Solving the Geodesic Equation: Raising Contravariant Indices

[Pacopag]

Homework Statement

I would like to manipulate the geodesic equation.

Homework Equations

The geodesic equation is usually written as
$$k^{a}{}_{;b} k^{b}=\kappa k^{a}$$ (it is important for my purpose to keep it in non-affine form).
It is clear that by contracting with the metric we may write alternatively
$$k_{a ;b} k^{b} = \kappa k_{a}$$.
What I would like to know is how to raise to a contravariant indices in the derivative on the left-hand side.

The Attempt at a Solution

If I had to guess, I would like to be able to write something like.
$$k^{a ;b} k_{b}=\kappa k^{a}$$.
Is this a valid form of the geodesic equation?

 

[Dick]

Sure. $$g^{a b} k_{;b}=k^{;a}$$.

 

[Pacopag]

Thank you for your reply Dick.
But how do you explain the lowering of the b index in the second factor on the left-hand side?

 

[cristo]

On the LHS you've got $${k^a}_{;b}k^b=k^{a;c}g_{cb}k^b=k^{a;c}k_c=k^{a;b}k_b$$

 

[Pacopag]

Excellent! Thank you both very much.

 

[Dick]

Excellent! Thank you both very much.

The crucial point is that the covariant derivative transforms as a tensor, unlike say, the partial derivative.