Hello, can somebody please help me understanding the following. Action of general relativity consists of two terms: action of gravitation, dependent on metric tensor and its derivatives; action of matter, say one freely moving point mass particle, dependent on particle coordinates and metric tensor. Stationary principle for this action gives all the eqs: variation w.r.t. metric tensor gives Einstein eqs with stress-energy tensor on r.h.s.; variation w.r.t. particle coordinates gives motion eqs for the particle, resulting that particle moves along a geodesic. Further, I read (in Dirac's lectures on General Theory of Relativity) that Einstein eqs are not independent. Due to Bianchi identity for Ricci tensor there are certain compatibility conditions on stress-energy tensor. And from these conditions it follows that the particle moves along a geodesic. It's OK, the system is closed and has a solution. Although it would be enough to consider only Einstein eqs to reproduce the whole dynamics. My first question comes from an attempt to count degrees of freedom. For 10 independent variations contained in metric tensor I would expect 10 independent gravitational eqs (PDEs); for 4 independent variations in coordinates - 4 independent eqs of motion (ODEs). Not just 10 eqs with dependencies from which 4 others follow. The related question: let me fix coordinate system and fix the world line of the particle, e.g. let it be a straight line in non-flat metric. The particle is now non-free, by intent. I consider variation of action w.r.t. metric tensor only and obtain 10 eqs. They must be compatible since the action must have a stationary point somewhere also for fixed particle's path. The eqs seem to be just the same Einstein eqs. Which are not compatible since the particle does not move along a geodesic. Do I miss something? Cheers, Igor.