Look. General Relativity is Weird.

It makes quantum mechanics seem practically staid by comparison.

If I asked you to name the early 20th century theory that has massive empirical support but is also one of the weirdest parts of physics, you would probably say Quantum Mechanics (QM). But once you get past the non-intuitive non-commuting operators you are left with a theory that is basically “preserve information through time using complex numbers”¹. There’s a sensible scale ℏ you can use in an expansion that keeps the non-intuitive aspects tidy and mostly out of view of our human senses. Particles can be waves — but at least we “get” waves².

The real answer is General Relativity (GR).

The problem of time

The root of the so called “problem of time” is that time is a parameter in QM but is a coordinate in GR. These are of course loaded, technical terms — but it is a conceptual chasm in terms of physics. And here GR is the weirder one. There’s no contest.

QM has our Newtonian sense of time — an abstract, external clock giving a real number t with units that parameterizes how a system changes between time t₁ and time t₂ via e.g. unitary operator U = exp(−𝒾 H (t₂ − t₁)/ℏ) which “solves” the Schrodinger equation:

\(\displaystyle \hat{H} | \psi \rangle = i \hbar \frac{\partial}{\partial t} | \psi \rangle\)

Even when you add special relativity to get Quantum Field Theory (QFT), this still “works” because all Special Relativity (SR) does is stretch time. SR is basically GR if you turn off gravity — a description of “flat” space with a constant metric that isn’t curved by matter. Once you couple the metric to mass-energy it can twist time into space.

In GR, time is on an equal footing with space (i.e. a coordinate) and therefore the Einstein field equation can bend one into the other. This is part of a symmetry that defines GR — diffeomorphism invariance. It’s a general bendiness of spacetime that only generally preserves the fact that if two worldlines winding through the Minkowski 4D spacetime M intersect before a diffeomorphism is applied, they’ll intersect after. It also means the spacetimes (Newtonian or Special Relativity) used in QM which break up M into a “stack” (foliation) made up of 3D spacelike slices Σ³ lined up along “time” T so that M = Σ³ × T is not an invariant way to describe things in GR.

There is a Hamiltonian approach to GR that does this in a “sensible” way that preserves the diffeomorphism invariance by adding a bunch of new equations that among other things allows you to do numerical simulations. It also provided a route to “quantize gravity” but following that primrose path yields

\(\displaystyle \hat{H} | \Psi \rangle = 0\)

This is the abstract operator form of the Wheeler-DeWitt equation which, in a way, says the universe doesn’t evolve in time or that “time drops out” (relative to the Schrodinger equation above).

That’s the primary irreconcilable conceptual issue between QM and GR. Some others follow from that — e.g. what does probability (key to QM) mean if time is generally observer dependent? There are lots of different ways people have tried to deal with this, but none of them have achieved that “eureka” moment. You might hear that QM and GR disagree about small scale physics — lengths on the order of the Planck length. But that’s just a guess based on dimensional analysis. Could be true. Hasn’t led to an answer in over 50 years though. And people thought string theory should have given us something by now³.

Regardless, the main point here is that GR is weirder than QM in its treatment of time. A unification of these two non-overlapping magisteria will upend one or both of these non-overlapping views of time.

Local physics depends on global constraints

I posted this footnote (from here) on bluesky the other day:

The fact that the conserved energy-momentum is not a local observable in GR follows from the principle of equivalence: there is always a frame of reference in which the gravitational field vanishes locally.

Well, when you put it like that …

GR is, in a compactified form, governed in part by the Einstein field equation⁴:

\(\displaystyle G_{\mu\nu} = \frac{8 \pi G}{c^{4}} T_{\mu\nu}\)

The LHS is the Einstein tensor Gμν which describes spatial curvature — the gravitational field. The RHS consists of some constants for units⁵ and the stress-energy tensor Tμν (i.e. energy-momentum). If you can transform away the Gμν, at least locally, you can transform away the Tμν. This is yet another problem from not only from the perspective of QM, but for our intuition. Diffeomorphism invariance is quite the unstable foundation for the fabric of reality.

Erik Curiel goes a bit further in describing the weirdness in his primer on energy conditions (more on those in a minute):

… one can always find a conformal transformation of the metric that is the identity outside the open set and non-trivial [inside the open set] such that at some point in the set the transformed stress-energy tensor will yield whatever one wants on contraction with a timelike or null vector

He continues a few sentences later:

Now, this fact poses a serious problem for any attempt to formulate a notion of dynamical evolution that would support any minimal notion of predictability or determinism. ... If one cannot give principled reasons for why exactly one of those spacetimes and no other is the natural result of dynamical evolution ... according to the Einstein field equation, then one has captured one sense in which the Einstein field equation may “have no content”.

This makes GR, in a sense, manifestly non-local in order to “have content”. It depends on global constraints (boundary conditions, or in a minute, energy conditions) to have meaningful observables.

The calculation of Hawking radiation discussed in my last post relies on the fact that the black hole exists in an extensive enough region of spacetime that you can’t transform it away completely. Hawking radiation doesn’t rely on energy conditions (we’ll get to in a moment) to obtain the black body spectrum, but it does rely on the Schwarzschild solution. The thermal occupation of the vacuum state technically occurs with just quantum field theory in an accelerated frame — the Unruh effect. But the Unruh effect is more “controversial” precisely because the “surface gravity” of a black hole (g = c⁴/4GM) isn’t frame-dependent — the Schwarzschild solution operates as a global constraint.

Again this makes GR is weirder than QM — the former is in a sense incomplete. Not incomplete as in it doesn’t describe everything. It is incomplete in the sense that it requires additional assumptions to make predictions about physical processes it is designed to handle — the changes to a spacetime containing purely classical mass-energy. Otherwise, the theory is like “sure that can happen” to just about anything. GR is the ultimate “yes, and” improv partner. I guess that’s what you get when you try to couple pure math (the LHS of the Einstein field equations) to incompletely understood physical reality (the RHS).

Energy conditions

The previous two extended quotes were from one of the more fascinating papers I’ve read recently. I wasn’t a GR person in grad school⁶. I was into quarks — there were still things that were relatively interesting and feasible to try to model as a grad student that could be seen in accelerator experiments⁷. But Erik Curiel’s review of energy conditions in GR was wild. It basically inspired this post. Those earlier quotes pale in comparison to his thesis:

The remarkable fact I will discuss in this paper is that such simple, general, almost trivial seeming propositions [i.e. energy conditions] have profound and far-reaching import for our understanding of the structure of relativistic spacetimes. It is therefore especially surprising when one also learns that we have no clear understanding of the nature of these conditions, what theoretical status they have vis-a-vis fundamental physics, what epistemic status they may have, when we should and should not expect them to be satisfied, and even in many cases how they and their consequences should be interpreted physically.

This is physicist talk for “we say this stuff because it works in calculations but we really don’t know what it means”. Any time the word “epistemic” comes up in a review paper it’s … quite something. So; an energy condition is a kind of constraint on the RHS of the Einstein field equations — the stress energy tensor “source” of the gravitational curvature tensor that’s on the LHS shown above. If your constraint is wrong it could permit e.g. “superluminal propagation of physically significant structure” (a wild euphemism from the same review). Sounds bad.

The diagonal components of the stress-energy tensor Tμν are⁸ energy density and three components of pressure (x, y, and z). Let’s play dimensional analysis. What can you make from density ρ (mass per volume) and pressure P (force per area)? Velocity:

\(\displaystyle v_{s} = \sqrt{\frac{dP}{d\rho}}\)

The subscript “s” is for sound — this is the “speed of sound” inside your mass-energy. Without an energy condition, there is no particular reason this doesn’t exceed the speed of light c. We don’t think it should! But without energy conditions (that we don’t understand) it very well could.

There’s a really simple pedagogical illustration of this issue here using the kind of simple system you’d encounter in Physics 101 — nothing exotic at all. However the assumptions required boil down to knowing the properties of every kind of energy in the universe (including, say, the vacuum energy of quantum fields) — and whether it will break before it allows “superluminal propagation”. It seems obvious this should follow from some kind of physical principle. Right? Right!?

Now QM (QFT) does allow you to break a classical symmetry of your theory — it’s called an “anomaly”. However anomalies are real, important, and lead to observable consequences; for example, the QCD energy scale is the result of a conformal anomaly. In GR, this is different — instead of telling you how it breaks the maximum speed limit, it just kind of lets you. If you’re game, it’s game.

Curiel believes understanding energy conditions might be a route to understanding quantum gravity and I’m inclined to agree — or at least I agree if you think of “quantum gravity” as a more abstract term like dark energy⁹ giving a name to something we don’t understand. In terms of “weirdness”, this is really a more specific sub-case of the previous one. GR is incomplete. But the way these energy conditions get tossed around in papers — it’s not so much GR but the culture around GR that’s weird¹⁰.

Asymptotic BMS symmetry

Penrose diagram of flat Minkowski space with the various infinities labeled. BMS group symmetry is at null infinity (ℐ ⁺, ℐ ⁻) — the place where massless particles that never encounter anything end up. There are also timelike infinities 𝒾₊ and 𝒾₋ (future and past) as well as spacelike infinity 𝒾₀. Light rays (and massless particles) always travel along 45 degree lines.

This particular “weirdness” has experienced a resurgence in interest over the past decade or so when it was connected to the “Carrollian” limit c → 0 after originally being discovered in the 1960s. Back then several physicists (Bondi, Metzner, and Sachs = BMS, but as always others were involved) tried to get a handle on the boundary conditions at null infinity for asymptotically¹¹ flat Minkowski space. Null infinity is, among other things, where gravity waves end up, and consists of all the infinite ends of light cones. It was expected to have Poincaré symmetry, but it turned out to have a much larger BMS symmetry¹² which has an infinite set of translation operators¹³ dubbed “supertranslations”. This also only happens in four spacetime dimensions¹⁴.

I mean technically a QFT for gravitons would also have this asymptotic symmetry, but it would really just inherit it via the correspondence principle from GR. Therefore I’m giving the weirdness of a massive, unexpected asymptotic symmetry group to GR.

Memory effects

Turns out this one is actually directly related to the previous “weirdness” via an interesting “triangle” relationship between asymptotic symmetries, “soft theorems”, and the subject of this section: memory effects. More on the triangle in a second.

First, a memory effect is precisely that — a set of masses in free fall (a memory effect “detector”) will “remember” that a gravitational wave passed through them by having their configuration permanently altered. The quadrupole moment of the set of masses will be different after the wave passes. Intuitively you might expect a passing gravity wave would cause the masses to oscillate, but as that oscillation dies down it would die down symmetrically — displacements in either direction relative to the initial condition would get smaller and smaller, eventually vanishing. Instead, it leaves a residual phase dependent on the relative orientation of the two quadrupole moments — those of the “source” and the “detector”. It’s weird!

Memory effects have recently been determined to also happen in electromagnetism, but I wouldn’t say that detracts from the weirdness of the gravitational memory effect. Classical EM and GR can both be weird and QM (QED) would just inherit this weirdness via the correspondence principle.

Now back to the triangle. The important parts for us are in this 2014 paper by Strominger and Zhiboedov. The detector state before and after the gravitational wave (the memory effect) are related by a BMS supertranslation¹⁵. The third vertex of this triangle depends on semi-classical gravity¹⁶, but basically says the memory effect and the BMS supertranslation are essentially represented by the addition of zero energy (i.e. “soft”¹⁷) gravitons to the vacuum after the wave passes.

I put this one last because I would say it’s “quirky” more than “weird”. It’s less weird than the EPR “paradox”, but I wanted to include it because a) it’s related to BMS symmetry, and b) I wanted to show GR has a “full spectrum” of weirdness from active contradiction of every other part of physics to aspects where understanding is completely lacking to strange non-intuitive effects.

In closing

I’m not sure, but I think the dominant view in the community of the future of reconciling GR with QM is one where GR loses out. It’s just a low energy effective theory, they’ll say. Sure it’s just dimensional analysis, but when we get to the Planck scale there will be QM but not GR. Susskind might disagree, telling us that GR = QM. With this post I’d like to add that, sure, QM is weird; however, GR is weirder so not only is GR = QM plausible but maybe QM ⊂ GR. From there, it’s only a short journey to SR = QM¹⁸.

1. This is essentially unitarity. As put by Weinberg (1979):

This remark is based on a “theorem”, which as far as I know has never been proven, but which I cannot imagine could be wrong. The “theorem” says that although individual quantum field theories have of course a good deal of content, quantum field theory itself has no content beyond analyticity, unitarity, cluster decomposition, and symmetry. This can be put more precisely in the context of perturbation theory: if one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order of perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles. As I said, this has not been proved, but any counterexamples would be of great interest, and I do not know of any.

2. They’re waves in configuration space so the intuition only helps for single particle wavefunctions. But it’s easy to generalize from N = 1, right?

3. In fact, depending on when you think “String Theory” got started as a purported description of fundamental physics (string theory aka Regge trajectories were first used to try and explain mesons in the 1950s but that’s not String Theory, capital S capital T) it has gone on about as long as all the semi-classical black hole stuff (~ 50 years from the 1970s).

4. I am writing it using the Einstein tensor Gμν since I am writing for a non-physicist general audience with some technical background if my engagement on bluesky or other blogs is any indication. There is almost nothing to be gained by breaking out the curvature tensor Rμν and the Ricci scalar R (which is just the contraction of Rμν with the metric gμν BTW) since I’m not going to go into those components. I’ve seen so many cases where people are talking about GR to a non-physics audience and write down the broken out version instead of Gμν and then never discuss any of the components. This is pure “look at this cool equation” nonsense. Makes me want to just write down the Einstein-Hilbert action density. (This is just R, BTW. The √−g is the Jacobian for the d⁴x and technically present in special relativity.)

5. It is the inverse of the “Planck force” and e.g. here could be considered (proportional to) the maximum tension (force). It is the only “Planck unit” that doesn’t have all three constants G, c, and ℏ in it. However, per my wisecrack, you might also consider the Planck action [ℏ = pₚ × ℓₚ = (ℏ/ℓₚ) × ℓₚ] or the Planck velocity [c = ℓₚ/tₚ] to also lack all three constants.

6. I did buy a copy of the Dover edition of Bergmann’s Introduction to the Theory of Relativity from my local Barnes & Noble when I was in high school that covers both SR and GR (and Kaluza-Klein theory (!) as well). While I probably didn’t really understand tensor analysis until I took the class in college, I can say mere exposure probably helped. The book is from 1942 (though updated in 1976) — but it probably contributed to my feeling that GR was “old fashioned”. I mean my tensor analysis textbook was from 1964 and even that seemed ancient.

7. The CQS curves in Figs 2 & 3 here are from my thesis project.

8. Imagine several caveats in this footnote.

9. Was going to link to Wikipedia in the main text of the post but the first line there is wrong. It is not “a proposed form of energy” but rather a name given to an effect that behaves like a negative pressure (in that aforementioned and not-well-understood stress-energy tensor in GR) everywhere in the universe that in its simplest explanation consists of a constant energy density everywhere in said universe including empty space.

10. I’ve read several papers that invoke the null energy condition and my response, given these have been published in respectable journals where no specific guidance was given around the assumption, was “I guess that’s fine”. Reading Curiel’s review changed my view of these assumptions to “you’re just making $#!T up now” unless it’s about modeling an observable effect for which there is empirical / observational data.

11. It is important that it is asymptotically flat which implies something is in the universe (and manifests particular sensible fall-offs for fields). Empty flat space has normal Poincare symmetry at infinity. Part of the weirdness is that you could add a single Hydrogen atom** to empty space and suddenly the symmetry at infinity goes from the Poincare group to the much, much larger BMS group. **Pretty sure you have to add something with a gravitational quadrupole moment (regardless how small) so it can’t just be a single particle — the asymptotics of the Schwarzschild solution do not get you the BMS group. You need propagating gravitational waves.

12. What’s interesting here is that one of the leading answers to the question “what is a particle?” is “it’s a representation of the Poincare group” — more recently physicists have been saying maybe the answer is “it’s a representation of the BMS group”.

13. If you have any experience with spherical harmonics, then you can think of the four ℓ = 0 (m = 0) and ℓ = 1 (m = 0, ±1) components as being the normal translations (t, x, y, and z) and the supertranslations being all the others. The BMS group has a representation in terms of these spherical harmonics (along with other components), so this isn’t just an analogy.

14. It doesn’t happen in d > 4 per the paper. It doesn’t happen in d < 4 since GR is trivial (d = 1 or 2) or doesn’t have waves (d = 3).

15. See the earlier footnote on spherical harmonics; it this case the element of the BMS group includes both the ℓ = 0 and ℓ = 1 “supertranslations” that represent the ordinary translations in space and time as well as ℓ > 1.

16. See my earlier post for my recent “thoughts” on semi-classical gravity. TL;DR = I’m skeptical now.

17. The soft theorems are basically Feynman diagram calculations where additional gauge bosons are attached to the outgoing particles in the interaction. They’re super boring in your QFT class which is almost certainly just talking about the soft photon theorem in QED.

18. Be warned. This is my own crackpot take.