Phase Transitions of Thermal Radiation in Anti-de Sitter Space

In this presentation we review some results pertaining to black holes in Anti-de Sitter space following Hawking and Page (1983). We use units in which G = c = h ̄ = kB = 1. The metric of the covering space of anti-de Sitter space is of the form: ds2 = −V dt2 + V −1dr2 + r2(dθ2 + sin2 θdφ2). (1) Here V = 1 + r 2 b2 and b = ( − 3Λ)1/2. Anti-de Sitter space is obtained by making the above space periodic in t, such that the point labeled by (t, r, θ, φ) is the same as that with the label (t + γ, r, θ, φ); the value of γ is 2pib. We state without proof Hawking and Page’s result that for zero mass parti-cles of a conformally invariant field, the energy momentum tensor is T µν = Aδ µ ν + f(T)V −2(δµν − 4δµ0 δ0ν) (2) where f(T) = pi 2 90 gT 4 + O(b−2T 2) and g is the effective number of spin states. The first term represents an anomaly (the conformal anomaly), a symmetry that is broken in a quantum theory that is unbroken in the corresponding classical theory. Its volume integral over all space is formally infinite; we treat it as an unobservable renormalization of the vacuum energy. 1 Einstein’s equations have solutions which describe a Schwarzchild black hole in a spacetime which is asymptotically Anti-de Sitter. The metric in this case is the same as in (1), but with V = 1 − 2Mr + r 2 b2. We note that for both forms of V we have Rµν = Λgµν and R = 4Λ. The horizon occurs where g00 = 0, namely at r = r+ where V (r+) = 0. Explicitly, r+ = b2/