So, I'm reading Wald, and in it he talks about how the divergence-free nature of the stress-energy tensor implies "a lot" of knowledge about how matter moves in a curved space time. I'm wondering, how much is "a lot"? Can we obtain the full equations of motion from this? Wald gives the example of a perfect fluid in which you can; however, I'm wondering if you can obtain the full equations of motion of an arbitrary distribution of matter from this condition. Wald says that this condition implies that small masses move on geodesics (so that the "geodesic hypothesis" is actually present within Einstein's equation itself.), he goes on to say that large masses which feel tidal forces do not move exactly on geodesics, but move according to divT=0. Thinking back, it's natural that conservation of energy (and momentum and stress) would imply a condition on the motion of particles. After all, in classical mechanics, one often uses conservation of energy and momentum to restrict a particle's motion. However, in classical mechanics, conservation of energy (and momentum, and angular momentum, etc) by itself is usually not sufficient to determine the full trajectory of a particle, it usually only gives 1 or 2 constants of integration (making the problem easier). But to get the full equations of motion, one must usually just solve the differential equations (Euler Lagrange eqns, or some such). In GR, is divT=0 sufficient to find the FULL trajectory of particles moving in curved space-time? Thanks.