23 February, 2021

Special look at Relativity 001: Spacetime - Setting the Stage

If we accept that as a consequence of Einstein's postulates that time and space become relative then we might be tempted to ask the question “can we construct a framework incorporating time and space where we all agree on the 'distance' between two events?”.

In order to incorporate both time and space into the same framework we need a way of converting between them. There is a saying loved by primary school maths teachers “you can't add apples to oranges”, but maybe they are missing a subtle point. You can add apples to oranges if you can find some way of converting between them. If a grocer is selling apples for 30 cents and oranges for 10 cents, then I can buy three oranges for the same price as 1 apple, or six oranges for the same price as 2 apples, or 9 oranges for the same price as 3 apples etc. The ratio of the value of apples to oranges is 3. If I were to construct a graph relating the number of apples I could exchange for oranges it would be a straight line, and the slope of this line would be 3. So that the slope of the apples-oranges curve is the exchange rate or conversion factor between the two.

What we need to find is an exchange rate between distance and time. Well the ratio of distance to time would be measured in metres per second so our exchange rate should be a velocity. This is promising because the principle of relativity implies that velocity is something we all agree on. If I see you moving 10 m s-1 relative to my position, then you see me moving at 10 m s-1 relative to your position. At the moment we don't know what the value of this exchange rate velocity is so we will just call it 'c' (note: at this stage we are making no assumptions that c is the speed of light). So now to convert distance, x, to time t we use this exchange rate c.

x t = c x = c t x over t = c newline newline x = c t

Actually, the idea of a mashup of space and time, with velocity as an exchange rate, is not as weird as it seems. In fact, we do it all the time. If you ask me how 'far' it is from Coffs Harbour to Sydney I might reply “5 hours”. By which I mean it takes about 5 hours which, using the average speed of a car as the exchange rate, is about 400 km. Ask me the distance to Alpha Centauri and I'll tell you it is about 4 light years. Still not comfortable? Then take a look at this video from minutephysics.

So now we have defined an exchange rate we can start adding apples to oranges, or more specifically, plotting events in spacetime. Now spacetime consists of four dimensions: three space dimensions and one time dimension but I can't draw 4 dimensions on a two dimensional screen (in fact if I am honest I cant really imagine what four dimensions would look like) but there is no problem describing it with maths. To represent it in two dimensions and to keep the maths simple we will roll up the space dimensions into one axis labelled x. This shouldn't worry you too much if you remember that our aim is to come up with a way of measuring the 'distance' between 'events' in spacetime, and if we know the location of any two points in space we can measure or calculate the distance between them to come up with a single value, x, for the space component of their distance. For example, if I constructed a three dimensional map of my office, the distance between my chair and the bin can be represented by a single value of 2.1 m.

Enough babble, lets see what a graphical representation of spacetime might look like. A spacetime diagram has one axis representing space, x, and another representing time, ct (remember that c is just the exchange rate we need to use to convert time into the same units as space). On this diagram we can plot some 'events' that occur at a particular locations in space and time. Our first spacetime diagram has three points, R, G and B. R lies directly to the left G so they both have the same value for their time component but a different value for the space component. In other words, events R and G occur at the same time at different places. Similarly, G lies directly below B so we can say that these two events occur at different times in the same place.

We can even start to determine the spacetime distance between these events. For example, I throw a screwed up bit of paper into the bin 2.1 m away and at the instance it lands in the bin (event G) I throw a second piece of paper (event R) which lands in the bin 1 second later (event B). R and G occur in different places at the same time so the spacetime distance between them is simply the spatial distance, 2.1 m. G and B occur at the same place but at different times so the spacetime distance between them is simply the time difference, 1 s. But what is the spacetime distance between R and B? That is a much trickier question and to answer that we need to develop the concept of spacetime in more detail. That will be the task of our next exciting installment.

21 February, 2021

Special look at Relativity 00: Prequel - How we do science

The journey ahead

At the end of the 19th century many people thought that physics was pretty much 'in the bag'. Newton had given us a seemingly universal set of laws describing motion and forces, while Maxwell had done the same for electromagnetism. It didn't really seem like there was much left for the apprentice boffins to get their teeth into.

There were some annoying little problems. One was a small incompatibility between the laws of Newton and Maxwell, and in 1905 a young patent clerk proposed a solution to this problem that was to usher in a new era of physics. That clerk was Albert Einstein and the solution was Special Relativity and it changed fundamentally the way we think about time and space.

If you look at any high school or undergraduate physics textbook and you will find an introduction to Special Relativity based around light clocks and a presumption that the speed of light is constant. I want to take a completely different approach which I hope will lead to deeper understanding while actually making it easier to solve relativity problems. This approach will be based on Minkowski diagrams, also called spacetime diagrams, and will hopefully lead you to an understanding of why the speed of light is constant for all observers, and why \[ E = mc^2 \]

How Science works

Before we develop the spacetime framework I want to indulge in a few words about how science works. Perhaps one of the most succinct summations of science at work is given by Richard Feynman in a lecture to class at Cornell University in 1964.


In summary, when looking for a solution to a problem we make an educated guess, compute the predictions of that guess, then test those predictions against experiment and observation.

In Einstein's case the problem was the incompatibility between the laws of Maxwell and Newton. I want to leave the discussion of the nature of this incompatibility until later. To begin, I just want to introduce Einstein's solution, which was to suggest that perceptions of time and space vary according to the relative velocity of the observers. In other words, if Prof Stick is moving relative to Albert, then they will measure space and time differently. Our quest in this series is to develop a framework which combines time and space in such a way that Prof Stick and Albert can agree on measurements of 'distance' regardless of their relative velocities. We'll call this framework spacetime.

To develop this framework we are going to make a series of educated and not-so-educated guesses and compute their consequences. This is going to take a fair bit of work but at the end we will have a set of predictions that we can compare observation and experiment.

All aboard!

Before we depart, let's pause, look around and get our bearings. Our starting position has 2 reference points:

  • Galileo's Principle of Relativity - which states that all motion is relative and that there are no special or absolute frames of reference. This is important because it means that if two observers are moving relative to each other they both agree on their relative velocity.

  • Einstein's solution to the Newton/Maxwell incompatibility that suggests that observers with relative velocities will measure space and time differently.

OK, so now we know where we are starting from, follow me...and watch your step!