Degenerate Solutions of General Relativity from Topological Field Theory

TL;DR

This paper constructs degenerate solutions to general relativity using 2D BF theory, revealing flux tube structures and exploring implications for quantum gravity quantization.

Contribution

It introduces a method to generate degenerate GR solutions from 2D BF theory, linking classical solutions to quantum gravity frameworks.

Findings

Solutions describe flux tubes with area in 3D slices.

Metric vanishes outside neighborhoods of embedded surfaces.

Framework applies to any signature and cosmological constant.

Abstract

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 Sigma embedded in M. In these solutions the cotetrad field e (and thus the metric) vanishes outside a neighborhood of Sigma, while inside this neighborhood the connection A and the field E = e ^ e satisfy the equations of 4-dimensional BF theory. Moreover, there is a correspondence between these solutions and certain solutions of 2-dimensional BF theory on Sigma. Our construction works in any signature and with any value of the cosmological constant. If M = R x S 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.