General formulae for the periapsis shift of a quasi-circular orbit in static spherically symmetric spacetimes and the active gravitational mass density

TL;DR

This paper derives comprehensive formulas for the periapsis shift in static spherically symmetric spacetimes, linking the shift to matter properties like active gravitational mass density, and explores implications for dark energy and observational constraints.

Contribution

It provides new full-order formulas for periapsis shift in general spacetimes and connects the shift to active gravitational mass density, including observational implications.

Findings

Shift can be backward due to extended mass effects.

Active gravitational mass density influences periapsis shift.

Constraints on mass density in Solar System and Galactic Centre.

Abstract

We study the periapsis shift of a quasi-circular orbit in general static spherically symmetric spacetimes. We derive two formulae in full order with respect to the gravitational field, one in terms of the gravitational mass $m$ and the Einstein tensor and the other in terms of the orbital angular velocity and the Einstein tensor. These formulae reproduce the well-known ones for the forward shift in the Schwarzschild spacetime. In a general case, the shift deviates from that in the vacuum spacetime due to a particular combination of the components of the Einstein tensor at the radius $r$ of the orbit. The formulae give a backward shift due to the extended-mass effect in Newtonian gravity. In general relativity, in the weak-field and diffuse regime, the active gravitational mass density, $\rho_{A}=(\epsilon+p_{r}+2p_{t})/c^{2}$, plays an important role, where $\epsilon$, $p_{r}$, and $p_{t}$ are the energy density, the radial stress, and the tangential stress of the matter field, respectively. We show that the shift is backward if $\rho_{A}$ is beyond a critical value $\rho_{c}\simeq 2.8\times 10^{-15} \mbox{g}/\mbox{cm}^{3} (m/M_{\odot})^{2}(r/\mbox{au})^{-4}$, while a forward shift greater than that in the vacuum spacetime instead implies $\rho_{A}<0$, i.e., the violation of the strong energy condition, and thereby provides evidence for dark energy. We obtain new observational constraints on $\rho_{A}$ in the Solar System and the Galactic Centre.