Binary System with different masses in the characteristic formulation

# Point Particle Binary System with Components of Different Masses in the Linear Regime of the Characteristic Formulation of General Relativity

C E Cedeño M , J C N de Araujo Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Av. Dos Astronautas 12227-010 Jardim da Granja, São José dos Campos SP, Brazil
###### Abstract

A study of binary systems composed of two point particles with different masses in the linear regime of the characteristic formulation of general relativity with a Minkowski background is provided. The present paper generalizes a previous study. The boundary conditions at the world tubes generated by the particles’s orbits are explored, where the metric variables are decomposed in spin-weighted spherical harmonics. The power lost by the emission of gravitational waves is computed using the Bondi News function. The power found is the well-known result obtained by Peters and Mathews using a different approach. This agreement validates the approach considered here. Several multipole terms contribution to the gravitational radiation field is also shown.

###### pacs:
95.30.Sf, 04.30.-w, 04.25.Nx, 04.30.Db
: Class. Quantum Grav.

Keywords: Characteristic formulation, linear regime, gravitational waves, Bondi News

## 1 Introduction

The production of gravitational waves by binary systems, in the limit in which mass and momentum transfer do not occur, has been well described with high accurate precision with both the Post-Newtonian and the Post-Minkowskian approaches. These are the most useful tools in general relativity to provide good estimates of the features of the gravitational wave sources. Recently, great advances in these methods have improved their precision, using different multipole expansions. For a review of these issues we refer the reader to [1].

On the other hand, the characteristic formulation of general relativity on outgoing null cones provides an interesting point of view for treating gravitationally radiating systems, since it is based on the radiation’s coordinates. Originally formulated by Bondi et. al. in the 1960s [2, 3], it has served as the basis for the development of recent and accurate characteristic numerical codes. These have evolved from the quasi-spherical regime, where the non-linear angular terms are neglected [4], to the full non-linear case [5]. With the inclusion of (perfect fluids) matter sources into the numerical codes [6], post hoc extensions were made, putting the system of equations in terms of only first order angular derivatives [7]. The development of the Cauchy-characteristic matching (CCM) method [8] allows the implementation of highly accurate gravitational wave propagation techniques. Great advances in the precision for the angular derivatives were made through the introduction of a gnomonic atlas to cover the angular manifold [9, 10]. Also, there has been impressive progress in spectral methods [11] and in mixed methods, combining finite-difference schemes and spectral representation of the angular part of the field equations, including use of the Runge-Kutta 4 method to perform the time integration [12, 13, 14].
The main purpose of such codes, after all, is to simulate the gravitational radiation patterns produced, for example, in stellar collapses or by coalescing black holeblack hole (BH-BH), black holeneutron star (BH-NS) or neutron starneutron star (NS-NS) binary systems. As calibration tools, some toy models have been constructed from analytical solutions of the Einstein field equations in the linear regime (see, e.g., [15, 16, 17, 18]).
These toy models allow one to study the boundary conditions that the metric and their derivatives should satisfy in different situations in a simple, clear and straightforward way. Bishop et. al. [15, 17] and Kubeka [18], provide a prescription to solve analytically the field equations for the vacuum in the linear regime. Then, they construct solutions for a variety of simple systems, such as a static (dynamical) thin shell, and a binary system composed of point particles of equal masses moving in circular orbits. However, the issue concerning the boundary conditions deserves additional studies, and most importantly, binaries of different masses in circular orbits have not been considered and therefore deserve closer scrutiny.
Therefore, the present paper generalizes a study presented in Sec. III of [17], which is related to binary systems. It is worth stressing that the approach considered here has been performed in a completely analytical way. The validity of this approach is supported by finding that the power lost in gravitational waves by the system agrees with the well-known result obtained by Peters and Mathews [19] using a different approach.
In the next sections, the case of a binary system composed of two point particles of different masses moving in circular orbits in a Minkowski background is studied. In Sec. II, we present a brief review of the formalism, the notation used and the Einstein field equations in the characteristic formulation on null cones oriented to the future. In Sec. III the specific solutions for the field equations are constructed by expanding the metric variables in terms of the spin-weighted spherical harmonics. A discussion on the boundary conditions satisfied by the coefficients in these expansions for arbitrary values of the and modes is provided. In Sec. IV the gravitational radiation emitted by the binary system is computed using the Bondi News function. In particular, the contribution to the gravitational radiation by the () odd modes due to the asymmetry (different masses) of the system is considered, and it is shown how these terms vanish if the system is symmetric (equal masses). Finally, in Sec. V other issues for further study are presented.

## 2 Formalism

The characteristic formulation based on future-oriented null cones, when the eth formalism is employed, is well known and has been used to explore a wide variety of problems [9, 13, 14, 12, 16, 20]. However, a brief review of some of its most important aspects is necessary to present the notation and conventions used here. We use the geometrised unit system, i.e. . The Greek indices run from 1 to 4, whereas the capital latin letters run from 3 to 4, labelling the angular indices. The coordinates used are , where the retarded time labels each null cone upon which the spacetime is foliated, is the luminosity distance and represents the angular coordinates. Here can denote or for the usual spherical coordinates or for the stereographic two patch representation, respectively. In these coordinates the Bondi-Sachs metric takes the form

where and the redshift are related to the square of the ADM lapse function [4]. The shift vector between two successive null cones is labelled by . The metric associated with the angular manifold (unit sphere) is represented by (). In addition, the Bondi gauge is imposed, i.e. . Further it is required that .
The metric of the unit sphere is decomposed as , where the overline indicates complex conjugation, the parenthesis symmetrization and the dyad are the null tangent vectors along the coordinate lines associated with the angular charts used to make the finite covering of the sphere. These vectors satisfy the relations: and . In stereographic coordinates the vectors take the form

 qA=(1+ζ¯¯¯ζ)(δA  3+iδA  4)2,ζ=tan(θ2)eiϕ,ζ=q+ip

(see e.g. [4, 5, 21, 7]) whereas in spherical coordinates (see e.g. [10, 17, 14]),

 qA=δA  3+icscθδA  4.

All tensor quantities, such as the metric or the Ricci tensors, are contracting with the dyad producing spin-weighted scalars. The metric for the angular manifold can be represented by and , and by . Also, the Bondi gauge constrains to satisfy . Thus depends only on . It is worth stressing that the case returns the symmetrical unit sphere metric.
Associated with the projections of the covariant differentiation referred to onto the vectors, the differential operators, labelled and , are constructed. These operators can raise or lower the spin-weight of any spin-weighted scalar function. The spin-weighted functions are given by

 sΨ=m∏i=1ΛBin∏j=1ΛAjΨB1⋯BmA1⋯An,

whereas is defined

 ð sΨ=qDm∏i=1ΛBin∏j=1ΛAjΨB1⋯BmA1⋯An|D,

and is defined

 ¯¯¯ð sΨ=¯¯¯qDm∏i=1ΛBin∏j=1ΛAjΨB1⋯BmA1⋯An|D,

where the vertical line in the indices indicates covariant differentiation associated with , and the symbols and can take the values or and or , respectively. Specifically, these operators take the following form

 ð sΨ=qD sΨ,D+sΩ sΨ,¯¯¯ð sΨ=¯¯¯qD sΨ,D−s¯¯¯¯Ω sΨ,

where . For a complete review of these operators and their properties, see [21, 22, 23].
The Einstein equations can be written as

 Eμν=Rμν−8π(Tμν−gμνT/2)=0,

and in this formalism, they are decomposed as

 E22=0,E2AqA=0,EABhAB=0, (2a) EABqAqB=0, (2b) (2c)

which corresponds to hypersurface, evolution and constraint equations, respectively [5, 15, 16, 17, 13].
In the linear regime, the following set of equations corresponding to a perturbation in the Minkowski background, as originally obtained by Winicour [24] and later by Bishop [15], is given by

 8πT22=4β,rr, (2ca) 8πT2AqA=¯¯¯ðJ,r2−ðβ,r+2ðβr+(r4U,r),r2r2, (2cb) 8π(hABTAB−r2T)=−2ð¯¯¯ðβ+ð2¯¯¯¯J+¯¯¯ð2J2+(r4(¯¯¯ðU+ð¯¯¯¯U)),r2r2 +4β−2w,r, (2cc) 8πTABqAqB=−2ð2β+(r2ðU),r−(r2J,r),r+2r(rJ),ur, (2cd) 8π(T2+T11)=ð¯¯¯ðw2r3+ð¯¯¯ðβr2−(ð¯¯¯¯U+¯¯¯ðU),u2+w,ur2+w,rr2r−2β,ur, (2ce) 8π(T2+T12)=ð¯¯¯ðβr2−(r2(ð¯¯¯¯U+¯¯¯ðU)),r4r2+w,rr2r, (2cf) 8πT1AqA=¯¯¯ðJ,u2−ð2¯¯¯¯U4+ð¯¯¯ðU4+12(ðwr),r−ðβ,u+(r4U),r2r2 −r2U,ur2+U. (2cg)

## 3 Binary system

It is worth stressing that one of our aims is to study the well-known problem of a circular binary system of point particles with different masses. We show that the Peters and Mathews result for the power radiated in gravitational waves (see A) can be obtained by using the characteristic formulation and the News function. Also, the present paper generalizes previous results [17] applying to particles with equal masses.
The particles held together by their mutual gravitational interaction are far enough from each other such that to first order, the interaction between them can be considered essentially Newtonian. This assumption is valid if one considers the weak field approximation, in which the Bondi-Sachs metric in stereographic null coordinates reduces to,

 ds2 = −(1−wr−2β)du2−2(1+2β)dudr−2r2(U+¯¯¯¯U)1+|ζ|2dqdu−2r2i(U−¯¯¯¯U)1+|ζ|2dpdu (2cd) +2r2(2+J+¯¯¯¯J)(1+|ζ|2)2dq2−4ir2(J−¯¯¯¯J)(1+|ζ|2)2dqdp−2r2(−2+J+¯¯¯¯J)(1+|ζ|2)2dp2.

Note that , where represents the Newtonian potential, as usual in this kind of approximation.
We take these two particles in a Minkowski background, analogously to Peters and Mathews [19] and Bishop et. al. in [17]. Such a system allows one to explore in full detail the boundary conditions across the hypersurfaces generated by their orbits shown in figure 1.

The density that describes the binary system is given by

 ρ=δ(θ−π/2)r2(M1δ(r−r1)δ(ϕ−νu)+M2δ(r−r2)δ(ϕ−νu−π)), (2ce)

where, are the orbital radius (mass) of each particle and .
The orbits of the masses generate world tubes, which are extended along the retarded time coordinate, allowing the separation of the spacetime into three empty and disjoint regions: inside, between and outside the matter distribution.
In order to solve equations (2ca)-(2cg) for the vacuum, the following expansion for the metric variables is used

 sf=∑l,mR(flmei|m|νu)ðs 0Zlm, (2cf)

where means , are the spin-weighted spherical harmonics, which are defined in [25] as and are eigenfunctions of the operator and represents the functions . The substitution of equation (2cf) into (2c) provides a system of ordinary differential equations for the coefficients in the above expansions. The families of solutions, for , satisfying this system of equations for the vacuum read

 β2m(r) = D1β2m, (2cga) J2m(r) = 2iD1β2mνr|m|−D1J2m(νr|m|−1)(νr|m|+1)6r3 (2cgb) −iD2J2me2iνr|m|(νr|m|+i)28ν5r3|m|5+D3J2m(νr|m|−3i)νr|m|, U2m(r) = 2D1β2m(νr|m|+2i)νr2|m|−D1J2m(2ν2r2|m|2+4iνr|m|+3)6r4 (2cgc) −D2J2me2iνr|m|(2νr|m|+3i)8ν5r4|m|5−iD3J2m(ν2r2|m|2+6)νr2|m|, w2m(r) = −10rD1β2m+6rD3J2m(2+iνr|m|)−3iD2J2me2iνr|m|4ν5r2|m|5 (2cgd) −iD1J2m((1+i)νr|m|−i)(1+(1+i)νr|m|)r2,

where the constants of integration are represented by ; here indexes the constant and corresponds to the Bondi metric function whose integration generated it. This set of families of solutions depends only on four constants, namely, , , and . This is so because the families of solutions for the coefficients , , and , resulting from (2ca)-(2cd), are constrained by (2ce)-(2cg). This fact is independent of , thus the set of families of solutions for any will have four degrees of freedom. A unique solution for the whole spacetime cannot be determined by only imposing regularity of the metric variables at the null cone vertices and at the null infinity. Therefore, additional boundary conditions must be imposed. In particular, this can be done by imposing boundary conditions on other hypersurfaces, such as in the case of the thin shells studied by Bishop [15], in which the additional conditions are imposed across the world tubes generate by the shell itself. Once the above constants are determined, one readily obtains the metric functions , , , and for the whole spacetime.
As divergent solutions are not expected at the vertices of the null cones, regularity conditions at these points must be imposed for the metric. In order to do so, an expansion of the metric variables around in power series of is made and the divergent terms are disregarded. This procedure establishes relationships between the coefficients, leading to a family of solutions for the interior that depends only on one undetermined parameter, where in particular . One obtains, for example, for ,

 β2m−(r)= 0, (2cgha) J2m−(r)= (2cghb) +6νr|m|e2iνr|m|+3ie2iνr|m|−3i), U2m−(r)= −iD2J2m−24ν5r4|m|5(2ν4r4|m|4+6ν2r2|m|2−6iνr|m|e2iνr|m| (2cghc) −12iνr|m|+9e2iνr|m|−9), w2m−(r)= D2J2m−4ν5r2|m|5(2iν4r4|m|4+4ν3r3|m|3−6iν2r2|m|2−6νr|m| (2cghd) −3ie2iνr|m|+3i).

For the intermediate region, the same structure of the general solutions is maintained, for the case given by (2cga)-(2cgd). That is so because there is no reason to discard any particular term, or to establish any relationship between the constants as occurs for the interior region. For the exterior region, since that is required regularity at the null infinity, the coefficient of the exponential factor must be null. This means that all constants of the form , with , must be zero. Therefore, the number of degrees of freedom for the exterior family of solutions is reduced in one parameter. Thus, a family of solutions to describe the whole spacetime for the field equations (2ca)-(2cg), with eight parameters to be determined is obtained. Now, in order to fix these eight constants, it is necessary to impose additional boundary conditions, in particular across the time-like world tubes generated by the orbits of the particles.
These boundary conditions across the world tubes, i.e. when , come from imposing discontinuities on the metric coefficients,

 [g11]ri=0,[g12]ri=Δg12|ri,[g1A]ri=0,[g22]ri=0, [g2A]ri=0,[g3μ]ri=0,[g4μ]ri=0, (2cghi)

and on their first derivatives,

 [g′μν]ri=Δg′μν,μ,ν=1,⋯4, (2cghj)

where the brackets mean .
From the linearised Bondi-Sachs metric (2cd), and from the two sets of jump conditions (2cghi) and (2cghj), the coefficients and are restricted to satisfy

 [wlm(rj)]=Δwjlm, [βlm(rj)]=Δβjlm, (2cghk) [Jlm(rj)]=0, [Ulm(rj)]=0,

and for their first derivatives

 [w′lm(rj)]=Δw′jlm, [β′lm(rj)]=Δβ′jlm, (2cghl) [J′lm(rj)]=ΔJ′jlm, [U′lm(rj)]=ΔU′jlm,

where , and , , , , and are functions to be determined.
Solving equations (2cghk) and (2cghl), simultaneously for both world tubes, the boundary conditions are explicitly obtained. We find that

 Δβjlm=bjlm,Δwjlm=−2rjbjlm, (2cghma) where bjlm are constants. Note that this last fact implies that Δβ′jlm=0. We determine that the jumps for the first derivative of the Jlm and Ulm functions are given by ΔJ′jlm=8ν2rjbjlm|m|2(l−1)l(l+1)(l+2), (2cghmb) ΔU′jlm=2bilm(1r2i−4iν|m|l(l+1)ri). (2cghmc)

The boundary conditions (2cghmb) and (2cghmc) fix all parameters of the families of solutions, providing the specific solutions for the coefficients and . These coefficients can be written as

 flm(r)= f1lm(r)(1−Θ(r−r1))+f2lm(r)(Θ(r−r1)−Θ(r−r2)) (2cghmn) +f3lm(r)Θ(r−r2),

where represents , , and , with the first subscript on the right hand side terms indicating the interior (1), the middle (2) and the exterior (3) solutions; and is the Heaviside function, namely

 Θ(r)={0r≤01r>0. (2cghmo)

These solutions depend explicitly on two specific parameters, namely , with , which are related to the density of matter. The specific form of these relationships is obtained by just integrating the first field equation (2ca) across each world tube. As a result one obtains

 bjlm=2πrjρjlm(1+v2j), (2cghmp)

where are given by

 ρjlm=1π∫Sd(νu)∫ΩdΩ∫Ijdr 0¯¯¯¯Zlme−i|m|νuρ, (2cghmq)

in which , is the physical velocity of the particle in the space, and is an interval around that is given by , with .
Before proceeding, it is worth noticing that the above procedure is a generalization of Section 3 of the paper by Bishop et. al. [17], in which the binary components have equal masses. In particular, the boundary conditions are also generalized since in the present case there exist two independent world tubes. Interestingly our solution is fully analytical.
In order to include the null infinity, which is reached when tends to infinity, a radial compactified coordinate is defined as follows

 s=rr+R0,

where is a compactification parameter. Thus, , where and corresponds to the null cone vertices and the null infinity, respectively.
Figure 2 shows some of the coefficients of the expansion of the metric variables in terms of the compactified coordinate for .

Here , , , and the radius of each orbit is referred to the center of mass of the system, namely

 rj=μMjd0,j=1,2, (2cghmr)

where is the reduced mass of the system and is the distance between the masses. The frequency of rotation is computed by means of Kepler’s third law, i.e.,

 ν=√M1+M2d30. (2cghms)

It is worth noting that the jumps in and functions are present at exactly and , whereas for and only their first derivatives present discontinuities, in agreement with the boundary conditions (2cghmb) and (2cghmc).
To illustrate the behaviour of , , and we present them in figure 3 as a function of and for a particular value of the retarded time . These functions are constructed by using equation (2cf), and the solutions for the coefficients for each and . In this case, we use .

As expected, the metric functions and and the first derivatives of and show jumps at and , which are just the positions of the masses, in agreement with the boundary conditions initially imposed.
Note that since the first field equation for the vacuum implies that are constants along , as sketched in figure 2, and that is a gauge term for the gravitational potential, then, can be redefined as . These facts make the choice of the frequency as obeying Kepler’s third law completely consistent and natural.

## 4 Gravitational Radiation from the source

Now, we proceed with the calculation of the power emitted by the binary system via gravitational wave emission. We show that the approach presented here is consistent with the well-known result obtained by Peters and Mathews [19].
Following Bishop [15], the Bondi News function in the weak field approximation is given by,

 N=limr→∞(−r2J,ur2+ð2ω2+ð2β).

Substituting the metric variables given in (2cf), one obtains the News function for and , namely

 N=limr→∞∑l,mR((−i|m|νr2(Jlm),r2+βlm+l(l+1)Jlm4)ei|m|νu) ð2 0Zlm. (2cghmt)

Now, substituting the coefficients of the metric variables for the exterior region, one obtains

 N= −iν3 2S21√6−4i√23ν3 2S22−iν4 2S31√30(ν−3i)−8i√215ν4 2S322ν−3i (2cghmu) −9i√310ν4 2S33ν−i−iν5 2S413√10(ν2−7iν−14)−8i√25ν5 2S423(2ν2−7iν−7) −81iν5 2S43√10(9ν2−21iν−14)−256i√25ν5 2S443(8ν2−14iν−7) +ν6 2S51√210(iν3+12ν2−54iν−90)+16√2105ν6 2S524iν3+24ν2−54iν−45 +27√370ν6 2S533iν3+12ν2−18iν−10+1024√2105ν6 2S5432iν3+96ν2−108iν−45 +625√542ν6 2S5525iν3+60ν2−54iν−18+⋯,

where we define the spin 2 quantity as

 2Slm=(R(D1Jlm+ei|m|νu) ð2 0Zlm+R(D1Jl−m+ei|m|νu) ð2 0Zl −m)√(l−1)l(l+1)(l+2). (2cghmv)

Since the binary system is confined to a plane, then a natural choice to simplify the problem of expressing the News function, without loss of generality, is to put the masses motion in the equatorial plane . This means symmetry of reflection for the density of matter and, consequently, for the spacetime. Thus, this choice restricts the components of the density, obtained from (2cghmq), to have the following form

 ρlm=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩~ρlmM2r21δ(r−r2)+M1r22δ(r−r1)r21r22l,m \ % even~ρlmM1r22δ(r−r1)−M2r21δ(r−r2)r21r22l,m \ odd, (2cghmw)

where are numerical constants. Therefore, for binaries of different masses, the News function (2cghmu) is simplified to

 N= −4i√23ν3 2S22−iν4 2S31√30(ν−3i)−9i√310ν4 2S33ν−i−8i√25ν5 2S423(2ν2−7iν−7) (2cghmx) −256i√25ν5 2S443(8ν2−14iν−7)+ν6 2S51√210(iν3+12ν2−54iν−90) +27√370ν6 2S533iν3+12ν2−18iν−10+625√542ν6 2S5525iν3+60ν2−54iν−18+⋯.

When the explicit solutions are used, the News functions for the binary system take the form

 N = 8√2π5 2L22(M21+M22)ν3+13i√π35 2L31(M31−M32)ν4 (2cghmy) − 9i√3π7 2L33(M31−M32)ν4+863√2π 2L42(M41+M42)ν5 − 1289√2π7 2L44(M41+M42)ν51180i√π154 2L51(M51−M52)ν6 − 2740i√3π11 2L53(M51−M52)ν6+62524i√5π33 2L55(M51−M52)ν6 + ⋯,

where

 Mlj=Mjrlj(v2j+1), (2cghmz)

and are defined as

 2Llm=( 2Zl −mR(ei|m|νu)−R(iei|m|νu) 2Zlm). (2cghmaa)

Note that, as consequence of (2cghmw), for the terms with odd disappear from the News function (2cghmy). Thus, as expected, one obtains immediately

 N = 16√2π5ν3M0r20(V20+1) 2L22+1663√2πν5M0r40(V20+1) 2L42 (2cghmab) −2569√2π7ν5M0r40(V20+1) 2L44+32√2π131485ν7M0r60(V20+1) 2L62 +⋯.

where is the physical velocity of the identical masses, which is obviously tangent to the circular orbit.
The energy lost by the system is related to the News function, via

 dEdu=14π∫ΩdΩ N¯¯¯¯¯N, (2cghmac)

which results for in

 dEdu = 325ν6(M21+M22)2+2734315ν8(M31−M32)2 (2cghmad) + 573763969ν10(M41+M42)2+4010276155925ν12(M51−M