Conceptual GR question (regarding orbit paths)

[Matterwave]

I'm by no means even adequate in the field of General relativity, so if my question is dumb, please excuse me.

Anyways, so I know that from the EFE's you can get the metric for the space-time you are considering, and from this metric you can get the geodesics of this space-time using calculus of variations. (Intensely tedious imo)

So, according to the GR, a freely moving body will move along to the geodesics, correct?

So, my question is, where does the velocity of that object come into play? I know that objects moving at different velocities will exhibit different trajectories, but this is not readily apparent if you just say all freely moving bodies will move along the geodesics of the space-time.

My guess on this matter would be that motion creates a Lorentz contraction (from Special relativity), which changes the space-time that you "see" in such a way that the geodesics are changed. Am I on the right track here? Or is there like a velocity term in the deriving of the geodesics, but it just doesn't look that way from the equations I've been looking at?

The equations for geodesics I'm looking at are (for the Schwarzschild metric):

$$\frac{d^2u}{d\phi^2}+u=\frac{m}{h^2}+3mu^2$$
$$r^2\frac{d\phi}{ds}=h$$

Where u=1/r, m and h are constants of integration (m is actually the mass of the sun). I was not told, in my textbook, what h actually is. Is h velocity dependent maybe?

 

[A.T.]

So, my question is, where does the velocity of that object come into play? I know that objects moving at different velocities will exhibit different trajectories, but this is not readily apparent if you just say all freely moving bodies will move along the geodesics of the space-time.

The initial velocity in space determines the initial direction in space-time. There are many geodesics passing the starting point in space time. But with the initial direction given you get a unique one.

 

[Matterwave]

Oh phoo, I had forgotten that the geodesics include time. <_<

Ok thanks. :)

 

[George Jones]

I was not told, in my textbook, what h actually is. Is h velocity dependent maybe?

Which book? Roughly, $h$ is angular momentum per unit mass, and is the conserved quantity associated with the $\phi$-independence of the metric. Yes, $h$ is velocity dependent.

If you want to play visually with relativistic orbits, try

https://www.physicsforums.com/showthread.php?p=1091901a#post1091901.

I calculate $h$ from initial conditions. I might expand on how in a later post.

 

[Matterwave]

Yea, h actually was explained in the book I have - I should not have called it a textbook (Llillian Lieber's The Einstein Theory of Relativity) - except it was in a footnote, so I didn't notice at the time of the posting. :P

Thanks :)

EDIT:
I have another question, sort of unrelated, but I don't want to start a new thread so...

So, the Schwarzschild metric was derived setting the stress energy tensor to 0, so I assume this solution only works in the space around the mass and not where the mass actually is, that is correct?

My question is then, when do you actually use the stress energy tensor? Are there solutions to the EFE's which include it?

 

[A.T.]

So, the Schwarzschild metric was derived setting the stress energy tensor to 0, so I assume this solution only works in the space around the mass and not where the mass actually is, that is correct?

Thers is also an interior Schwarzschild metric for the part where the mass is. Search this forum or google it.