Multipole expansion of energy-momentum tensor In this paper it is stated that the multipole moments of the stress-energy tensor $T^{\mu \nu}$ are given by
$$ \int_{x^0 = const} T^{\mu \nu} \delta x^{\alpha_1}\cdots \delta x^{\alpha_n} \sqrt{g} \: \mathrm d^3x $$
Can anyone explain where this equation comes from?
It seems to be very different to the multipole moments of e.g. an electrostatic potential $V(r)$ which is where I am most familiar with multipole expansions.
 A: These are 'cartesian' multipole moments, and you do find them in electrostatics ─ as an equivalent description to the standard one, and from time to time in explicit form. To be fully precise, the cartesian multipole moments as in your question can be easily reduced to the usual spherical multipole moments since they contain more information, but that extra information is irrelevant as to the actual fields produced.
The source of the disconnect probably comes from the standard spherical form of the multipole expansion, which for the electrostatic potential produced by a charge distribution $\rho(\mathbf r')$ (probed at some point $\mathbf r$ much further away from the origin than the support of $\rho$) reads
$$
V(\mathbf r) 
= \frac{1}{4\pi\epsilon_0}\sum_{l=0}^\infty\frac{4\pi}{2l+1} \sum_{m=-l}^l
\int
\frac{r'^l}{r^{l+1}} Y_{lm}(\theta,\phi)Y_{lm}^*(\theta',\phi')
\rho(\mathbf r')
\mathrm d\mathbf r',
$$
but this form obscures much of what's going on with the moment. Instead, it is much more instructive to write the expansion in terms of the solid harmonics $S_{lm}(\mathbf r)=r^lY_{lm}(\theta,\phi)$, so it reads
$$
V(\mathbf r) 
= \frac{1}{4\pi\epsilon_0}\sum_{l=0}^\infty\frac{4\pi}{2l+1} \sum_{m=-l}^l
\frac{ Y_{lm}(\theta,\phi)}{r^{l+1}}
\int
\rho(\mathbf r')
S_{lm}^*(\mathbf r')
\mathrm d\mathbf r',
$$
where the advantage now is that the solid harmonic $S_{lm}(\mathbf r')$ is a homogeneous polynomial of degree $l$ in the cartesian components of $\mathbf r'$. (See how much simpler the spherical harmonics just became?)
Just for flavour, here are the first few of these polynomials:


*

*$S_{00}(\mathbf r)=1/\sqrt{4\pi}$

*$S_{11}(\mathbf r)=-\sqrt{\frac{3}{8\pi}}(x+iy)$, $S_{11}(\mathbf r)=\sqrt{\frac{3}{4\pi}}z$, $S_{1,-1}(\mathbf r)=\sqrt{\frac{3}{8\pi}}(x-iy)$

*$S_{22}(\mathbf r)=\sqrt{\frac{15}{32\pi}}(x+iy)^2$, $S_{21}(\mathbf r)=-\sqrt{\frac{15}{8\pi}}(x+iy)z$, $S_{20}(\mathbf r)=\sqrt{\frac{5}{16\pi}}(2z^2-x^2-y^2)$, $S_{21}(\mathbf r)=\sqrt{\frac{15}{8\pi}}(x-iy)z$, $S_{2,-2}(\mathbf r)=\sqrt{\frac{15}{32\pi}}(x-iy)^2$


and so on. To be a bit more precise, these polynomials, when seen as functions from the unit sphere to $\mathbb C$, are orthonormal with respect to the even-weights solid-angle measure. This means that we need to drop some polynomials: there are in principle six independent polynomials of degree two ($x^2$, $y^2$, $z^2$, $xy$, $yz$ and $zx$) but they're not all linearly independent on the sphere, since $x^2+y^2+z^2\equiv 1$, so we need to reduce that sector down by one; thus, the basis $\{x^2-y^2, 2z^2-x^2-y^2\}$ is enough for that sector.
By now it should be clear how this fits with the cartesian moments in the paper you referenced: those give moments of the form
$$
M_{ij}=
\int
\rho(\mathbf r)
x_ix_j
\mathrm d\mathbf r,
$$
say, for the quadrupole sector, and those are enough to recover the spherical forms of the quadrupole moments via just explicit combinations of the $M_{ij}$. Thus, if you define the spherical moments as
$$
Q_{lm}
=
\int
\rho(\mathbf r)
S_{lm}^*(\mathbf r)
\mathrm d\mathbf r,
$$
and you only know the $M_{ij}$, then you can recover the $Q_{2m}$ as


*

*$Q_{22}=\sqrt{\frac{15}{32\pi}}(M_{xx}-M_{yy}+2iM_{xy})$,

*$Q_{21}=-\sqrt{\frac{15}{8\pi}}(M_{xz}+iM_{yz})$,

*$Q_{20}=\sqrt{\frac{5}{16\pi}}(2M_{zz}-M_{xx}-M_{yy})$,

*$Q_{2,-1}=\sqrt{\frac{15}{8\pi}}(M_{xz}-iM_{yz})$, and

*$Q_{2,-2}=\sqrt{\frac{15}{32\pi}}(M_{xx}-M_{yy}-2iM_{xy})$.


On the other hand, it seems like you've lost information, because the linear combination
$$
\sum_i M_{ii} = 
\int
\rho(\mathbf r)
r^2
\mathrm d\mathbf r
$$
cannot be represented from the $Q_{lm}$, but as far as electrostatics goes, this combination is irrelevant and it is never used in the calculation of any electrostatic field ─ for the deep reason that it is essentially a measure used to distinguish between spherical shells of equal charge but different radius, and those produce the same electrostatic fields.
That last bit, of course, is domain-dependent, and depending on what it is you want to do with your moments they may or may not contain superfluous information that you cannot use, in which case it may or may not be advantageous to reduce the representation to one that (like the $S_{lm}(\mathbf r)$) explicitly trims off the superfluous degrees of freedom. However, there is no question at all that the rightful name for the moments in that paper is the multipole moments of the $T_{\mu\nu}$.
