Classical GR as a topological theory with linear constraints

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. 1. Motivation As is well known, general relativity in three dimensions is topological, as there are no local degrees of freedom. Its first order formulation in terms of a connection ω and a triad E is given by S = 1 16πG ∫ Σ ǫABCE A ∧R [ω] . (1) The quantisation of such a theory is rather well understood and different possible quantisation procedures are known. This is of course in stark contrast to general relativity in four dimensions, whose quantisation is to a large extent still an open problem. One may try to exploit what is known in 3D to write general relativity in 4D as a constrained topological theory with action