Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole

Abstract

We present the first numerical calculation of the (local) metric perturbation produced by a small compact object moving on an eccentric equatorial geodesic around a Kerr black hole, accurate to first order in the mass ratio. The procedure starts by first solving the Teukolsky equation to obtain the Weyl scalar $ψ_4$ using semi-analytical methods. The metric perturbation is then reconstructed from $ψ_4$ in an (outgoing) radiation gauge, adding the appropriate non-radiative contributions arising from the shifts in mass and angular momentum of the spacetime. As a demonstration we calculate the generalized redshift $U$ as a function of the orbital frequencies $Ω_r$ and $Ω_φ$ to linear order in the mass ratio, a gauge invariant measure of the conservative corrections to the orbit due to self-interactions. In Schwarzschild, the results surpass the existing result in the literature in accuracy, and we find new estimates for some of the unknown 4PN and 5PN terms in the post-Newtonian expansion of $U$. In Kerr, we provide completely novel values of $U$ for eccentric equatorial orbits. Calculation of the full self-force will appear in a forthcoming paper.

Figures (6)

• Figure 1: Graphical representation of the method of extended homogeneous solutions. The field equations are solved in the interior and exterior vacuum regions in the frequency domain. To obtain the field at the particle the time domain solutions are constructed and analytically extended towards the worldline (either from the inside or outside).
• Figure 2: Convergence of the $l$-modes with increasing $n$-modes (for an orbit with spin $a=0.9$, semilatus rectum $p=3.32$, and eccentricity $e=0.2$). Each line represents the $n$-mode contributions to one $l$-mode with darkers shades representing higher values of $l$. The two horizontal lines represent the target accuracy and the regularization parameter $B$ to which the sum of the $n$-modes will (approximately) converge for each $l$-mode.
• Figure 3: The maximal contribution to the $l$-modes at each $m$ and $n$$n$-modes (for an orbit with spin $a=0.9$, semilatus rectum $p=3.32$, and eccentricity $e=0.2$). The horizontal plane represents the value of the accuracy goal.
• Figure 4: The l-modes $h_{uu}^{l,\pm}$ for an orbit with spin $a=0.9$, semilatus rectum $p=3.32$, and eccentricity $e=0.2$. The top lines are the values of the retard field before subtraction of the regularization parameters. The bottom lines give the values after subtraction of the regularization parameters. They follow the diagonal gridlines which represent $l^{-2}$ decay.
• Figure 5: The difference between the partial sums of the $l$-modes of $\langle h_{uu}^\pm\rangle$ (excluding completion terms) of an orbit with spin $a=0.9$, semilatus rectum $p=3.32$, and eccentricity $e=0.2$ . As $L$ increases the difference decays as approximately $l^{-4}$ (diagonal gridlines).