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Black hole thermodynamics The goal of this problem is to explore some interesting properties of Black Holes. The following equation was obtained by L. Smarr in 1973:

M^2 = \frac{1}{16\pi} A + \frac{4\pi}{A}\left(J^2 + \frac{1}{4}Q^4\right)+\frac{1}{2}Q^2

where

M

J

Q

and

A

are the mass, angular momentum, charge and area of the event horizon of a black hole.
To make contact with thermodynamics we write for the entropy of the Black Hole,

S = \frac{1}{4}k_B A

where

k_B

is the Boltzmann constant.
[*] Work in natural units

G = \hbar = c = 1

and show that the equation for the entropy is dimensionally correct. [/*] [*] Take

k_B = 1/8\pi

(by choosing units) and derive an expression for

S(M,J,Q)

. Is this expression unique? (Hint: What is the entropy of the Schwarzschild Black Hole which corresponds to

J=Q=0

?) \item We suppose the mass-energy

M

(since

c=1

) plays the role of internal energy. Show that

T,\Omega,\Phi

defined via,

dM = T dS + \Omega dJ + \Phi dQ

are given by, \begin{eqnarray*} & T = \frac{1}{M} \left[1- \frac{1}{16S^2}\left(J^2 + \frac{1}{4}Q^4\right)\right] \\ & \Omega = \frac{J}{8MS}\\ & \Phi = \frac{Q}{2M}\left[1+\frac{Q^2}{8S}\right]. \end{eqnarray*} This is the analog of the first law of thermodynamics. [/*] [*]Look at the expression for

M(S,J,Q)

closely and derive the analog of the Gibbs-Duhem Relation familiar from Thermodynamics. [/*] [*] Show that,

S \to \frac{1}{4}M^2 - \frac{1}{8}Q^2

T \to 0

. What does this say about the third law of thermodynamics? Give reasons to support your answer. \item An alternative statement to the third law is that "it is impossible to reach absolute-zero in a finite number of steps". What can we conclude from part (e)? [/*]

Problem Statement