Measuring curvature with parallel transport

In summary, parallel transport enables changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, by comparisons made at the same location.
  • #1
oldman
633
5
Parallel transport, as one means of quantifying the curvature of a coordinate space, enables
changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. Those changes which are independent of the size of the loop are measurable manifestations of local curvature, and can be coded into components of the Riemann tensor. Have I got this right?

Now spacetime has four dimensions, three of space and one of time. Transporting anything
around a loop takes time, so a one-way leg along the time dimension must in principle be part of any loop. It is therefore never quite possible --- especially in cosmology! --- to compare the original vector with its parallel-transported version at the same location in spacetime, as is possible with a loop on the 2-D Earth's surface (sometimes used to explain how parallel transport measures curvature).

How could the curvature of spacetime on say, a cosmological scale then be measured, even in thought experiments? And is the separation of spacetime curvature into that of space sections and of time thus moot?
 
Physics news on Phys.org
  • #2
oldman said:
How could the curvature of spacetime on say, a cosmological scale then be measured, even in thought experiments?

At any event in spacetime, spacetime curvature can be measured by looking at tidal forces. Take a collection of test particles that is slightly extended in space and measure how the shaped of the collection deforms over a small time interval.
And is the separation of spacetime curvature into that of space sections and of time thus moot?

Why?
 
  • #3
George Jones said:
At any event in spacetime, spacetime curvature can be measured by looking at tidal forces. Take a collection of test particles that is slightly extended in space and measure how the shaped of the collection deforms over a small time interval.

Yes. I agree. As with John Baez's clusters of coffee grounds. But I have cosmological puzzles in mind. For example, if one waited long enough, would the test particles move apart because 'the universe is expanding'? Or would they move together as Peacock described recently? I was hoping to avoid this kind of puzzle by thinking of curvature measured by parallel transport instead. But then I have difficulty with parallel transport, too, as I explained.

As to
why?
I just can't see how this method would work, even in principle, in spacetime and allow one to find components of the Riemann tensor and then talk of space foliations, etc.

But perhaps I'm just getting too muddled.
 

1. What is parallel transport?

Parallel transport is a mathematical concept used to describe the movement of a vector along a curve while keeping it in the same direction. It involves transporting the vector along the curve without rotating or stretching it.

2. How is curvature measured using parallel transport?

Curvature is measured using parallel transport by comparing the differences in the transported vector with the original vector at different points along the curve. The greater the difference, the higher the curvature of the curve.

3. Why is measuring curvature important?

Measuring curvature is important because it allows us to understand the geometric properties of a curve. It is particularly useful in fields such as physics and engineering where curved surfaces are common.

4. What are some real-world applications of measuring curvature with parallel transport?

Some real-world applications of measuring curvature with parallel transport include studying the curvature of space-time in general relativity, analyzing the shape of objects in computer graphics, and predicting the trajectory of objects in motion.

5. Are there any limitations to using parallel transport to measure curvature?

Yes, there are limitations to using parallel transport to measure curvature. It is only applicable to curves in a Euclidean space and may not accurately measure the curvature of non-smooth or discontinuous curves. Additionally, it can be computationally intensive to calculate for complex curves.

Similar threads

  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
2
Views
701
  • Special and General Relativity
Replies
8
Views
981
  • Special and General Relativity
Replies
3
Views
830
  • Special and General Relativity
2
Replies
63
Views
3K
  • Special and General Relativity
Replies
19
Views
2K
  • Special and General Relativity
Replies
26
Views
2K
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
14
Views
2K
  • Special and General Relativity
Replies
2
Views
1K
Back
Top