Conformal Transformation, Near Horizon Symmetry, Virasoro Algebra and Entropy

There are certain black hole solutions in general relativity (GR) which are conformally related to the stationary solutions in GR. It is not obvious that the horizon entropy of these spacetimes is also one quarter of the area of horizon, like the stationary ones. Here I study this topic in the context of Virasoro algebra and Cardy formula. Using the fact that the conformal spacetime admits conformal Killing vector and the horizon is determined by the vanishing of the norm of it, the diffemorphisms are obtained which keep the near horizon structure invariant. The Noether charge and a bracket among them corresponding to these vectors are calculated in this region. Finally, they are evaluated for the Sultana-Dyer (SD) black hole, which is conformal to the Schwarzschild metric. It is found that the bracket is identical to the usual Virasoro algebra with the central extension. Identifying the zero mode eigenvalue and the central charge, the entropy of the SD horizon is obtained by using Cardy formula. Interestingly, this is again one quarter of the horizon area. Only difference in this case is that the area is modified by the conformal factor compared to that of the stationary one. The analysis gives a direct proof of the earlier assumption.Comment: Minor comments added, to appear in Phys. Rev.

Thermodynamics of Sultana-Dyer Black Hole

The thermodynamical entities on the dynamical horizon are not naturally defined like the usual static cases. Here I find the temperature, Smarr formula and the first law of thermodynamics for the Sultana-Dyer metric which is related to the Schwarzschild spacetime by a time dependent conformal factor. To find the temperature ($T$), the chiral anomaly expressions for the two dimensional spacetime are used. This shows an application of the anomaly method to study Hawking effect for a dynamical situation. Moreover, the analysis singles out one expression for temperature among two existing expressions in the literature. Interestingly, the present form satisfies the first law of thermodynamics. Also, it relates the Misner-Sharp energy ($\bar{E}$) and the horizon entropy ($\bar{S}$) by an algebraic expression $\bar{E}=2\bar{S}T$ which is the general form of the Smarr formula. This fact is similar to the usual static black hole cases in Einstein's gravity where the energy is identified as the Komar conserved quantity.Comment: typos corrected, to appear in JCA

Entropy function from the gravitational surface action for an extremal near horizon black hole

It is often argued that {\it all the information of a gravitational theory is encoded in the surface term of the action}; which means one can find several physical quantities just from the surface term without incorporating the bulk part of the action. This has been observed in various instances; e.g. derivation of the Einstein's equations, surface term calculated on the horizon leads to entropy, etc. Here I investigate the role of it in the context of entropy function and entropy of extremal near horizon black holes. Considering only the Gibbons-Hawking-York (GHY) surface term to define an entropy function for the extremal near horizon black hole solution, it is observed that the extremization of such function leads to the exact value of the horizon entropy. This analysis again supports the previous claim that there exists a ``{\it holographic}'' nature in the gravitational action -- surface term contains the information of the bulk.Comment: Matches with the accepted version, to appear in EPJ

Vacuum condition and the relation between response parameter and anomaly coefficient in (1+3) dimensions

The role of Israel-Hartle-Hawking vacuum is discussed for anomalous fluid in presence of both the gauge and gravitational anomalies in ($1+3$) dimensions. I show that imposition of this vacuum condition leads to the relation $\tilde{c}_{4d}=-8\pi^2c_m$ between the response parameter ($\tilde{c}_{4d}$) and the anomaly coefficient ($c_m$). This establishes a connection between the coefficients appearing in a first order and a third order derivative terms in the constitutive relation.Comment: Comments added, to appear in JHE

Entropy corresponding to the interior of a Schwarzschild black hole

Interior volume within the horizon of a black hole is a non-trivial concept which turns out to be very important to explain several issues in the context of quantum nature of black hole. Here we show that the entropy, contained by the {\it maximum} interior volume for massless modes, is proportional to the Bekenstein-Hawking expression. The proportionality constant is less than unity implying the horizon bears maximum entropy than that by the interior. The derivation is very systematic and free of any ambiguity. To do so the precise value of the energy of the modes, living in the interior, is derived by constraint analysis. Finally, the implications of the result are discussed.Comment: Two new references and additional discussions added, to appear in Phys. Lett.

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.Comment: Minor modifications, to appear in EPJ