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:

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 [

Earlier [

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.

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

The EH action can be separated into two terms: one contains the squires of the Christoffel connections (i.e., it is in

The most important one is that these two terms are related by an algebraic relation, usually known as

Interestingly, all the previous features happened to be common even for the Lanczos-Lovelock theory [

Although an extensive study on the Noether current of gravity has been done starting from Wald et al. [

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.

This surface term calculated on the Rindler horizon gives exactly the Bekenstein-Hawking entropy [

Extremization of the surface term with respect to the diffeomorphism parameter whose norm is a constant leads to Einstein’s equation [

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:

In this paper we will proceed as follows. First a detailed derivation of the Noether current for a diffeomorphism

Before going into the main calculation, let us summarize the main features of the present analysis.

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 [

The important one is the simplicity of the criterion (

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.

Since our analysis will be completely based on

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

The organization of the paper is as follows. In Section

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 [

The variation of the right-hand side of (

Next we find the variation of left-hand side of (

Now we will calculate charge (

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 [

In the previous section, we have given the expression for the charge (see (

We will find the bracket following our earlier works [

To calculate the above bracket we need to know about the generators

Next we calculate

Now since the integration (

A couple of comments are in order. It must be noted that, in finding the expression for bracket (

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

To calculate the previous expressions (

It has already been observed that several interesting features and pieces of information can be obtained from the surface term without incorporating the bulk term of the gravity action. In this paper we studied the surface term of the Einstein-Hilbert (EH) action in the context of Noether current. So far we know that this has not been attempted before. First the current was derived for an arbitrary diffeomorphism by using Noether prescription. Then we showed that the charge evaluated on the horizon for a Killing vector led to the Bekenstein-Hawking entropy after multiplying it by

In this paper, we defined the bracket among the charges. It was done, in the sprite of our earlier works [

Let us now discuss in details what we have achieved in this paper. We first tabulate a couple of technical points.

To obtain the exact expression for entropy we did not need any hand waving prescriptions, like shifting of the value of the zero mode eigenvalue or the specific choice of the value of the parameter

The relevant diffeomorphisms for invariance of the horizon structure can be obtained by various ways. Here our idea was to impose minimum constraints so that the bracket led to Virasoro algebra. It is also possible to have other choices of constraints to find the vectors

Finally, we discuss several conceptual aspects. The analysis presents a nice connection between the horizon entropy and the degrees of freedom which are responsible for it. In the usual cases, one always find that the concepts of degrees of freedom and entropy are absolute. These do not have any observer-dependent description. But in the case of gravity, as we know, the notion of temperature and entropy is observer dependent, and hence one can expect that the degrees of freedom may not be absolute. Here we showed that a certain class of observers which can see the horizon and keep the horizon structure invariant always attribute entropy. This signifies the fact that, among all the diffeomorphisms, some of them upgraded to real degrees of freedom which were originally gauge degrees of freedom and they have observer-dependent notion. Also, everything we achieved here was done from surface term. This again illustrates the holographic nature of the gravity actions—

The author thanks T. Padmanabhan for several useful discussions.