Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 4 / d / ii

Table of contents
- Solution 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}

is

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 .

With the stated orientation and metric volume form,

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

The Background-subtracted Komar energy is consequently

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}}) .

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