Hello fellow mathematicians!
I was wondering why most metrics I have seen in Relativity are symmetric, also a property I have started to notice in my differential geometry book.
I guess the definition through the "dot-product" may be the reason why...
$$ g_{\alpha \beta}=\vec{x}_\alpha\cdot \vec{x}_\beta $$
But:
Are there non-symmetric metrics?
What can be said about them?
Also, how do we call those metrics which only have non-zero components in the main diagonal?
And perhaps the question that has been bothering me a lot: What can be said about a metric having non-zero components out off the main diagonal?
I have been having thoughts on physical implications but since I'm not that far into Relativity nor Diff. Geometry, I can't really come up with something yet.
My main book has been Kreysizg's Differential Geometry, which may not be suited for some questions I have been thinking off, yet it has been an awesome book to start with! Any further recommendations?
You usually want a metric to "measure" distance in some way. One property we want to be preserved is that the distance from A to B is the same as the distance from B to A. This property is exactly what symmetry in the form means. So if a metric is not symmetric then it doesn't really do anything useful.