Can we extend our inertial frame globally?

[johne1618]

People tell me that we cannot extend our inertial frame, as defined in special relativity, across the Universe because that would be in violation of general relativity.

The problem is that general relativity says that space-time can be curved whereas a global inertial frame assumes flat space-time.

However it has been observationally verified that the Universe is spatially very close to flat.

Thus only the time-component can be curved leading to the Universe's expansion rate either accelerating or decelerating.

If the Universe's expansion rate is constant then space-time is indeed flat.

But if one is only considering a small interval of cosmological time then the Universe's expansion rate is almost constant.

Thus it seems to me that one could extend our inertial frame globally provided that one is only considering a small interval of cosmological time.

Is that correct?

 

[johne1618]

Here is a better way of putting it.

General relativity says that inertial frames can only be defined locally in space-time.

But if we assume that space is flat then inertial frames can be defined globally in space provided they are only local in time.

Is that right?

 

[Ich]

If the Universe's expansion rate is constant then space-time is indeed flat.

Not necessarily. Generally, as you transform from expanding coordinates to "static" coordinates, space becomes positively curved. So for spacetime to be flat, the expanding space must be negatively curved by the exact right amount, which happens to be the case (surprise) only in an empty spacetime.
A spatially flat FRW Universe will always be positively curved in a static coordinate representation, therefore you can't use SR globally.

 

[johne1618]

A spatially flat FRW Universe will always be positively curved in a static coordinate representation, therefore you can't use SR globally.

But do I need to use a static coordinate representation?

I don't want to define a global inertial frame for all time.

I just want to define a global inertial frame for a small interval of cosmological time.

 

[Ich]

But do I need to use a static coordinate representation?

You want an inertial frame. Inertial frames are static.

I don't want to define a global inertial frame for all time.

"Static" doesn't mean "for all time", it means "doesn't change with time". This applies also to finite times or, strictly speaking, for infinitesimal short times. Further, what I really mean by "static" is: no radial expansion, a necessary requirement for inertial frames. That is defined also for an infinitesimal short time.

I just want to define a global inertial frame for a small interval of cosmological time.

That is impossible. Cosmological time is different from the time used in inertial frames, different notion of simultaneity there. Every global "inertial frame" will now spatially extend back to the Big Bang. Portions of the universe at every cosmological time until now are part of such a frame. Maybe you want to read this page which explains the different slicing in flat spacetime.

 

[Chalnoth]

Here is a better way of putting it.

General relativity says that inertial frames can only be defined locally in space-time.

But if we assume that space is flat then inertial frames can be defined globally in space provided they are only local in time.

Is that right?

No, it doesn't work this way. It's pretty easy to see that if you try to do this in an expanding universe, you quickly end up with nonsense (e.g. galaxies moving faster than light).