Unruh Effect (Script) (Patreon)

Worried about black holes? Consider this: Every time you accelerate - put your foot on the gas, quicken your step, get out of your chair - you generate an event horizon behind you. The more you accelerate away from it the closer it gets. Don’t worry, it can never catch up to you, but the Unruh radiation it generates sure can.

Around the same time that Stephen Hawking was demonstrating the existence of the black hole radiation that would bear his name, three other researchers, Stephen Fulling, Paul Davis, and William Unruh, were looking at an effect that now seems eerily similar. They were independently studying how the nature of quantum fields appears to change depending on whether or not an observer is accelerating. They found that simple act of acceleration cuts off your causal access to a region of the universe. It creates a type of event horizon. As we saw in our episode on horizon radiation, the presence of horizons distorts the quantum vacuum in a way that can create particles. 

This is the Fulling-Davies-Unruh effect, or sometimes just the Unruh effect. It tells us that accelerating observers find themselves in a warm bath of particles. To understand this we don’t need general relativity with its spacetime curvature and conflicts with quantum mechanics. We just need a little special relativity. And a spacetime diagram. We’ve talked about these before, but a quick rehash. A spacetime diagram has two axes: time, and, well, space, with time on the vertical axis. We can show an object’s path through space and time using worldlines on the diagram. For an object with a constant velocity - an inertial object - these worldlines are just straight lines and the slope gives their velocity. A particle not moving at all has a vertical worldline.

Einstein taught us that objects without mass – like a photon – can only travel at the speed of light, no slower. On the spacetime diagram this is a line with a 45 degree angle from the vertical axis. This angle isn’t anything special, it’s just determined by the units of space and time that we use. Objects with mass can never reach the speed of light, so the worldline of massive objects – which includes any observer – has to be less than 45 degrees from the vertical. Extending light-ray worldlines backwards from our observer defines what we call the past light cone: the region of spacetime that can have a causal influence on the observer. That’s because photons fired from anywhere in the past light cone can reach our observer, either at the current point or at some point on their past worldline.

As the observer moves forward in time, as long as they don’t travel faster than light, their past light cone should eventually contain the entire universe. Well, that’s if you ignore the expansion of the universe. This makes sense: if you wait long enough, photons from anywhere in the universe can catch up to you.

At least you’d think so, right? Actually, there IS a sub-light-speed worldline that can outpace light – or at least keep ahead of it. That’s the worldline of an observer undergoing constant acceleration. Acceleration means change in speed, so an accelerating worldline is curved – the slope changes. Here’s an example: imagine my friend is traveling towards me, initially at close to the speed of light. For some reason they change their mind. I don’t know, they suddenly remembered they had something better on? They fire their rockets and begin a constant acceleration in the opposite direction. To begin with, that just slows down their approach speed. Just before they reach my spacetime location that constant acceleration brings them to a halt and they start moving back in the opposite direction. They then accelerate back up to close to the speed of light, and keep accelerating. OK, whatever, I didn’t want to hang out anyway.

That constant acceleration worldline traces out a hyperbola, and it has a very interesting property. If I fire a photon at the point of closest approach, say, to send a message, that photon can never catch my “friend”, as long as they stay on that hyperbolic trajectory. The photon will always be inching closer - it’ll become asymptotically close - but will never overtake. Now this is only true as long as they continue to accelerate away. Slow down or stop and my message can catch up. In reality, eternal constant acceleration would take infinite energy, so after draining all the energy in the universe they’d finally have to stop accelerating and my message would overtake them.

But until that happens they stay just ahead of my photon. They also stay ahead of any photon emitted from this diagonal line OR any point on the other side of it. This means that any events happening to the left of this diagonal line will never affect the accelerating observer, which sounds pretty horizon-like. In fact the act of acceleration DOES create a type of event horizon called a Rindler horizon. It’s named after the coordinate system we use to describe a constantly accelerating observer in special relativity – Rindler coordinates, devised by Austrian physicist Wolfgang Rindler, who by the way also invented the term “event horizon”.

The Rindler horizon follows at a fixed distance behind an accelerating observer. Let’s call them Rindler observers from now on. The distance of the Rindler horizon is inversely proportional to acceleration. The larger the acceleration, the closer the horizon. All parts of the universe beyond that horizon are out of causal connection with the Rindler observer as they continue to accelerate.  But here’s the weird thing – even momentary acceleration generates a Rindler horizon. You don’t need to pinky-swear that you’ll keep accelerating forever: it’s like the projected future acceleration gives you a Rindler horizon in the present.

And just as with Hawking radiation, that horizon cuts off your access to certain fundamental frequency modes of the quantum vacuum. The derivation of this requires a switch between inertial or Minkowski and Rindler space via the beloved Bogoliubov transformations, which are also used in the Hawking derivation. And here they also lead to a mixing of positive and negative frequency modes in the accelerating frame of reference, which this leads to the creation of particles in the accelerating frame of reference. Those particles should have the same type of spectrum as Hawking radiation – a thermal spectrum. The vacuum should appear warm, with a temperature proportional to the acceleration. This is the Unruh effect.

Now there’s a big difference between the Unruh and Hawking effects. In the case of Hawking radiation, an inertial observer far from the black hole sees the radiation. This is because that distant point in spacetime is smoothly connected to the spacetime near the horizon - it’s all one big spacetime. The only observers who don’t see Hawking radiation are those plummeting in freefall towards the event horizon. 

But if an accelerating Rindler observer is in the same location as an inertial observer, the former will see that patch of space filled with radiation but the latter will see an empty vacuum in the same patch.  At first glance this disagreement seems like a huge conflict. What if the Rindler observer accelerates fast enough that they are burned to a crisp by Unruh radiation? Does the inertial observer see some sort of spontaneous combustion? Where does that energy appear to come from if not from particles? A little less gruesomely, imagine the Rindler observer has a particle detector. Every time an Unruh particle hits the detector it would click. And the inertial observer would agree that it clicked, but they wouldn’t see the particle that triggered it.

And this is actually the case. It’s been worked out with math and everything. The proof uses something called a Unruh-DeWitt detector. This is a fancy name for a particle in a box. This particle is coupled to the quantum field of interest, meaning it can exchange energy with that field. That means the particle can be excited into a higher-energy quantum state when it encounters a particle associated with that field.  So as the detector accelerates Unruh particles appear. The detector particle gets excited causing the detector to click.

But what does this look like for someone not accelerating but in the same patch of space? Well they also still see the accelerating detector ‘click’ but they argue that it’s for a very different reason. When they perform the relativistic field theory calculation to understand the coupling between the detector particle and the field, they get that there’s a sort of drag or friction term between the detector and the field that results from the acceleration. That causes energy to be dumped into the detector particle. The source of that energy is the acceleration itself. 

The upshot is that the existence of particles is observer dependent. Here's an example of this observer dependence of particle existence. A charged particle accelerating in a magnetic field emits radiation. Bremstralung radiation. An inertial observer sees the charged particle itself radiating, its energy extracted from the magnetic field. But an observer accelerating WITH that charged particle sees it absorbing Unruh particles and then spitting them out again. They disagree on the source of the energy, even if  they agree on the final result. 

So how strong is Unruh radiation? Don’t worry too much - you’d need to accelerate at a rate of 10^20 m/s^2 to increase the temperature by a single degree Kelvin. It’s difficult to directly observe Unruh particles, although analogies have be observed, even in classical systems, like this really cool study with water waves.

One more thing: According to Einstein’s equivalence principle, remaining stationary in a gravitational field is equivalent to acceleration in free space. That means you, right now, are bathed in a very tiny amount of Unruh radiation. But there’s one place in the universe where the gravitational acceleration can get that high – that’s right above the event horizon of a black hole. If you hover close enough to that event horizon you actually would be bathed in Unruh radiation. 

Here we get to a really interesting question: what’s the relationship between the Unruh particles seen by someone hovering at the event horizon and the particles of Hawking radiation seen by a distant observer? Great question, and one we’re going to have to come back to. Right now I have to jet. But not too fast, lest I combust in a Fulling-Davis-Unruh thermal bath as I accelerate to that future point in spacetime.