We can now add some more events. In August 1972, I was four kilometres away from Oldham Royal Infirmary in my paddling pool. The time on the Oldham watch at that point reads 4½ years. On 3 March 1989 I was in Florence, Italy, on a tour bus after playing a show with my band Dare while supporting the Swedish rock band, Europe. I know. That’s 21 years as measured on the Oldham watch, and I’m around 2000 kilometres from Oldham Royal Infirmary. One more. On 2 September 2009 I was in one of my favourite countries, Ethiopia, filming for Wonders of the Solar System at the Erta Ale lava lake with my friendly guard from the Afar tribe.
I could mark every event in my life this way, as measured by the time on the Oldham watches and the distance from Oldham Royal Infirmary. The resulting line on the spacetime diagram is called my worldline. It represents every moment in my life at the locations measured by the Oldham watches and the Oldham rulers. Remember Hermann Weyl’s evocative quote: ‘Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time’. This is what he meant.
There is another feature of the spacetime diagram that we must mention; the diagonal lines passing through 3 March 1968. These are known as a lightcone, and lightcones play a very important role in relativity. To understand what they are, imagine that someone decides to flash a beam of light out into the Universe from Oldham at the moment of my birth – perhaps in celebration, who knows? After one second, the light would have travelled a distance of 1 light second. We would mark the point in spacetime that the beam of light reached as an event at position 1 second x c on the time axis, and 1 second x c on the space axis. After 2 seconds the light beam would have travelled 2 light seconds, and so on. This light cone, therefore, is the worldline of a light beam that originates at the origin of the diagram – the event of my birth. It extends all the way across the spacetime diagram at an angle of 45 degrees. My worldline wanders around inside the lightcone, as it must because nothing can travel faster than the speed of light. To see this, look at the event marked ‘X’ on the diagram. It is something that happened far away. Let’s imagine that the event is a little alien boy on a planet 50 light years away, paddling in his swimming pool, and that, according to the Oldham watches, this event occurred at the same time as my pool adventure in 1972. We say that these events are simultaneous in the Oldham frame of reference. You’d have to travel much faster than light to get there if you were present at my birth – 50 light years in 4½ years, in fact, and this is not allowed because nothing can travel faster than light. You may have heard this many times, and wondered why. ‘It’s impossible to travel faster than the speed of light, and certainly not desirable as one’s hat keeps blowing off’, said Woody Allen. We’ll gain insight into why it’s not allowed in a moment.
The lightcone in the top half of the diagram is known as the future lightcone of my birth, because it marks out the region in spacetime that I could possibly visit or influence. I could not influence events outside of the lightcone in any way because I would have to travel faster than light to reach them.
There is also a lightcone in the bottom half of the diagram, which represents the time before my birth. This is called the past lightcone of my birth. My parents’ worldlines must be contained within the past lightcone, because they obviously influenced it. What’s more, every one of my ancestors’ worldlines, stretching back to the origin of life on Earth 4 billion years ago, must also be contained within the past lightcone. No events outside the past lightcone could have influenced my birth, because no signal could have made it from them to Oldham on 3 March 1968 without travelling faster than the speed of light.
Only to the gaze of my consciousness, crawling along the lifeline of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.
Let’s now ask a question. What does this diagram look like from the point of view of our intrepid French aviator, flying past at a constant speed? We already know that she would see the Oldham watches run slow and the Oldham rulers shrink, so she would place events on my worldline at different places on her spacetime diagram. For simplicity, let’s imagine that the aviator agrees to synchronise her watches with the Oldham watches at the moment of my birth, and agrees that my birth in Oldham Royal Infirmary also occurs at position zero on her space axis. In other words, the origins of the two diagrams coincide at t = 0.
My life events: Me in my paddling pool on that hot, sunny day in August 1972, and filming in Ethiopia on 2 September 2009.
The Spacetime diagram of my life from Albert perspective, sitting in Oldham Royal Infirmary.
The spacetime diagram from the perspective of the aviator, flying at high velocity in the +x direction relative to me. I’ve labelled the axis as ct’ and x’ to emphasise that the time and space co-ordinates of the events are different in the aviator’s frame of reference. The event at the origin – my birth – labelled 3 March 1968, remains at the origin because we agreed that both frames of reference have their origins coincident at t = t’ = 0.
Although the observer sitting diligently at Oldham Royal Infirmary will not agree with the aviator on the time difference and spatial distance between events on my worldline, they will both agree on the distances in spacetime between the events, given by Δs2 = c2Δt2 - Δx2. This means that Δx and Δt must change in a very specific way, such that Δs always remains the same. The aviator’s spacetime diagram is shown in the illustration opposite. Notice that the lightcones do not change, in accord with Einstein’s second postulate – both the aviator and the Oldham observer must agree on the speed of light. Now look at the position of the event that represents my twenty-first birthday. We know that the aviator’s clocks will tick at a different rate to the Oldham clock, and that the aviator’s rulers will be a different length to the Oldham rulers. But we also know that whatever distance and time difference she measures between the ‘3 March 1968’ event and my ‘twenty-first birthday’ event, they must obey the rule that Δs2 between the events remains the same. We’ve drawn all the possible positions of my twenty-first birthday on the aviator’s spacetime diagram as a curve. The actual position she marks will depend on how fast she flies by and in what direction. Here, we’ve assumed that she flies close to the speed of light in the direction of Oldham’s positive x direction. Something interesting is immediately obvious. My twenty-first birthday always stays in the future lightcone of my birth. This must be the case, because my birth caused my twenty-first birthday! We’d be in trouble if, from someone else’s point of view, my birthday drifted out of the lightcone of my birth and couldn’t have influenced it!
So far so good. Look now, however, at the event marked ‘X’ – the little alien boy in his paddling pool – that lies outside the lightcone of my birth. This event must also maintain its distance in spacetime from 3 March 1968, but to do that it has to move on a different curve. Crucially, it doesn’t have to stay in my future. For certain relative velocities between the aviator and Oldham, the event appears, from her perspective, to be in my past! This deserves an exclamation mark. The time-ordering of my birth and event X have been reversed from the perspective of the aviator. Is Einstein’s beautiful theory producing nonsense? Can it really be true that the time-ordering of events in spacetime is not agreed upon by all observers? Yes it is true, but this isn’t a problem, because event X always stays outside of my future and past lightcones. This means that my birth could not have influenced it, and it could not have influenced my birth. The two events are causally disconnected. This means that it doesn’t actually matter what time-ordering we ascribe to such events (which are called ‘spacelike separated’ events) because they cannot, even in principle, have anything to do with each other. Let’s give a specific example to make this clearer.
That summer’s day hasn’t happened yet. It’s out there in spacetime, in his future, albeit in a region of spacetime inaccessible to him.
Imagin
e that, at the exact moment of my birth in my frame of reference, a huge explosion occurred on the Sun. The Sun is eight light minutes away, which means that the explosion cannot influence anything on Earth for at least eight minutes, which is the time it takes a light beam to travel from the Sun to the Earth. These events are ‘spacelike separated’, so therefore an astronaut flying past us at high speed might see the explosion happen before, or after, my birth. The time-ordering would be changed. But who cares? What difference does it make? None at all, because the events cannot influence each other.
Notice, however, that after eight minutes the shockwave from the explosion could hit the Earth and destroy Oldham, which would, to use the local vernacular, piss on my chips. Remember, though, that we are talking about events in spacetime. My birth is an event, and the explosion is an event, and my birth is outside the lightcone of the explosion and therefore cannot be stopped by it. My unfortunate death eight minutes later is another event, and that event is in the lightcone of the explosion. Nobody will see the time-ordering of these events reversed. Events that are in each other’s past or future lightcones are known as ‘timelike separated’ events, and their ordering cannot be changed.
It is quite remarkable that everything works out, albeit in a rather subtle way. But there is a sting in the tail. Think about my birth event – ‘3 March 1968’ – and event X again. In the Oldham frame of reference, event X lies in my future. In another frame of reference, event X happens simultaneously with my birth, and in the aviator’s frame of reference it lies in my past. Events that happen simultaneously in one frame of reference are not simultaneous in another frame of reference. Whilst this doesn’t cause problems, as we’ve seen, it does raise an interesting question. If there is no clear distinction between the future and the past, and indeed if an event lies in someone’s future according to one observer and in their past according to another, then what do the concepts of future and past actually mean? When I was born, had event X happened or not? According to me, it hadn’t. According to the aviator, it had. This suggests that, in the theory of relativity, events have an existence in spacetime beyond our local concept of past, present and future.
Let’s make this more vivid. Recall that event ‘X’ represents a little alien boy playing in a paddling pool on a planet 50 light years away from Earth. In the Oldham frame of reference, this event happened simultaneously with my summer’s day in 1972. Now look at the illustration opposite, which shows how this event appears to the aviator travelling at high speed relative to me. There exist frames of reference in which the alien boy’s paddling pool day is in my past, and my entire life, including my paddling pool day, is in his future. My summer’s day hasn’t happened yet. It’s out there in spacetime, in his future, albeit in a region of spacetime inaccessible to him. From my perspective, my 1972 paddling pool day is in my memory. I remember it with fondness. Surely it’s gone, hasn’t it?
If we take Einstein’s theory of relativity at face value, there is no sense in which the past has happened and the future is yet to happen. A spacelike separated event can be in someone’s future from one perspective, and in their past from another. This doesn’t matter in the sense that such events can have no influence on each other, provided that nothing can travel faster than the speed of light. This is why the speed of light as a universal speed limit is so important in relativity. It protects cause and effect. But this behaviour does raise the question of whether all events that can happen and have happened in the history of the Universe are, in some sense, ‘out there’. This idea is known as the Block Universe. Spacetime can be pictured as a four-dimensional blob over which we move, encountering the events on our worldline as we go. We are forced to move over the blob at the speed of light, which from our own personal perspective means that we have to move into the future at a speed of 1 second per second.2 You have to get old because of the geometry of spacetime.
We should emphasise that, while the Block Universe is a consequence of relativity, it is not necessarily correct. We know that relativity is not fully consistent with quantum theory, and most physicists hope and expect that a quantum theory of spacetime will be developed at some point. Whether this will allow for a more intuitive picture of past and future is unknown. We must always remember that physical theories such as relativity are models of reality that produce predictions that agree with experiment – a test which both the Special and General theories have passed with flying colours for over a century. Is the Block Universe actually real, or just an artefact of Einstein’s model? Who knows? But I think its implications are at the very least worth thinking about. On the downside, there is no free will in the Block Universe. All the events in our future ‘exist’, waiting for us to barrel along our worldline to intersect them. I don’t care personally whether I have free will or not. It makes no difference to me. But I find the other side of the coin quite wonderful. In the Block Universe, the past is also out there. My idyllic summer’s day in 1972, with my Mum and Dad and sister, doesn’t exist only in my memory. It hasn’t gone, although I can never revisit it. It is still there; all those people, all those moments, always and forever, somewhere in spacetime. I love that.
Spacetime calculations
Monet and the aviator
Monet and the aviator.
We can use Einstein’s two postulates to show why it is that the aviator and Monet measure different intervals of time between any pair of events. This is surely one of the most bizarre ideas ever to come out of a human being’s head. It is all the more bizarre for being demonstrably correct. The argument is surprisingly simple. First let us imagine a special type of clock – at the end we will show that the argument must work for any type of clock, but for now we will consider a ‘light clock’. A light clock is made up of two parallel mirrors with a beam of light bouncing back and forth between them. Suppose that the two mirrors are a distance d apart. If light travels at a speed c it will take a time t=2d/c for the light to travel from one mirror to the other and back again, as determined by someone who is holding the clock (more formally, we might say ‘by someone who is at rest relative to the clock’). Let us refer to the person holding this clock as (and here we will not bother exercising our imagination) ‘person A’. Now let’s introduce a second person: ‘person B’. If person A and person B are both at rest relative to each other then both will clearly agree on how long the light clock takes to tick (let’s call one tick of the clock the time it takes for the light to make one round-trip, i.e. t=2d/c). Pre-Einstein, and according to common sense, we’d say that the clock takes this time t to tick, regardless of what it is doing or who is doing the measuring. But that is wrong, as we are about to show.
Hyperbola.
To see how time is not absolute, let’s put person A and their clock on a train (Einstein often used trains to explain his theories), and person B on the platform. Now let us consider how the clock is understood by person B. The top illustration shows the path taken by the light as it makes one tick of the clock.
According to person B, the clock moves a distance equal to υt' in one tick, where υ is the speed of the train and t' is the duration of the tick. At this stage we will resist the temptation to say that t' (the time of one tick of the light clock according to person B) is equal to t (the time of one tick according to person A). From the figure we can see the path that the light beam traces out as it moves up and down. Obviously, the light travels further according to person B than it does according to person A. Using Pythagoras’s Theorem, the distance the light travels according to person B is whilst for person A it is just 2d (notice that it would be just 2d for person B if υ=0, i.e. if the train isn’t moving). The fact that the light travels further according to person B is not by itself anything to get excited about, because the train is moving. The next step is the shocker.
Einstein’s second postulate states that the speed of light in a vacuum is the same in all inertial frames of reference. It follows that person B must agree that the light moves at a s
peed c. If the light moves at speed c according to both A and B, and if the light travels further according to person B than it does according to person A, then it follows that the light must take longer to make the round trip according to person B than it does according to person A. This is worth re-reading and thinking about, because it is surprising.
We have just proven that, if Einstein’s second postulate is correct, it logically follows that the light clock ticks more slowly according to person B (who is on the platform) than it does according to person A (who is on the train). Since we went to the trouble of invoking Pythagoras and a little algebra to write down how far the light travels in one tick according to person B, we can easily write down by how much the moving clock slows down according to person B. The time taken for one tick, according to person B, is the distance the light travels in one tick, divided by the speed of light c, i.e. Notice that the time we want to know (t') is on both sides of this equation, which means we have to re-arrange the equation using some low-level algebra. Squaring both sides of the equation gives t'2=4(υt'/2)2/c2+4d2/c2, which can be re-arranged to read t'2(1–υ2/c2)=4d2/c2. Now we can write down what t' is in terms of υ, d and c. It is just And since t=2d/c we can write down that And that is our final answer. So long as υ is smaller than c, the square root makes sense and t' is always bigger than t, which means that the person on the platform must conclude that the person on the train is holding a clock which is taking longer to tick than it would if the clock were not moving. As an aside, the factor appears very often in relativity, and is known as the Lorentz factor or Gamma factor, and given the symbol γ.
Forces of Nature Page 13