On quadrupole and octupole gravitational radiation in the ANK formalism

Abstract

Following the approach of Adamo–Newman–Kozameh (ANK) we derive the equations of motion for the center of mass and intrinsic angular moment for isolated sources of gravitational waves in axially symmetric spacetimes. The original ANK formulation is generalized so that the angular momentum coincides with the Komar integral for a rotational Killing symmetry. This is done using the Winicour–Tamburino Linkages which yields the mass dipole-angular momentum tensor for the isolated sources. The ANK formalism then provides a complex worldline in a fiducial flat space to define the notions of center of mass and spin. The equations of motion are derived and then used to analyse a very simple astrophysical process where only quadrupole and octupole contributions are included. The results are then compared with those coming from the post newtonian approximation.

Appendices

Appendix A: Legendre polinomials with spin weight

The associated Legendre polynomials $P_{l}^{s}(x)$ and $P_{l}^{-s}(x)$ are solutions to the associated Legendre differential equation, where l is a positive integer and $s=0\ldots l.$. They are implemented in Mathematica as Legendre P[l, m, x]. For positive s, they can be write in terms of the unassociated polynomials by

$$\begin{aligned} P_{l}^{s}(x)=(-1)^{s}\left( 1-x^{2}\right) ^{s/2}\frac{d^{s}}{dx^{s}} P_{l}(x), \end{aligned}$$

(95)

where $P_{l}(x)$ are the unassociated Legendre polynomials. To define the associated Legendre polynomials for negative s we follow the convention due to Abramowitz and Stegun [17],

$$\begin{aligned} P_{l}^{-s}(x)=(-1)^{s}P_{l}^{s}(x). \end{aligned}$$

(96)

They are orthogonal over the interval $\left[ -1;1\right]$ with respect to l

$$\begin{aligned} \int _{-1}^{1}P_{l}^{s}(x)\, P_{l'}^{s}(x)dx=\frac{2}{2l+1}\frac{\left( l+s\right) !}{(-s)!}\delta _{ll {\acute{}} }, \end{aligned}$$

(97)

and orthogonal over $\left[ -1;1\right]$ with respect to s with the weighting function $\left( 1-x^{2}\right) ^{-1}$,

$$\begin{aligned} \ \ \int _{-1}^{1}P_{l}^{s}(x)\ \ \ P_{l}^{s {\acute{}} }(x)\frac{dx}{\left( 1-x^{2}\right) }=\frac{2}{2l+1}\frac{\left( l+s\right) ! }{s\left( l-s\right) !}\delta _{ss {\acute{}}}. \end{aligned}$$

(98)

The associated Legendre polynomials also obey the following recurrence relations

$$\begin{aligned} \left( l-s\right) P_{l}^{s}(x)\ \ =x\left( 2l-1\right) P_{l-1}^{s}(x)\ -\left( l+s-1\right) P_{l-2}^{s}(x). \end{aligned}$$

(99)

It is also very useful to consider the identities

$$\begin{aligned} P_{l}^{l}(x)= & {} \left( -1\right) ^{l}\left( 2l-1\right) !! \left( 1-x^{2}\right) ^{1/2}, \end{aligned}$$

(100)

$$\begin{aligned} P_{l+1}^{l}(x)= & {} x\left( 2l+1\right) P_{l}^{l}(x). \end{aligned}$$

(101)

Letting $x=\cos \theta$ we obtain Legendre polynomials $P_{l}^{s}(\cos \theta )$ . Below we list the $s=0;1;2$ Polynomials together with sone useful products.

1.1 Legendre polynomials with spin weight s = 0,1,2

The $s=0$ (spin weigth zero) associated Legendre polynomials are

$$\begin{aligned} P_{0}^{0}= & {} 1, \, P_{1}^{0}=\cos \theta , \, P_{2}^{0}=\frac{1}{2}(3\cos \theta ^{2}-1), \end{aligned}$$

(102)

$$\begin{aligned} P_{3}^{0}= & {} \frac{1}{2}(5\cos \theta ^{3}-3\cos \theta ), \, P_{4}^{0}=\frac{1}{8}(35\cos \theta ^{4}-30\cos \theta ^{2}+3), \end{aligned}$$

(103)

$$\begin{aligned} P_{5}^{0}= & {} \frac{1}{8}(63\cos \theta ^{5}-70\cos \theta ^{3}+15\cos \theta ), \nonumber \\ P_{6}^{0}= & {} \frac{1}{16}\left( 231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5\right) , \end{aligned}$$

(104)

The polinomyals with spin weight s are obtained via the $\eth$ derivative, defined as

$$\begin{aligned} \begin{array}{cc} \eth F=-(\sin \theta )^{s}\frac{\partial }{\partial \theta }\left( (\sin \theta )^{-s}F\right) ,&{\bar{\eth }}F=-(\sin \theta )^{-s}\frac{\partial }{\partial \theta }\left( (\sin \theta )^{s}F\right) . \end{array} \end{aligned}$$

(105)

Using the above formula we then define the $P_{l}^{s}(\cos \theta )$ as

$$\begin{aligned} P_{l}^{s}(\cos \theta )= & {} \eth ^{s}P_{l}^{0}(\cos \theta ),0\le s\le l, \end{aligned}$$

(106)

$$\begin{aligned} P_{l}^{s}(\cos \theta )= & {} (-1)^{s}{\bar{\eth }}^{-s}P_{l}^{0}(\cos \theta ),-l\le s\le 0, \end{aligned}$$

(107)

$$\begin{aligned} P_{l}^{s}(\cos \theta )= & {} 0,l\le |s|. \end{aligned}$$

(108)

and use the recurrence relationship $\sqrt{(l-s)(l+s+1)}P_{l}^{s+1}(\cos \theta )=\eth P_{l}^{s}(\cos \theta ),$ so that $\sqrt{(l)(l+1)}P_{l}^{1}(\cos \theta )=\eth P_{l}^{0}(\cos \theta ),$ where $\eth P_{l}^{0}(\cos \theta )=-\frac{\partial }{\partial \theta }P_{l}^{0}(\cos \theta )$.

A few $s=1$ Legendre polynomials are listed below

$$\begin{aligned} P_{1}^{1}= & {} -P_{1}^{-1}=\frac{1}{\sqrt{2}}\sin \theta , \, P_{2}^{1}=-P_{2}^{-1}=\frac{3}{\sqrt{6}}\sin \theta \cos \theta , \end{aligned}$$

(109)

$$\begin{aligned} P_{3}^{1}= & {} -P_{3}^{-1}=\frac{1}{\sqrt{12}}\sin \theta \left( \frac{15}{2}\cos \theta ^{2}-\frac{3}{2}\right) , \nonumber \\ P_{4}^{1}= & {} -P_{4}^{-1}=\ \frac{1}{\sqrt{20}}\sin \theta \left( \frac{35}{2}\cos \theta ^{3}-\frac{15}{2}\cos \theta \right) ,\end{aligned}$$

(110)

$$\begin{aligned} P_{5}^{1}= & {} -P_{5}^{-1}=\sin \theta \frac{5}{8\sqrt{30}}(63\cos \theta ^{4}-42\cos \theta ^{2}+3) \end{aligned}$$

(111)

The polynomials with spin weigth $s=2$ can be calculated from the recurrence relation, i.e.,

$$\begin{aligned} \sqrt{(l-1)(l+2)}P_{l}^{2}(\cos \theta )=\eth P_{l}^{1}(\cos \theta ). \end{aligned}$$

(112)

We list a few polynomials,

$$\begin{aligned} \begin{array}{cc} P_{2}^{2}=P_{2}^{-2}=\frac{3}{2\sqrt{6}}(\sin \theta )^{2},&P_{3}^{2}=P_{3}^{-2}=\frac{15}{\sqrt{120}}(\sin \theta )^{2}\cos \theta , \end{array} \end{aligned}$$

(113)

1.2 Products with polynomials $P_{1}^{1}$ , $P_{2}^{1}$ , $P_{3}^{1}$

Using the above formulae together with their inverse relationships we can compute the following products with s $=$ 1,

$$\begin{aligned} P_{1}^{1}P_{1}^{1}= & {} \frac{1}{3}\left( P_{0}^{0}-P_{2}^{0}\right) , \, P_{1}^{1}P_{2}^{1}=\sqrt{3}\left( \frac{4}{5}P_{1}^{0}-\frac{1}{5} P_{3}^{0}\right) \end{aligned}$$

(114)

$$\begin{aligned} P_{1}^{1}P_{3}^{1}= & {} \frac{\sqrt{6}}{7}\left( P_{2}^{0}-P_{4}^{0}\right) , \, P_{2}^{1}P_{2}^{1}=\frac{1}{5}P_{0}^{0}+\frac{1}{7}P_{2}^{0}-\frac{12}{35}P_{4}^{0}\end{aligned}$$

(115)

$$\begin{aligned} P_{2}^{1}P_{3}^{1}= & {} \frac{\sqrt{2}}{4}\left( -\frac{20}{21}P_{5}^{0}+\frac{4}{15}P_{3}^{0}-\frac{177}{70}P_{1}^{0}\right) ,\nonumber \\ P_{3}^{1}P_{3}^{1}= & {} \frac{1}{12}\left( -\frac{300}{77}P_{6}^{0}+\frac{158}{385}P_{4}^{0}+\frac{1}{14} P_{2}^{0}+\frac{443}{140}P_{0}^{0}\right) \end{aligned}$$

(116)

1.3 Products with polynomials s = 2 $P_{2}^{2}$ and $P_{3}^{2}$

We list below some products of polynomials that are useful for our work.

$$\begin{aligned} P_{2}^{2}P_{2}^{2}= & {} \frac{3}{35}P_{4}^{0}-\frac{2}{7}P_{2}^{0}+\frac{1}{5}P_{0}^{0}, \nonumber \\ P_{2}^{2}P_{3}^{2}= & {} \sqrt{20}\left( \frac{1}{21}P_{5}^{0}-\frac{1}{15}P_{3}^{0}+\frac{3}{70}P_{1}^{0}\right) ,\nonumber \\ P_{3}^{2}P_{3}^{2}= & {} \frac{10}{77}P_{6}^{0}-\frac{3}{11}P_{4}^{0}+\frac{1}{7}P_{0}^{0}, \end{aligned}$$

(117)

1.4 Other useful products

Other products of polynomials that are also useful are

$$\begin{aligned} P_{1}^{1}P_{1}^{0}= & {} \frac{\sqrt{3}}{3}P_{2}^{1},\, P_{2}^{1}P_{1}^{0}=\frac{\sqrt{6}}{15}\left( \sqrt{12}P_{3}^{1}+\frac{3\sqrt{2}}{2}P_{1}^{1}\right) ,\end{aligned}$$

(118)

$$\begin{aligned} P_{3}^{1}P_{1}^{0}= & {} \frac{1}{7\sqrt{12}}\left( 3\sqrt{20}P_{4}^{1}+4\sqrt{6} P_{2}^{1}\right) ,\, P_{1}^{1}P_{2}^{0}=\frac{\sqrt{6}}{5}P_{3}^{1}-\frac{1}{5}P_{1}^{1},\end{aligned}$$

(119)

$$\begin{aligned} P_{1}^{1}P_{3}^{0}= & {} \frac{\sqrt{10}}{7}P_{4}^{1}-\frac{\sqrt{3}}{7}P_{2}^{1}, \, P_{2}^{1}P_{2}^{0}=\frac{3}{\sqrt{6}}\left( \frac{3\sqrt{20}}{35}P_{4}^{1}+ \frac{\sqrt{6}}{21}P_{2}^{1}\right) ,\end{aligned}$$

(120)

$$\begin{aligned} P_{2}^{1}P_{3}^{0}= & {} \frac{3}{\sqrt{6}}\left( \frac{4\sqrt{30}}{63}P_{5}^{1}-\frac{2\sqrt{12}}{315}P_{3}^{1}-\frac{3\sqrt{2}}{35}P_{1}^{1}\right) ,\nonumber \\ P_{3}^{1}P_{2}^{0}= & {} \frac{1}{\sqrt{12}}\left( \frac{2\sqrt{30}}{7}P_{5}^{1}+ \frac{9\sqrt{12}}{70}P_{3}^{1}+\frac{18\sqrt{2}}{35}P_{1}^{1}\right) ,\end{aligned}$$

(121)

$$\begin{aligned} P_{3}^{1}P_{3}^{0}= & {} \frac{1}{\sqrt{12}}\left( \frac{50\sqrt{42}}{231} P_{6}^{1}-\frac{15+66\sqrt{20}}{77}P_{4}^{1}\right. \nonumber \\&\left. +\left( \frac{4790}{1232}-\frac{15\sqrt{6}}{7}+\frac{3\sqrt{2}}{2}\right) P_{2}^{1}\right) , \end{aligned}$$

(122)

$$\begin{aligned} P_{2}^{2}P_{2}^{1}= & {} \frac{3}{4}\left( -\frac{2\sqrt{20}}{35}P_{4}^{1} +\frac{4\sqrt{6}}{21}P_{2}^{1}\right) ,\nonumber \\ P_{2}^{2}P_{3}^{1}= & {} \frac{\sqrt{2}}{8} \left( -\frac{4\sqrt{30}}{21}P_{5}^{1}+\frac{61\sqrt{12}}{105}P_{3}^{1}- \frac{12\sqrt{2}}{35}P_{1}^{1}\right) ,\end{aligned}$$

(123)

$$\begin{aligned} P_{3}^{2}P_{2}^{1}= & {} \frac{3\sqrt{20}}{8}\left( -\frac{8\sqrt{30}}{315} P_{5}^{1}+\frac{16\sqrt{12}}{315}P_{3}^{1}+\frac{4\sqrt{2}}{35} P_{1}^{1}\right) ,\end{aligned}$$

(124)

$$\begin{aligned} P_{3}^{2}P_{3}^{1}= & {} \frac{\sqrt{10}}{8}\left( -\frac{20\sqrt{42}}{231} P_{6}^{1}+\frac{\left( 180\sqrt{20}-315\right) }{350}P_{4}^{1}+\frac{44\sqrt{6}+119}{56}P_{2}^{1}\right) ,\nonumber \\ \end{aligned}$$

(125)

Appendix B: Komar integral and angular momentum

The Komar integral associated with the rotation Killing field $K _{(\varphi )}^{a}$ yields a suitable definition of angular momentum for vacuum axially symmetric spacetimes,

Using Stokes theorem and the fact that $K _{(\varphi )}^{a}$ is a Killing field we have

$$\begin{aligned} \underset{\partial \Sigma }{\oint }\nabla ^{a}K_{(\varphi )}^{b}dS_{ab}=2\int _{\Sigma }R_{ab}K_{(\varphi )}^{b}d\Sigma ^{a}=0, \end{aligned}$$

(126)

where $\partial \Sigma$ is the boundary of the hypersurface $\Sigma$. Using Bondi coordinates on an asymptotically flat spacetime one can choose $\Sigma$ as the $r=const.$ and $\partial \Sigma$ as two $u=const.$ 2-surfaces. It then follows from the above equations that

$$\begin{aligned} J^{z}=\frac{1}{16\pi }\underset{r\rightarrow \infty }{\lim }\oint _u \nabla ^{a}K _{(\varphi )}^{b}dS_{ab}= const., \end{aligned}$$

(127)

i.e., the value of the integral does not depend on the Bondi time u. One can explicitly integrate this equation in the N-P formalism to obtain a formula at $\mathscr {I}^{+}$ in terms of the spin coefficients. We first write the Killing vector field $K _{(\varphi )}^{b}$ as a combination of the null tetrad vectors as

$$\begin{aligned} K_{(\varphi )}^{b}=\frac{Im(\sigma ^{0}{\bar{\omega }}^{0})}{r}\cos \theta l^{b}-ir\cos \theta m^{b}+ir\cos \theta {\bar{m}} ^{b}, \end{aligned}$$

(128)

and the two-dimensional surface area can also be expressed as

$$\begin{aligned} dS_{ab}=-2n_{[a}l_{b]}r^{2}\sin \theta d\theta d\varphi , \end{aligned}$$

(129)

thus the Komar integral can be written as

$$\begin{aligned} J^{z}=-\frac{1}{16\pi }\underset{r\rightarrow \infty }{\lim } \int \nolimits _{0}^{\pi }\int \nolimits _{0}^{2\pi }\nabla ^{a}K _{(\varphi )}^{b}(n_{a}l_{b}-l_{a}n_{b})r^{2}\sin \theta d\varphi d\theta . \end{aligned}$$

(130)

Using Eq. (128), and writing this equation up to order $O(r^{-2})$ we get

$$\begin{aligned} \nabla ^{a}K _{(\varphi )}^{b}(n_{a}l_{b}-l_{a}n_{b})=\frac{\cos \theta }{r^{2}}Im(\psi _{1}^{0}+\sigma ^{0}{\bar{\omega }}^{0}) \end{aligned}$$

(131)

where ${\bar{\omega }}^{0}=-\eth {\bar{\sigma }}^{0}$ [8, 18]. Thus, the Komar integral can be written as

$$\begin{aligned} J^{z}=\frac{1}{8}\int \nolimits _{1}^{-1}Im(\psi _{1}^{0}-\sigma ^{0}\eth {\bar{\sigma }}^{0})\cos \theta d(\cos \theta ). \end{aligned}$$

(132)

where we have used the axial symmetry to integrate in the azimuth direction. Finally, inserting the appropriate constants back into the equation yields,

$$\begin{aligned} J^{z}= -\frac{c}{6\sqrt{2}G}\ Im(\psi _{1}^{0}-\sigma ^{0}\eth {\bar{\sigma }}^{0})|_{l=1} \end{aligned}$$

(133)

Appendix C: Gravitational radiation: gauge TT and postnewtonian methods

The equations of motion in the postnewtonian approximation are usually given in the Transverse Tracelees (TT) gauge [11, 12]. To compare our results with those coming from this approximation it is convenient to use the same gauge. We give a brief review below of the interaction of two point particles in the TT gauge.

Given a cartesian coordinate system, (x, y, z, t) with coordinate basis vectors will $e_{x}$, $e_{y}$ ,$e_{z}$, we introduce a standard spherical coordinate system $\left( r,\theta ,\phi \right)$, where $\theta$ its the angle inclination from the axis z and $\phi$ its the phase angle.

If the waves propagate in the direction of the radial unit vector

$$\begin{aligned} e_{r}=e_{x}\sin \theta \cos \phi +e_{y}\sin \theta sen\phi +e_{z}\cos \theta , \end{aligned}$$

(134)

a natural set of vectors orthogonal basis of which to build the basic tensors TT is

$$\begin{aligned} e_{\theta }= & {} e_{x}\cos \theta \cos \phi +e_{y}\cos \theta \sin \phi -e_{z}sen\theta \end{aligned}$$

(135)

$$\begin{aligned} e_{\phi }= & {} -e_{x}\sin \phi +e_{y}\cos \phi \end{aligned}$$

(136)

Our problem then is how to extract information from the radiation detected at null infinity in relation to the Weyl scalars, i.e., how to relate $\psi _{4}$ to the second time derivative $h_{ij}$, which can written as

$$\begin{aligned} h_{ij}=\ A_{ij}\frac{M}{r}+O(r^{-2}), \end{aligned}$$

(137)

where $A_{ij}$ does not depend on r and M is the total mass of the system.

It is convenient to separate $h_{ij}$ in two polarizations ($h_{+}$ and $h_{\times }$) modes in the wave zone.

One can show that $h_{ij}$ can be written as

$$\begin{aligned} h=\ \sum _{i,j}h_{ij}e_{i}\otimes e_{j}=\ \ h_{+}\left( e_{\theta }\otimes e_{\theta }-e_{\phi }\otimes e_{\phi }\right) +h_{\times }\left( e_{\theta }\otimes e_{\phi }+e_{\phi }\otimes e_{\theta }\right) \!, \nonumber \\ \end{aligned}$$

(138)

Linearized gravity then tell us that two time derivatives of (138) is related to the Weyl tensor.

Using The Newman Penrose formalism one can show that the gravitational radiation is contained in a particular component of the Weyl tensor, $\psi _{4}$,

$$\begin{aligned} \psi _{4}=C_{abcd}n^{a}\overset{\_}{m}^{b}n^{c}\overset{\_}{m}^{d}, \end{aligned}$$

(139)

where $n^{a}$ and $\overset{\_}{m}^{a}$are two vectors of a null tetrad constructed from the orthonormal spherical basis

$$\begin{aligned} l^{a}= & {} \frac{1}{\sqrt{2}}\left( e_{t}^{a}+e_{r}^{a}\right) , \nonumber \\ n^{a}= & {} \frac{1}{\sqrt{2}}\left( e_{t}^{a}-e_{r}^{a}\right) , \nonumber \\ m^{a}= & {} \frac{1}{\sqrt{2}}\left( e_{\theta }^{a}+ie_{\phi }^{a}\right) , \nonumber \\ \overset{\_}{m}^{a}= & {} \frac{1}{\sqrt{2}}\left( e_{\theta }^{a}-ie_{\phi }^{a}\right) , \end{aligned}$$

(140)

where $e_{t}^{a}$ is the unit timelike vector and $e_{r}^{a}$, $e_{\theta }^{a}$, $e_{\phi }^{a}$ are usual orthonormal basis induced by the spherical coordinates.

One can then show that

$$\begin{aligned} \psi _{4}=\ \overset{\cdot \cdot }{h}_{+}-i\overset{\cdot \cdot }{h}_{\times } \end{aligned}$$

(141)

At a sufficiently large distance from the source of the gravitational waves is write

$$\begin{aligned} \psi _{4}=\psi _{4}^{0}r^{-1}+\cdots =-\overset{\overset{\cdot \cdot }{\_}}{ \sigma _B }r^{-1}+\cdots \end{aligned}$$

(142)

where $\sigma _B$ is the Bondi shear of the null congruence of the geodesics. We are assuming that $\sigma _B$ (38) contains radiative quadrupole and octupole terms and that they are related with the two polarizations $h_{+}\$ and $h_{\times }$.

The above equations show that $h_{+}-ih_{\times }$ can be expressed in terms of the mass quadrupole and octupole moments $\left( Q_{+}^{\,\cdot \cdot }\text { and }O_{+}^{\,\cdot \cdot \cdot }\right)$ and current quadrupole and octupole moments $\left( Q_{\times }^{\,\cdot \cdot }\text { and }O_{\times }^{\,\cdot \cdot \cdot }\right)$, as it is written in the Eqs. (39, 40).

If one then write $h^{ij}$ in terms of the mass and current source multiple moments up to post-Newtonian order 1/2 [3] we then have

$$\begin{aligned} h^{ij}= & {} \frac{2}{r}\left\{ \overset{(2)}{I}^{ij}+\frac{1}{3}\overset{(3)}{I }^{ijk}N^{k}+\ldots \right. \\&\left. +\, \epsilon ^{kl(i}\left[ \frac{4}{3}\overset{(2)}{J}^{j)k}N^{l}+ \frac{1}{2}\overset{(3)}{J}^{j)km}N^{l}N^{m}+\ldots \right] \right\} _{TT} \end{aligned}$$

where $I^{ij..}$ and $J^{ij..}$ are respectively the mass and current multipole moments, D is the distance from the source to the observer, $N^{i}$ is the unit vector from the center of mass of the source to the observation point (in the axysimmetric case $N^{i}=e_{r}^{i}$ because of the center of mass is located in the origin of coodinates), the number in parentheses (n) indicates the number of derivatives with respect to the delayed time and $\epsilon ^{ijk}$ is the Levi-Civita symbol.

The source mass multipole moments can be written as

$$\begin{aligned} I^{L}(u)= & {} \int \left( x^{L}\right) ^{STF}\rho \left( \overset{\rightarrow }{x},u\right) d^{3}x \nonumber \\&-\frac{4\left( 2l+1\right) }{\left( l+1\right) \left( 2l+3\right) }\frac{d }{du}\int \left( x^{iL}\right) ^{STF}\rho ^{i}\left( \overset{\rightarrow }{x },u\right) d^{3}x \nonumber \\&+\frac{1}{2\left( 2l+3\right) }\frac{d^{2}}{du^{2}}\int \left| \overset{\rightarrow }{x}\right| ^{2}\left( x^{L}\right) ^{STF}\rho \left( \overset{\rightarrow }{x},u\right) d^{3}x \end{aligned}$$

(143)

where L is a multiple index, i.e. indicates that $x^{L}\equiv x^{i_{1}}x^{i_{2}}x^{i_{3}}\ldots x^{i_{l}}$, while STF (Symmetric Trace Free) indicates that this is a traceless symmetric tensor.

If we apply the above formulae to a binary system and in particular to the mass quadrupole moment one obtains [3]

$$\begin{aligned} I^{ij}= & {} \sum _{A}\left\{ m_{A}\left[ x_{A}^{i}x_{A}^{j}\ \right] ^{STF} \left[ 1+\frac{3}{2}v_{A}^{2}-\sum _{B\ne A}\frac{M}{r_{AB}}\right] \right. \nonumber \\&-\frac{20}{21}\frac{d}{dt}\left[ m_{A}\left[ x_{A}^{i}x_{A}^{j}\ x_{A}^{k} \right] ^{STF}v_{A}^{k}\right] \nonumber \\&+\frac{1}{14}\frac{d^{2}}{dt^{2}}\left[ m_{A}x_{A}^{2}\left[ x_{A}^{i}x_{A}^{j}\ \right] ^{STF}\ \right] +4\left[ x_{A}^{i}\left( \overset{\rightarrow }{v}_{A}\times \overset{\rightarrow }{S}_{A}\right) ^{j}\right] ^{STF} \nonumber \\&-\frac{4}{3}\frac{d}{dt}\left[ x_{A}^{i}\left( \overset{\rightarrow }{x} _{A}\times \overset{\rightarrow }{S}_{A}\right) ^{j}\right] ^{STF} \end{aligned}$$

(144)

We now apply the above equation to the head on collision of two different point masses $m_{1}$and $m_{2}$ along a z direction, keeping axisymmetric conditions.

As we deal with axisymmetric case the vectors are aligned then $\overset{ \rightarrow }{v}_{A}\times \overset{\rightarrow }{S}_{A}=0$ and $\overset{ \rightarrow }{x}_{A}\times \overset{\rightarrow }{S}_{A}=0$.

Using relative coordinates $x^{i}=x_{2}^{i}-x_{1}^{i}$; $v^{i}=v_{2}^{i}-v_{1}^{i}$, and post-Newtonian corrections [3], the mass quadrupole and octupole terms can be written as

$$\begin{aligned} I^{ij}= & {} \mu \left( x^{i}x^{j}\right) \ ^{STF}\left[ 1+\ \frac{29}{42}\left( 1-3\eta \right) v^{2}-\frac{1}{7}\left( 5-8\eta \right) \frac{m}{r}\right] +\ldots \end{aligned}$$

(145)

$$\begin{aligned} I^{ijk}= & {} -\mu \frac{\delta m}{m}\left( x^{i}x^{j}x^{k}\right) \ ^{STF}\left[ 1+\ \frac{1}{6}\left( 5-19\eta \right) v^{2}-\frac{1}{6}\left( 5-13\eta \right) \frac{m}{r}\right] +.\qquad \quad \end{aligned}$$

(146)

where $m=m_{1}+m_{2}$ ; $\mu =\frac{m_{1}m_{2}}{m}$ it is the reduced mass; $\delta m=m_{1}-m_{2}$ ; $\eta =\frac{\mu }{m}$.

To exhibit the postnewtonian corrections, it is useful to identify the newtonian quadrupole $(I_{_{N}}^{ij})$ and octupole $(I_{_{N}}^{ijk})$ moments in the above equations, i.e.

$$\begin{aligned} I_{_{PN}}^{ij}= & {} I_{_{N}}^{ij}\left[ 1+\ \frac{29}{42}\left( 1-3\eta \right) v^{2}- \frac{1}{7}\left( 5-8\eta \right) \frac{m}{r}+\ldots \right] , \end{aligned}$$

(147)

$$\begin{aligned} I_{_{PN}}^{ijk}= & {} I_{_{N}}^{ijk}\left[ 1+\ \frac{1}{6}\left( 5-19\eta \right) v^{2}- \frac{1}{6}\left( 5-13\eta \right) \frac{m}{r}+..\right] , \end{aligned}$$

(148)

where

$$\begin{aligned} I_{_{N}}^{ij}= & {} \mu \left[ 3x^{i}x^{j}-\delta ^{ij}x^{2}\right] , \end{aligned}$$

(149)

$$\begin{aligned} I_{_{N}}^{ijk}= & {} -\mu \frac{\delta m}{m}\left[ 15x^{i}x_{j}x^{k}-3x^{j}\delta ^{ik}x^{2}-3x^{i}\delta ^{jk}x^{2}-3x^{k}\delta ^{ij}x^{2}\right] . \end{aligned}$$

(150)

A direct calculation of the above components shows that teh only nontrivial terms are given by

$$\begin{aligned} I_{_{N}}^{zz}=2\mu x^{2}, \; \; \; I_{_{N}}^{zz}=-6\mu \frac{\delta m}{m}x^{3} \end{aligned}$$

(151)

Although in many astrophysical situations the spin effects can be very small and negligible [13], in our situation they give the only nontrivial contribution to the current moments.

To lowest orders, the current multipole moments are given by

$$\begin{aligned} J^{iL}=\left\{ \epsilon ^{iab}\int \sigma ^{b}x^{aL}d^{3}x\right\} ^{STF} \end{aligned}$$

(152)

The current quadrupole moment $J^{ij}$ has an orbital angular momentum as well as a spin contribution. Substituting $\sigma \left( x,t\right) =T^{00}+T^{ii}$; $\sigma \left( x,t\right) =T^{0i}$; $T^{0i}=\rho ^{*}v^{i}$ y $T^{ij}=\rho ^{*}v^{i}v^{j}+p\delta ^{ij}$, the current cuadrupole moment is given by

$$\begin{aligned} J^{ij}=\left\{ \sum _{A}\epsilon ^{iab}\int _{A}\rho ^{*}\left( \overset{ \rightarrow }{x}\right) v^{b}x^{a}x^{j}d^{3}x\right\} ^{STF}. \end{aligned}$$

(153)

Using $\int _{A}\rho ^{*}\left( \overset{\rightarrow }{x}\right) \overset{\_}{x}_{A}^{a}\overset{\_}{v}_{A}^{b}d^{3}x=\frac{1}{2}\epsilon ^{iab}S_{A}^{i}$ together with axial symmetry, the current quadrupole moment can be written as

$$\begin{aligned} J^{ij}=\frac{3}{2}\sum _{A}\left( x_{A}^{i}S_{A}^{j}\right) ^{STF}, \end{aligned}$$

(154)

and,

$$\begin{aligned} J^{ijk}=2\sum _{A}\left( x_{A}^{i}x_{A}^{j}S_{A}^{k}\right) ^{STF}. \end{aligned}$$

(155)

Writing the above equations written in relative coordinates and using the post-Newtonian correction for the quadrupole current moment of Wiseman [3], we obtain

$$\begin{aligned} J^{ij}=\ -\frac{3}{2}\eta \left( x^{i}\Delta ^{j}\right) ^{STF}, \end{aligned}$$

(156)

and

$$\begin{aligned} J^{ijk}=2\eta \mu \left( 1-3\eta \right) \left( x^{i}x^{j}\xi ^{k}\right) ^{STF}, \end{aligned}$$

(157)

where $\xi ^{z}=S^{z}+\frac{\delta m}{m}\Delta ^{z}$ ; $S^{z}=S_{1}^{z}+S_{2}^{z}$ and $\Delta ^{z}=m\left( \frac{S_{2}^{z}}{m_{2}}- \frac{S_{1}^{z}}{m_{1}}\right)$.

The only nontrivial components are given by

$$\begin{aligned} J^{zz}=\ -\frac{3}{2}\eta x\Delta ^{z} \; \; J^{zzz}=2\eta \mu \left( 1-3\eta \right) x^{2}\xi ^{z}. \end{aligned}$$

(158)

To compute the cuadrupole and octupole current moments we take two time derivatives on the spin terms. As the spins of the bodies are aligned with the angular momentum which is constant in the axisymmetric case we have that $\overset{\cdot }{J}=\overset{\cdot }{S^{z}}=0$. We can also assume that each individual spin does not change in magnitude. Thus the only allowed changes are its orientation, parallel or antiparallel, and they enter as parameters in the calculations.

Therefore, the quadrupole and octupole current moments depend on x and $x^{2}$ respectively but the parameter coefficients are much smaller that the mass counterparts [3]. Summarizing, in a typical head-on collision the mass quadrupole moment will be dominant in the emission of gravitational radiation, followed by the the mass octupole moment and even a smaller contribution from the current quadrupole moment. Since the current octupole moment is at least an order of magnitude smaller than the above mentioned terms and for typical rotating objects is negligible [3]; we can safely assume in our equations that the octupole current moment is null.