I'm not sure what the proper definition of the quadrupole moment tensor is. In the book on gravitational waves by Maggiore, the definition is $$M^{ij}=\int d^3x T^{00}x^ix^j. \tag{3.37}$$ ( Maggiore, Gravitational Waves, Volume 1 eq. 3.37.)

Meanwhile, in the books by Wald, Carroll and Weinberg, they say $$M_{ij}=\int d^3x T^{00}x^ix^j. \tag{7.138}$$ (For instance, see Spacetime and Geometry by Carroll, eq. 7.138.)

This last definition tensorially speaking seems very strange to me. You may say that since we're dealing with a small perturbation in analyzing gravitational waves, our metric $g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}$ with all $h^{\mu\nu}$ small. Then you could say that the position of the spatial indices doesn't matter.

However, when we're inside of the integral we have to raise/lower indices at each point in space. Then more specifically we also have to raise/lower indices at the location of the source, where $h^{\mu\nu}$ will certainly not be small. So it seems like the two definitions are not equivalent.

The typical analysis of the field far from a source in general relativity is contained entirely within the framework of the linearised theory. This means we always assume

$$h_{\mu \nu} \sim T_{\mu \nu} \sim \epsilon$$

and drop $\epsilon^2$ terms wherever they arise. If we lower indices on the $x^i$ in $(7.138)$, the extra terms that are generated are indeed of order $\epsilon^2$.

This is just as true at the source as elsewhere. We often make other simplifying assumptions in deriving weak-field results, such as assuming that we are very far from the source – these are also assumptions we make when analysing radiation in (exactly linear) electromagnetism. They are separate from the assumption that we make at the very start: that GR can be treated as linear.

One can ask how good an approximation this is. For a black hole or dense neutron star, the approximation is indeed poor, and one really needs to use numerical simulations to get accurate results. However, for other compact objects such as white dwarfs, the approximation is good.

Yes, OP is right. The spatial indices on the LHS. of eq. (7.138) should formally be upper (as opposed to lower). This does not make a difference if we raise and lower with the Minkowski metric in rectangular coordinates $x^{\mu}$ with $\eta_{\mu\nu}={\rm diag(-1,1,1,1)}$, which Carroll presumably assumes.

• But check the second part of my post please; there I give my thoughts on why this raising/lowering indices with the Minkowski metric seems strange, or not well-founded (at least to me). This is the part where I guess I'm misunderstanding something. – Physics Llama Aug 8 '18 at 18:09
• Eqs. (3.37) & (7.138) are merely definitions. That they are useful definitions follows from the linearized EFEs. – Qmechanic Aug 10 '18 at 8:56