Lorentz Boosted 2-Sphere: Solving the Problem

[Adam Lewis]

Hello,

I have a probability density which in its rest frame is evenly painted upon a 2-sphere. I need to figure out how that density transforms under a Lorentz boost.

Heuristically, this will consist of boosting the 2-sphere to obtain an ellipsoid of some sort, then doing a parallel projection along the boost axis back onto the sphere.

At least the first part of this, I think, is a fairly standard problem since it is similar to that of calculating the beaming of synchrotron radiation. I was just wondering if anyone knows of a source that works through it so I don't have to do the whole thing from scratch.

Thanks!

 

[MikeLizzi]

Hello,

I have a probability density which in its rest frame is evenly painted upon a 2-sphere. I need to figure out how that density transforms under a Lorentz boost.

Heuristically, this will consist of boosting the 2-sphere to obtain an ellipsoid of some sort, then doing a parallel projection along the boost axis back onto the sphere.

At least the first part of this, I think, is a fairly standard problem since it is similar to that of calculating the beaming of synchrotron radiation. I was just wondering if anyone knows of a source that works through it so I don't have to do the whole thing from scratch.

Thanks!

I'd like to help. I have a program that will transform any 3d object between any reference frames. But I'm not sure what you are really asking. What's a 2-sphere?

 

[Adam Lewis]

Hi,

I mean, a spherical shell (the surface of a ball). I need e.g. parametric equations or something for the resulting surface.

-Adam

 

[MikeLizzi]

Hi,

I mean, a spherical shell (the surface of a ball). I need e.g. parametric equations or something for the resulting surface.

-Adam

Spherical shell. Ok. I’m trying to think if my program will give you what you want, so please be patient with this next question. When you say you have “a probability density which in its rest frame is evenly painted upon” the shell,
Do you mean you have some evenly spaced set of points on the shell with associated density magnitude like;
P1 = (x1, y1, z1, density_1)
P2 = (x2, y2, z2, density_2)
P3 = (x3, y3, z3, density_3) etc.

Or do you have a set of “dots” like;
P1 = (x1, y1, z1)
P2 = (x2, y2, z2)
P3 = (x3, y3, z3) etc.
And the density at any area is determined by the number of dots?

Or do you mean something different?