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.

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