Black Hole Thermodynamics in a Box

Simple calculations indicate that the partition function for a black hole is defined only if the temperature is fixed on a finite boundary. Consequences of this result are discussed. 1 The Black Hole Partition Function From the work of Gibbons and Hawking in the late 1970’s came a very simple prescription for the computation of the temperature of a static black hole [1]: • Write the black hole metric in static coordinates; • Euclideanize (t → −it) and periodically identify t; • Adjust the period to remove conical singularities. The resulting period is the inverse temperature β. The origin of this prescription is a formal calculation of the partition function Z(β) as a functional integral over all Euclidean geometries g with period β and action I[g]. Some of the key features of this calculation can be captured in a ‘microsuperspace’ version based on the metric ansatz [2] (1) ds = N(r) dt + (1 − 2M/r)dr + rdΩ . Let t have the range 0 ≤ t < 2π and r have the range 2M ≤ r < ∞. Also restrict N so that 2πN(∞) = β and 2πN(r) ∼ 8πM √ 1− 2M/r near r = 2M . The first restriction fixes the proper period at infinty to the inverse temperature. The second restriction insures that the metric (1) describes a smooth geometry with no conical singularities, and with topology R × S2. The action for (1) is a function of M only, I(M) = Mβ − 4πM2. A ‘toy’ partition function can be constructed as the integral over M of exp(−I(M)). The extremum of the action satisfies 0 = ∂I/∂M = β − 8πM . (This is the classical equation of motion obtained by integrating (Nr/2)(1 − 2M/r)−3/2Gr over t and r, where Gr is the r-r component of the Einstein tensor.) The solution for the extremum is a Euclidean black hole with M = β/(8π), and the partition function is classically approximated by lnZ(β) ≈ −I(β/(8π)) = −β2/(16π). The expectation value of energy is 〈E〉 ≡ −(∂ lnZ/∂β) ≈ β/(8π), which equals the extremal value of the mass parameter M . An interpretation of these results is that Z(β) describes a system that contains a black hole of mass M and inverse temperature β = 8πM . What about pre–exponential factors in Z(β)? A simple calculation shows that the second derivative ∂I/∂M is negative at the extremum M = β/(8π). Therefore, the extremum lies along a path of steepest ascents, not along a path of steepest descents. The Euclidean black hole does not dominate the integral for Z(β), and should not be used to approximate Z(β). As a consequence, the conclusion 〈E〉 = β/(8π) is unfounded. Formally, the Eu∗ Departments of Physics and Mathematics, North Carolina State University, Raleigh, NC 27695–8202