d s^{2} = - f (r) d v^{2} + 2 d v d r + r^{2} d Ω_{2}^{2}

k = \partial_{v}

κ = f^{'} (r_{+}) /2

. Here

f^{'} (r) = \frac{2 r}{L ^{2}} + \frac{r _{+}}{r ^{2}} (1 + \frac{r _{+}^{2}}{L ^{2}}),

κ = \frac{1}{2 r _{+}} + \frac{3 r _{+}}{2 L ^{2}} .

T_{H} = \frac{κ}{2 π} = \frac{1 + 3 r _{+}^{2} / L ^{2}}{4 π r _{+}} .

ii

Solution

Let

μ = r_{+} (1 + \frac{r _{+}^{2}}{L ^{2}}), f (r) = 1 + \frac{r ^{2}}{L ^{2}} - \frac{μ}{r} .

The one-form dual to

k = \partial_{v}

k^{♭} = - f d v + d r

. For the

r_{+} \to 0

anti-de Sitter background,

\overset{ˉ}{k}^{♭} = - (1 + r^{2} / L^{2}) d v + d r

, so

k^{♭} - \overset{ˉ}{k}^{♭} = \frac{μ}{r} d v, d (k^{♭} - \overset{ˉ}{k}^{♭}) = \frac{μ}{r ^{2}} d v \land d r .

⋆ (d v \land d r) = - r^{2} sin θ d θ \land d ϕ .

M = - \frac{1}{8 π} \int_{S^{2}} ⋆ d (k - \overset{ˉ}{k}) = - \frac{1}{8 π} (- 4 π μ) = \frac{r _{+}}{2} (1 + \frac{r _{+}^{2}}{L ^{2}}) .

The horizon area is

A = 4 π r_{+}^{2}

, while

\frac{d M}{d r _{+}} = \frac{1}{2} (1 + \frac{3 r _{+}^{2}}{L ^{2}}), \frac{κ}{8 π} \frac{d A}{d r _{+}} = κ r_{+} = \frac{1}{2} (1 + \frac{3 r _{+}^{2}}{L ^{2}}) .

d M = \frac{κ}{8 π} d A = T_{H} d S_{B H}

S_{B H} = A /4

For equilibrium with an infinite fixed-temperature reservoir, local stability requires positive heat capacity. Since

d M / d r_{+} > 0

, its sign is the sign of

\frac{d T _{H}}{d r _{+}} = \frac{1}{4 π} (- \frac{1}{r _{+}^{2}} + \frac{3}{L ^{2}}) .

r_{+} > \frac{L}{3} .

The equality is the minimum-temperature, marginal case; smaller black holes have negative heat capacity and unstable thermal equilibrium.

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