The Area Metric Reality Constraint in Classical General Relativity

TL;DR

This paper develops a classical foundation for reality conditions in spin foam models of quantum gravity by imposing an area metric reality constraint, linking complex and real general relativity through a modified Plebanski formalism.

Contribution

It introduces a new classical reality condition based on the area metric, connecting complex and real general relativity within the Plebanski framework and discussing discretization in Barrett-Crane models.

Findings

All solutions with complex action correspond to real general relativity.

Half of the solutions with real action correspond to real general relativity.

Discretization of the area metric reality constraint is discussed in Barrett-Crane theory.

Abstract

A classical foundation for an idea of reality condition in the context of spin foams (Barrett-Crane models) is developed. I extract classical real general relativity (all signatures) from complex general relativity by imposing the area metric reality constraint; the area metric is real iff a non-degenerate metric is real or imaginary. First I review the Plebanski theory of complex general relativity starting from a complex vectorial action. Then I modify the theory by adding a Lagrange multiplier to impose the area metric reality condition and derive classical real general relativity. I investigate two types of action: Complex and Real. All the non-trivial solutions of the field equations of the theory with the complex action correspond to real general relativity. Half the non-trivial solutions of the field equations of the theory with the real action correspond to real general relativity. Discretization of the area metric reality constraint in the context of Barrett-Crane theory is discussed. In the context of Barrett-Crane theory the area metric reality condition is equivalent to the condition that the scalar products of the bivectors associated to the triangles of a four simplex be real. The Plebanski formalism for the degenerate case and Palatini formalism are also briefly discussed by including the area metric reality condition.