Category: physics
Type: argument
Associated thinker(s): Albert Einstein, John Earman, John Norton
Original publication or approximate period: Einstein developed the argument in 1913 and published a version in 1914; modern reconstruction by Earman and Norton (1987)
Difficulty: Advanced
Themes: spacetime, determinism, general covariance
Related experiments: Newton’s Bucket; Laplace’s Demon; Einstein’s Elevator
General relativity models spacetime with a metric field satisfying Einstein’s equations. General covariance means the equations retain their form under smooth coordinate transformations. In the active formulation relevant to the hole argument, a smooth diffeomorphism applied to the metric and other fields maps a solution to another mathematical solution. This active transformation is not merely a change of coordinate labels.
Einstein’s original version imagined a region containing no matter. In the modern construction, choose a region H and a smooth diffeomorphism that is the identity outside H but nontrivial inside. Transform the metric and other fields together: the resulting solution agrees with the original outside H and differs inside as a mathematical model.
If spacetime points have primitive identities and the two models count as distinct physical worlds, the same exterior data allow different interiors, and the equations do not select one. If diffeomorphism-related models count as the same physical possibility, this is representational non-uniqueness rather than physical indeterminism.
Which assumptions about spacetime points and equivalence between mathematical models would turn this underdetermination into physical indeterminism rather than two descriptions of one physical state?
Einstein developed the argument late in 1913 against generally covariant gravitational equations, then returned to general covariance. In 1916 he defended the theory using the point-coincidence argument. Earman and Norton’s 1987 reconstruction made the modern hole argument an explicit challenge to spacetime substantivalism.
The argument asks how mathematical symmetries relate to physical possibilities: are diffeomorphism-related solutions different worlds, or different descriptions of one world?
Some responses accept Leibniz equivalence: diffeomorphism-related models represent the same physical possibility. A sophisticated substantivalist can still regard spacetime as real while adopting that equivalence; it does not by itself entail relationism. Other responses accept a form of indeterminism or revise which structures count as physically significant. These positions differ, and the argument alone does not settle which metaphysics is right.
The hole is a region of a spacetime model, not a physical object or an actual place with missing matter. Einstein’s original version used a matter-free region; the modern argument does not depend on the hole being an empty physical cavity. The argument does not show that general relativity is wrong. Its force depends on how diffeomorphic models are interpreted.
Earman and Norton formulate the argument for a broad class of local spacetime theories. Related analyses use it as a template for examining gauge freedom, but the details depend on the theory and on which transformations represent the same physical state.
Norton’s SEP entry was consulted for the distinctions between passive coordinate covariance and active diffeomorphisms, the modern construction, and the response literature. Earman and Norton’s full 1987 paper was read at the relevant passages (pp. 515–525). Norton’s translated/transcribed account was consulted for Einstein’s 1914 §12, p. 1067, and his 1916 point-coincidence argument; the complete original German papers were not directly consulted. Stachel’s 2014 review was read at §§1, 1.1, and 1.3. The modern Earman–Norton argument is not presented as Einstein’s exact original formulation.
Research check: 2026-11-01. This is an original educational explanation, not a substitute for the cited primary and scholarly sources. Additional comparisons are teaching prompts, not established historical variations.