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The Isaacson-stress energy tensor is given by

$$T_{\alpha \beta}=\frac{1}{32 \pi}\left\langle\left\langle\bar{h}_{\mu \nu, \alpha}^{T T} \bar{h}^{T T \mu \nu}_{, \beta}\right\rangle\right\rangle$$

Now suppose we have a gravitational wave propagating in the +z direction with + polarization and amplitude A. I am interested in finding the Isaacson stress-energy tensor for this wave.

I believe the wave looks like this:

$$ \bar{h}_{\mu \nu}=A\left[\begin{array}{cccc}{0} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {-1} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right] e^{i(k z-\omega t)}$$

but I am unsure of what how exactly to compute the Isaacson stress-energy tensor and specifically how the TT gauge plays a role in this.

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