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

Growing skepticism of not just the Planck length but GR + QM

Because I am extremely cool, I spent part of the long Memorial Day weekend watching a recent talk (as in just last month) by Raphael Bousso for the Simons Foundation available on YouTube. I am something of a Bousso fanboy — and while it usually takes me the better part of a year to understand one of his papers nearly every one has some kind of side comment that sticks with me for decades. This series of posts (starting here and continuing to at least the present one) has been mulling one of his comments in the conclusion of a now 20+ year old paper (quoted e.g. here).

The comment this time comes in at the beginning — and I will paraphrase instead of transcribe: it is remarkable that classical gravity somehow “knows” about its quantum state. How? The Bekenstein-Hawking entropy. A bit of a quibble — it should be semi-classical gravity, but that doesn’t really matter because there are other (better) examples of a similar idea at work. The Dirac equation somehow “knows” about anti-matter. Relativistic quantum mechanics somehow “knows” about quantum field theory. Newton’s laws somehow “know” about the symmetry principles that organize all of modern physics. And in one that hasn’t been realized … the Standard Model somehow “knows” about supersymmetry¹.

My bold claim (crackpot theory) is that somehow special relativity “knows” about quantum mechanics². To riff on the proposition in Susskind’s dadaist open letter, SR = QM. So my mind immediately turned to the question of whether special relativity could know about ℏ — or at least its units. And it in fact does.

Let’s look at both the real space and momentum space volume elements d³x and d³p; we’ll perform a Lorentz boost on the latter given the “on shell” constraint E² = p² + m²:

\(\displaystyle \begin{eqnarray} d^{3}p& \rightarrow &\frac{p^{0}}{\tilde{p}^{0}} \; d^{3}\tilde{p}\\ & \therefore & \frac{d^{3}p}{p^{0}} \quad \text{is invariant} \end{eqnarray} \)

There are lots of ways you can work with the real space volume element d³x, but my favorite is kind of a cheat. We say 1) d⁴x is Lorentz invariant, 2) m is Lorentz invariant, and 3) d𝜏 (proper time interval) is Lorentz invariant — therefore:

\(\displaystyle \frac{m}{d\tau} d^{4}x = m\frac{dx^{0}}{d\tau} \; d^{3}x = p^{0} \; d^{3}x\)

There are of course other options. Combining these invariants, this means:

\(\displaystyle p^{0} \; d^{3}x \; \frac{1}{p^{0}} d^{3}p = d^{3}x \; d^{3}p\)

is invariant. Therefore an infinitesimal phase space volume is Lorentz invariant. In quantum mechanics, that “infinitesimal volume element” is ~ ℏ³. It’s non-trivial! And the only other fundamental invariants are things like the speed of light c or p² = −m² (really anything with all the indices contracted).

I was reminding myself of this LIPS³ exercise when it struck me that this was far more complicated in General Relativity (GR). On the momentum side, you start with

\(\displaystyle \frac{d^{4}p}{\sqrt{-g}} \delta (g_{\mu\nu}p^{\mu}p^{\nu} + m^{2}) \rightarrow d^{3}p \int \frac{dp^{0}}{\sqrt{- g}} \delta (g_{\mu\nu}p^{\mu}p^{\nu} + m^{2})\)

The real space integral gains a factor of the Jacobian determinant as well, so the invariant quantity is

\(\displaystyle \sqrt{-g\;}d^{4}x\)

those √−g pieces can cancel in the 4-volumes but in general for d³x d³p you get pieces involving the metric that only go away for diagonal / conformal / constrained metrics (see e.g. here). At a fundamental level, phase space, and therefore ℏ, is not invariant in GR⁴. We’ve stumbled upon a “pick two” trilemma:

A “pick two” trilemma between fundamental constants G, ℏ, and c. G and c are invariant in General Relativity, and c and ℏ are invariant in Special Relativity. However, ℏ is not invariant in General Relativity.

I am not 100% sure what the theory is where you pick G and ℏ, but selecting G, c, and ℏ simultaneously could result in some serious contradictions. The one that pops out like a slasher in a horror movie is ℓₚ = √Gℏ/c³. A fundamental length scale made from G, c, and ℏ is a problem⁵. First because ℏ isn’t invariant in the theory that brings in G, but also because a fundamental length scale like ℓₚ breaks Lorentz invariance⁶.

It’s not like this hasn’t been noticed before. There are attempts at rectifying this “paradox” — e.g. doubly special relativity (in fact related to loop quantum gravity). What grabbed me while I was watching Bousso’s talk was that his observation that semi-classical gravity “knows” about quantum gravity is entirely dependent on this (in my view) unjustified combination that produces ℓₚ.

A semi-classical expansion is (loosely) a treatment of effects starting at ~ o(ℏ), but willy-nilly dropping ℏ into a theory with G and c (i.e. GR) allows you to create ℓₚ and therefore of course the area of a black hole event horizon is measured in units of ℓₚ² … I mean, what else can it be?⁷

Personally, I can come to no other conclusions than:

• A semi-classical treatment of gravity (i.e. willy-nilly adding ℏ to GR) is nonsense

• The Planck length is an artefact of the erroneous semi-classical treatment of gravity and is therefore also nonsense

Or, another way …

A fortiori this would mean “quantum gravity” — any attempt to combine general relativity with ℏ — is in general nonsense. So what does that leave us? Well, entropic gravity is a start (e.g. here, here). If G (and thus gravity) is not fundamental but rather emergent as a thermodynamic quantity far from the Planck scale (talking ℏ, not ℓₚ) then putting in G with c and ℏ would be like putting in Avogadro’s number — something tied to measurements at our human scale, not fundamental physics.

This isn’t the same thing as saying string theory is a dead end⁸. This is saying anything based in QFT or GR that references the Planck length is a dead end. This encompasses the vast majority of black hole physics over the past 50 years. Y’all been counting Planck areas on the head of a pin.

I fully understand that the idea that “the Planck length isn’t Lorentz invariant” has been around for almost as long as the Planck length has. My “critique” is somewhat deeper than that — if combining GR with a “semi-classical” approach allows you to add an ℏ to G and c you are not only allowing the creation of a Lorentz-invariance violating physical constant (ℓₚ) but combining a general-covariance-violating quantity (ℏ) with your otherwise completely consistent theory (GR).

There are lots of other reasons you shouldn’t just take stuff from QM and plop it in GR (e.g. time is a parameter in the former but a coordinate in the latter). However, adding ℏ seems particularly problematic because it interferes with the “superpower” of physics — scaling and dimensional analysis. Regardless, from now on I am going to be on guard whenever I see G and ℏ together.

Update 30 May 2025:

I should add that none of the above constitutes a “proof” that “debunks physics” or whatever. It just 1) makes me want to re-read the stuff that claims entropic gravity has issues, and 2) nudges me in the direction of “black hole thermodynamics is probably right” to “unsure”. Also it was either autocorrect or a brain malfunction, but in one of the footnotes I wrote “election” instead of “electron”. Twice.

I also want to add a couple of “counter arguments” to my “Hawking is wrong because of dimensional analysis” claim above:

• Anomalies: In QFT there are symmetry breaking anomalies that are perfectly valid results. One of them goes by the name of “dimensional transmutation” (which is kind of an archaic term and we’d just say “conformal anomaly”) where an approximately scale-free theory (QCD) spontaneously generates an energy scale from renormalization.

• Emergent quantities: Temperature exists and as an energy derived from motion it may or may not be invariant special relativity (only the rest masses of the constituent particles are invariant). People go back and forth on it. Einstein and Planck were two such people. Here’s a more recent paper. (My view is that it is emergently Lorentz invariant at large 4-volumes where it just looks like an internal energy.)

• Dualities and holographies⁹: AdS/CFT relates a quantum field theory to gravity (albeit stringy supergravity) so it is entirely possible stuff like ℏ leaks through. In the sense that the horizon of a black hole is a “boundary theory”, it’s not completely implausible the bulk space or the boundary theory could import aspects from one another.

Again, instead of proving everything wrong (which I realize is my clickbait-y title), what I wrote in the post above should be taken more as a warning that doing dimensional analysis using G and ℏ together or their combination ℓₚ should come with some big caveats. Basically to say to Bousso — no, it’s not remarkable semi-classical gravity has ℓₚ in it because you put it there.

Update 31 May 2025:

Kinda feel like my footnote discussion of the Compton wavelength (a length scale made from ℏ, m, and c) and the Schwarzschild radius (a length scale made from G, M, and c) wasn’t quite as honed as I would have liked. I think I can put it more clearly and usefully promote it to the main text. The key bit is that c, G, and ℏ are framework physical constants whereas a mass (m or M) is a system model. The speed of light tells you about Minkowski space — the relationship of that 1 to the 3 in the SO(3,1) symmetry of Lorentz invariance in special relativity. But you can set c = 1 and get away from our human scale engineering units. Newton’s constant tells you about how much spacetime curves for a given mass (energy) in general relativity, but again you can set G = 1 and get away from our human scale. Similarly Planck’s constant tells you what the scale of quantum mechanics is — the commutator [x, p] = 𝒾 still makes things quantum without ℏ. These are all aspects of the universe that ostensibly exist¹⁰ even in empty space. When you add a particle to this space, you’re kind of already breaking a symmetry — it’s no longer uniform everywhere. “How far away is the particle from my point of observation in this theory?” is now a germane question and there are Lorentz invariant ways to answer it. In particular, the answers go as ~ ℏ/mc in QM, ~ 2GM/c² in GR, and ~ c 𝜏 in SR.

Putting G, ℏ, and c in the same theory gives you a fundamental framework length scale. I can ostensibly ask how far away two points are in a complete empty space before I put a particle in it. The answer is ~ ℓₚ. But what does that even mean? Quantum foam? Information limits? Fundamental discreteness of space? Seems like you’re positing some pretty bold claims just from dimensional analysis — QFT and GR have no empirical overlap (yet) so their numbers don’t just go together naturally. Maybe a good analogy is like combining two LEGO sets that result in piece combinations that allow you to employ “illegal” build techniques.

Also — I did get asked what about action (e.g. Einstein-Hilbert action) in GR. That has units of action and is Lorentz invariant, right? Yes, but it is an integral over the entire spacetime. I agree — it could be a basis for a fundamental constant with units of action that could be merged with G and c to create a fundamental scale with units of Length (or Length^N), but importantly only for a length scale on the order of the observable universe¹¹. (Hey, there is a such a scale and it is!) In contrast, the Lorentz-invariant 3-phase space d³x d³p is local and infinitesimal, and therefore could be a sensible basis for ℏ.

The more technical answer is that while time can change the rate it passes in flat space in SR based on velocity, there is still a monotonic foliation, and therefore a sensible (i.e. Lorentz invariant) way to separate time from d³x d³p. In GR, there isn’t a good prescription in general for a foliation over time, so splitting it out to get to d³x d³p is system-dependent. Asymptotically flat spacetime (or at least very nearly so) in GR basically is like saying at large enough scales you can approximate a d³x d³p with a sensible time foliation, but only in that aforementioned integral over cosmological distances — no way is that a basis for a fundamental constant at the scale ℏ. I’m pretty sure this is related to the fact that it’s hard to specify local operators in GR but am currently reading up on that [pdf].

1. One of the more remarkable aspects of supersymmetry, and why most physicists are rooting for it, is that it causes the running of the Electroweak and Strong interaction couplings to exactly match up at what is called the GUT (Grand Unified Theory) scale ~ 10^16 GeV.

2. As a side note, the rest of Bousso’s talk contained the construction of “the fundamental complement” of a wedge that is 100% elsewhere (“definitely out of reach”).

3. Lorentz Invariant Phase Space

4. You might ask: Why not just look at d⁴x d⁴p and get not only dx dp ~ ℏ but dt dE ~ ℏ? Well, time is a parameter in QM (and QFT) and a coordinate in GR (i.e. “the problem of time”) so that second “uncertainty principle” is not really an uncertainty principle (it’s instead this). Additionally, E is not independent of p, but rather constrained by the mass shell condition. Strictly speaking, it’s the x and p commutation relation (and its QFT generalization) that gives us “quantum theory” — so it’s d³x d³p that needs to be invariant.

5. You might object that in QM and QFT you can combine ℏ, m and c into the Compton wavelength — another length scale that should break Lorentz invariance. But 1) this is not fundamental (the theory constants of c, G, and ℏ aren’t quite the same as the mass of your system), and 2) the appearance of the Compton wavelength in electron-photon scattering is not as an observable itself. It’s just part of the equation that gives you the observable (scattering cross section). You can measure the three (Lorentz invariant) parts separately. You could potentially counter then that maybe ℓₚ isn’t an observable — but it is! At least according to the semiclassical gravity approach of the past 50 years. You just have to measure the temperature of a black hole. Hard? Sure. But not forbidden. (Side note: an example in pure GR is the Schwarzschild radius made from G, M, and c.)

6. For a really simple way to see this just think of length contraction.

7. I tried to look this up, but didn’t find any particular website reference that describes it. It is a fundamental aspect of physics education — related to dimensional analysis, sure, but it’s a bit deeper. When considering a problem, you generally only have a few fundamental or problem-specific scales available so the answers are almost certainly the proper combination of those scales. The example I like to give is the energy of a bound electron in a Hydrogen atom. The mass of an electron is so much smaller than a proton so it’s the only “energy” (E = m c²) scale. The electromagnetic interaction gives you factors of α per photon coupling so the energy level should be E₀ ~ α² mₑ with an o(1) factor out front. And it is! The o(1) factor is 1/2. (Side note: another answer to the “what else can it be?” question regarding the surface area of a black hole just using GR is ‘4π’ — that is to say you should measure the surface area in units of the Schwarzschild radius. Sometimes “what else can it be?” is ambiguous, but it’s almost always a good start.)

8. In fact, string theory gets around this problem by having a length scale (string tension) at the heart of the theory. It’s related to a duality R → 1/R so it doesn’t cause an issue with Lorentz invariance. In fairness, I should add loop quantum gravity, too. It says “fuck Lorentz invariance”.

9. This is pluralized purely for the alliteration.

10. Although as noted, the G and ℏ seem to be in tension, but GR and QM are “non-overlapping magisteria”, at least empirically speaking. Therefore their “engineering unit” function may not go together.

11. The action scale (call it Ħ?) we’d get is Ħ ~ 2.7×10^88 J-s which is where you might get weird, um, anti-quantum? hyper-quantum? effects. Effects based on not a small scale limit to the action, but a large scale one. Edge effects? I believe this is actually where we get the fact that stuff will eventually appear to lose energy and stop over cosmological distances but don’t quote me on that.