Unbounded perturbed geometry due to analyticity

[CharlesJQuarra]

I have a certain Ansatz for a gravitational wave perturbation of the metric $h_{\mu \nu}$ that is nonzero near an axis of background flat Minkowski spacetime

The Ansatz has the following form:

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

The Ansatz has the following property:

$h^{\mu}_{\mu}=0$

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

$\partial_{\mu} h^{\mu \nu} = 0$

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

$\frac{\partial U}{\partial x}= \frac{\partial V}{\partial y}$

$\frac{\partial U}{\partial y}=- \frac{\partial V}{\partial x}$

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 $x=0, y=0$ axis, $h_{\mu \nu}$ will become larger in magnitude than $\eta_{\mu \nu}$, 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 $z$ as well?

 

[bcrowell]

I wouldn't describe your ansatz as a gravitational wave. Gravitational waves are transverse, so if the perturbations to the metric are in the x and y elements, it needs to propagate in the z direction, and therefore U and V need to depend on z. In fact, they need to depend on the quantity z-t or z+t.

 

[CharlesJQuarra]

Hi Ben,

True, I've should've added it explicitly. In any case the fact that the only nontrivial perturbation components are on the $xx$, $xy$, $yx$ and $yy$ means that derivatives of $t$ and $z$ do not show up in the gauge conditions $\partial_{\mu} h^{\mu \nu} = 0$.

The issue is that, for example, I cannot have the $h_{ij}$ fields orthogonal to $z$ propagation to vanish after a maximum width (like it would be the case for example, with a Gaussian beam), because the holomorphic nature of the non-trivial gauge conditions, forces these components to either be constant or become too large far away from the origin, even while one would've expected the functions to taper and become zero as we move away from the region with nontrivial fields, as we should be approaching Minkowski spacetime.