kev said:
Yes. As far as I can tell, we only require the length contraction transformation of Special Relativity,
There is no such thing as "the length contraction transformation of Special Relativity".
The Lorentz transforms are
space and
time transforms and you need
both of them to solve this problem:
x'=\gamma(x-vt)
t'=\gamma(t-\frac{vx}{c^2})
y'=y
z'=z to work out that an object that is physically a sphere in its rest frame S, is physicaly an oblate spheriod in frame S' when it has relative inertial motion.[/quote]
You need to find all points at t'=k in S' (line of simultaneity in S'). In order to do that you will need the second Lorentz transform:
k=\gamma(t-\frac{vx}{c^2})
The above, solved for t:
t=k/\gamma+vx/c^2
Substitute t into the expression for x' :
x'=x/\gamma-vk
or:
x=\gamma(x'+vk)
Subsitute the above into the equation of sphere in S:
R^2=x^2+y^2+z^2
R^2=\gamma^2(x'+vk)^2+y'^2+z'^2
meaning that the object is an ellipsoid in S'. The cross-section of the ellipsoid for a viewer situated on the common x-axis is a circular disc. Indeed
x'=a means:y'^2+z'^2=R^2-\gamma^2(a+vk)^2
Note that :
y'^2+z'^2<R^2
For observers not situated along the common x axis, the situaton is more complicated, there is no obvious proof that such an observer obtains a circular disc as the photograph of the ellipsoid.
After that, all that is required is standard ray-tracing, taking into account the velocity of the oblate spheriod in S' and the finite speed of light, to work out that it can visually appear to be a sphere to observers at frame S'.
Can you prove the above? Mathematically, I mean.
We can also note that this apparent visual unobservability of the length contraction of a sphere is only aproximately true, very close to the object and at greater distances from the sphere, the length contraction is increasingly visually observable.
It is still unobservable if the sphere is not textured. Only if the sphere is textured, it is observable.
For non-spherical objects, the apparent inability to visually observe the length contraction is even less true.
Yet, no experimental proof exists (to date). No one has managed to photograph length contraction.
It is odd that the very special case of the inability to visually observe the length contraction of one specific shape of object at very limited distances, has led to the popular misconception / myth that length contraction of any object at any distance, is not visually observable.
There is no such "myth" amongst people who know physics.