Gravitational equation of motion of spherical extended bodies.

Abstract

The gravitational equation of motion of the nonrotating, spherically symmetric, extended celestial body is calculated explicitly to $\frac{1}{{c}^{2}}$ order within the general class of scalar-tensor theories of gravity. Corrections to the motion proportional to spherical extension are generally nonzero in such theories; but when both PPN coefficients $\ensuremath{\gamma}$ and $\ensuremath{\beta}$ separately take their general relativistic values, the extension-dependent corrections to the motion vanish. Starting from the rather unique $\frac{1}{{c}^{2}}$ order expression obtained for the extended body equation of motion in this perturbative, but otherwise general, case, an educated guess can be made for the fully covariant variational principle with resulting tensor equation of motion of a rigid, spherically symmetric, extended body. Such bodies couple their spherical extension to the Ricci tensor of the metric field rather than the full curvature tensor, and hence such coupling vanishes in general relativity.