Here is a picture of spacetime. Picture the events of your life. Your first day at school. Christmas with your grandparents. That night down the pub. A chronicle of moments from joy to despair and everything in between. Events are the atoms of experience. From our human perspective, events come with labels; we speak of them in terms of the place and time they happen. Imagine carefully laying out the events of your life one by one to form a line snaking over spacetime; an unbroken path charting your journey through the world. This is called your worldline.
Figure 2.1. The events of a life in space and time. The line through the events is known as a worldline. The cones at each event are known as light cones. They are the paths of a flash of light, emitted at the event. Because nothing can travel faster than light, only future events inside these cones can be influenced by the original event.
Figure 2.1 depicts a worldline winding its way over spacetime. It is known as a spacetime diagram. Imagine this is your worldline, your life laid out before you. Change the dates, add your own events and memories, construct the map of your experiences. Spacetime is an evocative thing. The collection of all events in your life, past, present and future. Your memories are of events in spacetime. The moments that make up your life – Christmases long ago, summer afternoons with school-friends, first kisses and last goodbyes – have not been lost forever. Those moments are still out there, somewhere in spacetime. Your future – everything that has yet to happen to you – every event including your death at the end of your worldline – is waiting for you to arrive, somewhere in spacetime. If we lay out all events in this way, we have created a map of spacetime, and the distance between the events is given by the interval. How wonderful it would be to have freedom of movement over this map, the ability to revisit every moment. We can move anywhere over a map of space, so why should we not have the same freedom over a map of spacetime? The reason is to be found in the interval.
Let’s recap. From a particular point of view, the distance in space between two events is measured to be Δx and the difference in time between the events is measured to be Δt. From a different point of view, Δx and Δt will be different, which is very counter-intuitive. But crucially, the interval (Δτ)2 will not depend on point of view:
(Δτ)2 = (Δt)2 – (Δx)2
We can use the idea of the interval to introduce the concept of the length of a worldline. To be specific, think of the worldline in Figure 2.1 as it goes from being born in 1968 to the enigmatic future event marked X. How long is this portion of the worldline? If event X occurs at precisely the same location as the birth, then the equation above informs us that the interval between the two events (birth and X) is just given by the time interval, i.e. Δτ = Δt because Δx = 0. This is the interval between the two events, but it is not the length of the worldline. Rather it is the length of the worldline that goes straight up the time axis (the vertical line on Figure 2.1). Just like the distance of a journey between Oldham and Wigan depends on the route taken, so it is with distances in spacetime. They depend on the spacetime path taken by the worldline. The way to compute the length of the snaking worldline in Figure 2.1 is to imagine chopping it into lots of tiny segments. Each segment being approximately a straight line.** Then we can compute Δτ for each segment, using the formula above, and add up all the Δτ’s to get the total length.
We can also make the important observation that there are three different sorts of intervaclass="underline" (Δτ)2 can be positive, negative or zero. We might say that there are three different sorts of ‘distance’ in spacetime, in contrast to one sort of distance in space.
If the time difference between the events is larger than the distance in space between them, the interval is positive. Such pairs of events are referred to as ‘timelike separated’. All the events on your worldline are timelike separated from each other. There is a simple physical interpretation for the interval in this case. If you had a perfect stopwatch that you started at the moment you were born, and carried it with you for your whole life, the watch would measure the length of your worldline, from your birth to the present moment. The length of your worldline is therefore your age. This is the meaning of the interval for timelike separated events. It is the time measured on a watch carried along a worldline between events.††
If the distance in space between the events is larger than the time difference, the interval is negative. We say these events are ‘spacelike’ separated. We can now no longer interpret the interval in terms of a watch moving between the events. A physical interpretation does exist, however. For the case where the two events occur at the same time, we can interpret the interval as recording the distance between these events as measured on a ruler. It turns out that for spacelike separated events, it is always possible to find an observer (i.e. a point of view) from whose perspective the events happen at the same time. This means it’s not possible for someone or something to be physically present at both events since that would require being in two places at the same time. That’s just another way of saying that we could not arrange to carry a watch between the events.
There are therefore two fundamentally different regions of spacetime surrounding any event: a region containing those events that could conceivably be on the worldline of a watch passing through that event, and a region containing events that could not. We’ll see the significance of this division in a moment.
The third possibility is that the time difference between a pair of events is exactly equal to the distance in space between them. This is the case if a worldline between the two events is the path taken by a beam of light. To see this, recall that we measure time in seconds and distance in light seconds. Light travels 1 light second in 1 second, 2 light seconds in 2 seconds, and so on. So, for any pair of events that lie on the path of a beam of light, (Δt)2 = (Δx)2 and the interval is zero. These events are known as ‘lightlike’ separated. If we draw the paths of light rays out over spacetime from an event, they form what is known as the future light cone of the event. In Figure 2.1, the light cones are depicted as small cones at each event. The light cones spread out at an angle of 45 degrees from each event. Inside the future light cone, all events are timelike separated from the original event while outside the future light cone, all events are spacelike separated from the original event. Since we were present at every event in our own lives, our worldline snakes along inside the light cones.‡‡