Periapsis shift in the Zipoy-Voorhees spacetime
Akihito Katsumata, Tomohiro Harada
TL;DR
The paper analyzes the periapsis shift of timelike bound orbits in the Zipoy-Voorhees γ-metric, identifying the γ-dependence (and thus quadrupole deformation) as entering at 2PN order. It derives a PN expansion for arbitrary eccentricity, relates it to Schwarzschild and Curzon-Chazy limits, and introduces a new, rapidly convergent series representation based on a parameter that blends eccentricity with proximity to ISCO, enabling accurate calculations in both weak and strong fields. Observational data from the S2 star bound Sgr A*’s deformation to γ ≳ 1.9×10^{-2} (i.e., tilde M2 ≲ 9.7×10^2), with a theoretical maximum observable factor f_SP ≈ 1.00091 in the CC limit; future measurements could decisively test the ZV exterior. The work provides practical tools for testing spacetime geometry around supermassive objects and highlights avenues for extension to rotating or inclined configurations and matter distributions.
Abstract
We study the periapsis shift of timelike bound orbits in the equatorial plane in the Zipoy-Voorhees spacetime, which is an exact, static, axisymmetric, and vacuum solution characterized by the deformation parameter $γ$, including the Schwarzschild spacetime as $γ= 1$. We newly derive the formula for the periapsis shift by the post-Newtonian (PN) expansion and show that the quadrupole moment contributes to the periapsis shift apparently as in the 2PN order. Applying this formula to observational data on S2, a star orbiting closely to Sagittarius A* (Sgr A*), we show that the parameter $γ$ is constrained to $γ\gtrsim 1.9 \times 10^{-2}$ for the gravitational field of Sgr A*. This is equivalent to $\tilde{M}_2 \lesssim 9.7 \times 10^2$ in terms of the dimensionless quadrupole moment $\tilde{M}_2$, that is, the supermassive compact object at Sgr A* cannot be very prolate, whereas $\tilde{M}_2 \geq -1/3$ is required from the solution. Finally, we derive a new series representation in this spacetime using a recently proposed prescription, which shows fast convergence not only in the weak-field regime with not necessarily small eccentricity but also in the strong-field regime with small eccentricity.