5
$\begingroup$

At the page 336 of Hawking, Ellis: The Large Scale Structure of Space-Time, the Gauss-Bonnet theorem is stated as

$$\int_H \hat{R}\ d\hat{S} = 2\pi \chi(H) \qquad (1)$$

with

$$\hat{R} = R_{abcd} \hat{h}^{ac} \hat{h}^{bd}$$

and induced metric on the horizon $\hat{h}_{ab}$,

$$\hat{h}_{ab} = g_{ab} + \ell_a n_b + n_a \ell_b \ ,$$

where $\ell^a$ and $n^a$ is a pair of future-directed null vectors on the horizon.

Is there a missing factor of 2 on the RHS of equation (1)?

The (2-dimensional) Gauss-Bonnet theorem in the literature is usually stated using "Gaussian curvature" $K = R/2$, so I'm suspecting in this "hidden factor" (compare it, for example, with Heusler: Black Hole Uniqueness Theorems, equations (6.23)--(6.26)).

$\endgroup$

1 Answer 1

2
$\begingroup$

Yes, the factor $2\pi$ in eqn 1 should be $4\pi$ – if the area integral is normalized conventionally. For example, for a sphere, the scalar curvature is $R=2/a^2$. When multiplied over the $4\pi a^2$ surface, we get $8\pi$ which is $4\pi$ times the Euler character of the sphere, $\chi=2$. Well, at least I hope that these considerations aren't affected by using the metric $\hat h$ instead of $g$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.