Does faster-than-light travel lead to a grandfather paradox?

Whatever you do, don’t f*ck with mom.

Fast track to wisdom: Not necessarily.

I stopped going to church around the same time I started reading science fiction. Because who really needs god if you can instead believe in alien civilizations, wormholes, and cell rejuvenation? Oh, yes, I wanted to leave behind this planet for a better place. But my space travel enthusiasm suffered significantly once I moved from the library’s fiction aisle to popular science, and learned that the speed of light is the absolute limit. For all we know. And ever since I have of course wondered just how well we know this.

Fact is we’ve never seen anything move faster than the speed of light (except for illusions of motion), and it is both theoretically understood and experimentally confirmed that we cannot accelerate anything to become faster than light. That doesn’t sound good for what our chances of visiting the aliens are concerned, but it isn’t the main problem. It could just be that we haven’t looked in the right places or not tried hard enough. No, the main problem is that it is very hard to make sense of faster-than-light travel at all within the context of our existing theories. And if you can’t make sense of it, how can you build it?

Special relativity doesn’t forbid motion faster than light. It just tells you that you’d need an infinite amount of energy to accelerate something which is slower than light (“subluminal”) to become faster than light (“superluminal”). Ok, the infinite energy need won’t fly with the environmentalists, I know. But if you have a particle that always moves faster than light, its existence isn’t prohibited in principle. These particles are called “tachyons,” have never been observed, and are believed to not exist for two reasons. First, they have the awkward property of accelerating when they lose energy, which lets them induce instabilities that have to be fixed somehow. (In quantum field theory one can deal with tachyonic fields, and they play an important role, but they don’t actually transmit any information faster than light. So these are not so relevant to our purposes.) Second, tachyons seem to lead to causality problems.

The causality problems with superluminal travel come about as follows. Special relativity is based on the axiom that all observers have the same laws of physics, and these are converted from one observer to another by a well-defined procedure called Lorentz-transformation. This transformation from one observer to the other maintains lightcones, because the speed of light doesn’t change. The locations of objects relative to an observer can change when the observer changes velocity. But two observers at the same location with different velocities who look at an object inside the lightcone will agree on whether it is in the past or in the the future.

Not so however with objects outside the lightcone. For these, what is in the future for one observer can be in the past of another observer. This means then that a particle that for one observer moves faster than light – ie to a point outside the lightcone – actually moves backwards in time for another observer! And since in special relativity all observers have equal rights, neither of them is wrong. So once you accept superluminal travel, you are forced to also accept travel back in time.

At least that’s what the popular science books said. It’s nonsense of course because what does it mean for a particle to move backwards in time anyway? Nothing really. If you’d see a particle move faster than light to the left, you could as well say it moved backwards in time to the right. The particle doesn’t move in any particular direction on a curve in space-time because the particles’ curves have no orientation. Superluminal particle travel is logically perfectly possible as long as it leads to a consistent story that unfolds in time, and there is nothing preventing such a story.

Take as an example the below image showing the worldline of a particle that is produced, scatters twice to change direction, travels superluminally, and goes back in time to meet itself. You could interpret the very same arrangement as saying you have produced a pair of particles, one of which scatters and then annihilates again.

No, there is no problem with the travel of superluminal particles in principle. The problems start once we think of macroscopic objects, like spaceships. We attach to their curves an arrow of time, pointing into the direction in which the travelers age. And it’s here where the trouble starts. Now special relativity indeed tells you that somebody who travels faster than light will move backwards in time for another observer, because a change of reference frame will not reverse the travelers’ arrow of time. This is what creates the grandfather paradox, in which you can travel back in time to kill your own grandfather, resulting in you never be born. Here, requiring consistency would necessitate that it is somehow impossible for you to kill your grandfather, and it is hard to see how this would be insured by the laws of physics.

While it’s hard to see what conspiracy would prevent you from killing your grandpa, it is fairly easy to see that closing the loop backwards in time is prevented by the known laws of physics. We age because entropy increases. It increases in some direction that we can, for lack of a better word, call “forward” in time. This entropy increase is ultimately correlated with decoherence and thus probably also with the restframe of the microwave background, but for our purposes it doesn’t matter so much exactly in which direction it increases, just that it increases in some direction.

Now whenever you have a closed curve that is oriented in the direction in which the traveler presumably experience the passage of time, then the arrow of time on the curve must necessarily run against the increase of entropy somewhere. Any propulsion system able to do this would have to decrease entropy against the universe’s thrust of increasing it. And that’s what ultimately prevents time-travel. In the image below I have drawn the same worldline as above with an intrinsic arrow of time (the direction in which passengers age), and how it is necessarily incompatible with any existing arrow of time along one of the curves, which is thus forbidden.

There is no propulsion system that would be able to produce the necessary finetuning to decrease entropy along the route. But even if such a propulsion existed it would just mean that time in the spaceship now runs backwards. In other words, the passengers wouldn’t actually experience moving backwards in time, but instead moving forwards in time in the opposite direction. This would force us to buy into an instance of a grandfather pair creation, later followed by a grandchild pair annihilation. It doesn’t seem very plausible, and it violates energy conservation, but besides this it’s at least a consistent story.

I briefly elaborated on this in a paper I wrote some years ago as a sidenote (see page 6). But just last month there was a longer paper on the arxiv, by Nemiroff and Russell, that studied the problems with superluminal travel in a very concrete scenario. In their example, a spaceship leaves Earth, visits an exoplanet that moves with some velocity relative to Earth, and then returns. The velocity of the spaceship at the both launches is the same relative to the planet from which the ship launches, which means it’s a different velocity on the return trip.

The authors then calculate explicitly at which velocity the curves start going back in time. They arrive at the conclusion that the necessity of a consistent time evolution for the Earth observer would then require to interpret the closed loop in time as a pair creation event, followed by a later pair annihilation, much like I argued above. Note that singling out the Earth observer as the one demanding consistency with their arrow of time is in this case what introduces a preferred frame relative to which “forward in time” is defined.

The relevant point to take away from this is that superluminal travel in and by itself is not inconsistent. Leaving aside the stability problems with superluminal particles, they do not lead to causal paradoxa. What leads to causal paradoxa is allowing travel against the arrow of time which we, for better or worse, experience. This means that superluminal travel is possible in principle, even though travel backwards in time is not.

That travel faster than light is not prevented by the existing laws of nature doesn’t mean of course that it’s possible. There is also still the minor problem that nobody has the faintest clue how to do it... Maybe it’s easier to wait for the aliens to come visit us.