Anomalous effective action, Noether current, Virasoro algebra and Horizon entropy

Several investigations show that in a very small length scale there exists corrections to the entropy of black hole horizon. Due to fluctuations of the background metric and the external fields the action incorporates corrections. In the low energy regime, the one loop effective action in four dimensions leads to trace anomaly. We start from the Noether current corresponding to the Einstein-Hilbert plus the one loop effective action to calculate the charge for the diffeomorphisms which preserve the Killing horizon structure. Then a bracket among the charges is calculated. We show that the Fourier modes of the bracket is exactly similar to Virasoro algebra. Then using Cardy formula the entropy is evaluated. Finally, the explicit terms of the entropy expression is calculated for a classical background. It turns out that the usual expression for entropy; i.e. the Bekenstein-Hawking form, is not modified.


I. INTRODUCTION AND MOTIVATION
One of the striking features of general relativity is its deep connection with the laws of thermodynamics. A "marriage" between general theory of relativity (GR) and quantum mechanics shows that black holes behave as thermodynamic objects with a Hawking temperature T H = κ/2π and intrinsic entropy given by Bekenstein-Hawking area law S = A h /4G, where κ is the surface gravity and A h is the area of the horizon [1] (for recent reviews see [2]). It should be noted that the interpretation of classical black hole laws with that of black hole thermodynamics become possible due to taking into account the quantum mechanical effects, leading to Hawking radiation. Also we could readily obtain the same area dependence of entropy and that leads to a kind of universality to the above result.
Although the thermodynamic interpretations were first observed in black hole solutions, such features are much more general. It has been observed that the accelerated observer on a Minkowski spacetime also associates temperature and entropy on the Rindler horizon. This has been originated from the Unruh effect [3]. This leads us to think that the static observer (black hole) and the Rindler observer (accelerated frame) are in the same footings. The reality is much more general. In a local region one can always has a null surface which is not a solution of the Einstein's equation and on that a entropy functional can be associated such that evaluation of it on the null surface leads to the entropy expression. Moreover, the extremization of it leads to the Einstein's equations. This is very important in the context of emergent paradigm of gravity [4]. Such a local concept implies that the notion of entropy is much more fundamental and it must be observer dependent quantity. Hence any analysis leading to it must have an observer dependent off-shell description.
For more general class of theories including higher curvature terms in the action the Bekenstein-Hawking area law no longer applies. However, assuming the Zeroth law, the first law can be derived in these class of covariant actions [5]. Here the Noether charge corresponding to the diffeomorphism symmetry, calculated over the spatial cross-sections of the horizon, plays the role of entropy. This mode of thinking has two important aspects, which are as follows: • These results implies that the connection between gravity and thermodynamics is valid well beyond Einstein theory of gravity. It only requires some general argument, like covariance and principle of equivalence [36].
• It is also possible to attribute an observer dependent entropy to any null surfaces (need not be event horizons).
So far the results are well valid in the classical regime. In a very small length scale, like the Plank length, the quantum effects has to be incorporated, particularly since the fluctuations of the metric and the external fields can not be ignored. Then the classical results have to be modified. One way to include these effects is to find the effective action corresponding to the fluctuations. It modifies the partition function and hence the entropy [7]. It turns out that in GR the leading order correction to entropy is logarithmic of the horizon area [8] and the coefficient of it is related to the trace anomaly. Similar results has been achieved in several methods (see for example [9], [10], [11]) but the value of the coefficient depends on the particular model for calculating the entropy. On the other hand, the entanglement of quantum fields between inside and outside of horizon approach leads to power law corrections [12]. Hence it is evident that existence of corrections is universal and one would expect identical situation in a method of computing entropy.
One of the interesting ways to understand this universality is by Noether charge prescription. This will be done for the anomalous effective action due to the fluctuations of the quantum fields which leads to the trace anomaly in four dimensions. The one loop effective action can be determined by the effective field theory technique. In the low energy limit, if one breaks the conformal symmetry the resulting theory leads to trace anomaly [13] - [16]. In [17], an attempt has been done by evaluating the Noether charge corresponding to the GR action plus this one loop effective action for a Killing vector. It must be noted that the charge was derived by using equation of motion and so the analysis is on-shell. Here we will present a completely off-shell analysis.
So far no attempts have been done to quantify the degrees of freedom which are responsible to such universality. This issue we will discuss in the context of Virasoro algebra and Cardy formula. The method was first introduced in the context of gravity by Brown and Hannueax [18] and later developed by Carlip [19]. Although the method has been followed up in several gravity theories [20] - [23], no discussion exists in the presence of the one loop effective action which leads to the trace anomaly. Here we will fill up this blank. Our analysis will be followed from a recent work of one of the authors [23]. The key features of the calculation are: (i) The whole analysis will be off-shell;, i.e. no equation of motion will be used. (ii) The derivation of the current corresponding to the total action (i.e. Einstein-Hilbert part plus the one loop effective action) is off-shell and it is off-shell conserved. These are essential since, as we have argued earlier, the notion of entropy has general sense beyond the black hole horizon.
Let us first summarize the methodology. The main step is to define a bracket among the charges. In a previous work of one of the authors [23], an off-shell definition of the bracket has been given for any general covariant Lagrangian in terms of the arbitrary diffeomorphism vector and Noether current. Here we will use this definition where explicit form of the current will be taken for the present theory. To evaluate it the diffeomorphism vectors will be chosen using Carlip's formalism [19]. This essentially tells that the vectors are chosen in such a way that they are asymptotically Killing vector near the horizon so that the asymptotic killing horizon remains invariant. It turns out that the Fourier modes of the bracket is similar to the standard form of the Virasoro algebra which has central extension. Then the central charge and the zero mode eigenvalue are automatically identified. Substituting these in Cardy formula [24] we obtain the expression for entropy. We will show that the entropy will incorporate no correction for a classical background to the usual Bekenstein-Hawking expression. Physically this analysis tells that some of the degrees of freedom (DOF), which were originally gauge DOF, raise to true DOF which leads to entropy. Since this is happening due to imposition of particular condition on the diffeomorphisms, the DOF responsible for the entropy are observer dependent. This has been elaborated earlier in more details in [21,22].
The organization of our paper is as follows. In section II we present an analysis for a general covariant Lagrangian. First the form of the Noether charge for a general covariant Lagrangian is given. Next we give the expressions for the Fourier modes of the charge and the central term corresponding to the diffeomorphisms which keep the Killing horizon structure invariant and are asymptotically Killing vector near the horizon. Section III is devoted to calculate these quantities explicitly for the anomalous effective action. We then find the entropy using Cardy formula. Finally, we conclude in section IV. For completeness, an appendix has been given at the end of the paper.

II. VIRASORO ALGEBRA AND CENTRAL TERM: A GENERAL APPROACH
The recent progress in black hole thermodynamics shows that the microscopic features may not be too sensitive to the details of quantum gravity as the derivations for black hole temperature and entropy only uses semiclassical gravity. In order to achieve this it was first pointed out by Strominger that quantum state calculation should rely on symmetry [25]. In an earlier work, Brown and Henneaux [18] showed that (2 + 1) dimensional gravity has a asymptotic symmetry consisting of Virasoro algebra with central extension, implying that any microscopic quantum theory should be a conformal field theory. Strominger noticed that use of the above central charge in the Cardy formula [24] leads to the usual Bekenstein-Hawking entropy. This implies that the asymptotic symmetries might shed some light to the calculation of the density of states leading to the statistical description of entropy. The limitations of the earlier works are that they use the asymptotic AdS symmetry which is insensitive to the structure of interior spacetime; i.e. the details of the horizon. Moreover the method is confined to (2+1)-dimensions. Later on, Carlip introduced an approach which relies on the Killing horizon structure preserving diffeomorphisms and is independent of dimension of the spacetime [19]. So it reflects the information of the horizon which is important because the entropy depends on the horizon structure. In this paper, we shall adopt this approach.
In this section, we shall present the general expressions for the Fourier modes of Noether charge Q and the central term corresponding to a generally covariant Lagrangian. This method not only applies to Einstein gravity but also to any theories of gravity whose Lagrangian is a function of metric, curvature tensor and some other external fields, like scalar field. The technique we shall follow from [19,23].
Let us first introduce the general expressions for the Noether conserved current J a and the charge Q corresponding to a generally covariant Lagrangian. The explicit variation of this general Lagrangian for the metric variation leads to the following form, where E ab = 0 leads to equation of motion and δv a is the surface term. Now if the variation is given by the Lie derivative due to the coordinate transformation x a → x a + ξ a , then δg ab ≡ £ ξ g ab = ∇ a ξ b + ∇ b ξ a . Therefore, using the generalized Bianchi identity ∇ a E ab = 0 [4], the first term on the right hand side of Eq. (1) can be casted to a total derivative form −2 √ −g∇ a (E a b ξ b ). On the other hand, since L grav √ −g is a scalar density, the Lie variation is given by √ −g∇ a (L grav ξ a ). Using all these in Equation (1) we obtain a conservation relation given by ∇ a J a = 0, where Here £ ξ v a represents boundary term arising from the Lie variation of the metric as stated earlier. The current J a is called Noether current. Since the current is covariantly conserved, it can be expressed as the covariant derivative of an antisymmetric tensor: J a = ∇ b J ab , where J ab is known as the Noether potential. For the general class of covariant gravitational theories, substituting the respective values in (2) one can obtain the explicit expression. This leads to the current and potential as (See, Page 394 of [26], for details): where four index tensor P abcd is defined as P abcd = ∂L grav /∂R abcd . This tensor has identical symmetry properties as that of the curvature tensor R abcd i.e. antisymmetric in interchange of a, b and c, d, symmetric in interchange of pairs (a, b) and (c, d) along with P a(bcd) = 0. The corresponding Noether charge is defined as: where dΣ ab corresponds to the surface element. For instance, using Eq. (4) one can show that the explicit expression for Noether potential corresponding to the Einstein gravity appears to be, J ab = (1/16πG) ∇ a ξ b − ∇ b ξ a . Now using this in Eq. (5) for a timelike Killing vector field ξ a = χ a , Noether charge Q, calculated on the horizon, can be shown to be related to the horizon entropy. More specifically, Q multiplied by the periodicity of the Euclidean time leads to Bekenstein-Hawking entropy expression: (2π/κ)Q = A h /4G, with κ being the surface gravity. The above procedure has also been extended for Lanczos-Lovelock models to get black hole entropy as Noether charge [23]. However note that in all these models we have ∇ a P abcd = 0. Having set the stage we shall now deal with the general expressions (3) and (4) to consider the situation where ∇ a P abcd = 0. Next we give the expressions for the Fourier modes of the charge and the central extension. Applying the method of Carlip [19] for stretched horizon scenario and using the definition of bracket among the charges, given in [23], we find the Fourier modes of the charge as, (6) while the Fourier modes for the central term comes out to be: The detailed derivation of the above quantities can be followed from [23]. But for the completeness and clarity, we have shown this in the appendix A. We will use these expressions to identify the zero mode eigenvalue and the central charge to calculate the entropy. To evaluate the above, the explicit expression for T m is needed. This will be chosen subject to the condition that the Fourier modes for the diffeomorphism vector ξ a obey the following subalgebra isomorphic to Diff S 1 , with { , } being the Lie bracket. Keeping this condition in mind, the form for T m can be taken as, where α being a constant and g(x) is a function which is regular at the Killing Horizon. Here p is an integer and x ⊥ is the (d− 2) dimensional transverse co-ordinates with t− x plane defining the null surface. The explicit expression Equation (9) for T m will be needed later to evaluate the charge and the central term. However, as we will see later, this will involve the choice for the parameter α. To obtain the correct expression for the entropy one has to consider the periodicity of the Euclidean time 2π/κ. Then for periodicity in time coordinate of Eq. (9), one must have α = κ. This will be needed at the end of the paper. For Lanczos-Lovelock gravity ∇ c P abcd vanishes and so the above expressions reduce to those obtained earlier in [23]. Here the results are much more general and can be applied for the covariant gravity theory for which the covariant derivative of P abcd does not vanish. Moreover, the expressions are off-shell and free of any ambiguity, as mentioned earlier.
Having obtained these general results we shall now apply these results for anomaly induced effective action arising from the fluctuations in the matter fields and background metric. The purpose of the next section is to calculate each terms of Eq. (6) and Eq. (7) for the anomalous effective action and use them in the Cardy formula to derive the corrections to the usual form of the entropy. The analysis will be done for the case where the four dimensional trace anomaly appears. The basic idea is that trace anomaly, being independent of renormalization schemes and quantum states, it has some effects on quantum correction to black hole area law. Furthermore, the explicit expression for the anomaly can be derived from the anomaly-induced effective action. Hence it would be interesting to find the Noether charge and then use it to find the entropy. Actually, to use Eq. (6) and Eq. (7), we need to find the explicit expression for P abcd for the action on which we are interested. Then substitution of it will lead to the final result.

III. ANOMALOUS EFFECTIVE ACTION AND ENTROPY
In this section we shall briefly review the anomalous effective action for the fluctuations of the fields and consequently the Noether charges are obtained thereof. The classical General Relativity does not take into account the microscopic effects of quantum matter on any scale. In order to discuss the wave like nature of a particle or phase correlation we need at least a semi classical treatment of effective stress energy tensor appearing in Einstein's equation. The one-loop effective action could be determined by the method called Effective Field Theory (EFT). It can be shown that non-local macroscopic coherence effects are actually self contained in low energy EFT provided the one-loop trace anomaly for massless fields are included in Einstein Theory [16]. The low energy EFT of gravity contains an expansion in derivatives of local term, while the higher order terms are suppressed by inverse ultraviolet cutoff scale M . Also the theory is not renormalizable, however this is not sensitive to all the microscopic details due to decoupling of short distance degrees of freedom [13]. When we broke a classical symmetry by quantum trace anomaly, the decoupling of short and long distance physical situations using standard technique fails. Thus an anomaly can have significant effect on low energy EFT. The necessity of trace anomaly for low energy EFT of gravity can be understood from the behavior of different terms in effective gravitational action under global Weyl rescaling. Thus addition of anomaly terms to low energy effective gravity action is consistent with both quantum theory and Equivalence principle [30].
This search for quantum corrections to black hole entropy takes place in the context of quantum field theory in curved spacetime [31]. The backreaction of quantum fields on curved spacetime is determined by the semiclassical Einstein equation G ab + Λg ab = 8π T ab (10) such that the quantum fields affect the curved background through the expectation value of the energymomentum tensor. This expectation value can be obtained using one loop correction due to the fluctuations in the quantum fields to the classical action and then taking the variation with respect to the metric. The trace of it is then given by ( [15], [16]), In four dimension there exists three geometric contribution to anomaly induced action, namely the Euler density (called Type-A), the Weyl tensor squared (called Type-B) and a R term that comes from variation of R 2 . Here we will pay attention only to type-A anomaly, since it is interesting in various ways for the study of black holes [30] and also it is simple compared to other types. This type of anomaly leads to trace anomaly: where E 4 = R abcd R abcd − 4R ab R ab + R 2 . The corresponding low energy effective action is given by, where the anomalous part of the action is of the form S anom = S 0 + S 1 + S 2 + S 3 with [17], The scalar field φ satisfies the following equation of motion: Note that S 0 does not contain any curvature part and so for that part derivative with respect to R abcd vanishes. Thus there is no Noether charge corresponding to this part. The standard way to get trace anomaly, as first pointed out by Polyakov [14], is to find the conformal primitive. In four dimensions we need two generalizations, firstly the conformal properties of Gaussian curvature where D'Alembertian are determined by the Q-curvature defined as E 4 − 2 3 R [32] and secondly the Paneitz operator [33] should be introduced (see also [34] and [35]).
The entropy tensors can be obtained from the above actions by the usual prescription. These are given by Along with the usual Noether charge due to Einstein-Hilbert action we have three extra charges which are obtained by substitution of the respective entropy tensors in Eq. (4) and using the definition Eq. (5) for the charge . These turn out to be, Note that these are identical to those obtained in [17] except the terms containing the covariant derivative of P abcd . However, it can be shown that these terms behave as O(χ 2 ) near the horizon for ξ a , chosen to be a timelike Killing vector χ a . For instance, let us consider P abcd 2 , presented in Eq. (20). Then ∇ c P abcd = −(a/96π 2 G) g ac g bd − g ad g bc ∇ c φ as covariant derivative of metric tensor vanishes. Now to evaluate this term explicitly, for simplicity we take a general spherically symmteric metric of the form ds 2 = −f (r)dt 2 + dr 2 /f (r) + r 2 dΩ 2 . Then one can easily show that the relevant quantity µ ab χ d ∇ c P abcd is O(f ) since the timelike Killing vector in this case is given by χ a = (1, 0, 0, 0). Similar conclusions hold for other parts also. Hence the terms, containing the covariant derivative of entropy tensor, in the above equations are all of O(χ 2 ) as χ 2 = −f . Wald [5] has first introduced the idea that entropy could be calculated from the Noether charge by choosing the diffeomorphism vector ξ a as the timelike Killing vector χ a . In the original formulation we have two spacetime boundaries, the asymptotic region and the horizon. The Noether charges on the asymptotic region determines mass and angular momentum, while that on the horizon determines the entropy. The boundary conditions are very important for calculation of Noether charge for they can get modified by modification of the boundary terms. Remember that to determine entropy, we need to calculate the charges (22), (23), (24) and (25) on the horizon defined by the relation g ab χ a χ b = 0. So the O(χ 2 ) terms do not contribute and hence one can drop those terms.
Following the similar arguments as above, it is also possible to show that the terms containing the covariant derivative of P abcd in Eq. (6) and Eq. (7) are of the order χ 2 . So they will not contribute near the horizon. Taking into account this fact and following the identical steps, employed in [23], we obtain from Eq. (6) and Eq. (7) as, where we have introduced the Wald entropy function quantity, which leads to the horizon area in GR. To achieve Eq. (26) and Eq. (27), Eq. (9) has been used and then the integration is done over the transverse coordinates. Collecting all these it is possible to find the following form, which is in the form of Virasoro algebra with central extension. C is known as the central charge and is given by The zero mode eigenvalue from Equation (26) turns out to be, Therefore, we obtain the central charges and zero mode eigenvalues from Eq. (30) and Eq. (31) for each term of the action as, where the respective terms due to S 1 vanishes. To obtain the above expressions the explicit forms of P abcd , given by (19), (20) and (21), have been used in (28). χ is the two dimensional Euler characteristics of the horizon and φ r h is the value of φ at the horizon. Also in the above expression A h is the value of the quantity defined in equation (28) with P abcd = 1 2 g ac g bd − g ad g bc . Now substituting all these in the standard Cardy formula [24], we find the expressions for entropy: where we have used the identification α = κ. The reason has been explained earlier below Eq. (7). This can also be followed from [23]. Hence the total expression for entropy corresponding to the anomaly induced action would be given by, The same expression was also obtained in [17] by evaluating the Noether charge for a Killing vector on the horizon. The charge obtained by the authors of [17] is on-shell. Here we derived it from the Virasoro algebra technique. The analysis, adopted here, is totally off-shell.
The remarkable fact about the above expression is, in contrary to [17], that the last two terms yield an additive constant to the entropy which can be neglected. This can be shown as follows. For simplicity, let us consider the classical background as the Schwarzschild metric. To obtain the explicit expression for φ in terms of metric coefficients, we need to solve Equation (18). The solution will be found out by imposing certain boundary conditions on the scalar field φ. Since our only concern is around the horizon we would require the finiteness of φ and its first derivative at the horizon. Such conditions lead the following solution [16]: where dilog(x) = x 1 ln (t) 1−t dt. Integrating the above we find, Note an important point that any constant term can shift the value of φ at horizon. But the respective contribution to the Noether charge and hence entropy would then vanish since the action reduces to a topological term, which is the integral of Euler density. Now it is easy to see that the term φ r h leads to a constant contribution and so the last term does not affect the Bekenstein-Hawking entropy. However dφ dr leads to a term which is proportional to inverse of square root of horizon area. This would get compensated by the r h term. Hence the final expression for entropy leads to usual expression of entropy. Therefore, from the above discussion, one can conclude that the type-A trace anomaly does not lead to any correction to entropy.

IV. CONCLUSIONS
The idea that horizon entropy can be obtained from certain class of diffeomorphism generators is pioneered mainly by the works of Brown, Henneaux [18] and Carlip [19]. However all these methods use on-shell criteria i.e. equations of motion have been used at one stage or another. Recently, one of authors of this paper has derived horizon entropy using Virasoro algebra and Cardy formula bypassing the use of equation of motion [23]. In the present work, as well we have given an off-shell description i.e. no equation of motion has been used. This again illustrates that the notion of entropy is beyond black hole horizon. We have started from the expressions for the Noether current and potential for an arbitrary generally covariant Lagrangian which could be function of metric tensor, curvature tensor and scalar combination of different fields. From these, using the off-shell definition of bracket among the charges [23], a general expression for the central term was derived. We then obtained the Fourier modes of the charge and the central term.
Next we have evaluated the general expressions explicitly for the anomaly induced action which produces the type-A trace anomaly. This action has a pure geometric meaning except the Euler characteristics and couplings appears in it. For this particular action we have shown that the terms containing derivatives of P abcd vanishes in the near horizon limit and the most dominating term is the one that appears in Lanczos-Lovelock models. It has been shown that the Fourier modes of the bracket is similar to the usual Virasoro algebra with central extension. Identifying the central charge and the zero mode eigenvalue and then using them in the Cardy formula we obtained the expression for the entropy. The result is identical to that obtained in [17]. Considering the classical background as the Schwarzschild metric, it has been shown that the entropy of the black hole is A h /4G, the usual result. Hence type-A trace anomaly does not lead to any correction to horizon entropy.
So far, we have observed that there is no correction term in the entropy. Our present analysis was based on the type-A trace anomaly. As we mentioned earlier, there also exists other types of trace anomalies, like type-B etc. It would be interesting to study the effective actions corresponding to these anomalies in the present direction and investigate if they lead to any correction to entropy. This we leave for future.
To evaluate the charge and the bracket we also need three identities which are valid upto O(χ 2 ). The detailed derivations are given in [23] (see, Eq. (B37), Eq. (B43) and Eq. (B38)). The identities are We shall now use the above expressions to find the charge and the bracket. For a covariant Lagrangian, we define the bracket among the charges as ] has been used. The above expression is off-shell and free of any ambiguity. The only fact needed is that, the Noether current can be expressed as the covariant derivative of a antisymmetric tensor (see Ref. [23] for more details). To evaluate it for the present setup, we first calculate the current using Eq. (3). It can be shown that the current, presented in equation (3), takes the following form: In the intermediate steps, the identities (A5), (A6) and (A7) have been used. Then using Eq. (A2) and Eq. (A4) we obtain, where the relation P abcd µ ab µ cd = 4 ρ 2 χ 2 P abcd ρ a χ b ρ c χ d has been used. Finally, substitution of Eq. (A10) in Eq. (A8) leads to, Next we shall obtain the expression for the charge Q[ξ] in the near horizon limit. The charge is given by Eq. (5). Use of Eq. (A2) and Eq. (A7) in Eq. (5) leads to Now the central term will be calculated. This is defined as where the quantity [Q 1 , Q 2 ] given in Eq. (A11) and Q[{ξ 1 , ξ 2 }] will be evaluated using (A12) in the following way. Using Eq. (A2), the Lie bracket near horizon turns out to be, where we have {T 1 , T 2 } = (T 1 DT 2 − T 2 DT 1 ) and so on. Substituting this in Eq. (A12) we find, Finally, substitution of Eq. (A11) and Eq. (A15) in Eq. (A13) yields the form of the central term as, Now the Fourier modes of the Noether charge and the central term will be evaluated. For this we need to first give the Fourier modes of the arbitrary function T . The Fourier decomposition of the functions T 1 and T 2 are taken as, with C m,n ≡ A m B n and so C * m,n ≡ C −m,−n . This manipulation in turn leads to the Fourier decomposition of the central term as presented by Eq. (7).