Classical general relativity as BF-Plebanski theory with linear constraints

TL;DR

This paper reformulates 4D classical gravity as a constrained BF theory using linear constraints, bridging continuum and discrete approaches, and clarifying the role of volume and simplicity constraints in the spin foam context.

Contribution

It introduces a linear constraint formulation of Plebanski gravity, connecting continuum and discrete models, and clarifies the relationship between simplicity and volume constraints.

Findings

Identifies continuum linear simplicity and volume constraints.

Shows discrete volume constraints derive from simplicity constraints under certain conditions.

Bridges continuum and discrete formulations of gravity in the BF-Plebanski framework.

Abstract

We investigate a formulation of continuum 4d gravity in terms of a constrained 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.