Parallel transport in flat polar coordinates

In summary: The geodesics of a curved surface are, again, straight lines, but they may be curved in various directions. Thanks for the clarification!In summary, the geodesics of a surface depend only on the metric, not the coordinate system.
  • #1
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If we have as a manifold euclidian R^2 but expressed in polar coordinates...
Do any circle centered at the origin constitute a geodesic?
Because I think it parallel transport its own tangent vector.
 
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  • #2
The geodesics of the manifold depend only on the metric, not the coordinate system. The geodesics are still straight lines; arcs of a circle are not geodesics in the standard Euclidean metric.
 
  • #3
Thanks for the reply.
But I still don't quite get it: the tangent vectors of the coordinate lines in polar coordinates do rotate about the origin.
Maybe the covariant derivative compensate for this rotation, but I can't figure out how to write the geodesic equation of R^2 in polar coordinates, I know it must give me a system of 2 differential equations but don't know how to get them.
Say, if I have point [tex]p=(r,\phi) = (0,0)[/tex] and vector [tex]v=(r,\phi) = (1, \pi/4)[/tex], how do I calculate it's geodesic (I know in cartesian coordinates it has to be the y=x line).
 
  • #4
I have calculated the christoffel symbols for polar coordinates and it gives me [tex]\Gamma^{\phi}_{r \phi} = \Gamma^{\phi}_{\phi r} = \frac{2}{r}[/tex], and all the other [tex]\Gamma = 0[/tex].

Now how can I calculate a geodesic starting at [tex](0,0)[/tex] and with initial tangent vector [tex](1, \pi/4)[/tex] ?
 
  • #5
Sorry there was a mistake in my calculation of the christoffel symbols.

They are [tex]\Gamma^{\phi}_{r \phi} = \Gamma^{\phi}_{\phi r} = \frac{1}{r} [/tex] and [tex]\Gamma^{r}_{\phi\phi} = -r [/tex]

The others [tex]\Gamma = 0[/tex].
 
  • #6
What differential equation must geodesics satisfy then?
 
  • #7
I calculated the 16 components of the riemann curvature tensor for flat space in polar coordinates and they all gave me 0 :smile:
It seems to work!

As to the geodesics differential equation for polar coordinates, I found the answer on MTW.
But I'm not very good at solving diffs eqs.
 
  • #8
In any case, the whole point is that geodesics are "intrinsic" to a surface- they depend on the surface, not what coordinate system you have. The geodesics of a flat plane are straight lines.
 

1. What is parallel transport in flat polar coordinates?

Parallel transport in flat polar coordinates refers to the movement of a vector along a curve in a polar coordinate system without changing its direction. It is a mathematical concept used in differential geometry to understand how vectors change as they move along a curved surface.

2. How is parallel transport different from regular transport?

Parallel transport is different from regular transport because it takes into account the curvature of the surface on which the vector is moving. In regular transport, the vector's direction may change as it moves along a curved surface, whereas in parallel transport, the vector's direction remains constant.

3. What is the significance of parallel transport in physics?

In physics, parallel transport is used to understand how quantities such as velocity, momentum, and force change as they move along a curved path. It is particularly important in understanding the behavior of objects in a gravitational field, as it helps to determine how their properties change as they move through space.

4. How is parallel transport calculated in flat polar coordinates?

To calculate parallel transport in flat polar coordinates, one must take into account the Christoffel symbols, which represent the curvature of the surface. These symbols are used in a mathematical formula to determine the change in the vector's direction as it moves along a given curve.

5. What are some real-world applications of parallel transport in flat polar coordinates?

Parallel transport in flat polar coordinates has applications in various fields, including physics, engineering, and computer graphics. It is used in general relativity to understand the motion of objects in a gravitational field, in computer graphics to simulate the movement of objects, and in engineering to design structures that can withstand external forces without changing their shape.

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