### Faster than a speeding photon

### How to outrun a photon

I thought it would be fun to try to give a readable account of Unruh effect. It's a surprising phenomenon, and there isn't universal agreement over what exactly the theory predicts, let alone whether the effect has ever been observed. It has important implications for physics and philosophy and may even give a way to test some aspects of quantum gravity in the lab.

One way to start the story is consideration of this problem: if a photon is speeding towards you, can you outrun it? Let's simplify things a bit so that we're considering motion in one dimension.

If we're confined to one dimension, we can't dodge the photon, we can only hope to remain ahead of it. As the only things that can travel at the speed of light, c, are massless things like photons, it seems that there is no hope for a massive thing like a person in a spaceship to avoid it. The photon will always be faster than you, and so it'll catch you.

But in theory you can outrun a photon! Do you see the flaw in the above reasoning that made it seem impossible?

The best way to make things clear is to draw a diagram. We'll plot some graphs of position vs. time for some photons and spaceships. We'll have time going up the vertical axis and position along the horizontal axis. Here's an example:

I've chosen units so that one second on the vertical axis is drawn the same size as one light-second on the horizontal axis. The net result is that photons always travel at 45 degree angles to the axes. Massive objects, that travel slower than light, are confined to travel on courses that have angles of smaller than 45 degrees with respect to the vertical axis. The path of the photon is the diagonal black line and the path of a spaceship is in red. It starts to the right of the photon but as we move up the time axis the photon eventually catches up with it.

If the spaceship travels faster then it will follow an angle closer to 45 degrees. Here are a pair of paths corresponding to faster spaceships:

The faster the ship is, the further it gets before the photon catches up. But we're just putting off the inevitable. It seems that whatever we do, the photon will always catch up.

But there's a hidden assumption in the above. By drawing straight lines for the spaceship I was assuming it was travelling at a constant velocity. But there's no reason for that to be true. Here's a different path the spaceship could follow:

At no point does the red path of the spaceship meet the black path of the photon. And yet at no point does the red path reach 45 degrees to the vertical axis. In other words, the spaceship never travels at the speed of light, and yet the photon never catches up with it. Spaceships can outrun photons!

So what kind of path is that? It's actually a hyperbola and it corresponds to a spaceship accelerating at a constant rate. You might wonder how it can be constant acceleration when the speed of the spaceship never exceeds that of light. From an external observer's point of view, after a while it does look like the ship is travelling at a more or less constant velocity close to the speed of light. But from the point of view of someone on the spaceship it feels exactly like constant acceleration. So that is the path that would be taken by a spaceship with its thrusters firing at a constant rate.

### An event horizon!

I chose that path so that the spaceship stays just in front of the photon. A photon that starts slightly to the right will eventually catch up with the ship. But photons starting further to the left of the ship will never reach it. This means that absolutely nothing starting to the left of the black photon path can ever catch the ship. That should sound familiar. It's exactly like a black hole. From the point of view of someone on the ship, the diagonal black line is exactly like the event horizon of a black hole. Nothing to the left of it can ever be seen by observers on the ship.

What does it look like if an observer in the ship watches something that crosses the event horizon? Here's another diagram:

Again, the black diagonal line is the path of a photon, which we know is a bit like an event horizon. The blue line is the path of an object at rest. In effect, it's falling over our apparent event horizon. Of course the blue object doesn't see any unusual phenomenon on approaching the event horizon because there's nothing really there - it's only something seen by observers in the ship. The blue object emits a series of photons (shown in green) at equal intervals. As long as these photons are emitted before the event horizon they eventually catch up with the red spaceship. But notice how they arrive at more and more widely spaced intervals. A photon released exactly at the event horizon never reaches the ship. So the viewers on the ship see the spacing between the photons get longer and longer. They'll never see the blue object cross the event horizon they'll just see it getting closer and closer until eventually it appears to freeze. Again, this is just like a black hole event horizon.

The universe looks pretty weird from the point of view of a constantly accelerating observer. Half of it is simply missing behind an event horizon. But that's just the start of the weirdness. When we throw Quantum Mechanics into the mix something much weirder happens.

### Matter from a vacuum

It's well known that physicists expect black holes to emit particles as Hawking radiation. But our accelerating observer sees something like a black hole, so we might expect them to see something like Hawking radiation. We also know that an observer at rest sees no event horizon. Which means that we might predict that accelerating observers see particles that observers at rest don't. Can we take such a prediction seriously?

Let's look a bit more closely at this. According to a popular view of quantum mechanics, the vacuum is teeming with vacuum fluctuations - ephemeral particle-antiparticle pairs that briefly come into existence and then annihilate each other. In the diagram below I've drawn some of these events:

As we follow up the time axis, pairs of (complementary coloured) particles come into existence and then annihilate each other. These events are so fleeting that they have no effect on our particle detectors and we see a vacuum. But note that I've drawn one of these events straddling our apparent event horizon. From the point of view of an accelerating observer this looks like a pair of particles coming into existence, but because of the argument I sketched above, they seem to freeze near the event horizon. In other words, to an accelerating observer these fleeting events are no longer fleeting, they look like real particles coming into existence and sticking around forever. Accelerating observers appear to see particles in a vacuum!

What I've described above is absolutely

*not*a rigourous argument. But amazingly, when you use the machinery of quantum field theory you end up making exactly the same prediction: accelerating observers see particles. This is known as the Unruh effect. When you do this properly you can compute a bit more detail. It turns out that the energies of the particles are random with exactly the same distribution as black body radiation. In other words, the vacuum looks like it has a glow corresponding to a particular temperature that is proportional to the acceleration. But it's not a bright glow. You need to accelerate at about 10

^{20}m/s

^{2}before the temperature appears to be 1K. Building a thermometer that can survive such accelerations is no mean feat. So it looks like the Unruh effect is a curiosity that might never be observed in the lab.

But it has been suggested that the Unruh effect has already been observed. There aren't many things that can survive that kind of acceleration, but an electron can, and an electron can behave like a thermometer. Electrons in circular particle accelerators routinely undergo the kinds of accelerations we're talking about. They do so because they are driven by a magnetic field. Now an electron has spin, so you can think of it as a bit like a little electric current running round in a loop. That means an electron is like a little dipole electromagnet. Magnets in magnetic fields tend to want to line up along the field - that's how a compass works. So electrons that spend long enough in a particle accelerator, eg. those in a storage ring, should eventually line up with the field. Lining up like this is known as polarisation, and in this particular case it's known as he Sokolov Ternov effect. But when we look at electrons in a storage ring it turns out they're not quite completely lined up, they're slightly depolarised. This is easily explained by Unruh radiation - they're constantly accelerating and so they feel themselves to be in a hot environment. The continual interaction with this hot environment causes the electron spins to be a bit randomised, so they don't all line up nicely.

Unfortunately this isn't definitive evidence for Unruh radiation because when we carry out the full calculation of the Sokolov-Ternov effect it turns out that it predicts partial depolarisation anyway. Now it looks like we don't have evidence for the Unruh effect. But it's not that simple. The Unruh effect isn't a new effect made up by a physicist. It's a prediction based on a new way of looking at fairly conventional physics. The Sokolov-Ternov effect is also predicted from standard physics, just in a different frame of reference. So maybe the partial depolarisation predicted by this effect is in fact the very same thing as the Unruh effect, just looked at from a different point of view.

What does this mean philosophically? We're used to the idea that looking at things from different angles changes how they look. Einstein extended this notion to spacetime so space and time seem different to moving observers. The Unruh effect goes one step further. Whether or not an individual particle exists depends on your point of view. Do particles not have any kind of existence independently of how we look at them? And how does it look to an observer at rest watching an accelerating observer fly by with a thermometer. Do they see a thermometer apparently responding to nothing? Or do they also see the particles once they've interacted with a thermometer? It's all so weird that some physicists take the view that the notion of the particle is outdated and we should only be talking about quantum mechanical wavefunctions.

There's another reason why the Unruh effect is important. The Unruh effect doesn't involve General Relativity, which is all about curved spacetime. But it does use the same mathematical machinery so it gives a way to test out that mathematics. So even if we don't have laboratory black holes to play with, we may still be able to investigate the mathematical framework that predicts phenomena like the Hawking effect. But also note that at least one paper claims it's all a mathematical error and there is no Unruh effect in reality.

In recent years there has been a dramatic increase in the number of papers on the Unruh effect. I expect this trend is going to keep going for a while

### References

- I learnt about the Unruh effect from Wald's book Quantum field theory in curved spacetime and black hole thermodynamics.
- The diagram showing particle creation/annihilation events straddling the event horizon came from Susskind and Lindesay's An Introduction to Black Holes, Information and the String Theory Revolution.
- I tried to catch up with recent developments by reading some of the paper The Unruh effect and its applications.