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Physics
Special and General Relativity
Solving Vanishing Tensor Eqn & Raising All Indices
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[QUOTE="George Keeling, post: 6396819, member: 647945"] [B]TL;DR Summary:[/B] Problem with vanishing tensor equation with all indices down. Does it still vanish when they are up? I have an equation $$ \chi_\nu\nabla_\mu\chi_\sigma+\chi_\sigma\nabla_\nu\chi_\mu+\chi_\mu\nabla_\sigma\chi_\nu=0 $$so we also have$$ g_{\nu\rho}g_{\mu\tau}g_{\sigma\lambda}\left(\chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho\right)=0 $$Does that mean that$$ \chi^\rho\nabla^\tau\chi^\lambda+\chi^\lambda\nabla^\rho\chi^\tau+\chi^\tau\nabla^\lambda\chi^\rho=0 $$as well? I can prove in two dimensions that $$ x_i=0\Rightarrow g_{ij}x^j=0\Rightarrow x^j=0 $$as long as the metric is not degenerate. It would be horrendous to extend the proof to four dimensions and three indices. I think there is some more intuitive way to get from the second equation to the third, but the intuition eludes me. [/QUOTE]
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Special and General Relativity
Solving Vanishing Tensor Eqn & Raising All Indices
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