Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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,

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Covariant derivative of coordinates

**Physics Forums | Science Articles, Homework Help, Discussion**