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

TL;DR

This work develops a relativistic formalism for disk–SCO interactions in Schwarzschild spacetime, combining self-force insights with Hamiltonian perturbation theory to derive a relativistic disturbing function and the secular evolution at Lindblad and corotation resonances. It provides analytic guarantees for resonance locations, torque magnitudes, and eccentricity evolution, and demonstrates that relativistic corrections can enhance torques by 1–2 orders of magnitude for orbits within $\sim 10M$ and can even reverse the torque near the ISCO. Using a parameterized disk model, the authors compare disk-driven energy exchange with gravitational-wave losses, finding disks generally subdominant for $M=10^5M_\odot$ but potentially competitive for $M=10^8M_\odot$ near ISCO, leading to meaningful dephasing in LISA-band EMRIs. The results underscore the necessity of relativistic environmental modeling for accurate EMRI waveform predictions and motivate extensions to spinning (Kerr) black holes and more realistic disk physics.

Abstract

We develop a relativistically accurate formalism to model the interaction between stellar mass compact objects embedded in thin accretion disks around a non-spinning supermassive black hole, using tools from self-force theory and Hamiltonian perturbation theory. We then apply this formalism to analyze the evolution of a compact object on a nearly circular and equatorial orbit interacting with a thin equatorial disk. We provide analytic and relativistically-accurate expressions for the rates of energy and angular momentum exchanged during interactions due to Lindblad and corotation resonances. Our results show that relativistic corrections can enhance the magnitude of the torque by 1-2 orders of magnitude compared to purely Newtonian expressions when the orbit of the compact object is smaller than $10$ Schwarzschild radii of the supermassive black hole. We also demonstrate that strong relativistic shifts the inner Lindblad resonances closer to the compact object than the outer Lindblad resonances when the compact object is closer than 4 Schwarzschild radii to the supermassive black hole, potentially leading to a reversal in the direction of the torque acting on the compact object. Finally, we provide a dephasing estimate and show that using the relativistic torque formula is crucial to obtain reliable estimates for extreme mass ratio inspirals in orbits closer than 5 Schwarzschild radii to the supermassive black hole. Our results highlight the importance of using relativistically-accurate models of environmental interactions in extreme mass-ratio inspirals close to a supermassive black hole.

Figures (5)

• Figure 1: Cartoon (not to scale) depicting the interaction between a small SCO (yellow) on an equatorial circular orbit interacting with a thin accretion disk (blue background) surrounding a supermassive black hole (black circle). The thin cyan disk surrounding the SCO is used to highlight the fact that the dominant source of energy and angular momentum exchange are resonances close to the SCO. We also highlight two important surfaces where relativistic effects are important. The green dashed circle shows the approximate location at which relativistic effects change the sign of differential Lindblad torque on the SCO. The relativistic effects become extremely large near the innermost stable circular orbit, shown by a solid red circle.
• Figure 2: Location of the $k=\pm 1$ corotation resonances as a function of the position of the SCO. In the left panel we show the distribution of the inner (red squares) and outer (blue dots) corotation resonances when the SCO position is fixed at $p'= r/M = 10$ using Eq. \ref{['eq:location-corotation-resonance']}. Observe that the resonances are approximately symmetrically distributed. In the right panel, we quantify the asymmetry in the location of inner and outer resonances by plotting $j \Delta_{\mathrm{cr}}$ [Eq. \ref{['eq:Delta-corot']}] as a function of $p'$. Observe that $j \Delta_{\mathrm{cr}}$ is always positive which indicates that the outer corotation resonances are closer to the SCO than the inner corotation resonances.
• Figure 3: Comparison between Newtonian and relativistic torque expressions. The left panel shows the absolute value of the ratio of the differential relativistic Lindblad torque [Eq. \ref{['eq:Ldot-Relativistic-expr-LR']}] to the Newtonian torque [Eq. \ref{['eq:Ldot-Newt-expr-LR']}] as a function of the position of the SCO for different values of $\Sigma_{p}$. Observe that the ratio steeply increases as we approach the innermost stable circular orbit. The ratio also changes sign around $p' \sim 8$ because of the transition in the distribution of the inner and outer Lindblad resonances. In the right panel, we show the percentage difference between the Newtonian and relativistic torques.
• Figure 4: Comparison between energy exchange due to disk-SCO interaction to gravitational-wave emission for a supermassive black hole of mass $10^5 M_{\odot}$ (left) and $10^8 M_{\odot}$ (right) for different values of $\Sigma_{0}$ and $h_0$. From the left plot, we see that for disk surface densities characteristic of $\alpha$-disks $\Sigma_{0} \sim 10^3$, the energy exchange due to disk-SCO interaction is 6-8 orders of magnitude smaller than the loss due to gravitational-wave emission. For $\beta$-disk like values $\Sigma_{0} \sim 10^7$, the ratio increases to $10^{-2}-10^{-5}$ . In right plot, we see that the ratio increases significantly because of the increase in supermassive black hole mass [Eq. \ref{['eq:scaling-relationship-Edot-Ldot-edot']}] and we see that for $\beta$-disk like surface densities the energy exchange due to the disk can compete with gravitational-wave loss.
• Figure 5: Total accumulated phase due to disk-SCO interactions for fiducial EMRI systems with parameters listed in Table \ref{['tab:EMRI-params']}. Observe that for system I (left panel) the total accumulated phase is large in a significant portion of the $\Sigma_{0},h_{0}$ parameter space due to the longer duration it spends in the LISA band. The total phase accumulated for system II (right panel) is greater than $\mathcal{O}(10)$ radians only when $\Sigma_{0} > \mathcal{O}(10^5)$. However, system II is highly relativistic due to its closeness to the supermassive black hole and we require the relativistic expression for the energy loss due disk-SCO to accurately model this system, see Eq. \ref{['eq:sys-II-diff']}.