Temperature and thermodynamic structure of Einstein's equations for a cosmological black hole

TL;DR

This paper investigates the thermodynamic structure of Einstein's equations for a cosmological black hole described by the Sultana-Dyer metric, confirming the validity of thermodynamic quantities and the first law in a time-dependent spacetime.

Contribution

It demonstrates that Einstein's equations encode thermodynamic relations for a cosmological black hole, clarifying the correct horizon temperature and validating thermodynamic expressions.

Findings

Identifies the correct time-dependent horizon temperature via tunnelling formalism.

Shows Einstein's equations lead to the first law of thermodynamics on the horizon.

Confirms the validity of thermodynamic quantities like entropy and energy for the Sultana-Dyer black hole.

Abstract

It is expected that the cosmological black holes are the closest realistic solutions of gravitational theories and they evolve with time. Moreover, the natural way of defining thermodynamic entities for the stationary ones is not applicable in the case of a time dependent spacetime. Here we confine our discussion within the Sultana-Dyer metric which is a cosmological black hole solution of Einstein's gravity. In literature, there exists two expressions of horizon temperature -- one is time dependent and the other does not depend on time. To single out the correct one we find the temperature by studying the Hawking effect in the tunnelling formalism. This leads to time dependent structure. After identifying the correct one, the Einstein's equations are written on the horizon and we show that this leads to the first law of thermodynamics. In this process the expressions for horizon entropy and energy, obtained earlier by explicit calculations, are being used. This provides the evidence that Einstein's equations have thermodynamic structure even for a cosmological black hole spacetime. Moreover, this study further clarifies the correctness of the expressions for the thermodynamic quantities; like temperature, entropy and internal energy.