# It’s hot when I accelerate!

Let us discuss one of the most intriguing predictions of theoretical physics. Picture yourself moving through empty space with fixed acceleration, carrying along a particle detector. Despite the fact that space is empty, your detector will click sometimes. The number of clicks will increase if you accelerate further, and stop completely if you bring your acceleration to zero. It is called Unruh effect, and was predicted in 1976.

That’s weird, isn’t it? Well, we have not even scratched the surface of weirdness!

So, more weirdness. The particles will be detected at random times, and will have random energies. But, if you plot how many particles you get at each energy, you’ll get a thermal plot. I mean: the same plot that you would get from a thermal bath of particles at a given temperature T. And what is that temperature?

$T = \hbar a / 2\pi c$

That is called the Unruh temperature. So nice! All those universal constants… and an unexpected link between acceleration and temperature. How deep is this? We will try to uncover that.

In our previous Physics Napkin we discussed the geometry of spacetime felt by an accelerated observer: Rindler geometry. Take a look at that before jumping into this new stuff.

Has this been proved in the laboratory?

No, not at all. In fact, I am working, with my ICFO friends, in a proposal for a quantum simulation. But that’s another story, I will hold it for the next post.

So, if we have not seen it (yet), how sure are we that it is real? How far-fetched is the theory behind it? Is all this quantum gravity?

Good question! No, we don’t have any good theory of quantum gravity (I’m sorry, string theoreticians, it’s true). It’s a very clear conclusion from theories which have been thoroughly checked: quantum field theory and fixed-background general relativity. With fixed background I mean that the curvature of spacetime does not change.

Detecting particles where there were none… where does the energy come from?

From the force which keeps you accelerated! That’s true: whoever is pushing you would feel a certain drag, because some of the energy is being wasted in a creation of particles.

It’s hot when I accelerate!! Ayayay!!!

I see $\hbar$ appeared in the formula for the Unruh temperature. Is it a purely quantum phenomenon?

Yes, although there is a wave-like explanation to (most of) it. Whenever you move with respect to a wave source with constant speed, you will see its frequency Doppler-shifted. If you move with acceleration, the frequency will change in time. This change of frequency in time causes makes you lose track of phase, and really observe a mixture of frequencies. If you multiply frequencies by hbar, you get energies, and the result is just a thermal (Bose-Einstein) distribution!

But, really… is it quantum or not?

Yes. What is a particle? What is a vacuum? A vacuum is just the quantum state for matter which has the minimum energy, the ground state. Particles are excitations above it. All observers are equipped with a Hamiltonian, which is just a certain “way to measure energies”. Special relativity implies that all inertial observers must see the same vacuum. If the quantum state has minimal energy for an observer at rest, it will have minimal energy for all of them. But, what happens to non-inertial observers? They are equipped with a Hamiltonian, a way to measure energies, which is full of weird inertial forces and garbage. It’s no big wonder that, when they measure the energy of the vacuum, they find it’s not minimal. And, whenever it’s not minimal, it means that it’s full of particles. Yet… why a thermal distribution?

Is all this related to quantum information?

Short story: yes. As we explained in the previous post, an accelerated observer will always see an horizon appear behind him. Everything behind the horizon is lost to him, can not affect him, he can not affect it. There is a net loss of information about the system. This loss can be described as randomness, which can be read as thermal.

Long story. In quantum mechanics we distinguish two types of quantum states: pure and mixed. A pure quantum state is maximally determined, the uncertainty in its measurements is completely unavoidable. Now imagine a machine that can generate quantum systems at two possible pure states A and B, choosing which one to generate by tossing a coin which is hidden to you. The quantum system is now said to be in a mixed state: it can be in any two pure states, with certain probabilities. The system is correlated with the coin: if you could observe the coin, you would reduce your uncertainty about the quantum state.

The true vacuum, as measured by inertial observers, is a pure state. Although it is devoid of particles, it can not be said to be simple in any sense. Instead, it contains lots of correlations between different points of space. Those correlations, being purely quantum, are called entanglement. But, besides that, they are quite similar to the correlations between the quantum state and the coin.

When the horizon appears to the accelerated observer, some of those correlations are lost forever. Simply, because some points are gone forever. Your vacuum, therefore, will be in a mixed state as long as you do not have access to those points, i.e.: while the acceleration continues.

Where do we physicists use to find mixed states? In systems at a finite temperature. Each possible pure state gets a probability which depends on the quotient between its energy and the temperature. The thermal bath plays the role of a hidden coin. So, after all, it was not so strange that the vacuum, as measured by the accelerated observer, is seen as a thermal state.

Thermal dependence with position

As we explained in the previous post, the acceleration of different points in the reference frame of the (accelerated) observer are different. They increase as you approach the horizon, and become infinite there. That means that it will be hotter near the horizon, infinitely hotter, in fact.

After our explanation regarding the loss of correlations with points behind the horizon, it is not hard to understand why the Unruh effect is stronger near it. Those are the points which are more strongly correlated with the lost points.

But from a thermodynamic point of view, it is very strange to think that different points of space have different temperatures. Shouldn’t they tend to equilibrate?

No. In general relativity, in curved spacetime we learn that a system can be perfectly at thermal equilibrium with different local temperatures. Consider the space surrounding a heavy planet. Let us say that particles near the surface at at a given temperature. Some of them will escape to the outer regions, but they will lose energy in order to do so, so they will reach colder. Thus, in equilibrium systems, the temperature is proportional to the strength of gravity… again, acceleration. Everything seems to come together nicely.