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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,

$$\int_S\epsilon_{abcd}\xi^d\nabla_e(\gamma^{ce}-g^{ce}\gamma)$$

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

$$W_{ab}=f\hat\epsilon_{ab}$$

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

$$-2\int_Sn_cN_d\xi^{[c}\nabla_e(\gamma^{d]e}-g^{d]e}\gamma)=\int_SN_d\nabla_e(\gamma^{de}-g^{de}\gamma)$$

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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