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Let $(M,g)$ be spacetime and consider a system composed of two particles of masses $m_1$ and $m_2$ connected by a massless rod whose lenght varies in time.

This constraint is quite easy to implement on Newtonian Mechanics. We simply consider that the two particles have position vectors $\mathbf{r}_1(t),\mathbf{r}_2(t)$ and then the constraint is that given $l(t)$ the lenght of the rod at time $t$ we have

$$|\mathbf{r}_1(t)-\mathbf{r}_2(t)|=l(t).$$

This isn't so simple on a general spacetime. There are several points. The first one is already obvious: which time should we choose?

The other are all related on how should we translate the constraint equation. The first and naive approach is to consider some arbitrary coordinate system $(x,U)$ with $\partial_{0}$ timelike and $\partial_i$ spacelike and then define the coordinate difference $R^\mu(\tau)=x^\mu(\alpha(\tau))-x^\mu(\beta(\tau))$ where $\alpha,\beta$ are the worldlines of the particles parametrized by proper time.

What bothers me the most in this is taking this "coordinate difference". This isn't a very natural operation to do in differential geometry anyway and I believe that not always $x^0$ has the meaning of time measured by some observer. One example of how this can be awkward is to consider coordinates $(t,r,\theta,\phi)$ on Schwarzchild spacetime. We would have $R^\mu=(\Delta t,\Delta r,\Delta \theta,\Delta \phi)$ which is not directly related to the lenght of the rod as seen by some observer.

One refinement would be to introduce the world function bi-scalar $\sigma(x,y)$ which gives half the geodesic distance squared between two events and impose $\sigma(\alpha(\tau),\beta(\tau))=\frac{1}{2}l(\tau)^2$. But I'm not sure this is correct, because I'm not sure that the geodesic joining $\alpha(\tau)$ and $\beta(\tau)$ is spacelike and more importantly because it seems to me that $l$ should actualy depend on some observer time $l(t)$ and not on proper time $\tau$.

To refine the first idea we could use reference frames as defined by Sachs & Wu on "General Relativity for Mathematicians". A reference frame is a future-directed timelike vector field $Q$ and a naturaly adapted coordinate system to $Q$ is a coordinate system $(x,U)$ as above with the spatial components of $Q$ zero. The issue then that, for example, $x^0$ only has meaning of time when $Q$ is synchronizable.

The important thing is that I believe the constraint is observer dependent: we should model the constraint in a general form that can be adapted for any observer, which will see the lenght of the rod changing differently.

So any of my ideas are correct? How should we model this constraint and use in practice? Is my intuition that the constraint is observer dependent correct?

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    $\begingroup$ Pretty sure the constraint is physically impossible; you cannot have rigid objects in relativity. That's not to say it's mathematically impossible, but it does sound fishy. $\endgroup$ – Javier Nov 17 '17 at 23:13
  • $\begingroup$ A related answer to a similar question may help : how-to-identify-a-measuring-rod-and-how-to-compare-separated-measuring-rods $\endgroup$ – StephenG Nov 17 '17 at 23:52
  • $\begingroup$ You could try connecting the points with an actual physical rod (which can bend and stuff). Then give it some physical properties, like tension. You will find yourself working with a relativistic string of the same kind string theoriests are working with, only classical (I.e. non-quantum) :) It is described mathematically by the Polyakov action (or equivalently by Nambu-Goto action). Then you could try taking a dynamical limit of sorts where the string has a fixed length... $\endgroup$ – Prof. Legolasov Nov 18 '17 at 16:17

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