Degenerate Solutions of General Relativity from Topological Field Theory

Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional $BF$ theory on any framed surface Σ embedded in M. In these solutions the cotetrad field e (and thus the metric) vanishes outside a neighborhood of Σ, while inside this neighborhood the connection A and the field satisfy the equations of 4-dimensional BF theory. Our construction works in any signature and with any value of the cosmological constant. If for some 3-manifold S, at fixed time our solutions typically describe “flux tubes of area”: the 3-metric vanishes outside a collection of thickened links embedded in S, while inside these thickened links it is nondegenerate only in the two transverse directions. We comment on the quantization of the theory of solutions of this form and its relation to the loop representation of quantum gravity. Received: 21 April 1997 / Accepted: 22 August 1997