# Hawking Radiation and Black Hole Entropy

UNDERSTANDING GRAVITATIONAL ENTROPY

Gravitational entropy is an extension to the concept of entropy, which is a measure of number of accessible configuration to the system, as an effect of gravitational field. It is a consequence of the semi classical theory of gravity and is our only clue towards an understanding of complete quantum theory of gravity. Gravitational entropy, on the one hand, rescues the second law of thermodynamics in the presence of a black hole, while on the other hand, also poses new questions by violating the principal of unitary time evolution of Quantum Mechanics.

According to Classical theory of Relativity, if the mass density is sufficiently high, then there can form regions in space-time where gravity is so intense that no particle or radiation can escape this region. Such a region is called black hole, a term coined by John Wheeler in 1967, and its boundary is called event horizon. As matter falls inside the black hole, its mass and gravitational pull increases which in turn increases the radius of event horizon (boundary of the region from which nothing can escape). Since matter cannot escape a black hole by any means in classical relativity, area of an event horizon always increases. Absence of radiation from black holes also implies that temperature of a black hole must be zero, as according to thermodynamics every object with non-zero temperature emits radiation. As an object with zero temperature, it was assumed that entropy of the black hole is also zero.

Now, imagine that we have a box of radiation with entropy S and we throw it in a black hole. This will reduce the entropy of the universe, as the entropy of black hole according to classical relativity is always zero. But this result is a direct contradiction to Second law of Thermodynamics, which states that entropy of universe can only increase.

In 1972, Bekenstein realized that area of the black hole is a measure of the black entropy as it always increases just like entropy. This was confirmed in 1974 when Hawking worked out semi-classical theory of gravity that predicted that black holes emit radiation, called Hawking radiation or Black hole radiation, through the process of quantum tunneling. So the concept of gravitational entropy also called black hole entropy was confirmed by semi-classical theory of gravity. Black hole entropy is the measure of degrees of freedom of black hole or number of internal configurations accessible to the system. However, where these degrees of freedom are located is still unknown. These degrees of freedom may reside inside the black hole or on the surface, that is, event horizon.

While Hawking radiation, on the one hand, reconciles thermodynamics with gravity, on the other hand, it also gives rise to information paradox. Hawking radiation is thermal in nature so it is characterized by only one parameter, namely the temperature of the black hole. Imagine configuration of a gas is completely known that is we know the positions and velocities of all the individual particles. Now if this gas collapses to form a black hole, which in turn evaporates through Hawking radiation, then the final state will be determined solely by temperature of black hole which depends only on the mass of black hole. No matter what the initial configuration was, final state would remain same as long as the mass of the gas remains same. So all the information about initial configuration of gas is destroyed after the evaporation of the black hole. Information loss in black holes also violates the principle of unitary evolution of Quantum Mechanics, which states that there is a one-to-one relationship between initial and final states and we can precisely determine the initial state if the final state is known. Understanding the gravitational entropy is still an active field of research and may lead to understanding of a complete quantum theory of gravity.

References:

1. Black Hole Thermodynamics, Hawking Radiation and Information loss Paradox. By Alexandor Vitanov. Pro-Seminar in Theoretical Physics.

2. http://www.iscid.org/encyclopedia/Gravitational_Entropy

3. Classical and Quantum Black Holes. Edited by P Fre, V Gorini, G.Magli, and U Moschella.