Solving the Geodesic Equation: Raising Contravariant Indices

In summary, the conversation discusses manipulating the geodesic equation, specifically how to raise and lower indices in the derivative terms. The geodesic equation is usually written in non-affine form, but can also be written with affine terms by contracting with the metric. One member suggests a possible form of the equation, which is confirmed by another member. The crucial point is made that the covariant derivative transforms as a tensor, unlike the partial derivative.
  • #1
Pacopag
197
4

Homework Statement


I would like to manipulate the geodesic equation.


Homework Equations


The geodesic equation is usually written as
[tex]k^{a}{}_{;b} k^{b}=\kappa k^{a}[/tex] (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
[tex]k_{a ;b} k^{b} = \kappa k_{a}[/tex].
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.
[tex]k^{a ;b} k_{b}=\kappa k^{a}[/tex].
Is this a valid form of the geodesic equation?
 
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  • #2
Sure. [tex]g^{a b} k_{;b}=k^{;a}[/tex].
 
  • #3
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?
 
  • #4
On the LHS you've got [tex]{k^a}_{;b}k^b=k^{a;c}g_{cb}k^b=k^{a;c}k_c=k^{a;b}k_b[/tex]
 
  • #5
Excellent! Thank you both very much.
 
  • #6
Pacopag said:
Excellent! Thank you both very much.

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

Related to Solving the Geodesic Equation: Raising Contravariant Indices

1. What is the geodesic equation?

The geodesic equation is a mathematical equation used to describe the shortest path between two points on a curved surface, such as a sphere or a curved space-time.

2. Why is it important to solve the geodesic equation?

Solving the geodesic equation allows us to find the shortest path between two points and understand the behavior of objects moving in curved spaces, which is crucial in fields such as physics, astronomy, and engineering.

3. What does "raising contravariant indices" mean in relation to the geodesic equation?

Raising contravariant indices is a mathematical operation used to transform a tensor from a covariant form to a contravariant form. In the context of the geodesic equation, this allows us to better understand the curvature of a space and make calculations easier.

4. How do you solve the geodesic equation?

The geodesic equation can be solved using various mathematical techniques, such as differential geometry and calculus of variations. It involves finding the shortest path between two points by calculating the geodesic curve, which is the path that minimizes the distance between the two points.

5. What are some real-world applications of solving the geodesic equation?

Solving the geodesic equation has various practical applications, such as predicting the motion of planets in space, designing efficient transportation routes, and understanding the behavior of light in gravitational fields. It also plays a crucial role in the theory of general relativity and has been used in the development of GPS technology.

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