There are two possible resolutions to this bizarre feature of Maxwell’s equations. The obvious one would be to modify Maxwell’s equations so that this is not the case, and light behaves like a cricket ball. Ultimately this is an experimental question; a question about what actually happens in Nature. Innumerable observations of disparate physical phenomena for well over a hundred years tell us that Maxwell’s equations are correct as they stand and therefore light always travels at the same speed.
The other, less obvious, resolution is to change the way that observers travelling at different speeds relative to each other account for distances and time differences such that everybody always measures the speed of light to be the same. Einstein chose this route, thus rejecting Newton’s notions of absolute space and time, and this choice led him to relativity.
Einstein’s theory of relativity
Einstein’s theory is a model, which is to say it is a mathematical framework that allows us to make predictions about how objects that exist in the natural world behave. The model is inherently geometrical, which lends itself to intuitive visual pictures which require very few equations – a good thing for a book such as this. We believe that the best approach to explaining relativity is to describe this geometrical picture, rather than attempt to present its evolution historically. Our justification is that the model works, and that is the only justification necessary. Einstein could have simply plucked his theory out of thin air without any reference to Maxwell’s theory or experiments, and it would be equally valid because it is a good model in the sense that its predictions have passed every experimental test to date.
If Einstein could have plucked one single idea out of thin air that would have led him directly to his theory, including the explanation of what happened in Hafele and Keating’s experiment and the most famous equation in all of physics, E = mc2, it would be a concept known as ‘the spacetime interval’. The idea is beautifully simple.
Let’s return to Pakistan against England in Cape Town and Shoaib Akhtar’s record-breaking delivery to Nick Knight. We are going to simplify things for now by switching gravity off – we’ll switch it back on at the end of this chapter. This means that when the ball leaves Akhtar’s hand it will travel to Knight in a perfect straight line at a constant speed – 100.2 mph relative to the ground.‡ Let’s further imagine that the cricket ball has a clock inside. At the moment the ball leaves Akhtar’s hand, the ball emits a flash of light and records the time on its internal clock. At the instant the ball reaches Knight’s bat, the ball emits another flash of light and records the time of arrival on its internal clock. We’ll call the time interval between the flashes as measured on the cricket ball clock Δτ – pronounced delta tau.
In the commentary box, Jonathan Agnew (Aggers), for the BBC, notes the arrival of the two flashes of light from the ball and calculates the time interval between the emission of the flashes from his point of view: ΔtAggers.§ He also measures the distance between the place where the ball leaves Akhtar’s hand and the place where the ball hits Knight’s bat: ΔxAggers.
In his Grumman F14 Tomcat flying over the wicket in a straight line between the stumps at 600 mph Tom, the pilot, also notes the two flashes of light and calculates the time interval between the emission of the flashes from his point of view: ΔtTom. Like Aggers, he also measures the distance between the place where the ball leaves Akhtar’s hand and the place where the ball hits Knight’s bat: ΔxTom.
The Hafele and Keating result tells us that the time differences between the emission of the flashes as measured by Aggers, Tom and the cricket ball will all be different. Likewise, the distance the ball travelled from bowler to batsman will also be different. For those who have never encountered Einstein’s ideas before, these differences should come as a tremendous shock. They are counter-intuitive because they mean that distances and time intervals are not something everyone can agree upon. However, here is a remarkable and important result. If Aggers calculates the quantity (ΔtAggers)2 – (ΔxAggers)2 and Tom calculates the quantity (ΔtTom)2 – (ΔxTom)2 then they will both get the same result, and the result will be equal to the square of the time interval measured using the cricket ball clock, (Δτ)2:
(Δτ)2 = (ΔtAggers)2 – (ΔxAggers)2 = (ΔtTom)2 – (ΔxTom)2
(Δτ)2 is known as the spacetime interval between the two events: event 1 is the ball leaving the bowler’s hand and event 2 is the ball striking the bat. You may well ask: ‘What does it mean to subtract a distance in space squared from a time difference squared?’ The answer is that we must specify the distance between two events as the time it takes for light to travel between those events, which means we should compute the distance in light seconds. The spacetime interval (or ‘interval’ for short) is important because it is a quantity on which everyone agrees, no matter what their point of view. In physics we call such a quantity an invariant. Since Nature doesn’t care about our point of view,¶ we should only seek to describe Nature in terms of invariant quantities. When we discover an invariant, it is a big deal because we learn a little more about the essential structure of the Universe.
In their book Exploring Black Holes, Taylor, Wheeler and Bertschinger describe the equation for the interval as ‘one of the greatest equations in physics, perhaps in all of science’. Kip Thorne and Roger Blandford, in Modern Classical Physics write that the interval is ‘among the most fundamental aspects of physical law’. The word ‘fundamental’ is important. You might reasonably ask: ‘Why is the interval like this?’ ‘Why does everyone agree on this particular combination of time and space?’ The answer, as Thorne and Blandford imply by their use of the word ‘fundamental’, is that this is the way the Universe is constructed. We know of no deeper explanation for the form of the interval.
A further question you may be asking is: ‘How should I think about the interval, this most fundamental aspect of physical law?’ This is a good question. Physicists usually endeavour to develop a mental picture of what’s happening in their equations: physical intuition brings equations to life. Fortunately, the interval does have a simple physical interpretation. It is related to what we will refer to as ‘the distance between two events’. This is not the usual distance between them in space, but the distance in spacetime. Let’s explore that idea.
Events and worldlines in spacetime
The concept of an event is fundamental to relativity. An event is something that happens somewhere and somewhen. You clicking your fingers is a good approximation of an event: it happens very quickly and in a well-defined location. The emission of a flash of light from our cricket ball is an event. Strictly speaking an event is an idealised concept, something that happens so fast and in such a small area that it corresponds to a single point in space and time. The theory of relativity is concerned with the relationship between events; how far apart they are in spacetime and whether they influence each other or not. This is a very intuitive way to think about the world, so much so that it’s how we speak in everyday life; ‘I’ll meet you tomorrow evening at eight o’clock at the pub.’ ‘I was born on 3 March 1968 in Oldham.’ Things that have happened to us and things that will happen to us are all events in space and time, and they happen somewhere and somewhen. A slight shift in wording, and we have the basis of the theory of relativity: things that have happened to us and things that will happen to us are all events in spacetime. What is spacetime? It’s the collection of all events. Everything that has ever been and will ever be in the Universe.