Ensembles of Classical Gravitational Fields

We define ensembles on configuration space for classical gravitational fields that obey the Einstein equations. Our starting point is the Hamilton–Jacobi formulation of general relativity. After a brief review of the Einstein–Hamilton–Jacobi equation in the metric representation, we introduce the additional mathematical structure that is needed to formulate the theory of configuration space ensembles; i.e., a measure over the space of metrics and a probability functional. Then we define an appropriate ensemble Hamiltonian for the gravitational field, show that it leads to the correct equations, and recover the Einstein equations in the usual formulation. In addition, we show that the formalism of ensembles on configuration space provides a novel approach to solving the reconstruction problem; i.e., the derivation of the full set of Einstein equations from a Hamilton–Jacobi formulation of gravity. Having derived the equations for the general case, we move on to the simpler case of spherical symmetric spacetimes and derive the corresponding equations for midisuperspace models of spherically symmetric gravity. We consider the example of classical ensembles of black holes in this midisuperspace approximation.