Expressing Locality in Vector Space for GR

[kent davidge]

How do we express the locality of a vector space in general relativity?

I mean, it's not clear what the boundaries of a given vector space are.

To put in another way, we could, in principle, blindly consider that we have the entire of $\mathbb{R}^4$ at our disposal to describe, say, the motion of a particle near Earth. The question which comes in is to what extent our calculations become wrong, at which point we should sort of split our reference frame in two, three.. etc reference frames as required by general relativity.

 

[fresh_42]

How do we express the locality of a vector space in general relativity?

I mean, it's not clear what the boundaries of a given vector space are.

To put in another way, we could, in principle, blindly consider that we have the entire of $\mathbb{R}^4$ at our disposal to describe, say, the motion of a particle near Earth. The question which comes in is to what extent our calculations become wrong, at which point we should sort of split our reference frame in two, three.. etc reference frames as required by general relativity.

The calculations become wrong the moment you leave the point of consideration. The question is how wrong? E.g. Newton works perfectly well, until you decide to measure with atomic clocks. Or as @PeterDonis has recently put it: As long as you don't want to establish a GPS for Mars, general relativity isn't needed. I think the crucial point is, that the local tangent space can be transported along curves in the manifold and it is still a tangent space at some new point. Of course this only works well as long as no singularities or different connection components come into play. An interesting example are fast myons in cosmic radiation. For an average lifetime of $2\cdot 10^{-6}$ seconds, how come we can detect them on the ground? So when calculations are wrong is what calculations are performed.

 

[Orodruin]

that the local tangent space can be transported along curves in the manifold and it is still a tangent space at some new point

This remains true also globally. The question is how much the parallel transports along different curves differ.

It should also be pointed out to the OP that space-time itself is a manifold, not a vector space.

 

[PeterDonis]

at which point we should sort of split our reference frame in two, three.. etc reference frames as required by general relativity.

I don't understand what you're referring to here.

 

[sweet springs]

If I am not mistaken, though I can not find its proof on internet now, in early 1800s, Friedlich Gauss proposed an idea that non-Eucledian geometry may be found by observation of stars in space far away. Discovery by Hubble might have been on this line. Evaluation of Hubble constant might give some numerical estimation to your problem, deviation from flat spacetime.

Hubble constant
67.15 km/s /Mpc=67.15 km/s / 3.086 E22 m = 2.18E-18 (m/s) /m
1m distance of two "still" bodies increases 2 attometer every second.