Why is it that in general geodesics are paths of stationary character

In summary, the speaker discusses their experience with the relativity forum and their decision to ask for help in a different forum. They then ask two questions about general geodesics and a mathematical equation. The expert summarizer explains that geodesics can be either a minimum or maximum, and clarifies the process of differentiating in the given equation. The speaker also mentions that the forum members were not mean, but simply gave them the silent treatment.
  • #1
Thrice
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Well since the denizens of the relativity forum don't like me, I thought I might ask here see if I get better replies.1) Why is it that in general geodesics are paths of "stationary character" rather than minimum?

2) http://img366.imageshack.us/img366/3280/math30016nx.jpg

I can't quite follow equation 11.16. Specifically how they differentiate dxm/ds in the denominator.
Shouldn't be necessary, but for reference, the following page is http://img353.imageshack.us/img353/6488/math30024mk.jpg" .
 
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  • #2
Thrice said:
Well since the denizens of the relativity forum don't like me, I though I might ask here see if I get better replies.
They have been mean?

1) Why is it that in general geodesics are paths of "stationary character" rather than minimum?
They impose that it's an extremum (functional derivative is zero) so it could be either a min or a max.

2) http://img366.imageshack.us/img366/3280/math30016nx.jpg

I can't quite follow equation 11.16. Specifically how they differentiate dxm/ds in the denominator.
Shouldn't be necessary, but for reference, the following page is http://img353.imageshack.us/img353/6488/math30024mk.jpg" .

They do not differentiate dxm/ds..they differentiate with respect to dxm/ds. You must treat the *entire* combination dxm/ds as your variable and differentiate with respect to it (So, calling the variable x, L is essentially [itex] {\sqrt{ g_{kn} x^k x^n}}[/itex]).

Pat
 
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  • #3
No they weren't mean. They just gave me the silent treatment.

Thanks. It's a lot clearer now.
 

1. Why are geodesics considered paths of stationary character?

Geodesics are paths that follow the shortest distance between two points on a curved surface. This means that any deviation from the geodesic path would result in a longer distance. Therefore, geodesics are paths of stationary character because they represent the path of minimum length.

2. What is the significance of geodesics in physics and mathematics?

Geodesics are important in physics and mathematics because they represent the path of least energy or action. In physics, geodesics are used to describe the path of a particle in the presence of gravity or other forces. In mathematics, geodesics are used to define the shortest distance between two points on a curved surface, which is crucial in fields such as geometry and differential equations.

3. How are geodesics related to the concept of curvature?

Geodesics are closely related to the concept of curvature. In fact, the curvature of a surface at a given point can be defined as the rate of change of the geodesic with respect to distance. This means that the curvature of a surface determines the shape of the geodesic paths on that surface.

4. Are geodesics always straight lines?

No, geodesics are not always straight lines. On a curved surface, the shortest distance between two points is often a curved path. If the surface is flat, then the geodesic will be a straight line. However, on a curved surface, geodesics can exhibit complex and non-linear behavior.

5. How do geodesics play a role in general relativity?

In general relativity, geodesics are used to describe the path of a particle in the presence of a gravitational field. In this theory, gravity is seen as the curvature of spacetime, and geodesics represent the path of least resistance for a particle moving in this curved spacetime. This allows for the prediction of the motion of objects under the influence of gravity.

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