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Thread: Parallel Transport

  1. #1 Parallel Transport 
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    I think I have got the hang of this concept

    I know it is useful in 4D space-time as a way of showing the intronsic curvature of a surface in it but I would like to ask a question about it on a simple 2D surface such as the surface of our own planet.

    Imagine we want to drive a car around an area of the Earth's surface and measure its intrinsic curvature.Is there a way of constructing the car so as to (grossly perhaps) enable it to be driven back to its starting point in a "parallel transport" fashion?

    At regular intervals I think the driver needs to make sure that the car is pointing in the same direction as at his last measurement but also for it to be parallel with the surface's tangent vector at that point.

    Have I got the hang of it and can one design a car to be driven that crazy crab like way?

    I assume each wheel would need to be able to be driven forwards or sideways and for them also to be raised or lowered.

    Can I parallel transport this "car"?

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  2. #2  
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    I am new to this so I hope someone else will jump in if I've got it wrong ... but I think the answer is 'yes you can'.

    I don't think you need to move the wheels up and down though - the car can remain standing on its four wheels in the usual way.

    Start at the north pole - drive the car sideways to the equator, without rotating left or right. It will start off pointing south (all directions from the north pole are south). It will reach the equator pointing (say) west.

    Then drive it a quarter of the way around the equator, keeping it facing west

    The drive it sideways back up to the north pole, again, keeping its position parallel to its immediately previous position at all times. It will end up at the north pole, inevitably pointing south, but at right angles to its original starting direction.


    I found that replicating this thought experiment in practice by sliding a pencil along such a path around the surface of a globe made everything intuitively clear.
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  3. #3  
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    Quote Originally Posted by lesaid View Post
    I am new to this so I hope someone else will jump in if I've got it wrong ... but I think the answer is 'yes you can'.

    I don't think you need to move the wheels up and down though - the car can remain standing on its four wheels in the usual way.

    Start at the north pole - drive the car sideways to the equator, without rotating left or right. It will start off pointing south (all directions from the north pole are south). It will reach the equator pointing (say) west.

    Then drive it a quarter of the way around the equator, keeping it facing west

    The drive it sideways back up to the north pole, again, keeping its position parallel to its immediately previous position at all times. It will end up at the north pole, inevitably pointing south, but at right angles to its original starting direction.


    I found that replicating this thought experiment in practice by sliding a pencil along such a path around the surface of a globe made everything intuitively clear.
    Yes that works for me .

    But how exactly is the car ,on a continuous basis kept aligned as it makes this journey ? What measurements are made to ensure that it follows this path exactly ? Would the car have to be aware of the direction of the tangent to the surface at all points? (maybe in this case by means of a plumb line pointing to the centre of the Earth)

    I f that were done would this allow the car the negotiate a surface that was not smooth like a sphere but hilly in places?

    Would the hills and valleys tend to cancel out so that the car might actually return at (eg) 89 degrees to its initial direction and so show an aggregate value of intrinsic curvature for the inside area of the path driven?

    Would this bring up the need for raisable wheels?
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  4. #4  
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    To keep the car aligned, I think you could simply ensure the front and back wheels remain parallel to and rotating at the same speed as each other at all times. Then as the car moves, the displacement of front and back of the car to the front and back of where it was a moment earlier, would stay the same, and from the (local) perspective of the driver, the car would always be parallel to where it was a moment earlier. On a perfectly spherical earth, no need for a plumb line - the car would inevitably remain pointing along a tangent to the surface.

    Parallel transport is a way of finding the curvature intrinsic to a surface without having to view it from 'outside' the system. It can find the curvature at a particular point by considering parallel transport around a small closed path on that surface. If that path passes over a variety of arbitrary hills and valleys, we no longer have a consistent curvature, and perhaps need to consider a smaller path to enable the local curvature to be found on a scale where the curvature is consistent. That analysis would then reflect the curvature in the vicinity of the path, incorporating whatever local variation might be present.

    If, by raising and lowering wheels, you managed to keep the car tangential to the earth's surface with constant orientation, ignoring local variations, and navigated your path in terms of spherical coordinates with a constant radius (i.e. ignoring the height variations and perceived distances along the undulating 'two dimensional' surface), I think (intuitively) that you would still end up with a valid result for the 'spherical' earth.
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  5. #5  
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    Quote Originally Posted by lesaid View Post
    To keep the car aligned, I think you could simply ensure the front and back wheels remain parallel to and rotating at the same speed as each other at all times. Then as the car moves, the displacement of front and back of the car to the front and back of where it was a moment earlier, would stay the same, and from the (local) perspective of the driver, the car would always be parallel to where it was a moment earlier. On a perfectly spherical earth, no need for a plumb line - the car would inevitably remain pointing along a tangent to the surface.

    Parallel transport is a way of finding the curvature intrinsic to a surface without having to view it from 'outside' the system. It can find the curvature at a particular point by considering parallel transport around a small closed path on that surface. If that path passes over a variety of arbitrary hills and valleys, we no longer have a consistent curvature, and perhaps need to consider a smaller path to enable the local curvature to be found on a scale where the curvature is consistent. That analysis would then reflect the curvature in the vicinity of the path, incorporating whatever local variation might be present.

    If, by raising and lowering wheels, you managed to keep the car tangential to the earth's surface with constant orientation, ignoring local variations, and navigated your path in terms of spherical coordinates with a constant radius (i.e. ignoring the height variations and perceived distances along the undulating 'two dimensional' surface), I think (intuitively) that you would still end up with a valid result for the 'spherical' earth.
    I see my plumb line defeats the purpose of the exercise as the measurements are no longer intrinsic.

    I think that regularly the car should lower its front/back wheels as required (and when moving sideways both its "side" wheels) to remain tangential to the surface.

    I am right ,am I to think that this car has to proceed sideways at some point in order to make a round trip in a "parallel" fashion?
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  6. #6  
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    yes - in order to go around a small closed path, the car would have to pass through a full rotation (not necessarily 360 degrees on curved surfaces) in direction of travel (though not in orientation!). This is ignoring paths that would get back to the start point by going right around the earth on a great circle rather than by 'turning back' to reach the start point. A trip all the way around a great circle would be a closed path which would not show any curvature.

    I find curvature to be deceptive - interestingly, the surface of a cylinder turns out not to have intrinsic curvature, unlike a sphere.

    Raising and lowering the wheels to remain tangential to the underlying spherical surface of a mountainous terrain is a way of implementing the orientation that the plumb line would give you. You have to decide whether you want to find the general curvature of the region (ignoring local undulations) or the local curvature in the vicinity of a point (including whatever the hills and valleys are doing at that point).

    If you're looking for the more general curvature, then adjusting the wheels and checking ground level with a plumb line might be a way of discarding the 'noise' in the path. Though I'm not quite certain of that.


    In practice, I'm not aware of a circumstance where a physical 'parallel transport' experiment would be done to evaluate curvature. This is really a thought experiment to parallel the mathematical procedure.

    If you want to detect the intrinsic curvature of a physical surface in practice, experimentally, try drawing a triangle - on a curved surface, the internal angles won't add up to 180 degrees. Or draw a circle and measure the diameter. The ratio between them will only be pi if the surface is flat. If you think about the journey I described (north pole to equator, go 1/4 of the way around it and then back up to the north pole) - that is a triangle containing three internal angles of 90 degrees each - adding up to 270 degrees. Were the surface not curved, the angles would add to 180 degrees.

    This link has some diagrams and animations illustrating all of this ... The Geodetic Effect: Measuring the Curvature of Spacetime

    I also like The Feynman Lectures on Physics Vol. II Ch. 42: Curved Space

    Others may have better links
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  7. #7  
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    Quote Originally Posted by geordief View Post
    I see my plumb line defeats the purpose of the exercise as the measurements are no longer intrinsic.

    I think that regularly the car should lower its front/back wheels as required (and when moving sideways both its "side" wheels) to remain tangential to the surface.

    I am right ,am I to think that this car has to proceed sideways at some point in order to make a round trip in a "parallel" fashion?
    The body of the car is perpendicular on meridians and parallel to the Equator. As such, when a car makes a trip from a pole to the Equator and back to the pole, the car will return rotated. In the example given by "lesaid", it returns rotated by 90 degrees. (you can also drive the car alongside a meridian, transverse to the Equator and back it up the other meridian and the result is the same, 90 degree rotation).
    If you do the above exercise along the sides of a closed polygon embedded in a planar surface, you will see that the car returns in the same exact position (zero angle) as the starting position. This is how you can measure intrinsic curvature.
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