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I am currently working on gravitational waves, and as in every lecture on general relativity I derived the symmetric trace-free perturbation of a Minkowski metric with the Lorenz gauge condition, known as the "quadrupole formula" :

$$h_{ij}^{STF}=\frac{2}{r}\frac{d^{2}I_{ij}^{STF}}{dt^{2}} $$

Where $ I_{ij}^{STF}$ is the symmetric trace-free part of the quadrupole momentum tensor of the source. (Correct me if mistaken or uncomplete)

The thing is: how can we calculate the quadrupole momentum tensor of an object, and particularly a compact binary one ? (one of those which emits the gravitational waves detected by LIGO). I have come across the direct formula:

$$I_{ij}=\int d^{3}x \rho x_{i} x_{j}$$

for the quadrupole momentum but I don't think we can use that directly, since e.g. we don't have accurate equation of state for neutron stars or other compact objects.

Some people seem to have proceeded differently (and in a more general way, using a multipolar expansion here, eq. 50), but that's very technical.

Are there clever tricks to compute this momentum ? Or at least some (massive?) approximations that would give at least a qualitative yet directly understandable physical result/expression ?

EDIT : It just stricked me that I am not familiar with the concept of quadrupole, so maybe that's my issue here... I am going to make some research and maybe re-edit this post.

EDIT 2 : I did some research on the quadrupole moment and multipole expansion in general. I think I got the physical meaning of this expansion for a single compact object (e.g. why it is meaningful to proceed that way for the ringdown oscillation of a BH). Still, I could not find anything directly related to my question. Any help would be highly appreciated, even if it is "You have to go brute force for this computation boy"

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  • $\begingroup$ For neutron stars, you do need to assume some equation of state (EOS) and integrate numerically. This is one of the ways that we could (eventually) begin to infer the EOS from GW observations (although other signatures will likely be easier). Still, even considering two point masses, these equations are good (and usually sufficient) approximations. $\endgroup$ – DilithiumMatrix Mar 13 '17 at 14:10
  • $\begingroup$ That's what I've seen in the book I read on numerical simulation of compact binaries, but it seems nobody tried to go with the order-of-magnitude/qualitative understanding for the GW emitted during the inspiral and merger of the binaries. Maybe it's been tried and now it's admitted to be pointless... Though I'd love to have this kind of point of view. (If you know any text attempting this approach I'd love to read it). Being curious : what are the signatures you're thinking about ? The Fourier spectrum of the GW ? $\endgroup$ – Naptzer Mar 13 '17 at 14:30
  • $\begingroup$ I couldn't find a good source, but perhaps take a look at the references in this paper: arxiv.org/abs/gr-qc/9910091. The point-mass formalism definitely breaks down during the actual merger: that requires numerical GR (solved in 2005), so for the estimates, perhaps look at stuff around then and earlier. In terms of the NS EOS: the strongest measurement is the highest frequency the waveform reaches. That indicates the point at which the NS is tidally disrupted---which measures the EOS. $\endgroup$ – DilithiumMatrix Mar 13 '17 at 15:54

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