Modification of the Palatini variational principle in general relativity

Abstract

A new classical action formulation of general relativity is presented. The aim is to develop a variational principle which derives both Riemannian geometry and Einstein's field equations in the general case of a gravitational field interacting with matter. The method is applied to the Hilbert action functional with a variable matter-gravity coupling. Surprisingly, the resulting theory is not general relativity with a variable G; instead, the method yields general relativity with a constant G and a cosmological constant. It is shown that the variational principle has a physical content different from that of the usual Hilbert and Palatini action formulations. A possible application of the new variational principle to the path-integral quantization of gravity is discussed.