Reducing vacuum Einstein field equations to wave equations using wave coordinates

Question

In a lot of research papers, authors claim that it is well know that in wave coordinates, Einstein field equations can be reduced to wave equations. But I did not find any reference. Can anybody suggest some good reference? Thanks a lot.

Answer 1

You could have a look, for instance, at the books by Straumann [1, Section 5.1 and 5.3], Sean Carroll [2, Chapter 7, in particular Eq. 7.91 and the text leading up to it], or his lecture notes [3, Chapter 6], but this is really treated in most standard textbooks on general relativity.

It is important to note that the claim is in general only true for the linearlized field equations. The idea is to write the metric tensor as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, where $|h_{\mu\nu}|\ll 1$ is a small perturbation of the flat Minkowski metric $\eta_{\mu\nu}$ (i.e. we consider only very small gravitational fields) and work to linear order in $h_{\mu\nu}$ and its derivatives. After working it all out, the linearlized Einstein equations then take the form.

\begin{align} \label{eq:linearized_EFEs2} G_{\mu\nu} = \tfrac{1}{2}\left(-\partial_\mu\partial_\nu h - \square h_{\mu\nu} + \partial_\nu\partial_\rho h^\rho_\mu + \partial_\mu\partial_\rho h^\rho_\nu + \eta_{\mu\nu}\left(\square h - \partial_\rho\partial_\sigma h^{\rho\sigma}\right)\right). \end{align}

Now the trick is to show that, given any $h_{\mu\nu}$, one can always perform a suitable coordinate transformation (this context usually called a gauge transformation) such that in the new coordinates $h_{\mu\nu}$ satisfies

\begin{align} \partial_\mu h^\mu{}_\lambda - \frac{1}{2}\partial_\lambda h = 0, \qquad h \equiv \eta^{\mu\nu}h_{\mu\nu}. \end{align}

This gauge (i.e. choice of coordinates) goes by many names (Lorenz gauge, Einstein gauge, Hilbert gauge, de Donder gauge, Fock gauge). You can confirm that a lot of the terms in $G_{\mu\nu}$ then cancel each other, and we are left with

\begin{align} G_{\mu\nu} = -\frac{1}{2}\left(\square h_{\mu\nu} -\frac{1}{2}\eta_{\mu\nu}\square h\right). \end{align}

The field equation $G_{\mu\nu} = \kappa T_{\mu\nu}$ thus reduces to

\begin{align} \square h_{\mu\nu} -\frac{1}{2}\eta_{\mu\nu}\square h = -2\kappa T_{\mu\nu}. \end{align}

In vacuum (i.e. $T_{\mu\nu}=0)$ the trace of this equation shows that $\square h =0$, and hence the vacuum field equation becomes

\begin{align} \boxed{ \square h_{\mu\nu} = 0,} \end{align}

which is a homogeneous wave equation for each component of the metric. If $T_{\mu\nu}\neq 0$ then one has to introduce the "trace-reversed" perturbation $\bar h_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}h$, in term of which the field equations become

\begin{align} \boxed{ \square \bar h_{\mu\nu} = -2\kappa T_{\mu\nu},} \end{align}

which is a classic inhomogeneous wave equation for each component of $\bar h_{\mu\nu}$.

[1] Straumann, Norbert, General relativity. With applications to astrophysics., Texts and Monographs in Physics. Berlin: Springer (ISBN 3-540-21924-2/hbk). xii, 674 p. (2004).

[2] Carroll, Sean, Spacetime and geometry. An introduction to general relativity, San Francisco, CA: Addison Wesley (ISBN 0-8053-8732-3). xiv, 513 p. (2004).

[3] Sean Carrol's lecture notes

EDIT: After posting the answer I realized there is also such a thing as nonlinear wave coordinates, which you might have meant. But well, let's hope this answer is useful for somebody nontheless.