Does the metric have to be symmetric? Why?

In summary, the reason for the equality of g_{\mu\nu} and g_{\nu\mu} is that any nonsymmetric part of the metric would not have any physical consequences. It appears to be a convention to make it symmetric because the metric determines the interval between events in 4-space and can always be replaced by a symmetric metric. The goal is to have a coordinate system in which the metric is locally diag(-1,1,1,1) and any arbitrary metric can be obtained from it by a change of basis. This also eliminates the possibility of torsion, which could potentially have observable effects.
  • #1
pellman
684
5
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 

1. What is a symmetric metric?

A symmetric metric is a measurement or scale that is balanced or equal on both sides. In other words, if you were to split a metric in half, the two halves would be identical.

2. Why is symmetry important in a metric?

Symmetry is important in a metric because it ensures that both sides of a measurement or scale are equally represented. This allows for fair and accurate comparisons and evaluations.

3. What happens if a metric is not symmetric?

If a metric is not symmetric, it can lead to biased or skewed results. This is because one side of the measurement or scale may be given more weight or importance over the other, resulting in an inaccurate representation of the data.

4. How can we determine if a metric is symmetric?

A metric can be determined as symmetric by visually examining the data or by conducting statistical tests such as the mean difference test or the t-test. These tests look for significant differences between the two halves of the metric.

5. Can a metric be considered valid if it is not symmetric?

It is generally preferred for a metric to be symmetric, as it ensures fair and accurate comparisons. However, depending on the context and purpose of the measurement, a metric may still be considered valid even if it is not symmetric. It is important to carefully consider the potential biases and limitations of a non-symmetric metric before using it for evaluation or decision-making.

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