r100_a0.05_lz0.5

The results given here are for an orbit at r/M = 100, a/M = 0.05. To make the orbit inclined, I started with the z-component of angular momentum needed to make a prograde, equatorial orbit and cut that angular momentum in half. I then solved for the energy and Carter constant needed to produce a stable circular orbit. The following table summarizes the resultant orbital characteristics, and the characteristic effects of radiation reaction on that orbit. These results include multipoles from l = 2 to l = 6.

Orbit 
quantities
Energy
0.9950390
z-component of 
angular momentum (Lz)
5.075963
Carter constant
77.31178
iota
60.00250 degrees
iota'
59.99754 degrees
Omega_phi
1.000025 × 10-3
Omega_theta
9.999249 × 10-4
 
Radiation 
reaction 
quantities
Energy flux to infinity
-6.237244 × 10-10
Energy flux down the horizon
+3.513815 × 10-17
Lz flux to infinity
-3.119053 × 10-7
Lz flux down the horizon
+3.469661 × 10-14
Rate of change of radius
-1.267633 × 10-5
Rate of change of Carter constant
-9.498841 × 10-6
Rate of change of inclination angle
+6.693633 × 10-12
 

All of these quantities are in non-dimensional units. To get to physical units, apply the following rules:

  • Multiply the energy by mu (where mu is the mass of the orbiting particle); multiply the angular momentum by mu M (where M is the black hole mass); multiply the Carter constant by mu2 M2.

  • Divide the frequencies by M.

  • Multiply rate of change of energy by (mu/M)2; multiply rate of change of angular momentum by mu2/M; multiply rate of change of Carter constant by mu3.

  • Multiply rate of change of radius by (mu/M).

  • Multiply rate of change of inclination angle by (mu/M2).

    Note that the inclination angle increases, as predicted by Fintan Ryan [Phys. Rev. D 52, R3159 (1995)]. In this case, since both 1/r and a are fairly small, we expect that Fintan's formula should work out well. Fintan's prediction for this orbit is 7.04 × 10-12, off by about 5%.

    In a different paper, Fintan gives the rate of change of radius for inclined orbits [Phys. Rev. D 53, 3064 (1995)]; his prediction for this orbit is -1.28 × 10-5, off by only 0.98% from the numerical result.

    Note also that the energy and angular momentum flux from the horizon is positive, i.e. the particle's energy and angular momentum increase due to radiation falling into the horizon. This is due to superradiant scattering: radiation is scattered by the hole's ergosphere, absorbing some of the energy stored in the spin of the spacetime, and then gives a "kick" to the orbiting particle.

    Here's what the waveform looks like at this point in the particle's evolution:

    The time axis is for a black hole of 107 solar masses. This waveform is observed in the hole's equatorial plane. The blue lines are h+, the red lines h×. The low-frequency amplitude modulation is due to Lense-Thirring precession, i.e. the dragging of the orbit's nodes by the black hole's spin. The frequency of this modulation is 2 × (Omegaphi - Omegatheta) = 1.001 × 10-7 M-1. This difference is quite small, so orbital precession is very slow: notice that in this plot we haven't even gotten very far out of the initial peak, even though the evolution shown is over roughly 6 years!

    Here's a zoom on the early portion of the signal.

    The radiated energy has the following distribution:

    This is rather narrow. Notice that the falloff slope has a sharp break, just as in the case r = 7M, a = 0.05M. The sharp break appears to be a feature associated with small spin.