Is Hawking radiation imaginary?

I mean obviously, in a practical sense, it's unobservable; however, what if it's not even an observable in the hand-wavy world of theoretical physics?

Introduction

Let’s recap this rabbit hole thus far. First, I was watching a recent lecture by Raphael Bousso where he said something interesting that, likely contrary to his intent, made me question the entire foundation of modern black hole physics. I wrote about it here:

• Has the past 50 years of black hole physics been a bunch of nonsense?

The TL;DR is that the Planck length is a dimensional analysis game and dimensional analysis is insufficient to support any concept of semi-classical physics in a gravitational setting — a paradigm on which the past 50 years of black hole physics is based. In meme form¹:

This inspired me to go back to the beginning and read Bekenstein (1972), which inspired this post:

• Why exactly do we need black holes to satisfy the 2nd law of thermodynamics in finite time?

Now if we forego nuance, the question in the title is very similar to a point made by Netta Engelhardt in a lecture from a few years ago — so I feel I’m allowed put this in the non-crackpot sections of this post. But the general point I was trying to make is that the “entropy equals surface area of a black hole in Planck areas” is a whole lot of dimensional analysis where the ℏ actually drops out on both sides of the entropy balance. And per the question in the title, it’s not even clear why the 2nd law needs to be satisfied by degrees of freedom that are inaccessible to the observer.

So I decided to move forward a couple years and read Hawking’s first paper on his eponymous radiation. Two things. I had to read too many journal articles from the 70s as a grad student² and the notation and style … ugh. But from a more enlightened standpoint, I also think criticism of a paper from 50 years ago should probably not focus on the paper itself, but rather a modern (“steelman’d”) version. To that end I am deeply indebted to this excellent lecture series by Suvrat Raju (youtube³, lecture notes) for not only helping this aging physicist refresh his quantum field theory but also helping me really understand Hawking radiation for maybe the first time. Unlike the earlier posts where I mixed things together willy-nilly, I’m going to break this one into a “non-crackpot” section (i.e. where I am trying to be even-handed and mainstream) and a “crackpot” section (i.e. where I make speculative points and give personal views).

Non-crackpot physics

Penrose diagrams to aid in understanding the Unruh effect and Hawking radiation. On the left is flat Minkowski space where we are using Rindler coordinates on the “Right Rindler Wedge” (RRW) in cyan. A system undergoing constant acceleration takes a hyperbolic path where they cannot receive signals from an observer in the brown region. On the right is the symmetric spacetime for a Schwarzschild black hole which looks very much like you’ve cut off future timelike infinity. There are of course some subtleties you could “well, actually” at me but suffice to say this is how a Rindler horizon and a black hole horizon look kinda similar. We only care about a very tiny* (more details about that later) region near the horizon where we have a local hyperbolic coordinate system (light cone Kruskal coordinates) using u and v (transverse coordinates y and z are suppressed).

By dint of the equivalence principle, Hawking radiation is basically the Unruh effect⁴ but in a scenario where it is both less controversial and more interesting. While you can easily transform between the different frames in the Unruh effect — both locally⁵ and globally in the derivation — you can only do so locally in the case of a stable black hole. The Unruh effect is “controversial” as to whether you’d see actual thermal radiation propagating from near the Rindler horizon to infinity because a) a system is probably not going to accelerate forever under either its own or even external power, and b) the aforementioned different frames with which you can freely use make it observer-dependent. Hawking radiation is less controversial for reasons that are extremely interesting but also kind of technical. So we’re going to need a bit of quantum field theory. However, it’s only scalar quantum field theory for a scalar field⁶ ϕ(x, t) so it’s not too brutal.

First, everything in physics is basically a harmonic oscillator, so there’s a standard way to expand a scalar quantum field into quantum harmonic oscillator modes:

\(\displaystyle \phi (\textbf{x}, t) = \sum_{k} \frac{1}{\sqrt{2 \omega_{k}}} \left( \hat{a}_{k} \;\phi_{k} (\textbf{x}) \; e^{-i\omega_{k}t} + \hat{a}_{k}^{\dagger} \;\phi_{k}^{\dagger} (\textbf{x}) \; e^{+i\omega_{k}t}\right)\)

where k is a collection of quantum numbers. Those a and “a dagger” (i.e. Hermitian conjugate) operators represent a “projection” onto a harmonic oscillator basis and are called, respectively, annihilation and creation operators⁷. When the latter “hits the vacuum” state |0⟩, it adds a particle to it, i.e.

\(\displaystyle \hat{a}^{\dagger} |0\rangle = \sqrt{1} \cdot |1\rangle = |1\rangle\)

Quantum field theory is critical to understanding Hawking radiation because this is where the “particle pops out of the vacuum” stuff comes from. Also, you can see where the quasi-standard vacuum state designation |0⟩ comes from — harmonic oscillator basis with zero particles (zero particle Fock state). I dropped the quantum number k, but the new state would have quantum number k (and energy aka frequency ωₖ) for operator aₖ. Two important points: 1) the creation and annihilation operators do not commute, and 2) when put together they give the number operator, or suitably normalized, the occupation number operator. Mathematically⁸,

\(\displaystyle \begin{eqnarray} [\hat{a}_{j}^{}, \hat{a}_{k}^{\dagger}] & =& \hat{a}_{j}^{} \hat{a}_{k}^{\dagger} - \hat{a}_{k}^{\dagger} \hat{a}_{j}^{} = \delta_{jk}\\ \hat{n}_{k} & = & \hat{a}_{k}^{\dagger} \hat{a}_{k}^{} \\ n_{k} & = & \langle n, k | \hat{a}_{k}^{\dagger} \hat{a}_{k}^{} |n, k\rangle \end{eqnarray}\)

This is a nice, fun, and easy to use basis for a lot of scenarios you might encounter. Two scenarios where it is not useful are non-inertial reference frames and non-trivial vacua — both happen to be the case when you are near the horizon of a black hole⁹. So we need to change our basis in order to make the calculations a bit easier.

One of the reasons I really liked Suvrat Raju’s lectures is because he puts this change of basis front and center. Lecture 1 is a general introduction. Lecture 2? The change of basis. He writes down a general basis for considering a scalar field correlator ⟨ϕ(x₁, t₁) ϕ(x₂, t₂)⟩ across a null hyperplane¹⁰ — albeit only to leading order, but that’s all we need. A black hole event horizon is (locally) a null hyperplane. So is a Rindler horizon. In fact, you could draw a null hyperplane anywhere you wanted in flat Minkowski space. The horizon temperature T ~ 1/β basically drops out of this new basis almost immediately — the number operator above becomes:

\(\displaystyle \langle 0 | \hat{b}_{k}^{\dagger} \hat{b}_{k}^{} | 0 \rangle \sim \frac{1}{e^{\beta \, \tilde{\omega}_{k}} -1}\)

where the b’s are the new creation and annihilation operators that are combinations of the a’s. The RHS is the well-known occupation number in Bose-Einstein statistics that appears in Planck’s law for blackbody radiation. But wait, you ask — didn’t you just say you could draw an arbitrary null hypersurface in flat Minkowski space and use this basis? How is that thermal factor not everywhere — even in empty space?

Those three examples I gave help illuminate the answer:

• Arbitrary null hypersurface in flat Minkowski space: You (an observer) have access to both sides of the arbitrary null hypersurface and adding up both sides gives you zero. It cancels out.

• Rindler horizon: The accelerated observer does not have access to both sides of the horizon, but one that isn’t accelerating does. No cancellation for the former means a thermal occupation number — but it is observer-dependent. That makes the Unruh effect somewhat controversial because in general relativity you can generally transform away a local acceleration (i.e. the acceleration one of the observers is experiencing). On the other hand, the temperature turns out to be so incredibly miniscule for any reasonable acceleration that maybe it’s always been there and we just didn’t notice. I once traveled for work so much in one year that I calculated I had aged about a nanosecond more relative to the average American. Reality is weird.

• Black hole event horizon: The only observers that might disagree with your choice of basis are on their way to their doom at the singularity. Everyone in what’s left of the rest of the universe sees a stable black hole with surface gravity g that can’t just be transformed away because a black hole is a real thing that persists for a long time¹¹.

I don’t want to go into the gory mathematical details of getting those bₖ operators above from the aₖ ones and constructing a new correlator (see the linked lecture notes above), but I do want to point out three major elements of the process. These elements are not “orthogonal” — there’s a sense in which doing one involves the others. The elements are:

• Bogoliubov transformation: This is what literally creates those bₖ operators from a linear combination of the aₖ and aₖ† operators¹². There are standard empirically-validated applications of this to condensed matter physics (invented for superfluidity, but also has application to superconductivity).

• Hyperbolic/light cone coordinates: This gives us a well-nigh rectilinear coordinate system near the horizon but also means “time and (one dimension of) space” switch meaning as you cross the null hypersurface. These are the u and v coordinates in the figure above. They have application in particle physics (including papers I myself have written¹³).

• Wick rotation: If you take t → 𝒾 t the 4D metric goes from Lorentzian signature (−, +, +, +) to Euclidean signature (+, +, +, +) and Minkowski space becomes Euclidean space. Among other things, this ‘one weird trick’ allows you to do some integrals that have led to some of the most empirically accurate theoretical predictions humans have ever produced. Also, if you do this at the outset, quantum field theory turns into thermal field theory (TFT). This is basically where that thermal occupation number comes from — that correlator effectively turns into the Planck blackbody factor.

All of these are perfectly acceptable operations in quantum field theory that independently have wide use in empirically validated applications. Since we are using the black hole as simply a classical background metric that gives us a source of a constant acceleration — no Planck scale physics here¹⁴ — everything should be very uncontroversial. I mean the controversial part, per above, is the Unruh effect where the question is observer-dependence. Not the case for Hawking radiation! The only additional key (and also uncontroversial) assumptions are that the horizon is smooth (which is playing the role of thermodynamic equilibrium here) and the black hole persists for a long time (which creates an approximate time-translation invariance that allows you transition between the near horizon “Bogoliubov” modes and propagating Minkowski space modes). Nothing to see here!

Crackpot physics

Wick rotation of Rindler time coordinate 𝜏 transforms the hyperbolic paths in the right and left Rindler wedges (RRW, LRW in cyan) into circles. The distance to the Rindler horizon is given by the inverse of the acceleration, 1/g. This “radius” of the hyperbolic sheet becomes the radius of a Euclidean circle. Orbiting this circle requires you to obey periodic boundary conditions (Matsubara frequencies) β = 2π/g if the acceleration in the Rindler frame is g. Formally, β ~ 1/T in the thermal field theory obtained after Wick rotation, so T = g/2π. This is the Unruh temperature for a general accelerated frame, and the Hawking temperature if g is the surface gravity of a black hole.

Ok this isn’t crackpot so much as questioning the “mainstream” view. It’s more along the lines of the “stop doing math” meme in the sense of “hello I would like a 30𝒾 minute massage please”. I mean look at the graphic above (you could also check out Hawking’s own lecture notes from 1994). Transformations dreamed up by the utterly deranged, indeed. But then there’s nothing wrong with this mathematically. Analytic continuation to imaginary time is fine. It’s fine.

Still.

Seems a bit … cheat-y.

Wick rotation just gives you thermal physics. That thermal physics requires a Matsubara frequency. Since we have a massless scalar field theory, the only length scale is ξ = c²/g where g is the acceleration (i.e. black hole surface gravity) so the only frequency scale is c/ξ = g/c. To get an energy scale, add Planck’s constant: ℏg/c. To get a temperature scale, Boltzmann’s: ℏg/(ck). The 2π comes from the periodic boundary condition (i.e. the circle in the picture) for the Matsubara frequency. So T = ℏg/(2πck). That’s the Hawking temperature (no really, that’s it). Stuff with a temperature radiates black body radiation. In a sense you could “explain” Hawking radiation by this diagram alone: the Wick rotation combines the left and right Rindler wedges¹⁵ (cyan in the graphic above) to give you thermal physics and obliterates the spacetime outside them (black triangle regions along the time axis) — but in the case of a black hole (and maybe only in the case of a black hole) no information can come from the obliterated spacetime so this is fine¹⁶. In the Unruh case or arbitrary null hypersurface case, there are observers there — so maybe you have to be a bit more careful in those cases. But not here!

So you basically just apply a couple transformations, including “QFT → TFT”, and use the fact that parts of those transformations that usually cancel are hidden behind an event horizon so don’t cancel.

You know what else happens when you Wick rotate?

For one, you kill quantum mechanics. The Schrodinger equation turns into a diffusion equation. No more interference. For another: you kill special relativity. No more light cones. No more causal structure. Both of these pieces describe what’s happening when you perform the “QFT → TFT” transformation via Wick rotation.

Don’t get me wrong — there are lots of empirically validated applications where Wick rotation is used to simplify a calculation. Lots of integrals resulting from Feynman diagrams are evaluated using this ‘one weird trick’. However, in those cases you have to be extremely careful with the analytic structure of your integral (see e.g. here for a random example of what’s involved compared to other approaches; here’s another short discussion). And Hawking radiation is part of the whole modern black hole physics paradigm that appears to be completely consistent even down to the cancellation of divergences in the higher order terms of the Generalized Second Law (see Bousso’s talk linked at the top of this post). But the subsequent consequences — including black hole evaporation — makes me think Hawking radiation might actually be a reductio ad absurdum. See what happens when we put quantum field theory in non-inertial reference frames!¹⁷

On the one hand, Hawking radiation is purely a consequence of standard quantum field theory transformations in a particular non-inertial frame. On the other hand we are saying an object from which not even light can escape somehow radiates thermally at a well-defined temperature that just falls out of some standard quantum field theory transformations¹⁸. This is part of the joke in the title: Wick rotation, a transformation to imaginary time, yields Hawking radiation.

I don’t think I can emphasize enough how acceptable this result is in the physics community, nor how fringe (“crackpot”) questioning e.g. Wick rotation in this application is. But still, the best evidence we have for black hole thermodynamics is “numerology” per Andrew Strominger (youtube video, referenced to the quote) so I’m considering myself hardened in my skepticism of this whole … idiom.

1. Modification of an XKCD comic.

2. E.g. I refer you to citations [24] and [25] here. I would also like to point out that the light cone (light front) coordinates used in that paper (and subsequent complications of interpretations of the quantum field theory modes) is of direct relevance to the present post. This is just to say I know what I am talking about. Or at least, I know what I am talking about more than some rando — which I admit is a low bar.

3. You only really need lectures 1-4 (of 26!) for what I am talking about here.

4. I would say the “scholarpedia” article I link is far better than the wikipedia article.

5. Locally in spacetime — i.e. centered on the neighborhood of a point {t, x, y, z} in Minkowski space.

6. It works for other types of particles, it just is messier in terms of indices, charges, and the like. Plus any massive particles (even neutrinos!) are basically too massive to realistically contribute much at the extremely low temperature obtained.

7. Or in less metal parlance raising and lowering operators or the completely mundane ladder operators.

8. Operators can “act” to the left or to the right. When they act to the right, they have their normal sense. When they act to the left, everything gets complex conjugated so it reverses creation and annihilation. Or you can say they become antiparticle creation and annihilation operators. However in our scalar field context, each particle is its own antiparticle. As a side note, Hawking’s paper also use scalar fields so the whole “particle-antiparticle pair popping out of the vacuum and one falls in the black hole” interpretation of Hawking radiation is, um, not what is going on.

9. Condensed matter physics is an exceedingly normal case where there is also a sense of a non-trivial quasi-particle vacuum. This includes superfluidity for which the Bogoliubov transformation method I will mention in a second was created.

10. A tangent plane to a light cone.

11. I mean the current consensus in physics is that the black hole will eventually evaporate due to the side effects of the Hawking radiation described by that occupation number operator expectation value. However, that very long time is important to make it observable.

12. Again, as I mentioned in the other footnote, this kind of shows how the “particle-antiparticle pair popping out of the vacuum and one falls in the black hole” interpretation of Hawking radiation is not entirely accurate. The thermally occupied state is made from the bₖ operators but those are made from bₖ = u aₖ + v aₖ† which in the pre-Bogoliubov picture is a linear combination of a particle and anti-particle … and so is bₖ† = u* aₖ† + v* aₖ. Again, we are looking at scalar fields (so did Hawking!) so particle = antiparticle, but nevertheless you need this alternative basis — not the “usual” one.

13. This paper mentions a lot of items in this post! In addition to light cone coordinates, it has the mode expansion and scalar fields. In a bit of foreshadowing, the paper concerns a transformation (Soper-Yan) that is uncontroversial in some contexts can actually lead to incorrect conclusions when it is not used carefully. The prior paper proved a relativistic version of an old many body theorem from the ‘50s and the Soper-Yan transformation would cause that theorem to be violated if it ended up in observables. This paper extended that result to include, essentially, “edge effects”.

14. In fact, it is explicitly cut off.

15. A bit of subtlety here as in the Penrose diagram there is a 2-sphere at every point (except x = 0) representing the other two dimensions of space that aren’t pictured.

16. I do want to say that ostensibly the quantum gravity we’re supposed to be getting hints of here (as well as AdS/CFT-like views) say unitarity is preserved and “akshually” that information isn’t inaccessible so maybe obliterating it is a problem? Maybe? Also I keep thinking of this (youtube) whenever I say “it’s fine”.

17. I should note that the purported applications of the Unruh effect to observed phenomena are usually alternative explanations of effects (or sometimes just poor first order approximations).

18. Another note: the Bogoliubov transformations only have empirical validation for non-relativistic systems (condensed matter physics). I am unaware of relativistic applications outside of Hawking radiation and neutron star physics (neither of which have observations to corroborate the use of those transformations).