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

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, ρ A = (ϵ + p r + 2 p t) / c 2 , plays an important role, where ϵ , 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 ρ A is beyond a critical value ρ c ≃ 2. 8 × 1 0 − 1 5 g/cm3 (m / M ⊙) 2 (r / a.u.) − 4 , while a forward shift greater than that in the vacuum spacetime instead implies ρ A < 0 , i.e. the violation of the strong energy condition, and thereby provides evidence for dark energy. We obtain new observational constraints on ρ A in the Solar System and the Galactic Centre. [ABSTRACT FROM AUTHOR]