Local Inertial Frame: Understanding Riemann Curvature & Metric Tensor

In summary, the conversation discusses the concept of local inertial frames of reference in arbitrary space-time regions. It is mentioned that experiments in these frames are the same as in special relativity. However, if a device is used to measure the Riemann curvature tensor values, it would show non-zero values. It is suggested that such a device may not be possible to build, and the concept of the equivalence principle in general relativity is also discussed.
  • #1
Neitrino
137
0
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 ?
 
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  • #3
DaleSpam said:
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
 
  • #4
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.
 
  • #5
DaleSpam said:
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.
 
  • #6
Neitrino said:
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.
 
  • #7
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.
 
  • #8
Rasalhague said:
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.
 
  • #9
Rasalhague said:
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.

JesseM said:
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).
 

1. What is a local inertial frame?

A local inertial frame is a reference frame in which the laws of physics can be described by the special theory of relativity. In this frame, the effects of gravity can be neglected and objects move in straight lines at constant speeds. It is also known as a freely falling frame, as an observer in this frame would feel weightless.

2. How is the Riemann curvature tensor related to a local inertial frame?

The Riemann curvature tensor is a mathematical object used to describe the curvature of spacetime. In a local inertial frame, the Riemann curvature tensor is zero, meaning that there is no curvature in that specific region of spacetime. This is because in a local inertial frame, the effects of gravity are negligible and spacetime is considered flat.

3. What is the role of the metric tensor in understanding local inertial frames?

The metric tensor is a mathematical object that describes the distance between two points in spacetime. In a local inertial frame, the metric tensor is constant and has a diagonal form, indicating that spacetime is flat. It is used to calculate the interval between two events, which is an important concept in understanding the special theory of relativity.

4. How can understanding local inertial frames help in understanding general relativity?

Local inertial frames play a crucial role in understanding general relativity. In this theory, gravity is described as the curvature of spacetime caused by the presence of mass and energy. By understanding local inertial frames, we can understand how gravity affects the motion of objects in spacetime and how it is related to the curvature of spacetime.

5. How does the concept of local inertial frames help in practical applications?

Understanding local inertial frames is important in practical applications such as space travel and satellite navigation. In these scenarios, the effects of gravity must be taken into account in order to accurately predict the motion of objects. By using the concept of local inertial frames, scientists and engineers can make precise calculations and ensure the success of these applications.

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