"The Hole Argument and Some Physical and Philosophical Implications"
John Stachel 
1 Introduction
1.1 Why should we care?
1.2 Summary: Where we are headed
1.3 Outline of the article
2 Early History
2.1 From the special theory to the search for a theory of gravity
2.2 From the equivalence principle to the metric tensor
2.3 From the metric tensor to the hole argument
2.4 From the hole argument back to general covariance
2.5 The point coincidence argument
2.6 From general covariance to Kretschmann’s critique
2.7 The Cauchy problem for the Einstein equations: from Hilbert to Lichnerowicz
3 Modern Revival of the Argument
3.1 Did Einstein misunderstand coordinate transformations?
3.2 Einstein’s vision and fiber bundles
4 The Hole Argument and Some Extensions
4.1 Structures, algebraic and geometric, permutability and general permutability
4.2 Differentiable manifolds and diffeomorphisms, covariance and general covariance
4.3 Fiber bundles: principal bundles, associated bundles, frame bundles, natural and gauge-natural bundles
4.4 Covariance and general covariance for natural and gauge-natural bundles
5 Current Discussions: Philosophical Issues
5.1 Relationalism versus substantivalism: Is that all there is?
5.2 Evolution of Earman’s relationalism
5.3 Pooley’s position: sophisticated substantivalism
5.4 Stachel and dynamic structural realism
5.5 Relations, internal and external, quiddity and haecceity
5.6 Structures, algebraic and geometric
5.7 Does “general relativity” extend the principle of relativity?
6 Current Discussions: Physical Issues
6.1 Space-time symmetry groups and partially background-independent space-times
6.2 General relativity as a gauge theory
6.3 The hole argument for elementary particles
6.4 The problem of quantum gravity
7 Conclusion: The Hole Argument Redivivus
A Nijenhuis on the Formal Definition of Natural Bundles
B Appendix to Section 4: Some Philosophical Concepts
B.1 Things: Synchonic states and diachronic processes
B.2 Open versus closed systems: Determinism versus causality
B.3 Properties: Intrinsic and extrinsic
B.4 Relations: Internal and external
B.5 Quiddity and haecceity

6 Current Discussions: Physical Issues90

6.1 Space-time symmetry groups and partially background-independent space-times

As noted in Section 2.6, in the case of a generic metric (i.e., one having no symmetries), the Kretschmann–Komar coordinates may be used to individuate the points of space-time. While non-generic metrics constitute a subset of measure zero, all known solutions to the Einstein field equations belong to this subset. Only the a priori imposition of some fixed, background symmetry group on the pseudo-metric tensor enables construction of such solutions (see, e.g., Stephani et al., 2003Jump To The Next Citation Point). The symmetry group, also called the isometry group of the metric, determines a portion of the metric field non-dynamically; the remaining portion obeys a reduced set of dynamical Einstein equations. One must examine each symmetry group to see how much freedom remains in the class of solutions to these reduced equations; in particular, whether enough freedom remains for a restricted version of the hole argument to apply to these solutions.

The possible isometries of a four-dimensional pseudo-Riemannian manifold have been classified: they are characterized by two integers (m, o) (see, e.g., Stephani et al., 2003; Hall, 2004):91

  1. The dimension m ≤ 10 of the isometry group.
  2. The dimension o ≤ min (4,m ) of the highest-dimensional orbits of this group.

The two extreme cases are:

The maximal symmetry group (m = 10, o = 4). Minkowski S-T is the unique Ricci-flat space-time in this class. Its isometry group is the Poincaré or inhomogeneous Lorentz group, which acts transitively on the entire space-time manifold. The field equations of special-relativistic field theories must be invariant under this group; they provide the most important physical example of background-dependent theories.

At the other extreme (m = 0,o = 0) is the class of generic metrics already mentioned. Their isometry group is trivial, i.e., consists of only the identity element. Field theories that are covariant under the group of all diffeomorphisms of the underlying four-dimensional differentiable manifold (see Section 4), will include a subclass of the generic metrics among their models. If such a theory is generally covariant, so that all diffeomorphically-related models represent one physical model, it is called a background-independent theory.

6.1.1 Non-maximal symmetry groups and partially-fixed backgrounds

In between these two extremes lie all models of a background-independent theory that are restricted by the further requirement that they have some fixed non-maximal, non trivial symmetry group. If this symmetry group is a function group, we shall say the theory has a partially-fixed background. As noted above, all known exact models of general relativity fall into this category.92 Considerable work has been done on two classes of such models:

  1. The mini-superspace cosmological solutions, in which so much symmetry is imposed that only functions of one parameter (the “time”) are subject to dynamical equations (see, e.g., Ashtekar et al., 1993a,b; Ashtekar, 2010Jump To The Next Citation Point). Quantization here resembles more the quantization of a system of particles than of waves, and does not seem likely to shed too much light on the generic case.
  2. The midi-superspace (see Torre, 1999) solutions, notably the cylindrical wave metrics (see Ashtekar and Pierri, 1996; Ashtekar et al., 1997a,b; Bičák, 2000). Here, sufficient freedom remains to include both degrees of freedom of the gravitational field. Diffeomorphisms of a two-dimensional manifold with pseudo-metric are still possible, so a two-dimensional version of the hole argument can be formulated.

Taking advantage of fact that any two-dimensional pseudo-metric is conformally related to two-dimensional Minkowski space, one can adopt a coordinate system in which the two degrees of freedom are represented by a pair of scalar fields obeying non-linear, coupled wave equations in this flat space-time (Stachel, 1966). In addition to static and stationary fields, the solutions include gravitational radiation fields having both states of polarization. Their quantization can be carried out formally as if they were two interacting two-dimensional fields (see Kouletsis et al., 2003). But, of course, the invariance of any results under the remaining diffeomorphism freedom must be carefully examined, as well as possible implications for the generic case.

6.1.2 Small perturbations and the return of diffeomorphism invariance

As noted in Section 4.3, the space of all four-geometries forms a stratified manifold, partitioned into slices. Each slice consists of all geometries having the same isometry group. If a metric in one such slice is then perturbed, unless the perturbation is restricted to lie within the same (or some other) symmetry group, it will move the geometry into the generic slice of the stratified manifold. This observation is often neglected when perturbation-theoretic quantization techniques, developed for special relativistic field theories, are applied to perturbations of the Minkowski metric. Diffeomorphisms of such perturbations cannot be treated as pure gauge transformations on a fixed background Minkowki S-T; they modify the entire causal and inertio-gravitational structure of space-time (see, e.g., Doughty, 1990, Chapter 21). This seems to be the fundamental reason behind the problems that plague the application of special-relativistic quantization techniques to such perturbations.

6.1.3 Asymptotic symmetries

An important class of solutions to the field equations lacks global symmetries, but does have asymptotic symmetries as infinity is approached in null directions. This results in the possibility of separating kinematics from dynamics, and the asymptotic quantization of such solutions (see Komar, 1973, Section VI, and Ashtekar, 1987Jump To The Next Citation Point). The imposition of certain conditions on the behavior of the Weyl (conformal curvature) tensor in the future (past) null limit allows conformal compactification of a large class of space-times (Penrose, 1963) by adjoining boundary null hypersurfaces scri plus ℐ+ (scri minus ℐ −) to the space-time manifold. Both boundaries have a symmetry group that is independent of particular dynamical solutions to the field equations in this class.

Thus, there is a separation of kinematics and dynamics on ℐ ±, and a quantization based on this asymptotic symmetry group can be carried out in more or less conventional fashion.

“More or less” because the asymptotic symmetry group, the Bondi–Metzner–Sachs (BMS) group, is not a finite-parameter Lie group, as is the Poincaré group; it includes four so-called supertranslation functions, which depend on two “angular” variables. Nevertheless, asymptotic gravitons may be defined as representations of the BMS group, independently of how strong the gravitational field may be in the interior region (Ashtekar, 1987).

6.1.4 Generalization of this classification to other geometric object fields

This classification of metrics can be extended to other fields. Consider a natural or gauge-natural fiber bundle involving a geometric object field Φ with base manifold M, and a covariant theory T that picks out a valid class of sections σ of the Φ-bundle. Since the theory is assumed covariant, if σ is a model of the theory, then so is d ∗ σ, where d ∗ is the unique fiber-preserving bundle diffeomorphism corresponding to a base diffeomorphism dM if the bundle is natural; or to any member of the class of such diffeomorphisms if the bundle is gauge-natural.

Consider the equation £v Φ = 0, where v is a vector field generating a one-parameter family of base diffeomorphisms. We say that such a family of diffeomorphisms is a symmetry of Φ generated by v. The class of all models of the theory is divided into equivalence classes by the following equivalence relation: Two models σ and ′ σ are equivalent if any v that generates a symmetry of σ also generates a symmetry of σ ′ and vice versa.

If there is more than one generator of the symmetries in such an equivalence class, they form a group under addition with constant coefficients. The Poisson bracket of two symmetry generators: ′ [v, v ] is always a generator of a symmetry; so the generators form a Lie algebra.93

If we assume the theory based on Φ to be background-independent, i.e., generally covariant, we can impose a further condition on models of the theory: that they all have the symmetries generated by some particular Lie sub-algebra. This results in a class of partially background-dependent theories, each based on the background independent theory.

6.2 General relativity as a gauge theory

There are a number of ways to treat general relativity as a gauge theory (see Sections 4.3 and 4.4; for a survey that includes some generalizations of general relativity, see Hehl et al., 1995). We shall follow the discussion in Trautman (1980):

For me, a gauge theory is any physical theory of a dynamic variable which, at the classical level, may be identified with a connection on a principal bundle. The structure group G of the bundle P is the group of gauge transformations of the first kind; the group 𝒢 of gauge transformations of the second kind may be identified with a subgroup of the group AutP of all automorphisms of P. In this sense, gravitation is a gauge theory. The basic gauge field is a linear connection ω (or a connection closely related to the linear connection). In addition to ω, there is a metric tensor g which plays the role of a Higgs field. The most important difference between gravitation and other gauge theories is due to the soldering of the bundle of frames LM to the base manifold M. The bundle LM is constructed in a natural and unique way from M, whereas a noncontractible M may be the base of inequivalent bundles with the same structure group. …The soldering form 𝜃 leads to a torsion which has no analogue in nongravitational theories. Moreover, it affects the group 𝒢, which now consists of the automorphisms of LM preserving 𝜃. This group contains no vertical automorphism other than the identity; it is isomorphic to the group Di ffM of all diffeomorphisms of M (ibid., p. 306).

Trautman contrasts such theories with a “pure gauge” theory:

In a gauge theory of the Yang–Mills type over Minkowski space-time, the group 𝒢 is isomorphic to the semi-direct product of the Poincaré group by the group 𝒢0 of vertical automorphisms of P …. In other words, in the theory of gravitation, the group 𝒢 of “pure gauge” transformations reduces to the identity; all elements of 𝒢 correspond to diffeomorphisms of M (ibid., p. 306).

Trautman points out that, even for a given theory, the choice of structure group G is not unique.

Since space-time M is four-dimensional, if P = LM then G = GL (4,R ). But one can equally well take for P the bundle AM of affine frames; in this case G is the affine group. There is a simple correspondence between affine and linear connections which makes it really immaterial whether one works with LM or AM. If one assumes – as usually one does – that ω and g are compatible, then the structure group of LM or AM can be restricted to the Lorentz or Poincaré group, respectively. It is also possible to take, as the underlying bundle for a theory of gravitation, another bundle attached in a natural way to space-time, such as the bundle of projective frames or the first jet extension of LM. The corresponding structure groups are natural extensions of GL (4,R ),O (1,3) or the Poincaré group (ibid., p. 306).

As indicated earlier (see Section 4.2), for all relevant physical theories there are good arguments for considering SL (4,R ) as the frame automorphism group, with the unimodular diffeomorphism group as the corresponding structural group (see Stachel, 2011; Bradonjić and Stachel, 2012).

6.3 The hole argument for elementary particles

To recapitulate the main results of Section 4.1: The essence of the hole argument is independent of the differentiability or even the continuity properties of a manifold. When one abstracts from these properties,94 a manifold diffeomorphism becomes a permutation of the members of the set; so one can use fibered sets to formulate the hole argument for permutable theories. The covariance of a theory defined on a fibered manifold – any valid model of it is turned into another valid model by a diffeomorphism acting on its base manifold – becomes the permutability of a theory defined on a fibered set: A theory is permutable if any valid model of it is turned into another valid model by a permutation of the elements of its base set. The theory is generally permutable if an equivalence class of such mathematically distinct models corresponds to a single model of the theory. These concepts can be applied to elementary particles.

Consider a system consisting of N so-called identical, or better, indistinguishable elementary particles, sharing a common quiddity. By nature, these particles lack an inherent haecceity; but in order to formulate a dynamical theory for the system (e.g., in order to write down a Lagrangian or Hamiltonian for it in non-relativistic quantum mechanics), one needs to enumerate them (i.e., assign a number from 1 to N to each of the particles). This is a discrete example of coordinatization (see Section 4.1), and one must ‘undo’ this individuation by requiring invariance under all possible permutations of the initial enumeration.

As discussed in Section 4.1 for sets in general, such permutations can be done in either of two ways:

  • Active: Fix the enumeration and permute the particles; or
  • Passive: Fix the particles and permute the enumeration.

Either way, it is obvious that each of the N particles will have each of the integers from 1 to N assigned to it in some of these permutations.

Like space-time points, particles of the same natural kind (quiddity) can only be individuated (to the extent that they are) by their position in some relational structure in a theory that is permutable in both the active and passive senses: If some state of affairs is possible for a system that includes N indistinguishable particles, then the state of affairs resulting from the action of any element of the permutation group Perm (N ) on these N particles must be an identical state of affairs. In other words, the theory must be generally permutable.

What is a “possible state of affairs” in quantum mechanics? As discussed in the Appendix B, one may adopt a synchronic or diachronic approach to such questions. The term “state of affairs” is often interpreted synchronically, as referring to the instantaneous state of the system (e.g., its state vector or wave function). But quantum theory can only treat open systems (again see the Appendix B); and its task is diachronic: to compute the probability amplitude for a complete process (or “phenomenon” in Bohr’s terminology).

In short, only a complete process constitutes a possible state of affairs for a quantum-mechanical system (see Stachel, 1997). If the system is generally permutable and a certain value of the probability for such a process is calculated, then the same value must be calculated for any process that results from this one by a permutation of the indistinguishable particles in either the initial act of preparation or (inclusive “or”) the final act of registration. In order to verify these probabilities, some macroscopic device must register the results of the preparation and registration.

The macroscopic counter is assumed to be inherently individuated. It seems that, for such individuation of an object, a level of structural complexity must be reached, at which it can be uniquely and irreversibly “marked” in a way that distinguishes it from other objects of the same nature (quiddity). My argument is based on an approach, according to which quantum mechanics does not deal with quantum systems in isolation, but only with processes that such a system can undergo. …. A process (Feynman uses process, but Bohr uses “phenomenon” to describe the same thing) starts with the preparation of the system, which then undergoes some interaction(s), and ends with the registration of some result (a “measurement”). In this approach, a quantum system is defined by certain essential properties (its quiddity); but manifests other, non-essential properties (its haecceity) only at the beginning (preparation) and end (registration) of some process. (Note that the initially-prepared properties need not be the same as the finally-registered ones.) The basic task of quantum mechanics is to calculate a probability amplitude for the process leading from the initially prepared values to the finally-registered ones. (I assume a maximal preparation and registration – the complications of the non-maximal cases are easily handled) (Stachel, 2006aJump To The Next Citation Point).

Consider a scattering process for a system of indistinguishable particles, for example. The process consists of the transition from some initial free in-state of the particles to some final free out-state of the particles after their interaction with the target producing the scattering. The cross section for this process depends on the choice of initial in- and out-states.95 For a permutable theory, if some value for such a cross section is a possible result, then the same value must result for the processes that arise from separate permutations of the particles in both the in-state and the out-state.

6.3.1 The analogue of the hole argument for a permutable theory

Suppose the theory were permutable but not generally permutable. Then specification of a unique initial preparation for some process (an experiment) could never result in a unique prediction of the outcome of the registration: Every possible permutation of the particles in the registration result would be equally probable.

Conversely, registration of a unique result (an observation) could never produce a unique retrodiction of the preparation that led to this outcome: Every possible permutation of the particles in the act of preparation would be equally probable.

The way out of this non-uniqueness paradox, of course, is to require that any permutable theory involving indistinguishable particles be generally permutable. Then all such permutations belong to the same process.

Similar arguments apply to special-relativistic quantum field theory. Here the basic entities are elementary particles (fermions) and field quanta (bosons).

I reserve the term “elementary particles” for fermions and “field quanta” for bosons, although both are treated as field quanta in quantum field theory. I aim thereby to recall the important difference between the two in the classical limit: classical particles for fermions and classical fields for bosons. …[I]n the special-relativistic theories, a preparation or registration may involve either gauge-invariant field quantities or particle numbers.

At the level of non-relativistic quantum mechanics for a system consisting of a fixed number of particles of the same type, this [difference] is seen in the need to take into account the bosonic or fermionic nature of the particle in question by the appropriate symmetrization or anti-symmetrization procedure on the product of the one-particle Hilbert spaces …. At the level of special-relativistic quantum field theory, in which interactions may change particle numbers, it is seen in the notion of field quanta, represented by occupation numbers (arbitrary for bosons, either zero or one for fermions) in the appropriately constructed Fock space; these quanta clearly lack individuality (Stachel, 2006aJump To The Next Citation Point).

As discussed in Section 4.3, in background-independent theories such as general relativity, an analogous principle of general covariance holds for the points of space-time. This common lack of haecceity suggests that, whatever their nature, the fundamental entities of any theory purporting to explain a deeper physical level should satisfy the principle of maximum permutability.

6.3.2 Principle of maximal permutability

Thiemann has pointed out that

The concept of a smooth space-time should not have any meaning in a quantum theory of the gravitational field where probing distances beyond the Planck length must result in black hole creation which then evaporate in Planck time, that is, spacetime should be fundamentally discrete. But clearly smooth diffeomorphisms have no room in such a discrete spacetime. The fundamental symmetry is probably something else, maybe a combinatorial one, that looks like a diffeomorphism group at large scales (Thiemann, 2001).

It is hard to believe that, having had to renounce intrinsic individuality at the level of field quanta in QFT and at the level of events in GR, that it will reemerge as we go to a deeper level, from which both QFT and GR will emerge in the appropriate limits.

[T]he way to assure the inherent indistinguishability of the fundamental entities of the theory is to require the theory to be formulated in such a way that physical results are invariant under all possible permutations of the basic entities of the same kind …I have named this requirement the principle of maximal permutability. …The exact content of the principle depends on the nature of the fundamental entities. For theories, such as non-relativistic quantum mechanics, that are based on a finite number of discrete fundamental entities, the permutations will also be finite in number, and maximal permutability becomes invariance under the full symmetric group. For theories such as general relativity, that are based on fundamental entities that are continuously, and even differentiably related to each other, so that they form a differentiable manifold, permutations become diffeomorphisms. For a diffeomorphism of a manifold is nothing but a continuous and differentiable permutation of the points of that manifold. So, maximal permutability becomes invariance under the full diffeomorphism group. Further extensions to an infinite number of discrete entities or mixed cases of discrete-continuous entities, if needed, are obviously possible (Stachel, 2006aJump To The Next Citation Point).

Current versions of string theory, for example, do not meet this criterion, and it has been suggested that an ultimately satisfactory version of that theory will have to be background independent (see the discussions in Stachel, 2006aJump To The Next Citation Point,b; Greene, 2004). On the other hand, various discretized space-time theories, such as causal set theory, do seem to meet this criterion; but have other problems (see the discussion in Stachel, 2006aJump To The Next Citation Point).

The next Section 6.4 deals with a much more modest problem: attempts to quantize the field equations of general relativity based on the application of some variant of the standard techniques for the quantization of field theories.

6.4 The problem of quantum gravity

There is a well known tension between the methods of quantum field theory and the nature of general relativity. The methods of quantization of pre-general-relativistic theories are based on the existence of some fixed, background space-time structure(s) with a given symmetry group.96 The space-time structure is needed both for the development of the formalism and – equally importantly – for its physical interpretation (see Dosch et al., 2005). It provides a fixed kinematical background for the dynamical theory to be quantized: The dynamical equations for particle or fields must be invariant under all automorphisms of the symmetry group.

As we have seen, general relativity does not fit this pattern. It is a background-independent theory, with no fixed, non-dynamical structures. To recapitulate, its field equations are generally covariant under all differentiable automorphisms (diffeomorphisms) of the underlying manifold, the points of which have no haecceity (and hence are indistinguishable) until the dynamical fields are specified. In a background-independent theory, there is no kinematics independent of the dynamics. This applies to both the homogeneous Einstein equations, and to the inhomogeneous Einstein equations coupled to the dynamical equations for any non-gravitational fields. And this is still the case if the automorphism group is restricted to the unimodular diffeomorphisms (see Section 4.2).

However general relativity and (special-relativistic) quantum field theory do share one fundamental feature that often is not sufficiently stressed: the primacy of processes over states. The four-dimensional approach, emphasizing processes in regions of space-time, is basic to both (see, e.g., Stachel, 2006a; Reisenberger and Rovelli, 2002Jump To The Next Citation Point; DeWitt, 2003Jump To The Next Citation Point). The ideal approach to quantum gravity would be a diffeomorphism-invariant method of quantization that takes process as primary. However, the most successful approach so far, loop quantum gravity, only goes a certain way in this direction: It singles out a preferred fibration and foliation (see Section 5.7); and by adopting a (3 + 1) Hamiltonian approach and a set of redundant variables subject to constraints on the initial hypersurface, it effects a certain separation between kinematics and dynamics. But given these limitations, it has produced a mathematically rigorous – and surprisingly a unique (but see Nicolai and Peeters, 2007Jump To The Next Citation Point) – kinematic Hilbert space (see Ashtekar, 2010, for a brief review).

Even in non-relativistic quantum mechanics, the basic goal is to calculate a probability amplitude for a process connecting some initial preparation to some final registration. However, the existence of an absolute time allows one to choose a temporal slice of space-time so thin that it is meaningful to speak of “instantaneous measurements” of the initial and final states of the system (see Micanek and Hartle, 1996Jump To The Next Citation Point). But this is not so for measurements in (special-relativistic) quantum field theory, nor in general relativity (see, e.g., Bohr and Rosenfeld, 1933Jump To The Next Citation Point, 1979Jump To The Next Citation Point; Bergmann and Smith, 1982Jump To The Next Citation Point; Reisenberger and Rovelli, 2002Jump To The Next Citation Point; Stachel and Bradonjić, 2013). The breakup of a four-dimensional space-time region into lower-dimensional sub-regions – in particular, into a one parameter family of three-dimensional hypersurfaces – raises a problem for measurements in both quantum field theory (see Bohr and Rosenfeld, 1933, 1979; DeWitt, 2003) and general relativity (see Bergmann and Smith, 1982Jump To The Next Citation Point). A breakup of a process into the evolution of instantaneous states on a family of space-like hypersurfaces is a useful, perhaps sometimes indispensable, calculational tool. But no fundamental significance should be attached to such breakups, and mathematical results obtained from them should be carefully examined for their physical significance from the four-dimensional, process standpoint (see, e.g., Nicolai and Peeters, 2007).

An important criterion, called “measurability analysis” (see Bergmann and Smith, 1982Jump To The Next Citation Point), based on “the relation between formalism and observation” (see Reisenberger and Rovelli, 2002), sheds light on the physical implications of any formalism. The possibility of the definition of some physically significant quantity within a theoretical framework should coincide with the possibility of its measurement in principle; i.e., by means of idealized measurement procedures consistent with that theoretical framework. This is true at the classical level, at which any complete classical set of physical observables should be measurable in principle by a single compound procedure.97 This criterion is easily met by unconstrained systems, such as a set of non-relativistic particles interacting via two-body potentials, or a scalar field obeying the classical Klein–Gordon equation. But delicate problems arise in applying it to constrained dynamical systems, and in particular to gauge field theories, including general relativity (see Section 6.2).

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