What is the actual importance of Dixon's extended bodies theory for fundamental physics? The point here is: in 1970 W. G. Dixon developed three papers explaining one approach to describe the dynamics of extended bodies in GR. These papers are:

*

*Dynamics of extended bodies in general relativity. I. Momentum and angular momentum - In this paper, considering one extended body of energy-momentum tensor $T_{ab}$ on a spacetime $(M,g)$ with electromagnetic field $F_{ab}$ with Killing vector fields $K_a$ preserving $F_{ab}$ Dixon proposes, on the analysis of the conservation laws induced by $K_a$, definitions for momentum and angular momentum for the extended body.

*Dynamics of extended bodies in general relativity - II. Moments of the charge-current vector - In this paper, Dixon develops one general definition for multipole moments of tensor fields describing properties of extended bodies. He discusses uniqueness and existence. The main result is that there is one set of multipole moments for the charge-current vector $J_a$ called the reduced set of moments, which encodes in the simplest possible manner the conservation condition $\nabla_a J^a =0$.

*Dynamics of extended bodies in general relativity III. Equations of motion
 - In this paper, Dixon reviews the result for the existence and uniqueness of the reduced multipole moments of $J_a$ and develops the analogous construction for $T_{ab}$ finding reduced moments for the energy-momentum tensor. In terms of these moments the conservation condition $\nabla_{a}T^{ab}=J_a F^{ab}$ yields the definitions from the first paper, together with the equations of motion that can be expanded to any desired multipole order.

The issue is: I can't simply find any real importance, specially from a fundamental physics point of view, in any of this. The amount of effort to construct this theory is enormous, and in the end, what it provides is just a way to analyze extended objects, that most of the time we have seem imagined as point particles, giving already good results.
Some people try to support that this is important by talking about e.g., the swimming effect, but this for me is just a mathematical curiosity. My advisor even just wants me to use all this to solve one exercise about a dumbell falling in Schwarzschild spacetime.
I don't actually see much physical motivation behind all this, nor any real need from observations to build this whole theory, which seems just to give some corrections due to considering the size of an object.
What is actually the importance of developing this dynamics of extended bodies in GR? Does it have any important implications in fundamental physics and in the understanding of GR? Is there some actual system in nature which requires one to develop this to explain observations?
Or in the end, it is just as I'm seeing it now, a mathematical curiosity which just yields some corrections to the point-particle results and that can be used just to solve some examples (that were imagined just to be examples, instead of being motivated for occurring in nature)?
 A: Dixon's formalism is of considerable importance for the modelling of the gravitational waves emitted by so-called "extreme mass-ratio inspirals" (EMRIs). EMRIs are binary black holes consisting of a supermassive black hole and a stellar mass black hole, and form a key science goal for the planned space based gravitational wave observatory LISA.
One of the interesting aspects of EMRIs, is that unlike the binary black holes observable by LIGO, there evolution (controlled by the ratio of the masses) is very slow, meaning that we observe many ($\sim m_1/m_2$) gravitational wave cycles while the binary is in the sensitivity band of the detectors. As a result, the properties of the binary can be measured to very high accuracy ($\sim m_2/m_1$). This also means that EMRIs can be used as very sensitive probes of alternative theories of gravity. (e.g. by probing whether the primary black hole is described by the Kerr metric).
However, any such precise measurement requires any equally precise model of the evolution of the EMRI. This is where Dixon's formalism (and lot of theory built from it) comes into play. Corrections to the equations of motion, due to the (spin) dipole moment of the secondary (small) black hole come in at linear order, while corrections due to the quadrupole moment of the secondary (the "dumb bell" your advisor wants you to work on) appear at second order. Generally, we expect to need the equations of motion to at least second order, to provide waveforms of sufficient accuracy.
The subject of providing accurate modelling of EMRIs has been an active field of research for the last two decades or so, and is still on going. Currently, the field is on the verge of being able to calculate the second order corrections due to the gravitational field produced by the secondary. This means that the comparative "low hanging fruit" of the quadrupole correct is actually extremely timely.
There is an annual workshop of the people active in this field called the "Capra meetings on Radiation Reaction on General Relativity". As it happens I am organizing the 21st edition this year at the Albert Einstein Institute in Potsdam (capra21.aei.mpg.de) in June. This could be a good opportunity for you to learn about some of the motivation for the subject you are working on.
A: The theory of relativistic elasticity is one approach to extended bodies in GR. Carter & Quintana (1972) list two motivating applications:

(a) the interaction of gravitational radiation with planetary type bodies such as the Earth, and (b) the vibrations and deformations of neutron star crusts.

For rigid bodies, even the simplest examples are very instructive conceptually. The rotating disc or "Ehrenfest paradox" has been largely understood for a century (e.g. Einstein 1915; Rizzi & Ruggiero 2002, 2004), but I think there is more to be written on arbitrary reference frames, and considering other physically-motivated affine connections. Similarly for Rindler acceleration, the relation to quantum effects is still being researched.
In philosophy of physics, debate continues over the "ruler hypothesis", rulers being extended objects that measure. Also whether relativity is a "constructive" or "principle" theory (Brown 2005): do length-contraction and time-dilation arise from properties of matter? From the perspective of unifying with quantum physics, we should know what GR can and cannot do, hence what it brings to the table. The work of Dixon and others suggest that those who claim GR cannot describe extended bodies are mistaken.
These examples may be somewhat tangential to Dixon's papers specifically, but I hope they suggest the importance of extended bodies to foundational physics.
