"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

5 Current Discussions: Philosophical Issues

On the basis of the analysis developed in the previous sections,69 I shall re-examine some of the issues currently being discussed in the philosophy of science. Rather than attempting to cover the vast literature on this subject, the discussion is limited to a few representative samples of what I consider to be the most important trends, and will show that they are converging towards variations around a common denominator.

5.1 Relationalism versus substantivalism: Is that all there is?

Since Earman and Norton (1987Jump To The Next Citation Point) (see Section 3), philosophical discussion of the hole argument has centered largely around the issue of space-time absolutism – now often called substantivalism70 – versus the opposing viewpoint, usually denominated relationalism or relationism. Einstein summarized an earlier version, the age-old controversy over the nature of space:

Two concepts of space may be contrasted as follows:

  1. space as positional quality of the world of material objects;
  2. space as container of all material objects.

In case (a), space without a material object is inconceivable. In case (b), a material object can only be conceived as existing in space; space then appears as a reality which in a certain sense is superior to the material world (“Foreword” to Jammer, 1954).

Relativity theory metamorphosed the object of controversy from space to space-time, and Einstein made is his own viewpoint quite clear:

On the basis of the general theory of relativity …space as opposed to ‘what fills space’ …has no separate existence. If we imagine the gravitational field …to be removed, there does not remain a space of the type [of the Minkowski space of SR], but absolutely nothing, not even a ‘topological space’ [i.e., a manifold] …There is no such thing as an empty space, i.e., a space without field. Space-time does not claim existence on its own, but only as a structural quality of the field (Einstein, “Relativity and the Problem of Space,” in Einstein, 1952Jump To The Next Citation Point).

Here are a couple of recent statements on the nature of the controversy:

Substantivalists understand the existence of spacetime in terms of the existence of its pointlike parts, and gloss spatiotemporal relations between material events in terms of the spatiotemporal relations between points at which they occur. Relationists will deny that spacetime points enjoy this robust sort of existence, and will accept spatiotemporal relations between events as primitive (Belot and Earman, 2001, p. 227).

A modern-day substantivalist thinks that spacetime is a kind of thing which can, in consistency with the laws of nature, exist independently of material things (ordinary matter, light, and so on) and which is properly described as having its own properties, over and above the properties of any material things that may occupy parts of it (Hoefer, 1996).

What is space? What is time? Do they exist independently of the things and processes in them? Or is their existence parasitic on these things and processes? Are they like a canvas onto which an artist paints; they exist whether or not the artist paints on them? Or are they akin to parenthood; there is no parenthood until there are parents and children? That is, is there no space and time until there are things with spatial properties and processes with temporal durations? These questions have long been debated and continue to be debated. The hole argument arose when these questions were asked in the context of modern spacetime physics. In that context, space and time are fused into a single entity, spacetime, and we inquire into its status. One view is that spacetime is a substance, a thing that exists independently of the processes occurring within spacetime. This is spacetime substantivalism (Norton, 2011Jump To The Next Citation Point).

In the light of the hole argument, I find it more fruitful to frame discussion in terms of two other distinctions, leading to a point of view about space-time distinct from either substantivalism or relationalism as traditionally defined. These are the distinctions between:

  1. internal and external relations, and between
  2. quiddity and haecceity.

These concepts are discussed in Appendix B and briefly reviewed in Section 5.5. When applied to mathematical structures, they lead to succinct discussions of algebraic and geometric structures and the nature of coordinatization in Section 5.6, which establishes a correspondence between the two (for a fuller discussion, see Section 4.1).

These concepts lead to a viewpoint on the nature of space-time that has been given various names, such as structural spacetime realism71 and sophisticated substantivalism (see Pooley, 2000Jump To The Next Citation Point, summarized in Section 5.3). I have called it dynamic structural realism (see Stachel, 2006aJump To The Next Citation Point), which has several advantages. It avoids use of the words “substantivalism” and “relationalism,” fraught with so many unwanted implications; it places emphasis on diachronic aspects of structure; and its application is not confined to theories of space-time structure (see Stachel, 2005). The fiber bundle approach, motivated in Section 3 and treated in more detail in Section 4.3, allows a rigorous formulation of this viewpoint in Section 5.4.

But first, I shall give a brief account of the controversy between relationalists and substantivalists provoked by the hole argument and how it has led a number of participants from each camp to adopt this new viewpoint. Rather than attempting a (necessarily superficial) review of the vast philosophical literature on the controversy, I shall focus on an account of the views of one important relationalist and one important substantivalist.

5.2 Evolution of Earman’s relationalism

Earman (1989Jump To The Next Citation Point) is a standard reference, so I shall employ its terminology and notation in discussing his views. Discussing a “modified form” of absolutism, he states that

The only plausible candidate for the role of supporting the nonrelational structures [of a physical theory] is the space-time manifold M …(ibid. p. 125).

Calling “manifold substantivalism” the view that “M is a basic object of predication,” he sets out to show that this view “lays itself open to Leibniz’s argument” (p. 126). In his formulation of the problem, Earman uses the standard pre-bundle approach to theories (see Sections 3.2 and 4.3): A model ℳ of a theory consists of the manifold M, together with “[geometric] object fields on M”, which he denotes by A i and P j, “characterizing respectively the space-time structure and the physical contents of space-time.” Symbolically, ℳ = ⟨M, Ai,Pj⟩, with i and j each running through a finite sequence of integers. A manifold diffeomorphism d : M → M then results in a different model ℳd = ⟨M, d ∗ Ai,d ∗ Pj⟩, where d ∗ Ai, d ∗ Pj denote the pull backs or the push forwards of Ai, Pj.

It is important to note that Earman (1989Jump To The Next Citation Point) defines “general covariance” in a way that is equivalent to my definition of “covariance” (see Section 4.4):

Let us say that the laws of [a theory] T are generally covariant just in case whenever ℳ [is a model of the theory] then also ℳ Φ [is a model] for any manifold diffeomorphism …(ibid., p. 47).

In his treatment of the hole argument (ibid., Chapter 9, pp. 175–208), Earman applies this concept of model to the formulation of general relativity in terms of the metric tensor and its first and second derivatives. Thus his Ai is restricted to “the metric field gik;” while the Pj correspond to components of the stress energy tensor Tik. He presents a version of “Einstein’s Hole Argument,” involving a diffeomorphism d, such that

d = id outside the hole H but ⁄= id inside H and such that the two pieces join smoothly on the boundary …The upshot is that we have produced two solutions, ⟨M, g,T ⟩ and ⟨M, d ∗ g,T ⟩, which have identical T fields but different g fields – an apparent violation of the Kausalgesetz that the T field determines the g field (ibid., p. 176).

He then presents a version of Hilbert’s Cauchy problem argument (see Section 2.7). Assuming the existence of a Cauchy surface, parameterized by t = 0, he considers:

a diffeomorphism d such that d = id for all t ≤ 0 and ⁄= id for t > 0 and such that there is a smooth join at t = 0 (ibid., p. 179).

One can then construct two solutions, ⟨M, g, T⟩ and ⟨M, d ∗ g,T ⟩, that do not differ for t ≤ 0 and sharing the same initial data to any finite order of differentiability on t = 0.

[This provides] a seeming violation of the weakest form of Laplacian determinism …indeed, any nontrivial form of determinism suffers equally (ibid., p. 179).

The discussion of the range of applicability of the hole argument in Earman (1989Jump To The Next Citation Point) differs significantly from that in Earman and Norton (1987Jump To The Next Citation Point), which maintained that the hole argument applied to “every classical spacetime field theory [that] can be formulated as a local space-time theory.” They included all special-relativistic field theories, which they maintained were made local by adjoining the Riemann tensor Rabcd(g) to the set of geometric objects included in any model, and adding the equation Rabcd = 0 to the set of field equations defining acceptable special-relativistic models.

5.2.1 Confusion between the trivial identity and general covariance

I must linger a bit longer on Earman and Norton (1987Jump To The Next Citation Point), because their discussion of general covariance has led to much confusion. They divide geometric object fields O1,...On into two classes: O1, ...,Ok− 1 and Ok, ...,On. The second, dynamical class is assumed to obey field equations

Ok = 0, Ok+1 = 0 ..., On = 0, (4 )
while the first class may include non-dynamical, fixed background fields.72 When applied to special relativistic field theories, the Minkowski metric and the associated flat affine connection are included among these non-dynamical fields. In order to demonstrate that the hole argument applies to all such theories (and in contrast to the definition of general covariance in Earman (1989Jump To The Next Citation Point), cited above), Earman and Norton (1987Jump To The Next Citation Point) prove a “Gauge Theorem (General covariance)”:

If ⟨M, O1, ...,On ⟩ is a model of a local spacetime theory and h is a diffeomorphism from M to M, then the carried along n-tuple ⟨M, h ∗ O ,...,h ∗ O ⟩ 1 n is also a model of the theory (ibid., p. 520).

Notice that all the O1, ...,On fields of both classes are subject to arbitrary diffeomorphisms, which need not be symmetries of the non-dynamical fields. Thus, this “theorem” is essentially the trivial identity (see Sections 2.5, 4.2 and 5.7). Indeed, we can reformulate the trivial identity in their notation:

If ⟨M, O1, ...,On⟩ is a model of a local spacetime theory and h is a diffeomorphism from M to M, taking a point p ∈ M into a point ′ ′ p ∈ M : p → p , then the carried along n-tuple is ⟨M, h ∗ O1,...,h ∗ On ⟩. If we now carry out a coordinate transformation x → x ′ , such that x (p ) = x ′(p′), i.e., the new coordinates of the new point equal the old coordinates of the old point; then [h ∗ O ]′ = O i i , i.e, the new components of the new geometric objects are numerically equal to the old components of the old objects, then clearly nothing has changed. So it is not clear why the authors feel any:

need to establish that the vanishing of the field equations Ok = 0,Ok+1 = 0...,On = 0 is preserved under a diffeomorphism (ibid., p. 520).

While Earman (1989Jump To The Next Citation Point) avoids this confusion by silently renouncing this position, much of the later literature on the hole argument still falls into this error.

5.2.2 Back to Earman’s evolution

Earman now argues that one cannot simply

take a special relativistic theory of motion and rewrite the equations using covariant derivatives with respect to an undetermined Lorentz metric g. Then write the “field equation” for g, namely, Rijkl(g ) = 0, where Rijkl is the Riemann curvature tensor (Earman, 1989Jump To The Next Citation Point, p. 183),

and then apply the hole argument to show that the theory is non-deterministic. He introduces

a distinction between absolute and dynamical objects …[T]his distinction corresponds to the distinction between the [geometric] object fields (Ai) that characterize the structure of space-time and those (Pj) that characterize the physical contents of space-time (ibid., p. 184).

He then requires that,

for any two dynamically possible models of the theory ℳ = ⟨M, Ai,...Pj ...⟩ and ′ ′ ′ ′ ℳ = ⟨M ,A i,...P j ...⟩ there is a diffeomorphism ′ d : M → M such that d ∗ Ai = A′i for all i (ibid., p. 184)

Letting M = M ′, ones sees that the condition d ∗ A = A i i singles out those diffeomorphisms d of the manifold that are symmetries of the Ai fields.73

Without going into further detail (see ibid., p. 184), Earman essentially argues that the hole argument does not apply if the symmetry group makes the absolute-space time structures sufficiently rigid. It is also clear (although Earman does not make the point) that the trivial identity (Earman and Norton’s “Gauge Theorem”) is of no help in an attempt to apply the hole argument to such cases.

Earman (2004) continues the line of reasoning in Earman (1989Jump To The Next Citation Point), but with some further evolution: It emphases from the start the difference between finite-parameter Lie symmetry groups, covered by Noether’s first theorem; and symmetry groups that are function groups, covered by Noether’s second theorem. Earman (2006Jump To The Next Citation Point) starts off in a way reminiscent of Earman and Norton (1987):

It will be assumed that the spacetime theories to be discusses have been formulated in such a way that (a) their models have the form (ℳ ,O1,O2, ...,ON ), where ℳ is a differentiable manifold and the On are geometric object fields that live on M, and (b) their laws of motion/field equations have the form &tidle; &tidle; &tidle; F(O1, O2,...,OK ) = 0, where F is some functional and the O&tidle;k are geometric object fields constructed from the On (ibid., p. 446).

The difference is that now he separates the field equations F = 0 from the quantities defining the model, and relaxes the demand that they be tensorial equations. He introduces the concept of gauge symmetry as a transformation, in which:

the physical situation is not being changed; rather different but equivalent descriptions of one and the same physical situation are being generated. This is the characteristic feature of a gauge symmetry (ibid., p. 447).

He then defines substantive general covariance (SGC):

The equations of motion/field equations of the theory display diffeomorphism invariance; that is, if (ℳ ,O1,O2, ...,ON ) is a solution, then so is (ℳ ,d ∗ O1, d ∗ O2, ...,ON ) for any d ∈ diff (ℳ ). And this diffeomorphism is a gauge symmetry (ibid., p. 447).

In other words, the two solutions are mathematically distinct descriptions of the same physical solution.

He now states explicitly that “formal general covariance” is “a rather trivial gauge symmetry” (ibid., p. 447).

Earman’s “formal general covariance” is what Bergmann (1957Jump To The Next Citation Point) calls “weak covariance”, Stachel (1993Jump To The Next Citation Point), following a suggestion by Bergmann,74 calls “trivial general covariance;” and Stachel and Iftime (2005Jump To The Next Citation Point) call “covariance” tout court. Earman’s definition of “substantive general covariance” now corresponds to the Stachel and Iftime (2005) definition of “general covariance” – but what’s in a name?75 The two positions are now “substantively” the same.

The remaining difference is mathematical: Rather than using fiber bundles, Earman still works with fields on a manifold, so his formalism is still vulnerable to the substantivalists’ attack. As he noted in another context, “Formalism generated the problem and formalism is needed to resolve it” (Earman, 1989Jump To The Next Citation Point, p. 184). Or perhaps it would be better to say: “If you adopt a certain philosophical stance, you should adopt the formalism best suited to it.”

5.3 Pooley’s position: sophisticated substantivalism

Pooley describes “sophisticated substantivalism” succinctly as a “combination of anti-haecceitism and realism about spacetime points” (Pooley, 2006Jump To The Next Citation Point, p. 103).

A frequent response [to the argument from Leibniz equivalence] is that one can regard all isomorphic models of general relativity as representing the same physical possibility (Leibniz Equvalence) AND regard spacetime as a basic, substantival and concrete entity. …

Sophisticated substantivalism:

Isomorphic models ℳ and ℳ0 represent the same physical possibility (= L[eibniz] E[quivalence]) AND spacetime points exist as fundamental entities.

  • LE accords with the practice of physics
  • the metric (plus manifold) gets its natural interpretation as spacetime
  • ℳ and ℳ0 can only be regarded as representing distinct possible worlds if spacetime points have primitive identity. Denying that they do is good metaphysics independently of the hole argument (Pooley, 2000).

Sophisticated substantivalism may be compatible with taking seriously physicists’ concerns, but does it have a coherent motivation? The obvious thing to be said for the position is that one thereby avoids the indeterminism of the hole argument. This motivation is, of course, rather ad hoc. A less ad hoc motivation would involve a metaphysics of individual substances that does not sanction haecceitistic differences, perhaps because the individuals are individuated by – their numerical distinctness is grounded by – their positions in a structure. … Stachel has recently sought to embed his response to the hole argument in exactly this type of more general framework. I hope enough has been said …to indicate the coherence of such a point of view; it is perhaps a modest structuralism about spacetime points, but it is a far cry from the objectless ontology of the ontic structural realist (Pooley, 2006Jump To The Next Citation Point, p. 102).

Again, there is “sophisticated substantial” agreement between Pooley’s viewpoint and those of Earman and Stachel (see Pooley, 2013, for a more recent account of his position).

5.4 Stachel and dynamic structural realism

My earliest discussions of the hole argument were based on a purely relationalist approach to space time, which denied any physical significance to points of the four-dimensional manifold M; they only became elements of space-time after a metric tensor field was specified. This was largely in response to mathematical formulations of physical field theories in terms of geometric object fields on a given M. If one conceded that the points of this manifold represented elements of space-time, this seemed to hand victory to the absolutists (subsequently metamorphosed into substantivalists). When I realized the full implications of the fiber bundle approach, which allows the definition of M as the quotient of the total manifold of the bundle by the equivalence relation defining the fibration (see Section 4.2); and of Schouten’s (1951) observation that, in contrast to mathematical tensor fields, physical tensor fields have physical dimensions; I came to recognize that the points of M, so defined, do have the physical character of elements of space-time even before the choice of a particular field (cross section of the bundle).76 What they lack is individuality, or haecceity as I put it after adopting Teller’s (1998) terminology (see Section 5.5). This led me to a structuralist account of physical theories, but not the kind of structuralism espoused by Ladyman and French (see, e.g., Ladyman, 1998; French and Ladyman, 2003) which they call “ontic structural realism;” but which is really a kind of hyper-relationalism.77 Stachel (2006aJump To The Next Citation Point) espouses a form of traditional realism as a philosophical position, and also stresses the priority of processes over states, hence it names this position “dynamic structural realism.”

To summarize the last three sections, starting from various relationalist or substantivalist positions, Earman, Pooley and Stachel have been led to a third position, which Earman calls “substantive general covariance,” Pooley calls “sophisticated substantivalism,” and Stachel calls “dynamic structural realism”; but all three positions are essentially the same. The major difference is Stachel’s emphasis on the utility of the fiber bundle approach for the mathematical expression of this position.

After this lengthy historical-critical excursus, I shall turn to some philosophical arguments for this approach, starting with the definition of some terms already given in Section 4 and Appendix B, but repeated here for the benefit of those who did not read that section.

5.5 Relations, internal and external, quiddity and haecceity

A relation is said to be internal if one or more essential properties of the relata78 depend on the relation. It is said to be external if no essential property so depends.79 This distinction is in turn based on the distinction between intrinsic and extrinsic properties of an entity. Some of its intrinsic properties serve to characterize what has been variously called the essence, nature or natural kind of the entity; if any of these essential intrinsic properties depend on its relation(s) to other entities, then these relations are internal. No extrinsic property can depend on an internal relation.

Whether a relation is internal or external is theory-dependent, and hence may depend on the theoretical level at which the objects are treated. In any physical theory, for example, a set of units must be adopted before a mathematical form can be given to any physical quantity. Its numerical expression is actually a relation – the ratio of the quantity to its unit. At this level, it is an external relation based on the properties of the quantity and its unit. Whether these properties themselves are intrinsic or extrinsic may depend on the theoretical level considered. In the system of units adopted, is the unit of this quantity “basic” or “derived”?

A second important distinction is that between quiddity and haecceity. Quiddity is what characterizes all entities of the same nature. Haecceity refers to those properties of the relata that enable us to individuate entities of the same quiddity. Up until the last century, it was assumed that entities of the same quiddity could always be individuated by some of their intrinsic properties, independently of any relations, into which they entered. This is Leibniz’s principle: the identity of indiscernibles.80

Any further individuation due to such relations was supposed to supervene on this basic individuation.81

With the advent of quantum statistics, it was argued that there are entities – the elementary particles – that have quiddity (any particle with charge e, mass me and spin 1/2 is an electron) but no inherent haecceity (one cannot distinguish one electron from another by any intrinsic property). And the refutation of hole argument can be similarly formulated: The points of space-time have quiddity but no inherent haecceity. So theoretical physics led to the introduction of a new category: entities having quiddity but no inherent haecceity.

5.6 Structures, algebraic and geometric

An important example of the utility of this category in mathematics is the fundamental distinction between geometric and algebraic structures.

deals with elements that have (the same) quiddity but lack inherent haecceity; a set of internal relations between these elements then defines a particular geometric structure. The group of permutations of these elements preserving the defining internal relations is the symmetry or automorphism group of the geometry. Each geometry has such a group of transformations of its elements, under which all geometrical relations of that geometry remain invariant. 82
deals with elements that possess both quiddity and haecceity; a set of external relations between these elements defines a particular algebraic structure. 83
of a geometry by an (appropriate) algebra is the assignment of a unique element of this algebra to each point of the geometry; one can carry out certain algebraic operations and then give the result a geometric interpretation.
Coordinate transformations:
Any coordinatization of a geometry gives each of its elements a haecceity, thus negating their homogeneity. This is restored by negating in turn any individual coordinatization: A group of coordinate transformations between all admissible coordinate systems is introduced.

An admissible coordinate transformation is one that corresponds to an element of the automorphism group of the geometry. It follow that each point of the geometry will have every element of the algebra as its coordinate in (at least) one admissible coordinate system.

5.7 Does “general relativity” extend the principle of relativity?

To talk about a principle of relativity only makes sense if one has first defined a frame of reference. One then asserts that the laws of physics take the same form in all members of some class of frames of reference. In special relativity, this class of frames (actually a group in the case) consists of the inertial frames of reference. Given the Minkowski metric and its associated flat inertial connection, such a frame may be defined by taking any time-like autoparallel (“straight”) line, and constructing the family of such lines, one through each point of the manifold (i.e., a fibration of the space-time), each of which is parallel to the initial line.84 One may then pick a fiduciary point on each such line, and use the proper time τ along this world line, counted forwards and backwards starting from that point τ = 0, to individuate the points along the line. Assuming that each line is itself physically individuated (given haecceity) in some way, all the points of the space-time are now individuated. It is customary to choose all the fiduciary points to lie on the same space-like hyperplane orthogonal to the time-like fibration (Einstein convention for defining distant simultaneity). Then the entire group of inertial frames may be generated from the initial one by the action of the Poincaré group on the points of that inertial frame.

Of course, the trivial identity holds: if we move everything together with some diffeomorphism of the manifold, nothing has changed. But given that we move only the world lines with respect to the metric and connection, the Poincaré group is the automorphism group of the inertial frames. The inertial frames thus form a rigid structure, individuating the points of Minkowski space-time, and the hole argument fails, as it will for any finite-parameter Lie group.

In general relativity, a spatial frame of reference also corresponds to a fibration of the four-dimensional manifold M with the stipulation that, when a metric tensor field g is introduced, the fibration consist of curves with a unit time-like vector field v tangent to the fibration: v = dx∕dτ.85 We may then define projection operators ′P along the foliation and ′′P orthogonal to it. The vector field v represents the four-velocities of observers in the chosen reference frame and the orthogonal projection of the metric field 3 g represents the instantaneous spatial rest-frame of each observer. Again, one may pick a fiduciary point on each time-like world line and use the proper time, forwards and backwards starting from that point, to individuate the points along the line.

Evolution of any geometric object field Φ along the congruence will be represented by its Lie derivatives with respect to v,£v Φ. One will usually pick the fiduciary points so that they fit together smoothly to form a space-like hypersurface that transvects the fibration. Now there are two possibilities:

Holonomic case: If one has chosen a congruence, the tangent field v of which has vanishing rotation ω, there will be a foliation of space-time consisting of a one-parameter family hypersurfaces orthogonal to the fibration. The fiduciary points can be chosen to lie on one hypersurface of the foliation, and the local spatial rest-frames of each observer will fit together to form a global spatial rest-frame; so that the local spatial rest-frames of each observer fit together to form a one-parameter family of global spatial rest-frames. This is the geometric basis of the traditional approach to the Cauchy problem in general relativity (see Section 2.7).86

Non-holonomic case: But there is no need to impose this requirement. It is customary to introduce a triad of orthonormal space-like vectors ei(i = 1,2,3 ) that, together with v, span the tangent space at each point of the manifold. Then, the components with respect to this tetrad of any geometric object field Φ, called the physical components by Pirani, are assumed to be the physically measurable quantities by an observer in that frame at that point. On the assumption that each curve in the three-parameter fibration is physically individuated (given haecceity) in some way, and that some foliation is introduced to provides the fourth individuating quantity, the hole argument still fails, because a fibration and foliation provide an individuating field (see Section 4.4), whether or not the rotation ω of the foliation vanishes. Indeed, one does not even need a foliation: Just as in the case of SR, if one hypersurface intersecting all the fibers is chosen as the origin for the proper time τ on each fiber (i.e., τ = 0 on this hypersurface); then the proper time on each fiber provides the fourth individuating quantity. The transformation from one fibration with associated proper times to another is merely a change of labeling of the individuation.87 This individuation evades the hole argument and allows the formulation of the Cauchy problem for the Einstein field equations in terms of Lie derivatives of the tetrad components of the appropriate quantities with respect to any time-like congruence, holonomic (see Stachel, 1969) or non-holonomic (see Stachel, 1980).88

Thus, the principle of relativity has been extended beyond inertial frames in Minkowski space-time: the laws of any physical theory based on a geometric object field Φ, or indeed the laws governing any particle world-lines introduced into the theory, can be formulated with respect to any reference frame based on any such fibration and the associated proper times.

The automorphisms of these reference frames now form a function group, which can be defined by its action on the orthonormal tetrad field (v, ei) (i = 1,2,3) characterizing some initial frame. At any point x, an element of the group SO (1,3) will take one such tetrad into another ′ ′ ( v,ei); such an element depends on six position-dependent parameters (three rotations and three pseudo-rotations). Since any smooth vector field is holonomic, the resulting ′v field will generate a new fibration. Four “translations” (i.e., diffeomorphisms of the manifold M that are transitive and effective) will take the “origin” of the first reference frame into the origin of the second.89

In this sense, the general theory does extend the principle of relativity from inertial frames in Minkowski space-time to arbitrary orthonormal tetrad frames in pseudo-Riemannian space-times, either given a priori (background-dependent theories) or constructed from a solution to the Einstein equations (background-independent theories such as general relativity).

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