# How do physicists mathematically define 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.

• What you are writing about and asking about is only one type of gravitational waves: the case of weak (perturbative) gravitational waves propagating in Minkowski spacetime. These can be described by linearized Einstein's equations and using the D'Alembert wave equation. - But this does not handle the other types. The strong gravitational waves and the waves generated and propagating in the background of curved spacetime. Commented Mar 6 at 16:10

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.

• Do you have a source in which this procedure is described and in which the claims about the TT gauge are proven? Commented Mar 5 at 21:23
• @Moguntius It's quite standard. For example, it's contained in the first chapter of Maggiore's GW textbook Commented Mar 6 at 0:36
• Thanks, I'll check that out! Commented Mar 8 at 1:33

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.

• Do you have any source in which that is mentioned? If I recall correctly, Carroll does not mention that. Commented Mar 4 at 20:40
• David Tong's GR course is an example I recall, see section 5.2.2 "Bobbing on the waves:" damtp.cam.ac.uk/user/tong/gr.html. Commented Mar 4 at 20:43
• So, after looking into it, it could definitely be used as a criterion but is it actually being used as such? Commented Mar 4 at 21:14

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.

• But couldn't the linearised Weyl tensor also not vanish if the metric is a vacuum solution that doesn't represent gravitational waves such as the Schwarzschild metric? Commented Mar 6 at 22:35
• Electric fields also don’t vanish when there is point charge, but we have no problem distinguishing Coulomb electric fields from electromagnetic waves. Why wouldn’t the same approach apply to gravitational waves? Commented Mar 8 at 6:02
• Because gravitational waves, in contrast to electromagnetic waves, carry the charge they couple to. Gravity couples to mass/energy/stress and grav. waves carry energy, whereas elm. waves, which couple to electric charge, are not charged themselves and, therefore, do not (directly, i.e. if we neglect high order quantum effects) interact with each other. Commented Mar 13 at 19:12
• I don’t see the relevance of nonlinearities in determination of the existence of gravitational waves. If the Weyl tensor doesn’t vanish and if it has wavelike propagating modes (w.r.t. the background metric) what would force us to conclude that gravitational waves are not actually present? Commented Mar 14 at 20:12
• If you are thinking of full nonlinear solutions of GR then yes, it’s a completely different beast, but apart from numerical simulations almost all calculations are done in the weak field regime where linearised equations make sense. Commented Mar 14 at 20:20