I’m currently working through Wald’s “Quantum Field Theory in Curved Spacetimes and Black Hole Thermodynamics” and am stuck at deriving equation (4.3.8) (which is claimed to be verifiable by “direct calculation”).
The situation is as follows. We’ve got a globally hyperbolic stationary spacetime with timelike Killing field $\xi$. We now consider (complex) classical solutions to the Klein-Gordon equation $\nabla^a\nabla_a\psi - m^2\psi = 0$ whose initial data (i.$\,$e. value and normal derivative) have compact support on some (and therefore any) Cauchy surface. For two such solutions $\psi_1,\psi_2$, we define $$B(\psi_1,\psi_2) := \int_\Sigma\left(\psi_2 n^a\nabla_a\overline{\psi_1} - \overline{\psi_1} n^a\nabla_a \psi_2\right)\sqrt{h}\mathrm{d}^3x,$$ which is, apart from a missing factor of $\mathrm{i}$, just the usual “Klein-Gordon inner product” (which is not positive definite on the space of all solutions), and $$\langle\psi_1,\psi_2\rangle := \int_\Sigma \xi^a n^b T_{ab}(\psi_1,\psi_2) \sqrt{h}\mathrm{d}^3x$$ with $$T_{ab}(\psi_1,\psi_2) := \nabla_{(a}\overline{\psi_1}\nabla_{b)}\psi_2 - \frac{1}{2}g_{ab}\left(\nabla^c\overline{\psi_1} \nabla_c\psi_2 + m^2 \overline{\psi_1} \psi_2\right),$$ where $\Sigma$ is any (smooth spacelike) Cauchy surface and $n$ its future-directed unit normal. Since $T_{ab}(\psi,\psi)$ is the energy-momentum tensor of the solution $\psi$, Wald calls $\langle\cdot,\cdot\rangle$ the “energy inner product” (which can be shown to be positive definite for $m\ne0$).
Equation (4.3.8), which I want to derive, claims, up to notational differences, that $$B(\psi_1,\xi^a \nabla_a \psi_2) = 2 \langle\psi_1,\psi_2\rangle$$ for all $\psi_1,\psi_2$. After a few manipulations of the definitions, I get (using that $\psi_2$ solves the KG equation and $\xi$ is Killing) $$B(\psi_1,\xi^a \nabla_a \psi_2) - 2 \langle\psi_1,\psi_2\rangle = \int_\Sigma n^b\nabla^a\left(\overline{\psi_1}\xi_b\nabla_a\psi_2 - \overline{\psi_1}\xi_a\nabla_b\psi_2\right)\sqrt{h}\mathrm{d}^3x.$$ But why does this expression vanish? Maybe I’m missing something really easy here, but I just don’t see it.
EDIT: Contrary to what I wrote first, boundary terms we would get by applying the divergence theorem / Gauß’ theorem do vanish, because either (a) $\Sigma$ is non-compact and $\psi_i,n\cdot\nabla\psi_i$ have compact support or (b) $\Sigma$ is compact and has no boundary.