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Theory of relativity (view source)
Revision as of 13:49, September 30, 2016
, 13:49, September 30, 2016Added additional paradox
The Ehrenfest Paradox considers a rigid wheel or disc rotating a bout its axis at high speed (somewhat like a bicycle wheel spinning freely on its axle). The rim of the wheel travels at close to the speed of light and therefore undergoes length contraction, whereas the radius (the spokes, for the bicycle wheel) does not. Hence the circumference is no longer equal to 2<big><math>\pi</math></big>r, which is paradoxical.
The Ehrenfest Paradox considers a rigid wheel or disc rotating a bout its axis at high speed (somewhat like a bicycle wheel spinning freely on its axle). The rim of the wheel travels at close to the speed of light and therefore undergoes length contraction, whereas the radius (the spokes, for the bicycle wheel) does not. Hence the circumference is no longer equal to 2<big><math>\pi</math></big>r, which is paradoxical.
The apparent paradox was finally resolved in 1975 by the Norwegian scientist [[Øyvind Grøn]].<ref>http://www.physicsforums.com/showthread.php?t=224955</ref>
The apparent paradox was finally resolved in 1975 by the Norwegian scientist Øyvind Grøn.<ref>http://www.physicsforums.com/showthread.php?t=224955</ref>
==Speed paradox==
One apparent inconsistency involves two spacecraft approaching each other. Suppose an observer on earth sees two spacecraft moving towards each other at half the [[speed of light]]. One travels in the positive x direction, the other in the negative. Therefore they should each see the other approach them at the speed of light, an apparent contradiction given that no object with mass may travel at the [[speed of light]].
However, this is easily resolved by realising that adding the [[speed]]s is correct for [[Galilean relativity]]. Since the spacecraft are traveling at a significant fraction of the speed of light, it in not a valid approximation. Therefore, the [[Lorentz transformation|velocity Lorentz transformations ]] must be used. Suppose the observer is in the undashed few and measures a speed <math>v_x</math>, then on the spacecraft traveling in the positive x direction, they measure speed <math>v_x^'</math>. The relevant equation is:
<math>v_x^' = \frac{v_x - u}{1- \frac{uv_x}{c^2}}</math>
Where <math>u</math> is the speed between the [[inertial frame of reference|inertial frames of reference]], in this case half the speed of light. <math>v_x</math> is also half the speed of light (but negative), and substituting in gives:
<math>v_x^' = -\frac{4}{5} c</math>
and so the paradox is resolved. If the observer on earth observes a beam of light, then the spacecraft also observes light traveling at the same speed, agreeing with the second postulate, that all observers in [[inertial frames of reference]] measure the same value for the speed of light.
== Variable Speed of Light ==
== Variable Speed of Light ==
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