General relativity, geodesic, KVF, chain rule covariant derivatives

In summary: In normal calculus, the order in which you take the derivatives doesn't matter. This is because in normal calculus, the derivatives commute with each other. In other words, the derivatives of a function can be taken in any order and the result will be the same. However, in differential geometry, the covariant derivative does not commute with the tangent vector, so the order in which you take the derivatives does matter. In this case, taking the derivative with respect to the tangent vector first, and then with respect to the affine parameter, gives the correct result.
  • #1
binbagsss
1,259
11

Homework Statement



To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0##

Homework Equations


see above

The Attempt at a Solution



Proof:

So the proof is to use the chain rule that ##\frac{d}{ds}= ## , where ##s## is some affine parameter:

## \implies \frac{dK}{ds}=V^{\alpha}\nabla_{\alpha}(K_uV^u)= V^u V^{\alpha}\nabla_{\alpha}K_u + K_u V^{\alpha}\nabla_{\alpha}V^u## ; first term is zero from KVF equation - antisymetric tensor multiplied by a symmetric tensor, and the second term is zero from the geodesic equation

MY QUESTION - this may be a stupid question, but concerning the order of the chain rule application since the covariant derivative operates on everything to the right...

How do you know to write ##\frac{d}{ds}=V^{u}\nabla_u## as a pose to ##\frac{d}{ds}=\nabla_uV^u##

In normal calculus when you use the chain rule, the order doesn't matter does it? For e.g ## \frac{d}{ds}=\frac{dx}{ds}\frac{d}{dx} = \frac{d}{dx}\frac{dx}{ds} ## ?

But if i try to apply the above proof writing ##\frac{d}{ds}=\nabla_uV^u## I get an extra non-zero term : ## K_uV^u\nabla_{\alpha}V^{\alpha}##

so the proof fails.

Many thanks
 
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  • #2
binbagsss said:
In normal calculus when you use the chain rule, the order doesn't matter does it?
Yes it does.
 
  • #3
Orodruin said:
Yes it does.

I never knew this, what is the order?
 

Related to General relativity, geodesic, KVF, chain rule covariant derivatives

1. What is general relativity?

General relativity is a theory of gravitation that was developed by Albert Einstein in the early 20th century. It describes how gravity affects the motion of objects in the universe and how the structure of space and time is affected by the presence of mass and energy.

2. What is a geodesic in general relativity?

In general relativity, a geodesic is the path that a freely moving particle follows in a curved space-time. It is the shortest path between two points in space-time and is determined by the curvature of space-time caused by the presence of mass and energy.

3. What is the KVF in general relativity?

KVF stands for Killing vector field, which is a vector field that describes a symmetry of a space-time. In general relativity, KVF plays an important role in determining the conservation laws of energy and momentum.

4. What is the chain rule for covariant derivatives in general relativity?

The chain rule for covariant derivatives is a mathematical tool used in general relativity to calculate how a vector field changes along a curve in a curved space-time. It takes into account the effects of the curvature of space-time on the vector field.

5. How are covariant derivatives used in general relativity?

Covariant derivatives are used in general relativity to describe how vector fields change as they move along a curved space-time. They take into account the effects of the curvature of space-time and help to calculate important quantities such as energy and momentum in the theory.

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