Forced motion near black holes

Abstract

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 gravitationalradiation-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 non-vacuum 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 [1] for planar forces in a Schwarzschild background.

Census of transient orbital resonances encountered during binary inspiral

“…More precise understanding of how a system evolves through orbital resonances is almost certainly necessary to ensure that future observatories like eLISA will be able to measure these waves. Our plan for going beyond this analysis is to compute components of the instantaneous dissipative self force and the self consistently evolve the system as a "forced geodesic" [19]. Results describing the instantaneous dissipative self force for scalar fields are given in Ref.…”

“…Eq. (2.14)], and plan to couple this to a prescription for computing forced motion near Kerr black holes [19]. This work will be reported later [9].…”

Extreme mass ratio inspiral data analysis with a phenomenological waveform

“…[9]. Alternatively, one could have a separate evolution of the orbital motion using self-force computed across various geodesic orbits and employing the osculating elements approach [8,14]. The waveform at infinity could be obtained from the Teukolsky equations [15] in time or in frequency domain [16,17].…”

“…Under the self-force the motion is not geodesic anymore, however, it can still be accurately described as slow drift from one geodesic to another. In the oscillating element approach, we evolve three constants defining initial position of the compact object (due to conservative part of the self-force) as well as fE; L z ; Qg or equivalently ff r ; f ; f g [8,14]. We evolve fE; L z ; Qg according to PN expressions suggested in Ref.…”

“…There are also questions concerning the calculation of the orbital evolution under the self-force: the self-force depends on the past history of the compact object (which is usually assumed to be a geodesic in the background spacetime). To compute the motion under the self-force, one can use the osculating elements approach [8] or the self-consistent approach of direct integration of the regularized equations [9]. For more details on this subject, we refer to Ref.…”