This debate concerns the ontological status of spacetime as described by general relativity. The standard geometric interpretation holds that spacetime, represented by a four-dimensional manifold equipped with a metric field, exists as a real physical entity in its own right, independently of the matter and events within it. The relationist viewpoint holds that spacetime is not an independently existing substance but is instead constituted by, or reducible to, the relations among material bodies, fields, and events. The dispute traces back to the Leibniz-Clarke correspondence over Newtonian absolute space, but it takes a distinctive form in general relativity because the theory's diffeomorphism invariance raises the question of whether spacetime points have any individual identity apart from the physical fields defined on them. The disagreement is not over the predictions or mathematical content of general relativity, which both sides accept, but over what that mathematical content tells us about what fundamentally exists.

Advocates of the standard geometric interpretation hold that general relativity is best understood, in keeping with its standard formulation, as describing a four-dimensional Lorentzian manifold whose points are connected by a dynamical metric field. On this view the manifold and metric are not bookkeeping devices for tracking relations among matter; they are themselves the physical entity called spacetime, and matter and energy are embedded within it and curve it according to the Einstein field equations. The metric field carries physical degrees of freedom, propagates, carries energy in the form of gravitational waves, and can exist where there is no matter at all, as in vacuum solutions such as Schwarzschild spacetime or in gravitational wave propagation through empty regions. Advocates argue that the very intelligibility of general relativity's central claim, that gravity is the curvature of spacetime, presupposes that spacetime is the kind of thing that can be curved, which requires it to be a real geometric object rather than a mere shorthand for relations among bodies. They point to the existence of vacuum solutions with no matter content whatsoever, including the empty Minkowski spacetime of special relativity and gravitational radiation propagating through otherwise empty space, as evidence that spacetime structure outruns material content and therefore cannot be reduced to it. Proponents further hold that treating the metric field as a physical field on a manifold, the same conceptual category as the electromagnetic field, gives the most natural and mathematically faithful reading of the formalism physicists actually use, with the manifold serving as the field's domain.

Advocates of relationism hold that spacetime is not a fundamental, independently existing entity but is instead grounded in the network of relations among physical events, bodies, and fields. On this view, talk of spacetime points and a continuum existing “underneath” or “containing” physical processes is a useful mathematical device rather than a description of a further layer of reality over and above those processes. Proponents argue that general relativity itself, properly understood, supports this reading: the theory's diffeomorphism invariance means that any two models related by a smooth coordinate transformation represent the same physical situation, so that individual spacetime points, considered apart from the matter and fields located at them, have no independent physical identity. This is taken to show that what is physically real are the coincidences and relations between events, such as the intersection of particle worldlines, rather than the bare points of the manifold that merely provide a coordinate scaffolding for representing those relations. Some relationists hold that the gravitational field itself, rather than being one more field on a pre-existing spacetime, is to be identified with spacetime structure, so that there is no spacetime at all distinct from the dynamical field content of the theory; on this variant, often called the dynamical or structuralist approach, what exists is the network of metrical relations among events, not a manifold of points that those relations happen to occupy. Advocates also point to algebraic reformulations of general relativity that dispense with reference to an underlying point manifold while remaining mathematically equivalent to the standard formulation, arguing that this equivalence shows the manifold of points is dispensable to the physics and therefore not warranted as a fundamental ontological commitment.

In response, advocates of the standard geometric interpretation argue that the hole argument, often cited in support of relationism, shows at most that spacetime points lack primitive individual identity independent of the metric field, not that spacetime as such fails to exist independently of matter. They argue that a substantivalist can accept that two diffeomorphic models represent the same physical possibility, a position known as Leibniz equivalence, while still holding that the manifold-plus-metric structure as a whole is a real, mind-independent entity; what is given up is haecceitism about bare points, not substantivalism about spacetime itself. They further argue that the existence of vacuum solutions remains the central difficulty for relationism: if spacetime is nothing but relations among material bodies, it is unclear what the metric field is a field of, or what curves, in a region containing no matter at all. The claim that the gravitational field simply is spacetime, they argue, concedes the substantivalist's main point under a different name, since a dynamical entity that carries energy, propagates, and exists in the absence of matter is doing the explanatory work substantivalists attribute to spacetime regardless of what it is called. They also caution that the existence of alternative, manifold-free mathematical formulations does not show that the manifold picture is dispensable as a matter of physical ontology, since formal equivalence between two formulations does not by itself settle which, if either, correctly describes what is fundamentally real.

In response, relationists argue that the move to drop haecceitism about spacetime points while retaining substantivalism about the manifold as a whole concedes most of what relationism was arguing for in the first place, since it abandons the idea that spacetime points are independent individuals and instead makes the identity of spacetime structure parasitic on the metric field defined on it. They argue that the appeal to vacuum solutions mistakes the absence of ordinary matter for the absence of physical content, since on the dynamical view, the metric field present in a vacuum solution is itself the relevant physical field, not evidence of a separate substantival container beneath it; a vacuum solution shows that the gravitational field can exist without other matter fields, not that geometric structure exists without any field content whatsoever. They further argue that the standard interpretation's reliance on the manifold as the natural or default formalism reflects a contingent feature of how general relativity is typically taught and written, rather than a discovery about fundamental ontology, and that the existence of equivalent manifold-free formulations is precisely what shows the point manifold is not owed a fundamental ontological commitment, since a theory's content should not be read off of whichever among several equivalent formalisms happens to be most familiar.

Both sides agree that general relativity is generally covariant and that the theory's empirical and mathematical content does not by itself adjudicate the dispute, since both interpretations can be made compatible with the full predictive content of the theory. Both sides also agree that bare manifold substantivalism, the view that spacetime points are individuals with primitive identity independent of any field defined on them, faces the hole argument and is not widely defended in its original form, with most substantivalists instead defending some variant of metric-field substantivalism or structuralism. Both sides further agree that the gravitational field plays a distinctive physical role unlike that of matter fields embedded in a fixed background, whatever the correct ontological characterization of that role turns out to be.

• General Relativity
• General Relativity - History
• General Relativity
• General Relativity - Standard Geometric Interpretation Viewpoint
• General Relativity - Relationism Viewpoint
• Dark Energy - Debate

1. John Earman and John Norton, “What Price Spacetime Substantivalism? The Hole Story,” The British Journal for the Philosophy of Science 38, no. 4 (1987): 515-525.
2. Tim Maudlin, “The Essence of Space-Time,” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 2 (1988): 82-91.
3. Jeremy Butterfield, “The Hole Truth,” The British Journal for the Philosophy of Science 40, no. 1 (1989): 1-28.
4. Harvey R. Brown and Oliver Pooley, “Minkowski Space-Time: A Glorious Non-Entity,” in The Ontology of Spacetime, ed. Dennis Dieks (Amsterdam: Elsevier, 2006), 67-89.
5. John D. Norton, “The Hole Argument,” in The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta and Uri Nodelman (Stanford: Metaphysics Research Lab, Stanford University, 2015), https://plato.stanford.edu/archives/win2015/entries/spacetime-holearg/.
6. James Owen Weatherall, “Regarding the 'Hole Argument,'” The British Journal for the Philosophy of Science 69, no. 2 (2018): 329-350.
7. Dean Rickles, “Symmetry, Structure, and Spacetime” (Amsterdam: Elsevier, 2008).