Classical GR as a topological theory with linear constraints

TL;DR

This paper reformulates 4d continuum gravity as a constrained topological BF theory using linear constraints, clarifying the connection between spin foam models and classical general relativity.

Contribution

It introduces a linear constraint formulation of continuum gravity, linking discrete spin foam constraints to a continuum action principle for general relativity.

Findings

Identifies continuum linear simplicity constraints used in quantum spin foam models.

Derives a linear version of quadratic volume constraints for gravity.

Shows conditions under which discrete volume constraints follow from simplicity constraints.

Abstract

We investigate a formulation of continuum 4d gravity in terms of a constrained topological (BF) theory, in the spirit of the Plebanski formulation, but involving only linear constraints, of the type used recently in the spin foam approach to quantum gravity. We identify both the continuum version of the linear simplicity constraints used in the quantum discrete context and a linear version of the quadratic volume constraints that are necessary to complete the reduction from the topological theory to gravity. We illustrate and discuss also the discrete counterpart of the same continuum linear constraints. Moreover, we show under which additional conditions the discrete volume constraints follow from the simplicity constraints, thus playing the role of secondary constraints. Our analysis clarifies how the discrete constructions of spin foam models are related to a continuum theory with an action principle that is equivalent to general relativity.