3.6 The Kerr Black Hole and the Laws of Thermodynamics |
69 |
The hole is lighter and rotates slower. The important thing is that as a consequence of (3.6.20) we have:
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δm |
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δJ < |
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(3.6.22) |
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||
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ΩH |
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3.6.2 The Bekenstein Hawking Entropy and Hawking Radiation
Inserted into the identity (3.6.5) the inequality (3.6.22) implies that, for the Penrose process, we have:
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1 |
δAH ≥ 0 |
|
κ |
8π δAH ≥ 0 |
(3.6.23) |
and the thermodynamical interpretation is consistent since both the first and the second law are respected. It is obviously important to establish that such conditions hold true for any physically conceivable process. This was advocated with many arguments and in 1971 a very important result in classical differential geometry was rigorously proved by Hawking [1], stating that in any time development, governed by Einstein field equations and involving black-holes, the total sum of all the horizon areas can never decrease.
Hence the interpretation of the horizon area as an entropy got momentum and in 1974 it was proposed by Bekenstein [2] that the formula:
SBH = |
1 |
(3.6.24) |
8π AH |
should be interpreted as stating that in all thermodynamical processes of the universe the black-hole entropy takes part as an addendum to the total statistical entropy.
This stimulated the hunt for the statistical interpretation of the horizon area. Indeed, if this latter behaves as a true entropy, it means that classical black holes actually correspond to a very large number Nmicro of quantum microstates and we have
AH log Nmicro |
(3.6.25) |
Which microstate and in which quantum theory was not clear for a long time and it is not completely clarified to the present time. Yet the statistical interpretation of blackholes obtained further evidence from the parallel discovery of the phenomenon of Hawking radiation [3], which gave an independent argument to identify the above introduced function κ with a temperature. The actual intrinsic definition of κ is the following. Let us simply name χ the Killing vector χ (ΩH ) which is null-like and future directed on the horizon. Since the horizon is a null surface, χ is both tangential and orthogonal to it. The norm of χ vanishes on the horizon and as such it is constant on it. The gradient of this norm is therefore normal to the horizon and as such it is proportional to χ . In other words we necessarily have:
μ(χ , χ ) = −2κχ μ |
(3.6.26) |