Velocity and Position Vectors

1. Aug 3, 2013

Philosophaie

I would like to have a Star orbiting a Schwarzschild Black Hole if the the velocity and position vectors in the galaxy are given. The only thing that comes to mind is the Newtonian method.

The velocity and position vectors of the Star are:

$$v = {v_x, v_y, v_z)$$
$$r = {x, y, z)$$

where (no acceleration)

$$x = v_x * t +x_0$$
$$y = v_y * t +y_0$$
$$z = v_z * t +z_0$$

convert to spherical

What is the method for GR?

Note:The Schwarzschild metric produces a non zero Riemann Tensor and a Ricci Flat as you well know.

Last edited: Aug 3, 2013
2. Aug 3, 2013

3. Aug 3, 2013

Staff: Mentor

Note, in GR position is not a vector.

4. Aug 3, 2013

tiny-tim

Hi Philosophaie!
You can ignore that fact that it's a black hole, just use an ordinary star of the same mass …

outside the event horizon, the gravitation is the same.

5. Aug 3, 2013

ProfDawgstein

Oh yes, sorry.
GR is non-affine.
But you can have a displacement vector, right?

6. Aug 3, 2013

Philosophaie

I am unclear how you derive the Position Vector in Spherical coordinates and how to incorporate the velocity vector into Spherical coordinates as well. Please explain.

7. Aug 3, 2013

ProfDawgstein

Plug in your $x$ $y$ $z$ and you get $r$ $\theta$ $\phi$.
What are you trying to do anyway?
Do you want to create a computer simulation or just solve it on paper?

8. Aug 4, 2013

Staff: Mentor

Only locally. I.e. Infinitesimal displacements form vectors in the local tangent space.

9. Aug 4, 2013

Staff: Mentor

There is no position vector in GR.

10. Aug 4, 2013

ProfDawgstein

How would you keep track of the objects position?
Just assign $x, y, z$ or $r, \theta, \phi$ to it?

I know.
It's mostly a problem with expressing things using normal language.

11. Aug 4, 2013

Staff: Mentor

Yes, a given point in the manifold can be uniquely identified by a list of it's coordinates. But that list is just a list, not a vector.