- #1

George Keeling

Gold Member

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- TL;DR Summary
- 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.

\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.