The Hole Argument and Some Extensions default38 38See Stachel and Iftime (2005).

4 The Hole Argument and Some Extensions ³⁸

After defining geometric and algebraic structures, a space is defined as a set of points with a geometric structure that is invariant under some group of transfomations of its points. Then I discuss product and quotient spaces, fibered spaces, and theories based on these spaces, in particular permutable and generally permutable theories (Section 4.1).

Up to this point, the only structures discussed have been relations between the elements of a set. But all definitions are still applicable – appropriately modified, of course – when additional structures are introduced. In particular, the case of most physical interest is that of geometric object fields defined on a differentiable manifold (Section 4.2). They provide the framework for coordinate-independent definitions of covariant and generally covariant theories, followed by a precise formulation of the of the original hole argument against general covariance and of the way to avoid its conclusion, discussed informally in Section 2.

Then I discuss fiber bundles, which consist of a total space, a base space, and a projection operator. Under certain circumstances, the base space may be defined as the quotient of the total space divided by an equivalence relation defining its fibers (Section 4.3) This approach allows a more precise formulation of Einstein’s vision of general relativity, discussed informally in Section 3.2.

Finally, the distinction between natural and gauge natural bundles is discussed, and between the concepts of covariance and general covariance when applied to theories defined on each type of bundle (Section 4.4).

A number of philosophical concepts used but not defined in this section, such as: intrinsic and extrinsic properties, internal and external relations, and quiddity and haecceity, are discussed in the Appendix B.

4.1 Structures, algebraic and geometric, permutability and general permutability

Consider a set $S$ of elements $x$ , $y$ , $z$ , etc., together with a set of relations $R$ between its elements.³⁹ There is a major distinction between a geometry and an algebra:

Geometry:

In a geometry, the elements of $S$ (hereafter called points and symbolized by $p$ , $q$ , etc.)⁴⁰ all are of the same quiddity (i.e., of the same nature) but lack haecceity (i.e., are not inherently individuated): The only distinctions between the points arise from a set of internal relations $R$ between them. If we abstract from these relations, the set $S$ is invariant under $Perm (S )$ , the group of all permutations of the points of $S$ .⁴¹ The set of relations defining a geometry structure or geometry $G$ on $S$ will be symbolized by $RG$ . The maximal subgroup $AutR (S ) G$ of $Perm (S)$ that preserves all these relations between the points of $S$ is called the symmetry or automorphism group of this geometry, and could just as well be used to define it.⁴²

Obviously, $Perm (S)$ is the maximal possible automorphism group; so a study of its subgroups and their relation to each other is equivalent to a study of all possible geometries on $S$ and their relation to each other.

Algebra:

In contrast to a geometry, in an algebra each element (symbolized by $a$ , $b$ , etc.) in addition to having the same quiddity also has an intrinsic haecceity (individuality). An algebraic structure or algebra $A$ on a set is also defined by a set of relations $RA$ between its elements; but these are external relations, which do not affect the intrinsic individuality of each element.⁴³

Coordinatization:

Since Descartes introduced analytic geometry, it has proved convenient and often necessary to apply algebraic methods in the solution of geometrical problems. This is done by a coordinatization of the geometry (see Weyl, 1946 , for this term): A one-one correspondence is set up between the points of the geometry and certain elements of an appropriately chosen algebra. This coordinatization assigns to each point $p$ of the geometry an element a of the algebra, called its coordinate and symbolized by $a(p).$ Now certain algebraic operations can be given a geometrical interpretation, and vice versa. But, by individuating the points of a geometry, a coordinatization negates their homogeneity, turning the geometry into an algebra. The only way to restore their homogeneity is to negate the coordinatization as follows: Introduce the class of all admissible coordinatizations of the geometry⁴⁴ based on the given algebra, so that each point of the geometry will have every admissible element of the algebra as its coordinate in at least one admissible coordinate system. Transformations between two admissible coordinate systems are called admissible coordinate transformations; they usually form a group that includes a subgroup isomorphic to the automorphism group of the geometry.

Permutations and the basic identity.

There are two distinct ways in which the assignment of all admissible coordinates to each point of a geometry may be accomplished.

Active point transformations: Keep the coordinate system fixed, $a → a$ , and permute the points of the geometry: $p → q$ , $a(p) → a(q)$ .
Passive coordinate transformations: Keep the points of the geometry fixed, $p → p$ , and carry out an admissible coordinate transformation of the elements of the algebra: $a → b$ , $a(p) → b(p)$ .

The terms “active” and “passive” refer to the effects of a transformation on the points of the geometry. A passive coordinate transformation is an active transformation of the elements of the algebra.⁴⁵

Two active permutation groups of any geometry have already been introduced:

$Perm (S)$ , the group of all permutations of the elements of $S$ ;
$AutRG (S)$ , the subgroup of $Perm (S)$ consisting of the permutations that belong to the automorphism group $RG$ of a particular geometry $G$ .

Relations may also be permuted. Let $R (p)$ symbolize a relation between the set $x$ of all points of $S$ .⁴⁶

Consider a permutation $Px$ of the elements of $S$ . Define the permuted relation $PR$ as follows: $P R (x )$ holds iff $R(P x)$ does. When a permutation $Px$ is carried out, the relation $R$ will be said to be “carried along” if it is also permuted into the relation $P R$ .⁴⁷

It follows that, if $R (x )$ is valid, then so is $P R (Px )$ .

The basic identity for the group of all permutations $Perm (S )$ .

By virtue of the intrinsic homogeneity of its points, a geometry $G$ remains unchanged if, for each Permutation $P$ of $Perm (S)$ , the corresponding permutation $P RG$ of the set of relations $RG$ defining $G$ is also carried out. For any relation $RG ∈ RG$ , it is clear that $P RG (P p)$ holds if and only if $RG (p )$ holds, so $P RG ∈ RG$ ; thus $RG$ and $P RG$ describe the same geometry. I shall refer to this result as the basic or trivial identity for the group $Perm (S )$ : it holds for any geometry based on a subgroup of $Perm (S )$ .

Equivalence relations and quotient maps.⁴⁸

An equivalence relation $Req$ on any set $S$ is a two-place relation having the following three properties: For all $x, y,z$ in $S$ , it is reflexive: $Req(x,x)$ holds; symmetric: if $Req (x,y)$ holds, then so does $Req (y, x)$ ; and transitive: if $Req (x, y)$ and $Req(y,z )$ both hold, then so does $Req(x,z )$ . If the context is clear, one often abbreviates $Req (x, y)$ by $x ≈ y$ . An equivalence relation divides $S$ into equivalence classes $SR$ , often also called its orbits (see Neumann et al., 1994

, Chapter V). Every element of $S$ belongs to one and only one such equivalence class. The quotient set of $S$ by $R eq$ , often called the orbit space and abbreviated $S = S∕R Q eq$ , is defined by the condition that each element of the quotient set corresponds to one and only one such equivalence class.

Product sets and quotient sets.

Given two sets $A$ and $B$ , we can form the product set $A × B$ , consisting of all pairs of elements $(x,y)$ , with $x ∈ A$ and $y ∈ B$ . $A$ mapping $φ$ from the domain A to the range or codomain $B$ (see Lawvere and Schanuel, 1997 , pp. 13–14), often symbolized $φ : A → B$ , is defined as a subset of $A × B$ , such that for each $x$ in the domain there is one and only one $y$ in its range.

In various contexts, “mappings” might also be called functions, transformations, or operators. Homomorphisms, isomorphisms, homeomorphisms, diffeomorphisms, continuous or differentiable maps will be more attached to certain classes of mappings, which “preserve” certain structures on the sets which are their domains and ranges (Hermann, 1973 , p. 3).

The mapping is surjective if, for every element $y ∈ B$ , there is at least-one element $x ∈ A$ that maps onto $y$ . If both mappings $A → B$ and $B → A$ are surjective, the mapping is called bijective. If the bijective map preserves all structures on $A$ and $B$ , $A$ and $B$ are said to be isomorphic. If the mapping $φ : A → B$ is surjective, the set $B$ is isomorphic to the quotient set $A ∕φ$ ; so $B$ can actually be defined as the quotient set.

This “passing to the quotient” is a way of defining new spaces and mappings that is very important in all of mathematics, particularly in algebra and differential geometry. (Hermann, 1973 , p. 5).

This possibility allows us to realize Einstein’s vision of general relativity (see Section 3.2), which in this context is simply: If no $A$ , then no $B$ . We can define the mapping or morphism $φ$ from $S$ to $SQ$ , $φ : S → SQ$ , which projects each element $x$ of $S$ into the element $φ(x )$ of $SQ$ corresponding to the equivalence class that includes $x$ . Conversely a section of $S$ is an inverse mapping $−1 φ$ from the point $φ (x )$ of $SQ$ to a unique point $y$ in the equivalence class of $S$ that maps into that point of $SQ : φ− 1(x ) = y$ , with $x ≈ y$ .

So far, these concepts can be applied to any set. If $S$ is a geometry with automorphism group $G$ , it is referred to as a $G$ -space (see Neumann et al., 1994 , pp. 74ff). The equivalence relation is said to be $G$ -invariant if, whenever $p ≈ q$ holds for two points of $S$ , then it follows that $g(p) ≈ g(q)$ for all $g ∈ G$ . Consequently, the action of an element $g ∈ G$ on $S$ preserves the orbits of $S$ but permutes them; so all orbits, henceforth called fibers, must be isomorphic to what is called a typical fiber. The quotient set $SQ = S∕Req$ is itself a $G$ -space called the quotient space.

Fibered or fiber spaces.

A fibered space consists of a total space $E$ ; a base space $B$ ; and a projection operator $π : E → B$ that is a surjective mapping, as defined above. The fiber $F b$ over each point $b ∈ B$ is the set of all inverse elements $−1 π (b) ∈ E$ ; that is, all elements $p ∈ E$ such that $π (p ) = b$ . A section $σ$ of a fiber space is a choice of one element on each fiber $Fb$ for every $b ∈ B$ . To convert a homogeneous set $S$ with an equivalence relation $ρ$ into a fibered space, let $S$ constitute the total space $E$ ; then $SQ$ forms the base space $B$ , and the mapping $φ$ becomes the projection operator $π$ . If $S$ is a geometry with automorphism group $G$ , then $G$ preserves the equivalence classes; so all the fibers are isomorphic, resulting in a fiber bundle:

A fiber bundle $(E, B, π)$ consists of: 1) a total space $E$ , divided into fibers by an equivalence relation $ρ$ , all of these fibers being isomorphic to a typical fiber $φ$ ; 2) a base space $B$ that is isomorphic to the quotient $E ∕ρ$ ; and 3) a projection operator $π : E → B$ that takes each point of its domain $E$ into the point $p$ of its range $B$ that corresponds to the fiber including $p$ . A section of the bundle is a mapping $σ$ that takes each point $p$ of its domain $B$ into a unique point of its range, consisting of the set of fibers of $E$ . The point of $σ$ on the fiber $φp$ over $p$ is symbolized by $φp (σ )$ . If $E$ has the automorphism group $G$ , the action of an element $g ∈ G$ on the points of any section $σ$ will result in a new section $′σ$ ; symbolically: $g (σ ) = ′σ$ . So, given one section $σ$ , the action of the elements of $G$ produces a whole equivalence class of sections $g (σ )$ .

Theories, permutable and generally permutable.

A theory of a certain type is a procedure for producing models of that type. A particular theory of that type is a rule for selecting a subset of these models. One type of theory is defined by the choice of a fiber bundle with automorphism group $G$ ; its models are the sections of this bundle. A particular theory is a rule for choosing a subset of sections of the bundle as models. If the rule is such that, when $σ$ is a model, then so is the entire equivalence class of sections $g (σ )$ , the theory is permutable. It is generally permutable if this entire equivalence class is interpreted as a single model of the theory. In terms of the distinction between syntax and semantics, one may say: While each section of a theory is always syntactically distinguishable from the others, in a permutable theory they may also be semantically distinguishable. However, in a generally permutable theory they are not; only an entire equivalence class of sections has a unique semantic interpretation. Take Euclidean plane geometry, for example. All assertions about geometric figures, such as right triangles, rectangles, circles, etc., are invariant under its automorphism group, which consists of translations and rotations; so it is certainly a permutable theory. But these assertions actually apply to the whole equivalence class of geometric figures satisfying any of these definitions; so it is a generally permutable theory.

On the other hand, plane analytic geometry includes a choice of origin, unit of length, and a pair of rectangular axes. So all of its assertions are still permutable; but some of them include references to the origin, axes, etc. We can distinguish, for example, between a circle of radius $r$ centered at the origin, and one of the same radius centered at some other point. So plane analytic geometry is a permutable theory, but not generally permutable. The reason, of course, is that the choice of a unique preferred coordinate system converts the Euclidean plane from a geometry into an algebra.

4.2 Differentiable manifolds and diffeomorphisms, covariance and general covariance⁴⁹

For the space-time theories forming the main topic of this review, $S$ is often a four-dimensional differentiable manifold $M$ ; and the analogue of $Perm (S )$ is $Diff(M )$ , the diffeomorphism group consisting of all differentiable point transformations of the points $x$ of $M$ . Any given, fixed geometric structures defined on $M$ , such as a metric tensor field, will be symbolized by $Λ(x )$ ; they represent the analogue of the relations $RG$ . The $Λ$ -geometry of $M$ is also defined by the invariance of these $Λ (x$ ) under the action of some Lie subgroup $G$ of $Di ff(M )$ . In other words, $G$ plays the role, analogous to that of $Aut (RG )$ , of the automophism group $Aut (M, Λ )$ of the $Λ$ -geometry of $M$ . And just as in that case, here every $G$ -space can also be defined as a quotient or orbit space:

Every $G$ -space can be expressed as in just one way as a disjoint union of a family of orbits. (Neumann et al., 1994 , p. 51)

Just as in analytic geometry, one may set up ordinary and partial differential equations for various particles and fields on $M$ . Denote a set of such geometric object fields on $M$ collectively by the symbol $Φ(x)$ , and consider the effect of an element $g (x ) ∈ G$ on $Φ (x)$ .⁵⁰ From the definition of a geometric object (see Schouten, 1954 , pp. 67–68) it follows that if $′ x → g(x) = x$ , then $Φ (x) → ′Φ (′x)$ .

Geometric quantities form an important subclass of geometric objects which includes all tensor fields. Their transformation law under $g(x)$ is linear and homogeneous in the components of $Φ (x )$ and homogeneous in the derivatives of $g(x)$ .

In both Galilean space-time (see Yaglom, 1979 ) and in special relativistic space-time (Minkowski space), $G$ is a ten parameter Lie group. Four of these parameters generate spatial and temporal translations of the points, making these space-time geometries homogeneous. And in both, the six remaining parameters act at each point of space-time: three generate spatial rotations and three generate “boosts”, making both space-times non-isotropic. But they do so in different ways because their “boosts” differ: For Galilean space time, they are Galilei transformations that preserve the invariance of the absolute time. For Minkowski space-time, they are Lorentz transformations that combine spatial and temporal intervals into an invariant, truly four-dimensional space-time interval. Both of these groups are subgroups of $SL (4,R )$ , the group of four-volume-preserving transformations.⁵¹ And both theories have a homogeneous, flat affine connection in common that is the mathematical expression of the Newton’s first law of inertia. Its invariance group is $AL (4,R )$ , which is a subgroup of $SL (4,R )$ .

Newtonian gravitational theory, in the form which incorporates the equivalence principle, preserves the global space-time structure of Galilean space-time, but abandons the homogeneous flatness of the affine connection in favor of a non-flat affine connection that is the mathematical expression of the dynamical inertia-gravitational field. This field is non-homogeneous, but its compatibility with the space-time structure requires that locally it remain invariant under $AL (4,R )$ , which means that its globally automorphism group must be $SDiff (M )$ , the group of unimodular diffeomorphisms.

General relativity similarly abandons the homogeneous flatness of the affine connection in favor of a non-flat affine connection that is the mathematical expression of the dynamical inertia-gravitational field. But, in order to preserve its compatibility with the special-relativistic chrono-geometry expressed by the metric tensor, the latter must also become a dynamical field. It preserves the local space-time structure of the special theory at each point. But globally both dynamical fields must have automorphism groups consisting of diffeomorphisms of $M$ , the space-tme manifold, now itself no longer globally fixed. Traditionally, $Diff (M )$ , the full diffeomorphism group, has been assumed to be the correct automorphism group for general-relativistic theories. However, there are good arguments for restricting this group to $SDi ff (M )$ , the group of unimodular diffeomorphisms with determinant one.

But first some definitions are needed (see, e.g., Wikipedia: Group action ). The action of $G$ is said to be effective if its identity element is the only one that takes each point into itself: That is, if $g ∈ G$ , $x ∈ M$ and $π : x → g(x)$ is such that $g(x) = x$ for all $x$ , then $g = e$ , the identity element of $G$ . The action is transitive if $π$ is a map onto $M$ that connects any two of its points: That is for any two points $x, y ∈ M$ , there is always a $g ∈ G$ for which $g(x ) = y$ .

The stabilizer group $Hx$ at a point $x$ of $M$ is the subgroup of transformations of $G$ that leave the point $x$ invariant: That is, $g ∈ Hx$ if and only if $g(x ) = x$ .⁵² Since $G$ is a Lie group, $Hx$ is a closed subgroup at each point of $G$ and these stabilizer groups are conjugate subgroups of $G$ . Indeed, $M$ is isomorphic to $G ∕Hx$ ; so one may actually define a geometry by the pair $(G,H )$ , where $H$ is a closed subgroup of $G$ .

The action of $G$ is free or semiregular if its stabilizer group is the identity: That is, if $gx = x$ for some point $x$ , then $g = e$ , the identity element of $G$ ; equivalently, if $gx = hx$ for some $x$ , then $g = h$ . For example, the translation groups discussed above act freely on Galilean and special relativistic space-times.

Now we are ready to return to the question of automorphism groups for general relativistic theories. The action of the stabilizer of $Di ff(M )$ on the tangent space at each point $x$ of $M$ is $Lx = GL (n)$ , the group of all linear transformations at $x$ . But consider the objects defining the geometry of a general-relativistic space-time with $n = 4$ : Again, if one wants to preserve the four-volumes of space-time, which are needed to formulate meaningful physical averages, one must restrict these transformations to $SL (4)$ , the group of 23 special linear transformations with unit determinant. The linear affine connection at $x$ , which represents the inertio-gravitational field, is only invariant under the subgroup $ASL (4)$ , the group of affine transformations with unit determinant. And the invariance group of the metric tensor, which represents the chrono-geometry, is even further restricted to the pseudo-orthogonal subgroup $SO (3,1)$ of $SL (4 )$ . In short, globally physical considerations suggest the need to start from $SDi ff(M )$ as the automorphism group of general-relativistic theories.

So physically, $Diff(M )$ overshoots the mark by allowing non-unimodular transformations, i.e, transformations with any value of their determinant at a point. Geometrically, they correspond to similarity transformations, which preserve the shape but not the size of four-volumes in space-time. Usually, one “compensates” for this unwanted change of size by introducing tensor densities: When appropriate weights are introduced for various tensors, these densities can undo the effects of the size changes produced by non-unimodular transformations.

However, one may simply start from $SDi ff(M )$ as the automorphism group. The action of its stabilizer on the tangent space at each point of $M$ is $SLx$ , the maximal symmetry group that preserves the size of four-volumes, thus avoiding the need to introduce densities, among its many other advantages (see Stachel, 2011 ; Bradonjić and Stachel, 2012 ). For much of the following discussion, however, the distinction between $Di ff(M )$ and $SDi ff (M )$ is inessential, so I shall continue to discuss $Di ff(M )$ , and only point out the distinction at some places where it is really important.

Symmetry transformations:

By definition, $AutRG (S )$ , the group of permutations of the points of $S$ defining the geometry $G$ , leaves the relations $RG$ (which equally well define the geometry of $S$ ) unchanged; so the $RG$ do not need to be permuted when the points of $S$ are. Whichever Lie subgroup $G$ of $Di ff(M )$ is chosen as the automorphism group defining the geometry of a differentiable manifold $M$ , similar comments hold for it. As we shall see, the important difference for the hole argument is that between geometries based on finite-parameter Lie groups and those based on Lie groups depending on one or more functions (functional Lie groups).

Passive coordinate transformations and the trivial identity:

Since it is no more than a re-labeling of its points, any admissible passive coordinate transformation has no effect on a geometry (see Section 4.1). However, if one restricts the group of coordinate transformations to a subgroup of those corresponding elements of the automorphism group of the geometry, then there is an isomorphism between this subgroup of passive coordinate transformations and the group of active point transformations defining the geometry. Hence, it is possible to reformulate any statement about the geometry in terms of relations between the coordinate components of the geometric object fields $Λ$ that are invariant under this subgroup of restricted coordinate transformations. In the past, this is how coordinate-dependent techniques were used to arrive at geometric results; and many contemporary treatments still utilize this technique. If one permutes the points of $M$ by a diffeomorphism, carries along the $Λ$ fields defining its geometry and the $Φ$ fields defining the theory, and also carries out the corresponding coordinate transformations, then clearly the new fields at the new points will have the same coordinate components in the new coordinate system as the old fields at the old points in the old coordinate system. This observation is another, coordinate-dependent variant of the basic or trivial identity. It holds for any $Φ$ fields, quite independently of any theory, or any field equations that these fields may obey.

Basis vectors and a coordinate-independent formulation of the trivial identity:

Geometrically, a coordinate system corresponds to the choice of a holonomic basis $ei$ at each point of $M$ : That is, there is a local coordinate system such that $i ei = ∂ ∕∂x$ . But the essential element geometrically is the choice of a basis, not its holonomicity. So, introduce an ordered set of basis vectors $ei(x)(i = 1,2, ...,n)$ , holonomic or not, at each point $x$ of $M$ together with the associated dual basis of covectors or one-forms $ej(x)$ , such that $⟨eiej⟩ = δji$ .⁵³ Associated with the geometric object fields $Φ$ and $Λ$ on $M$ are their components with respect to such a pair of basis vectors, which will be symbolized by $Φ[e(x)]$ and $Λ [e (x)]$ : This is a set of coordinate-independent scalars that result from saturating all the free covariant and contravariant indices of $Φ$ and $Λ$ with the $ei$ and $ej$ respectively. Of course, under a change of basis: $e(x) → ′e(x )$ these scalars transform appropriately. A diffeomorphism $′ D : x → x$ induces such a change of basis: $′ e(x) → eD (x)$ , and corresponding changes in the geometric object fields $′ Λ (e) → ΛD (x)$ and $′ Φ (x) → ΦD (x$ ). However, the values of these scalars remain unchanged if one carries out the associated push forwards and pull backs of $Φ$ and $Λ$ , as well of the basis vectors and covectors. That is, if we take the new basis vectors at the new point: $′eD (′x )$ ; then the new components with respect to the new basis vectors at the new points will equal the old components with respect to the old basis vectors at the old points: $Φ [e (x)] = ′ΦD [′eD (′x )]$ and $Λ [e(x)] = ′ΛD [′eD (′x )]$ . This observation is a coordinate-independent formulation of the basic identity. Since any model of a physical theory can only fix the values of such coordinate-independent scalars with respect to some basis for all geometric objects in that model, this identity cannot fail to hold for any theory based on the $Λ$ -geometry of $M$ .

Covariance and general covariance:

Suppose we perform the push forwards and pull packs on the geometric object fields $Φ$ , but not on the $Λ$ -geometry or the basis vectors and convectors. That is, let $e(x ) → e(x)$ and $Λ[e(x)] → Λ [e (x )]$ , but $′ Φ [e(x )] → ΦD [e(x)]$ . In general, $′ Φ [e (x)] ⁄= ΦD [e(x)]$ , so this results in a set of scalars that is distinct from $Φ [e(x )]$ at each point $x$ of $M$ . A theory is covariant under the the $Λ$ -geometry’s automorphism group if, whenever $Φ [e (x)]$ is a model of the theory, then so is $′Φ [e (x)] D$ . Covariance clearly defines an equivalence relation between models of the theory; so covariance divides all models of a theory into equivalence classes.⁵⁴

A covariant theory is generally covariant under the $Λ$ -geometry’s automorphism group if an entire equivalence class of its mathematically distinct models corresponds to a single physical model of the theory.

4.3 Fiber bundles: principal bundles, associated bundles, frame bundles, natural and gauge-natural bundles ⁵⁵

An ordered set of basis vectors $ei(x)$ at a point of $M$ is called a linear frame, and the set of all such linear frames at a point of $M$ constitutes one fiber of the bundle of linear frames over $M$ . As Kobayashi explains, the bundle concept can be used to formulate any geometry on $M$ as a $G$ -structure:⁵⁶

Let $M$ be a differentiable manifold of dimension $n$ and $L (M )$ the bundle of linear frames over $M$ . Then $L (M )$ is the principal fibre bundle over $M$ with group $GL (n;R )$ . Let $G$ be a Lie subgroup of $GL (n;R )$ . By a $G$ -structure on $M$ we shall mean a differentiable subbundle $P$ of $L (M$ ) with structure group $G$ . (Kobayashi, 1972 , p. 1).

Such a fiber bundle formulation of geometries has several crucial advantages:

It makes evident the fundamental distinction between vertical geometrical quantities, such as metric tensors, that live on the fibers of the bundle; and horizontal geometrical objects, such as linear affine connections, that serve to connect these fibers. This is the case whether the metric and/or connection are fixed and given components of $Λ$ ; or are components of $Φ$ , themselves subject to dynamical field equations.

It enables us to go from global to local formulations of background-independent theories, such as general relativity, in which the global topology of the base manifold $M$ cannot be specified a priori, because it differs for different solutions to the field equations.⁵⁷

Fibered manifolds

The concept of fibered spaces for a set, discussed in Section 4.1, can now be applied to differentiable manifolds (see Section 4.2). After a fibered manifold is defined, the important cases of principal bundles, vector bundles, natural bundles and gauge-natural bundles and their physical applications are discussed, stressing the importance for general relativity of quotient bundles and local considerations.

A fibered manifold $(E, M, π )$ consists of a total manifold $E$ , a base manifold $M$ , and a projection operator $π : E → M$ . $E$ is a differentiable manifold, the points of which, $u$ , $v$ , etc., are grouped into fibers by an equivalence relation $ρ$ between its points. $M$ is also a differentiable manifold, the points of which are symbolized by $x$ , $y$ , etc. The fiber over $x$ is symbolized by $F x$ . Note that, if the relation $ρ$ is given initially, sometimes the base manifold $M$ may be defined as the quotient of the total manifold $E$ by $ρ : M = E ∕ρ$ ; in other words as the orbit space of $G$ (see Sections 4.1 and 4.2). But the situation is generally somewhat more complicated:

Usually, when symmetries and invariance groups are considered, a problem reduces to the corresponding orbit space, and therefore the structure of these spaces has to be investigated. This structure theory is quite complicated in general, since these spaces usually are singular spaces and not again manifolds. In fact, only if the action of the Lie group is free (i.e., all isotropy subgroups of single points are trivial), the resulting orbit space bears a manifold structure and forms together with the manifold and the quotient map a principal fiber bundle, whose structure is well known. More often, the orbit space admits a stratification into smooth manifolds with an open and dense largest stratum, the set of principal orbits…. This stratified space can then be treated almost like a manifold when taking special care. The existence of such a stratification is usually shown by proving the existence of slices at every point for the group action (Schichl, 1997 , p. 1).

I shall assume that – as in general relativity – in any theory considered, the quotient space is either a manifold or a stratified manifold; and that any local solution to its field equations can be extended to a global solution.⁵⁸

A fiber bundle is a fibered manifold in which all its fibers are isomorphic to a typical fiber $F$ , itself a manifold; that is, for all $x$ , $Fx ≈ F$ . Suppose $F$ is $q$ -dimensional and $M$ is $p$ -dimensional One can always introduce a local trivialization of the bundle: Let $X$ be an open subspace of $M$ . Locally, the total space $E$ is a product space $(F × X$ ), and one can introduce $p$ variables $(x1,x2 ...,xp)$ as local coordinates of a point $x$ of $X$ , and $q$ variables $(u1,u2,...,uq )$ as coordinates of a point $u$ of $F$ . So $(F × X$ ) is coordinatized by the $(p + q )$ coordinates $(x,u$ ) of a point $u x$ of $E$ lying on the fiber $Fx$ over the point $x$ . Let $G$ be a Lie group of diffeomorphisms that acts on $E$ .⁵⁹ The action of an element $g ∈ G$ on the point $(x,u )$ of $E$ is symbolized by $(x,u ) → g(x,u ) = (′x,′u) = [χ(x,u),ψ (x,u)]$ .

Two subgroups of $G$ are especially important:

The base transformations (diffeomorphisms of $X$ ) that do not affect the fibers:
$x → ′x = χ (x), ′u = u$ .
The pure fiber or pure gauge transformations on the fiber at each point:
$′ ′ x = x, u = ψ (x, u)$ .

Both of these are included in a third subgroup:

The fiber-preserving transformations:
$′ ′ (x,u) → (x, u) = [χ(x),ψ (x,u)]$ .

If $G$ is a connected Lie group, all of its actions can be constructed from iterations of the action of its Lie algebra, composed of its infinitesimal generators: The vector fields $V$ on $E$ , each of which generates a one-parameter group of point transformations, or flow, on $E$ . Locally $V$ may be written in terms of the coordinates $(x,u)$ :

The generator $V$ is called:

Horizontal if $φα = 0$ , i.e., it generates only base transformations.

Vertical if $i ξ = 0$ , i.e., if it generates only pure fiber or pure gauge transformations.

The flow generated by $V$ will be fiber preserving if and only if $ξi = ξi(x)$ .

Natural bundles:

A fiber-preserving diffeomorphism projects naturally into a unique diffeomorphism of the base manifold $M$ ; but generally the converse does not hold. If it does, i.e., if a base diffeomorphism of $M$ lifts uniquely to a fiber-preserving diffeomorphism of $E$ , then the bundle is a natural bundle. A geometric object defined on such a bundle is called a natural object.⁶⁰ This is the fiber bundle version of the definition of geometric objects in Section 4.2.

Principal bundles, associated vector bundles:

If the the typical fiber $F$ is isomorphic to the structure group $G : F ≈ G$ , then the bundle is a principal fiber bundle $P$ with structure group $G : P = (E, M, π; G )$ . Corresponding to any $P$ with structure group $G$ , there is a class of associated vector bundles. In such an associated bundle, each fiber forms a vector representation of $G$ . This vector representation need not be irreducible, so the class of associated vector bundles includes all tensor fields.

4.4 Covariance and general covariance for natural and gauge-natural bundles

The use of fibered manifolds allows a precise formulation of the concepts of covariance and general covariance for any physical theory; and of the hole argument for background-independent theories, and even – with appropriate modifications – for some partially-background-dependent theories.

4.4.1 Fiber bundles needed in physics

Every natural physical theory can be formulated in terms of some natural geometric object(s)⁶¹ that lives on an appropriate fibered differentiable manifold,⁶² the nature of which depends on these geometric object(s). If the theory is defined on a differentiable manifold $M$ that is the quotient of the fibered manifold $E$ divided by the equivalence relation $ρ$ defining the fibration, $M = E ∕ρ$ ; then there is an operator $π$ , projecting each fiber onto the corresponding point of $M$ : $π : E → M$ . Since the fibered manifold represents a natural object, there is a one-one correspondence between fiber-preserving diffeomorphisms of $E$ and diffeomorphisms of $M$ .

A number of most important gauge natural theories cannot be so formulated, but require the broader concept of gauge natural bundles for their precise formulation. Indeed, every classical physical theory can be reformulated as the jet prolongation of some gauge natural bundle by adjoining the derivatives of the geometric object fields to the original bundle.⁶³

4.4.2 Background-dependent theories

A theory based on such fixed $Λ$ -fields on $M$ is called background-dependent, with $Aut(M, Λ )$ as its symmetry group.⁶⁴ Any geometric object fields $Φ(x)$ can then be introduced on this fixed-background space-time together with a set of field equations governing their dynamics, which generally involve some or all of the $Λ (x)$ .

In many theories, the fixed geometric object fields on $M$ consist of a vertical chrono-geometric metric tensor on each fiber and the corresponding horizontal inertio-gravitational linear connection. Any non-gravitational theory can be formulated on a fiber bundle associated with the principal bundle determined by the metric and connection: The $Φ (x)$ break up into two subclasses: The fields of massive objects (such as charged bodies) are represented by geometric quantities living on the vertical fibers; and the gauge fields transmitting the forces between these objects (such as the electromagnetic field) are represented by verical connections along the fibers; these connections are only fixed up to some group of gauge transformations.

4.4.3 Background-independent theories

In the case of general relativity and other background-independent theories (such as the coupled Einstein–Maxwell equations), $Λ$ reduces to the identity and there are no fixed background space-time structures on $M$ . $Diff(M )$ is chosen as $Aut (M )$ in the usual formulations; but, as suggested in Section 4.2, $SDi ff(M )$ , the unimodular subgroup, may be chosen. In that case, the space-time structures subdivide further: The pseudo-metric splits into a conformal metric with determinant $− 1$ and a scalar field, both of which live on the vertical fibers; while the linear affine connection splits into a trace-free projective connection and a one form, both of which serve to connect the fibers.

4.4.4 Gauge symmetries

To define the gauge symmetries of a certain type of theory, one must consider the sections of the corresponding fiber bundle. A local section $σ : M → X ∈ E$ or global cross section $σ : M → E$ is a map taking each point $p$ of $X$ or $E$ , respectively, into a unique point of the fiber $Fp$ over $p$ .⁶⁵

For each type of physical theory, a section represents a particular configuration of the corresponding physical field. However, in theories of the gauge-field type, this representation is not unique. There is a group of gauge transformations, each element of which maps one mathematical representative of some field configuration into another representative of the same configuration. A gauge symmetry is an equivalence relation on the set of sections: Two sections $σ$ and $′σ$ are gauge equivalent if there is a gauge transformation taking one into the other. This equivalence relation divides the set of all sections into equivalence classes, the gauge orbits; each section belongs to one and only one such orbit. If the gauge group of some type of theory consists entirely of fiber-preserving transformations, then the theory can be formulated on a natural bundle. But if its gauge group includes non-fiber-preserving transformations, then a gauge-natural bundle is needed to formulate this type of theory correctly.

The field equations of a particular gauge field theory serve to pick out a class of preferred sections consisting of the solutions to these equations. For a gauge theory, these equations must be of such a form that, if one section is a solution, then so are all members of the entire gauge orbit of that section. In other words, the gauge transformations must form a symmetry group of the field equations. This group is the automorphism group $Aut (P )$ of the principal gauge-natural bundle $P$ corresponding to the theory (see, e.g., Fatibene and Francaviglia, 2003 , p. 223). Fatibene and his collaborators explain the distinction between the two types of theory well:

The main technical difference between natural and gauge natural theories is that [base] diffeomorphisms are completely replaced by gauge transformations. In gauge natural theories spacetime diffeomorphisms do not act at all on fields, since the only action one can define in general is that of gauge transformations. This is due to the fact that although pure gauge transformations are canonically embedded into the group of generalized gauge transformations, there is no canonical “horizontal” complement to be identified with $Diff(M )$ . …“Horizontal” symmetries, in fact, are generally associated to physically relevant conservation laws, such as energy, momentum and angular momentum. The definition of such quantities is almost trivial in natural theories; on the contrary, in gauge natural theories pure gauge transformations are easily associated to gauge charges (e.g., the electric charge in electromagnetism), while the absence of “horizontal” gauge transformations is a problem to be solved to appropriately define energy, momentum and angular momentum. For this reason, in gauge natural theories the dynamical connection plays an extra role in determining horizontal infinitesimal symmetries as the gauge generators which are horizontal with respect to the principal connection (Fatibene et al., 2001 , pp. 3–4).

While bundle formulations of the hole argument originally dealt only with natural bundles, Lyre (1999 ) develops a generalized version that can be applied to gauge-natural bundles:

The generalized hole argument is motivated and extended from the spacetime hole argument …. [It] rules out fiber bundle substantivalism and, thus, a relationalistic interpretation of the geometry of fiber bundles is favored (Lyre, 1999 , p. 1).

Healey (2001 ) also argues that “fiber bundle substantivalism …is subject to an analogue of the ‘hole’ argument against space-time substantivalism.”

4.4.5 Four-geometries and stratified manifolds

To recapitulate: The choice of a bundle $(E, M, π )$ selects a certain type of physical theory but does not picked out a particular theory of that type, nor introduce any space-time structures on $E$ or $M$ . The points of $M$ form a geometry (see Section 4.1). As points of the space-time manifold, they have quiddity but they lack haecceity: a priori there is nothing to distinguish one such point from the others. Their automorphism group is the diffeomorphism group of $M$ or some appropriate subgroup, such as the unimodular group (see Stachel, 2011 ; Bradonjić and Stachel, 2012 ). For example, a metric-free formulation of electromagnetic theory can be based on a bundle of one-forms.

A particular theory is a rule for choosing a preferred class of cross sections of the fiber bundle. This rule generally includes specification of some space-time structures on $M$ . For example, in addition to a bundle of one-forms, source-free Maxwell electromagnetic theory, requires the specification of a conformal structure on $M$ .

In general relativity, an equivalence class of diffeomorphically-equivalent pseudo-metrics on a four-dimensional manifold,⁶⁶ often referred to as a four-geometry, is regarded as corresponding to a single inertia-gravitational field. While the fiber space consisting of all four-metrics over a given manifold forms a manifold,⁶⁷ the space of all four-geometries does not form a simple manifold, but a stratified manifold. That is, it is partitioned into slices, each of which is itself a manifold, consisting of all four-geometries having the same symmetry or isometry group. The largest slice is the manifold of generic geometries having no nontrivial symmetries; it contains the vast majority of geometries. Thence one descends slice by slice down to the slice consisting of all four-geometries having the maximal symmetry group (see Stachel, 2009 and Section 6.1).

The rule specifying the choice of a preferred class of space time structures may or may not include some restriction on $Diff(M )$ , the maximal possible automorphism group of $M$ . Obviously, diffeomorphisms always remain unrestricted in the sense of the trivial identity. A true restriction on the theory arises with the imposition of a finite-parameter Lie group as the symmetry group of the class of space-time structures picked out by the rule.⁶⁸ If there are no such restrictions, the theory is background independent. If the rule includes a Lie group involving some functions as well as parameters, the theory is partially background dependent. If the Lie group is maximal (ten-parameters in four dimensions), then the theory is totally background dependent. If the rule restricts the preferred class maximally, i.e., to the identity, then the theory specifies an individuating field on the space-time, turning it into an algebra. In this formulation, the symmetry group is included in the rule defining a physical theory, rather than being imposed a priori on the space-time structures defined on $M$ . This change enlarges the class of physical solutions: For example, not fixing the global topology of $M$ allows several possibilities for the global topology associated with a given local metric. But this does not alter the fact that the symmetry group of the space-time structures must be preserved by all such solutions. To what extent the hole argument applies to a non background-independent theory depends on the degree of background dependence that has been imposed (see Stachel, 2009 ); but if a theory is background independent, the hole argument certainly applies.

	Abstract
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
	References
	Footnotes

4 The Hole Argument and Some Extensions 38