http://arxiv.org/abs/2210.07516
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 other in terms of the orbital angular velocity $\omega_{\phi}$. These formulae reproduce the well-known ones for the prograde shift in the Schwarzschild spacetime. In a general case, the shift deviates from that in the Schwarzschild spacetime due to a particular combination of the components of the Ricci tensor at the radius $r$ of the orbit. The formulae give a retrograde shift due to the extended-mass effect in Newtonian gravity. In the post-Newtonian regime of general relativity near a massive compact object, a retrograde shift implies that the energy density is beyond a critical value $\epsilon_{c}=3Gm^{2}/(2\pi r^{4})\simeq 3r^{2}\omega_{\phi}^{4}/(2\pi G)$, whereas a prograde shift greater than that in the Schwarzschild spacetime implies the violation of the weak energy condition there. Implications to the Galactic Centre are also discussed.
T. Harada, T. Igata, H. Saida, et. al.
Mon, 17 Oct 22
8/56
Comments: 18 pages