How do compute an energy momentum tensor, given some equations of motion This problem can be found in a paper called "Gravitational Radiation From Point Masses In A Keplerian Orbit", but I do not have access to this, so cannot see how to do it.


I have been given a problem where we have two (point-like, non-relativistic) masses $m_1,m_2$, which are in some kind of binary orbit. They each trace out an ellipse (of different sizes), and we are given two formulae / equations of motion:

*

*One for the distance between the two masses, $r(t)$, for all time (given in terms of the eccentricity)


*One for the angular velocity $\dot\phi(t)$ in terms of $r(t)$
Then I am asked to compute the energy momentum tensor for this system.


I don't know exactly where to start. I had some ideas - it would be helpful if I knew a particular action to apply a variation to to obtain the energy-momentum tensor, but I don't know which action to use. I don't know what form of metric to use either (if I did, I could perhaps compute the Ricci tensor and Ricci scalar, then use Einstein's equations to find $T_{\mu\nu}$). But again, I don't have this starting point.
Any ideas or hints?
 A: Let us consider the energy-momentum tensor $T^{\mu \nu}$ of a perfect fluid. Then we specialize to the configuration in question.
$T^{\mu \nu} = (\rho + p) U^\mu U^\nu + p g^{\mu \nu}$
where:
$c = G = 1$ natural units
$\rho$ energy density in the rest frame
$p$ pressure in the rest frame
$U^\mu$ four-velocity
$g^{\mu \nu}$ inverse metric tensor  
A nonrelativistic system means:
1) Energy density close to mass density in the rest frame
2) Pressure negligible. In fact $p = (1/3) v^2 \rho$ where the Newtonian velocity $v$ is negligible compared to $1$ (speed of light in natural units).
3) $U^\mu$ close to $(1, \vec v)$. Again the spatial part is negligible compared to the time component.  
Hence the energy-momentum tensor of a binary system with point-like nonrelativistic masses $m_1$ and $m_2$ can be approximated in a polar coordinates system $(t, r, \theta, z)$ centered in one of the focal points as
$T^{t t} = m_1 (1/r) \delta(r - r_1) \delta (\theta - \theta_1) \delta (z) + m_2 (1/r) \delta(r - r_2) \delta (\theta - \theta_2) \delta (z)$
Other components negligible
where:
$\delta$ Dirac delta function
The factor $(1/r)$ allows for the unity when integrating the $\delta$ function in polar coordinates
$r_1, \theta_1, r_2, \theta_2$ as functions of time $t$ are the laws of motion of the masses and are given  
Note: The delta functions allow for a straightforward integration to get the quadrupole moment which in turn allows for the gravitational perturbation.
