Time Travel Paradoxes

Introducing Grandad
With all the excitement over the possibility of faster-than-light (FTL) neutrinos, every so often someone mentions that FTL opens up the possibility of time travel and this leads to the Grandfather Paradox. Hence the whole thing is impossible.

In case anyone has been living on Pluto until now, the Grandfather Paradox is this. If I go back in time and kill my own grandfather before my parents meet, then I don’t get born. But then I don’t grow up and decide to travel back in time. So I don’t get to shoot my grandfather. Hence I both kill and don’t kill my grandfather. Which is a paradox.

This paradox creates a fair amount of heated discussion considering that time-travel is purely science fiction. However there is no obvious logical reason why it should not be possible, regardless of whether physics allows it. Of course “I kill and I don’t kill” is no problem to quantum physicists, who never object to systems being in a superposition of states. However, the resolution of the paradox does not depend on quantum mechanics.

Nor, you will be relieved to know, does it depend on arbitrary “Timey-Wimey” stuff from Doctor Who or “alternative time-lines” which seem to be taken for granted in most discussions. The only deviation from simple, known, physics will be the time-travel itself.

Let’s begin with a picture. I hop into the TARDIS, set off and arrive in my own past. Whoosh!



I walk out of the TARDIS and, by a stroke of luck, there’s my old Grandad (whom I always hated) back there in the prime of his life. I see red! I reach for my gun. BANG! And my grandfather lies dead. Dead men don’t beget grandchildren. Which means I cannot exist. Oops.

Self-Consistency
This is self-contradictory. If time-travel exists and I am going to refrain from ad-hoc science fiction, the only thing left is to look deeper into self-consistency.

According to the omniscient Wikipedia, ”Stated simply, the Novikov consistency principle asserts that if an event exists that would give rise to a paradox, or to any "change" to the past whatsoever, then the probability of that event is zero”.

This can probably be proved formally in some areas of physics but it is basically common-sense.

The Unchangeable Past
In addition, we cannot just assume I am free to kill when I get there. I am free to try, but we can’t assume I’ll succeed, as that is the very point of the paradox. All we can reasonably say is “What happens if someone goes back in time and tries to kill their own grandfather?”

So, the smarty-pants answer is “Therefore we know that if you do try, something always stops you. Perhaps you miss. Or your grandfather just happens to have a coin in his breast pocket which stops the bullet. Or perhaps he makes an amazing recovery. One way or another, he won’t be killed.”

This seems to be enough for some people.

The trouble is, it means that every time anyone goes into the past and tries to change it, something will stop them. But that, of course, sounds just as crazy as the original paradox. Why on earth should unlikely things happen automatically every time? You may as well invoke magic – a big hand will come out of the sky and push the gun to one side.

Logic
Let’s see what happens when we try to create a system that contradicts itself without time-travel. Here’s some simple electronics. Digital circuitry uses the “language” of logic. A high voltage (say 3v) represents “true” and a low voltage (say 0v) represents “false”. The paradox is therefore like the following circuit.


The triangle and circle symbol are what is called an inverter: "high" (voltage) in produces "low" out and vice versa. Or in logical terms:
OUTPUT = NOT INPUT
The connection completes the paradox by forcing the input to be the same as the output.
Thus OUTPUT = NOT OUTPUT

This is an exceedingly dangerous circuit! According to received wisdom, if we construct a paradox, the whole of spacetime will collapse into a chaotic black hole. Well, engineers make circuits like this all the time – frequently by accident. We’re still here. In reality it settles to a value midway between high and low. (It may also oscillate but that’s not relevant here!) This is shown in the graph.


The lazy Z is the characteristic of the inverter, the diagonal line is the equation “in=out”. They are both true at a single point where they cross. The universe does not disintegrate.

Knife Edge States
The Grandfather Paradox can be resolved the same way. There will be some state half-way between what the paradox assumes are the only two possibilities. Half-killed? Half-born? It sounds crazy but it’s the right answer and I’ll show you how it all works.

First, a simplification. Here are two particles bumping into each other. If you don’t play billiards or snooker, let me explain what happens in this “ideal” world. If A hits B head-on then B shoots forward. If A hits slightly to the right, B goes off to the left. And vice versa. (Thank you, Helen Barrett, for correcting my orginal diagram!)


In the next diagram, some cruel physicist has decided to play a trick on the particles. He knows that by turning on the Star Trek tachyon field, particle A is slightly affected by the outcome of the collision, a tiny bit of backward causation. Get things right and, if the particle veers off to the left, backward causation can force it to the right. And vice versa. Left implies right implies left. Paradox!

By now it should be obvious what’s going to happen. The influence from the future is simply feedback. The system will find a self-consistent state. No magic: just the system settling instantly to a fine balance.

Of course such a precise set-up would be impossible to achieve with plain engineering. But add backwards causation and the system is interacting with its own future state. This is feedback very similar to the inverter. The only possible outcome is the one that the feedback automatically and painlessly creates: the finely balanced “knife-edge” state that is consistent all round.

Macroscopic Systems
So a simple system seems to permit a nice self-consistent resolution. What about big systems?

Any system has a certain number of degrees of freedom – for a macroscopic system there are several associated with each particle so the list can grow very long. But as a result, the state can be represented by a vector in a space of many dimensions called phase space. The laws of physics are deterministic in the classical model (and even in some interpretations of QM), so the system evolves from one point in phase space to the next in a predictable manner.

The whole succession between that fateful day 60 years ago when I try to shoot my grandfather can thus represented as a point moving through phase space; in other words, a line.

The next diagram starts as the ”normal” time-line in red/orange. It goes through a region of phase space in which I am definitely born. The slight waviness I have given it is a concession to the fact it is actually thrashing about in something like a googol dimensions, not just one.

Now we need to add my TARDIS. The TARDIS operates at two points in time, the point at which it takes off, and the point at which it arrives. When it arrives I divert the trajectory into the “Me not born” region in which there is no TARDIS – at least none with me at the helm.

However, this paradox is not the complete picture: there is no allowance for my aim being affected. There is no account of the many other possibilities that depend on precisely what “footprints” I leave behind in addition to actually killing or not killing Grandad. The slightest trace I leave behind grows exponentially with time. This is commonly called the Butterfly Effect.

You can't even be sure I’ll behave myself if I’m not born! A single molecule of my breath “back then” might result in atoms leaping into the TARDIS through a thermal fluctuation, assembling a killer me as they do. Unlikely? Oh yes. Impossible? No. And we need to consider all possibilities. So here they are. The outcome is a collection of possible state trajectories before we add the requirement for consistency.


A Map
The diagram shows a slice through the collection as a blue line. In the next diagram the requirement for consistency is shown by examining the slice in detail. I do this by drawing the same slice twice in order to separate the A side (past facing) and the B side (future facing). I have deliberately shown them in the reverse order, B before A, not in order to make it hard to follow but so that all the shenanigans of time-travel and shooting Grandad can be summarised by a “map” between the two versions of the slice. This is drawn as the criss-crossing arrows.


The arrow from the top states – which would normally be extremely likely to evolve to “me born” – shows the paradoxical mapping of most “me born” states to “me not born”. Likewise the arrow from the bottom “me not born” states maps to the paradoxical “me born”. Nearly all states create paradox! After all, there’s nothing special about killing Grandad, one atom out of place and there’s a contradiction. The blue arrow represents a "born" state that does map to "born". That's not in itself contradictory. However, for proper consistency even that is not enough. The only states that are possible are ones which map precisely to themselves, like the thick black line.   

Self–Consistency Again
So the big question is, can we be sure that there are points on the slice that map to themselves?

If the phase space were one-dimensional it would be easy. If “born” maps to “not born” and vice versa there would be a value which maps to itself. We saw this with the transfer function for the inverter. The system automatically settles to an intermediate value between two extremes. This is why the two-particle system is easy to accept intuitively. It’s obvious the system can find a believable solution. And it can be proved for physically realistic functions. It’s called the Intermediate Value Theorem.

It’s not so obvious for a system with such a high dimensionality as Grandad’s phase space.
In fact there is no Intermediate Value Theorem for vectors. Nevertheless I believe the existence of the required self-mapping states can be demonstrated for physically realistic states.
 
Johannes Koelman has kindly suggested that Brouwer's fixed point theorem is the proof I require.

Needless to say, the Wikipedia article is a mass of impenetrable mathematical jargon though the introductory statement is encouraging: "for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. Of course, the "certain properties" is a little worrying, but I think we can take Johannes' word that it applies here. Interestingly, there is no mention of time-travel at all!  

This, then, is the solution to the Grandfather Paradox. The system must find a self-consistent solution and the phase-space model suggests that such solutions do actually exist. They may be very unusual, "unlikely freak events" according to our everyday reckoning, but the fact that the trajectory is joined up by the time travel means that it is constrained at two points, one in the past as is normal, but also one in the future. This is so important that it will bear repeating:

The trajectory is constrained at two points, one in the past and one in the future. Ordinarily, the state is free to go wandering off unconstrained but with time travel it has to come back to one particular state - the very state where it started. This alters the range of possibilities beyond all recognition.

Finally, these very freaky trajectories - what would they be like? Well some of them would simply mean that although Grandad is named as my biological grandfather, in fact there was another man somewhere. Other, less prosaic, possibilities include very rare thermal fluctuations that bring him back to life again or even assemble a complete "me" ready to jump into the TARDIS.

Why on earth should such stupidly unlikely events happen? Well, they're only "unlikely" in a freely evolving state trajectory!. If the trajectory is pinned down in the future then the highly unlikely events are simply the result of the trajectory being forced to return to its previous state. The system has no choice but to undo all the Butterfly Effects right down to the microscopic level. The physical means for this to happen is that I carry my microstate back in time and impose it on the system I find there.