Before we get into the more exciting aspects of Einstein’s General Relativity, it’s important that we understand how we can visualise spacetime on a diagram. I included a few spacetime diagrams in my last post – here I will elaborate more on these, and introduce the concept of light cones.
Space is made up of 3 dimensions, and time is made up of 1 dimension. To keep our drawings simple (because drawing in 4 dimensions is incredibly difficult) we’ll stick to 1 dimension of space (denoted by r for radius) and 1 dimension of time (denoted by t). Let’s put these on an axis. Then each individual point is called an “event” – it has a time and a place.
Events can be connected by a line called a “worldline”. We can think of this as a time line of events from our GCSE history class. The events near the bottom of the worldline happen first, and the events near the top of the worldline happen later.
Now let’s plot a worldline of a person. I’m going to try to plot my future. Currently, I’m sat in my living room on my laptop, but what could I be doing in 8 minutes and 20 seconds?
The first possibility (and probably the most likely) is that I’ll still be sat here. But I could easily realise I’ve run out of milk and make a quick dash to the supermarket. Or, if I had a fast enough aeroplane, I could get to Blackpool in 8 minutes and 20 seconds.
It takes light 8 minutes and 20 seconds to reach us from the Sun. So, if I could travel at the speed of light, I could be on the Sun in 8 minutes and 20 seconds. If I could travel faster than the speed of light, then I could be on Pluto in 8 minutes and 20 seconds. Let’s put these events on our spacetime diagram.
Now, some of these events are possible for me to get to, some of these are physically impossible. I cannot travel at the speed of light, and I definitely can’t travel faster than it, so there is no way that in 8 minutes and 20 seconds I could be stood on either the surface of the Sun or the surface of Pluto. We can draw lines to separate the possible from the impossible. This leaves us with a triangle, or, if we remember that space is made up of several dimensions, a cone. This is called our future light cone.
Every event in the spacetime diagram has a future light cone. All events inside the light cone can be reached by travelling at less than the speed of light, all events outside the light cone can only be reached by travelling at more than the speed of light, and events on the surface of the light cone can only be reached by travelling at exactly the speed of light. Similarly, every event has a past light cone.
Now, let’s look at drawing light cones on events on a worldline. First, let’s look at my worldline. The worldline stays inside the light cones at all times. This is because I’m travelling at less than the speed of light. We call this type of worldine “timelike”.
If instead we were to consider a particle of light, called a photon, we would see that the worldline is always touching the edge of the light cone because the photon is travelling at the speed of light. This worldline is called “lightlike” or “null”.
Similarly, an observer travelling faster than the speed of light would have his worldline outside the lightcones, and this worldline is called “spacelike”.
These are important concepts that will be revisited when we start discussing more about black holes. The cool outer space stuff is coming, but this is important preliminary work… stick with me! We’ll get to the black holes soon!
