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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