Joseph O’Rourke’s question at MathOverflow touched on an interesting characterization of geodesics in pseudo-Riemannian geometry, which was apparently originally due to Einstein, Infeld, and Hoffmann in their analysis of the geodesic hypothesis in general relativity^{1}. (One of my two undergraduate junior theses is on this topic, but I certainly did not appreciate this result as much when I was younger.) Sternberg’s book^{2} has a very good presentation on the theorem, but I want to try to give a slightly different interpretation in this post.

## Geodesics and variation

One of the classical formulation of the criterion for a curve to be geodesic is that it is a stationary point of the *length functional*. Let $(M,g)$ be a Riemannian manifold, and let $\gamma:[0,1]\to M$ be a $C^1$ mapping. Define the length functional to be
\begin{equation}
L: \gamma \mapsto \int_0^1 \sqrt{g_{ab}(\gamma(s)) \dot{\gamma}^a(s) \dot{\gamma}^b(s)} ~\mathrm{d}s
\end{equation}
A geodesic then is a curve $\gamma$ that is a critical point of $L$ under perturbations that fix the endpoints $\gamma(0)$ and $\gamma(1)$.

One minor annoyance about the length functional $L$ is that it is invariant under reparametrization of $\gamma$, and so it does not admit unique solutions. One way to work around this is to instead consider the *energy functional* (which also has the advantage of also being easily generalizable to pseudo-Riemannian manifolds)
\begin{equation}
E: \gamma \mapsto \int_0^1 g_{ab}(\gamma(s)) \dot{\gamma}^a(s) \dot{\gamma}^b(s) ~\mathrm{d}s.
\end{equation}
It turns out that critical points of the energy functional are always critical points of the length functional. Furthermore, the energy functional has some added convexity: a curve is a critical point of the energy functional if it is a geodesic and that it has constant speed (in the sense that $g_{ab} \dot{\gamma}^a \dot{\gamma}^b$ is independent of the parameter $s$).

The standard way to analyze the variation of $E$ is by first fixing a coordinate system $\{ x^1, x^2, \ldots, x^n\}$. Writing the infinitesimal perturbation as $\delta \gamma$, we can compute the first variation of $E$: \begin{equation} \delta E[\gamma] \approx \int_0^1 \partial_c g_{ab}(\gamma) \cdot \delta\gamma^c \cdot \dot{\gamma}^a \dot{\gamma}^b + 2 g_{ab}(\gamma) \cdot \dot{\gamma}^a \cdot \dot{\delta\gamma}^b ~\mathrm{d}s. \end{equation} Integrating the second term by parts we recover the familiar geodesic equation in local coordinates.

There is a second way to analyze the variation. Using the diffeomorphism invariance, we can imagine instead of varying $\gamma$ while fixing the manifold, we can deform the manifold $M$ while fixing the curve $\gamma$. From the point of view of the energy functional the two should be indistinguishable. Consider the variation $\delta\gamma$, which can be regarded as a vector field along $\gamma$ which vanishes at the two end points. Let $V$ be a vector field on $M$ that extends $\delta \gamma$. Then the infinitesimal variation of moving the curve in the direction $\delta \gamma$ should be reproducible by flowing the manifold by $V$ and *pulling back the metric*. To be more precise, let $\phi_\tau$ be the one parameter family of diffeomorphisms generated by the vector field $V$, the first variation can be analogously represented as
\begin{equation}
\frac{1}{\tau} \lim_{\tau\to 0} \int_0^1 \left[(\phi_\tau^\ast g)_{ab}(\gamma) - g_{ab}(\gamma)\right] \dot{\gamma}^a \dot{\gamma}^b ~\mathrm{d}s
\end{equation}
By the definition of the Lie derivative we get the following characterizing condition for a geodesic:

## The Einstein-Infeld-Hoffmann theorem

The EIH theorem follows immediately from the discussion in the previous section and the following lemma.

Notice that the proof here uses the positive definiteness of $\Xi_{ab}\Xi^{ab}$. In the pseudo-Riemannian setting, this is generally not satisfied. However, if $\gamma$ were a *time-like* geodesic, then $\hat{\Xi}$, being the projection to the orthogonal complement of $\dot{\gamma}$, will automatically be space-like, and the same proof holds.

- Einstein, Infeld, Hoffmann, "Gravitational Equations and the Problem of Motion", Annals of Math (39) 65--100.
^{^} - Schlomo Sternberg,
*Curvature in Mathematics and Physics*^{^}