I Why Does GR Not Allow Global Reference Frames?

[lmoh]

Under SR, we can talk about inertial frames that apply globally. However apparently, under GR, this concept only applies locally, because it breaks down on larger scales.

Can anyone provide an explanation as to why this is? Is this due to the fact that space-time is warped in GR?

 

[Orodruin]

Can anyone provide an explanation as to why this is? Is this due to the fact that space-time is warped in GR?

No, I would say it has more to do with the (possibly) non-trivial topology of space-time. A cylinder is not curved (with the standard metric) but does not allow a global coordinate system.

 

[PAllen]

Also, for the same reason accelerated frames in SR are non-unique and limited in coverage.

1) For inertial frames in SR, all 'reasonable' simultaneity conventions are equivalent (radar, surfaces 4-orthogonal to 4-velocity, coordinates that manifest conservation laws in simplest form, etc.). In GR, even for an inertial observer, these produce different results in general, so you have a uniqueness problem even for a given inertial world line. However, the more local your coordinate coverage, the more these choices converge to the same thing (specifically, radar = Einstein convention, and 4-orthogonal criterion approach the same result sufficiently locally).

2) Even if you are willing to adopt one choice for simultaneity as preferred (e.g. Fermi-Normal coordinates based on some inertial world line), you find that the coordinates can only be defined over a limited world tube around the world line without 1-1 mapping breaking down.

I would say that both of these are due to curvature. They happen even over scales in which the topology is trivial. Topology limits the ability to have any single global coordinate system, in general. Curvature limits the ability to have unique coordinates similar to SR for inertial observer beyond a small region.

 

[Orodruin]

I would say that both of these are due to curvature. They happen even over scales in which the topology is trivial. Topology limits the ability to have any single global coordinate system, in general. Curvature limits the ability to have unique coordinates similar to SR for inertial observer beyond a small region.

This requires the existence of a metric (of course, in GR you do have a metric). However, the issue of presenting a global coordinate system in itself does not and you do not even need to have an affine connection. (Again, of course you do if you want to make sense out of things in terms of some basic physics, but generally speaking I would say the topology is the crucial obstacle.) I agree with the "similar to SR" part and that such coordinates are related to the curvature.

In the end, I think we do agree on things, but I just want to make the differences in assumptions clear here.

 

[stevendaryl]

No, I would say it has more to do with the (possibly) non-trivial topology of space-time. A cylinder is not curved (with the standard metric) but does not allow a global coordinate system.

The original poster was talking about inertial frames, rather than coordinate systems. GR has a problem with global inertial frames even when the topology is trivial, although you can still have a global coordinate system.

 

[alw34]

Would a correct answer for the OP also be to say:
"In GR spacetime is not homogeneous." as it is in SR.

For example, time passes differently here versus over there.

 

[lmoh]

Thanks for the replies guys. So the general idea is that the curvature of space makes the comparison of inertial frames difficult, due to the fact that the space between two observers is "bumpy", is that right? Since the concept of inertial frames applies only under local conditions, would it make sense to say that "local" here only means when the space within a region is mostly flat?

 

[Ibix]

Yes. Formally, you can always pick "free-fall" coordinates where the second derivatives of the metric vanish at a chosen point. "Local" means an area around that point where the second derivatives are small enough that you can ignore them within the precision of your measurements.

 

[Ibix]

Bumpy is probably the wrong word. An analogy from the surface of the Earth is that you can always define Cartesian coordinates in a neighbourhood on the Earth. That's like an area where special relativity works ok - but the curvature you are neglecting will come back to bite you if you push it too far. In particular you can find odd things happening like that there are multiple "straight line" routes between two points (you can go straight from A to B or straight away from B - on a sphere you'll end up at B eventually). That'll eventually mess up any attempt to define a global inertial frame.

 

[lmoh]

Thanks again guys. Just an additional question. I've been following a thread earlier on this forum that discusses gravity as a force instead of as the curvature of spacetime. There are apparently models of GR, such as teleparallel gravity and GTG that are described in a flat spacetime. Since gravity is treated as a force in those those models, would it be possible to define a global frame in them?

 

[alw34]

So the general idea

Sounds good to me. See if the experts erupt before deciding.

Here is a bit of a physical insight I found described in these forums:

In SR, 'curved worldlines' [paths in space and time] such as those which appear during acceleration between observer and object are of a simple type: they would appear on flat graph paper as smooth plots. This is a frame dependent curvature (observer dependent), a variable, observer dependent, overlay on top of this flat fixed background [flat graph paper]. edit: This is the result of space and time being interchanged between different observers... akin to looking at an object in 3D space from an angle...where some of my space you may see as time, or vice versa, when we are in relative motion. ]

"Curvature" in GR is a special type in which the spacetime background itself is 'curved' [distorted] in such as way that the graph paper representation itself would be crumpled in such a way that it could not be flattened without undistorting the crumples. A key is that gravitational curvature IS observer independent, all observers see it because it is the curvature of the space time manifold [“graph paper”] itself . The spacetime background in GR is also dynamic. Gravitational curvature of spacetime is an intrinsic property of spacetime and does not depend on the observer.

This contrasts with the type of curvature of a cylinder or cone, for example, which can be flattened [unrolled] and so is not GR type curvature.

Einstein's measure of GR type 'curvature is encoded in his 'stress energy tensor'.

edit: I see Ibex posted a similar description while I was composing.

 

[pervect]

It may be worth reading chapter XXIV in Einstein's "Relativity, the special and general theory" for some insight, see for instance https://www.ibiblio.org/ebooks/Einstein/Einstein_Relativity.pdf

I'll do a quick outline of the argument.

Suppose you have a marble slab. If the marble slab is flat, you can cover it with tiny square tiles - in Einstein's argument, made up of little rods - because squares "tile the plane". Einstein used the term "The arrangement is such, that each side of a square belongs to two squares and each corner to four squares." which is what I mean to summarize by saying "tile the plane".

If you try this on a sphere, it doesn't work. You can't tile a sphere, not in the sense Einstein describes.

Einstein goes on to consider a thought experiment, where the marble slab is heated, and the heat expands the rods. And he supposes that every rod, no matter what material it is made from, expands or contracts by exactly the same amount, so that there is no experiment that can recover the "unshrunk" length of the rod.

Then in terms of these rods, the marble slab is no longer flat in the sense Einstein described earlier, because you can't tile it with square arrangements of rods. I'll need to go back and add something Einstein mentioned earlier, that a "square" consists of four rods of equal length, joined together so that the diagonals are equal as well. This is pretty much the way a carpenter is typically taught to build a square frame, for instance. So we've made a bit of a jump from "square tiles" to "squares made from rods" here. Rods (which can be thought of as rulers) are the fundamental entity of concern. Einstein's description is more thorough and well laid out logically than my summary, but it's also much longer. It's definitely worth a read if you have the interest and patience to do so.

So an alternative to creating a curved space with real rulers to represent our marble slab, would be to imagine how we need to distort rulers (via the temperature field) to make the space-time flat with these imagined rulers. Because our assumptions, these imaginary rulers are completely non-physical, there is no physical substance unaffected by gravity we can use to make them experimentally. Then in terms of these imaginary and idealized rulers that don't actually exist, we might be able to describe space as being flat.

I say "space" here because that's the analogy Einstein is using with the marble slab, but abstractly Einstein is applying the same ideas to space-time rather than space, he's using the analogy to make things less abstract. . And in the analogy of space-time to space, it is the principle of equivalence that says that all the rods behave the same way due to the effects of gravity (here gravity replaces the temperature field), so that there is no sort of "little rod" we can imagine that's not affected by gravity.

 

[Cruz Martinez]

N A cylinder is not curved (with the standard metric) but does not allow a global coordinate system.

This is wrong, the cylinder does admit a homeomorphism with an open set of R^2.

 

[micromass]

No, I would say it has more to do with the (possibly) non-trivial topology of space-time. A cylinder is not curved (with the standard metric) but does not allow a global coordinate system.

A cylinder does allow for a global coordinate system actually.

 

[Orodruin]

A cylinder does allow for a global coordinate system actually.

You are right of course, for some reason I was thinking only of simply connected subsets.