In linearized gravity we can one sets(adsbygoogle = window.adsbygoogle || []).push({});

$$(1) \ \ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$

where h is taken to be a small perturbation about the flat space metric. One common decomposition of h is to write the spatial part as

$$ h_{i j} = 2 s_{ij} - 2\psi \delta_{ij} \ h_{0i} \equiv w_i \ h_00 = -2\phi$$

There are certain gauge transformations that leave (1) invariant which can be used to simplify Einstein's equation; one choice, "the transverse gauge", makes ##\nabla \cdot w = 0## and ##\partial_i s^{ij} =0##. One can show that by expressing the (time-time) part Einstein field equations in terms of these fields in the transverse gauge yields for empty space yields

$$\nabla^2 \psi = 0.$$

Now at the beginning of section 7.4 "Gravitational Wave solutions" in Carroll's "Spacetime and Geometry", he states that the above equation with "well behaved boundary conditions" implies

$$\psi = 0.$$

I'm not sure what to make of this. What does he mean by well behaved boundary conditions, and why are these relevant for the gravitational wave solutions?

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# Gravitational wave solution boundary conditions

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