A Geometric Derivation of the Einstein Equations from the Causal Action Principle

Abstract

The causal action principle for causal fermion systems is analyzed for a minimizing measure whose support is assumed to have the structure of a smooth manifold $\tilde{M}$. The concept of osculating vacua is introduced. It is shown that the Lagrangian induces on $\tilde{M}$ a Lorentzian metric. Moreover, the Euler-Lagrange equations of the causal action imply that the Ricci tensor must satisfy the Einstein equations of general relativity for an energy momentum tensor given in terms of a power expansion in the regularization length. The gravitational coupling constant is found to be the square of the regularization length. Our methods provide a systematic procedure for deriving corrections to the Einstein equations. The paper includes a self-contained introduction to causal variational principles and the causal action principle. Most geometric structures (connection, Riemannian metric and curvature) are introduced and analyzed in the general setting of causal variational principles for an arbitrary dimension of $\tilde{M}$. The Lorentzian setting works only for causal fermion systems and is worked out only in four spacetime dimensions.