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

In summary, the Christoffel symbols are the same, but the equation for incorporating them into the covariant derivative changes sign depending on whether you are taking the covariant derivative of a vector or a one-form.
  • #1
Skhaaan
6
0
Hello all,

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]
 
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  • #2
Skhaaan said:
Hello all,

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?

One minor appearance tip. To make the Christoffel symbols vertical alignment proper, the latex looks llike this: \Gamma^{\alpha}{}_{\beta\mu}, i.e. [itex]\Gamma^{\alpha}{}_{\beta\mu}[/itex]. The extra empty {} at the end does the trick.

The Christoffel symbols are the same, but the equation for incorporating them into the covariant derivative changes sign depending on whether you are taking the covariant derivative of a vector or a one-form.

I'm having a bit of trouble following your notation, the way I"m used to writing this is:

eq1 [tex]\nabla_a t^b = \partial_a t^b + \Gamma^{b}{}_{ac} t^c[/tex]
eq2 [tex]\nabla_a \omega_b = \partial_a \omega_b - \Gamma^{c}{}_{ab}\omega_c[/tex]

where eq1 represents taking the covariant derivative of a vector [itex]t^b[/itex], and eq2 represents takig the covariant derivative of a one form [itex]\omega_b[/itex].

In the above notation [itex]\partial_a[/itex] represents an ordinary derivative, i.e [tex]\frac{\partial}{\partial a}[/tex]

[itex]\nabla_a[/itex] represents the corresponding covariant derivative. Thus eq1 and eq2 tell you how to construct the covariant derivative out of the ordinary derivative in some coordinate basis and the Christoffel symbols.
 
Last edited:
  • #3
It looks to me like what the OP is calculating are the Ricci rotation coefficients. They are antisymmetric in the two lower indices rather than symmetric.
 
  • #4
Hey Guys, thanks a lot for replying

Thanks for the tip with arranging the indices using latex.

After some thought I think I have figured out what I was asking

Cheers guys
 
  • #5
Shouldn't these equations in the OP read

[tex]
\nabla_{\mu} e_{(\nu)} \equiv \Gamma_{\mu\nu}^{\rho} e_{(\rho)}
[/tex]
stating that the covariant derivative on basis vectors e are linear sums of basis vectors? See e.g. Nakahara.
 

1. What are Christoffel symbols in polar coordinates?

Christoffel symbols are a set of coefficients used in differential geometry to analyze curves and surfaces. In polar coordinates, they represent the change in direction of a vector or one-form as it moves along a curved surface.

2. How are Christoffel symbols calculated in polar coordinates?

In polar coordinates, the Christoffel symbols can be calculated using the metric tensor, which describes the distance between points on a curved surface. The formula for calculating Christoffel symbols involves taking derivatives of the metric tensor and then performing some algebraic manipulations.

3. What is the significance of Christoffel symbols in polar coordinates?

Christoffel symbols play a crucial role in differential geometry as they allow us to understand the intrinsic properties of a curved surface. In polar coordinates, they help us to calculate the curvature and geodesic equations of a surface, which are important for various applications in physics and engineering.

4. How do Christoffel symbols relate to the Christoffel symbols in other coordinate systems?

Christoffel symbols are coordinate-independent, meaning they have the same values regardless of the chosen coordinate system. However, the equations for calculating them may vary depending on the coordinate system being used. In polar coordinates, the Christoffel symbols are expressed in terms of the metric tensor and its derivatives.

5. Can Christoffel symbols be zero in polar coordinates?

Yes, it is possible for some Christoffel symbols to be zero in polar coordinates. This occurs when the metric tensor is diagonal, meaning the surface is flat in that direction. However, in curved surfaces, most Christoffel symbols will not be zero, indicating the non-Euclidean nature of the surface.

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