Metric tensor derived from a geodesic

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Discussion Overview

The discussion revolves around the derivation of a metric tensor from geodesic equations within the context of a 2D manifold. Participants explore the relationship between geodesics, connections, and metrics, as well as the complexities involved in solving partial differential equations (PDEs) to restore the metric.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether it is sufficient to derive a metric solely from the geodesic equation or if additional definitions are necessary for the manifold.
  • It is noted that geodesic curves are generated by a connection, and a connection can exist without a corresponding metric, complicating the restoration of a metric compatible with a symmetric connection.
  • Participants discuss the complexity of solving PDEs to restore the metric from a known connection, with some indicating that these PDEs may not have suitable or unique solutions.
  • There is a query about using a curve equation not related to the affine parameter to find connection coefficients and whether this would yield a correct metric in the given coordinate system.
  • Some participants assert that the geodesic equation does not account for the anti-symmetric part of the connection coefficients, suggesting that torsion must be specified to find connection coefficients.
  • A question is raised regarding whether a chosen family of curves can provide information about torsion and if a second derivative is necessary for torsion to exist.
  • Participants express uncertainty about the existence of definitive answers to the posed questions, indicating ongoing exploration of the topic.
  • Links to external resources are shared for further reading on the topic of recovering a metric from geodesics.
  • One participant inquires if defining geodesics over a subset of a manifold allows for the assumption of no torsion, leading to the possibility of solving the geodesic equation to obtain a metric for that subset.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the relationship between geodesics, connections, and metrics. The discussion remains unresolved, with no consensus on the sufficiency of deriving a metric from geodesic equations alone or the implications of torsion.

Contextual Notes

Limitations include the dependence on specific definitions of connections and metrics, the unresolved nature of the PDEs involved, and the implications of torsion on the geodesic equations.

VladZH
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Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
 
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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.
 
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.
Thanks for reply! Could you tell me what is the complexity? As I see I need to solve partial differential equations. That's it?
 
Yes, you must solve the PDE to restore metric by known connection; these PDE are not obliged to have a suitable solution as well as they are not obliged to have a unique solution
 
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?
 
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?
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).
 
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).
Can my chosen curve/curves family give me information about torsion? Should it have second derivative to have a torsion?
 
Is there really no answer?
 
  • #10

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