# Notes: Generalized harmonic form of Einstein’s equations from a gauge-fixed action

This is another note in the series of “if I don’t write it down in an easy-to-find place, then I’m going to have to keep re-deriving this result.”

The Einstein-Hilbert action for the theory of general relativity is

\begin{align} I_{EH} = \frac{1}{16\pi} \int R \sqrt{-g} d^4x \,. \end{align}

This is a beautiful diffeomorphism-invariant integral, so it leads to the diff-invariant equations of motion (the Einstein field equations) $G_{ab} = 8\pi T_{ab}$ (once you add the matter action).

Because of diff-invariance, 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,

\begin{align} \label{eq:ghcond} \newcommand{\cd}{\nabla} \cd_b\cd^b x^{(a)} = g^{ab}H_b(x, g) \,. \end{align}

Here the four functions $H_b$ can depend on the coordinate functions $x^{(a)}$ and metric, but not derivatives of the metric. This gauge choice is a generalization of the one that Yvonne Choquet-Bruhat used to prove well-posedness of the Einstein field equations. The generalization was introduced by Friedrich1 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

\begin{align} \newcommand{\pd}{\partial} 0 = g^{cd}\pd_c\pd_d g_{ab} + 2\cd_{(a}H_{b)} + \text{L.O.T.s} + 16\pi(T_{ab} - \frac{1}{2}g_{ab} T^c{}_c) \,, \end{align}

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 $H_b$ is allowed to depend on $g$ but not on $\pd g$ – 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 momentum-space cousin to showing well-posedness of the PDEs). For examples see 4 and 5.

So, let’s try to add a gauge-fixing 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,

\begin{align} I_{g.f.} = \frac{1}{16\pi} \int \xi (\Gamma_{acd} g^{cd} + H_a)g^{ab}(\Gamma_{bcd}g^{cd} + H_b) \sqrt{-g} d^4x \,. \end{align}

This tries to impose the GH condition \eqref{eq:ghcond} because $\square x^{(a)} = - \Gamma^a = -\Gamma^a{}_{cd}g^{cd}$. Note that this type of gauge-fixing term has long been known (for the special case of $H=0$) to post-Newtonian practitioners.4 Now if we vary6 $I_{EH}+I_{g.f.}$, we get the principal part

\begin{align} & g^{cd} \pd_{c}\pd_{d}g_{ab} + (1 + 2 \xi) g^{cd} \pd_{a}\pd_{b}g_{cd}\nonumber\\ &{}- (1 + 2 \xi) g^{cd} \pd_{a}\pd_{d}g_{bc} - (1 + 2 \xi) g^{cd} \pd_{b}\pd_{d}g_{ac} \,. \end{align}

By making the choice $\xi=-\frac{1}{2}$, 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 $I_{EH}+I_{g.f.}$, with the choice $\xi=-\frac{1}{2}$, is “on-shell” equivalent to Eq. (7) of Ref.3 (that is, you have to do some “on-shell” replacements of $H_a$ with $\Gamma_a$ to get the exact same equation, but only in lower-order-terms).

The punchline is that the generalized harmonic formulation of Einstein’s equations comes from adding the above gauge-fixing term (with $\xi=-\frac{1}{2}$) to the Einstein-Hilbert action.

# References

1. Friedrich, H. On the hyperbolicity of Einstein’s and other gauge field equations, Commun. Math. Phys. 100, 525-543 (1985)

2. Pretorius, F. Evolution of Binary Black-Hole Spacetimes, Phys. Rev. Lett. 95, 121101 (2005)

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

4. Appendix C of Damour & Schäfer, Lagrangians for n point masses at the second post-Newtonian approximation of general relativity, Gen. Rel. Grav. 17, 879-905 (1985) 2

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

6. Maybe the variation is subtle… I treated $H_a$ as fixed when varying the metric.

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