Aspects of the Black Hole Interior Volume and Entropy

Abstract: The basic and conceptual notion of this work starts from the recent investigations of Marios Christodoulou and Carlo Rovelli (CR) in their paper entitled ''How big is a black hole?''. This work is related to the black hole interior volume and the entropy by Baocheng Zhang. In CR work, a spherically symmetric Schwarzschild black hole is considered and defines the interior volume, as the volume inside a sphere $S$ is the maximal proper volume of a space-like spherically symmetric $3d$ hypersurface bounded by the sphere $S$. Using this definition, they found the interior volume of a black hole is proportional to the advanced time. This is the main characteristic of this formulation. Which means that a black hole could store a large amount of infalling information. Using this special property, one can prob the nature of Hawking radiation emitted by a black hole. They also extend their result to the charged static black hole and found a consistent result. Their numerical analysis showed that the special character of this formulation is that the volume linearly increases with advancing time. Later, the CR work is followed by Baocheng Zhang, who investigated the entropy of scalar quantum modes in the interior of the Schwarzschild Black hole. He found the entropy of scalar quantum modes in the interior of Black hole is proportional to the lack hole surface area. The proportionality constant is found to be less than unity, which means that the interior entropy is less as compared to the Bekenstein Hawking entropy (horizon entropy). Note that these analyses are only applicable for black holes with mass greater than the Planck mass. If massless than Planck's mass, then one needs to understand the uncertainty relation.