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suppose i have a space-time where we have one point-like object* which we will call movement space probe or $\mathbf{M}_{A}$ for short, and it will be moving with constant velocity $V^A_{\mu}$ in some referential frame. We don't know (so far) how flat or twisted is the overall space-time, but we know that as far as our probe goes, it seems to be moving with total uniformity (its velocity is constant in our coordinates)

Goal: understand the true physical degrees of freedom in which we have to express the space-time and matter kinematics, assuming space-time is not necessarily flat, but momentarily forgetting to demand strictly $G_{\mu \nu} = T_{\mu \nu}$.

in the case where we demand that the geometry is completely flat and Minkowski everywhere, it seems we have to require momenta for the probe to be of the form $M^A V^A_{\mu}$. We call this description the flat description or representation $W_0$

but for our premise, space will be allowed to expand and/or contract arbitrarily in any geometrically meaningful way that comply with the observed movement of our probe, even if such geometries are not required to satisfy the Einstein equations dynamics.

As an example of the above, lets consider another interesting representation (lets call it $W_1$), which is a geometry description where all the displacement i observe of our probe is due to space-time geometry to be expanding behind the probe and contracting in front of it. In this representation, if i consider geodesic curves joining my origin of coordinates and the probe along its worldline, and then i proceed to parallel-transport the velocity vector $V_{\mu}$ of the probe along these geodesics, the coordinates of the velocity of the probe transported in such way are zero.

As a part of my stated goal, i want to consider both representations $W_0$ and $W_1$ to belong to the same equivalence class under some (unknown to me) equivalence relation, which i'll refer as the geomorphism, in lack of a cuter name

Notice that this is NOT a simple diffeomorphism, since i'm stating that space-time might actually be curved differently in each representation of the geomorphism, instead a diffeomorphism is a lot more strict than that, demanding that covariant geometric quantities transform "nicely" everywhere. The geomorphism only cares about keeping invariant across representations my actual observables (the coordinates of my probe), leaving the rest of space-time (i.e: its curvature and metric) to vary otherwise arbitrarily.

In any case, a trivial corollary of the above is that two diffeomorphically equivalent space-times are geomorphically equivalent. However, two geomorphically equivalent space-times are not necessarily diffeomorphically equivalent

Questions

1) is there a way to represent the above equivalence relation? what group structure does it have? how does it relate to diff(M)? I'm hoping to get an idea about what sort of techniques i can learn and exercise to compute the group structure given the above definition

2) can i split or recast the Einstein equations in such a way that each component transforms differently under the above equivalence relation? obviously, if the whole Einstein equations do not change when making such equivalence transformations, then in this case the answer is that we do not need any such split.


*. for simplicity i am considering a single object with uniform movement, although everything said should apply to many point-like moving masses as well, not necessarily with uniform movement

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