On length contraction (Special Relativity)

In summary: 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 ??
  • #1
physikamateur
12
0

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.
 
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  • #2
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?
 
  • #3
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
 
  • #4
physikamateur said:
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.
 
  • #5
physikamateur said:
The equatorial radius (a) of the prolate spheroid would be contracted by a factor of γ. (i.e a=r/γ).

Why?
 

1. What is length contraction in special relativity?

Length contraction is a phenomenon described by Einstein's theory of special relativity, where the length of an object appears to decrease when it is moving at high speeds relative to an observer. This is due to the distortion of space and time at high speeds.

2. How does length contraction occur?

Length contraction occurs because of the principle of relativity, which states that the laws of physics are the same for all observers in uniform motion. When an object is moving at high speeds, its measurements of time and distance are different from those of a stationary observer, resulting in the appearance of a shortened length.

3. What is the formula for calculating length contraction?

The formula for calculating length contraction is L = L0 / γ, where L is the contracted length, L0 is the rest length of the object, and γ is the Lorentz factor, which depends on the relative velocity between the object and the observer.

4. Can length contraction be observed in everyday life?

Yes, length contraction can be observed in everyday life, but the effects are only noticeable at very high speeds. For example, in particle accelerators, particles traveling at close to the speed of light appear to be shorter than they would at rest. However, for everyday objects and speeds, the effects of length contraction are negligible.

5. How does length contraction relate to time dilation?

Length contraction and time dilation are two sides of the same coin in special relativity. Time dilation is the stretching of time at high speeds, while length contraction is the shortening of length at high speeds. Both are consequences of the distortion of space and time at high velocities, and they are mathematically related through the Lorentz factor.

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