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Hole Argument


The original "hole argument" (lochbetrachtung ) was created by Albert Einstein. The point of the argument may be put as follows: If a physical theory's equations are generally covariant (that is, invariant under a wide group of continuous coordinate transformations) then the theory is in a certain specific sense indeterministic. Einstein put the argument to two different uses. First before the discovery of his final field equations for the General Theory of Relativity (GTR), the argument was put forward as a justification for accepting non -generally covariant field equations, namely those of the 1913 Einstein-Grossman Entwurf theory. Einstein was not fully satisfied with that theory, in part because he believed that general covariance was necessary if a theory were to capture a fully general relativity of motion, and so the hole argument served to help Einstein reconcile himself (temporarily and only partially) to the Entwurf theory. The second use of the hole argument came in 1915 when Einstein came to see the argument, taken in its first form, as a mistake. From his second point of view the argument rests on a mistaken interpretation of the mathematics of general covariance. The indeterminism allegedly shown by the hole argument is spurious, and the argument cuts no ice in favor of any particular theory or interpretation of the nature of space-time.

Seven decades later, after the rediscovery of Einstein's argument by John Stachel and John Norton, history repeated itself. A close cousin of Einstein's hole argument was put forth by John Earman and John Norton (1987) as an argument claiming to show that, if one embraces a substantival view of space-time, then in a generally covariant theory such as the GTR, one is committed to an unpleasant form of indeterminism. Earman and Norton argued that the problem is reason enough to justify rejecting a substantival view of space-time in GTR. But within a few years this view of the argument's significance was widely rejected. Instead most philosophers came to think that the hole argument's indeterminism is merely an artifact of a particular interpretation of the mathematical structure of GTR that we are not logically compelled to accept.

Regardless of which viewpoint is better supported, it is indisputable that Earman and Norton's hole argument led to a huge resurgence of interest in the interpretation of space-time in GTR, and lies at the core of much of the philosophy of space-time theories published since 1987. Subsequently philosophers have explored the status of general covariance, and therefore of the hole argument, in the domain of quantum gravity theories.

The 1987 Hole Argument

GTR describes the dynamical interaction of material substances in space-time with other material substances, as well as their interactions with the variably-curved structure of space-time itself. Einstein's field equations describe these interactions, and delimit the set of models, or physically possible worlds, corresponding to the theory.

A model of GTR is usually presented as a triple <M, g, T > consisting of a four-dimensional, continuously differentiable manifold M, a metric-field tensor g (representing the geometry of space-time) defined everywhere on the manifold, and a stress-energy tensor T representing the material substances in space-time. Like g, T is defined everywhere in the space-time, but unlike g, T may be exactly equal to zero at some or even all points of space-time. (In the latter case we say the space-time is "empty," but it may still have an interesting structure as encoded in g .) Notice that each of these objects is four -dimensional, representing not just how things are at a specific time but rather how things are over the entire history of the (model-) universe.

The manifold is a collection of points with a local and global topology built-in. For example some models of GTR have M structurally identical to 𝕽4, which means that space-time can be coordinatized (all the points labeled) with four-dimensional Cartesian coordinates. The metric tensor defines the metric and geometric structure of the space-time: distances between points A and B, whether points A, B and C are collinear, whether line L is a straight line (geodesic) or curved, and so on. Note that M by itself does not have such geometric structure; there are no distances between points in M alone, no straight lines, and so forth. Finally T represents the matter, the energy-momentum, existing in space-time.

Physicists and philosophers confront a set of interpretational issues regarding GTR and its model worlds, and one prominent issue is this: Should space-time be thought of as an object existing in its own right, that is as a substantial entity ? To answer this question in the affirmative is to take GTR as vindicating space-time substantivalism, a close cousin of Newton's absolutism. But what exactly is space-time according to GTR? Earman and Norton (1987) argued that the manifold M, by itself, is what deserves the name of substantival space-time in GTR:

We take all the geometric structure, such as the metric and derivative operator, as fields determined by partial differential equations. Thus we look upon the bare manifoldthe "container" of these fieldsas space-time

The advent of general relativity has made most compelling the identification of the bare manifold with space-time. For in that theory geometric structures, such as the metric tensor, are clearly physical fields in space-time. The metric tensor now incorporates the gravitational field and thus, like other physical fields, carries energy and momentum whose density is represented by the gravitational field stress-energy pseudo-tensor

If we do not classify such energy-bearing structures as the [gravitational] wave as contained within space-time, then we do not see how we can consistently divide between container and contained.
(pp. 518551)

If space-time substantivalism is understood as the thesis that (a) the manifold M by itself represents space-time and (b) its points are substantial entities themselves, then the ground is prepared for the hole argument. The general covariance of the Einstein field equations, interpreted in an active sense, allows one to take a given model M1 = <M, g, T > and construct a second via an automorphism h on the manifold. The automorphism maps points of M to other points of M in a smooth fashion. The effect of this re-arranging of the points is the production of a new model: M2 = <M, h*g , h*T > which also satisfies the field equations, and in which the "contents" of space-time, g and T , have been "slid around" on the manifold. The kind of automorphism employed in the hole argument is usually called a "hole diffeomorphism." Think of M2 as obtained from M1 by sliding the metric and matter fields around on the point-manifold in the region of M called "the Hole," leaving everything unchanged elsewhere. (Equivalently one can think of the hole diffeomorphism as a kind of shifting-around of the manifold points, moving the points around "underneath" the metric and material contents of the space-time.)

If M2 and M1 agree or match for all events before a certain time t, but differ for some events afterward (inside the Hole), then we have a form of indeterminism, at least on the most straightforward way of defining determinism in the context of GTR. Relative to our chosen substantial entities, space-time points considered as the elements of M, we can say: In GTR, what happens at what space-time locations is radically underdetermined. Earman and Norton (1987) presented this indeterminism as an argument against the kind of substantivalism (manifold substantivalism) they see as most natural in GTR:

Our argument does not stem from a conviction that determinism is or ought to be true. There are many ways in which determinism can and may in fact fail Rather, our point is this. If a metaphysics, which forces all our theories to be deterministic, is unacceptable, then equally a metaphysics, which automatically decides in favor of indeterminism, is also unacceptable. Determinism may fail, but if it fails it should fail for reasons of physics, not because of a commitment to substantival properties which can be eradicated without affecting the empirical consequences of the theory.
(p. 524).

Substantivalism about space-time is thus, according to this argument, ruled out as an acceptable interpretive option for GTR. Before we consider responses to the hole argument, we need to note three points. First this indeterminism is unobservable: M1 and M2 are qualitatively indistinguishable. Second, a tacit assumption of the hole argument is that the identities of the manifold points may be taken as given or specified, in some sense, independently of the material/observable processes occurring in space-time (represented by g and T ). In fact one way of thinking of a hole automorphism is as a (continuous) permutation of the points underlying physical processes, or (equivalently?) as a re-labeling of the points. Third, a manifold is a collection of spacetime points, not space points. In other words the points do not have duration;

each one is an ideal point-event, a representative of a spatial location at a single instant of time. They do not exist over time and hence serve as a structure against which motion may be defined, as Newton's space points did. In light of the second point just above, the indeterminism at issue is not a failure of the determination of future events at pre-existing spatial locations, but rather a failure of the mathematics to specify which individual points would pop into and out of existence underneath specified physical events.

Not surprisingly most responses to Earman and Norton's hole argument have departed from these three points, arguing either that the indeterminism is innocuous, or that substantivalism can be reinterpreted in ways that do not lead to the apparent indeterminism.

Responses to the Hole Argument

Two authors, Cartwright and Hoefer, have responded to the hole argument by denying that it has any prima facie force at all. Their response attacks the logic of Earman and Norton's reasoning. Since the indeterminism is both unobservable and peculiarly metaphysical (involving as it does only questions of which points, considered as identity-bearing individuals, will underlie which physical events, it is not properly speaking a physical indeterminism at all and hence not something that ought to be ascribed physical/ontological importance. Most other authors however have not questioned the logical force of Earman and Norton's argument, agreeing with them that determinism must "be given a fighting chance" (Earman 1989, p. 180). But most authors have also rejected the hole argument's anti-substantivalist conclusion. They argue either for a different understanding of substantivalism, a different definition of determinism, or both.

The first to respond to Earman and Norton's argument were Tim Maudlin (1988) and Jeremy Butterfield (1989). Both accepted the prima facie reasonableness of the hole argument but then argued that a metaphysical mistake was nevertheless being committed in the course of the hole argument. For Butterfield, the mistake lay in (a) taking the identities of manifold points between models as an unproblematic given and hence (b) defining indeterminism in too direct and unsubtle a fashion. Butterfield argued that we should avail ourselves of something like David Lewis's apparatus of counterpart theory in order to decide which points in a given model are identical with which points in a different model and accordingly revise the definition of determinism in terms of counterpart relationships. The technical details are too complicated to present here, but the upshot is that GTR turns out not to be indeterministic after all once both point trans-world identity and determinism are properly understood.

Maudlin rejected Earman and Norton's claim that the manifold by itself represents space-time. Instead he argued that the manifold plus metric is what represents space-time, and moreover that we should consider the spatio-temporal, geometric properties ascribed to points of M by g to be essential properties in a strong metaphysical sense. In support of the former point, Maudlin (1988) and Hoefer (1996) adduce the following points:

1) A manifold by itself has few of the paradigmatic spatiotemporal properties we would expect space-time to have: Distance relations between points, collinearity on a straight line, and so forth. In fact there is not always even a distinction to be found between space-like directions and time-like directions! So it is odd to think of M alone as representing space-time.

2) There is an easy way to separate between space-time (the "container") and its contents (the "contained"): it is the distinction between M + g and T . Mathematically the distinction is clear. Moreover, as was true for classical substances in the Newtonian tradition, T can vanish at some, or even all, space-time locations. g cannot vanish anywhere, in any genuine part of a GTR model space-time.

3) If it is accepted that g can carry genuine (stress-) energy content, that only makes it even more substantial than Newtonian space-time's structure was; it is hardly a reason for considering g not to be part of the characterization of space-time in GTR.

4)"[I]f the metric is classified as a physical field in space-time, rather than as representing part of space-time itself, the following odd situation emerges. Space-time itself is not appealed to in explaining the motions of material things; they are explained by relations to a different kind of physical field. Even distances and other geometric relations have nothing to do with space-time, but instead with the relations between two kinds of physical fields in space-time. When substantivalism starts to sound like relationism, something is wrong!" (Hoefer 1996, p. 13).

Most authors now seem to agree that Earman and Norton's identification of M as the sole representor of space-time is questionable. This alone does not block their hole argument, though it points the way toward various different versions of substantivalism, incorporating the metric as part-representor of spacetime, which may avoid the hole argument's indeterminism.

Maudlin's essentialism about the metrical properties and relations of space-time points blocks the hole argument by making the metrical properties of individual points be (metaphysically) essential properties: Thus, if model M1 = <M, g, T > represents a genuine physically possible world, then model M2 = <M, h*g , h*T > cannot in general do so, since it ascribes metaphysically impossible properties to the points of M. Thus, properly interpreted, GTR does not allow a determinism-violating plethora of indistinguishable space-times.

In only slightly different ways, Maidens (1993), Stachel (1993) and Hoefer (1996) diagnose the hole argument as resting on an interpretive mistake: The mistake of considering models such as M1 and M2 as representing (meta-) physically distinct space-times. As noted above, the differences between such models concern only which substantial individuals (manifold points) underlie which material happenings and relations. Without exaggeration one can put the distinction like this: While M1 says that the point Larry underlies my fingertip at this moment, M2 says that the (qualitatively identical, in all respects) point Fred does so instead. Earman and Norton's interpretation of substantivalism therefore ascribes primitive identity to the points of space-time, and models such as M1 and M2 differ only in what philosophers call haecceitistic ways, that is, in which properties are ascribed to which individuals, where the individuals are mere "bare particulars".

General relativists routinely deny the significance of such alleged differences, and say that diffeomorphic models like M1 and M2 represent just one physically possible world (thereby advocating Leibniz Equivalence ). We should do the same, urge these authors; when we do, the hole argument evaporates, and we are nevertheless left with a strong form of substantivalism, one that takes M + g to represent space-time and considers any two diffeomorphic mathematical models as representing one and the same physically possible world. The disadvantage of taking this interpretive route is that one loses the ability to describe certain metaphysical possibilities that were accepted by Newton and Samuel Clarke, that is, the possibility that every event in the world's history could have taken place five meters to the East of its actual location. Some philosophers maintain that these metaphysical possibilities are an essential part of any substantivalist view.

Not all those inspired by the hole argument to work on spacetime issues try to shore up substantivalism. The hole argument inspired those with relationist leanings to revive the idea, advocated by Reichenbach earlier in the twentieth century but effectively killed by Earman (1989) and Friedman (1983), that GTR can be interpreted as fully compatible with relationism. Teller (1991), Huggett (1999), and Saunders (2003) are examples of this approach. What makes this position possible is the adoption of a liberal attitude toward the idea of relations between material things. If the manifold is viewed as only representing the continuity, dimensionality, and topology of spacetime (as some substantivalists would agree anyway), then what is really indispensable is the metric. Can it be interpreted relationally? Those philosophers who argue that it can are not espousing a Machian reduction of metrical structure to material relations. Instead they claim that the metric itself can be interpreted as merely giving the structure of actual and possible spatiotemporal relations between material things. g is not a thing or substance. Where matter is present, it is crucial to the definition of local standards of acceleration and non-acceleration; the Einstein field equations record just this relationship. In many ways the desires of traditional relationists (especially Leibniz, Huygens, and Mach) arearguablymet by GTR when interpreted this way.

Further Developments

By the late 1990s a broad consensus was reached among philosophers of spacetime that there are acceptable interpretations of spacetime in the GTR context that do not run afoul of the hole argument indeterminism (both substantival and relational interpretations). In a series of papers, however, John Earman and Gordon Belot (2001) argued that consideration of the extension of GTR to a quantum theory vindicates the importance of the hole argument, and reveals the vacuity of certain philosophical responses to it. The issues and arguments involved in this new broaching of the hole argument are complicated and technical; only a cursory review can be attempted here.

One approach to quantizing GTR begins by recasting the theory in the Hamiltonian formalism, wherein a three-dimensional configuration representing the state of the physical world evolves in accordance with the Hamiltonian equations of motion. This is a natural way of formulating GTR preparatory to attempting to quantize it, since there are established recipes for quantizing theories starting in the Hamiltonian framework. But the (active) general covariance of GTR makes for a resulting indeterminism in the Hamiltonian presentation of the theory, just as it did for the standard theory when interpreted as a theory about what happens at individual manifold points. Various ways of dealing with this indeterminism can be linked conceptually to respective philosophical responses to the hole argument, and they appear to lead to genuinely different theories after quantization is done. So Earman and Belot (2001) claim:

There is a correspondence between interpretations of the general covariance of general relativity and approaches toand interpretations ofquantum gravity One demands that one's interpretation of general relativity should underwrite an approach to quantization which leads to a viable theory of quantum gravity, and that one's understanding of quantum gravity should lead to a way of viewing general relativity as an appropriate classical limit.
(p. 249)

Different responses to the hole argument make for different interpretations of space-time in GTR, and according to Earman and Belot these correspond to different approaches to quantum gravity. The "relationist" response of Teller, Saunders, and others which rejects the idea that diffeomorphic models are distinct physical possibilities, corresponds to the "gauge invariant" approach to quantum gravity. But the quite similar interpretation of classical GTR that is offered by Maidens, Stachel, Hoefer, and others is pejoratively labeled "sophisticated substantivalism" by Earman and Belot, and they find it the one view unworthy of even entering the playing field of the interpretive game. They claim that these philosophers are obliged to produce a gauge invariant mathematical treatment of classical GTR in the Hamiltonian framework, which may or may not be possible:

[W]e maintain that there is one sort of response to the hole argument which is clearly undesirable: the sort of sophisticated substantivalism which mimics relationalism's denial of the Leibniz-Clarke counterfactuals. It would require considerable ingenuity to construct an (intrinsic) gauge-invariant [LE-based] substantivalist interpretation of general relativity. And if one were to accomplish this, one's reward would be to occupy a conceptual space already occupied by relationalism. Meanwhile, one would forgo the most exciting aspect of substantivalism: its link to approaches to quantum gravity.
(2001, p. 248)

Earman and Belot evidently still characterize substantivalism as essentially a matter of believing in space-time points as individuals with primitive identity (a key conceptual part of manifold substantivalism, as we saw above), and relationism as the denial of such points-as-primitives. Most philosophers would reject both viewpoints, in line with points 14 above that argue for including the metric field in our characterization of space-time. It is also unclear why the sophisticated substantivalist faces a technical difficulty of constructing a "gauge-invariant" interpretation of Hamiltonian GTR, unless the relationist that Earman and Belot cites approvingly does toogiven that they occupy the same conceptual space.

Earman categorizes the loop quantum gravity approach of Rovelli and others as lined up with relationism. A crucial aspect of such a gauge invariant approach should be its evasion of hole argument-style indeterminism problems. But Rickles (2005) claims that a perfect analog of the hole argument can be constructed within the framework of loop quantum gravity. Rickles argues that indeterminism-via-surplus-structure can infect either relationist or substantivalist interpretations of GTR, whether in classical or quantized form, and therefore that the two issues should henceforth be kept apart.

As for the original hole argument itself, we should note that it was an argument that used general covariance to argue against a certain ontological view (manifold substantivalism), in the context of classical GTR. Regardless of what view of general covariance and determinism/indeterminism issues is eventually vindicated in the realm of quantum GTR (if one view isthere is no guarantee this will happen), it will not alter the dialectic of the hole argument itself, or the philosophical issue of whether GTR as a self-standing theory does or does not give us a picture of space-time deserving the label "substantival." Compare with the absolute/relational debate in the context of classical Newtonian mechanics. Greater mathematical rigor and conceptual clarity in the foundations of Newtonian mechanics did have a bearing on that philosophical debate, and led Earman (1989) and Michael Friedman (1983) to declare a hands-down victory for absolute space. But the nature and status of space-time in General Relativity has not been taken to be relevant to that earlier debate (or the correlate debate about space-time in Special Relativity), even though both of these earlier theories are "appropriate classical limits" of GTR. The moral would appear to be that philosophers should tackle interpretive issues one theory at a time. If and when a successful quantum gravity theory emerges, the substantival/relational debate can be addressed anew in light of its particular mathematical structure.

See also Clarke, Samuel; Determinism, A Historical Survey; Earman, John; Einstein, Albert; Leibniz, Gottfried Wilhelm; Logic, History of; Mach, Ernst; Newton, Isaac; Reichenbach, Hans; Relativity Theory; Space; Time.


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Carl Hoefer (2005)

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