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

I've been going through Bernard Schutz's A First Course In General Relativity, On Chapter 5 questions atm.

Should the Christoffel Symbols for a coordinate system (say polar) be the same for vectors and one-forms in that coordinate system?

I would have thought yes, but If you calculate the Christoffel Symbols using the Basis vectors of basis one forms then you get different Christoffel symbols?

I am asking because when you do the Covariant Derivative of a vector or a one form are the Christoffel Symbols the same of different?

For Vectors i.e.

[itex]\vec{V} = v^{\alpha}\vec{e}_{\alpha}[/itex]

We can calculate the Christoffel symbols using

[itex]\frac{∂ \vec{e}_{\alpha}}{∂x^{\beta}} = \Gamma^{\mu}_{\alpha \beta} \vec{e}_{\mu}[/itex]

Where the Basis vectors for polar coordinates are

[itex]\vec{e}_{r} = Cos(\theta)\vec{e}_{x} + Sin(\theta)\vec{e}_{y}[/itex]

[itex]\vec{e}_{\theta} = -r Sin(\theta)\vec{e}_{x} + r Cos(\theta)\vec{e}_{y}[/itex]

For One-Forms i.e.

[itex]\tilde{P} = p_{\alpha}\tilde{e}^{\alpha}[/itex]

We can calculate the Christoffel symbols using

[itex]\frac{∂ \tilde{e}^{\alpha}}{∂x^{\beta}} = \Gamma^{\alpha}_{\beta \mu} \tilde{e}^{\mu}[/itex]

Where the Basis vectors for polar coordinates are

[itex]\tilde{e}^{r} = Cos(\theta)\tilde{e}^{x} + Sin(\theta)\tilde{e}^{y}[/itex]

[itex]\tilde{e}^{\theta} = - \frac{Sin(\theta)}{r}\tilde{e}^{x} + \frac{Cos(\theta)}{r}\tilde{e}^{y}[/itex]

**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!

# Christoffel Symbols of Vectors and One-Forms in say Polar Coordinates

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