The definition of ADM mass is

$$M=\frac{1}{16\pi}\lim_{r\rightarrow\infty}\int \left(\frac{\partial h_{\mu\nu}}{\partial x^\mu}-\frac{\partial h_{\mu\mu}}{\partial x^\nu} \right)N^\nu dA$$

according to Wald. $h_{\mu\nu}$ is the perturbation of the metric at far distance $r\rightarrow\infty$ from the center of, say, a black hole. $N^\nu$ is the normal vector to the 2 dimensional compact surface enclosing everything, the black hole included. $dA$ is the area element.

In proving the 1st law of thermodynamics of black hole, Wald derived the following expression,


in section 12.5, page 336. Here, the abstract index notation is used, and $\gamma^{ce}$ is the same as the $h^{\mu\nu}$ in the defining expression for ADM mass. $\xi^a$ is a timelike Killing vector field.

To evaluate the second expression, let me call $n_a$ the unit normal to a spacelike hypersurface and the 2 dimensional compact surface $S$ is embedded in it, to which $N_a$ is the unit normal. So the induced volume element of $S$ is $\hat\epsilon_{cd}=n^aN^b\epsilon_{abcd}$. Call the integrand of the second integral $W_{ab}$, then there is a function $f$ such that


and $f$ can be solved for, which is $f=-2n_cN_dW^{cd}$, by contracting both sides with $\hat\epsilon^{ab}$. So the second integral is


Here, I assume $\xi^a$ is normal to $S$ at infinity, and $\xi^a n_a=-1$. This expression is the same as the first integral module a factor of $16\pi$, so it represents $16\pi\delta M$, but Wald states that it is $8\pi\delta M$. I really did not get it. Would you please help me with this?

P.S.: Sorry that Google book does not provide the preview for chapter 11. Hopefully, someone who read this part will help me.


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