Sphere-Ellipsoid Metric Transformation at u = 0.99c

[ChrisVer]

Hello,
Suppose you have a sphere, traveling at $u = 0.99 c$ along the $x$ axis in our frame S.
I was wondering how the sphere's metric changes due to Lorentz length contraction. Obviously at the rest frame of the sphere, it will appear as a sphere. However on S the sphere will appear as an ellipsoid.

How would the induced metric transform however, so that it changes from describing a sphere to describing an ellipsoid?

 

[WannabeNewton]

What do you mean by "appear"? Do you mean the intersection of the worldtube of the sphere with the past lightcone of S? If so then you're mistaken, the sphere will still appear as a sphere. If on the other hand you mean the intersection of the worldtube with the simultaneity surfaces of S then yes it will be ellipsoidal; in this case it will be an ellipsoid of revolution since the $y$ and $z$ axes of the ellipsoid will be equal semi-major axes and the $x$ axis will be the smaller semi-minor axis.

The easiest way to get the new metric is to simply take the parametrization of the 2-sphere $x = \cos u \cos v, y = \cos u \sin v, z = \sin u$, where $-\pi/2 < u < \pi/2, -\pi < v < \pi$ is a frame independent specification, and perform a Lorentz boost at $t = t' = 0$ so that $x' = \gamma x$ giving the parametrization of an ellipsoid of revolution $x' = \gamma\cos u \cos v, y' = \cos u \sin v, z' = \sin u$ from which the new induced metric can be calculated in the usual way for 2-surfaces embedded in $\mathbb{R}^3$.