I have a certain Ansatz for a gravitational wave perturbation of the metric [itex]h_{\mu \nu}[/itex] that is nonzero near an axis of background flat Minkowski spacetime(adsbygoogle = window.adsbygoogle || []).push({});

The Ansatz has the following form:

[itex]

g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 + V(x,y,t) & U(x,y,t) & 0 \\ 0 & U(x,y,t) & 1 -V(x,y,t) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}

[/itex]

The Ansatz has the following property:

[itex]h^{\mu}_{\mu}=0[/itex]

I want the Ansatz to be also in the Transverse-Traceless gauge, which implies

[itex]\partial_{\mu} h^{\mu \nu} = 0[/itex]

When I apply this condition on the Ansatz, I'm left with two nontrivial conditions:

[itex]\frac{\partial U}{\partial x}= \frac{\partial V}{\partial y}[/itex]

[itex]\frac{\partial U}{\partial y}=- \frac{\partial V}{\partial x}[/itex]

Oh by Thor Almighty! these are the Cauchy-Riemann equations!

Now, is well known that *analytic complex functions are either constant or unbounded*.

I am trying to interpret this correctly:

The Ansatz geometry does not seem to be able to become asymptotically Minkowski, if one asks that the metric is in the Transverse-Traceless gauge. For any far away region from the [itex]x=0, y=0[/itex] axis, [itex]h_{\mu \nu}[/itex] will become larger in magnitude than [itex]\eta_{\mu \nu}[/itex], which seems that is not our linear regime anymore, and would produce some large deformations

Is there an intuitive reason why the Transverse-Traceless gauge is not consistent with a perturbed metric that has this form? what if I would've tried a compact set, bounded on [itex]z[/itex] as well?

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# Unbounded perturbed geometry due to analyticity

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