JesseM said:
I think he basically means to find a light cone such that every position on the ellipsoid is also the position of some event on the light cone, then translate the positions and times of these events into the primed frame using the LT. Of course it's easier to work backwards--assume a light cone in the primed frame starting from x'=y'=z'=t'=0, consider the set of events at t'=r/c which all satisfy x'2 + y'2 + z'2 = r2, then translate these events to the unprimed frame and show their positions form an ellipsoid.
Okay.
I don't get why this is being discussed though. I would think this phenomenon is a consequence of not including the relativity of simultaneity. The OP has been asking for mathematics, so I will now give him mathematics.
Frame S : A^{\mu}=(ct,x,y,z)
Frame S' : B^{\nu}=(ct^\prime,x^\prime,y^\prime,z^\prime)
We assume the origins coincide at t=t'=0 and that the orientation of axes are equal. We also assume that S' frame moves with velocity v in the x-direction in the S frame. We apply the Lorentz transform:
B^{\nu}=\Gamma^{\nu}_{\mu}A^{\mu}= \left( \begin{array}{cccc}\gamma & -\gamma\frac{v}{c} & 0 & 0 \\ -\gamma\frac{v}{c} & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array} \right)\left(\begin{array}{c} ct \\ x \\ y \\ z \end{array}\right)=\left( \begin{array}{c} \gamma\left(ct-\frac{vx}{c}\right) \\ \gamma (x-vt) \\ y \\ z \end{array}\right)
So we have
t^\prime = \gamma \left(t-\frac{vx}{c^2}\right)
x^\prime = \gamma (x-vt)
y^\prime = y
z^\prime = z
The metric of flat space-time is \eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1) which by the definition of the metric gives us for frame S:
ds^2=\eta_{\mu\nu}dA^{\mu}dA^{\nu}=c^2dt^2-dx^2-dy^2-dz^2
and for frame S':
ds^2=\gamma^2c^2dt^2-\gamma^2\frac{v^2}{c^2}dx^2-\gamma^2dx^2+\gamma^2v^2dt^2-dy^2-dz^2=\left( \gamma^2(c^2+v^2) \right)dt^2-\left( \gamma^2\left(\frac{v^2}{c^2}+1\right) \right)dx^2-dy^2-dz^2
The OP can check this for himself if he wishes.
Light travels along null geodesics, meaning ds=0, and these equations then represent light cones in the two frames. Now, note the following:
(1) If we look at 3D slices of constant t in S, we get spheres.
(2) If we look at slices of constant t' in S', we get spheres (Remember that x^\prime \neq x).
(3) If we look at slices of constant t in S' or at constant t' in S, we get ellipsoids.
If the OP doubts this, by all means try to disprove it.
These three points confirm that the light sphere/light ellipsoid discussion is simply nonphysical. It is a result of combining measurements from different frames, which is illegal. The OP should now be convinced that his earlier claims about Euclidean spaces and whatnot are flawed.
