Gravitational field strength and generalized Komar integral

Abstract

We define a “gravitational field strength” in theories of the Einstein-Cartan type admitting a Killing vector. This field strength is a second rank, antisymmetric, divergence-free tensor, whose (“Komar”) integral over a closed 2-surface gives a physically meaningful quantity. We find conditions on the Lagrange density of the theory which ensure the existence of such a tensor, and show that they are satisfied forN=2-supergravity and for a special case of the bosonic sector ofN=4-supergravity. We discuss a possible application of the generalized Komar integral in the theory of stationary black holes. We also consider the “field strength problem” in Kaluza-Klein theory, where the application to black holes is particularly interesting.