The measurable impact of the 2pN spin-dependent accelerations on the jet precession of M87$^\ast$

TL;DR

The work derives spin-dependent accelerations for a spinless test particle in Kerr spacetime using harmonic coordinates, extending the analysis to formal second post-Newtonian order and allowing arbitrary spin orientations. It decomposes the accelerations into velocity- and position-dependent components up to $O(1/c^4)$, including novel $O(S/c^4)$ and $O(S^3/c^4)$ terms with no direct material-body analogues, and computes their long-term effects on orbital elements via a perturbative scheme. Applying the results to M87$^ imes$, the study shows that the $O(S/c^4)$ term can contribute up to about 20% of the Lense-Thirring precession, with $O(S^2/c^4)$ contributing a few percent and $O(S^3/c^4)$ being largely subdominant; these contributions shift the permissible spin-orbit parameter regions inferred from disk/jet precession measurements. Overall, the paper demonstrates that higher PN spin-dependent accelerations in Kerr spacetime can leave measurable imprints on jet precession, potentially enabling new tests of General Relativity in the strong-field regime with SMBH observations like M87$^ imes$.

Abstract

Motivated by recent accurate measurements of disk/jet coprecessions around some galactic supermassive black holes, the accelerations experienced by an uncharged, spinless object in the Kerr metric, written in harmonic coordinates, are analytically calculated up to the formal second post-Newtonian order. To such a level, some new accelerations make their appearance. They are proportional to even and odd powers of the hole's angular momentum. Their counterparts are not known where the primary is a material body. After expressing them in a coordinate-independent, vector form valid for any orientations of the hole's spin axis in space, their orbital effects are perturbatively worked out in terms of the particle's Keplerian orbital elements. The resulting expressions, averaged over one orbital revolution, are valid for generic shapes and inclinations of the orbit. The orbital plane's precession proportional to the first power of the hole's angular momentum and to the reciprocal of the fourth power of the speed of light amounts to about twenty per cent of the corresponding Lense-Thirring effect. The latter is believed to be the cause of the accurately measured disk/jet precessional phenomenology, currently measured to a few per cent accuracy. Although at a lesser extent, also the precession proportional to the second power of the hole's spin and to the reciprocal of the fourth power of the speed of light is measurable. Allowed domains in the parameter space of the jet precession around M87$^\ast$ are displayed.

Figures (2)

• Figure 1: Allowed regions in the $\left\{a^\ast,r_0\right\}$ parameter space of M$87^\ast$ obtained by imposing that the absolute values of the theoretical frequencies of the precession of the orbital plane of a fictitious test particle moving along a tilted, circular effective orbit around the supermassive black hole lie within the experimental range of Equation (\ref{['Op']}). The labels ‘ LT’, ‘ $Q_2$’ and ‘ $Q_4$’ denote the standard Lense-Thirring and the formally Newtonian even zonal harmonic frequencies of degree $\ell=2,4$, respectively, while ‘ $S/c^4$’, ‘ $S^2/c^4$’ and ‘ $S^3/c^4$’ refer to the contributions proportional to $S/c^4$, $S^2/c^4$ (the total one) and $S^3/c^4$, respectively. See Figure \ref{['Fig:2']} for an overview of all permitted regions.
• Figure 2: Joint representation of all the allowed regions in the $\left\{a^\ast,r_0\right\}$ parameter space of M$87^\ast$ obtained by imposing that the absolute values of the theoretical frequencies of the precession of the orbital plane of a fictitious test particle moving along a tilted, circular effective orbit around the supermassive black hole lie within the experimental range of Equation (\ref{['Op']}). The labels ‘ LT’, ‘ $Q_2$’ and ‘ $Q_4$’ denote the standard Lense-Thirring and the formally Newtonian even zonal harmonic frequencies of degree $\ell=2,4$, respectively, while ‘ $S/c^4$’, ‘ $S^2/c^4$’ and ‘ $S^3/c^4$’ refer to the contributions proportional to $S/c^4$, $S^2/c^4$ (the total one) and $S^3/c^4$, respectively.