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

Gravitational Lagrangians as derived by Fock for the Einstein-Infeld-Hoffmann approach, and by Kennedy assuming only a fourth rank tensor interaction, contain long range interactions. Here we investigate how these affect the local dynamics when integrated over an expanding universe out to the Hubble radius. Taking the cosmic expansion velocity into account in a heuristic manner it is found that these long range interactions imply Mach’s principle, provided the universe has the critical density, and that mass is renormalized. Suitable higher order additions to the Lagrangians make the formalism consistent with the equivalence principle. 1. Introduction and Outline We start by presenting the gravitational Lagrangians that form the basis of the present formalism in Section 2. After that we point out how the local equations of motion for a particle will be affected by the long range interactions with the other particles in the universe. Consistency demands that the only quantities that enter are velocities and accelerations relative to the rest of the universe. This is Mach’s principle. After that, in Section 4, the long range effects are calculated by integration over the universe as a whole out to the Hubble radius, where the expansion velocity reaches the speed of light. In Section 5 Mach’s principle is found to be obeyed provided the density is the critical density ( ). The original masses of the theory then, however, turn out to be renormalized. This problem is dealt with in Section 6 where it is shown that the addition of certain higher order terms in the gravitational coupling constant restores the usual interpretation of mass and the gravitational constant. In this way the formalism as a whole is consistent with both Mach’s principle and the equivalence principle. 2. Gravitational Lagrangians Fock [1] found the Lagrangian that yields the Einstein-Infeld-Hoffmann (EIH) equations of motion [2–4]. Modern derivations and discussions of this approach can be found in Landau and Lifshitz [5], Hirondel [6], Nordtvedt [7], Brumberg [8], and Louis-Martinez [9], among others. These are all based on general relativity and Hirondel’s is the shortest. Here we will, however, focus on the profound work by Kennedy [10] on approximately relativistic interactions and their Lagrangians. Kennedy first derives the (special) relativistic Lagrangian for one particle interacting with another particle with constant given velocity. In a second step one then wishes to combine such Lagrangians into a single two-body Lagrangian, symmetric in the particle indices. To do this it