Changes

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==

==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]].

This is not a paradox.  It arises from failure to know or understand the "speed addition" formula.  The formula tells how fast each observer measures the speed of the other observer when they are both traveling toward each other, as seen by some outside observer, at speeds <math>u</math> and <math>v</math>.  Under classical mechanics, the result is just <math>u+v</math>. But the formula under special relativity is

However, this is easily resolved by realising that adding the [[speed]]s is correct for [[Galilean relativity]]. Since the spacecraft are travelling at a significant fraction of the speed of light, it in not valid to use [[Galilean relativity]]. Therefore, the [[Lorentz transformation|velocity Lorentz transformations]] of special relativity must be used. Suppose the observer is in the undashed few and measures a speed <math>v_x</math>, then on the spacecraft travelling in the positive x direction, they measure speed <math>v_x^'</math>. The relevant equation is:

<math>\frac{u + v}{1 + \frac{uv}{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:

So, if two spacecraft are approaching each other, each traveling at what an outside observer would measure as half the speed of light, a person ignorant of how relativity works might think they each see the other coming toward them at the speed of light. But the formula shows that they actually see each other moving at <math>\frac{4}{5}</math> of the speed of light.

<math>v_x^' = -\frac{4}{5} c</math>

If <math>u</math> and <math>v</math> are both less than the speed of light, one can show that the result of the addition formula will be also.

 

and so the paradox is resolved. If the observer on earth observes a beam of light, then the spacecraft also observes light travelling 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 ==