How do physicists mathematically define gravitational waves?

Question

When one first encounters gravitational waves in a standard GR lecture or a standard textbook like Carroll's "Spacetime and Geometry", they are often "defined" as follows: The metric $g$ can be split up into $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$, where $\eta$ is the standard Minkowski metric and $h$ is a perturbation upon that flat background which satisifies a wave equation.

This is all nice and dandy but when one approaches the following metric this naïvely, one might think that it constitutes a gravitational wave:
$$g = \left(\eta_{\mu\nu} + h_{\mu\nu}\right)dx^\mu dx^\nu = -d\tau^2 + dx^2 + dy^2 + dz^2 ++\left\lbrace-\cos\left(\tau - x\right)\left[2+ \cos\left(\tau - x\right)\right]d\tau^2 + \cos\left(\tau - x\right)\left[1+\cos\left(\tau-x\right)\right]\left(d\tau dx + dxd\tau\right) +\\ - \cos\left(\tau -x\right)^2 dx^2\right\rbrace ,$$ after all $h$'s coefficients do satisfy a wave equation.
But psych! It's actually just Minkowski space hiding with weird coordinates as we get from the standard metric $\eta$ to this "wavy" metric by the coordinate transformation $\tau' = \tau + \sin\left(\tau - x\right)$ as one can easily check.

I am aware that in mathematics there are very precise and rigorous definitions of what constitutes a spacetime with gravitational waves; the spacetime having to be asymptotically of Petrov type $N$ and possessing a certain, 5-dimensional isometry group$^1$. While this is all very neat and tidy, I have yet to hear/read about it in any physics lecture or physics textbook. So my questions are:
How would a working theoretical physicist go about showing that a certain spacetime is in fact one containing gravitational waves? Or is it just that these mathematical definitions are known and used by everyone dealing with gravitational waves on the theoretical side but never put down in any paper or ever mentioned? If so, why?

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$^1$ An excellent, concise source for those who want to know more about this, is this paper which goes into detail about the different mathematical conditions for a spacetime to contain gravitational waves.

Answer 1

The most straightforward way is to simply take the transverse-traceless (TT) part of $h_{ij}$. The TT part of the metric, denoted $h^{\mathrm{TT}}_{ij}$, contains precisely the two propagating degrees of freedom, which correspond to the two polarizations of gravitational waves. This enables coordinate effects to be removed and exposes the true propagating gravitational waves. It is possible to find a gauge transformation in which the only nonzero part of $h_{\mu\nu}$ is $h^{\mathrm{TT}}_{ij}$. This is known as the TT gauge.

The wavevectors can then be found by taking the Fourier transform of $h^{\mathrm{TT}}_{ij}$. If there is just one single gravitational wave with propagation direction $n^i$, it is possible to find $h^{\mathrm{TT}}_{ij}$ by defining $P_{ij} = \delta_{ij} - n_i n_j$. Then, given $h_{kl}$ in the Lorenz gauge, $$h^{\mathrm{TT}}_{ij} = \left(P_{ik}P_{jl} - \frac{1}{2}P_{ij}P_{kl}\right)h_{kl}.$$

In your example, since there is only one spatial term in your $h_{\mu\nu}$ and it is on the diagonal, it is immediately obvious that its TT part is zero.

Answer 2

Without going to full mathematical GR like the very nice paper you've linked, one naive criterion in the weak field limit that physics textbooks mention is that nearby test particles should experience oscillating proper distance amongst themselves.

I.e, Jacobi fields should have to them some character of simple harmonic oscillation.

Answer 3

A quick answer: check whether the (linearised) Weyl tensor vanishes or not. This is similar to using $F_{\mu\nu}$ instead of $A_\mu$ to check presence of electromagnetic fields in electromagnetism, where the latter is subject to gauge choices and may contain unphysical degrees of freedom.