A Characterization of Geodesics

2017-05-27 Last updated on 2019-05-24 5 min read expository

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 relativity¹. (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 book² 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:

Theorem

A curve $\gamma$ is an affinely parametrized geodesic if and only if for every vector field $V$ vanishing near $\gamma(0)$ and $\gamma(1)$, the integral \[ \int_0^1 (L_V g)_{ab} \dot{\gamma}^a \dot{\gamma}^b ~\mathrm{d}s = 0 \]

Noticing that $(L_Vg)_{ab} = \nabla_a V_b + \nabla_b V_a$, where $\nabla$ is the Levi-Civita connection, we have that the above integral condition is equivalent to requiring \begin{equation} \int_0^1 \langle \nabla_{\dot{\gamma}} V, \dot{\gamma}\rangle_g ~\mathrm{d}s = 0. \end{equation} Using the boundary conditions and integrating by parts we see this also gives us, without passing through the local coordinate formulation, the geodesic equation \[ \nabla_{\dot{\gamma}} \dot{\gamma} = 0.\]

The Einstein-Infeld-Hoffmann theorem

Theorem [Einstein-Infeld-Hoffmann]

A curve $\gamma$ is geodesic if and only if there exists a non-vanishing contravariant symmetric two tensor $\Xi$ along $\gamma$ such that for every vector field $V$ vanishing near $\gamma(0)$ and $\gamma(1)$, the integral \ \int_{\gamma} (L_V g)_{ab} \Xi^{ab} ~\mathrm{d}\sigma \equiv 0 \.

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

Lemma

A contravariant symmetric two tensor $\Xi$ that satisfies the assumptions in the previous theorem must be proportional to $\dot{\gamma}\otimes\dot{\gamma}$.

Proof:

Choose an orthonormal frame along $\gamma$ for $(M,g)$ such that $e_n$ is tangent to $\gamma$. Write $\hat{\Xi} = \Xi - \langle \Xi, e_n \otimes e_n\rangle e_n \otimes e_n$. Suppose $\hat{\Xi} \neq 0$. Then there exists a vector field $W$ such that $W|_{\gamma}= 0$ and the symmetric part of $\nabla W$ is equal to $\hat{\Xi}$. (We can construct $W$ by choosing a local coordinate system in a tubular neighborhood of $\gamma$ such that $\partial_i|_{\gamma} = e_i$. Then $W$ can be prescribed by its first order Taylor expansion in the normal direction to $\gamma$.) Let $\psi$ be a non-negative cut-off function and setting $V = \psi W$ we note that $\nabla V |_{\gamma} = \psi \nabla W$ since $W$ vanishes along $\gamma$. Therefore we have that the desired integral condition cannot hold.

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 ^{^}

math.dg general-relativity geodesics

Willie WY Wong

Associate Professor

My research interests include partial differential equations, geometric analysis, fluid dynamics, and general relativity.