Changes

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]].

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:

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>v_x^' = \frac{v_x - u}{1- \frac{uv_x}{c^2}}</math>

<math>v_x^' = \frac{v_x - u}{1- \frac{uv_x}{c^2}}</math>

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

<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.

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 ==