Abstract

A derivation of Noether current from the surface term of Einstein-Hilbert action is given. We show that the corresponding charge, calculated on the horizon, is related to the Bekenstein-Hawking entropy. Also using the charge, the same entropy is found based on the Virasoro algebra and Cardy formula approach. In this approach, the relevant diffeomorphisms are found by imposing a very simple physical argument: diffeomorphisms keep the horizon structure invariant. This complements similar earlier results (Majhi and Padmanabhan (2012)) (arXiv:1204.1422) obtained from York-Gibbons-Hawking surface term. Finally we discuss the technical simplicities and improvements over the earlier attempts and also various important physical implications.

1. Introduction

The thermodynamic properties of horizon arise from the combination of the general theory of relativity and the quantum field theory. This was first observed in the case of black holes [1, 2]. Now it is evident that it is much more general, and a local Rindler observer can attribute temperature and entropy to the null surfaces in the context of the emergent paradigm of gravity [3–6]. Such a generality might provide us a deeper insight into the quantum nature of the spacetime. So far several attempts have been made to know the microscopic origin of the entropy, but every method has its own merits and demerits. Among others, Carlip made an attempt [7, 8] in the context of Virasoro algebra to illuminate this aspect which is basically the generalisation of the method by Brown and Henneaux [9]. In brief, in this method one first defines a bracket among the Noether charges and calculate it for certain diffeomorphisms, chosen by some physical considerations. It turns out that the algebra is identical to the Virasoro algebra. The central charge and the zero mode eigenvalue of the Fourier modes of the charge are then automatically identified in which after substituting in the Cardy formula [10–12] one finds the Bekenstein-Hawking entropy. (For a complete list of works which lead to further development of this method, see [13–53].) In all the previous attempts, the Noether current was taken in relation to the Einstein-Hilbert (EH) action, and the analysis was on shell; that is, equation of motion has been used explicitly. Later an off shell analysis and a generalization to Lanczos-Lovelock gravity have been presented in [54].

Earlier [55], based on the Virasoro algebra approach, we showed that the entropy can also be obtained from the Noether current corresponding to the York-Gibbons-Hawking surface term. But it is not clear if the same can be achieved from the surface term of the Einstein-Hilbert action since they are not exactly identical. So it is necessary to investigate this issue in the light of Virasoro algebra context, particullary because both surface terms lead to the same entropy on the horizon. This will complement our earlier work [55].

In this paper we will use the Noether current associated with the surface term of EH action. Before going into the motivations for taking the surface term only, let us first highlight some peculiar facts of EH action which are essential for the present purpose. (i)It is an unavoidable fact that to obtain the equation of motion in the Lagrangian formalism one has to impose some extra prescription, like adding extra boundary term (in this case York-Gibbons-Hawking term). This is because the action contains second-order derivative of metric tensor . But unfortunately the choice of the surface term is not unique. This is quite different from other well-known field theories. (ii)The EH action can be separated into two terms: one contains the squires of the Christoffel connections (i.e., it is in structure) and the other one contains the total derivation of ( structure). We will call them and , respectively. Interestingly, Einstein’s equation of motion can be obtained solely from by using the usual variation principle where no additional prescription is required [56]. (iii)The most important one is that these two terms are related by an algebraic relation, usually known as holographic relation [57, 58].

Interestingly, all the previous features happened to be common even for the Lanczos-Lovelock theory [59]. For a recent review in this direction, see [60].

Although an extensive study on the Noether current of gravity has been done starting from Wald et al. [61–63], discussion on the current derived from is still lacking. To motivate why one should be interested, let us summarise the already observed facts as follows.(i)It is expected that the entropy is associated with the degrees of freedom around or on the relevant null surface rather than the bulk geometry of spacetime. (ii)This surface term calculated on the Rindler horizon gives exactly the Bekenstein-Hawking entropy [56]. (iii)Extremization of the surface term with respect to the diffeomorphism parameter whose norm is a constant leads to Einstein’s equation [57].(iv)Another interesting fact is that, in a small region around an event, EH action reduces to a pure surface term when evaluated in the Riemann normal coordinates.

All these indicate that either the bulk and the surface terms are duplicating all the information or the actual dynamics is stored in surface term rather than in bulk term. To illuminate more on this issue, one needs to study every aspect of the surface term.

In this paper, we will discuss the Noether realization of the surface term of the EH action; particularly we will examine if the Noether current represents the Virasoro algebra for a certain class of diffeomorphisms. This is necessary to have a deeper understanding of the role of the surface term in the gravity. Also it will give a further insight into the earlier claim: the actual information of the gravity is stored in the surface. To do this explicitly, we will consider the form of the metric close to the null surface in the local Rindler frame around some event. This is given by the Rindler metric. The reasons for choosing such metric are as follows. According to equivalence principle, gravity can be mimicked by an accelerated observer, and an uniformly accelerated frame will have Rindler metric. Apart from that, it is a relevant frame for an observer sitting very near to the black hole horizon. Hence any thermodynamic feature of the null surface can be attributed by this metric, and it provides a general description which was originally obtained only for the black hole horizon. Moreover, all the quantities will be observer dependent.

In this paper we will proceed as follows. First a detailed derivation of the Noether current for a diffeomorphism , corresponding to , will be given. This is important because it has not been done earlier, and therefore the properties of the current have not been explored. Here we will show that the corresponding charge , calculated on the null surface for to be Killing, yields exactly one-quarter of the horizon area after multiplying it by , where is the acceleration of the observer or the surface gravity in the case of a black hole. Next, a definition of the bracket among the charges will be given. This will be done by taking variation of the charge for another transformation . Finally, we need to calculate all these quantities for a particular diffeomorphism. To identify the relevant diffeomorphisms from which the algebra has to be constructed, following our earlier work [55], we use the criterion that the diffeomorphism should leave the near horizon form of the metric invariant in some nonsingular coordinate system. This will lead to a set of diffeomorphism vectors for which the Fourier components of the bracket among the charges will be exactly similar to Virasoro algebra. It is then very easy to identify the zero mode eigenvalue and the central extension. Substitution of all these values in the Cardy formula [10–12] will yield exactly the Bekenstein-Hawking entropy [1, 2]. A similar analysis was done in [53] based on the Noether current corresponding to [64–66]. In this calculation, to obtain the correct value of the entropy, a particular boundary condition (Dirichlet or Neumann) was used. But its physical significance is not well understood.

Before going into the main calculation, let us summarize the main features of the present analysis. (i)The first is the technical aspect. To obtain the correct entropy, in most of the earlier works, one had to either shift the zero mode eigenvalue [8] or choose a parameter contained in the Fourier modes of as the surface gravity [53] or both [54]. Here we will show that none of the ad hoc prescriptions will be required. (ii)The important one is the simplicity of the criterion (near horizon structure of the metric remains invariant in some nonsingular coordinate system) to find the relevant diffeomorphisms for which we obtain the Virasoro algebra. This was first introduced by us [55] in this context. The significance of this choice is that the full set of diffeomorphism symmetry of the theory is now reduced to a subset which respects the existence of horizon in a given coordinate system. Hence it may happen that some of the original gauge degrees of freedom (which could have been eliminated by certain diffeomorphisms which are now disallowed) now being effectively upgraded to physical degrees of freedom as far as a particular class of observers are concerned. So all the thermodynamic quantities, attributed to the horizon, become observer dependent. (iii)In the present analysis we will not need any use of boundary condition like Dirichlet or Neumann to obtain the exact form of the entropy. (iv)Since our analysis will be completely based on , where no information about is needed, it will definitely illuminate the emergent paradigm of gravity, particularly the holographic aspects in the action.

We will discuss later more on different aspects and significance of our results.

The organization of the paper is as follows. In Section 2, the derivation of the Noether current for the will be presented explicitly. Next we will give the definition of the bracket among the charges and the relevant diffeomorphisms based on the invariance of horizon structure criterion. Section 4 will be devoted to show that the Fourier mode of the bracket is exactly like the Virasoro algebra which by the Cardy formula will lead to Bekenstein-Hawking entropy. Finally, we will conclude.

2. Derivation of Noether Current from the Surface Term of Einstein-Hilbert Action

In this section, a detailed derivation of the Noether current and the potential corresponding to the surface term of EH action will be presented. Then we will calculate the charge on the Rindler horizon.

The Lagrangian corresponding to the surface term is given by [56] where Here the normalization is omitted and it will be inserted where necessary. Now our task is to find the variations of both sides of (1) for a diffeomorphism and then equate them. The variation we will consider here is the Lie variation which is defined, in general, as where and and are evaluated in two different coordinate systems and , respectively. In the following, for the notational simplicity, we will denote as .

The variation of the right-hand side of (1) is given by Since is a tensor, for the Lie variation, is expressed by the Lie derivative and is given by Therefore, On the other hand, since is not a tensor, its variation cannot be expressed by simple Lie derivative. To find we will use the general definition (3). Let us first calculate . Under the change we have Here we considered infinitesimal change, and so the terms from have been ignored. This will be followed in later analysis. Hence, Substitution of these in leads to Another one is given by Therefore, according to (3), the Lie variation of due to the diffeomorphism is where Substituting this in (6) we obtain the variation of right-hand side of (1) as

Next we find the variation of left-hand side of (1), that is, . For this we will start from the following relation: where with . Since is a scalar, by the definition of Lie derivative, . Therefore using (5) we find To find , we will proceed as earlier. Under the change , is calculated as where (8) has been used. This can be expressed in terms of in the following way. The second term on the right-hand side can be expressed in the following form: where in the above we used . The third term of (17) reduces to Similarly, the last term can be expressed as where in the last line has been used. Substituting all these in (17) we obtain On the other hand, Hence, Substituting this in (16) we obtain Now equating (13) and (24) we obtain , where the conserved Noether current is given by Finally, using in the above, we can express the current as the divergence of an antisymmetric two-index quantity: It is evident that the anti-symmetric object is not a tensor and it is usually called the Noether potential. Therefore, inserting the proper normalization, the charge is given by where is the surface element of the 2-dimensional surface and is the determinant of the corresponding metric. Since our present discussion will be near the horizon, we choose the unit normals and as spacelike and timelike, respectively.

Now we will calculate charge (27) explicitly on the horizon. This will be done by considering the form of the metric near the horizon: where represents the transverse coordinates. The metric has a timelike Killing vector and the Killing horizon is given by ; that is, . The nonzero Christoffel connections are For metric (28) we find and hence . Also, (2) yields Therefore, Now if is a Killing vector, then , and so calculating charge (27) explicitly we find where is the horizon cross-section area. Multiplying it by the periodicity of time coordinate we obtain exactly the entropy: one-quarter of horizon area. Moreover, the above can be expressed as , where is the temperature of the horizon and is the entropy. Therefore one can call it the Noether energy. Such interpretaion was done earlier in [67, 68].

So far we found that the Noether charge corresponding to the surface term of EH action alone led to the entropy of the Rindler horizon. This was shown earlier for the charge coming from the total EH action [61–63]. Therefore, the present analysis revealed that it may be possible that the information is actually encoded in the surface term rather than the bulk term. Then the natural question arises: what are the degrees of freedom responsible for this entropy? So far they are not known. In the next couple of sections we will give an idea on the nature of the possible degrees of freedom in the context of Virasoro algebra and Cardy formula.

3. Bracket among the Charges and the Diffeomorphism Generators

In the previous section, we have given the expression for the charge (see (27)) for an arbitrary diffeomorphism. Here we will define the bracket among the charges. The relevant diffeomorphisms will be chosen by imposing a minimum condition on the spacetime metric. The charge and the bracket will be then expressed in terms of these generators.

We will find the bracket following our earlier works [54, 55]. For this let us first calculate the following: Using and in addition the expression for , given by (2), we obtain For the present metric (28), , , and hence . Therefore Finally we define a bracket as which for the present metric (28) reduces to

To calculate the above bracket we need to know about the generators . We will determine them by using the condition that the horizon structure remains invariant in some nonsingular coordinate system. For that let us first express metric (28) in Gaussian null (or Bondi like) coordinates: In these coordinates the metric reduces to the following form: Now impose the condition that the metric coefficients and do not change under the diffeomorphism; that is, where is the Lie derivative along the vector . These lead to The solutions are The condition is automatically satisfied near the horizon, because use of the above solutions leads to . These conditions appeared earlier in [69] in the context of late time symmetry near the black hole horizon. Finally expressing (44) in the old coordinates () we find where .

Next we calculate from (37) for the present case. Since , we find

Now since the integration (39) will ultimately be evaluated on the horizon, we will find the value of each term of the above very near to the horizon. Therefore, using (29), (31), and the form of the generators (45) we obtain the values of each term of the above expression near the horizon as So near the horizon (46) reduces to Substituting this in (39) and inserting the normalization factor, we obtain the expression for the bracket Similarly, (27) yields

A couple of comments are in order. It must be noted that, in finding the expression for bracket (49), no use of boundary conditions (Dirichlet or Neumann) has been used. Earlier this was used for the case of to throw away the noncovariant terms in the bracket without giving any physical meaning [53]. Also, we did not use the condition (see (34)) which was adopted in earlier works. For instance, see [8, 53]. This is logically correct since contradicts the algebra among the Fourier modes of the diffeomorphisms (see (52), in next section).

4. Virasoro Algebra and Entropy

In this section, the Fourier modes of the bracket and the charge will be found out. We will show that for a particular ansatz for the Fourier modes of the generators will lead to the Virasoro algebra. Finally using the Cardy formula, the entropy will be calculated.

Consider the Fourier decompositions of and : where and . The Fourier modes will be chosen such that the Fourier modes of the diffeomorphisms (44) obey one subalgebra isomorphic to Diff. : where is the Lie bracket. Now with the use of (51), let us first find the Fourier modes of bracket (49) and charge (50). Substitution of (51) in (49) yields where and so . Next defining the Fourier modes of as we find Similarly from (50), the Fourier modes of the charge are given by where . It must be noted that the present expression (56) is exactly identical to that obtained in [55] for the York-Gibbons-Hawking surface term, whereas the other expression (55) is different by a total derivative term. This may be because these two surface terms are not exactly the same. But we will show that final result for the bracket is identical to the earlier analysis, because the total derivative term will not contribute to an ansatz for .