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MakiseKurisu
MakiseKurisu
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I think your definition is not clear. Because "global" intuitively means "for every place", how can a global frame exclude the local part?

And the term "local" means for some certain region (always infinitesimal) so how can it be global?

If we use the usual definition of these concepts,then

Only for flat spacetime, we have global inertial reference frame. where global means everywhere and inertial means no gravity,so only flat spacetime has no gravity everywhere.

For curved spacetime, we can always construct locally inertial reference frame.Physically corresponding to the equivalence principle and mathematically corresponding to the Fermi coordiante (also called Riemann normal coordinate).

EDIT:

I see. But I guess your definition is not quite good. Suppose A is a free falling observer, by the equivalence principle, A don't know whether A is free falling in a gravitational field or just a observer without acceleration in flat spacetime.

But, the previous discussion only works for a small lab. Mathematically, the lab should exactly be 1 point or 1 line on the spacetime manifold which has no volume, this means for very real lab (which must have a finite volume), we can always, if ignore the experimental condition, clarify whether this lab is in a gravitational field or in flat spacetime. If we find the physical law in our lab exactly satisfy the special relativity, then our spacetime must be flat and we can define the global inertial frame. But if we find there exists a difference caused by gravity, then we can claim there is no global inertial frame.

Back to your comment, we can just perform the experiment: send a light beam in this lab and check its behavior, if the behavior perfectly fits special relativity, then you can claim that every frame attached with an inertial observer is a global inertial frame, although you can just design a lab with finite volume, but in principle it can be as large as possible.

If the light not fits SR, then we can claim there is no global inertial frame, also no physically local inertial frame because it always has a finite volume. The only thing we can do is to design the volume of lab as small as possible and all experiment in this lab will not be strongly affected by gravity. And if the lab is attached to a free falling observer, then the result of the experiment will almost behave like in a inertial frame.

I think your definition is not clear. Because "global" intuitively means "for every place", how can a global frame exclude the local part?

And the term "local" means for some certain region (always infinitesimal) so how can it be global?

If we use the usual definition of these concepts,then

Only for flat spacetime, we have global inertial reference frame. where global means everywhere and inertial means no gravity,so only flat spacetime has no gravity everywhere.

For curved spacetime, we can always construct locally inertial reference frame.Physically corresponding to the equivalence principle and mathematically corresponding to the Fermi coordiante (also called Riemann normal coordinate).

I think your definition is not clear. Because "global" intuitively means "for every place", how can a global frame exclude the local part?

And the term "local" means for some certain region (always infinitesimal) so how can it be global?

If we use the usual definition of these concepts,then

Only for flat spacetime, we have global inertial reference frame. where global means everywhere and inertial means no gravity,so only flat spacetime has no gravity everywhere.

For curved spacetime, we can always construct locally inertial reference frame.Physically corresponding to the equivalence principle and mathematically corresponding to the Fermi coordiante (also called Riemann normal coordinate).

EDIT:

I see. But I guess your definition is not quite good. Suppose A is a free falling observer, by the equivalence principle, A don't know whether A is free falling in a gravitational field or just a observer without acceleration in flat spacetime.

But, the previous discussion only works for a small lab. Mathematically, the lab should exactly be 1 point or 1 line on the spacetime manifold which has no volume, this means for very real lab (which must have a finite volume), we can always, if ignore the experimental condition, clarify whether this lab is in a gravitational field or in flat spacetime. If we find the physical law in our lab exactly satisfy the special relativity, then our spacetime must be flat and we can define the global inertial frame. But if we find there exists a difference caused by gravity, then we can claim there is no global inertial frame.

Back to your comment, we can just perform the experiment: send a light beam in this lab and check its behavior, if the behavior perfectly fits special relativity, then you can claim that every frame attached with an inertial observer is a global inertial frame, although you can just design a lab with finite volume, but in principle it can be as large as possible.

If the light not fits SR, then we can claim there is no global inertial frame, also no physically local inertial frame because it always has a finite volume. The only thing we can do is to design the volume of lab as small as possible and all experiment in this lab will not be strongly affected by gravity. And if the lab is attached to a free falling observer, then the result of the experiment will almost behave like in a inertial frame.

Source Link
MakiseKurisu
MakiseKurisu
  • 184
  • 9

I think your definition is not clear. Because "global" intuitively means "for every place", how can a global frame exclude the local part?

And the term "local" means for some certain region (always infinitesimal) so how can it be global?

If we use the usual definition of these concepts,then

Only for flat spacetime, we have global inertial reference frame. where global means everywhere and inertial means no gravity,so only flat spacetime has no gravity everywhere.

For curved spacetime, we can always construct locally inertial reference frame.Physically corresponding to the equivalence principle and mathematically corresponding to the Fermi coordiante (also called Riemann normal coordinate).