Does the metric have to be symmetric? Why?

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SUMMARY

The discussion centers on the necessity of having a symmetric metric tensor, denoted as g_{\mu\nu} = g_{\nu\mu}, in the context of general relativity. It is established that a nonsymmetric metric would not yield any physical consequences and is merely a convention for mathematical consistency. The metric tensor determines the interval between events in four-dimensional spacetime, and any nonsymmetric component could be transformed into a symmetric one. Additionally, the presence of torsion is incompatible with a symmetric metric, which raises questions about observable effects in physical theories.

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  • Understanding of metric tensors in general relativity
  • Familiarity with the concept of spacetime intervals
  • Basic knowledge of coordinate transformations
  • Awareness of torsion in differential geometry
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This discussion is beneficial for physicists, mathematicians, and students of general relativity who seek to deepen their understanding of metric properties and their implications in theoretical physics.

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