Notes: Generalized harmonic form of Einstein’s equations from a gaugefixed action
This is another note in the series of “if I don’t write it down in an easytofind place, then I’m going to have to keep rederiving this result.”
The EinsteinHilbert action for the theory of general relativity is
This is a beautiful diffeomorphisminvariant integral, so it leads to the diffinvariant equations of motion (the Einstein field equations) (once you add the matter action).
Because of diffinvariance, the solutions to Einstein’s equations are not unique. There are an infinite number of coordinate transformations that you can perform on a solution. Physicists think of these as the same (they “mod out gauge”) but from the simpler PDEs point of view, they’re different solutions. So, to pose GR as an initial value problem, you need to fix a gauge.
One pretty gauge choice is the “generalized harmonic” gauge,
Here the four functions can depend on the coordinate functions and metric, but not derivatives of the metric. This gauge choice is a generalization of the one that Yvonne ChoquetBruhat used to prove wellposedness of the Einstein field equations. The generalization was introduced by Friedrich^{1} and used to first successfully evolve a binary black hole merger by Pretorius.^{2} This is also the choice we make in SpEC, see Lindblom et al.^{3}
The reason this gauge choice is nice is that when you write out the Einstein equations in this gauge, the principal part is manifestly that of the scalar wave operator on the manifold, plus lower order terms. That is, the Einstein equations become
where L.O.T.s stands for lower order terms (that do not affect the manifest hyperbolicity of the equations). Aside: here you can see why is allowed to depend on but not on – the latter would affect the principal part.
Now, usually these gauge conditions are imposed at the level of the equations of motion to show hyperbolicity. However sometimes we want to see the gauge getting fixed in the action. This is important for e.g. diagrammatic methods like using Feynman diagrams (inverting to find a unique propagator is the momentumspace cousin to showing wellposedness of the PDEs). For examples see ^{4} and ^{5}.
So, let’s try to add a gaugefixing term to the action to get the generalized harmonic formulation of GR. The easiest guess is something like a Feynman’t Hooft gauge fixing term,
This tries to impose the GH condition \eqref{eq:ghcond} because . Note that this type of gaugefixing term has long been known (for the special case of ) to postNewtonian practitioners.^{4} Now if we vary^{6} , we get the principal part
By making the choice , we get the principal part to agree with the scalar wave operator.
It is only a little bit of work to show that the equations you get by variation of , with the choice , is “onshell” equivalent to Eq. (7) of Ref.^{3} (that is, you have to do some “onshell” replacements of with to get the exact same equation, but only in lowerorderterms).
The punchline is that the generalized harmonic formulation of Einstein’s equations comes from adding the above gaugefixing term (with ) to the EinsteinHilbert action.
References

Friedrich, H. On the hyperbolicity of Einstein’s and other gauge field equations, Commun. Math. Phys. 100, 525543 (1985). ↩

Pretorius, F. Evolution of Binary BlackHole Spacetimes, Phys. Rev. Lett. 95, 121101 (2005). ↩

Lindblom, Scheel, Kidder, Owen, and Rinne. A new generalized harmonic evolution system, Class. Quantum Grav. 23 S447 (2006). ↩ ↩^{2}

Appendix C of Damour & Schäfer, Lagrangians for n point masses at the second postNewtonian approximation of general relativity, Gen. Rel. Grav. 17, 879905 (1985). ↩ ↩^{2}

Goldberger and Rothstein, Effective field theory of gravity for extended objects, Phys. Rev. D 73, 104029 (2006). ↩

Maybe the variation is subtle… I treated as fixed when varying the metric. ↩