Black hole entropy

Carlos Castro. 2001. An Elementary Derivation of the Black Hole Entropy in any Dimension Entropy. Entropy 107-110. 

Remark 1 Black Hole Entropy. In the early 1970s Stephen Hawking (Hawking 1976 and references therein) showed that a black hole of mass M should emit radiation possessing the spectral distribution of black body radiation corresponding to the temperature... 

Our most important insight into the connection between thermodynamics and black holes comes from a celebrated result obtained by Bardeen, Carter and Hawking [bard73], that the four laws of black hole physics can be obtained by replacing, in the first and second laws of thermodynamics, the entropy and temperature of a thermodynamical system by the black hole event horizon (or boundary of the black hole) and surface gravity (which measures the strength of the gravitational field at the black hole s surface). 

Churazov et al. (2002) developed a model to explain the self-regulation of the accretion process around supermassive black holes. In this simple model, the supermassive black hole lies at the bottom of the gravitational potential surrounded by the lowest entropy gas. When cooling dominates, the gas entropy decreases and accretion increases. The higher accretion rate increases the energy ouput of the central black hole and the energy input to the gas increases, its entropy decreases, and the accretion declines. 

A detailed account of these problems is beyond the scope of this book, despite their fascination. Many aspects of the arguments are accessible only to specialists, but even a superficial reading of the above sources makes it clear that most cosmic events such as nuclear fusion, element formation, and formation, coalescence, and decay of black holes actually generate enormous amounts of entropy, relative to processes familiar to us on Earth. The major factor responsible for this, omitted in simplified accounts such as given above, is that we must take into account not only neutrons, protons, etc., but the enormous number of massless particles generated, such as photons and neutrinos. When this is done, the entropy balance is profoundly changed. 

Even in classical general relativity, there is a serious difficulty with the ordinary second law of thermodynamics when a black hole is present [19]. If matter is swallowed in a black hole, the entropy initially present in the matter is lost. No compensating gain of entropy occurs elsewhere. Therefore, the total entropy in the universe decreases. The resort is to continue to count the entropy in a black hole. However, the second law cannot be verified experimentally, because there is no access to the black hole. 

Therefore, an isolated black hole with no feed of matter will evaporate completely within a finite time. If the correlations between the inside and outside of the black hole are not restored during the evaporation process, then when the black hole has evaporated completely, the information with respect to these correlations is lost. Alternatives to this loss of information, e.g., the formation of a high entropy region elsewhere are implausible. This issue is addressed as the black hole information paradox [19]. 

According to Eq. (1.52) the black hole should posses entropy, which we can calculate by integrating this equation (S = dE T E ) + const) ... 

We have used E = Mc dE = c dM) and assumed that const = 0. Notice that Eq. (1.52) holds if the other variables (V,...) are held fixed. In the case of the black hole the corresponding quantities are its charge and angular momentum. The entropy in Eq. (2.47) may be cast in a different form, i.e. 

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