Key Points from this Appendix:

• We derived the Lorentz Transformations. These mean that if you know your space and time coordinates in one inertial frame of reference, you can figure out what they would be in any other frame. These transformations are the key to advanced work in special relativity, including its formulation by Einstein. 

• We used the Lorentz transformations to derive the velocity transformation.  This means that if you know your velocity in one reference frame, you can work out what it would be in any other frame. This also means that you can add up velocities measured in different reference frames. 

• We considered whether you can choose a reference frame such that your speed exceeds light speed.  Our example suggested that the answer is no, you can't. Using the relativistic velocity transformation showed that this could not work. 

•  We looked at the twin paradox:  if I'm moving relative to you then you're moving relative to me, so time dilation means that each of us will think the other experiences less time than we do.  Can we both be right? We saw that we can only really ask this question if we introduce a direction change so that we both end up at the same place. In this case we found that if we agree which one of us was actually moving (because they experienced the necessary accelerations) we will actually agree that this person experienced less time. So - at least in this case - there is actually no paradox.