Does the metric have to be symmetric? Why?

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Discussion Overview

The discussion centers on the symmetry of the metric tensor in the context of spacetime geometry, exploring the implications of having a nonsymmetric metric. Participants examine whether the condition g_{\mu\nu}=g_{\nu\mu} is necessary and what physical consequences might arise if this condition were not met.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants question the necessity of the symmetry condition g_{\mu\nu}=g_{\nu\mu}, suggesting that any nonsymmetric part of the metric might not have physical implications and could be seen as a convention.
  • One participant argues that the metric determines the interval between events in 4-space and is a quadratic form, implying that a nonsymmetric metric could always be replaced by a symmetric one.
  • Another point raised is the desire for a coordinate system where the metric is locally diagonal, indicating that the metric in arbitrary coordinates should be derivable from a symmetric form through a change of basis.
  • A later reply introduces the concept of torsion, questioning whether a symmetric metric allows for torsion and whether torsion would have observable effects, indicating a lack of familiarity with these concepts.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of metric symmetry, with multiple competing views presented regarding its implications and the role of torsion.

Contextual Notes

Limitations include potential assumptions about the nature of the metric and its implications for physical theories, as well as unresolved questions regarding the relationship between metric symmetry and torsion.

pellman
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Why must we have [tex]g_{\mu\nu}=g_{\nu\mu}[/tex]? What are the physical consequences if this did not hold?
 
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pellman said:
Why must we have [tex]g_{\mu\nu}=g_{\nu\mu}[/tex]? What are the physical consequences if this did not hold?

Perhaps someone knows of some subtle consequences, but on the surface it appears that any nonsymmeyric part of the metric would not do anything physical. Hence, it appears to be a convention to make it symmetric.
 
Because the metric determines the interval between events (points in 4-space), and it is a quadratic forum. If it were not symmetric, it could always be replaced by a metric that is symmetric.
 
We'd like a coordinate system in which the metric is locally diag(-1,1,1,1). The metric in an arbitrary coordinate system should be something that you can obtain from that by a change of basis.
 
Thanks, all.

There cannot be torsion with a symmetric metric, can there? And wouldn't torsion have an observable effect?

This isn't a rhetorical question. I am not very familiar with these concepts.
 

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