The Third (Unsettling) Option. Not Determined. Not Random.

Summary

This video challenges the common claim that Einstein’s general relativity (GR) is a deterministic theory. While GR is locally deterministic — knowing conditions in a small region lets you predict the immediate future — it fails at global determinism. Many physically valid solutions to Einstein’s field equations lack a property called “global hyperbolicity,” meaning you can’t even define “the state of the entire universe at one moment.” Beyond certain boundaries (Cauchy horizons), the equations produce infinitely many equally valid futures with no way to choose between them. Unlike quantum mechanics, which at least gives you probabilities, GR’s indeterminism offers no odds at all — just raw ambiguity.

Key Takeaways

• GR is a five-fold package — not just Einstein’s field equations, but also the equivalence principle, pseudo-Riemannian geometry, the geodesic equation, and the interpretation that curvature is gravity
• Local vs. global determinism — local determinism (small regions evolve uniquely) holds in GR; global determinism (entire universe at time t determines all future) does not always hold
• Cauchy surfaces are spatial snapshots where every worldline and light ray crosses exactly once — they let you define “the state of the universe at one time”
• Global hyperbolicity means a spacetime admits a Cauchy surface; without it, you can’t even formulate global determinism
• Many real solutions lack global hyperbolicity — charged black holes, rotating black holes, Anti-de Sitter space, and Gödel universes all violate it
• Cauchy horizons are boundaries beyond which initial data breaks down — new information can “leak in” from nowhere, with no cause
• GR’s indeterminism is worse than quantum mechanics — QM gives probabilities; GR gives no distribution, no selection principle, just multiple incompatible futures
• “Pathological” is not rigorous — attempts to dismiss non-deterministic solutions as unphysical (unstable, energy condition violations, cosmic censorship) all have counterexamples
• Quantum gravity may not help — many approaches (canonical QG, loop QG, string theory S-matrix) assume global hyperbolicity from the start, which is circular
• Determinism may be a property of specific solutions, not of the theory itself — whether your future is determined could depend on where you are in spacetime

Detailed Notes

GR as a Package Deal

General relativity is not just one equation. It comprises: (1) theoretical principles like the equivalence principle, (2) pseudo-Riemannian geometry on a 4-manifold, (3) Einstein’s field equations, (4) the geodesic equation for free particle motion, and (5) the physical interpretation that spacetime curvature is gravity. Wheeler’s phrase “spacetime tells matter how to move, matter tells spacetime how to curve” is misleading — these are dynamically coupled equations, not sequential causal steps.

The Determinism Distinction

• Local determinism: initial data in a small open set uniquely determines the immediate future of that region. This holds in GR via PDE theorems.
• Global determinism: initial data across the entire universe at one moment uniquely determines the entire future. This requires a Cauchy surface — a spacelike slice every causal curve crosses exactly once. In curved spacetime, such slices may provably not exist.

When Determinism Breaks Down

In spacetimes without global hyperbolicity, beyond Cauchy horizons, Einstein’s equations become ill-posed as initial value problems. You can extend the spacetime, but there are infinitely many inequivalent ways to do so — all equally valid mathematically. Information emerges from nowhere, without cause or probability.

Example: In a charged black hole, an observer would see the entire future history of the outside universe compressed into finite time. Beyond that point, the equations offer infinite equally valid continuations.

Why You Can’t Just Dismiss These Solutions

• Calling solutions “pathological” has no rigorous definition
• Black holes and the Big Bang were once called pathological
• Unstable solutions aren’t always dismissible — Cauchy horizons can be stable (e.g., charged black holes with cosmological constant)
• Quantum fields routinely violate energy conditions; dark energy violates the strong energy condition
• Cosmic censorship (Penrose’s conjecture) is unproven with potential counterexamples

The Quantum Gravity Hope

Many approaches to quantum gravity assume hyperbolicity from the start — canonical QG, loop QG, and S-matrix formulations of string theory all require Cauchy surfaces. This amounts to assuming determinism to prove determinism, paralleling Norton’s observation that Newtonian physics also has indeterminism unless you assume Lipschitz continuity.

Quotes / Notable Moments

“Schrödinger’s cat doesn’t know if it’s alive or dead, but it at least knows the odds. An observer crossing a Cauchy horizon, on the other hand — God doesn’t even play dice. He just shrugs.”

“Quantum mechanics has indeterminism, yes, but it’s domesticated. It’s random, but it’s predictable in distribution. It’s like a good boy. Einstein’s general relativity, on the other hand, has genuine indeterminism, which is feral… It’s like a Torontonian raccoon. There’s no rules. There’s no remorse.”

“Einstein said, ‘God doesn’t play dice.’ Turns out, in Einstein’s own theory, God sometimes doesn’t even show up to the table.”

“Perhaps the better way to define determinism isn’t as a property of theories, but as a property of specific solutions.”