A relativistic treatment of accretion disk torques on extreme mass ratio inspirals around spinning black holes

TL;DR

This work develops a fully relativistic framework for disk–SCO interactions in extreme mass-ratio inspirals around spinning black holes by combining self-force theory with Hamiltonian perturbation theory. It derives analytic expressions for the energy and angular-momentum exchange at Lindblad resonances in Kerr spacetime and explores how SMBH spin modifies resonance locations and torque reversal, showing relativistic torques can greatly exceed Newtonian estimates in the strong-field regime. The study finds that torque reversal is largely insensitive to spin in many regimes but is sensitive to disk density gradients, and it demonstrates that disk torques, while smaller than gravitational-wave flux, can still influence EMRI dynamics over the many cycles observed by LISA, especially for retrograde configurations. Overall, the results highlight the importance of incorporating relativistic environmental effects in EMRI waveform modeling and motivate future work on more realistic disk physics and generic orbital configurations.

Abstract

We model the motion of a small compact object on a nearly circular orbit around a spinning supermassive black hole, which is also interacting with a thin equatorial accretion-disk surrounding the latter, through tools from self-force and Hamiltonian perturbation theory. We provide an analytical and relativistically-accurate formalism to calculate the rate of energy and angular momentum exchanged at Lindblad resonances. We show that strong relativistic effects can potentially cause a reversal in the direction of the torque on the small compact object if the surface density gradient is not too large. We analytically explore the dependence of the torque reversal location on the spin of the supermassive black hole and demonstrate that the ratio of the reversal location to the innermost stable circular orbit is approximately insensitive to the spin of the supermassive black hole. Our results show that relativistic torques can be 1--2 order of magnitude larger than the Newtonian torque routinely used in the literature to model disk/small-compact-object interactions close to the supermassive black hole. Our results highlight the importance of including relativistic effects when modeling environmental effects in extreme mass-ratio inspirals.

Figures (5)

• Figure 1: Location where the inner Lindblad resonances begin to get closer to the orbit of the SCO as a function of the spin of the SMBH. Observe that the transition occurs in the inner regions of the SMBH where EMRI are expected to be detected in the LISA band.
• Figure 2: The torque reversal location as a function of the spin of the SMBH. The solid blue curve shows the approximate location where torque changes sign as a function of the spin of the SMBH obtained by solving $\mathcal{G}(p') = 0$ [Eq. \ref{['eq:G-def']}]. The orange dash-dotted curve is the relativistically accurate location obtained by solving $\mathcal{G}_1(p')$ [Eq. \ref{['eq:G1-location']}]. The green dashed curve shows the location of the ISCO.
• Figure 3: The dependence of the torque reversal location on the surface density gradient. Observe that the location approaches the ISCO for large and negative values of $\Sigma_{p}$ and the location roughly remains constant with respect to the ISCO as a function of the spin of the SMBH.
• Figure 4: The percentage difference between the Newtonian and the relativistic torque formula for different values of the spin of the SMBH as a function of the location of the SCO from the ISCO.
• Figure 5: Ratio of the rate of energy exchange due to disk-CO interactions to gravitational wave loss to infinity for a few different values of the aspect ratio and spin of the SMBH. The left panel shows the ratio for $\Sigma_0 = 10^3$ and $\Sigma_p=-3/2$, which are $\alpha$ disk-like values. The right panel uses $\Sigma_0 = 10^5$ and $\Sigma_p=-3/5$, which are consistent with $\beta$ disk-like values. Observe that for both panels, the ratio decreases with the spin of the SMBH because the ISCO is closer to the horizon of the SMBH, and gravitational wave energy loss increases drastically as we approach the horizon.