The Chase

They cannot escape! Our time machine will soon follow them. They will be exterminated! Exterminated! Exterminated!

So… how exactly do you chase someone through time and space?

There’s a quite straightforward way of going about this, which is unfortunately entirely unlike what we actually see in this story. However, it’s worth looking at anyway, as it will illustrate some ideas that will be useful for understanding the more complicated answer later on.

Space and time are not separate things – they are combined into a four-dimensional environment called spacetime. (Not the most imaginative name, I’ll admit.) As you travel through spacetime, you trace out what is called a world line, which is just a trajectory consisting of all the points you move through. Even sitting on your backside reading this blog you trace out a world line: as you orbit round the Sun and move forward in time, your worldline is a segment of a spiral.

Worldline of the Earth (red) orbiting the Sun (blue). Time is on the vertical axis. (Image credit: www.cleonis.nl)

Worldline of the Earth (red) orbiting the Sun (blue). Time is on the vertical axis. (Image credit: http://www.cleonis.nl)

So it seems that chasing someone through time is much the same as chasing someone through space. You just need to find the path they’ve taken – their worldline – and follow it a bit faster.

Except that isn’t really going to work, is it? You can’t go faster than me and still stay on the same worldline, as the worldline is defined by the speed of my movement through spacetime. Change your velocity, and you start travelling along a different worldline.

And anyway, a worldline isn’t like a trail of footprints. If a point on my worldline is “At the front door of the British Museum, 2:30 pm on Sunday 2 March 2014”, then if you go to that point on my worldline there’s no need for any chasing – I’m already there, that’s what a worldline means. You’ve got me, and the chase is over before it has even begun.

No, the way to catch me is to find out some point in spacetime where I am going to be, and arrange to be there at the same time. In other words, to make our worldlines intersect. If, for example, you know I am in the habit of popping into the British Museum around 2:30 on a Sunday afternoon, you hang around there as unobtrusively as possible until I show up. No chasing involved.

So that about wraps it up for The Chase. Except… what we’ve just described isn’t how the Tardis travels through time and space at all. Far from tracing out a single worldline in four-dimensional spacetime, it disappears here and appears there, travelling in between through some exotic other space, sometimes called a vortex.

To get an idea of how that might work, look back at the post on the very first episode, An Unearthly Child. There we saw the idea that two points in four-dimensional space-time can be connected by a five-dimensional tunnel. (Admittedly this involves manipulations of unknown complexity with forms of exotic matter that are not currently known to exist, but we can presume the Doctor’s people have long since mastered such implementation details.) The Tardis travels in space and time, not by following a continuous worldline from one spacetime event to another, but by creating a corridor through the fifth dimension and taking a short-cut through that.

So can the Daleks chase the Tardis through the same 5-d tunnel? Again, this brings up the same problem as in the 4-d case. The opening to the tunnel is a spacetime event, and to enter the tunnel you have to go through that spacetime event. As Tardis and its crew are also at that event, you’ve already caught them. Or to put it another way, the tunnel presumably opens and closes just long enough for the Tardis to vanish into it and zoom off to its new destination. If the Daleks turn up five minutes later, they’re too late and just have to whirl around bitching at each other.

Or do they? Perhaps there’s something extra we can add to this model that makes The Chase plausible [1] after all.

Any change in the geometry of spacetime creates ripples – usually called gravitational waves. These are very faint, and have only ever been observed indirectly. Some exquisitely sensitive instruments have been built over the last couple of decades to try to detect these waves as they pass across the Earth, with a view to seeing the signs of distant astronomical events, like the collision of black holes, that are invisible to telescopes. So far these efforts have been unsuccessful, but physicists continue to strive.

It’s reasonable to suppose that opening up this 5-d tunnel would also create such waves. Indeed, it would be surprising if it did not. We don’t know anything about this fifth dimension that the Tardis apparently travels through, but let’s assume it’s geometrically well-behaved, and that opening this corridor through it similarly creates ripples in the fifth dimension. The Doctor’s Time Path Detector can evidently register these disturbances, and presumably the Daleks have something similar.

Now we have all we need. The Daleks can detect these ripples, and set up their own 5-d corridor close to the one the Tardis is using. These ripples will die off the further away they get from their source, just like the ripples on a pond when you chuck a stone in, so the Daleks will want to keep their corridor as close as possible to the Doctor’s, so that they can be sure of arriving close to the end-point of his journey. For practical purposes, it’s natural to focus on the time displacement between the two end points, although obviously there is some spatial displacement as well, hence the Doctor’s repeated remarks that the Daleks are only so many minutes behind them. The more often the Daleks do this, the more precisely they will be able to calibrate their instruments, the more they can minimise the displacement between their arrival point and the Doctor’s – and so they effectively catch up with every trip the Tardis takes. It all seems to fit quite nicely.

There’s just one problem. In our model, there’s nothing to prevent the Daleks turning up a few minutes before the Tardis, and setting up an ambush, ready to exterminate our intrepid time travellers the moment they step out of their box. So why don’t they?

Perhaps the answer can be found, not in relativity, but in quantum mechanics.

One of the most strange and perplexing issues in quantum mechanics – the theory that describes the behaviour of microscopic objects such as atoms, electrons and so on – is what’s called the quantum measurement problem. It goes something like this.

The reason why quantum mechanics is called quantum mechanics is that, when you get right down to the microscopic level, variables like energy and spin that describe a particle’s motion cannot take on a continuous range of values, but can only have a limited number of very precise values. It’s as if your car could go at 40 or 50 miles per hour, but not any speed in between – and if you wanted to accelerate you would have to jump instantly from 40 to 50 without ever going at 41, 42 or 43 miles per hour. These discrete units were historically called quanta, hence quantum mechanics.

Now, whenever you measure a quantum mechanical system, you find it in one or other of its possible quantum states. In our quantum car, whenever you look at the speedometer you find it sitting at 40 or 50 mph. But very often systems exist in some mixture of the possible states. Our car might, for example, be in a state of one half 40 mph, one half 50 mph. That doesn’t mean the speedometer shows 45. It means that, on average, half the time you look at the speedometer you see it reading 40, the other half of the time you see it reading 50.

Now if the car goes past a speed camera set to trigger if the car is going over 45 mph, you will get a speeding ticket half the time, on average – as long as you don’t look at the speedometer. If you do look at the speedometer just as the car enters the speed trap, if you see it reading 40 you definitely won’t get a ticket, if you see it reading 50 you definitely will.

This is a bit weird, but it has been confirmed by a multitude of precise experiments, so it’s the way the world is. (Experiments on beams of electrons and the like, not improbable motor vehicles.) Somehow, when we measure a quantum system, we change it, forcing it out of its mixed state and into one of its pure states.

Exactly what is going on here has intrigued and baffled physicists and philosophers for more than eighty years. One interpretation – the most orthodox – has it that the act of measurement itself causes the quantum state to randomly assume one of its pure values. Looking at the speedometer forces the car to be going at either 40 or 50 mph, and then you either get a ticket or you don’t. Another interpretation, known as the many-worlds interpretation, holds that all the possible results of the measurement happen simultaneously, but in different universes, with each act of measurement spawning a new set of parallel worlds. Looking at the speedometer results in two versions of you in two versions of the car, one going at 40, the other at 50, and in one of these worlds you get a speeding ticket while your more fortunate self in the other world drives on without a care in the world. Lucky bastard.

We’ll have a lot more to say about parallel worlds in future episodes. For now, suffice to say that, in the universe of Doctor Who, the many-worlds interpretation is definitely correct. And since quantum interactions are happening all the time, every instant is spawning new sets of alternate universes.

So what does this lengthy digression have to do with The Chase? Well, remember the question we had a few paragraphs back – why don’t the Daleks arrange to arrive a few minutes before the Doctor and his chums and ambush them? The answer is now clear. For every set of universes in which the Tardis arrives at a particular place and time, there is a set of universes in which it does not. The Daleks have to arrive after the Doctor so they can be sure he is going to arrive at all. Otherwise they risk ending up in one of those other universes, hanging around in awkward silence until the embarrassment gets too much and they pile back into their spacetime craft to give it another go.

So there you have it. Should you ever find yourself tasked by an implacable manager with chasing after an old git in a phone box that can travel anywhere in time and space, I hope you will find this elementary guidance useful. No warranty is expressed or implied. Good luck.

1. You know what I mean.

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3 thoughts on “The Chase

  1. I’ve encountered almost all these concepts before, but never in such a clear and (at least seemingly) comprehensible way. Bravo! And I could stare at that GIF for ages.

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