Horizon Radiation Script (Patreon)

The most successful theory in physics combines the weirdness of quantum mechanics with, well, the weirdness of special relativity to give quantum field theory. This theory tells us that particles can be created and destroyed during interactions. Even so, every observer agrees on whether a particle exists or not, right? Yeah...about that...

Both theories of relativity, special and general, tell us that many things are observer dependent. Different observers might disagree about the speeds, lengths, or times, but the laws of physics should be the same for everyone. And for two observers with very different, but constant speeds - inertial observers - the vacuum itself should appear the same. 

But relativity can throw some weirder things in the mix. Specifically the idea of a horizon. For example the event horizon of a black hole, out of which no information can travel, or the cosmological horizon that limits the observable universe. A strange type of apparent event horizon even appears when we accelerate. Generally speaking, a horizon is a boundary in spacetime from beyond which no influence can pass. It limits an observer’s causal connection to part of the universe.

As weird as these spacetimes with horizons may be, the statement from before holds: the fundamental laws of physics shouldn’t change if we go near a black hole or if we start accelerating. But enforcing this isn’t automatic, and something has to give. As it turns out what gives is that nature of the vacuum, and in fact the notion of what a particle is, becomes observer dependent. 

In coming episodes we’ll be delving into Hawking radiation and the Unruh effect, both of which are very similar and are results of this observer-dependent vacuum result. There are a couple of compelling, but inaccurate, explanations of Hawking radiation around and here at spacetime we want to be as simple as we can, but no simpler. So we’re going to take this episode to lay the final stepping stone we need to examine these effects in detail, and accurately.

To get at this idea of observer dependent particles and vacua we’re going to need some quantum field theory, and we’re also going to draw heavily on this recent episode. In QFT we think about each particle type as having its own quantum field that exists at all locations in space. If the field vibrates with a single quantum of energy we see a particle. That oscillation can be distributed over some region of space representing the possible positions of the particle. The properties of the particles are encoded in the properties of these fields. The laws of physics as we know them are the rules defining how particles interact. For the laws of physics to be consistent, the fundamental properties of these fields must be the same for all observers. 

Imagine I fire a pair of photons which annihilate to produce an electron-positron pair. All observers, be they floating in empty space or accelerating or orbiting a black hole, should agree on the basic results of that interaction – two photons in, one electron, one positron out. That means everyone has to agree on the fundamental nature of the quantum fields that describe these particles and the way they interact. There’s no conflict for constant-speed, inertial observers. In fact quantum field theory is what we call Lorentz invariant - special relativity is built in, so the equations describing an interaction transform cleanly between inertial reference frames in a way that leaves the laws of physics and the nature of the vacuum intact. But when an observer who sees a horizon tries to write down these equations, in order to preserve the laws of physics, they find they have to redefining the nature of the vacuum itself.

To see how this happens we need to think about how particles, interactions, and vacuums are described in quantum field theory. Imagine the simplest type of quantum field - it comprises oscillators at all locations, a bit like a bed of springs that are attached to each other, or even better an infinite, extremely flexible drum skin. Now imagine an oscillation in one of these locations - a particle perfectly localized in space - a single spring or a single point on the drum skin. That oscillator can increase in energy in discrete chunks - in quanta - and we interpret each quantum of energy as representing a single particle. 

Now every point in this field - this drum skin - is connected to neighboring points. The equations we write to describe the oscillations are coupled to each other. This coupling allows the oscillation - the particle - to evolve through space. It also makes it devilishly difficult to solve those equations, because the individual oscillations in space can’t be solved independently.

But there’s a neat trick to get around this. As we saw in our recent episode on Fourier transforms, it’s possible to describe any vibration or wave in two ways. Sound waves can be described in terms of variation over time OR variation over frequency. Quantum wavefunctions and quantum fields can be described in terms of variation with position OR variation with momentum.  So instead of writing the field as having a value at every possible position in space, we can write it as having a value for every possible momentum. We switch between them with a Fourier transform. 

So let’s take our spatial quantum field - our drum skin - with its single localized particle and transform to momentum space. That “momentum field” also has infinite oscillators, but now each represents a different possible momentum for the particle. Bizarrely, in momentum space that single perfectly localized “position” oscillation can also be described as an infinite number of unlocalized “momentum” oscillations. Each of these “momentum modes” exists at all spatial points in the universe. If we add these momentum oscillations together with the right weightings, they cancel out everywhere except at the spatial location of the original particle. The superposition of infinite universe-sized momentum oscillators - "momentum particles" can represent a single spatial oscillator: a particle at one point in space!

Let’s go back to our slightly less abstract infinite drum skin. It starts out with no oscillations – analogous to vacuum state in quantum field theory. If we want to make an excitation in a particular spot – make a “particle” – we hit it in that spot with a drumstick and set an oscillation going. Most of the amplitude will be where we hit it, but really the vibration will extend everywhere on the drum. We could, in principle, make the same oscillation by simultaneously hitting the drum *everywhere* with an infinite drumstick, adding global vibrations with different frequencies that cancel out everywhere except in one spot.

But why exchange a single spatial equation for infinite equations in momentum space? Because they actually make the math much easier! Those momentum oscillations have two important qualities - they behave as simple harmonic oscillators - so their value over time is like a simple sine wave - and even more importantly, they are uncoupled. The field at each momentum point oscillates independently of its neighboring momenta. Dealing with uncoupled equations allows you to add and subtract oscillations without affecting the neighboring momenta. However changing the momentum modes DOES affect the superposition – the sum of all oscillations – for example by creating or destroying particles.

OK, so a single particle can be described as many oscillations in momentum space. Those oscillations can be reconfigured with our infinite drumstick to add new particles or remove old ones - for example to describe a particle interaction like those two photons annihilating into an electron-positron pair. There’s a nice mechanism in quantum field theory for doing this. It’s called the field operator, and it is our infinite drumstick. It’s comprised of creation and annihilation operators that can raise or lower the number of particles one at a time by changing the number of particles or oscillations in each momentum mode. 

Here the field operator represents the field properties, the laws of physics. This is the thing that needs to be the same for everyone - but there's no such restriction for the creation and annihilation operators it's made of, and it's these operators that define the vacuum. As discussed in a previous episode we can think of the vacuum as a sea of virtual particles. In momentum space we can think of it as a superposition of infinitely many momentum modes - infinite spatially undefined particles with defined momenta, and these just happen to cancel each other out leaving zero particles - a vacuum.

OK, so what happens when we add horizon to our infinite quantum field? What if say there’s an edge to the drum skin, or cut a hole in the middle? Now our skin responds very differently to our drumstick. For example, we no longer have access to part of the skin, and oscillations can reflect from the boundary. If we want to make the same particle as before we need to strike the remaining part of the skin in a very different way - with different forces at each point. The same is true of the universe. If we introduce an event horizon then we lose access to some momentum modes, while other modes behave differently. 

That means we have to reconfigure our field operator - our infinite drumstick - in order to create or annihilate the same particles as we had in an infinite, horizonless universe. We need to rejig the field operator for laws of physics to be consistent. If you’ll forgive a little QFT jargon: the new annihilation operator needs to be a mix of both the old annihilation operator *and* the old creation operator. Now when you use the new, rejigged field operator to describe the vacuum, some momentum modes that once canceled out no longer cancel out. What was once a vacuum now has particles. 

You CAN beat that horizoned drum skin to produce oscillations that are consistent with the infinite, un-horizoned drum skin. But doing so leaves additional oscillations. What does that look like? It looks like heat. The vacuum acquires a non-zero temperature. It appears to be bathed in thermal particles - particles that don’t exist for an observer who doesn’t experience the same horizon.

And in some cases, changing the boundaries of spacetime actually reduces the number of particles - for example in the Casimir effect. We’ve already seen how this effect reduces the energy of the vacuum between conducting plates. In upcoming episodes we'll apply these ideas to finally understand Unruh and Hawking radiation. So this year keep your eye on the horizon - the event horizon, and the strange things it does to the quantum contents of space time.