Black hole thermodynamics: general relativity and beyond
Abstract
Black holes have often provided profound insights into the nature of gravity and the structure of space–time. The study of the mathematical properties of black objects is a major research theme of contemporary theoretical physics. This review presents a comprehensive survey of the various versions of the first law and second law of black hole mechanics in general relativity and beyond. The emphasis is to understand how these laws can constrain the physics beyond general relativity.
1 Introduction
It is appropriate to start a review of black hole thermodynamics with the above quotation by S. Chandrasekhar. The quote brings out the fundamental characteristics of a black hole: The Universality. The properties of a black hole are (almost) independent of the details of the collapsing matter, and this universality is ultimately related to the fact that black holes could be the thermodynamic limit of underlying quantum gravitational degrees of freedom. Therefore, the classical and semiclassical properties of black holes are expected to provide important clues about the nature of quantum gravity. A significant obstacle in constructing a theory of quantum gravity is the absence of any experimental or observational result. The only “test” we can imagine is the theoretical and mathematical consistency of the approach. The understanding of the fundamental laws of black hole mechanics could be a necessary (if not sufficient) constraint on the theory of quantum gravity.
The modern understanding of the properties of black hole starts with the resolution of the “Schwarzschild Singularity” using Kruskal–Szekeres coordinates [1, 2]. These coordinates cover that entire spacetime manifold of the maximally extended vacuum spherically symmetric solution of the Einstein’s field equation and are wellbehaved everywhere outside the physical singularity at the origin, in particular at the position \(r = 2M\). The next important step is the discovery of the rotating asymptotically flat vacuum black hole solution by Roy Kerr [3]. The solution exhibited various interesting and generic properties of a stationary black hole in general relativity. The existence of Ergosphere and Superradiance show how to extract energy and angular momentum from the black hole. The study of these phenomena lead to a significant result; the area of the black hole can never be decreased using these processes. For example, using the Penrose process, it is possible to extract energy from the black hole, and as a result, the mass of the black hole decreases. At the same time, the process slowed down the rotation, and the net effect only increases the area.
Then comes the famous work by Hawking [4] which analyzes the general properties of a black hole, independent of the symmetry of a particular solution. This work contains several important theorems: the topology theorem, the strong rigidity theorem and most importantly, the area theorem. Area theorem is a remarkable result which asserts that the area of the event horizon can not decrease as long as the matter obeys a specific energy condition. This is a highly nontrivial statement related to the dynamics of black holes in general relativity. Consider the collision of two black holes which generated a burst of gravitational waves extracting energy from the black holes to infinity. The area theorem constrains the efficiency of this process and limits the amount of radiated energy so that the area of the final black hole is always greater than the sum of the individual black hole areas before the collision [5]. In this sense, the area theorem is a statement of the limitation of converting the black hole mass into energy; akin to the second law of thermodynamics.

How far the laws of Black Hole mechanics can be generalized beyond General Relativity?

Can we constrain possible extensions of general relativity using the black hole (BH) mechanics?

What exactly we have learned so far about Quantum Gravity from BH Mechanics?
The discussion in the review will be mostly classical, and we will assume the applicability of the classical energy conditions, in particular, the null energy condition. The primary focus is a comprehensive discussion of the physical process law and the second law. We will not consider the issues related to the semiclassical gravity; in particular, Hawking radiation and trans Planckian problem. Another vital omission will be the information loss paradox. We will also restrict ourselves to the mechanics and thermodynamics of the event horizon only.
2 The various versions of the first law
The first law of black hole mechanics has several avatars, and we need to distinguish the different formulations of the first law. In ordinary thermodynamics, the first law is the statement of the conservation of energy. The total energy can not be destroyed or created, but can always be converted into another form of energy. The statement is mathematically described by the difference equation \(\Delta U = Q  W\). The change of the internal energy U of the system is equal to the difference of the heat supplied Q, and the work done W by the system. The conservation of energy is builtin into the dynamics of general relativity. So, what we mean by the first law is the Clausius theorem which involves the notion of the entropy. Consider a system under quasistatic change which is subjected to an infinitesimal amount of heat Open image in new window from the surrounding. The heat change is an inexact differential, and therefore the total heat Q is not a state function. It is then assured that there exists a state function called the “entropy” S such that the temperature of the system acts as an integrating factor relating the change in entropy to the heat supplied as Open image in new window . The Clausius theorem ensures the existence of the state function entropy associated with a thermodynamic equilibrium state of the system. Note that all changes are considered to be quasistationary, always infinitesimally close from an equilibrium state.
In the case of a black hole, we need to be careful before applying these concepts. To begin, the obvious choice of an equilibrium state is a stationary black hole. So, let us first define the notion of the stationary black hole in general relativity.
To define the event horizon of a black hole, we require information about the asymptotic structure. Suppose we consider an asymptotically flat space–time such that the asymptotic structure is the same as that of the flat space–time. Then, the event horizon is defined as the complement of the past of the future null infinity. This is a global definition and to find the location of the event horizon, we require the knowledge about the entire space–time. This is not a very convenient concept. For example, if one is looking for the signature of the formation of the event horizon in the computer codes of numerical relativity, she has to wait for infinite time! As a result, alternative notions like apparent horizon and quasilocal horizons may suit much better for such an analysis. Nevertheless, the event horizon can be very useful because it is a null surface,^{2} and the causal boundary between two regions of space–time called inside and outside of the black hole. As a result, at least intuitively it makes sense to assign an entropy to the null event horizon.
The definition of the event horizon does not need any symmetries of the underlying space–time. Now, consider the particular case when the space–time is stationary and contains a time like Killing vector. Such a time like Killing vector provides a related concept called the Killing horizon. A Killing horizon is a surface where the time like Killing field becomes null. An example of such a surface would be the Rindler horizon in the flat space–time. It is easy to check that the boost Killing field indeed becomes null at the location of the Rindler horizon. This example shows that the Killing horizon may be entirely unrelated to the event horizon. The Rindler accelerated horizon is a Killing horizon but not an event horizon.
Next, consider an event horizon in a stationary space–time. Then, it is the Strong Rigidity theorem [4] which asserts that the event horizon in a stationary space–time is also a Killing horizon. The strong rigidity theorem is a powerful result, and the proof requires Einstein field equation and some technical assumptions like the analyticity of the space–time. Generalizing the proof beyond \(3+1\) dimensions needs more sophisticated mathematical machinery [11, 12]. The strong rigidity theorem is only proven for general relativity. Therefore for black holes in various modified gravity theories, we have to consider this as an assumption.
This derivation can be generalized in several ways. If we include matter, e.g., an electrovacuum solution, there will be additional work terms. But the most interesting generalization is for theories with higher curvature terms in the action. The area law fails generically for higher curvature gravity [18, 19, 20, 21, 22] and the entropy is proportional to a different local geometric quantity evaluated on the horizon. In fact, the black hole entropy in any diffeomorphism invariant theory of gravity turns out to be the Noether charge of the Killing isometry which generates the horizon [21, 22]. Before discussing the derivation of this “Wald entropy”, we will first try to understand intuitively why and how the area law fails beyond general relativity, using a generalized version of the original argument by Bekenstein [7, 8].
The detailed structure of these terms will depend on the specifics of the underlying quantum gravity theory. If we turn on these higher curvature corrections, the field equation will get modified, and the area theorem may not hold anymore. But, for specific higher curvature terms, we can still obtain exact black hole solutions as in case of GR. Now, consider the simplest case of spherical symmetry and assume that a set of identical particles with the same mass m is collapsing in D dimensions to form a black hole of mass M. If each of these particles contains one bit of information (in whatever form, may be information about their internal states, etc.), then the total loss of information due to the formation of the black hole will be \( \sim M/m\). Classically, this can be as high as possible, but quantum mechanically there is a bound on the mass of each constituent particle because we want the Compton wavelength of these particles to be less than the radius of the hole \(r_h\). Then, the maximum loss of information will be \(\sim M r_h\), and this is a measure of the entropy of the hole. Note that, we have not used any information about the field equation yet. So, this is completely an offshell result. The field equation will provide a relationship between the mass M and the horizon radius.
Let us now treat the specific case of general relativity. If we solve the vacuum Einstein’s equations for spherical symmetry, we obtain the usual Schwarzschild solution with \(M \sim r_{h}^{D3}\), and this lead to black hole entropy proportional to \(r_{h}^{D2}\), the area of the horizon.
Next comes the modified gravity, with higher curvature terms and we will have new dimensionful constants in our disposal. Therefore, there could be a complicated relationship between mass and horizon radius. For example, if we restrict ourselves up to only curvature square correction terms with a coupling constant \(\alpha \), we could have a relationship like \( M \sim r_{h}^{D3} + \alpha \, r_{h}^{D5}\), and the second term results in a subleading correction to black hole entropy. This simple illustration shows how the presence of new dimensionful constants in modified gravity theories leads to a possible modification of the black hole entropy.
This simple derivation can be made more rigorous by using the Noether charge formalism of Wald and collaborators [18, 19, 20, 21, 22]. The crucial input to the derivation is the diffeomorphism invariance in the presence of an inner boundary. The bulk part of the Hamiltonian vanishes onshell, and the two boundary terms (one at the horizon and other at the outer boundary) are related to each other. Then for variations in the space of stationary solutions, we get the first law as the Clausius theorem.
Having discussed the equilibrium state version of the first law, let us now focus on another version of the first law for black holes: The physical process law. This version of the first law involves the direct computation of the horizon area change when a flux of matter perturbs the horizon [27, 28, 29] (henceforth referred to as PPFL). Unlike the equilibrium state version, PPFL is local and does not require the information about the asymptotic structure of the space–time and is therefore expected to hold for a wide class of horizons (see Fig. 1). Consequently, after some initial debate regarding the applicability of PPFL in the context of Rindler space–time [29], it was later demonstrated, following [30, 31], that the physical process version of first law indeed holds for Rindler horizon in flat space time, or for that matter, any bifurcate Killing horizon.
For PPFL, the variation \(\delta A_{\mathrm{H}}\) represents the physical change in the area of the black hole due to the accretion of matter. As a result, here we are considering a genuinely dynamical situation. The physical process first law, therefore, relates the total change of entropy due to the matter flux from the bifurcation surface to a final state. If we assume that the black hole horizon is stable under perturbation, then the future state can always be taken to be stationary with vanishing expansion and shear, and the initial crosssection can be set at the bifurcation surface (\(\lambda =0\)). The choice of these initial and final states are necessary for this derivation of the physical process first law, to make some boundary terms vanish. The derivation can be generalized to obtain the expression of the entropy change between two arbitrary nonequilibrium cross sections of the dynamical event horizon. The additional boundary terms appearing in Eq. (13) are then related to the energy of the horizon membrane arising in the context of the black hole membrane paradigm [32].
To elaborate on the derivation of the PPFL, let us start by describing the horizon geometry and set up the notations and conventions. We will follow the derivation as presented in [32].
3 General structure of PPFL
\(\theta _k\) and \(\sigma _k\) are of first order in perturbation, i.e., \({\mathcal {O}}(\epsilon )\), while \(\theta _l\) and \(\sigma _l\) are of zeroth order, with \(\epsilon \) referring to the strength of perturbation everywhere on the future event horizon. However, since \(\theta _l\) and \(\sigma _l\) vanishes at the bifurcation surface of the stationary black hole, they both must be of at least \({\mathcal {O}}(\epsilon )\) only at \({\mathcal {B}}\). This result is a property of the bifurcation surface itself, independent of the physical theory one considers. Hence it also generalizes beyond general relativity and holds for higher curvature theories as well. In summary we have, \(\theta _k,\, \sigma _k,\, R_{ab}k^{a}k^{b} \backsim {\mathcal {O}}(\epsilon )\) and \(\theta _l,\sigma _l \backsim {\mathcal {O}}(\epsilon )\) at \({\mathcal {B}}\). As a result, terms like \(\theta _k \theta _l \backsim {\mathcal {O}}(\epsilon ^2)\) only at the bifurcation surface (Fig. 2).

The horizon possesses a regular bifurcation surface in the asymptotic past, which is set at \(\lambda = 0\) in our coordinate system.

The horizon is stable under perturbation and eventually settle down to a new stationary black hole. So, all Lie derivatives with respect to horizon generators vanish in the asymptotic future.
The validity of Eq. (25) is an important requirement for the validity of the physical process law. As of now, there is no general proof of the condition Eq. (25). In case f(R) gravity, the condition Eq. (25) holds as an exact identity leading to the physical process law for such a theory [33]. Same can be established for Einstein Gauss–Bonnet and Lovelock class of theories [34, 35, 36]. But, there is still no general proof of this condition. We will discuss more on this in the later sections.
In comparison with the equilibrium state version of the first law, the PPFL is local and independent of the asymptotic structure of the space–time. The relationship between these two versions is not straightforward. In the next section, we would like to understand how these two approaches are related to each other.
4 Equilibrium state version and physical process law
To understand the relationship, we consider the Vaidya spacetime, as a perturbation over a stationary black hole of ADM mass m. Therefore, we assume \(M(v) = m + \epsilon \, f(v)\). The parameter \(\epsilon \) signifies the smallness of the perturbation. Note that, the background spacetime with ADM mass m is used only as a reference; it does not have any physical meaning beyond this. In the absence of the perturbation, the final ADM mass would be the same as m. Therefore, we may consider the process as a transition from a black hole of ADM mass m to another with ADM mass \(M_{\text {ADM}}\), and this allows us to relate the PPFL to the equilibrium state version.
In the case of an ordinary thermodynamic system, the entropy is a state function, and its change is independent of the path. Therefore, we can calculate the change of entropy due to some non equilibrium irreversible process between two equilibrium states by using a completely different reversible path in phase space. In black hole mechanics, the equilibrium state version can be thought as the change of entropy along a reversible path in the space of solutions, whereas the PPFL is a direct irreversible process. The equality of the entropy change for both these processes shows that the black hole entropy is indeed behaving like that of a true thermodynamic entropy [28].
Having understood the relationship between these two versions of the first law for black holes, we will now study the ambiguities of Wald’s construction and how PPFL is affected by such ambiguities.
5 Physical process first law and ambiguities in black hole entropy
In summary, given a particular theory, if there is a choice of entropy density \(\rho \) which obeys the condition Eq. (25), then \(\rho + \Omega \) will also obey the same condition. So, Eq. (25) is independent of the ambiguities as long as we the integrating from a past bifurcation surface to a stationary future crosssection. If it holds for \(\rho _w\), it will hold for \(\rho \) also.
This result, however, doesn’t hold when secondorder perturbations are considered. Unlike first order, the difference in the change in black hole entropy and Wald entropy is given by a boundary term and a bulk integral. As a result, any conclusion about the change of black hole entropy beyond linearized perturbation requires the resolution of these ambiguities.
Similarly, if we demand an instantaneous second law, such that the entropy is increasing at every crosssection, to hold beyond general relativity, we need to fix the ambiguities and find the appropriate black hole entropy [38, 39]. Then, it is also possible to study the higher order perturbations and obtain the transport coefficients related to the horizon [40].
6 Linerized version of the second law
This can be generalized to any theory, and it is always possible to fix the ambiguities from the linearized second law so that we have a local increase theorem at every crosssection of the nonstationary event horizon [38, 39, 47].
If we now consider that the black hole is in an asymptotically Anti De Sitter space–time, after fixing the ambiguities, the black hole entropy becomes identical in form to the holographic entanglement entropy of the boundary gauge theory [38, 39]. The holographic entanglement entropy [43] is a proposal which relates the entropy of the boundary gauge theory with the area of certain minimal surfaces in the bulk (which obeys Einstein’s equation) within the context of gaugegravity duality. The original principle has been generalized to higher curvature theories [44, 45, 46] and the entanglement entropy density of the boundary theory is given as, \(\rho = \rho _w + a\, \theta _k \theta _l + b\, \sigma _k\sigma _l \),. The part \(\rho _w\) is of the same form of Wald entropy for black holes, and the coefficients a and b depends on the choice of gravity theory in the bulk; for general relativity \( a = b = 0\). The expansions and shears correspond to that of a codimension two surface which is anchored to a region of the boundary. Note, a priory, this entanglement entropy is not related to the entropy of the black hole in the bulk. Also, this entropy has no ambiguities, and the coefficients a and b can be calculated using AdSCFT [46]. Our calculations show, if we consider a nonstationary black hole in the bulk and demand that the black hole entropy obeys linearized second law, we will have \( p = a\) and \( q = b\) [39]. It is indeed remarkable that the entropy for black holes in AdS spacetime which obeys linearized second law turns out to be related with the holographic entanglement entropy. It seems that the validity of black hole thermodynamics is already encoded in the holographic principle; the holographic entanglement entropy satisfies the linearized second law while the Wald entropy does not.
Let us summarise the main results: a theory of gravity which has black hole solutions will obey the integrated version of the physical process law if Eq. (25) holds. Given a theory and an expression of black hole entropy, we can always verify the validity of this condition. Also, the condition Eq. (25) is independent of the ambiguities of the Noether charge construction, as long as we are integrating from initial bifurcation surface to a future stationary crosssection. Therefore, in any theory, if Wald entropy \(\rho _w\) satisfied this condition, so does the black entropy \(\rho \). On the other hand, the local increase law depends on the validity of Eq. (38) which is sensitive to the ambiguities. Hence, there is only a particular choice of the ambiguity coefficients p and q for which the local increase law for linearized fluctuations holds. Remarkably, such a choice makes the black hole entropy identical in form to holographic entanglement entropy.
7 Beyond the linearized second law
Another interesting case corresponds to the hyperbolic horizon. In this case, the intrinsic scalar is negative, and if we also assume that \(\gamma > 0\), then there is an obvious bound on the higher curvature coupling beyond which the entropy itself becomes negative and thereby loses any thermodynamic interpretation. This bound in general D dimension is found as \(\lambda _{GB} < D(D4) / 4 (D2)^2\). If the analysis of the second law has any usefulness, it must provide a more stringent bound for the coupling \(\gamma \), and that is indeed the case. Also, to analyze the case for hyperbolic horizons, we will only consider the socalled zero mass limit. In the context of holographic entanglement entropy, these topological black holes play an important role as shown in [44, 50, 51]. One can relate the entanglement entropy across a sphere to the thermal entropy in \(R \times H^{D2}\) geometry by a conformal transformation.
In principle, it is possible to repeat this analysis for any higher curvature gravity theory to obtain similar bounds on the higher curvature couplings provided we have an exact stationary black hole solution as the background [39]. These bounds will be necessary if we demand that the second law of thermodynamics holds for an observer outside the horizon. Any quantum theory of gravity which reproduces such higher curvature corrections and also aims to explain the microscopic origin of black hole entropy must satisfy these bounds. We can constrain various interesting gravity theories in 4 dimensions by our method. In 4 dimensions, our method is the only one to constrain these theories where the causality based analysis [55] is insufficient. For example, for critical gravity theories in \(D=4\) [56] analyzing black holes in AdS background we obtain the bound on the coupling (\(\alpha _{c}\)), \(\frac{1}{2}\le \alpha _{c} \ \le \frac{1}{12}.\) Also, for New Massive gravity in \(D=3\) [57, 58] we obtain the bound on couplings (\(\sigma \)) as, \(3\le \sigma \le \frac{9}{25}\).^{5}
In conclusion, these results show that the validity of a local increase law of black hole entropy can constrain the parameters of the higher curvature terms. Any theory of gravity which does not obey these bounds will have a severe problem with the second law in the presence of a black hole.
Interestingly, there are works which suggest that the higher curvature gravity does not make sense as a standalone classical theory. Consider Einstein–Gauss–Bonnet gravity in dimensions greater than four. The theory has exact shock wave solutions which can lead to a negative Shapiro time delay. This can be used to create a time machine: closed timelike curve without any violation of energy conditions [55]. As a result, such higher curvature theories have badly behaved causal properties for either sign of the higher curvature coupling. Hence, it is proposed that these theories can only make sense as an effective theory and any finite truncation of the gravitational action functional will lead to pathological problems. This result is criticized in [59] where gravitons propagating in smooth black hole spacetimes are considered. It is shown that for a small enough black hole, the gravitons of appropriate polarisation, and small impact parameter, can indeed experience negative time delay, but this can not be used to build a time machine. This is because the required initial data surface is not everywhere space like and therefore the initial value problem is not wellposed. Nevertheless, the result of [55] is quite significant and needs careful understanding.
Similar conclusions can be obtained about the validity of the classical second law for black hole mergers in Lovelock class of theories [60, 61]. In such theories, it is possible to construct scenarios involving the merger of two black holes in which the entropy instantaneously decreases. But, it is also argued that the second law is not violated in the regime where Einstein–Gauss–Bonnet theory holds as an effective theory and black holes can be treated thermodynamically [62].
8 Conclusions and open problems
Black hole thermodynamics provides a powerful constraint on any proposal to understand the quantum gravitational origin of black hole entropy. The area law has motivated significant progress in theoretical physics; most importantly the holographic principle. Similarly, the pioneering work by Jacobson [63] where he considered the concept of local Rindler horizons and showed that Einstein field equations could be derived from thermodynamic considerations hints a deep thermodynamic origin of the full dynamics of gravity.
Similar results are proven in a more general context by Padmanabhan and collaborators. They have shown that the field equations of any higher curvature gravity theory admits an intriguing thermodynamic interpretation [64, 65]. Interestingly, the result is also valid beyond black hole horizons and for any null surface in space–time [66]. These fascinating results lead an alternative approach “the emergent gravity paradigm” to understand the dynamics of gravity [67]. There is also a local gravitational first law of thermodynamics formulated using the local stretched light cones in the neighbourhood of any event [68]. This result indicates that certain geometric surfaces—stretched future light cones—which exist near every point in every spacetime, also behave as if they are endowed with thermodynamic properties. All these results seems to suggest that the thermodynamic properties of space time transcends beyond the usual black hole event horizon.
The derivation of a full second law beyond general relativity remains an important open problem. Ideally, we would like to follow a nonperturbative approach and find a suitable generalization of the area theorem with some restriction on the higher curvature parameters. This requires understanding the thermodynamic Raychaudhuri equation like Eq. (45) for an arbitrary theory of gravity. This is a formidable but straightforward problem. We also like to understand the relationship between holographic entanglement entropy and black hole entropy. The area theorem may have some interesting holographic interpretations. The Holographic Entanglement Entropy was shown to obey various nontrivial inequalities. One of these is the strong subadditivity condition (SSA) which is a fundamental property of entanglement entropy in any quantum field theory and a central theorem of quantum information theory. It is known that the violation of SSA for the boundary theory is connected with the violation of the null energy condition in the bulk spacetime [69, 70, 71]. Since null energy condition is a requirement for the validity of the Hawking area theorem, it is expected that there exists a strong connection between the area theorem for black holes and SSA for holographic entanglement entropy. This relationship may provide us a better understanding of the scope and applicability of the holographic principle.
We end this review with a quotation by Arthur Eddington,
“The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations  then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation  well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.”
Same can be said for any theory of gravity which has a black hole solution.
Footnotes
 1.
As a side remark, let me point out that the Chandrasekar mass formula also contains all the fundamental constants. But we do not associate that result with the quantum gravity.
 2.
 3.
This requires the generalized expansion \(\Theta _k\) goes to zero in the future faster than the time scale \(1/\lambda \) [29].
 4.
We thank Shiraz Minwalla to suggest this.
 5.
The lower bound for both these two cases are coming from demanding the positivity of the entropy.
Notes
Acknowledgements
This review is based on the work done in collaboration with Aron Wall, Srijit Bhattacharjee, Arpan Bhattacharyya, Aninda Sinha, Fairoos C, Akash K Mishra, Avirup Ghosh, Sumanta Chakraborty, and Maulik Parikh. The author thanks Amitabh Virmani for his encouragement to write this review. Special thanks to Ted Jacobson, Aron Wall, Aninda Sinha and Maulik Parikh for sharing their deep insights about black hole physics. SS also acknowledge many constructive comments from the referees on the previous draft of this review. The research of SS is supported by the Department of Science and Technology, Government of India under the Fast Track Scheme for Young Scientists (YSS/2015/001346).
References
 1.Kruskal, M.D.: Phys. Rev. 119, 1743 (1960). https://doi.org/10.1103/PhysRev.119.1743 ADSMathSciNetCrossRefGoogle Scholar
 2.Szekeres, G.: Publ. Math. Debr. 7, 285 (1960)Google Scholar
 3.Kerr, R.P.: Phys. Rev. Lett. 11, 237 (1963). https://doi.org/10.1103/PhysRevLett.11.237 ADSMathSciNetCrossRefGoogle Scholar
 4.Hawking, S.W.: Commun. Math. Phys. 25, 152 (1972). https://doi.org/10.1007/BF01877517 ADSCrossRefGoogle Scholar
 5.Hawking, S.W.: Phys. Rev. Lett. 26, 1344 (1971). https://doi.org/10.1103/PhysRevLett.26.1344 ADSCrossRefGoogle Scholar
 6.Bardeen, J.M., Carter, B., Hawking, S.W.: Commun. Math. Phys. 31, 161 (1973). https://doi.org/10.1007/BF01645742 ADSCrossRefGoogle Scholar
 7.Bekenstein, J.D.: Lett. Nuovo Cim. 4, 737 (1972). https://doi.org/10.1007/BF02757029 ADSCrossRefGoogle Scholar
 8.Bekenstein, J.D.: Phys. Rev. D 7, 2333 (1973). https://doi.org/10.1103/PhysRevD.7.2333 ADSMathSciNetCrossRefGoogle Scholar
 9.Hawking, S.W.: Commun. Math. Phys. 43, 199 (1975) Erratum: [Commun. Math. Phys. 46, 206 (1976)]. https://doi.org/10.1007/BF02345020 ADSMathSciNetCrossRefGoogle Scholar
 10.Townsend, P.K.: arXiv:grqc/9707012
 11.Hollands, S., Ishibashi, A., Wald, R.M.: Commun. Math. Phys. 271, 699 (2007). https://doi.org/10.1007/s0022000702164. [arXiv:grqc/0605106]ADSCrossRefGoogle Scholar
 12.Moncrief, V., Isenberg, J.: Class. Quant. Grav. 25, 195015 (2008). https://doi.org/10.1088/02649381/25/19/195015. [arXiv:0805.1451 [grqc]]ADSCrossRefGoogle Scholar
 13.Kay, B.S., Wald, R.M.: Phys. Rep. 207, 49 (1991). https://doi.org/10.1016/03701573(91)90015E ADSMathSciNetCrossRefGoogle Scholar
 14.Foster, B.Z.: Phys. Rev. D 73, 024005 (2006). https://doi.org/10.1103/PhysRevD.73.024005. [arXiv:grqc/0509121]ADSMathSciNetCrossRefGoogle Scholar
 15.Smarr, L.: Phys. Rev. Lett. 30, 71 (1973) Erratum: [Phys. Rev. Lett. 30, 521 (1973)]. https://doi.org/10.1103/PhysRevLett.30.521, https://doi.org/10.1103/PhysRevLett.30.71
 16.Kastor, D.: Class. Quant. Grav. 25, 175007 (2008). https://doi.org/10.1088/02649381/25/17/175007. [arXiv:0804.1832 [hepth]]ADSCrossRefGoogle Scholar
 17.Liberati, S., Pacilio, C.: Phys. Rev. D 93, no. 8, 084044 (2016) https://doi.org/10.1103/PhysRevD.93.084044. [arXiv:1511.05446 [grqc]]
 18.Visser, M.: Phys. Rev. D 48, 583 (1993). https://doi.org/10.1103/PhysRevD.48.583. [arXiv:hepth/9303029]ADSMathSciNetCrossRefGoogle Scholar
 19.Jacobson, T., Myers, R.C.: Phys. Rev. Lett. 70, 3684 (1993). https://doi.org/10.1103/PhysRevLett. [arXiv:hepth/9305016]. 70.3684ADSMathSciNetCrossRefGoogle Scholar
 20.Visser, M.: Phys. Rev. D 48, 5697 (1993). https://doi.org/10.1103/PhysRevD.48.5697. [arXiv:hepth/9307194]ADSCrossRefGoogle Scholar
 21.Wald, R.M.: Phys. Rev. D 48, no. 8, R3427 (1993) https://doi.org/10.1103/PhysRevD.48.R3427.[arXiv:grqc/9307038]ADSMathSciNetCrossRefGoogle Scholar
 22.Iyer, V., Wald, R.M.: Phys. Rev. D 50, 846 (1994). https://doi.org/10.1103/PhysRevD.50.846. [arXiv:grqc/9403028]ADSMathSciNetCrossRefGoogle Scholar
 23.Deser, S., van Nieuwenhuizen, P.: Phys. Rev. D 10, 401 (1974). https://doi.org/10.1103/PhysRevD.10.401 ADSCrossRefGoogle Scholar
 24.Zwiebach, B.: Phys. Lett. 156B, 315 (1985). https://doi.org/10.1016/03702693(85)916168 ADSCrossRefGoogle Scholar
 25.Boulware, D.G., Deser, S.: Phys. Rev. Lett. 55, 2656 (1985). https://doi.org/10.1103/PhysRevLett.55.2656 ADSCrossRefGoogle Scholar
 26.Jacobson, T., Kang, G., Myers, R.C.: Phys. Rev. D 49, 6587 (1994). https://doi.org/10.1103/PhysRevD.49.6587. [arXiv:grqc/9312023]ADSMathSciNetCrossRefGoogle Scholar
 27.Hawking, S.W., Hartle, J.B.: Commun. Math. Phys. 27, 283 (1972). https://doi.org/10.1007/BF01645515 ADSCrossRefGoogle Scholar
 28.Wald, R.M.: Quantum field Theory in curved SpaceTime and black hole thermodynamics. In: Chicago Lectures in Physics, 1st edn. University of Chicago Press, Chicago (1994)Google Scholar
 29.Jacobson, T., Parentani, R.: Found. Phys. 33, 323 (2003). https://doi.org/10.1023/A:1023785123428. [arXiv:grqc/0302099]MathSciNetCrossRefGoogle Scholar
 30.Amsel, A.J., Marolf, D., Virmani, A.: Phys. Rev. D 77, 024011 (2008). https://doi.org/10.1103/PhysRevD.77.024011. [arXiv:0708.2738 [grqc]]ADSMathSciNetCrossRefGoogle Scholar
 31.Bhattacharjee, S., Sarkar, S.: Phys. Rev. D 91, no. 2, 024024 (2015) https://doi.org/10.1103/PhysRevD.91.024024. [arXiv:1412.1287 [grqc]]
 32.Mishra, A., Chakraborty, S., Ghosh, A., Sarkar, S.: JHEP 1809, 034 (2018). https://doi.org/10.1007/JHEP09(2018). [arXiv:1709.08925 [grqc]]. 034ADSCrossRefGoogle Scholar
 33.Jacobson, T., Kang, G., Myers, R.C.: Phys. Rev. D 52, 3518 (1995). https://doi.org/10.1103/PhysRevD.52.3518. [arXiv:grqc/9503020]ADSMathSciNetCrossRefGoogle Scholar
 34.Chatterjee, A., Sarkar, S.: Phys. Rev. Lett. 108, 091301 (2012). https://doi.org/10.1103/PhysRevLett. [arXiv:1111.3021 [grqc]]. 108.091301ADSCrossRefGoogle Scholar
 35.Kolekar, S., Padmanabhan, T., Sarkar, S.: Phys. Rev. D 86, 021501 (2012). https://doi.org/10.1103/PhysRevD.86.021501. [arXiv:1201.2947 [grqc]]ADSCrossRefGoogle Scholar
 36.Sarkar, S., Wall, A.C.: Phys. Rev. D 88, 044017 (2013). https://doi.org/10.1103/PhysRevD.88.044017. [arXiv:1306.1623 [grqc]]ADSCrossRefGoogle Scholar
 37.Vaidya, P.C.: Phys. Rev. 83, 10 (1951). https://doi.org/10.1103/PhysRev.83.10 ADSCrossRefGoogle Scholar
 38.Bhattacharjee, S., Sarkar, S., Wall, A.C.: Phys. Rev. D 92, no. 6, 064006 (2015). https://doi.org/10.1103/PhysRevD.92.064006. [arXiv:1504.04706 [grqc]]
 39.Bhattacharjee, S., Bhattacharyya, A., Sarkar, S., Sinha, A.: Phys. Rev. D 93, no. 10, 104045 (2016). https://doi.org/10.1103/PhysRevD.93.104045. [arXiv:1508.01658 [hepth]]
 40.Fairoos, C., Ghosh, A., Sarkar, S.: Phys. Rev. D 98, no. 2, 024036 (2018). https://doi.org/10.1103/PhysRevD.98.024036. [arXiv:1802.00177 [grqc]]
 41.Bhattacharyya, S., Hubeny, V.E., Loganayagam, R., Mandal, G., Minwalla, S., Morita, T., Rangamani, M., Reall, H.S.: JHEP 0806, 055 (2008). https://doi.org/10.1088/11266708/2008/06/055. [arXiv:0803.2526 [hepth]]ADSCrossRefGoogle Scholar
 42.Wald, R.M.: General Relativity. https://doi.org/10.7208/chicago/9780226870373.001.0001
 43.Ryu, S., Takayanagi, T.: Phys. Rev. Lett. 96, 181602 (2006). https://doi.org/10.1103/PhysRevLett. [arXiv:hepth/0603001]. 96.181602ADSMathSciNetCrossRefGoogle Scholar
 44.Casini, H., Huerta, M., Myers, R.C.: JHEP 1105, 036 (2011). https://doi.org/10.1007/JHEP05(2011). [arXiv:1102.0440 [hepth]]. 036ADSCrossRefGoogle Scholar
 45.Bhattacharyya, A., Sharma, M.: JHEP 1410, 130 (2014). https://doi.org/10.1007/JHEP10(2014). [arXiv:1405.3511 [hepth]]. 130ADSCrossRefGoogle Scholar
 46.Dong, X.: JHEP 1401, 044 (2014). https://doi.org/10.1007/JHEP01(2014). [arXiv:1310.5713 [hepth]]. 044ADSCrossRefGoogle Scholar
 47.Wall, A.C.: Int. J. Mod. Phys. D 24, no. 12, 1544014 (2015). https://doi.org/10.1142/S0218271815440149. [arXiv:1504.08040 [grqc]]ADSCrossRefGoogle Scholar
 48.Buchel, A., Myers, R.C., Sinha, A.: \(\text{ Beyond } \text{ eta }/\text{ s } = 1/4 \text{ pi },\). JHEP 0903, 084 (2009). [arXiv:0812.2521 [hepth]]ADSCrossRefGoogle Scholar
 49.Buchel, A., Myers, R.C.: Causality of holographic hydrodynamics. JHEP 0908, 016 (2009). [arXiv:0906.2922 [hepth]]ADSMathSciNetCrossRefGoogle Scholar
 50.Myers, R.C., Sinha, A.: Phys. Rev. D 82, 046006 (2010). [arXiv:1006.1263 [hepth]]ADSCrossRefGoogle Scholar
 51.Myers, R.C., Sinha, A.: JHEP 1101, 125 (2011). [arXiv:1011.5819 [hepth]]ADSCrossRefGoogle Scholar
 52.Hofman, D.M., Maldacena, J.: Conformal collider physics: energy and charge correlations. JHEP 0805, 012 (2008). [arXiv:0803.1467 [hepth]]ADSCrossRefGoogle Scholar
 53.Brigante, M., Liu, H., Myers, R.C., Shenker, S., Yaida, S.: The viscosity bound and causality violation. Phys. Rev. Lett. 100, 191601 (2008). [arXiv:0802.3318 [hepth]]ADSCrossRefGoogle Scholar
 54.Buchel, A., Escobedo, J., Myers, R.C., Paulos, M.F., Sinha, A., Smolkin, M.: Holographic GB gravity in arbitrary dimensions. JHEP 1003, 111 (2010). [arXiv:0911.4257 [hepth]]ADSCrossRefGoogle Scholar
 55.Camanho, X.O., Edelstein, J.D., Maldacena, J., Zhiboedov, A.: Causality constraints on corrections to the graviton threepoint coupling. arXiv:1407.5597 [hepth]
 56.Lu, H., Pope, C.N.: Critical gravity in four dimensions. Phys. Rev. Lett. 106, 181302 (2011). [arXiv:1101.1971 [hepth]]ADSCrossRefGoogle Scholar
 57.Bergshoeff, E.A., Hohm, O., Townsend, P.K.: Massive gravity in three dimensions. Phys. Rev. Lett. 102, 201301 (2009). [arXiv:0901.1766 [hepth]]ADSMathSciNetCrossRefGoogle Scholar
 58.Grumiller, D., Hohm, O.: AdS(3)/LCFT(2): correlators in new massive gravity. Phys. Lett. B 686, 264 (2010). [arXiv:0911.4274 [hepth]]ADSMathSciNetCrossRefGoogle Scholar
 59.Papallo, G., Reall, H.S.: JHEP 1511, 109 (2015). https://doi.org/10.1007/JHEP11(2015). [arXiv:1508.05303 [grqc]]. 109ADSCrossRefGoogle Scholar
 60.Liko, T.: Phys. Rev. D 77, 064004 (2008). https://doi.org/10.1103/PhysRevD.77.064004. [arXiv:0705.1518 [grqc]]ADSMathSciNetCrossRefGoogle Scholar
 61.Sarkar, S., Wall, A.C.: Phys. Rev. D 83, 124048 (2011). https://doi.org/10.1103/PhysRevD.83.124048. [arXiv:1011.4988 [grqc]]ADSCrossRefGoogle Scholar
 62.Chatterjee, S., Parikh, M.: Class. Quant. Grav. 31, 155007 (2014). https://doi.org/10.1088/02649381/31/15/155007. [arXiv:1312.1323 [hepth]]ADSCrossRefGoogle Scholar
 63.Jacobson, T.: Phys. Rev. Lett. 75, 1260 (1995). https://doi.org/10.1103/PhysRevLett. [arXiv:grqc/9504004]. 75.1260ADSMathSciNetCrossRefGoogle Scholar
 64.Padmanabhan, T.: AIP Conf. Proc. 1241, 93 (2010). https://doi.org/10.1063/1.3462738. [arXiv:0911.1403 [grqc]]ADSCrossRefGoogle Scholar
 65.Padmanabhan, T.: Rep. Prog. Phys. 73, 046901 (2010). https://doi.org/10.1088/00344885/73/4/046901. [arXiv:0911.5004 [grqc]]ADSCrossRefGoogle Scholar
 66.Chakraborty, S., Parattu, K., Padmanabhan, T.: JHEP 1510, 097 (2015). https://doi.org/10.1007/JHEP10(2015). [arXiv:1505.05297 [grqc]]. 097ADSCrossRefGoogle Scholar
 67.Padmanabhan, T.: Mod. Phys. Lett. A 30, no. 03n04, 1540007 (2015). https://doi.org/10.1142/S0217732315400076. [arXiv:1410.6285 [grqc]]ADSMathSciNetCrossRefGoogle Scholar
 68.Parikh, M., Sarkar, S., Svesko, A.: arXiv:1801.07306 [grqc]
 69.Allais, A., Tonni, E.: JHEP 1201, 102 (2012). https://doi.org/10.1007/JHEP01(2012). [arXiv:1110.1607 [hepth]]. 102ADSCrossRefGoogle Scholar
 70.Callan, R., He, J.Y., Headrick, M.: JHEP 1206, 081 (2012). https://doi.org/10.1007/JHEP06(2012). [arXiv:1204.2309 [hepth]]. 081ADSCrossRefGoogle Scholar
 71.Caceres, E., Kundu, A., Pedraza, J.F., Tangarife, W.: JHEP 1401, 084 (2014). https://doi.org/10.1007/JHEP01(2014). [arXiv:1304.3398 [hepth]]. 084ADSCrossRefGoogle Scholar