The Hole Argument

The substantivalism–relationism debate was transformed when it moved from Newton's fixed absolute space to Einstein's dynamical spacetime. The sharpest modern argument against substantivalism is the hole argument, first stumbled upon by Einstein in 1913 (it delayed his completion of general relativity by two years), and revived as a philosophical bombshell by John Earman and John Norton in 1987. It purports to show that manifold substantivalism — taking the points of the spacetime manifold to be a real substance — commits one to a radical and objectionable indeterminism, and so should be rejected. It is the point at which the classical Leibniz–Newton dispute re-emerges, in a much more technical form, at the foundations of our best theory of space and time.

This page reconstructs the argument and surveys the responses. It presupposes the general-relativity page's account of the dynamical metric and the notion of a manifold and diffeomorphism.

The setup: diffeomorphism invariance

In general relativity a physical world is represented by a model $⟨M,g_{\mu \nu },T_{\mu \nu }⟩$: a differentiable manifold $M$ of spacetime points, a metric field $g_{\mu \nu }$ encoding geometry and gravity, and a matter field $T_{\mu \nu }$, related by the Einstein field equations. The theory has a crucial symmetry: diffeomorphism invariance. If $\varphi :M\to M$ is a diffeomorphism (a smooth, smoothly invertible point-shuffling of the manifold), and we "drag along" the fields to get ${\varphi }^{\ast }g$ and ${\varphi }^{\ast }T$, then $⟨M,{\varphi }^{\ast }g,{\varphi }^{\ast }T⟩$ is also a solution of the field equations whenever the original was. The equations cannot tell the two apart.

An active diffeomorphism does not merely relabel coordinates; it takes the geometry that was at point $p$ and puts it at point $\varphi (p)$, redistributing the fields over the same manifold points. If the manifold points are real, independently existing things (manifold substantivalism), then $⟨M,g⟩$ and $⟨M,{\varphi }^{\ast }g⟩$ describe genuinely different distributions of the metric over the same substance — two distinct physical worlds.

The hole construction

Now the twist. Choose a diffeomorphism $\varphi$ that is the identity everywhere outside some region $H$ (the "hole" — which need contain no matter; it is just any bounded open region) and differs from the identity inside $H$, smoothly. Then:

• Outside $H$, the two models $⟨M,g⟩$ and $⟨M,{\varphi }^{\ast }g⟩$ are point-by-point identical.
• Inside $H$, they assign different metric values to the same manifold points.

Both are solutions of the field equations. So the physics outside $H$ — including everything on and before any moment prior to $H$ — fails to fix the physics inside $H$: fix all the data outside the hole, and there remain infinitely many equally lawful continuations inside it, differing by the choice of $\varphi$.

This is a radical indeterminism. It is not the ordinary indeterminism of an underdetermined initial-value problem; it says that even a complete specification of the state everywhere outside the hole leaves the state inside the hole undetermined — and the hole can be placed to the future of any Cauchy surface. For the manifold substantivalist, who counts $⟨M,g⟩$ and $⟨M,{\varphi }^{\ast }g⟩$ as different worlds, general relativity comes out as violently, and gratuitously, indeterministic.

Earman and Norton's dilemma

Earman and Norton press this as a dilemma against the substantivalist:

1. If manifold points are substantival, then the two hole-related models represent distinct physical possibilities.
2. Those possibilities agree on all data outside the hole but differ inside it.
3. So determinism fails — and fails not because the world is indeterministic (an empirical question) but as an artifact of a metaphysical doctrine about points.
4. A metaphysics that makes determinism fail a priori, for reasons having nothing to do with physics, should be rejected.
5. Therefore reject manifold substantivalism.

The key premise is (1): only if the points have identities independent of the metric can "the metric being here rather than there" name a real difference. The relationist — for whom there are no bare points, only the structure of relations — sees no difference between the models at all, and so escapes untouched. The hole argument is thus a modern, GR-powered version of Leibniz's kinematic-shift / PII argument: a "difference" that no observation, and no law, can detect is no difference at all.

The responses

The literature since 1987 is large; the main lines:

• Relationism. Take the moral at face value: reject substantival points and read general relativity relationally (or structurally). The models are one physical world redescribed. The burden is then to say what the relata are once the metric itself is dynamical — the field is not obviously "matter," so this is relationism about a very abstract structure.
• Sophisticated substantivalism (the standard response). Keep spacetime as a substance but deny premise (1): adopt Leibniz equivalence — diffeomorphic models represent the same physical possibility. Points are real, but they have no identity across possibilities independent of the metric (an "anti-haecceitism" about spacetime points). Determinism is restored because the hole-related models are the same world. Critics object that this quietly concedes the relationist's core point: spacetime points are not robust individuals after all.
• Structural realism. What is real is neither the bare manifold nor primitive individuals but the structure — the invariant, diffeomorphism-independent pattern of the metric and its relations. The hole argument then simply reveals that individual points carry no physical content; only the structure does. Many regard this as the argument's true lesson and the natural ontology of general relativity.
• Haecceitist bite-the-bullet. A few substantivalists accept that the models differ but deny that this is objectionable — the "indeterminism" concerns only which primitive point plays which role, a difference beneath all possible observation, so no genuine predictive failure. Critics find this a Pyrrhic victory that reintroduces exactly the undetectable distinctions Leibniz banished.

What the argument does and does not show

The hole argument does not refute substantivalism outright: sophisticated substantivalism and structural realism keep a spacetime that exists independently of matter while blocking the indeterminism. What it does show is that the naïve picture — spacetime as a manifold of individually identifiable points over which the metric is spread like paint — is untenable. The identity of a spacetime point cannot outrun the metrical structure it carries. In this sense the argument delivers a qualified victory to the Leibnizian side: not "there is no spacetime substance," but "spacetime points are not the robust, self-identifying individuals the container picture imagined."

Where this sits

The hole argument is the modern climax of the ontological question and of the substantivalism–relationism debate — the static and kinematic shifts reborn in general relativity, with the PII doing the same work against manifold points that it once did against absolute positions. It depends entirely on the theory's dynamical metric and diffeomorphism invariance, and it pushes many toward the structural-realist middle way described in the method page. It also connects to the problem of time: diffeomorphism invariance, including invariance under time reparametrisation, is what makes time so elusive in canonical quantum gravity. The final page of this subfolder considers the opposite extreme — supersubstantivalism, which makes spacetime all there is.