I read Chris Hirata's paper on gravitational waves (http://www.tapir.caltech.edu/~chirata/ph236/lec10.pdf) where he performs a multipole expansion of the gravitational source.(adsbygoogle = window.adsbygoogle || []).push({});

I got most of it, apart from the part where he expands the inverse distance function into a series :

More specifically the second term in the last line, which should be positive according to me (product of 2 negative factors -1/2 and -2). Hirata finds a negative sign.

The sign is pretty important, since at the end of the paper he proves that the resulting gravitational wave is transversal by finding the vector product of the wave and the propagation vector to be zero. Which is indeed the case ... provided that the second term above is negative. Which, according to me, it isn't.

In no way I consider myself smarter than Chris Hirata, but here he lost me. And literature does not give me an answer.

Can anyone explain this sign ?

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# A Multipole expansion of linearized field equations

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