Combining the Einstein field equations with the geodesic equation

Question

I've seen this question How does Einstein field equations interact with geodesic equation?, but it doesn't make any sense to me. If spacetime is a Lorentzian manifold, then surely one thing general relativity tells us is what the possible manifolds are when gravity is the only "force". And in that context, the field equations themselves don't restrict the manifold at all -- any manifold has an Einstein tensor $G_{\mu\nu}$ that represents a possible matter distribution in that it is "conserved" (zero divergence).

So the form of the manifold is restricted only by the geodesic equation. How so? Well, here's my thought process:

First of all, we can say that $G(\vec u, \vec u)$, for a timelike unit vector $\vec u$, is the mass density flowing along $\vec u$. And all we require is that that mass follows a geodesic. So, if we take the geodesic along $\vec u$, the same mass should remain on it the whole time. But since the mass can spread out spatially over time, we actually have to consider a tight family of initially parallel geodesics, and say that the density integrated over their spatial cross-section (volume) $V$ is what remains constant. So we would basically get

$$\nabla_{\mu} (G_{\mu\mu} \cdot V_{geodesic}) = 0.$$

Now, the variation of the cross-sectional volume along a tight family of initially parallel geodesics is exactly what the Ricci curvature describes. Except that it is the second derivative of that volume, because the first derivative is identically zero, because they are initially parallel. But we wouldn't want the $\nabla_{\mu}$ to be ${\nabla_{\mu}}^2$, because that would allow the total mass to change linearly. So clearly something is off with the above equation. Or maybe I'm barking up the wrong tree, but the conceptual statement makes sense to me. Can anyone clarify this, and/or point me to an online reference that explains conceptually how the two equations are combined?

Answer 1

I feel there is something interesting to explore here, but it is not physical to assert "any manifold ... represents a possible matter distribution". The equivalent statement in Newtonian gravity would be to assert that there are no constraints on mass, not even that it cannot be negative. But such a constraint is an important aspect of the theory as a physical theory; it cannot be dropped or ignored. In G.R. the constraints on $T^{ab}$ (going by such names as weak energy condition, strong energy condition and so on) are part of the story of general relativity considered as a physical theory, as opposed to merely some mathematical tools to deal with manifolds. But it has never been quite fully settled which of these constraints one should insist upon. It is an open area of research at the boundary of quantum theory and general relativity.

A widely discussed aspect of G.R. is that the geodesic equation, with its interpretation as an equation of motion for particles, follows from the field equation (see e.g. MTW book). This is a bit like the way you can deduce the Lorentz force equation from the Maxwell equations plus energy conservation. So there are connections between geodesic equation, field equation and conservation. But that's about as far as this answer is going to take it. It is perhaps more of a long comment than strictly an answer, but I hope you may find it helpful.