# The Unruh Effect

The Unruh Effect is very interesting and not very well known outside of the field of physics. Simply put, and accelerated observer will notice a thermal “bath” of particles arising from the vacuum. The “Unruh Temperature” of these particles will be

where c is the speed of light, k is Boltzmann’s constant, h is planck’s (reduced) constant and a is the acceleration of the observer.

There is a famous sort of cartoon picture that explains where this effect comes from. According to this “cartoon,” particle-antiparticle pairs are constantly being produced in the vacuum, appearing for a brief time and annihilating each other soon after. However, because our observer is accelerating, they see an event horizon beyond which they can see no events, exactly analogous to a black hole. In the case of the black hole, this event horizon arises from a curvature of spacetime so intense that not even light can escape it. In this case, our observer is accelerating so light from too far away will never catch up, as at a certain point the observer will be approaching near-light speeds. Because of this horizon, if a particle-antiparticle pair is created right on the the horizon, the observer will only see one of them, intercepting the particle before it has time to be annihilated.

This cartoon picture, however, has nothing to do with the actual calculation of the effect. In fact, it’s not clear at all, looking  at the calculation, what it has to do at all with this picture.

Here is how the calculation actually plays out.

First, one must transform into the coordinates of the moving observer. In a particularly nice choice of coordinates, this is a conformal transformation, making calculations nice. One ten can calculate how the observer’s annihilation and creation operators of the field are transformed from the vacuum ground state. Finding the Bogloliubov coefficients of this transformation, one finds that the spectrum of particles produced is exactly equal to black body radiation with an analogous temperature as expressed above.

But this calculation makes it very unclear why the final spectrum should look so much like black body radiation. It just sort of pops out, and you are free to interpret it as some sort of an “effective temperature,” if you like. However, it really has nothing to do with thermodynamics, as we did not even define entropy, or anything like that. It is simply not a calculation from statistical mechanics.

Where, then, is the origin of this effect? Why must it take the form it does? Well, another explanation, doing this calculation with path integrals, does a sort of transformation of the Lagrangian being integrated over and makes it take the same algebraic form as the partition function from statistical mechanics. So that at least motivates it.

I think it’s very interesting how subtle this discussion of Unruh radiation is. It’s rarely explained well, and often not made clear what interpretations are helpful and which are a little misleading.