Is a falling elevator a valid inertial frame?

[Austin0]

Is a falling elevator a valid inertial frame?

The title about says it. Is a body, (frame) free falling in a gravitational field a valid inertial frame as defined by SR?


Thanks

 

[DrGreg]

Is a falling elevator a valid inertial frame? The short answer is "yes".

The longer answer is "locally, approximately yes".

If the elevator is small enough, and you consider it over a short enough period of time, such that you can consider the "acceleration due to gravity" to be constant in magnitude* and direction throughout the elevator, then, yes, within those limits it is a very good approximation of an inertial frame. Any free-falling object within the elevator will move at constant velocity relative to the elevator, and the laws of physics, locally, look identical to an inertial frame in the absence of gravity. This is the Equivalence Principle of general relativity.

But if you try to extend your elevator coordinates to cover a large distance or a very long time, you'll find that distant free-falling objects don't move at constant velocity relative to you.

In general relativity we therefore say that the frame of a free-falling observer is locally inertial.

The rigorous mathematical treatment of this considers the calculus limit as the size of the elevator tends to zero.

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*Actually, to be pedantic, "constant in magnitude" isn't quite true either. It really should be "uniform" in the same sense of "indistinguishable from Born rigid acceleration". But "locally" the difference is as negligible as the true variation due to "real" gravity.

 

[Austin0]

Is a falling elevator a valid inertial frame? The short answer is "yes".

The longer answer is "locally, approximately yes".

If the elevator is small enough, and you consider it over a short enough period of time, such that you can consider the "acceleration due to gravity" to be constant in magnitude and direction throughout the elevator, then, yes, within those limits it is a very good approximation of an inertial frame. Any free-falling object within the elevator will move at constant velocity relative to the elevator, and the laws of physics, locally, look identical to an inertial frame in the absence of gravity. This is the Equivalence Principle of general relativity.

But if you try to extend your elevator coordinates to cover a large distance or a very long time, you'll find that distant free-falling objects don't move at constant velocity relative to you.

In general relativity we therefore say that the frame of a free-falling observer is locally inertial.

If the elevator is extended to cover a large distance vertically wouldn't this also bring in a factor of relative acceleration due to the local gravitational difference at the extremes?
And also tend to cause stress and length elongation somewhat comparable to the Born hypothesis wrt acceleration?
From your responce can I assume that the measurement of light would be invariant and isotropic in relatively small systems?

 

[DrGreg]

If the elevator is extended to cover a large distance vertically wouldn't this also bring in a factor of relative acceleration due to the local gravitational difference at the extremes?
And also tend to cause stress and length elongation somewhat comparable to the Born hypothesis wrt acceleration?
From your responce can I assume that the measurement of light would be invariant and isotropic?

I stressed everything is a local approximation for small elevators, so you are right that for very tall elevators the approximation breaks down due to the tidal effects of gravity.

However, for "smallish" elevators, where tidal effects (due to gravitational variation) can be ignored but some other effects can't, you'll find the analogous "Born rigid" effects work the "wrong way round". The elevator is approximately inertial, and someone stationary on the ground is approximately undergoing Born rigid acceleration upwards. So the elevator measures approximately constant and isotropic light speed but the ground observer reckons light speed varies with height (relative to his own local coordinates), and "gravitational time dilation" that varies with height.

(By the way I made a minor edit to my last post since you read it.)

 

[A.I.]

So does that mean that the only true inertial reference frame is a mathematical point undergoing acceleration due to a uniform gravitational field?

 

[DrGreg]

So does that mean that the only true inertial reference frame is a mathematical point undergoing acceleration due to a uniform gravitational field?

Well if you won't accept any approximations, in the real Universe there aren't any truly inertial frames, only local approximations.

Yes, an observer falling freely in a uniform gravitational field would define an inertial frame, but there aren't any truly uniform gravitational fields in the real universe (as far as I know). An observer moving freely in the complete absence of gravity would define an inertial frame, but there's nowhere in the universe you can completely escape gravity. Nevertheless, there are lot of places where gravity can be considered either negligible or almost uniform, so in those places you can certainly set up a local inertial frame.

(Note that you could escape gravity inside a hypothetical hollowed out planet, but that would still only be local to inside that planet.)

 

[Austin0]

I stressed everything is a local approximation for small elevators, so you are right that for very tall elevators the approximation breaks down due to the tidal effects of gravity.

However, for "smallish" elevators, where tidal effects (due to gravitational variation) can be ignored but some other effects can't, you'll find the analogous "Born rigid" effects work the "wrong way round". The elevator is approximately inertial, and someone stationary on the ground is approximately undergoing Born rigid acceleration upwards. So the elevator measures approximately constant and isotropic light speed but the ground observer reckons light speed varies with height (relative to his own local coordinates), and "gravitational time dilation" that varies with height.

Is the stress and length dilation we are talking about here [for sufficiently large systems ]
due to tidal effects or the difference in the g factor due to relative distance from the center of the field??
Or am I incorrect in my understanding that the tidal forces are due to the ricci tensor (forces?) transverse to the orientation of the field center and that the g (acceleration?) was due to weyl forces radial to the field to that center?
I guess what I am asking in simple graphic terms is; would it get longer because it was being squeezed or because it was being accelerated faster at one end ,,,or both?

I have other questions relative to Born but I will wait for another opportunity
Thanks

 

[Austin0]

I stressed everything is a local approximation for small elevators, so you are right that for very tall elevators the approximation breaks down due to the tidal effects of gravity.

However, for "smallish" elevators, where tidal effects (due to gravitational variation) can be ignored but some other effects can't, you'll find the analogous "Born rigid" effects work the "wrong way round". The elevator is approximately inertial, and someone stationary on the ground is approximately undergoing Born rigid acceleration upwards. So the elevator measures approximately constant and isotropic light speed but the ground observer reckons light speed varies with height (relative to his own local coordinates), and "gravitational time dilation" that varies with height.

( .)

How is this perspective? From an inertial frame, the elevator is accelerating , with velocity, doppler shift and dilation changing dynamically.
From this viewpoint it would seem to be the same as the Born hypothesis.
For the elevator to retain its proper length would require additional acceleration at the back/top. In principle the exact right amount would just compensate for the elongation without adding to the velocity at the front. I guess in principle the increased velocity could be expected to cause Lorentz contraction also .
What about gravitational length contraction , does EP apply in that regard as well?
Thanks

 

[DrGreg]

To make this easier to discuss, let's assume the elevator is freely falling down the elevator shaft of a skyscraper. (Of course we are assuming there is no air resistance or friction.)

From an inertial frame, the elevator is accelerating , with velocity, doppler shift and dilation changing dynamically. From this viewpoint it would seem to be the same as the Born hypothesis. For the elevator to retain its proper length would require additional acceleration at the back/top. In principle the exact right amount would just compensate for the elongation without adding to the velocity at the front.

No, this is why I said it's the "wrong way round". The elevator is the inertial frame, and it's the skyscraper that is Born-rigidly accelerating upwards. Accelerometers in the skyscraper determine that the skyscraper is undergoing proper acceleration and that the bottom of the skyscraper is accelerating a little more than the top in order to maintain a constant proper height.

You can't pick your own choice of which frame is inertial. It has to be the one in which you are "weightless", the elevator frame.

(Note: throughout the above, "inertial" is a shorthand for the technically more correct "locally inertial".)

 

[DrGreg]

Is the stress and length dilation we are talking about here [for sufficiently large systems ]
due to tidal effects or the difference in the g factor due to relative distance from the center of the field??

These are really just two different ways of describing the same thing.

"Tidal effects" arise because the acceleration due to gravity doesn't vary in quite the same way as the "uniform acceleration due to pseudo-gravity" in Born-rigid accelerating frames in deep space. These tidal effects are of a smaller order of magnitude than the "Born rigid" effects, so over shortish distances they become negligible. The Born-rigid-like effects are "first order" effects (varying approximately proportional to distance), but the tidal effects are "second order" (varying approximately proportional to square distance, or smaller).