[47] K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan (in preparation).
[48] T. Padmanabhan, Phys.Rev. D 81, 124040 (2010) [arXiv:gr-qc/1003.5665]
47
