The Gravitational Radiation Emitted by Two QuasiParticles around a Schwarzschild Black Hole

We model analytically a relativistic problem consisting of two quasi-particles each with mass m in close orbit around a static Schwarzschild black hole with mass M = 1 situated at the center of mass of the system. The angular momentum l of the system is taken to be 2. We model the mass density of the orbiting particles as a δ-function and we assume that there are no deformations. To model the system, we apply the second-order differential equation obtained elsewhere for a dynamic thin matter shell on a Schwarzschild background. As it is the case in this paper, the framework on which the equation was obtained is Bodi-Sachs. The only change in the equation is that now the quasi-normal mode parameter represents the particle’s orbital frequency from which we are able to analytically compute the gravitational radiation emitted by the system at null infinity. We note that in a real astrophysical scenario the dynamics of the particles paths will be very dynamic and complicated and that the analytical methods used here will have to be developed further to accommodate that.


Introduction
Until recently, all sorts of relativistic binary systems have been studied only theoretically and on the 14 September 2015 a team of LIGO and Virgo collaborators announced their first detection of a gravitational wave signal from a binary black hole system of about 36 and 29 solar masses.This announcement reaffirmed the predictions of the existance of gravitational waves as predicted by

Formalism
The Bondi-Sachs formalism uses coordinates ( ) based upon a family of outgoing null hypersurfaces.We label these hypersurfaces by const.u = , null rays by A x ( ) , and the surface area coordinate by r.In this coordinates system the Bondi-Sachs metric [28] [29] takes the form

AB AB det h det q =
, with AB q being a unit sphere metric, U is the spin-weighted field given by A A U U q = .For a Schwarzschild space-time, 2 W M = − . We define the complex quantity J by

2.
A B AB J q q h = (2) For the Schwarzschild space-time, we have J and U being zero and thus they can be regarded as a measure of the deviation from spherical symmetry, and in For spherical harmonics we use s lm Z rather than s lm Y as basis functions as follows [27] ( ) ( ) The 0 s = will be omitted in the case 0 s = , i.e.
. The s lm Z are orthonormal and real.We assume the following ansatz ( ) where r 0 is the position of the matter shell, and σ the complex frequency mode which is physical damped and which further means that ( ) 0 Im σ > .In the Bondi frame, the field equations splits into;  the hypersurface equations and the evolution equations given by ( ) ,  and the constraint equations for off the matter shell in the case of vacuum given by ) ) Ref. [27] got the following second order differential equation when solving the above systems of ordinary differential equations for the Schwarzschild background; ( ) where , x is the compactification factor in this language.Bishop et al. [31] solved Equation (12) numerically and obtained interesting quasi-normal modes results of a Schwarzschild white hole.However in this paper, we are going to solved it for a different problem since we can apply the same physical settings in the Bondi-frame to model our problem with σ having a different physical meaning as we shall see later.

An analytic Algorithm for Calculating the Gravitational News
We shall use the following algorithm to calculate the gravitational radiation from the system. First we use Equation ( 12) and the constraints Equations ( 9)- (11) to get the junction conditions for the Bondi-Sachs matric variables U, ω and J at the boundary i.e. shell, J J U U , and ω are smooth across the boundary and if this is true, we then  Calculate the News function at +  .

The Problem
We consider a system consisting of two point-particles with equal mass m in quasi-orbit around a stationary Schwarzschild black hole with mass M situated at the center of mass r of the particles when l is 2. We take the orbital radius to be at r 0 which means that the distance between the particles is 2r 0 .We take the initial position of particle 1 to be at r 0 with θ and ϕ given by π/2 and νu respectively, ν is the orbital frequency and u the orbital period of the particles.We also take the initial position of a particle 2 to be at r 0 with θ and ϕ given by π/2 and νu + π respectively.This imply that the rotation in the following figure is in the yz plane.The initial positions of the objects on the figure should not be confused with the actual initial positions just outline which in actual sense should be along the y axis with the particle 1 on the right and the particle 2 on the left.
The dynamics of this problem is governed by Equation (12) and for our numerical calculation purposes we shall use its Ricatti form [31] ( ) ( ) ( ) where v is the orbital period of the system.


We model the problem as follows, we start by applying Equation ( 5) with rr T given by where the matter density ρ in the background space-time is given by ( ) ( ) ( ) Inside the particles orbital radius 0 r r < we set and outside the particles orbital radius 0 r r > we set .
Now integrating with respect to r we get By multiplying Equation (18) with l m Z ′ ′ we get and integrating over the sphere it simplifies to From Equation (20), for 0 m′ ≠ we the gravitational radiation otherwise we don't, and that l m β ′ ′ are generally non-zero for even l and m′ .We now consider the case 2 l′ = and we note that We note that 20 β mode does not vary in time and hence it does not contain the emitted gravitational radiation.
and similarly ( ) and then finally we write Now taking

The Gravitational Radiation
We assume that the orbit is at the innermost stable circular orbit (ISCO), so that where are the Bondi metric functions, and 0 β + , 0 β − are the values of the expansion of the light rays β given by Equation (32) in the exterior and interior domains respectively.Bishop [27] has indicated that the derivatives of J should not be worked out numerically, but should be worked out analytically in terms of 1 J , 2 J and v from Equation (13) with 0.0680 ν = .
We define the general solutions for ( ) x outside and inside the orbital radius respectively as ( ) ( ) where c 4 , c 1 , c 2 , c 9 , c 6 and c 7 are constants to be determined numerically.The which then results in Equations ( 36) and (39) being analytic near mn x .We used Matlab ode45 solver to find numerical solutions of the above derivatives in Equations ( 38) and (39).We used stringent numerical conditions to get the results to about seven significant figures with RelTol of 10 −12 , AbsTol of 10 −12 , and the MaxStep of 5 0.2 10 − × and the results we found to be ( ) ( ) and ( ) ( ) We have tested for the consistency of the above results by using other Matlab solvers; ode23 and ode15s (which uses the Gears method i.e. backward differentiation formulas) and also observed the accuracy of about 15 significant figures.We went further with the test using ode23t which uses the trapezoidal rule, ode23s which is a modified Rosenbrock formula of order 2, and ode23tb which is an implicit Runge Kutta as opposed to ode45 and ode23 and found the consistency of about 8 significant figures and as opposed to 15 significant figures which is also accurate enough.This illustrate how accurate and valid the results are.These results are very crucial in obtaining the emitted gravitational radiation and hence determining the extent of their convergence is of most paramount importance.
From the hypersurface equation Equation ( 7) rewritten as we are able to the Bondi metric function . But to find the solution the integration should be done analytically where possible.We only need a solution which is valid in a neighborhood of 0 x x = .Henceforth, it is convenient to make the coordinate transformation 1 x r x → = .Equation (46)   can further be rewritten as where for 2 l = we have 2 6 L = − .The constraints equations Equations ( 9), ( ) which we then apply in the domains D + and D − .Since these constraints are not completely analytic, this means that we should only evaluate them at the ISCO.We use them among others to eliminate the constants c 1 , c 2 , c 6 , and c 7 .We now assume that we end up with the solutions Thus, from the constraints ( ) which means that for large r, 0 0 β + = at +  imply that the coordinate time is the same as proper time and that the regularity at +  require 4 0 c = .We also impose the following junction conditions at 0 r : Physically the metric functions J and U have the smooth asymptotic expansion characteristic through out the entire computational domain and this property is confirmed in Figure 1 and Figure 2. The metric function ω do not have this physical property as can be confirmed in Figure 3  With further improvement, the method can be develop to look at two unequal orbiting black holes or neutron stars or a combination of both with efficiency and accuracy as demonstrated in [24] for single orbiting black hole/neutron star.
) )) ( 2) The Bondi metric variables computed at r 0 ( ) )) ( ) ) )) addition, they contain all the dynamic content of the gravitational field in the linearized regime[30].Usually we can describe this space-time by 0

.−
We then found the change in the Schwarzschild coordinate time  for one complete revolution of 92.3436 from which we found the orbital frequency ν of 0.0680.To now find the numerical solutions to continue Equation (13) we make the spatial coordinate transformation of 1respectively.We start the calculation with the transformed Equation(12) given by

.
and the hypersurface Equation (47), we found the metric variables From which the expressions of the constants c 9 , c 7 , c 5 , and c 10 , were found.We now impose the Bondi gauge conditions: conditions, we were able to find the exact numerical values of the constants c 1 , c 2 , and c 6 at 0 6 r = .The exact numerical values of the constants c 9 , c 7 , c 5 , and c 10 were then found by substituting the values of c 1 , c 2 , and c 6 back into their expressions.From here we were then able to plot the graphs of the Bondi metric functions observed in the following graphs.

Figure 1 .
Figure 1.The graph of

4.1. The Linear Expansion of the Light Rays From the System to +
Thus we are only interested in 22