It is important to understand the meaning of the light cones and what they tell us about the relationships between events in spacetime. They will be central to understanding black holes and the paradoxes they create. Let’s zoom in on a particular event on our worldline to get more of a feel for the light cone and the relationships between neighbouring events in spacetime.
Christmas in spacetime
Let’s imagine we’ve zoomed in on the region of spacetime in the vicinity of ‘Christmas 1974’ on our worldline. Your family are sitting around the TV arguing about whether to watch Bruce Forsyth and The Generation Game on BBC1 or Laurence Olivier in Henry V on BBC2. Presciently concerned about an incipient culture war, Granny springs to her feet and knocks a glass of Harvey’s Bristol Cream onto the electric fire.§§ This causes the main fuse to blow, rendering the debate meaningless.
Figure 2.2 shows the spacetime region around ‘Christmas 1974’, drawn from the point of view of someone sitting in your house. Event A is ‘Granny making contact with the glass of sherry’ and event D is ‘the fuse blows’. From this perspective, events A and D happen at very nearly the same place in space but at different times; D is in the future of A. The diagonal lines heading upwards and outwards from A trace out the future light cone of A. We’ve also drawn diagonal lines heading out into the past from A. These are known as the past light cone of A. All events in the shaded region inside the future light cone are timelike separated from A, which means that anyone who is present at A could also be present at any event inside the future light cone. All events inside the past light cone are also timelike separated from A. This means that anyone who is present at any event inside the past light cone could also be present at A. The expression for the interval between A and D is particularly simple: (Δτ)2 = (Δt)2, where Δt is the time difference between A and D as measured by a watch in your house.¶¶
Figure 2.2. An event ‘A’ in spacetime and its neighbouring region. The diagonal lines are the lines traced out by beams of light that pass through A. They form the future and past light cones of event A.
We’ve also marked two other events on the diagram, labelled B and C. From the house perspective, these happen at the same time as event A, but in a different place. Let’s say they are an alarm clock going off at the other end of the street and a car starting its engine in the neighbouring town. The interval between A and B is (Δτ)2 = – (ΔxAB)2, and the interval between A and C is (Δτ)2 = – (ΔxAC)2. The interval is negative, which means that events B and C are spacelike separated from event A; ΔxAB and ΔxAC are distances that could be measured on a ruler.
Here is the key point. Event A caused event D (Granny knocked over a glass and that caused the fuse to blow). However, event A could not have caused events B and C. For that to happen, some influence would have to travel instantaneously from A to B and C because these things all happened at the same time. This delineating of causal relationships is why light cones are so important. Events inside each other’s light cones can have a causal relationship because it is possible that some signal or influence could have travelled between them. Events outside each other’s light cones cannot have a causal relationship. The interval therefore contains within it the notion of cause and effect. Certain events can cause others, and the light cones at each event tell us where in spacetime the dividing lines lie.
Let’s now look at the same events in spacetime from two different perspectives. Figure 2.3 is a spacetime diagram constructed using measurements of distance and time made by an observer moving at constant speed past your house towards the car in the neighbouring town. As we have already discussed, such an observer will measure different times and different distances between events, but the intervals between events must remain the same because the interval is invariant. Nature doesn’t care about your point of view, and the interval is a fundamental property of Nature. For this to be the case, something quite surprising happens. Events B and C happen after event A according to this observer.
Figure 2.4 shows a spacetime diagram constructed by an observer moving in the opposite direction at constant speed past your house. This observer says that events B and C happen before event A.
Figure 2.3. Events A, B, C and D as described in the text, as seen by an observer moving past event A at constant speed travelling from left to right on the diagram.
At first sight, it seems that the spacetime picture has led to disaster. How can we countenance a theory that allows for the reversal in the time ordering of events? What if those events were your birth and death? Would someone be able to see you die before you were born?
Figure 2.4. Events A, B, C and D as described in the text, as seen by an observer moving past event A at a constant speed travelling from right to left on the diagram.
The resolution to this apparent paradox can be seen by looking at the light cones. The light cones are in precisely the same place on all three diagrams, as they must be because all observers agree on the speed of light. Notice that although events B, C and D all move around on the spacetime diagram with respect to event A as we switch between different points of view, event D always remains inside the future light cone of A and events B and C always remain outside both the future and past light cones of A. To see that this must be the case, remember that the interval between two events is invariant: if the interval is timelike from one perspective, it’s timelike from all perspectives. This means that events that can influence each other have their time-ordering preserved from all perspectives. Events that can’t influence each other do not have their time-ordering preserved, but that doesn’t matter because it does not mess with cause and effect. There is no contradiction if someone sees a house alarm going off or a car starting in the next town before or after Granny knocks over the glass, because these events could never have influenced each other – they are spacelike separated. There would of course be a contradiction if the lights fused before Granny knocked over the glass which caused them to fuse. But that can’t happen for events A and D because D is always in A’s future light cone, regardless of the point of view.
The future light cone of an event therefore tells us which regions of spacetime are accessible from that event and which regions are forbidden. Likewise, the past light cone of an event tells us which events in spacetime could have possibly had any influence on that event. If you look back at the worldline in Figure 2.1, you’ll see that travelling to revisit moments in your past, the people and memories left behind, is impossible because we can never move from inside to outside the light cone at any event in our lives. To do so, we would have to travel faster than light. But the interval is invariant, so we can’t do that. In a sense, our memories are out there, somewhere in spacetime, but we can never revisit them.
The picture of spacetime we’ve described above is that contained in Einstein’s Special Theory of Relativity, first published in 1905. It describes a Universe without gravity, which is why we took the unconventional step of switching gravity off when discussing England versus Pakistan in Cape Town. Incorporating gravity into the spacetime picture is the concern of Einstein’s General Theory of Relativity, published in 1915.
From special relativity to general relativity
The central idea of general relativity is that spacetime has a geometry that can be distorted. As we’ll see, this corresponds to changing the rule for the interval between events. Matter and energy distort spacetime in their vicinity, and Albert Einstein worked out the equations that allow us to calculate how it is distorted. This is shown schematically in Figure 2.5. Objects like the International Space Station moving close to the Earth will be travelling through a region of distorted spacetime, and if we were Newtonians, we would interpret its motion as being due to a force deflecting it from a straight line and into orbit. But there is no force in Einstein’s picture: gravity is to be understood purely as geometry.