Entropy of a Generic Null Surface from Its Associated Virasoro Algebra

Abstract

Null surfaces act as one-way membranes, blocking information from those observers who do not cross them. These observers associate an entropy (and temperature) with the null surface. In this spirit the black hole entropy can be computed from the central charge of an appropriately defined, local, Virasoro algebra on the horizon. In this chapter, we demonstrate that one can extend these ideas to a general class of null surfaces, all of which possess a Virasoro algebra and a central charge, leading to an entropy density which is just (1/4). All the previously known results arise as special cases of this very general property of null surfaces and the result represented here provides the derivation of the entropy-area law in the most general context.

Appendix

For the sake of completeness, we derive the bracket for the Noether charge associated with the vectors as given in Eq. (8.4).

The normalized normal to $r=\text {constant}$ surface corresponds to,

$$\begin{aligned} N_{a}&=\left( 0,\frac{1}{\sqrt{2r\alpha +r^{2}\beta ^{2}}},0,0\right) \end{aligned}$$

(8.14)

$$\begin{aligned} N^{a}&=\left( \frac{1}{\sqrt{2r\alpha +r^{2}\beta ^{2}}},\sqrt{2r\alpha +r^{2}\beta ^{2}},\frac{r\beta ^{A}}{\sqrt{2r\alpha +r^{2}\beta ^{2}}}\right) \end{aligned}$$

(8.15)

The extrinsic curvature of the $r=0$ surface can be computed easily using the above normalized normal $N_{a}$ as,

$$\begin{aligned} K&=-\nabla _{a}N^{a}=-\partial _{a}N^{a}-N^{a}\partial _{a}\ln \sqrt{-g} \nonumber \\&=-\partial _{u}\left( \frac{1}{\sqrt{2r\alpha +r^{2}\beta ^{2}}}\right) -\partial _{r}\sqrt{2r\alpha +r^{2}\beta ^{2}} \nonumber \\&=-\frac{\alpha }{\sqrt{2r\alpha +r^{2}\beta ^{2}}} \end{aligned}$$

(8.16)

The corresponding non-zero component of the Noether potential associated with the Noether charge on $r=0$ surface yield,

$$\begin{aligned} J^{ur}&=K\xi ^{u}n^{r}-K\xi ^{r}n^{u} \nonumber \\&=-\left( \alpha -\frac{\partial _{u}\alpha }{2\alpha }\right) \left( F+\frac{\partial _{u}F}{2\alpha }\right) \end{aligned}$$

(8.17)

Finally the relevant component of the Noether current for calculation of the bracket between Noether charges becomes

$$\begin{aligned} J^{r}&=\partial _{u}J^{ru}+\partial _{A}J^{rA}+J^{rA}\partial _{A}\ln \sqrt{-g} \nonumber \\&=\partial _{u}\left[ \alpha \left( F+\frac{\partial _{u}F}{2\alpha }\right) \right] \nonumber \\&=\alpha \partial _{u}F+\frac{1}{2}\partial _{u}^{2}F \end{aligned}$$

(8.18)

The other normalized vector (the time evolution vector) corresponds to,

$$\begin{aligned} M^{a}&=\left( \frac{1}{\sqrt{2r\alpha }},0,0,0\right) \end{aligned}$$

(8.19)

$$\begin{aligned} M_{a}&=\left( -\sqrt{2r\alpha },\frac{1}{\sqrt{2r\alpha }},-\frac{r\beta ^{A}}{\sqrt{2r\alpha }}\right) \end{aligned}$$

(8.20)

Hence one arrives at on the $r=0$ surface,

$$\begin{aligned} d\Sigma _{ab}J^{ab}&=-d^{2}x\sqrt{q}\left( n_{a}M_{b}-n_{b}M_{a}\right) J^{ab} \nonumber \\&=2d^{2}x\sqrt{q}\alpha \left( F+\frac{\partial _{u}F}{2\alpha }\right) \end{aligned}$$

(8.21)

Thus the Noether charge becomes,

$$\begin{aligned} Q[\xi ]=\frac{1}{2}\int d\Sigma _{ab}J^{ab}=\int d^{2}x\sqrt{q}\alpha \left( F+\frac{\partial _{u}F}{2\alpha }\right) \end{aligned}$$

(8.22)

The commutator reads,

$$\begin{aligned}{}[Q_{1},Q_{2}]&=\int d^{2}x \sqrt{q}\left\{ \left( F_{1}+\frac{\partial _{u}F_{1}}{2\alpha }\right) \left( \alpha \partial _{u}F_{2} \frac{1}{2}\partial _{u}^{2}F \right) -\left( 1\leftrightarrow 2\right) \right\} \nonumber \\&=\int d^{2}x\sqrt{q}\Bigg [\alpha \left( F_{1}\partial _{u}F_{2}-F_{2}\partial _{u}F_{1}\right) +\frac{1}{2}\left( F_{1}\partial _{u}^{2}F_{2}-F_{2}\partial _{u}^{2}F_{1}\right) \nonumber \\&+\frac{1}{4\alpha }\left( \partial _{u}F_{1}\partial _{u}^{2}F_{2}-\partial _{u}F_{2}\partial _{u}^{2}F_{1}\right) \Bigg ] \end{aligned}$$

(8.23)

Then with the ansatz for F(u) as in Eq. (8.8) will lead to the Fourier space commutator bracket Eq. (8.10).