How does the Unruh effect create particles?

Question

I've read that the Rindler horizon cuts off access to fundamental quantum fields and leads to a mixing of positive and negative frequencies via the Bogoliubov transformations. But here is where I have questions. How does this mixing happen? Does every horizon mix quantum fields?

Answer 1

The first point that should be made is that particles are not a fundamental notion, fields are. The notion of particle can be useful in some situations to comprehend what is going on, but it is not in any way essential to Quantum Field Theory. The Unruh effect is pretty much an expression of this fact.

In Minkowski spacetime, there is a somewhat intuitive interpretation of what is going on, so I'll be a bit silly and focus on the ideas instead of the math (I'll provide references to the math later on). We have a quantum field and we would like to attribute a notion of particles to it. How do we know what is a particle and what is not?

In a general spacetime, we can't really answer this question. However, we are in Minkowski spacetime, which provides a time translation symmetry due to Poincaré invariance. This time translation symmetry yields a notion of energy via Noether's theorem, and we can use this notion of energy to distinguish which parts of a quantum state are particles (positive frequency excitations) and which are antiparticles (negative frequency excitations). By doing this trick we can define what we call particles and what we call antiparticles and hence define a vacuum as well.

Let us assume then that the field is in this vacuum. For an inertial observer, whose four-velocity is parallel to the timelike Killing field we chose to define our notion of time translation symmetry, there are no particles to be seen.

However, there is another notion of time translation we could have chosen. On the region $x > |t|$ of spacetime (known as the right Rindler wedge), the Killing field associated with boost symmetries is also timelike. Hence, in this region, we could choose a different notion of time translation symmetry, which leads to a different notion of energy, which leads to a different notion of particles. Hence, what is going on is simply that different observers have different notions of time and energy, and this difference leads to different notions of particles.

While I'm writing this answer in a more intuitive approach, this can, of course, be put in solid mathematical background. See, for example, Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, the review arXiv: 0710.5373 [gr-qc], or pretty much any reference on QFTCS (see Suggested reading for quantum field theory in curved spacetime, for example).

The effect can be generalized to other spacetimes containing bifurcate Killing horizons. A Killing horizon is a null hypersurface orthogonal to a Killing field. A bifurcate Killing horizon is a pair of intersecting Killing horizons orthogonal to the same Killing field. I like to thing of them as the X-like structures that occur on spacetime diagrams.

As shown on Wald's text, for example, pretty much the same arguments that lead to the Unruh effect on flat spacetime can be generalized to show that the unique (if it exists) Killing-invariant state of the quantum field is thermal state at the Hawking temperature $T_H = \frac{\kappa}{2\pi}$, where $\kappa$ is the horizon's surface gravity. This means, for example, that an analogue of the Unruh effect holds at spacetimes such as de Sitter spacetime and Schwarzschild spacetime.

This is not the same thing as Hawking radiation, though. The Unruh effect predicts a thermal bath coming from all around the observer and it holds for spacetimes with bifurcate Killing horizons. The Hawking effect predicts a thermal bath coming from a black hole and it holds for black hole spacetimes that arise from gravitational collapse. In particular, as mentioned in Wald's text, there is no Unruh effect for the Kerr black hole, but there is Hawking effect.