How Does General Relativity Affect Star Orbits Around Schwarzschild Black Holes?

[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.

 

[ProfDawgstein]

You could just use http://en.wikipedia.org/wiki/Spherical_coordinate_system#Cartesian_coordinates for your position vector (the black hole stays at 0,0,0).
Then do the same for your velocity vector.
Now create 4-vectors out of your 2 vectors and solve your geodesic equation.

Note: The mass of the star should be a lot less than the mass of the black hole.

 

[Dale]

Note, in GR position is not a vector.

 

[tiny-tim]

Hi Philosophaie!

I would like to have a Star orbiting a Schwarzschild Black Hole if

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. ​

 

[ProfDawgstein]

Note, in GR position is not a vector.

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

 

[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.

 

[ProfDawgstein]

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.

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?

 

[Dale]

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

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

 

[Dale]

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.

There is no position vector in GR.

 

[ProfDawgstein]

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

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

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

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

 

[Dale]

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

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.