Quadratic metric-affine gravity

Abstract

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature and study the resulting system of Euler-Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi-Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with metric of a pp-wave and parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non-Riemannian solutions. We define the notion of a "Weyl pseudoinstanton" (metric compatible spacetime whose curvature is purely Weyl) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non-Riemannian solution which is a wave of torsion in Minkowski space. We discuss the possibility of using this nonRiemannian solution as a mathematical model for the graviton or the neutrino.

pp-waves with torsion and metric-affine gravity

“…(b) The double duality ansatz in its basic [6] or modified [14,15,16] forms does not work for the most general 16-parameter actions introduced in [3,4,2] and considered in our current paper. It works only for more special actions with up to 11 coupling constants.…”

“…Our notation follows [5,6,2]. In particular, we denote local coordinates by x µ , µ = 0, 1, 2, 3, and write ∂ µ := ∂/∂x µ .…”

“…We denote Weyl curvature by W; here, as in [6,2], Weyl curvature is understood as the irreducible piece of curvature defined by conditions (20), (21) and Ric = 0.…”

“…the b's being coefficients from formula (51) of [2]. The LHS's of equations (26) and (27) are the components of tensors A and B from the formula δS = (2A λν δg λν + 2B κµ λ δΓ λ µκ ) .…”

“…the given irreducible subspace of the vector space of curvatures is not isomorphic to any other irreducible subspace. See Section 6 of [2] for details. According to formula (44) of [2] the (symmetric) trace-free Ricci piece of curvature is not simple, hence the version of the double duality ansatz from [2] works for a pp-space, classical or generalised, only when curvature is purely Weyl.…”