Propagation equations for deformable test bodies with microstructure in extended theories of gravity

We derive the equations of motion in metric-affine gravity by making use of the conservation laws obtained from Noether’s theorem. The results are given in the form of propagation equations for the multipole decomposition of the matter sources in metric-affine gravity, i.e., the canonical energy-momentum current and the hypermomentum current. In particular, the propagation equations allow for a derivation of the equations of motion of test particles in this generalized gravity theory, and allow for direct identification of the couplings between the matter currents and the gauge gravitational field strengths of the theory, namely, the curvature, the torsion, and the nonmetricity. We demonstrate that the possible non-Riemannian spacetime geometry can only be detected with the help of the test bodies that are formed of matter with microstructure. Ordinary gravitating matter, i.e., matter without microscopic internal degrees of freedom, can probe only the Riemannian spacetime geometry. Thereby, we generalize previous results of general relativity and Poincaré gauge theory.

Figure 1

Sketch of the hypersurface $\Sigma$, i.e., the world tube of the test particle. A continuous curve through the tube is parametrized by $Y^a$. Coordinates within the world tube with respect to a coordinate system centered on $Y^a$ are labeled by $x^a$.