Forced motion near black holes

We present two methods for integrating forced geodesic equations in the Kerr spacetime. The methods can accommodate arbitrary forces. As a test case, we compute inspirals caused by a simple drag force, mimicking motion in the presence of gas. We verify that both methods give the same results for this simple force. We find that drag generally causes eccentricity to increase throughout the inspiral. This is a relativistic effect qualitatively opposite to what is seen in gravitational-radiation-driven inspirals, and similar to what others have observed in hydrodynamic simulations of gaseous binaries. We provide an analytic explanation by deriving the leading order relativistic correction to the Newtonian dynamics. If observed, an increasing eccentricity would thus provide clear evidence that the inspiral was occurring in a nonvacuum environment. Our two methods are especially useful for evolving orbits in the adiabatic regime. Both use the method of osculating orbits, in which each point on the orbit is characterized by the parameters of the geodesic with the same instantaneous position and velocity. Both methods describe the orbit in terms of the geodesic energy, axial angular momentum, Carter constant, azimuthal phase, and two angular variables that increase monotonically and are relativistic generalizations of the eccentric anomaly. The two methods differ in their treatment of the orbital phases and the representation of the force. In the first method, the geodesic phase and phase constant are evolved together as a single orbital phase parameter, and the force is expressed in terms of its components on the Kinnersley orthonormal tetrad. In the second method, the phase constants of the geodesic motion are evolved separately and the force is expressed in terms of its Boyer-Lindquist components. This second approach is a direct generalization of earlier work by Pound and Poisson [A. Pound and E. Poisson, Phys. Rev. D 77, 044013 (2008).] for planar forces in a Schwarzschild background.

Figure 1

Comparison between the exact and adiabatic approaches to evolving the orbit, both from Sec. 2b. The top panel shows the exact solution $x(t)$ (solid lines), as well as the difference between the exact and adiabatic $x(t)$ (dashed lines). The insets show close-up views of the first/last 100s of the same curves. The bottom panel shows the disagreement between the exact and adiabatic predictions for the phase $u$ (solid line) and time-offset $t_0$ (dashed line).

Figure 2

Evolution of the orbit under the influence of the “gas-drag” force. The panels show, as functions of Boyer-Lindquist time $t$, the particle coordinates $r$ (top left), $\theta$ (top right) and $\varphi$ (middle left) and the orbital constants $p$ (middle right), $e$ (bottom left) and $\iota$ (bottom right). In each panel the solid curve was computed using the exact evolution, while the dashed curve was computed using the adiabatic evolution. The plots showing $\varphi$, $p$, and $\iota$ have insets which show close-up views of the same data so that the difference between the adiabatic and exact results can be seen.