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 pathintegral quantization of gravity is discussed.

• Received 24 September 1982

DOI:https://doi.org/10.1103/PhysRevD.27.719