Can Newton's Theory Apply in a Small Volume?

[kent davidge]

Consider a place where General Relativity is the appropriate theory, but suppose we consider just a tiny volume, so to speak, of space-time.. which is to say we are observing a phenomenum for a very short amount of time and the "laboratory" is also very small in space. Under these circumstances can we shift from Relativity to Newton's theory?

I know such a situation would probably have no pratical use, because we are usually interested in, say, motion of a particle, and for this end we need more than an infinitesimal amount of space and time to study the particle, so this question is more like a theoretical question.

 

[jbriggs444]

Consider a place where General Relativity is the appropriate theory, but suppose we consider just a tiny volume, so to speak, of space-time.. which is to say we are observing a phenomenum for a very short amount of time and the "laboratory" is also very small in space. Under these circumstances can we shift from Relativity to Newton's theory?

For a region of sufficiently small extent in space and time, the rules of general relativity simplify to the rules of special relativity. As long as the objects in the volume have low speeds relative to one another, the rules of special relativity further simplify to the rules of Newtonian mechanics.

But if the objects in the volume have high speed relative to one another, their impact energies will not conform to the Newtonian predictions. Merely restricting attention to a small region is not enough to simplify GR all the way down to Newton.

 

[Paul Colby]

For a region of sufficiently small extent in space and time, the rules of general relativity simplify to the rules of special relativity.

Yet, the curvature is in general non vanishing and independent of how small the length and time scale of the region of interest.

 

[PeterDonis]

the curvature is in general non vanishing and independent of how small the length and time scale of the region of interest

But the curvature will be undetectable over a small enough extent of space and time, because any measurement will only have finite accuracy. This is all well-traveled ground, let's not belabor it.

 

[sweet springs]

So I would like to summarize the condition taught ;
1. small spacetime region around the observer
2. v/c<<1 for all the bodies within
moreover if we eliminate Newton's gravity law also
3. GMm/ (space size) is negligible than other energy of concern where M,m is all the combination of body mass in the region

 

[stevendaryl]

Yet, the curvature is in general non vanishing and independent of how small the length and time scale of the region of interest.

Yes, but a lot (all?) of the empirical ways to measure curvature get less and less accurate as the size of the region gets smaller. For example, in a 2D Riemannian manifold, a way that you can detect curvature is by creating a triangle whose sides are all geodesics. In a flat plane, the angles will add up to 180. But even if the manifold is curved, small triangles will have the property that their angles will approximately add up to 180. So if your ability to measure angles has a finite precision---say 0.1^o--then you will be unable to demonstrate curvature for sufficiently small regions via this method.

The same is true of geodesic deviation, another way of measuring curvature. You draw two line segments (geodesics) on the surface that are initially parallel. Then you continue them a certain length and measure the departure from being parallel at the end of the segments. If the geodesics are sufficiently short, then the departure will be very small.

Of course curvature is a derivative-like object: It doesn't matter, theoretically, how small the deviations from flatness are, because you compute ratios by dividing the deviation by another small quantity related to the size of the region. However, dividing two very tiny imprecise numbers produces a ratio that has large uncertainties. So in a small enough region, the uncertainty in the curvature will tend to be large enough that the empirical evidence will be consistent with zero curvature.

I've asked experts before whether there can be measurable curvature effects that don't vanish in small regions. The idea I had, which I'm not absolutely certain about, is that the Lense-Thirring effect might effect the trajectory of a spinning object in a way that is detectable in very small regions. I don't know the mathematical details, though.

 

[Paul Colby]

Yes, but a lot (all?) of the empirical ways to measure curvature get less and less accurate as the size of the region gets smaller.

I think this is a good question. Sweeping statements about what is measurable because of experimental error don't always hold up so well. I would site LIGO as a case in point. Took'm 40 years and like a third of of the physicist population but they did it and it's an amazing achievement.

Having said that, I would make a less sweeping statement about curvature measurements made using space and time interval measurements. Measuring curvature using the logical equivalent of clocks and meter sticks is, as you've said, like trying to form a (second order?) derivative from experimental data. This can be done in principle and it has a connection with the size of the measurement region relative to the size of the expected curvature. As mention, the size of the curvature is fixed by the situation of interest while the measurement volume up to the experimenter, their budget and, background estimates. One is virtually guaranteed that shrinking the measurement size will put the desired answer in the noise.

I've asked experts before whether there can be measurable curvature effects that don't vanish in small regions.

I think this is a good and interesting question. The general answer sticking to the bounds of this thread are tidal force effects. For myself the question is then how accurately one might measure these?

 

[Paul Colby]

The idea I had, which I'm not absolutely certain about, is that the Lense-Thirring effect might effect the trajectory of a spinning object in a way that is detectable in very small regions.

This sent me to the Lense-Thirring wiki. Lense–Thirring_precession which looks like a good read. Doesn't Gravity Probe-B qualify? Didn't see it mentioned.

 

[PeterDonis]

Doesn't Gravity Probe-B qualify?

As a test of the Lense-Thirring effect? Yes, it does.