Six-Layer Hawking Radiation Cake

I was going to give an explanation about how black holes evaporate via Hawking radiation, but to do that, I'll have to give an explanation of virtual pair creation-annihilation. That means I'll have to give an explanation of virtual particles. To do that, I'll have to give an explanation of the uncertainty principle. That'll require an explanation of eigenstates. For that to make any sense, I'll have to explain wave functions.

So while we wade through all those layers, just remember: You didn't ask for it.

6. Wave Functions

In physics, things are expressed by means of mathematical expressions and relations that have the parameters of the problem built into them. For example, near the surface of a planet, over short distances, for macroscopic objects, in a vacuum, given a measure of height above the ground x where x=0 is ground-level, given gravitational constant g and initial height h0, the height of a dropped object is given by x = h0 - 1/2 g t^2, where t is the time since you dropped it, for t > 0 and t < sqrt(2h0/g).

x = h0 - 1/2 g t^2 is a very simple equation, but look at all the definitions and assumptions that went into it! It's not really very simple at all, is it?

In quantum mechanics--which already implies that we're talking about very small phenomena--things are expressed in terms of wave functions. This is a complex function (i.e., it involves i, the square root of -1) whose magnitude indicates the probability of measuring the phenomenon in a particular way. For instance, you can write the wave function of an electron orbiting a hydrogen nucleus, and the wave function will give you the probability of finding the electron at a particular location around the nucleus.

Note one of the fundamental features of quantum mechanics: We're talking about the probability of measuring something, not the certainty. This contrasts with most large-scale phenomenon like Newtonian or Einsteinian mechanics, where you can determine the measurement with nearly arbitrary accuracy. Quantum mechanics is inherently probabilistic, not deterministic, something we'll return to later when we discuss the uncertainy principle.

Wave functions often have continuous or discrete parameters built into them. For example, the energy of an electron in an isolated hydrogen atom is -13.6/n^2 (electron volts). n is a discrete parameter, a quantum number, that can take any value from 1 to infinity. Other values for energy are forbidden by the constraints of the phenomenon that the electron's wave function describes.

So to sum up, a wave function is the way things are expressed in quantum mechanics. The form of the wave function is specified by the phenomenon being described, and it usually contains quantum numbers whose values can be measured.

5. Eigenstates

"Eigenstate" is a word from matrix algebra. All you need to know about it is that an eigenstate is, in some sense, a "solution" of a matrix. (Technically, if M a = c a for some matrix M, vector a, and constant c, then a is an eigenvector of M.)

Quantum mechanics imposes this rule: If you measure something, you force the wave function into an eigenstate of the thing you're measuring. Take that isolated hydrogen atom I mentioned before: Its eigenstates for energy have energy = -13.6/n^2. Before you measure the energy, the wave function can be anything. But you can mathematically express the wave function as a linear combination of all the possible eigenstates. E.g. if the wave function ψ is a quarter n=1 and three-quarters n=3, then you can say ψ = sqrt(1/4) ψ(n=1) + sqrt(3/4) ψ(n=3).

When you actually measure the energy, the wave function collapses into one of its constituent eigenstates. Which eigenstate it collapses into is probabilistic. For the wave function above, 1/4 of the time, it'll collapse to ψ(n=1) and you'll measure -13.6 eV; and 3/4 of the time it'll collapse to ψ(n=3) and you'll measure -1.51 eV.

In other words, to get the original wave function, you did something to the system that put it into a mixed state where it might have one energy and it might have another energy*; you don't know what state it's in until you actually measure it, at which point, you know that it's not in a mixed state any more.

*For instance, if you put a photon with 12.09 eV of energy into the ground-state system, it might be absorbed by the electron (jumping it to n=3) and it might not (leaving it at n=1). If you don't look for the photon to see whether or not it was absorbed, the electron's state is a wave function with both n=1 and n=3 energy eigenstates in it.

To sum up, an eigenstate is a wave function with a particular value for something you can measure, and the wave function is in that eigenstate once you've measured it.

4. The Uncertainty Principle

Before, I was talking about discrete quantum numbers, but you can also have continuous quantum numbers. Position is one such. A particle can be at x=1, but it can also be at x=1.1, x=1.001101, and x=0.9197. If you measure a particle's position, you collapse the wave function into a position eigenstate.

The same is true of momentum; it's continuous, and measuring momentum results in a momentum eigenstate.

Here's where the problem starts getting complex. It turns out that the position eigenstates and the momentum eigenstates are incompatible. There's no position eigenstate that's also a momentum eigenstate and vice versa. A position eigenstate can only be expressed as a linear combination of momentum eigenstates and vice versa. Remember that an eigenstate is equivalent to saying that something has a specific value.

So if you measure position, you collapse the wave function to a position eigenstate and you don't know the momentum because you're not in a momentum eigenstate; and if you then measure the momentum, you convert the wave function to a momentum eigenstate so you're no longer in a position eigenstate and you've lost your position information.

Once you dig into the math and take into account things like your inability to actually measure position accurately enough to collapse the wave function to a pure position eigenstate, and so on, you can actually reduce the whole argument to a simple inequality, which is Heisenberg's Uncertainty Principle: Δx Δp ≥ h-bar. h-bar (or ℏ if your browser supports it) is Planck's constant, a very small number which pretty much determines the granularity of the universe. A lot of things come in units of h-bar or are scaled in reference to h-bar.

What the uncertainty principle means is that if you measure position and momentum simultaneously, striking whatever balance for accuracy in one or the other you can, the absolute best the universe permits you to do is uncertainty of h-bar. If you measure x with absolute perfect precision (Δx → 0), you cannot measure momentum at all (Δp → ∞).

The uncertainty principle applies in some way to any incompatible eigenstates. For our purposes, the most important "conjugate pair" is energy and time: ΔE Δt ≥ h-bar. The more exactly you measure something's energy, the less accuracy you have about its duration.

3. Virtual Particles

The Uncertainty Principle lets nature cheat. Anything that happens below the level set by the Uncertainty Principle can't be measured and doesn't count. Einstein's relation specifies the relationship between energy, mass, and momentum, specifically E^2 = m^2 c^4 + c^2 p^2. (This reduces to the familiar E = m c^2 for particles at rest.) A particle always satisfies this relation--except that you can't tell if it's wrong if the error is below the h-bar limit.

Even simple existence or non-existence isn't certain, because for a particle with a small energy and a small lifetime, below the h-bar limit, you can't say it exists or doesn't exist, because you can't measure it that well.

That brings us to virtual particles. The simplest definition is that a virtual particle has an energy and lifetime below the h-bar limit. Because of that, it doesn't have to obey Einstein's relation.

The fundamental forces (electromagnetic, strong and weak nuclear forces) are mediated by virtual particles. The way this works for, e.g., electromagnetism is that an electron emits a virtual photon that carries a little bit of momentum across to another electron, which absorbs it. The photon is virtual because it's not massless, it doesn't obey Einstein's relation, and it exists only beneath the h-bar limit in terms of energy and time. In fact, because it exists below the h-bar limit, we can't say specifically that it was emitted by one electron and absorbed by the other; it's necessarily mathemtically equivalent to say that it was absorbed by one electron and emitted by the other. As a result of the emission and absorption of the virtual photon, the two electrons have a small change in momentum, which is the definition of a force (i.e., force is the time-derivative of momentum).

I could go into far more detail about coupling constants, vertices, loops, and the like, but without the facility to start drawing Feynmann diagrams, it becomes rapidly too hard to follow. Suffice it to say that it's important to believe what I'm wildly handwaving about is completely supported by the actual math of the actual quantum mechanics, which tells you how to write the wave functions and matrices of various kinds of virtual and real particles and their interactions.

To sum up, virtual particles are nature cheating by having all sorts of goings on below the limit determined by the Uncertainty Principle. Virtual particles aren't real because they don't obey the rules of real particles.

2. Virtual Pair Creation-Annihilation

In particle physics, an electron is a line with an arrowhead in its middle, and a positron is also a line with an arrowhead in its middle pointing backwards. This is because the math is absolutely equivalent between an electron going backwards in time and a positron going forwards in time. An electron and a positron can annihilate each other because they're the same thing only backwards.

Imagine a high-energy photon zipping along through space. If it passes near a nucleus, it can spontaneously transform into an electron and a positron. (It can only do this near a nucleus because it needs to dump some momentum into the nucleus so that the e/p pair can satisfy Einstein's relation.) You can freely create a particle and its anti-particle simultaneously without violating any quantum mechanical rules and conserving all necessary quantum numbers, provided you correctly conserve energy and momentum. The electron and positron can zip off in different directions and lead separate careers. They can even recombine, annihilating each other and producing a photon again (two, actually, for the same reason that the photon needed a nucleus).

Let's take another look at that. A photon moves along and splits into a pair of particles, which then recombine. It has to involve other particles at both ends of the interaction in order to satisfy the constraints of reality. But we already know that virtual particles don't care about the constraints of reality!

So now let's take that photon. It pair-produces an electron and a positron, but this time, let's leave them virtual; the photon doesn't interact with anything else, and the electron and the positron are both violating Einstein's relation. We're OK to do that as long as we stay below the Heisenberg limit; in this case, it limits the lifetime of the electron and the positron, because we know the energy of the photon. When they recombine, they recreate the original photon.

This is an example of a virtual pair creation-annihilation event, and it's one example of what physicists call a loop diagram, because when you draw the event in schematic form, the electron and the positron form a loop. Loop diagrams are nasty because they contain certain infinities, which require a branch of mathematics called renormalization to fix.

Because the electron and positron form a loop, a virtual pair creation-annihilation event can take place even in the vacuum of space without any real photon required. But let's suppose that you've got a virtual pair in progress and another particle (real or virtual) comes by. The interloper particle can interact with the virtual pair, and if it does so in a way that moves enough energy and momentum around that the electron and positron start obeying Einstein's relation, all of a sudden the virtual electron and positron can become real. (This is essentially what's happening in my original high-energy photon case--the nucleus interacts with the virtual pair and promotes them to reality.)

1. Hawking Radiation

Hawking radiation can be summed up simply now. You have virtual creation-annihilation events all the time, including at the event horizon. Some of the time, the virtual loops can be broken by another interaction, turning the virtual pair real. Some of the time, when that happens, one of the newly real pair is outside the event horizon. In this way, energy and momentum can escape the black hole.

It turns out that this process happens more efficiently the smaller the black hole is, because the smaller the black hole is, the bigger gradient you've got at the event horizon and the easier it is for a particle to become newly-real on the right side of the event horizon with enough energy to escape. So large black holes have a lifetime orders of magnitude longer than the lifespan of the universe. Only micro black holes have an observable lifetime.

And now you know too much of the story.

Posted by Greg at November 27, 2006 1:55 PM