General formulae for the periapsis shift of a quasi-circular orbit in a
static spherically symmetric spacetime and the active gravitational mass
density
We study the periapsis shift of a quasi-circular orbit in a general static
spherically symmetric spacetime. 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, ΟAβ=(Ο΅+prβ+2ptβ)/c2,
plays an important role, where Ο΅, prβ, and ptβ are the energy
density, the radial stress, and the tangential stress of the matter field,
respectively. We show that the shift is backward if ΟAβ 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 ΟAβ<0, i.e., the violation of
the strong energy condition, and thereby provides evidence for dark energy.Comment: 24 pages, minor revision, title and terminology modified, references
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