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I am trying to understand what the differences would be in replacing the symmetry equation:

g_mn = g_nm

with the Hermitian version:

g_mn = (g_nm)*

In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about moving to a fully complex space. We still have the 4 real values of ct, x, y, and z. I have searched the interweb for help, but cannot find anything talking about this particular issue. I only know differential geometry (basically, calculus), so the articles using groups etc. a far over my head, and most articles posted that I found just move to the more complicated "complex space" which I am trying to avoid.

What I really want to know is how the Christoffel Symbols and the Ricci Tensor would differ for this kind of metric. Any help at all would be greatly appreciated!

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# Hermitian Metric - Calculating Christoffel Symbols

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