The Gravitational Radiation Emitted by a System Consisting of a Point Particle in Close Orbit around a Schwarzschild Black Hole

We analytically model a relativistic problem consisting of a point-particle with mass m in close orbit around a stationary Schwarzschild black hole with mass M = 1 using the null-cone formalism when l = 2. We use the δ-function to model the matter density of the particle. To model the whole problem, we apply the second order differential equation obtained elsewhere for a dynamic thin matter shell around a Schwarzschild black hole. The only thing that changes on the equation is the quasi-normal mode parameter which now represent the orbital frequency of the particle. We compare our results with that of the standard 5.5 PN formalism and found that there is a direct proportionality factor that relates the two results, i.e. the two formalisms.


Introduction
To date, numerically speaking black hole systems have been studied only theoretically and this means that all areas of mathematics (both pure and applied) and the computational sciences are heavily utilized in this field.As a result at the moment the research into the formation and the evolution of compact binaries i.e. white dwarfwhite dwarf, neutron star-neutron star, black hole-black hole and colliding black holes is progressing very rapidly and important results are being published [1].The research into a black hole-neutron star binary system in quasi-equilibrium or in full dynamic motion in either Newtonian (see [2][3][4][5][6] for quasi-equilibrium and [7][8][9][10][11][12][13] for dynamic motion) or relativistic theory (see [14][15][16][17][18][19] for quasi-equilibrium and [20][21][22][23][24][25] for dynamic motion) is as challenging as that of a black hole-black hole binary system or two colliding black holes.Relativistically speaking, neutron stars binaries and black hole binaries are thought to be the primary sources of the gravitational radiation to be hopefully detected by the ground-based LIGO [26] and for white dwarf binaries by the spacebased LISA [27].In this paper we analytically study in the Bondi-frame, a binary system consisting of a pointparticle in quasi-orbit around a stationary Schwarzschild black hole.Our main aim shall be to determine the emitted gravitational radiation by the system at   .The null-cone formalism have also been used numerically to study quite extensively other systems consisting of black hole binaries [28].The PN formalism has should to be accurate for modeling gravitating systems at the Newtonian regime.So, by comparing the results from these two formalisms for the same physical problem is vitally important in validating our final results.This paper is structured as follows: Section 2 gives the background material.Section 3 define the physical problem to be studied.Section 4 calculates the emitted gravitational radiation at   .

The Null-Cone Formalism
The Bondi-Sachs formalism uses coordinates based upon a family of outgoing null hypersurfaces.We label these hypersurfaces by  , null rays by , and the surface area coordinate by r.In this coordinates system the Bondi-Sachs metric [29,30]

W r h U U u u r r r h U u x r h x x
where , with being a unit sphere metric, U is the spin-weighted A. S. KUBEKA 1504 field given by A U U . For a Schwarzschild spacetime, . We define the complex quantity J by

A B AB
q q h  J (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 addition, they contain all the dynamic content of the gravitational field in the linearized regime [31].Usually we can describe this space-time by 0 , or by and .
For spherical harmonics we use s lm Z rather than s lm Y as basis functions as follows [32]   The will be omitted in the case  , i.e.
. The s lm Z are orthonormal and real.We assume the following ansatz where 0 is the position of the matter shell, and r  the complex frequency mode which is physical damped and which further means that Im .In the Bondi frame, the field equations splits into;    > 0  the hypersurface equations and the evolution equations given by , 4 : 8π  and the constraint equations for off the matter shell in the case of vacuum given by Ref. [32] got the following second order differential equation when solving the above systems of ordinary differential equations for the Schwarzschild background; x is the compactification factor in this language.Bishop et al. [33] 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  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

The Physical Problem
We consider a system consisting of a point-particle with mass m in quasi-orbit around a stationary Schwarzschild black hole at 0 with mass M for is 2. We simplify the coordinate dynamics of the center of mass of the system by doing a mathematical trick.That is, we place a second point-particle directly opposite the first pointparticle at 0 .That means the center of mass will remain at the origin i.e. at the black hole during the duration of the orbit.The total distance between the point-particles is r  r A. S. KUBEKA 1505 0 .This trick has the consequence that the emitted gravitational radiation will be amplified by a factor of two which in the final analysis we divide the final result by two.This is as a result of the introduced point-particle.This procedure is physical correct as long as the pointparticles are equal and in equidistance in a quasi-circular orbit.We take the initial position of the first particle to be at 2 and u  for the  and  respectively.
We also take the initial position of the second particle to be at π 2 and π u   for  and  respectively.We define  as the orbital frequency and u as the or- bital period of the particles.
The dynamics of this problem is governed by Equation (12) and for our numerical calculation purposes we shall use its Ricatti form [33] where  is the orbital period of the system.

The Linear Expansion of the Light Rays
We start by applying Equation ( 5) with given by where the matter density  in the background spacetime is given by Inside the orbital radius  we set 0,   lm (16) and outside the orbital radius we set 0 r r Now integrating with respect to r we get i.e.

   
By multiplying Equation (19) with Z   we get ) and integrating over the sphere it simplifies to 21), for we the gravitational radiation otherwise we don't, and that l m    are generally non-zero for even and m .We now consider the case l 2 l  and we note that we get and Thus from Equation ( 21) Copyright © 2012 SciRes.JMP A. S. KUBEKA 1506 and similarly   and then finally we write Now taking M  , Equation ( 35) then becomes Finally, we divide Equation ( 34) by a factor of 2 to appropriate it for a single point-particle in orbit around a Schwarzschild black hole.We are able to do this because for π 2 0 . Therefore from here onwards, the calculations will be that of a single pointparticle.Equation ( 34) now becomes

The Gravitational Radiation
We assume that the orbit is at the innermost stable circular orbit (ISCO), so that 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  of 0.0680.To now find the numerical solutions to continue Equation (13) we make the spatial coordinate transformation of 1 x r  which then imply that the ISCO is now at with numerical solutions and  respectively.We start the calculation with the transformed Equation (12) given by where , are the Bondi metric functions, and 0 U    , 0   are the values of the expansion of the light rays  given by Equation (35) in the exterior and interior domains respectively.N.B the derivatives of J should not be worked out numerically, but should be worked out analytically in terms of 1 J , 2 J and from Equation ( 13) to be found with We define the general solutions for 2 J x mn at x outside and inside the orbital radius respectively as where , , c , c c and 7 c are constants to be determined numerically.The functions nd therefore can be Taylor expand as which then results in Equations ( 38) and (41) being analytic near mn x .We used Matlab ode45 solver to find numerical solutions of the above derivatives in Equations ( 40) and (41).We used stringent numerical conditions to get the results to about seven significant figures with RelTol of 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 consistency is of the most paramount importance to obtaining accurate final results.
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 . Equation (48) can further be rewritten as where for l we have 2 .The constraints equations Equations ( 9), (10), and (11) which we then apply in the domains  and  .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 , , , and .We now assume that we end up with the solutions Thus, from the constraints , 0 , We now impose the Bondi gauge conditions:  which means that for large , 0   at   imply that the coordinate time is the same as proper time and that the regularity at . We also impose the following junction conditions at : From the junction conditions, we were able to find the exact numerical values of the constants , , and at  .The exact numerical values of the constants , , , and were then found by substituting the values of , , and back into their expressions.From here we were then able to plot the graphs of the Bondi metric functions as observed in the following graphs; 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 Figures 1 and 2. The metric function  do where implemented correctly and that our numerical methods and the analytical algorithms we implemented to calculating the gravitational radiation worked properly as intended.
Then finally, since we are in the Bondi gauge, we found the gravitational news to be with the Bondi mass loss .We compare our results with that of 5.5 PN formalism by Poisson [34] and Sasaki et al. [35] which they found the gravitational radiation of the same system like ours with the same physical conditions as in this paper of about

Conclusion
The work presented here provides us with further future research opportunities to apply the analytic method presented here in the Bondi-frame, to real astrophysics problems involving all sorts of relativistic objects to calculate and analyze the emitted gravitational radiation at null infinity.The next step will be to apply this method to a real relativistic astrophysics problem involving a Kerr background.

20  22 
is that it does not vary in time so this mode does not contain the gravitational radiation.So we are more interested in and , and the hypersurface Equation (49), we found the metric variables , ,