Does the metric have to be symmetric? Why?

[pellman]

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

 

[stevenb]

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.

 

[goedelite]

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.

 

[atyy]

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.

 

[pellman]

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.