Thanks for reply! Could you tell me what is the complexity? As I see I need to solve partial differential equations. That's it?zwierz said:Geodesic curves are generated by connection; you can have a connection but do not have metric. Restoring metric compatible with symmetric connection is not a simple problem.
In general, no. The geodesic equation does not depend on the anti-symmetric part of the connection coefficients. You can only find the connection coefficients if you also specify the anti-symmetric part (i.e., you need to specify the torsion).VladZH said:There is a question, though. Let we have a curve equation not w.r.t. the affine parameter but a 2D chart. Can I use the geodesic equation with this curve equation to find connection coefficients? If so, will solving further PDE give me a correct metric in this coordinate system?
Can my chosen curve/curves family give me information about torsion? Should it have second derivative to have a torsion?Orodruin said:In general, no. The geodesic equation does not depend on the anti-symmetric part of the connection coefficients. You can only find the connection coefficients if you also specify the anti-symmetric part (i.e., you need to specify the torsion).
Thanks for reply. I have seen this question. For now my question is: if we define geodesics over a subset of a manifold, can we just say that there is no torsion, put geodesic curves into geodesic equation, solve PDE and get a metric or bunch of metrics for the subset?robphy said:possibly interesting reading:
https://mathoverflow.net/questions/132244/can-one-recover-a-metric-from-geodesics
A metric tensor derived from a geodesic is a mathematical object that describes the intrinsic curvature of a space. It is derived from the geodesic equation, which is a fundamental equation in the study of curved spaces.
In physics, a metric tensor derived from a geodesic is used to describe the geometry of spacetime in general relativity. It is also used in other areas of physics, such as in the study of fluid dynamics and quantum field theory.
A metric tensor is a mathematical object that describes the distance between two points in a space. A metric tensor derived from a geodesic is a specific type of metric tensor that is derived from the geodesic equation. This means that it takes into account the curvature of the space, rather than just the distance between points.
A metric tensor derived from a geodesic is calculated using the geodesic equation, which takes into account the curvature of the space. It involves solving a set of differential equations, which can be quite complex depending on the specific space being studied.
A metric tensor derived from a geodesic has many real-world applications, including in GPS navigation systems, which use general relativity to accurately measure distances. It is also used in the study of cosmology, where it helps to describe the large-scale structure of the universe.