- #1

- 649

- 3

I am familiar with the covariant derivative of the tangent vector to a path, [itex]\nabla_{\alpha}u^{\beta}[/itex] and some interesting ways to use it. I am wondering about

[tex]

\nabla_{\alpha}x^{\beta}=\frac{\partial x^\beta}{\partial x^\alpha}+\Gamma^{\beta}_{\alpha\gamma}x^{\gamma}=\delta_{\alpha}^{\beta}+\Gamma^{\beta}_{\alpha \gamma}x^{\gamma}

[/tex]

Then if we let this equal some arbitrary (1,1)-tensor we can manipulate to get

[tex]

\frac{d\tau}{dx^{\alpha}}\frac{dx^{\beta}}{d\tau}+\Gamma^{\beta}_{\alpha\gamma}x^{\gamma}=\Omega^{ \beta}_{\alpha}

[/tex]

which can be rewritten as

[tex]

(\delta_{\alpha}^{\beta} +\Gamma^{\beta}_{\alpha\gamma}x^{\gamma}-\Omega_{ \alpha}^{\beta})u^{\alpha}=J_{ \alpha}^{\beta}u^{\alpha}=0

[/tex]

which looks like a classic homogeneous linear algebra problem (that's the simplification I made in the last equality, just aesthetic). Does this equation have a good physical meaning, or is this just non-sense?

Thanks,