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my understanding of moments is that they refer to distributions about an expected value, which allows us to make the multipole expansion. I read that:

  1. the zeroth moment of mass refers to the mass of the body itself, first moment refers to the distribution about some centre of mass, and the second moment refers to how skewed the distribution is. I also read in some texts that the second moment refers to the moment of inertia (do correct me if I'm wrong)

  2. the monopole refers to the mass of the body, the quadrupole refers to how skewed the distribution of mass is.

Thus, my question is — is the gravitational quadrupole moment the same as the moment of inertia? If they aren't, what distinguishes them?

Qmechanic
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user107224
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3 Answers3

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They are not the same, but they have a relation. If the mass density at a point $\vec r = (x_1,x_2,x_3)$ is given by $\rho(\vec r)$, the components of the moment of inertia $I_{ij}$ and those of the quadrupole moment $Q_{ij}$ are defined by

$$ I_{ij}= \int_V (r^2 \delta_{ij}-x_ix_j)\rho(\vec r) dV,\quad Q_{ij}= \int_V (3x_ix_j-r^2 \delta_{ij})\rho(\vec r) dV. $$

In general, $Q=\operatorname{Tr}(I)\operatorname{id} -3I$, and in the reference frame of principal axes, their relation is the following

$$ I=\begin{pmatrix} \mathcal A&0&0\\ 0&\mathcal B& 0\\ 0&0&\mathcal C \end{pmatrix}, \quad Q=\begin{pmatrix} -2\mathcal A+\mathcal B +\mathcal C&0&0\\ 0&\mathcal A-2\mathcal B +\mathcal C& 0\\ 0&0&\mathcal A+\mathcal B -2\mathcal C \end{pmatrix}. $$

mauricio misquero
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A non-zero dipole term merely indicates that the central point in the multipole expansion is not the center of mass. The standard approach is to make the central point of the expansion be at the center of mass, in which case the dipole terms are zero.

All but the monopole term vanish in the multiple expansion of the gravitational potential of an object with a spherical mass distribution (with the expansion about about the object's center of mass). A non-zero quadrupole term indicate deviations from a spherical mass distribution.

The quadrupole tensor is quite distinct from the moment of inertia. Any non-point mass object will necessarily have a non-zero moment of inertia tensor. On the other hand, any object with a spherical mass distribution looks just like a point mass gravitationally, which means that its quadrupole tensor (expressed with respect to the center of mass) is identically zero.

David Hammen
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The first moment would be the mass dipole moment. Because the mass density is positive (i.e., we're assuming reasonable energy conditions in GR), we really don't have any interesting concept of a mass dipole. The reason we can have interesting charge dipoles is that a nontrivial charge distribution can have a zero total charge, and when the total charge is zero the charge dipole moment is independent of the choice of the origin of the coordinate system. For mass, we can't have a nontrivial mass distribution with zero total mass.

The gravitational quadrupole moment is a tensor quantity. The moment of inertia can be determined from the quadrupole moment, given a particular choice of axis. I'm not aware of a generally accepted definition of "the" second moment in more than one dimension. The quadrupole moment and moment of inertia are clearly second moments of some kind.