Covariant and Contravariant Vectors

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Discussion Overview

The discussion revolves around the concepts of covariant and contravariant vectors in the context of Schwarzschild geometry and general relativity. Participants explore the implications of these concepts on the constants of motion along geodesics, particularly focusing on the momentum 4-vector and its components.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about why covariant vectors are constant while contravariant vectors may not be, particularly in the context of Schwarzschild geometry.
  • Another participant provides a definition from a textbook, emphasizing the geometric interpretation of the Schwarzschild coordinate r and its relation to the area of a sphere.
  • A participant suggests that the momentum 4-vector's components depend on the radius r, indicating that E and p are constants of motion only for fixed radius orbits.
  • There is acknowledgment of a crucial point regarding the conditions under which covariant and contravariant vectors can be considered constant, highlighting the ambiguity in understanding these concepts.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding the behavior of covariant and contravariant vectors, with some agreeing on the definitions while others remain uncertain about the implications. The discussion does not reach a consensus on the interpretation of these vectors.

Contextual Notes

Participants note that the momentum 4-vector's behavior is contingent on the radius in the Schwarzschild metric, which introduces complexities in understanding the constancy of covariant versus contravariant vectors.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of general relativity, particularly those interested in the nuances of vector behavior in curved spacetime and the implications for constants of motion.

vaibhavtewari
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Dear friends,

while reading about schwarzschild geometry, I learned that E=-p_0 and L=p_{\phi} are constant along a geodesic or are constant of motion. I further read that p^0=g^{00}p_0=m(1-2M/r)^{-1}E and p^{\phi}=g^{\phi\phi}p_{\phi}=m(1/r^2)L, 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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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 r has a simple geometric interpretation arising from spherical symmetry. It is not the distance from any "center". Rather it is related to the area A of the two dimensional sphere of fixed r and t by the standard formula r=(A/4\pi)^{1/2}."

This is from Hartle's book.
 
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.
 
If the momentum 4-vector is p^\mu=(p^0,0,0,p^\phi) then dr/d\tau 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.
 
Thankyou very much for explaining, I relaize I was missing such a crucial point. Thanks again.
 
Thankyou for adding, it did help more..
 

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