Why is the metric invertible ?

We raise, and provide an (unsatisfactory) answer to, the title's question: why, unlike all other fields, does the gravitational " metric " variable not have zero vacuum? After formulating, without begging it, we exhibit additions to the conventional action that express existence of the inverse through a field equation. The metric variable's invertibility is (on a par with its dimensionality and signature) a tacit, but basic, assumption of gravitational theories. Unlike all other fields' dynamical variables, it does not vanish in the ground (or any other) state. This property is taken for granted (but see [1] for a recent attempted explanation) because it underlies existence of geometry and because the background-independence of covariant models does not single out any natural " zero ". Nevertheless, even if spacetime is but an emergent property of some substrate, one should still seek an intrinsic explanation of " why there is something rather than nothing ". Ours will exhibit additions to gravitational actions that embody invertibility as a field equation. The result will be far less satisfactory (or at least less familiar) than the Higgs effect's construction of a non-vanishing VEV. The first difficulty is just to establish a framework where invertibility is not presupposed ; we invoke the Palatini, first-order, approach where metric and affinity are independent variables. For concreteness, consider ordinary GR in D=4, L E = g µν R µν (Γ). (1) Here g µν is a symmetric contravariant density, Γ α µν is likewise (µν) symmetric, and R µν (Γ) is the usual affine Ricci tensor constructed from Γ; note that (at D=4) √ − det g µν is a scalar density. The Palatini procedure requires solving the field equation D α (Γ)g µν ∼ ∂ α g µν + gΓ = 0 (2) for Γ(g). This is where invertibility of g µν comes in: without it, Γ remains undetermined. Second order, Hilbert-Einstein, actions where Γ is already the metric affinity obviously cannot even be written, absent invertibility. Having pinpointed the requirement's origin, we provide the most elementary formal enlargement of the action (1) so that invertibility becomes a consequence of the field equations. The simplest way to ensure that g µν is nonsingular is of course that its determinant not 1