The self force is an approximation to General Relativity in limit of either an extreme or intermediate ratio of masses between the small and large black hole. The smaller blackhole's orbit would be a geodesic if it were massless, but since it contains a small amount of mass relative to the central blackhole, this creates a back-reaction on the spacetime that distorts the spacetime around the smaller blackhole and causes a force on the smaller blackhole as it orbits, called the self force of the smaller blackhole.

We fed this force into the geodesic equation as a source term, and this produces a wave equation for orbital evolution using a numerical differential equation solving method.  In our case we use the discontinuous galerkin method along a the x axis, with spherical harmonics filling out the y and z axes. We considered a Schwarzchild blackhole, an uncharged, spherically symmetric, and non-rotating blackhole.

We use a toy model that is an approximation to the General Relativistic form, called the scalar self force. This approximation allowed us to develop the tools for creating precise waveforms quickly, but ultimately General Relativity will be necessary (for someone else to carry out). A scalar particle is a spin 0 particle, while gravity is spin 2. Another way of looking at this is that a scalar field has only strength, while a tensor field distorts space in two directions. Water waves, waves on springs, or sound waves are vector fields, because they distort space in one direction-- up and down, side to side, or front to back respectively. 

In this work I ported an existing Fortran code to C++. I also made use of that existing Fortran code to perform some analyses on the best way to extrapolate better data from that code. 

First I tested the generalized wave equation solver with sinusiodal and gaussian initial data on a Minkowski (flat) spacetime. The gaussian evolved wave, and the convergence tests of our sinusiodal discontinuous galerkin resolution of our spatial grid as well as the convergence tests of the resolution of the lm oder of our spherical harmonic sum are shown.

Next we used initial data resembling a post-merger black hole, with a gaussian perturbation at the center of a Schwarzchild spacetime. This resulted in the appropriate ringing as a function of time, called the Quasi--Normal-Modes (QNM) and the power-law tails of the amplitude after the ringing stops.

After that, I implemented a toy model of a stellar mass black hole on a circular orbit about a supermassive black hole, using the scalar self force and a Schwarschild spacetime.  This code is governed by a wave equation where the singular field at the blackhole (psi^S) is subtracted from the retarded field that is delayed due to light propagation times as it moves away from the stellar mass blackhole (psi^ret), and the resulting field is subject to a wave equation (box is the differential equation operator). The right hand side that provides the forcing function to the differential equation is the effective source of the stellar mass blackhole, S_eff. This effective source is due to the backreaction of the stellar mass blackhole's curvature interacting with the curvature of the supermassive blackhole's spacetime, causing it's path to deviate from a geodesic. The self force itself depends on a window function about which the singular field is removed, w. That window function is called the world tube. 

For this model, I maintained a circular orbit, and measured the resulting self force, to determine the precision of the comparison between the C++ code I had implemented, and the original FORTRAN code, as well as to aid implementation. The resulting precision was 1 part in 10^10 to  10^12, or roundoff error! The three plots below show the absolute and relative error scaled by a large factor ranging from 10^10 to 10^12 for modes l=0 m=0, l=1 m=1, and l=2 m=2. The shape of the original self force curve is also shown. Note that it approaches a constant for l= 1 m=0, oscillates about 0 for l=1 m=1, and approaches a constant for l=2 m=2.

The width and shape of the world tube, that removes the singular field, is shown below. The result of the l=2 m=0 (as opposed to l=2 m=2) self-force is also shown below, though without relative and absolute error. The oscillations of radius versus time are also shown, and it is clear that it is moving in a periodic fashion, like a circle. 

All of the plots  from here forward depict the results of the original FORTRAN code from which I ported my C++ code, and my analysis of that code. In the plots immediate below, one can see the precession of the orbit of the stellar mass black hole about the supermassive black hole (at least our toy model for it) in the first figure, as the orbit spirals around the center. This is explained in the second figure, where the angular frequency, chi, is higher than the radial frequency, so that it completes one angular period faster than it completes one radial period. The third plot depicts the self force as a function of time. The self force converges with lm mode. The first few lm modes have the highest amplitude, and oscillate about a constant value. As the mode number decreases, the oscillations become centered about zero, and the amplitude decreases. There is an initial transient decreasing in amplitude that forms the diagonal sloping downward on the left, due to initial conditions.