Geodesic deviation equation

[ledol83]

Hi...does anyone have a good description (or a link to it) on geodesic deviation equation?Most of the references i have are in a setting of relativity, which make me all at sea.

Please help me if you know a mathematical characterization of how geodesics from one point deviate (which just involves the Riemannian curvature tensor).

Thanks a bunch!

 

[Chris Hillman]

That's the Jacobi geodesic deviation formula, which is discussed in many textbooks on Riemannian geometry (the Lorentzian version is almost identical).

 

[tacman]

The geodesic deviation equation is a fundamental equation in Riemannian geometry that describes the behavior of nearby geodesics. It is also known as the Jacobi equation, after the mathematician Carl Gustav Jacob Jacobi who first derived it. The equation is used to study the curvature of a space and its effect on the paths of objects moving through it.

In simple terms, the geodesic deviation equation describes how the distance between two nearby geodesics changes as they travel through a curved space. Geodesics are the shortest paths between two points in a curved space, and the deviation equation tells us how these paths may diverge or converge due to the curvature of the space.

The equation involves the Riemannian curvature tensor, which is a mathematical object that characterizes the curvature of a space. It is a tensor field that describes how the curvature of a space varies from point to point. In the geodesic deviation equation, the Riemannian curvature tensor is used to calculate the acceleration of one geodesic relative to another.

One way to think about the geodesic deviation equation is to imagine two ants walking on the surface of a sphere. As they walk, their paths will eventually diverge due to the curvature of the sphere. The geodesic deviation equation can be used to calculate the rate at which their paths are diverging and how this rate changes depending on the curvature of the sphere.

If you are interested in learning more about the geodesic deviation equation, I recommend checking out some resources on Riemannian geometry or differential geometry. These subjects can be quite complex, so it may be helpful to have a strong understanding of calculus and linear algebra before diving into this topic. Good luck!