general relativity - How is the approximate gravitational wave stress energy momentum tensor not 0?

In Section 35.7 of Misner, Thorne, and Wheeler, p. 955, an "effective" stress energy momentum tensor for gravitational waves is defined:

$$T^{\text{GW}}_{\mu \nu} = \frac{1}{32 \pi} \left< \bar{h}_{\alpha \beta, \mu} \bar{h}^{\alpha \beta}_{\space\space,\nu} - \frac{1}{2}\bar{h}_{,\mu}\bar{h}_{\nu} - \bar{h}^{\alpha \beta}_{\space \space ,\beta} \bar{h}_{\alpha \mu, \nu} - \bar{h}^{\alpha \beta}_{\space \space, \beta} \bar{h}_{\alpha \nu,\mu} \right>.$$

The brackets indicate an averaging over a region of space much larger than a wavelength of the wave. The text then says that on a background space time with Einstein tensor $G^{\text{B}}_{\mu \nu}$,

$$G^{\text{B}}_{\mu \nu} = 8 \pi \left(T^{\text{GW}}_{\mu \nu} + T^{\text{other fields}}_{\mu \nu} \right).$$

But for a gravitational wave propagating through empty space, with $R^{\text{B}}_{\mu \nu} = 0$, wouldn't this then imply that $T^{\text{GW}}$ is $0$? What am I missing?

Answer

The treatment of this in section 35.7 of MTW says they're only presenting the result, whereas the actual derivation is given in section 35.13, p. 964. The basic idea is as follows. First you take the vacuum field equation $R_{\mu\nu}=0$, and expand it in powers of the wave's amplitude $A$. They argue that the term linear in $A$ must vanish, because any effect of gravitational waves on the background curvature must be a second-order effect. Turning next to the $A^2$ term, they break it up into a fluctuating part and a smooth background. Setting the sum of these equal to zero, we get something that looks like the ordinary Einstein field equations, for the background, with the wave term interpretable as a source, i.e., as an effective stress-energy.

[EDIT] In a comment, the OP asks:

Thank you! But how does that address my question, exactly? Wouldn't this still mean that a gravitational wave propagating in vacuum carries no energy/momentum?

Locally (i.e., in any experiment on scales smaller than a wavelength), there is no energy-momentum, because the stress-energy tensor is zero. But in GR it is not true that you can simply integrate the stress-energy tensor over some volume and find the total energy-momentum inside. This is why, for example, we say that a Schwarzschild black hole has mass, even though the stress-energy tensor is zero everywhere. Integration of a tensor with rank>0 fails because of the ambiguities introduced by the path-dependence of parallel transport.