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I want to know what exactly are practical applications of multipole expansion in some problems concerning gravitation. I have also read in Subtle is the Lord by Abraham Pais that Einstein had published a paper on gravitational quadropule radiation .

Please explain these applications in detail in layman terms or provide a resource to look them up.

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  • $\begingroup$ I have limited it to gravitational quadruple $\endgroup$ – Kutsit Mar 20 '19 at 16:57
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One possible aplication of the study of multipole electric and magnetic moments is in the study of the structure of nuclei.

The reason for this is that most of the characteristics of the motion and distribution of nucleons inside the nucleus are determined by the Strong Interaction between them and with other nuclei, therefore the study of their electromagnetic properties allows to perform measurements without perturbing the system too much.

In this context however a quantum mechanical approach is required, so the electric and magnetic moments will be observables for which you can compute expectation values related to different nuclear states. The utility of this approach is given by the fact that the measure of multipole moments gives you informations about the simmetry of the nuclear state in a way analogous to the simmetry considerations of charge distributions in classical Electrostatics.

A more technical explanation is the following.

Every multipole moment has a certain Parity associated with it. Quantum mechanically, this means that the operators associated with electric and magnetic moments have simmetric behaviours under parity transformations. A Parity Operator $\hat P$ acts on a wavefunction $\psi (\vec r)$ like

$$ \hat P\psi (\vec r) = \psi (-\vec r)$$

and its easy to verify that it has eigenvalues $\pm 1$. A wavefunction $\psi(\vec r)$ which is an eigenstate of $\hat P$ with eigenvalue $1$ is said to be even, and odd otherwise. One usually refers to the Parity Operator eigenvalue as the parity of the wavefunction, i.e. the parity of the state.

The parity of the electric moment of order $l$ is given by $(-1)^l$, and by $(-1)^{l+1}$ for the magnetic moment of order $l$ (so for example $l = 1$ stands for the electric dipole moment).

If nuclear (strong) interactions preserve parity, expectation values for odd multipole moment operators vanish, therefore you expect for example to measure a zero electric dipole moment. Since this is the case, you can conclude that strong interactions in the nuclei preserve the parity of nuclear states.

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