Differential constants of motion for systems of freely gravitating particles in general relativity are first defined and then determined. It is shown that they are all consequences of the local simultaneity conservation property of general relativity. It is proved, further, that the restriction to vacuum conditions does not change the set of differential constants of motion, excluding the nonphysical cases of space-time of dimension 2 or 3. Another consequence is that nothing can be inferred from local (in space and time) measurements about the orientation of a laboratory in free fall relative to Fermi transported axes. A similar property exists in Newton's theory.