It is proved that a plausible definition of the total energy of a body in Newtonian cosmology can be given by the condition that the energy is conserved, if the body is participating in the general cosmic expansion. The same formula for the total energy results also from a certain metric describing a relativistic model of a universe with a uniform and isotropic distribution of matter and distinguished by an interesting property: that the differences of space-like coordinates have in it an immediate metric meaning as the lengths measured by rigid rods. It is found that the cosmic expansion has either no effect, or quite imperceptible effects on the motion of planets in our solar system, but during the formation of local systems such as clusters of galaxies, the deviations from the exact validity of the conservation law of energy are of considerable magnitude. This process is to be studied on the basis of McVittie's model.

• Received 10 August 1964

DOI:https://doi.org/10.1103/PhysRev.137.B1379