Covariant and Contravariant Vectors

In summary, the conversation revolved around the concept of covariant and contravariant vectors in relation to Schwarzschild geometry. It was mentioned that E=-p_0 and L=p_{\phi} are constants along a geodesic, and that p^0 and p^{\phi} depend on the radius r. The question of understanding this ambiguity was raised, and it was clarified that E and p are only constants of motion for a fixed radius in an orbit. Additional resources were also provided for further understanding.
  • #1
vaibhavtewari
65
0
Dear friends,

while reading about schwarzschild geometry, I learned that [tex]E=-p_0[/tex] and [tex]L=p_{\phi}[/tex] are constant along a geodesic or are constant of motion. I further read that [tex]p^0=g^{00}p_0=m(1-2M/r)^{-1}E[/tex] and [tex]p^{\phi}=g^{\phi\phi}p_{\phi}=m(1/r^2)L[/tex], which I can see depends on radius r. This made me think that I don't really understand covariant and contravariat vectors well as I though they ought both be constant of motion.

I will be glad if someone can give a insightful description on how to understand this so that I don't run into conflicts again. I am sure this will help other physicist too.

Thank You
 
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  • #2
Hi!,

It's cool someone else is cruisin PF on Sunday afternoon...

As I am looking over my GR textbook through the Schwarzschild Metric chapter, i find the following definition:

"The Schwarzschild coordinate [tex]r[/tex] has a simple geometric interpretation arising from spherical symmetry. It is not the distance from any "center". Rather it is related to the area [tex]A[/tex] of the two dimensional sphere of fixed [tex]r[/tex] and [tex]t[/tex] by the standard formula [tex]r=(A/4\pi)^{1/2}[/tex]."

This is from Hartle's book.
 
  • #3
Thankyou for pointing out that, though all I pointed out was why covariant vector is a constant and contravariant not. I believe we can have frame work when contavariant is constant but contravariant not.

So I was sort of confused how to truly understand this ambiguity.
 
  • #4
If the momentum 4-vector is [itex]p^\mu=(p^0,0,0,p^\phi)[/itex] then [itex]dr/d\tau[/itex] is zero and the 'r' in your formulae is a constant. To put it another way, E and p are only constants of motion for fixed radius in this orbit.
 
  • #5
Thankyou very much for explaining, I relaize I was missing such a crucial point. Thanks again.
 
  • #7
Thankyou for adding, it did help more..
 

Related to Covariant and Contravariant Vectors

1. What are covariant and contravariant vectors?

Covariant and contravariant vectors are two types of vectors used in multivariable calculus and tensor analysis. They represent different ways of transforming and measuring vectors in different coordinate systems.

2. What is the difference between covariant and contravariant vectors?

The main difference between covariant and contravariant vectors is in how they transform under a change of coordinates. Covariant vectors transform in the same way as the coordinate system, while contravariant vectors transform in the opposite way. This can also be seen in their components, where covariant vectors have the same indices as the coordinate system and contravariant vectors have indices that are the inverse of the coordinate system.

3. How are covariant and contravariant vectors used in physics?

In physics, covariant and contravariant vectors are used to describe physical quantities that are independent of the coordinate system. This allows for the formulation of physical laws and equations that are valid in any coordinate system.

4. Can covariant and contravariant vectors be interchanged?

No, covariant and contravariant vectors cannot be interchanged. They represent different mathematical objects and have different transformation properties. Interchanging them would result in incorrect calculations and equations.

5. How are covariant and contravariant vectors related to tensors?

Covariant and contravariant vectors are the building blocks of tensors. Tensors are mathematical objects that can represent physical quantities that have both magnitude and direction in multiple dimensions. Covariant and contravariant vectors are used to define the components of a tensor, and their transformation properties determine how the tensor changes under a change of coordinates.

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