I'm looking for the linearized metric in the far field for a pair of mutual orbiting bodies that are emitting gravitational waves (GW's). I gather finding this (approximate) metric should be possible using the quadrupole moment of the source.(adsbygoogle = window.adsbygoogle || []).push({});

From Landau & LIfshitz "Classical theory of Fields", pg 377 $110.8. I gather that the metric pertubation ##h_{\alpha\beta}## should be proportional to

$$ h_{ab} \propto \frac{1}{r} \, \frac{\partial^2}{ \partial t^2 } D_{ab}$$

(If I'm reading R_0 correctly? See more below.) where D_{ab} is the (reduced) quadrupole moment. (I'll note in passing that L&L use a different notation than MTW for the reduced quadrupole moment, and while I'm preferring MTW"s notation, I think I can translate L&L's notation on this point).

What I don't think I"m getting is some talk in L&L of "projecting" the quadrupole moment. As a result when I check my attempts at a solution it's not meeting the expected gauge conditions. Is the assumption being made that GW's are emitted radially, and the projection orthogonal to this null outgoing ray?

Some additional questions came up from my reading. I gather that the TT gauge condition won't be satisfied by non-GW fields, so I gather those parts of the field are simply not modeled. (for instance, no 1/r term in ##g_{00}##) in the solution, so it won't actually be a metric with terms up to order of 1/r in the approach I'm using. Additionally, I was reviewing the PPN formalism as an alternative to calculating the desired metric, and was surprised to see that it didn't seem to have any terms of the appropriate order (I'm expecting 1/r, as I said before) in the metric. So I'm wondering if I"m interpreting this all correctly - the terms of order 1/r due to GW's exist in the far field, but the PPN metric doesn't include them? And likewise, the static term of order 1/r in ##g_00## is not included in the proposed solution? And I'd need to linearize the GW's around a non-flat background metric (rather than the flat metric) to get these terms?

While a better understanding of the issues would be good, the main goal is to find the linear-order metric valid in the far field for the purposes of defining the geometry, along with an idea of what approximations were made.

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# A Linearized metric for GW emitting orbiting bodies

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