In the gravitational wave calculation for binary systems: what is the effect of rotation of two BH (or neutron stars, BH-NS,...) on the usual calculations? Is there any EXACT result known?
There are two questions; I will first answer the simpler one
is there an exact solution to two-body problem in GR:
- Currently, there is no exact solution to the two-body problem in GR, as pointed out by Qmechanic.
The second question is
what is the effect of rotation of two BH (or neutron stars, BH-NS,...) on the usual calculations?:
This is a bit more complicated.
First, let me point out that there are approximate solutions to binary systems in the limit of roughly equal mass binaries up to the innermost stable circular orbit using Post-Newtonian (PN) Expansion. The approximation breaks down after the innermost stable circular orbit because the system becomes highly non-linear and the binaries move at relativistic velocities (PN expansion is in v/c, where v is the effective velocity of the binary). The merger part itself is computed using numerical relativity or semi-analytical models based on numerical relativity.
The spin-orbit coupling has contributions at higher post-Newtonian terms. What do I mean by higher post-Newtonian terms? Each post-Newtonian term is computed by solving the Einstein Field Equations (EFE) recursively to next order (0th Order is Newtonian, plugging this into EFE will yield 0.5 PN correction terms, which you can recursively insert back to EFE -- solving the equations up to 4 PN order takes a few years and so we only have PN expansion up to 4 PN for binaries of equal masses). The effect itself is somewhat known from Newtonian physics already, which is that the spin wobbles around the orbital angular momentum vector (see e.g. this simulation for demonstration).
This shows up as small wiggliness in the gravitational wave strain. See the following figure of the absolute value of plus-polarized gravitational wave strain from (5, 5) solar mass binary without spin (red) and with near extremal spin perpendicular to the orbital angular momentum (black) as a function of the gravitational wave frequency:
In addition to showing up in the gravitational wave strain amplitude, it shows up in the phase of the gravitational wave. I'll leave solving the specifics of that as homework, since I don't think explaining it will be very informative. I'll just say that the phase of the gravitational wave is directly related to the phase of the orbit.
If you want to learn more about the math, I would suggest taking a look at this reference. It takes a bit of effort to read.
Hope that helps!
No, there is no exact solution for the 2-body problem in GR; only approximative & numerical solutions.