There is a calculation that I had been thinking for a long time of working out to my own satisfaction, both because of its intrinsic importance and because it seemed like it would be fun. This was to calculate the intensity of a gravitational wave for a given amplitude. E.g., for LIGO events we have amplitudes expressed as a fractional change in the metric, and these can be related to the mass-energy released in the event that created the waves. Misner, Thorne, and Wheeler have the following expression (I've changed the notation slightly) in section 36.7:
$$ T_{cd}=\frac{1}{32\pi}\langle g_{ab,c}g^{ab}{}_{,d} \rangle , $$
where $g_{\mu\nu}-\eta_{\mu\nu} \ll 1$. This is for the effective stress-energy tensor for a gravitational wave, in the transverse traceless (TT) gauge, averaged over several wavelengths.
MTW's derivation of this is complex and spread around in little chunks in different places in the book. I thought I would go through the features of this equation and see how many of them I could understand or give heuristic derivations for. Dimensionally, it makes sense, and this occurs iff we have two derivatives. It depends on the square of the amplitude, which makes sense in the small-amplitude limit for any wave. The derivatives are partial derivatives, not covariant derivatives, because the covariant derivative of the metric is zero. The average is required because by the equivalence principle, we can never have a local expression for the energy density of the gravitational field. Next I started worrying about the factor of $1/32\pi$, wondering if there was some heuristic way to produce it.
What occurred to me was to work out an expression, in similar notation, for the energy density of the gravitational field in Newtonian mechanics. The energy density is $-(1/8\pi)\textbf{g}^2=-(1/8\pi)(\nabla\phi)^2$, where the units are such that $G=1$, and $\nabla$ is a 3-gradient. In the static case, in the semi-newtonian limit, the metric is $g_{tt}=1+2\phi$. So translating the above expression for the energy density into GR-style notation, we have
$$\textbf{g}=-\frac{1}{2}\nabla g_{tt}$$
$$T_{tt} = -\frac{1}{32\pi}g_{tt,\mu}g^{tt,\mu},$$
where $\mu$ runs only over the spacelike coordinates.
So these two expressions have some obvious similarities, but they are also different in certain ways. It's nice that the $1/32\pi$ is the same in the two cases, although the sign is opposite. However, we're contracting over different indices. I'm also not very convinced by my own comparison of these expressions because the one for a gravitational wave is specific to a certain gauge, and I never actually used that assumption.
Is it possible to give a semi-heuristic derivation -- one that's more convincing than mine -- for the expression for the effective stress-energy tensor of a gravitational wave?