General Relativity Is Not Deterministic

What is the proper domain of (in)determinism? What technically makes a theory non-deterministic?

You’ve probably heard that general relativity is a “deterministic theory.” Technically speaking, this is false.

The reasons are quite subtle, but physicists like Penrose and others know about it intimately. Today we’ll learn:

1. What is general relativity precisely?

   • Beyond the bowling ball on a rubber sheet

2. What is determinism precisely?

   • And then local vs. global determinism

3. What is global hyperbolicity precisely?

   • And why it’s not just overscrupulous math

4. (and finally) What do all of these have to do with one another?

I’m going to speak at two levels simultaneously. One for the person who wants meticulous detail (that’s where the sources come in), and one for the person who just wants the crux of the argument. For the latter person, you have to keep in mind that nearly every claim, when simplified, is replete with nuances that challenge it. That’s why the technical rigorous podcast format is an attempt to ameliorate the predicaments of lossily compressed messages, and you can hear more about my thoughts on that here against “explain like I’m five”-explanations.

So… why is saying “general relativity is deterministic” technically a false claim that needs to be caveated? Not “false in some punctilious sense that only philosophers care about.” False in the sense that there exist perfectly valid solutions to Einstein’s field equations where specifying the complete state at one time does not uniquely determine the future. And I’m not talking (only) about singularities. I’m talking about regular, smooth regions of spacetime where your future simply… isn’t determined yet.

I find this super interesting because it’s unlike quantum mechanics where you at least get probabilities. In some sense, it’s worse!

What is General Relativity?

You've heard of EFE, but what is the relationship between them and general relativity?

General relativity is a five-fold package. It comprises:

1. Theoretical principles: The equivalence principle (physics in a freely falling frame locally reduces to special relativity) and general covariance (physics doesn't depend on your choice of coordinates).

2. Mathematical scaffolding: Pseudo-Riemannian geometry on 4-dimensional manifolds with Lorentzian signature (eg. (-,+,+,+)).

3. The field equations:

\(R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} \)

4. Geodesic equation: This tells you that free particles follow the “straightest possible” paths in curved spacetime…

\(\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \left(\frac{dx^\alpha}{d\tau}\right)\left(\frac{dx^\beta}{d\tau}\right) = 0 \)

5. Physical interpretation (sometimes): The curvature of spacetime is gravity… not that gravity “causes” curvature, but that they’re identical.

The above five form a package deal. You can’t just take the equations without the rest.

Or in coordinate-free notation (for those who like their Sashimi-physics):

• The metric g is a section of S²Tˣℳ (symmetric 2-tensors).

• The field equation becomes: G = Ric - ½ scal ⋅ g = 8π T, where Ric is the Ricci curvature endomorphism, and scal is the scalar curvature function.

• Geodesics satisfy: ∇γᵢγᵢ = 0, where ∇ is the Levi-Civita connection.

• Everything lives on the tangent and cotangent bundles… no coordinates needed.

This makes it manifest that GR is about intrinsic geometry, and not these coordinate gymnastics. Anyhow…

What is Determinism?

You probably have an intuitive sense of determinism. It’s something like if you know everything about the universe right now (whatever that means), you can predict everything about the future. But let’s be precise about what physicists and philosophers actually mean by this term.

Deterministic Theory: A theory is deterministic if the complete state of a system at any given time, combined with the laws governing it, uniquely specifies all future states.

Notice the key words: “complete state,” “uniquely specifies,” and “all future states.” Each of these matters.

So then… what counts as the “complete state”? And what if there’s no such thing as a “complete state at a given time t”?

This brings us to a distinction that most discussions gloss over:

• Local determinism: If you specify initial data in a small region of spacetime (technically an “open set” U), the equations uniquely determine what happens in the immediate future of that region. Mathematically: data on U at time t₀ uniquely determines the solution in the domain of dependence D⁺(U) (the region that can be causally influenced by U).

• Global determinism: If you specify initial data across the entire universe at one moment, the equations uniquely determine the entire future of the universe. Mathematically: data (hᵢⱼ, Kᵢⱼ) (the induced metric and extrinsic curvature) on a Cauchy surface Σ (a spacelike slice that every causal curve hits exactly once) uniquely determines the spacetime (ℳ, g) (the manifold and its metric) in the future D⁺(Σ) (all points causally influenced by Σ).

You may think these two are the same. After all, isn’t the global just made up of the local just taken as a totality? In flat spacetime, this is trivial, but it becomes tricky with curvature.

There are solutions to the field equations of Einstein (that #3 above) where you literally can’t define a “moment across the entire universe.” Not because you’re not clever enough (you’re an intellectual beast!), but because you can prove they don’t exist. In other words, in general relativity, there may not be a way to “slice” spacetime into “all of space at time t.” And without that, you can’t even formulate what “global determinism” means, even though you do retain “local determinism.”

Which brings us to our third concept…

What is Global Hyperbolicity?

You’re likely thinking: “Okay, so some spacetimes don’t have some manner of slicing them into space-at-a-time. Big deal. Just work with the ones that do.”

Great! That’s exactly what physicists mostly do. They give this condition a name: global hyperbolicity.

Global hyperbolicity: A spacetime (ℳ, g) is globally hyperbolic if it “admits” (a mathematician’s word for “allows for the existence of”) a Cauchy surface, which is a spacelike slice Σ that every inextendible causal curve (timelike or null) intersects exactly once.

Yes, that sounds like gibberish. Yes, it still somehow makes sense. And yes, I’ve lost my mind. All three can be true…

Anyhow, let me unpack that.

A Cauchy surface is a “snapshot” of the entire universe at one instant. Every particle’s worldline, every light ray’s path, they all cross this surface exactly once. Not zero. Not twice. Just once. No looping back. And no missing it entirely. The existence of this guy is what even allows you to say “the state of the universe at time t.”

If your spacetime is globally hyperbolic, you’re golden, yo. The Choquet-Bruhat–Geroch theorem guarantees that initial data (hᵢⱼ, Kᵢⱼ) on your Cauchy surface uniquely determines the maximal globally hyperbolic development (try saying that three times fast).

So what’s the problem? Not all solutions to Einstein’s equations are globally hyperbolic.

And I’m not talking about exotic mathematical curiosities that only sadists who read Counterexamples in Topology would find prepossessing. Some of the most physically interesting spacetimes violate global hyperbolicity:

• Reissner-Nordström black holes (charged black holes): Have Cauchy horizons (surfaces beyond which initial data loses predictive power) inside where multiple inequivalent futures exist.

• Kerr black holes (rotating black holes): Same story. Cauchy horizons where infinitely many distinct extensions of spacetime are all equally valid.

• Anti-de Sitter space: Has timelike infinity you can reach in finite time. This means predicting the future requires specifying boundary conditions at that boundary! Thus, evolution isn’t determined by the initial state alone. You may think this is like Newtonian mechanics, but this timelike boundary can actively send new signals into the system's future, unlike the causally inert spatial boundary of Newton.

• Gödel universe: Contains closed timelike curves where you can travel to your own past. This is something I referenced in my Gödel video.

In these spacetimes, knowing everything about “now” decidedly does NOT tell you everything about “later.” Not because you’re missing information, but because the information literally doesn’t exist yet!

So what happens at a Cauchy horizon? It seems like your future would just “stop,” but the Einstein equations actually have multiple, dissimilar solutions beyond that surface. It’s as if the universe reaches a point and says, “I have no idea what happens next, pick any of these infinite options.” No probability distribution. No selection principle. Just… ambiguity.

Quantum mechanics is of course infamous for its uncertainty principle, but it at least gives you probabilities! General relativity, which is supposedly a paragon of determinism, can leave you with genuine ambiguity. In other words, Schrödinger’s cat doesn’t know if it’s alive or dead, but at least it knows the odds! An observer crossing a Cauchy horizon? God doesn’t even play dice. He just shrugs…

Putting It All Together: Why General Relativity Isn't Deterministic

Now you see how these three concepts play together.

General relativity has the field equations. These are perfectly deterministic locally (that’s a theorem in PDEs on this), which means in any “small patch” of spacetime, if you know the conditions there, you can evolve them forward uniquely (up to a choice of coordinates).

However, when you zoom out to the global picture, then there can be problems. Without caveating by restricting yourself to global hyperbolicity, you can’t even define what “the state of the universe at time t” means. Further, when global hyperbolicity fails (such as when Cauchy horizons exist), the equations give you various non-equivalent answers for what occurs beyond that surface.

Let’s take a specific example: At a Cauchy horizon in a Reissner-Nordström black hole, an observer would see the entire future history of the outside universe compressed into finite time (infinite blueshift, if you’re keeping track). Beyond that point, the Einstein equations become underdetermined as an initial value problem. You can extend the spacetime, yes, but the cost is that there are not just many ways to do it. There are infinite ways. All of them equally valid mathematically.

What would this feel like physically? I don’t know. I’ll let you know when Musk sends some minions there. As far as I can tell from safely behind my LCD screen, information would emerge from nowhere. Not quantum uncertainty where at least you get probabilities. Just new information appearing without cause.

The “Pathological” Cop-Out

You’ll hear (some) physicists dismiss these examples as “pathological” or “unphysical.” Only some do this though. Most relativists I know are sharp enough to realize there’s no rigorous definition of “pathological!” It’s basically saying “I don’t like this solution and when I look out my window, I don’t see this sort of spacetime, thus you’re a pathology.”

Some attempts at rigor:

• “Unstable solutions are pathological”: Gödel universes are unstable under perturbations. However, some Cauchy horizons can be stable in certain contexts (eg., in charged black holes with a cosmological constant, Cardoso & Costa et al.). Oops.

• “Solutions violating energy conditions are pathological”: Except quantum fields violate these routinely. Dark energy violates the strong energy condition. Double oops.

• “Cosmic censorship”: Ah, the universe’s OnlyFans paywall… This is Penrose’s conjecture that nature “censors” naked singularities behind event horizons. It’s unproven and has potential counterexamples. He knows this and spoke to me about it personally here.

The truth is that we have no principled way to exclude these non-deterministic solutions. The space of solutions to Einstein’s equations is infinite-dimensional, and (as far as I know) we have no natural measure to say something quantitative like “‘most’ solutions are globally hyperbolic.”

But who cares about global determinism? Local determinism is all we need, no?

Well, the problem is that in a spacetime with closed timelike curves, even local determinism becomes suspect. You could have a region where the future loops back to influence the past. This creates a consistency condition that constrains your “free” initial data. The Gödel universe has CTCs through every point! It’s quite trippy to see visualizations of this (here’s one from M Buser, E Kajari, and W P Schleich). In this Gödel universe, global determinism fails and local predictability is severely compromised by global consistency conditions (though keep in mind that local PDE well-posedness still does holds on small acausal patches).

Another problem is that if nature allows naked singularities, then information can appear from the singularity with no prior cause. This violates determinism in a finite region, not at some abstract infinity.

TLDR: If you have to place a qualifier on general relativity (such as “only look at globally hyperbolic spacetimes”), then this is a tacit admission that general relativity, unqualified, is indeed not entirely deterministic; else, you’d have no need to qualify.

The Quantum Gravity Cop-Out

One may hope quantum gravity saves determinism. After all, these classical solutions wouldn’t survive quantization, correct? Well, firstly, quantum gravity has quantum indeterminacy but even disregarding that, the irony is that most approaches to quantum gravity assume global hyperbolicity from the start!

It would be like assuming determinism to prove determinism.

Much like how John Norton shows Newtonian physics has indeterminacy in it as well, unless you assume the Lipschitz continuity condition which amounts to assuming the very determinism you’re attempting to prove.

Canonical quantum gravity, loop quantum gravity, and even many string theory formulations require Cauchy surfaces to even define the theory (technically, yes, worldsheet amplitudes can be computed on non-globally hyperbolic backgrounds like Gödel spaces or orbifolds with CTCs… it’s specifically the S-matrix formulation and unitarity requirements that want global hyperbolicity, not the worldsheet consistency conditions themselves).

So Is General Relativity Deterministic or Not?

My view is that by the strict definition of what a deterministic theory is, namely that we always have the future being entailed uniquely by the past, then the answer is “no, general relativity is not a deterministic theory (in this sense).”

A more comprehensive answer would be that general relativity is a theory whose solution space contains both deterministic and non-deterministic evolutions. Also, the physical realizability of these non-deterministic solutions is so far empirically undetermined. In other words, it’s an open empirical question.

What we can say definitively:

• The Einstein equations are locally deterministic

• Global determinism requires global hyperbolicity

• Many physically interesting solutions lack global hyperbolicity

Perhaps the better way to define determinism isn’t a property of theories but a property of specific solutions. And in a universe described by general relativity, whether your future is determined could depend on where you are in spacetime.

Now that’s a plot twist even a five-year-old could appreciate. Well, a very precocious five-year-old who happens to know differential geometry…

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—Curt Jaimungal

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