Why does matter follow geodesics?
In the very first lessons on General Relativity, we learn that free particles follow geodesics — the equivalent of straight lines in curved spacetimes. Why? Well, it’s easy to show that point particles must follow geodesics, if Einstein’s Field Equations are respected. But if you’re like me, you might have had this lingering suspicion:
Real matter is not made of point particles. That might have lead you to ask: why does real matter follow geodesics?
Here’s a very interesting answer, provided by Geroch and Jang (1975):
Theorem: Let (M, gab) be a space-time. Let γ be a worldline satisfying the following condition: For any neighborhood U of γ, there exists a nonzero, symmetric, conserved tensor field Tab that satisfies the strong energy condition, and whose support is in U. Then γ is a timelike geodesic.
The idea is this: if there’s a any kind of field of matter following a worldline through spacetime, which a) behaves like the matter we’re used to, and b) is small compared to the curvature near the worldline, then that matter follows a geodesic.
Now, another lingering question: is there anything about this matter field that guaranteees the geodesic is unique?
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Sorry, I don’t get it. Doesn’t one need a relation between T_ab and gamma to draw that conclusion?
It is actually not correct that point particles must follow geodesics. This is only the case if the particles are “test” particles, meaning their distortion of the background is negligible. Sorry for the nitpicking, but that’s essential.
Regarding the uniqueness: if you specify the necessary initial conditions geodesics are unique. If you only say e.g. the geodesic goes through point x, then it won’t. You also need the tangential vector in this point. You can see that from the differential equation. Once you have this initial data, you can integrate it forward.
Thanks for the correction, Bee — I should have been more specific!
The relationship between T and γ is in the support condition; the whole proof hinges on this. In effect, Geroch and Jang just prove the proposition in Minkowski spacetime, and then show that getting closer to γ gets you closer to that result.
On uniqueness: we’re in agreement, as long as a spacetime is assumed to be analytic. However, if a spacetime fails to be analytic — and, in particular, if the first derivatives of the metric fail to satisfy the Lipschitz condition — then the usual ODE uniqueness theorem fails. Consequently, there exist examples of (admittedly weird) metrics for which the geodesic equation has non-unique solutions.
Now, one might discard such spacetimes outright, for having ill-posed initial value problems. But what I’d like to know is: might there be a stronger argument, which guarantees the uniqueness of geodesics, whenever the matter field on the initial data surface is sufficiently reasonable? I’m still working on providing a precise proposition to this effect, and would be interested to hear your thoughts.