Demystifying autoparallels in alternative gravity

Abstract

The equations of motion of test bodies in relativistic gravity are tightly linked to the conservation laws of the theory [1–3]. The explicit derivation of these equations has been intertwined with the development of approximation schemes within relativistic gravity [4–6]. As it is well known, geodesic curves arise as trajectories of structureless test bodies in Riemannian spacetimes with the metric gij as the gravitational field potential, that determines the metric-compatible Christoffel connection Γ̃ij k = 1 2g kl (∂igjl + ∂jgil − ∂lgij). In alternative gravity theories the set of the gravitational field variables is extended to (gij , Γij ) where the connection and the metric are no longer compatible, so that the torsion Tij k := Γij k −Γji k and the nonmetricity Qkij := −∇kgij = − ∂kgij + Γki glj + Γkj gil are nontrivial, in general. Autoparallel curves have been postulated on several occasions in the literature as candidates for the equations of motion of test bodies in alternative gravity theories. Such ad-hoc postulates, unsubstantiated by the conservation laws, usually lead to inconsistencies with the field equations. Consequently one should abstain from the practice of postulating equations of motion instead of deriving them [7, 8]. With this warning in mind, we here report on two special cases, in which autoparallel curves actually do emerge in theories with post-Riemannian spacetime structure. Let us consider the dynamics of massive particles under the action of the gravitational gij and a scalar φ field. We demonstrate that it is possible to recast the latter into a geometric property of the underlying spacetime, and construct an effective torsion Tij k and nonmetricity Qkij from this scalar field. As a preliminary step, we recall that the deviation of the spacetime geometry from the Riemannian [9] one is described by the distortion tensor