On length contraction (Special Relativity)

[physikamateur]

Homework Statement

Show that the length contraction deforms a sphere in motion to an oblate rotational ellipsoid whose volume decreases by a factor of γ (gamma)

Homework Equations

x=x'/γ v=4∏r^3/3 (volume for sphere) v=4∏(a^2)b/3 (volume for prolate and oblate spheroid)

The Attempt at a Solution

The question is taken from Special relativity for beginners by Jurgen Freund. Firstly, I am confused as to why the question claims that the sphere would deform into an oblate spheroid rather than a prolate spheroid since length contraction does not contract transversely. Anyway by length contraction, the radius (r) of the sphere would contract by a factor of γ, Therefore:
r=a/γ where a is the equatorial radius and b is the polar radius. By subbing r=a/γ into the formula of sphere, through some algebraic manipulation, I get a spheroid whose radius decreased by a factor of γ^3 instead. How should I go about doing it and where are my errors ?

Please help.

Thank you for viewing.

 

[clamtrox]

So what happens to the sphere with the length contraction? The direction parallel to motion is contracted, and the two perpendicular to it are not, right? So which of a and b are transformed?

 

[physikamateur]

The equatorial radius (a) of the prolate spheroid would be contracted by a factor of γ. (i.e a=r/γ). Therefore the volume of the sphere is V=4∏(r^3)/3 and the volume of the prolate spheroid would be V'=4∏(r^2)b/3(γ^2). Finally, taking the volume of the sphere and divide it by the volume of the prolate spheroid gives (V/V')=r(γ^2)/b ??

Sorry if I am slow at catching ideas. Please bear with me.

Thank you very much

 

[nrqed]

The equatorial radius (a) of the prolate spheroid would be contracted by a factor of γ. (i.e a=r/γ). Therefore the volume of the sphere is V=4∏(r^3)/3 and the volume of the prolate spheroid would be V'=4∏(r^2)b/3(γ^2). Finally, taking the volume of the sphere and divide it by the volume of the prolate spheroid gives (V/V')=r(γ^2)/b ??

Sorry if I am slow at catching ideas. Please bear with me.

Thank you very much

Watch out. It is b which is contracted whereas "a" remains equal to the initial radius.

 

[clamtrox]

The equatorial radius (a) of the prolate spheroid would be contracted by a factor of γ. (i.e a=r/γ).

Why?