Physical intuition behind geodesics and parallel transport

In summary: In a curved spacetime, there is no obvious notion of "parallel" but it is defined by the derivative and acceleration.4) The connection in GR is chosen to be the Levi-Civita connection for the geodesic to be equivalent to demanding that the unit tangent vector be parallel transported.5) A geodesic is a curve of zero acceleration in a coordinate-independent way.In summary, parallel transport is a way of defining a notion of "parallel" in a curved spacetime, and it is equivalent to demanding that a geodesic curve have zero acceleration or that a gyroscope in a box maintain its orientation when transported along a path. This concept is important in understanding the curvature of spacetime.
  • #1
quasar_4
290
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Hi all,

Sorry if this is a dumb question, but what exactly do we mean by the term parallel transport? Is it just the physicist's way of saying isometry?

Also, in my class we have just defined geodesics, and we're told that having a geodesic curve cis equivalent to demanding that the unit tangent vector be parallel transported along c. This confuses me. I can read the proof of this, and it's fine, but I don't feel any physical intuition developing here. Can someone explain, as physically as possible, why these two conditions are equivalent?

Thanks.
 
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  • #2
I don't think it is related to isometry.

In a curved spacetime, there is no obvious notion of "parallel". But you need one, so you just define it, by defining the notion of a derivative, a rate of change, an acceleration. No acceleration = no change in velocity = no change in tangent vector = parallelly transported (by definition, but hopefully it will seem like a graceful generalization of the terms we use for flat spacetime). There are many possible dervatives, and hence many possible notions of parallel. In GR, the derivative is chosen by specifying that the connection be the Levi-Civita connection http://en.wikipedia.org/wiki/Connection_(mathematics)
 
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  • #3
quasar_4 said:
Sorry if this is a dumb question, but what exactly do we mean by the term parallel transport?

Put a gyroscope in a box. Transport it along some path. Observe the orientation of the gyroscope at the end.
 
  • #4
  • #5
1) Geodesic is free fall.
2) Parallel transport reveals curvature.
 

1. What is a geodesic?

A geodesic is the shortest path between two points on a curved surface. In other words, it is the path that minimizes the distance traveled between two points, taking into account the curvature of the surface.

2. How is the physical intuition behind geodesics related to gravity?

The concept of geodesics is closely related to the theory of gravity. In Einstein's theory of general relativity, massive objects cause the fabric of space-time to curve, and the path of least resistance for objects moving through this curved space-time is a geodesic.

3. What is parallel transport and how is it related to geodesics?

Parallel transport is the process of moving an object along a path without changing its orientation. In the context of geodesics, parallel transport is used to define the path along which a vector can be transported without changing its direction. This path is known as the geodesic.

4. How does parallel transport differ from regular transport?

In regular transport, an object is moved from one point to another without any consideration for the curvature of the space it is moving through. However, in parallel transport, the object is moved along a path that takes into account the curvature of the space, resulting in a minimal change in the object's direction.

5. Why is understanding geodesics and parallel transport important in physics?

Geodesics and parallel transport are important concepts in physics as they help us understand how objects move in the presence of curved space-time. They are essential for understanding the behavior of massive objects in the universe, as well as for developing theories such as general relativity.

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