Post-Newtonian parameters $\gamma$ and $\beta$ of scalar-tensor gravity for a homogeneous gravitating sphere

TL;DR

This paper calculates the PPN parameters  and  for scalar-tensor gravity with an arbitrary potential, revealing their dependence on source size and distance, and compares results with Solar System measurements.

Contribution

It provides a conformally invariant calculation of  and  in scalar-tensor gravity for a homogeneous sphere, including long-distance corrections and their observational implications.

Findings

PPN parameters depend on source radius and observer distance.

 does not approach 1 at large distances due to gravitational self-energy effects.

Measurements of  tighten bounds on scalar-tensor gravity parameters.

Abstract

We calculate the parameters $\gamma$ and $\beta$ in the parametrized post-Newtonian (PPN) formalism for scalar-tensor gravity (STG) with an arbitrary potential, under the assumption that the source matter is given by a non-rotating sphere of constant density, pressure and internal energy. For our calculation we write the STG field equations in a form which is manifestly invariant under conformal transformations of the metric and redefinitions of the scalar field. This easily shows that also the obtained PPN parameters are invariant under such transformations. Our result is consistent with the expectation that STG is a fully conservative theory, i.e., only $\gamma$ and $\beta$ differ from their general relativity values $\gamma = \beta = 1$, which indicates the absence of preferred frame and preferred location effects. We find that the values of the PPN parameters depend on both the radius of the gravitating mass source and the distance between the source and the observer. Most interestingly, we find that also at large distances from the source $\beta$ does not approach $\beta = 1$, but receives corrections due to a modified gravitational self-energy of the source. Finally, we compare our result to a number of measurements of $\gamma$ and $\beta$ in the Solar System. We find that in particular measurements of $\beta$ improve the previously obtained bounds on the theory parameters, due to the aforementioned long-distance corrections.