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

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 [Formula: see text] 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 [Formula: see text] 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, [Formula: see text], plays an important role, where [Formula: see text], [Formula: see text] and [Formula: see text] are the energy density, the radial stress and the tangential stress of the matter field, respectively. We show that the shift is backward if [Formula: see text] is beyond a critical value [Formula: see text][Formula: see text]g/cm3[Formula: see text], while a forward shift greater than that in the vacuum spacetime instead implies [Formula: see text], i.e. the violation of the strong energy condition, and thereby provides evidence for dark energy. We obtain new observational constraints on [Formula: see text] in the Solar System and the Galactic Centre.