- #1

Mike Karr

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- TL;DR Summary
- It seems that the stress-energy tensor is both symmetric and non-symmetric.

I am a beginner in GR, working my through Gravitation by the above authors. If there is a better place to ask this question, please let me know.

I understand (from section 5.7) that the stress-energy tensor is symmetric, and from equation 5.23 (p. 141), it is explicitly symmetric. But evaluating equation 5.22 in a Lorentz coordinate frame, the last term is clearly symmetric, but the first term involves ##F^{\nu}_{\alpha}##, which is non-symmetric, at least according to equation 5.3 (p. 73). In particular, it is symmetric in the 0th column and row and anti-symmetric in the rest of the matrix. Multiplied by the anti-symmetric ##F^{\mu\alpha}## does not help the situation. So I can't get from 5.22 to 5.23.

I suppose my problem is understanding "in a Lorentz frame", but I thought all definitions of F and ##\nu## *are* in a Lorentz frame. Where am I going wrong?

I understand (from section 5.7) that the stress-energy tensor is symmetric, and from equation 5.23 (p. 141), it is explicitly symmetric. But evaluating equation 5.22 in a Lorentz coordinate frame, the last term is clearly symmetric, but the first term involves ##F^{\nu}_{\alpha}##, which is non-symmetric, at least according to equation 5.3 (p. 73). In particular, it is symmetric in the 0th column and row and anti-symmetric in the rest of the matrix. Multiplied by the anti-symmetric ##F^{\mu\alpha}## does not help the situation. So I can't get from 5.22 to 5.23.

I suppose my problem is understanding "in a Lorentz frame", but I thought all definitions of F and ##\nu## *are* in a Lorentz frame. Where am I going wrong?

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