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?

$^1$ 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.

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?

$^1$ 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.

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 the 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?

$^1$ 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.

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?

$^1$ 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.