1 Introduction and mental preparation
1.1 What is special relativity and why should it interest me?
The special1 theory of relativity concerns the constancy2 of the speed of light and how this results in modifications to the rules of classical mechanics with which you have become so familiar. Practically, it allows us to understand high speed phenomena and to perform related calculations. Perhaps more importantly, it also requires us to think differently about the fundamentals of the way time and space work. Indeed, the development of special relativity required physicists to reassess much of what we thought we already knew.
1 The general theory of relativity (GR) is a more advanced topic, involving gravity and the curvature of spacetime — hopefully you will be inspired to take a GR course later in your degree.
2 What we mean by ‘constant’ here will be explained in more detail later.
You have surely already seen some of the results of special relativity — the mass-energy equivalence formula, \(E = mc^2\), is difficult to avoid! You may have also heard of strange sounding effects, such as time dilation or length contraction, and have been told about the twin paradox. I hope that hearing about these counter-intuitive effects left you intrigued to learn more, though you have perhaps not had the chance to do so yet. These lectures are that chance.
1.2 The use of special relativity
I hope that a sense of intrigue is enough to interest you in special relativity. However, for those of you who are more practically minded, special relativity is one of the foundations of modern physics. It is an essential prerequisite if you wish to have a modern understanding of electromagnetism, gravity, and particle physics, among other topics.
1.3 What mathematics will I need
To learn the basics of special relativity requires very little mathematical background — just a little school-level algebra. There will also be a Taylor series and some matrices, but nothing too complex. So this should be easy, right?
1.4 So why does it have the reputation of being difficult?
Special relativity is counter-intuitive. You might find it mind-boggling at first. I fully expect you to read parts of these notes, only for your brain to rebel and say “that goes against all common sense”3.
3 Of course, the seemingly fantastical nature of special relativity might be part of the appeal.
But ask yourself the following questions.
Question: Where does your intuition come from? Who gave you your “common sense”?
Answer: I would expect4 that most of your intuition about the physical world comes from your past experience, from childhood to the present5. Some of your common sense might have been taught to you by your parents.
4 I am no neuroscientist, but…
5 Might we have some intuition of the physical world at birth? Perhaps, in which case this must be inherited. However, I see little evolutionary advantage to having, at birth, an intuition about near light-speed travel
Followup question: As you grew up, what experience did you (or your parents) have of travelling, or of objects travelling, near the speed of light?
Answer: None. (You might argue that you have experienced light for all of your life, but have you ever seen it move? You detect it when it hits your retina, but that is all.)
Final question: So, can you expect your intuition or common sense to be a reliable guide to the kinematics and dynamics of objects travelling near the speed of light?
Answer: No.
So be prepared for it to take some time (and effort) before special relativity begins to “make sense”. You will need to unlearn some things6, replacing formulae that look correct with new versions that look strange7. But the effort is worth it and you will discover a new way to view the fundamentals of how the universe works.
6 Which is always challenging.
7 The velocity addition formula immediately springs to mind — you will see this later.
1.5 The reality of special relativity
Before we really get started, I wish to dispel one common misunderstanding about special relativity, perhaps resulting from reading popular science accounts or as a reaction to some of the counter-intuitive results. I wish to confirm that the effects of special relativity, e.g. that time moves slowly on moving objects, are real effects, not merely optical illusions.
It is true that, without relativity, a clock receding from us at half the speed of light will appear to run slow due to the Doppler effect, since light from the clock will take longer to reach the eye as the clock gets further away. This is not a relativistic effect8. In contrast, we will soon see that time dilation9 means that a moving clock actually runs slow, and this is independent of the direction of motion of the clock.
8 Indeed, it might be best if you forget about it for a bit!
9 Which is a relativistic effect.
This difference between what happens and what appears to happen means that I ought to be careful in the language I use. However, it is inevitable that I will, at some point, state that “\(A\) observes that the clock runs slowly” or even “\(A\) sees the clock run slowly”. In such cases, assume that I mean “the clock runs slowly in \(A\)’s frame of reference”, unless I very clearly state otherwise.
1.6 Examinable material
With the exception of the historic introduction in the next chapter, all of what follows in these lecture notes is examinable.