- #1
faklif
- 18
- 0
Homework Statement
The problem concerns how to transform a covariant differentiation. Using this formula for covariant differentiation and demanding that it is a (1,1) tensor:
[tex]
\nabla_cX^a=\partial_cX^a+\Gamma^a_{bc}X^b
[/tex]
it should be proven that
[tex]
\Gamma'^a_{bc}=
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^e}{{\partial}x'^b}
\frac{{\partial}x^f}{{\partial}x'^c}
\Gamma^d_{ef}
-
\frac{{\partial}x^d}{{\partial}x'^b}
\frac{{\partial}x^e}{{\partial}x'^c}
\frac{{\partial}^2x'^a}{{{\partial}x^d}{{\partial}x^e}}
[/tex]
Homework Equations
[tex]
\partial'_cX'^a=
\frac{{\partial}x^f}{{\partial}x'^c}\frac{\partial}{{\partial}x^f}
(
\frac{{\partial}x'^a}{{\partial}x^d}X^d
)=
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\frac{\partial}{{\partial}x^f}
X^d
+
\frac{{\partial}^2x'^a}{{\partial}x^f{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
X^d
[/tex]
[tex]
X^d
=
\frac{{\partial}x^d}{{\partial}x'^b}
\frac{{\partial}x'^b}{{\partial}x^d}
X^d
=
\frac{{\partial}x^d}{{\partial}x'^b}
X'^b
[/tex]
[tex]
X^e
=
\frac{{\partial}x^e}{{\partial}x'^b}
X'^b
[/tex]
The Attempt at a Solution
I thought I'd simply look at the primed coordinates from two directions and that that should do it... didn't quite happen as planned though.
In primed coordinates
[tex]
\nabla_cX^a'
=
{\partial}'_cX'^a
+
\Gamma^a_{bc}'X'^b
[/tex]
Transformed using the equations under 2.
[tex]
\nabla_cX^a'
=
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\frac{\partial}{{\partial}x^f}
X^d
+
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\Gamma^d_{ef}X^e
=
{\partial}'_cX'^a
-
\frac{{\partial}^2x'^a}{{\partial}x^f{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
X^d
+
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\Gamma^d_{ef}X^e
=
{\partial}'_cX'^a
-
\frac{{\partial}^2x'^a}{{\partial}x^f{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\frac{{\partial}x^d}{{\partial}x'^b}
X'^b
+
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^f}{{\partial}x'^c}
\Gamma^d_{ef}
\frac{{\partial}x^e}{{\partial}x'^b}
X'^b
[/tex]
So my thought was that anything related to X'^b would be the what I seek which gives
[tex]
\Gamma'^a_{bc}=
\frac{{\partial}x'^a}{{\partial}x^d}
\frac{{\partial}x^e}{{\partial}x'^b}
\frac{{\partial}x^f}{{\partial}x'^c}
\Gamma^d_{ef}
-
\frac{{\partial}x^d}{{\partial}x'^b}
\frac{{\partial}x^f}{{\partial}x'^c}
\frac{{\partial}^2x'^a}{{\partial}x^f{\partial}x^d}
[/tex]
This i kind of close but it's different in the last part where I have f:s instead of e:s. I would really appreciate some help, I've been stuck for a long time.