José P. S. Lemos (University of Lisbon): Black hole entropy from matter entropy
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Europe/Budapest
Tanacsterem
Tanacsterem
Description
Black hole entropy S is one of the most fascinating issues
in contemporary physics, as one does not yet strictly know what are
the degrees of freedom at the fundamental microlevel, nor where are
they located precisely. In addition, extremal black holes, in contrast
to non-extremal ones, present a conundrum, as there are two mutually
inconsistent results for the entropy of extremal black holes. There is
the usual Bekenstein-Hawking S = A/4 value, where A is the horizon
area, obtained from string theory and other methods, and there is the
prescription S = 0 obtained from Euclidean arguments. In order to
better understand black hole entropy in its generality, we exploit a
matter based framework and use a thermodynamic approach for an
electrically charged thin shell. We find the entropy function for such
a system. We then take the shell radius into its gravitational radius
(or horizon) limit. We show that: (i) For a non-extremal shell the
gravitational radius limit yields S=A/4. The contribution to the
entropy comes from the pressure. (ii) For an extremal shell the
calculations are very subtle and interesting. The horizon limit gives
an entropy which is a function of the horizon area A alone, S(A), but
the precise functional form depends on how we set the initial
shell. The values 0 and A/4 are certainly possible values for the
extremal black hole entropy. This formalism clearly shows that
non-extremal and extremal black holes are different objects. In
addition, the formalism suggests that for non-extremal black holes all
possible degrees of freedom are excited, whereas in extremal black
holes, in general, only a fraction of those degrees of freedom
manifest themselves. We conjecture that for extremal black holes the
entropy S is restricted to the interval between 0 and A/4. Since an
extremal shell has zero pressure, the contribution to the entropy
comes from the shell's electricity. In this case, the contribution to
the entropy comes from the mass and electric potential. (iii) There is
yet another possibility that interpolates between the two previous
ones: to take the extremal limit concomitantly with the gravitational
radius limit. Remarkably, in this case, the mass, the pressure and the
electricity on the shell contribute to the entropy to give S=A/4.
