Covariant and Contravariant Vectors

[vaibhavtewari]

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

 

[jfy4]

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.

 

[vaibhavtewari]

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.

 

[Mentz114]

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.

 

[vaibhavtewari]

Thankyou very much for explaining, I relaize I was missing such a crucial point. Thanks again.

 

[atyy]

Take a look at George Jones's

https://www.physicsforums.com/showthread.php?t=261473 (post #5)
https://www.physicsforums.com/showthread.php?t=263012 (post #12)

 

[vaibhavtewari]

Thankyou for adding, it did help more..