Thermodynamic properties. As in the previous sections, we consider Dirichlet boundary conditions whereby the proper size of the boundary circle βT and the boundary value of the dilaton ϕb are fixed. We take ϕb > 0 here since non-positive ϕb reproduces the near-Nariai black hole setup described in Section 1.2. In general, there are two solutions obeying the boundary conditions which we call rD and rA, as shown in Figure 1.3. The temperatures of the interpolating geometry and the AdS2 geometry are T |ϕ˜ − 2ϕ | b D βD = 4π qr2 + ϕ2 + ϕ˜(r − ϕ D ) , (1.44) D b T |ϕ˜ + 2ϕ | b A b βA = 4π qr2 − ϕ2 A + ϕ˜(r − ϕ A ) , (1.45) where we have introduced the notation ϕD = rD to be the value of the dilaton at the dS2 black hole horizon and ϕA = rA to be the value of the dilaton at the AdS2 black hole horizon. We must set the temperatures equal to each other βD = βA = βT to compare their T T thermodynamic properties at a given temperature. Case 1: ϕ˜ ≥ 0 Let us begin by taking ϕb to be large compared to ϕ˜ and ϕA,D. The thermodynamics for the Euclidean AdS2 solution (1.40) with ϕ˜ = 0 are reviewed in Appendix A. For non-vanishing ϕ˜ we find a similar result for ϕA > 0, namely T AdS2 −β F = 2π ϕ − ϕ˜! + 1 2ϕ + ϕ˜ qβ2 + 4π2 . (1.46) b T The specific heat follows readily CAdS = ϕb + ϕ˜! 4π2β2 , (1.47) T T 2 2 (β2 + 4π2)3/2 which is positive for all ϕb > 0. The other solution has ϕD < 0 such that the region of the geometry near and including the horizon has positive curvature. For this interpolating solution, we have T interp −β F = 2π ϕ + ϕ˜! + β2 − 4π2 s(ϕb + χ+)(ϕb − χ−) , (1.48) T 2 T β2 − 4π2 where for convenience we have defined χ± ≡ √2±1 ϕ˜. In order to ensure the above expres- sion is real we must lie in one of two regimes. The first, which connects to parametrically ϕ˜ q large ϕb, is ϕb ≥ χ− and 4π ϕb(ϕb + ϕ˜) ≥ βT ≥ 2π, where the upper bound for βT ensures the negativity of ϕD. Given Finterp, we can compute the specific heat of the interpolating saddle for this range: − ×b + b − Cinterp = − (ϕ + χ )(ϕ χ ) T . (1.49) (β2 − 4π2)3/2 √ 4π2β2 T ϕ˜ q For this case the specific heat Cinterp is negative, and the only thermodynamically stable saddle is the Euclidean AdS2 black hole. We thus retrieve, within this range of boundary conditions a near-Nariai black hole horizon with negative specific heat. The second regime is 0 < ϕb < χ− and 4π ϕb(ϕb + ϕ˜) < βT < 2π. For this case the specific heat Cinterp is − ×b + b − Cinterp = (ϕ + χ )( ϕ + χ ) T , (1.50) (4π2 − β2 )3/2 √ 4π2β2 T which is positive....
