orbiting in the gravitational field of the earth with mass . The law of gravitation can be written where is the average mass density inside a sphere of radius . Gravitational force in terms of the 00 component of the stress–energy tensor Newton's law can be written . where is the volume of a sphere of radius . The quantity will be recognized from special relativity as the rest energy of the large body, the earth. This is the sum of the rest energies of all the particles that compose earth. The quantity in the parentheses is then the average rest energy density of a sphere of radius about the earth. The gravitational field is proportional to the average energy density within a radius r. This is the 00 component of the stress–energy tensor in relativity for the special case in which all the energy is rest energy. More generally where and is the velocity of particle i making up the earth and in the rest mass of particle i. There are N particles altogether making up the earth. Relativistic generalization of the energy density There are two simple relativistic entities that reduce to the 00 component of the stress–energy tensor in the nonrelativistic limit and the trace where is the 4-velocity. The 00 component of the stress–energy tensor can be generalized to the relativistic case as a linear combination of the two terms where 4-acceleration due to gravity The 4-acceleration due to gravity can be written . Unfortunately, this acceleration is nonzero for as is required for circular orbits. Since the magnitude of the 4-velocity is constant, it is only the component of the force perpendicular to the 4-velocity that contributes to the acceleration. We must therefore subtract off the component of force parallel to the 4-velocity. This is known as Fermi–Walker transport. In other words, . This yields . The force in the local frame is . Einstein field equation We obtain the Einstein field equation by equating the acceleration required for circular orbits with the acceleration due to gravity . This is the relationship between curvature of spacetime and the stress–energy tensor. The Ricci tensor becomes . The trace of the Ricci tensor is . Comparison of the Ricci tensor with the Ricci tensor calculated from the principle of least action, Theoretical motivation for general relativity#Principle of least action in general relativity identifying the stress–energy tensor with