General relativity (GR) is often called deterministic, but this is technically false. While the theory is locally deterministic—meaning small patches of spacetime evolve uniquely from initial data—it can fail globally. In several physically relevant solutions to Einstein’s field equations, specifying the complete state of the universe at one time does not uniquely determine the future. This is not a quantum effect; it is a structural feature of classical GR itself, and it is in some ways worse than quantum indeterminism because it offers not even probabilities—just genuine ambiguity.
What General Relativity Actually Is
GR is a five-fold package, not just “Einstein’s field equations”:
Theoretical principles such as the equivalence principle (physics in a free-falling frame reduces to special relativity) and general covariance (physics doesn’t depend on coordinate choice).
Mathematical scaffolding: pseudo-Riemannian geometry on a four-dimensional manifold.
Einstein’s field equations: the core dynamical equations relating spacetime curvature to matter-energy content.
The geodesic equation: describes how free particles move along the straightest possible paths in curved spacetime.
Physical interpretation: curvature of spacetime is gravity—not that gravity causes curvature, but that they are identical. This makes the common Wheeler slogan (“spacetime tells matter how to move; matter tells spacetime how to curve”) somewhat misleading, since the equations are dynamically coupled rather than causally sequential.
Two Kinds of Determinism
Local determinism: specifying initial data in a small open region of spacetime uniquely determines what happens in the immediate future of that region. GR satisfies this—it is a theorem from the theory of partial differential equations.
Global determinism: specifying initial data across the entire universe at one moment uniquely determines the entire future. This requires being able to define “the state of the universe at time t,” which in turn requires a Cauchy surface—a spacelike slice that every causal (inextendable) curve intersects exactly once.
In flat spacetime, local and global determinism trivially coincide. In curved spacetime, they can come apart.
There exist valid solutions to Einstein’s equations where no Cauchy surface exists at all—not because of insufficient cleverness, but because they can be proven not to exist. In such spacetimes, global determinism cannot even be formulated.
Global Hyperbolicity
A spacetime is globally hyperbolic if it admits a Cauchy surface. This is the condition that makes global determinism well-defined.
A Cauchy surface is analogous to a snapshot of all of space at one instant of time—every particle worldline and light ray crosses it exactly once, with no looping back or missing it.
If a spacetime is globally hyperbolic, a theorem guarantees that initial data on the Cauchy surface uniquely determines the maximal globally hyperbolic development—the largest region that can be predicted from that data.
The problem: many physically important spacetimes are not globally hyperbolic:
Charged black holes (Reissner-Nordström) and rotating black holes (Kerr) contain Cauchy horizons.
Anti-de Sitter space has a timelike infinity reachable in finite proper time, where worldlines simply end.
Gödel universes contain closed timelike curves (CTCs)—paths through spacetime that loop back to their own past.
Cauchy Horizons: Where Determinism Breaks Down
A Cauchy horizon is the boundary of the region predictable from a given Cauchy surface. It is distinct from the Cauchy surface itself (both named after the same mathematician).
At a Cauchy horizon, the Einstein equations become ill-posed as an initial value problem: there are infinitely many mathematically valid ways to extend spacetime beyond it, all equally compatible with the prior data.
No probability distribution or selection principle picks among them. The theory simply does not say what happens next.
Physical interpretation: an observer crossing a Cauchy horizon (e.g., inside a charged black hole) would see the entire future history of the external universe compressed into a finite time. Beyond that point, new information can emerge without any prior cause—not quantum uncertainty with Born probabilities, but genuinely uncaused information.
The speaker’s analogy: quantum indeterminism is “domesticated” (a “good boy”—random but predictable in distribution); GR’s indeterminism is “feral” (like a “Torontonian raccoon”—no rules, no remorse).
Why These Solutions Can’t Be Easily Dismissed
A common objection is that such solutions are “pathological” or “unphysical.” But there is no rigorous definition of “pathological”—it often amounts to “I don’t like this solution” or “I don’t observe it astrophysically.”
Black holes themselves were once dismissed as pathological. The Big Bang was a solution Einstein rejected.
Some argue that unstable solutions should be excluded. Gödel universes are indeed unstable under small perturbations (like a pencil balanced on its tip). But stable Cauchy horizons exist in certain contexts, such as charged black holes with a cosmological constant (work by Cardoso).
Some argue that solutions violating energy conditions are unphysical. But quantum fields routinely violate energy conditions, and dark energy violates the strong energy condition.
Cosmic censorship (Penrose’s conjecture that nature hides singularities behind event horizons) is sometimes invoked to save determinism, but it remains unproven and has potential counterexamples.
The space of solutions to Einstein’s equations is infinite-dimensional, and there is no known natural measure to say that “most” solutions are globally hyperbolic.
Even local determinism can fail in spacetimes with closed timelike curves, because the future looping back to the past imposes consistency constraints that reduce freely specifiable initial data. In Gödel universes, CTCs pass through every point, so both local and global determinism fail everywhere.
If naked singularities exist, information can emerge from them with no prior cause, violating determinism in a finite region of spacetime—not just at some abstract infinity.
Implications for Quantum Gravity
One might hope that quantum gravity resolves these classical pathologies. But:
Quantum gravity will still carry its own quantum indeterminacy.
More ironically, many approaches to quantum gravity (canonical quantum gravity, loop quantum gravity, many formulations of string theory) assume global hyperbolicity from the start in order to define the theory—requiring Cauchy surfaces to even formulate the problem. This risks assuming determinism in order to prove determinism, analogous to how John Norton showed that Newtonian physics has indeterminism unless one assumes Lipschitz continuity (which amounts to assuming what one hopes to derive).
It is worth noting that worldsheet amplitudes in string theory can be computed on non-globally hyperbolic backgrounds (including Gödel spacetimes and orbifolds with CTCs); it is specifically S-matrix formulations and unitarity requirements that typically demand global hyperbolicity, not the worldsheet consistency conditions themselves.
The Bottom Line
Three definite claims:
Einstein’s equations are locally deterministic.
Global determinism requires global hyperbolicity.
Many physically interesting, perfectly valid solutions lack global hyperbolicity.
GR is not a deterministic theory in the strict sense that the future is always uniquely entailed by the past. More precisely, GR’s solution space contains both deterministic and non-deterministic solutions.
Whether determinism holds may be a property of specific solutions, not of the theory as a whole—meaning that in a universe described by GR, whether your future is determined could depend on where you are in spacetime.
The physical realizability of non-deterministic solutions remains an open empirical question.
As the speaker puts it: Einstein said “God doesn’t play dice.” In Einstein’s own theory, God sometimes doesn’t even show up to the table.
