Gravitational Lagrangians, Mach’s Principle, and the Equivalence Principle in an Expanding Universe

Figure 1

This figure illustrates the central point particle of mass in a homogeneous isotropic Hubble expanding universe with constant rest mass/energy density . The radial speed is and at the Hubble radius the speed is , the speed of light. In the Lagrangian we assume that the particle has velocity and that the expanding sphere of density has overall translational velocity .