# About global inertial frames in GR

• I
Summary:
About the definition to use for a global inertial frame in GR
Hi,

starting from this thread Principle of relativity for proper accelerating frame of reference I'm convincing myself of some misunderstanding about what a global inertial frame should actually be.

In GR we take as definition of inertial frame (aka inertial coordinate system or inertial coordinate chart) the following:
a frame (aka coordinate chart) is inertial -- by definition -- if accelerometers everywhere at rest in it measure zero proper acceleration.

Now consider a finite region of spacetime outside the Earth: take a family of free-falling bodies (a geodesic congruence) foliating it: if we choose a frame (coordinate chart) in which those free-falling bodies are at rest (i.e. having constant spatial coordinates in that frame) then it should match the above definition of (global) inertial frame -- global at least for that finite region of spacetime.

What is wrong in the above claim ? (sure...I'm aware of global inertial frames actually do not exist in GR).

Thank you.

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## Answers and Replies

martinbn
You are using the word global when you mean local. If it is not the whole space-time, but just an open set, which is part of a coordinate chart, then it is local.

You are using the word global when you mean local. If it is not the whole space-time, but just an open set, which is part of a coordinate chart, then it is local.
So even if the coordinate chart of post #1 is inertial in a finite region of spacetime (an open set) we cannot claim of it as global, right ?

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martinbn
So even if the coordinate chart of post #1 is inertial in a finite region of spacetime (an open set) we cannot claim of it as global, right ?
This is just terminology. Call it what i want.

PeroK
PeterDonis
Mentor
What is wrong in the above claim ? (sure...I'm aware of global inertial frames actually do not exist in GR).
You're aware that global inertial frames do not exist in GR, yet you're asking about a proposed definition of a global inertial frame in GR. That's what's wrong with your claim.

Grasshopper, phinds, Prishon and 1 other person
PeterDonis
Mentor
even if the coordinate chart of post #1 is inertial in a finite region of spacetime (an open set)
Any coordinate chart covers an open set. Saying the chart "is inertial in an open set" is redundant. The question is whether the chart is inertial.

Strictly speaking, no chart in a curved spacetime can be inertial at more than a single point, since nonzero spacetime curvature will cause second order errors at every other point besides the single point you choose to center the chart on. In practice, we call a chart "locally inertial" over a finite-sized patch of spacetime if the second order errors are smaller than the accuracy of our measurements on the patch.

Grasshopper and vanhees71
So the definition quoted in post #1 actually make no sense in GR (or simply it does not apply in GR) ?

PeterDonis
Mentor
So the definition quoted in post #1 actually make no sense in GR (or simply it does not apply in GR) ?
No, the definition itself is fine (with, if you want to be precise, a qualifier that "measures zero proper acceleration" should really be "measures zero proper acceleration to within the measurement accuracy of the accelerometer"). It just is impossible for any global coordinate chart in a curved spacetime to meet it.

Grasshopper and vanhees71
Ibix
No, the definition itself is fine (with, if you want to be precise, a qualifier that "measures zero proper acceleration" should really be "measures zero proper acceleration to within the measurement accuracy of the accelerometer"). It just is impossible for any global coordinate chart in a curved spacetime to meet it.
Perhaps I'm missing something, but it seems to me that the definition as written isn't complete. I think you can produce a set of non-crossing inertial timelike worldlines that fill spacetime, at least in some cases - FLRW co-moving observers are an example. The definition in the OP is a frame (aka coordinate chart) is inertial -- by definition -- if accelerometers everywhere at rest in it measure zero proper acceleration. Co-moving accelerometers read zero and are at rest in the usual coordinates, so meet this definition. I think the definition needs something like "...and the distance between those inertial accelerometers doesn't change".

Dale and vanhees71
PeterDonis
Mentor
I think the definition needs something like "...and the distance between those inertial accelerometers doesn't change".
Ah, yes, you're right, this needs to be an additional qualifier, otherwise, as you note, standard FRW coordinates would qualify as an inertial frame.

cianfa72 and Ibix
Ibix
And we probably need to define "distance" formally in terms of radar pulses, since one could argue that co-moving distance is a distance and doesn't change between co-moving observers.

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Dale
PeterDonis
Mentor
And we probably need to define "distance" formally in terms of radar pulses
Yes, or use the more specific term "proper distance".

Ah, yes, you're right, this needs to be an additional qualifier, otherwise, as you note, standard FRW coordinates would qualify as an inertial frame.
A said in post #1 I think the same argument applies as well for any timelike geodesic congruence that foliate a finite region of spacetime (choosing a coordinate chart accordingly).

PeterDonis
Mentor
A said in post #1 I think the same argument applies as well for any timelike geodesic congruence that foliate a finite region of spacetime
No, it doesn't, because not every timelike geodesic congruence has zero expansion, i.e., not every timelike geodesic congruence has the proper distance between worldlines remaining constant.

In fact, examples of timelike geodesic congruences in a curved spacetime in which the proper distance between worldlines is constant are very rare; the only one I can think of off the top of my head is the congruence of worldlines at rest in the standard coordinate chart on Godel spacetime. Certainly there is no such congruence in FRW spacetime (except the edge case of the "empty" FRW universe, which is just Minkowski spacetime in disguise), or in Schwarzschild or Kerr spacetimes.

cianfa72
In fact, examples of timelike geodesic congruences in a curved spacetime in which the proper distance between worldlines is constant are very rare;

Certainly there is no such congruence in FRW spacetime (except the edge case of the "empty" FRW universe, which is just Minkowski spacetime in disguise), or in Schwarzschild or Kerr spacetimes.
Sorry, maybe I was unclear: that was actually my point; so even in Schwarzschild spacetime does not exist a timelike geodesic congruence such that the proper distance between geodesics is constant.

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PeterDonis
Mentor
so even in Schwarzschild spacetime does not exist a timelike geodesic congruence such that the proper distance between geodesics is constant
Yes.

Ibix
The "constant distance" requirement is Euclid's parallel postulate translated for Minkowski geometry. That it doesn't hold is one way to define curved-but-locally-Minkowski space.

The "constant distance" requirement is Euclid's parallel postulate translated for Minkowski geometry. That it doesn't hold is one way to define curved-but-locally-Minkowski space.
Is the concept of costant proper distance well defined for a pair of geodesics ?

Ibix
Is the concept of costant proper distance well defined for a pair of geodesics ?
Nearby ones, yes. You construct a perpendicular to one and measure the distance along that to reach the other. I prefer using radar because it fits better with my physical intuition, but it amounts to the same thing.

Nearby ones, yes. You construct a perpendicular to one and measure the distance along that to reach the other. I prefer using radar because it fits better with my physical intuition, but it amounts to the same thing.
From a spacetime point of view, to construct a perpendicular from an event that belong on the first geodesic, we're actually considering the tangent space at that event -- AFAIK any metric notion (including orthogonality) makes sense just in the tangent space.

Then, from a physical operational point of view, we can use radar distance, I believe.

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PeterDonis
Mentor
The "constant distance" requirement is Euclid's parallel postulate translated for Minkowski geometry.
No, it isn't. The parallel postulate is about how many geodesics through a point not on a given geodesic will never intersect the given geodesic (one in Euclidean geometry; zero in spherical geometry; more than one in hyperbolic geometry). Requiring all geodesics in a timelike geodesic congruence to maintain constant distance from neighboring geodesics is a different thing.

That it doesn't hold is one way to define curved-but-locally-Minkowski space.
If we restrict attention to spacetimes with zero cosmological constant, I think it is correct that the parallel postulate not holding is equivalent to the manifold being curved.

However, as the example of Godel spacetime shows, if there is a nonzero cosmological constant, one can have a timelike geodesic congruence that meets the "constant distance" requirement (and therefore satisfies the parallel postulate), even though the spacetime is curved.

cianfa72
Nearby ones, yes.
What if two timelike geodesics have 'finite' separation (not nearby) ? Can we define as well a proper distance between them ?

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Ibix
What if two timelike geodesics have 'finite' separation (not nearby) ? Can we define as well a proper distance between them ?
If there's a single spacelike geodesic connecting them, yes, but if there are multiple ways to connect them with spacelike geodesics that are initially perpendicular to one worldline then which one should you pick?

PeterDonis
Mentor
What if two timelike geodesics have 'finite' separation (not nearby) ? Can we define as well a proper distance between them ?
As @Ibix said, in general there will not be a unique spacelike geodesic between the worldlines. However, in one very specific case, there will be: if the timelike geodesic congruence is hypersurface orthogonal, and if geodesics of any hypersurface of the orthogonal foliation to the congruence are also geodesics of the spacetime. The only case I'm aware of that meets all these requirements is FRW spacetime.

vanhees71
PAllen