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Hi everyone.
At the end of the gravitational analogue video, we started from the more general Taylor expansion approach rather than the Laplace expansion and found a correction term. It is called a monopole shift: but why exactly? Earlier it is mentioned that the monopole and dipole terms are identical for the two approaches, and that the first difference arises at the quadrupole term: so why isn’t it in the final table in the column for “quadrupole”? Or alternatively: how can one see that this is a monopole contribution?
Also, on slides 14 and 15, in the given matrices, what is the meaning of the {} brackets? Is it just for clarity?
Thanks for the help and kind regards,
Thibeau WoutersStijnParticipantHi Thibeau,
Unfortunately I don’t know the answer to you first question, but I can adress your second question.
I believe the brackets {} are used as shorthand notation for an integral. Compare for example the first and second expression on slide 14. You will notice that {3x_1y_1} is shorthand notation for the integral over dr_1 with as integrand 3x_1y_1*rho_1(r_1).
I hope this helps.
Best,
StijnClaraParticipantHi!
As a reply on your first question: I wonder if this has to do with the fact that the contribution to the shift from the two distributions are scalars. The size of m_1 is a scalar and the mass contribution of m_2 at the origin as well.
This is then similar to the case of the monopole moment and the monopole field both being scalar in the monopole term. So perhaps this is the standard to call the resulting energy(shift) a monopole shift?Thanks for the answers!
Regarding the {} notation: I completely missed that, but I guess you are indeed right with the integral shorthand. Thanks Stijn!
For the first question: that could be true, but it still seems a little bit weird to me. After all, it should be a scalar since it’s an energy, and we also compute additional corrections for other multipole modes, e.g. there is also a correction coming from the dipole. So that would imply all corrections are monopole corrections. Perhaps it’s just definition, I would just expect it to be called a ‘quadrupole’ correction since it comes from quadrupole objects…

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