Local Inertial Frame: Understanding Riemann Curvature & Metric Tensor

[Neitrino]

Dear PF could you advise me
Whether I understand properly or not:

In an arbitrary space-time (with an arbitrary curvature) in any sufficiently little region we can go to Local Inertial Frame of Reference - sit into the free falling lift. Being there our experiments are the same as we have in SR.

But if I have some mysterious device like a voltmeter where instead of volts there are Riemann curvature tensor valus measuered along its scale...

What this device will show me? Of course it should show me non-zero curvature tensor values?

So on the one hand I think that I am in SR (in flat space-time) since my experiments in free falling lift are the same as in SR, but when I look on my "curvature-meter" device it shows me that there are non-zero tensor components...

If I have similar device called "metric-tensor-meter" it should show me the Minkovski values... correct ?

 

[Dale]

How would you build such a device?

 

[JesseM]

How would you build such a device?

On p. 400 of Gravitation by Misner/Thorne/Wheeler they discuss ways one might measure the Riemann curvature tensor:

To measure the Riemann curvature tensor near an event, one typically studies the geodesic deviation (relative accelerations) that curvature produces between the world lines of a variety of neighboring test particles; alternatively, one makes measurements with a "gravity gradiometer" (Box 16.5) if the curvature is static or slowly varying; or with a gravitational wave antenna (Chapter 37) if the curvature fluctuates rapidly.

The section on the gravity gradiometer can be read on a google book search if you skip forward to p. 401 (click 'Contents' at the top of the page to go directly to page 399):

http://books.google.com/books?id=w4Gigq3tY1kC&printsec=frontcover&source=gbs_v2_summary_r&cad=0

 

[Dale]

To measure the Riemann curvature tensor near an event, one typically studies the geodesic deviation (relative accelerations) that curvature produces between the world lines of a variety of neighboring test particles

Such a device would necessarily be large, not at all what the OP is thinking about. I don't think it is possible to build a device like what the OP is describing, something to measure the curvature at a point.

 

[JesseM]

Such a device would necessarily be large, not at all what the OP is thinking about. I don't think it is possible to build a device like what the OP is describing, something to measure the curvature at a point.

Not possible in practice, or theoretically impossible even for a device whose parts could be arranged with arbitrary levels of precision, and their movements measured arbitrarily precisely too? (and ignoring quantum physics which places limits on precision) I got the idea that the question in the OP was more theoretical than practical.

 

[Rasalhague]

In an arbitrary space-time (with an arbitrary curvature) in any sufficiently little region we can go to Local Inertial Frame of Reference - sit into the free falling lift. Being there our experiments are the same as we have in SR.

But if I have some mysterious device like a voltmeter where instead of volts there are Riemann curvature tensor valus measuered along its scale...

What this device will show me? Of course it should show me non-zero curvature tensor values?

In Spacetime Physics Taylor and Wheeler define such a local inertial frame of reference as a region of spacetime sufficiently small for tidal effects of gravity not to be detectable by whatever measuring devices you have at your disposal. So if I've understood this right, your curvature-meter would show no curvature in such a frame because, by definition, it wouldn't be sensitive enough. And if you found some way of improving its sesitivity to the point where you could detect some curvature in this region of spacetime, then, by that same definition, it would no longer be in an inertial reference frame.

 

[Naty1]

In an arbitrary space-time (with an arbitrary curvature) in any sufficiently little region we can go to Local Inertial Frame of Reference - sit into the free falling lift. Being there our experiments are the same as we have in SR.

I think Rasal's reference answers it...although it's a bit difficult to tell what the OP is most interested in...the chacteristic of the spacetime curvature or the device.

So on the one hand I think that I am in SR (in flat space-time)

You defined it so via "sufficiently little region"...in a free falling frame. You will see flat space in arbitrarily small observations and if your meters are good they will reflect that.

 

[JesseM]

In Spacetime Physics Taylor and Wheeler define such a local inertial frame of reference as a region of spacetime sufficiently small for tidal effects of gravity not to be detectable by whatever measuring devices you have at your disposal. So if I've understood this right, your curvature-meter would show no curvature in such a frame because, by definition, it wouldn't be sensitive enough. And if you found some way of improving its sesitivity to the point where you could detect some curvature in this region of spacetime, then, by that same definition, it would no longer be in an inertial reference frame.

I think the definition of the equivalence principle in GR is actually pretty subtle if you want to define it technically as opposed to heuristically, see this thread and this one for some previous discussions. In particular, in the second thread atyy suggests in the opening post that the equivalence principle may only apply to "first order in Taylor series", suggesting that if you have a device which can measure second-order effects that would allow you to tell you were in a curved spacetime, even though these effects should be measurable in an arbitrarily small region. Also, if I'm understanding Taylor and Wheeler's definition above, they're saying that if you have a device that can measure tidal effects within its limits of accuracy (which I think would be equivalent to second order effects in the Taylor series) then by definition you're in a region of spacetime that's not "sufficiently small" for the equivalence principle to hold; if your device can't measure them then it is "sufficiently small", even though in principle it would always be possible to build a more sensitive device that could measure tidal effects in the same region of spacetime.

 

[DrGreg]

In Spacetime Physics Taylor and Wheeler define such a local inertial frame of reference as a region of spacetime sufficiently small for tidal effects of gravity not to be detectable by whatever measuring devices you have at your disposal. So if I've understood this right, your curvature-meter would show no curvature in such a frame because, by definition, it wouldn't be sensitive enough. And if you found some way of improving its sesitivity to the point where you could detect some curvature in this region of spacetime, then, by that same definition, it would no longer be in an inertial reference frame.

I think the definition of the equivalence principle in GR is actually pretty subtle if you want to define it technically as opposed to heuristically, see this thread and this one for some previous discussions. In particular, in the second thread atyy suggests in the opening post that the equivalence principle may only apply to "first order in Taylor series", suggesting that if you have a device which can measure second-order effects that would allow you to tell you were in a curved spacetime, even though these effects should be measurable in an arbitrarily small region. Also, if I'm understanding Taylor and Wheeler's definition above, they're saying that if you have a device that can measure tidal effects within its limits of accuracy (which I think would be equivalent to second order effects in the Taylor series) then by definition you're in a region of spacetime that's not "sufficiently small" for the equivalence principle to hold; if your device can't measure them then it is "sufficiently small", even though in principle it would always be possible to build a more sensitive device that could measure tidal effects in the same region of spacetime.

I have to agree with both of the quotes above.

Bear in mind that a "locally inertial frame" is something that approximates a truly inertial frame over a "small enough" region. "Approximation" does not have a precise definition; you are neglecting the tidal effects as being too small to be of significance, but mathematically those effects are never exactly zero.

Curvature is defined in terms of derivatives. Mathematically you evaluate a derivative via Δy/Δx as both Δy and Δx shrink to zero. But that method won't work in the real world because both Δy and Δx are corrupted by errors ("noise"), so you have to stop shrinking the value before the noise gets too great. Any curvature measuring device's accuracy is therefore limited by its physical size and it will become too inaccurate if you make it too small, or don't allow it to operate over a long enough time (it's space-time curvature, not just space curvature).