List of Footnotes
1 | For a historical review of the disputes over general covariance, see Norton (1993). Throughout this paper, unless there is some indication to the contrary, “metric” is to be understood to mean “pseudo-metric tensor field with Lorentz signature;” and “connection” is the be understood to mean “linear affine connection with vanishing torsion.” | |
2 | See Mach (1988, 1986). | |
3 | This the semi-direct product of the homogeneous Lorentz group and translation group T_{4}. | |
4 | Indeed, even the then-future quantum theories were based on such fixed backgrounds: Galilei–Newtonian space-time for non-relativistic quantum mechanics, and Minkowski space-time for quantum field theory. Much of the debate about the form of an acceptable theory of quantum gravity that somehow reconciles quantum field theory with general relativity revolves around this question. See Section 6.4. | |
5 | The use of this term here is anachronistic: It was introduced much later. Weyl was the first to recognize the fundamental significance of this concept; he called it a “Führungsfeld” (see Weyl, 1923), which was translated as “guiding field” (see Weyl, 1950). | |
6 | In coordinates adapted to the static situation. Again, these coordinates have no direct physical significance. The measured local speed of light does not vary. | |
7 | The modern definition of a differentiable manifold and the distinction between diffeomorphisms of the manifold and coordinate transformations in and between regions of (the -dimensional Cartesian space) had not been formulated (see Stachel, 2014). | |
8 | The concept of parallel transport was only developed after completion of the general theory and largely in response to it. The only available geometrical interpretation of the Riemann tensor was in terms of the Gaussian curvature of two-dimensional sections, which was of little help in relating the Riemann tensor to anything in gravitation theory. Its interpretation in terms of geodesic deviation, so easily related to gravitational tidal forces, again lay in the future. | |
9 | He would have to subtract one-half of the Ricci scalar times the metric tensor (see Einstein’s Zurich Notebook, and the commentary on it in Renn, 2007). | |
10 | As we shall see later, it might have been more natural mathematically to formulate it in terms of an initial-value problem (see Section 2.7). | |
11 | In 1918 he named this requirement “Mach’s principle”: “The -field is determined completely by the masses of bodies. Since according to results of the special theory of relativity, mass and energy are the same, and since energy is formally described by the symmetric energy tensor , this therefore entails that the -field be conditioned and determined by the energy tensor of matter.” (Einstein, 1918) | |
12 | Contrary to the claim by Pais and others that he did not then understand coordinate transformations (see Section 3.1). | |
13 | To avoid future confusion, let me emphasize that the terminology introduced in Section 4 differs from that of Einstein, and indeed that of many contemporary accounts. In this review, the requirement that, if is a solution to the field equations, then must also be a solution is called covariance. General covariance is the requirement that and represent one and the same physical solution. Another difference is that Section 4 these conditions are formulated coordinate independently. | |
14 | See Iftime and Stachel (2006) for a modern treatment of the hole argument using the language of categories, functors and natural objects, and generalizing it to other covariant theories; and Section 4 of this review. | |
15 | See Stachel (1989), which introduced this name. There is evidence that discussions with Moritz Schlick influenced Einstein’s formulation of this argument: See Engler and Renn (2013) and references therein. | |
16 | See Rynasiewicz (1999), Section 4, pp. 443–448. | |
17 | In a hole, that is. Naturally, if other physical fields are present in a region of the manifold, it is possible that they may be utilized to individuate the points. | |
18 | Einstein argued that: “If one were to put Newtonian gravitational mechanics into the form of absolutely covariant (four-dimensional) equations, one would undoubtedly be convinced that [the] principle …rules them out, granted not theoretically, but practically” (Einstein, 1918, p. 242; transl. from Rynasiewicz, 1999, p. 458). Within a few years, Cartan and Friedrichs proved Einstein wrong by providing simple, four-dimensional formulations of Newtonian theory that incorporate the equivalence principle (see, e.g., Ehlers, 1973). | |
19 | The four-dimensional form of Newton’s theory (see the previous note) satisfies the first criterion, but not the second: its degenerate spatial and temporal metrics restrict the symmetry group of the theory (see, e.g., Ehlers, 1973). | |
20 | Kretschmann states: “G[ustav] Mie suggested this possibility of fixating the coordinate directions in an absolute way just by using the general theory of relativity in a letter of February 1916.” | |
21 | Kretschmann’s discussion of the bearing of these results on the relativity group of the theory fails to make a distinction between the group of transformations under which the theory is invariant (i.e., the nature of the equivalence classes of physically equivalent solutions), and the symmetry or isometry group of a particular solution to the equations. | |
22 | But see Section 6.4 and Section 7 for some further comments on this matter. | |
23 | For details, with references to Hilbert’s lectures and papers, see Renn and Stachel (2007). | |
24 | For example, while stating that physically meaningful results should not depend on the coordinate system, Hilbert actually adopted Gaussian normal coordinates in order to get uniqueness. An even more serious fault: he failed to mention the constraints on the initial data. | |
25 | See Stachel (1992) for a review of the early history of the Cauchy problem in general relativity. | |
26 | Lichnerowicz (1955) was particularly influential. See, e.g., Bruhat (1962). | |
27 | Even as astute a student of relativity as Cornelius Lanczos, who had worked with Einstein, misunderstood the issue. Explaining Einstein’s mistake, he writes: “It would be fatal to obtain an infinity of solutions of the field equations in one and the same reference system. In fact, however, we have merely the freedom of introducing curvilinear coordinates at will. In every given coordinate system the solution is determined, under the proper initial and boundary conditions” (Lanczos, 1970, p. 236). | |
28 | For a list of some other accounts based on this assumption, and a critique of them, see Norton (1984, especially Section 3). | |
29 | “John Stachel, who …was the first to understand the significance of Einstein’s reappraisal of the ‘hole’ argument …conjectures that Einstein’s self-criticism …rests on the tacit assumption that the points of spacetime where no matter is present cannot be physically distinguished except by the properties and relations induced by the spacetime metric” (Torretti (1983), p. 167) | |
30 | The Zurich notebook contains Einstein’s earliest preserved research notes on general relativity. Renn (2007), a critical edition of the Zurich Notebook with extensive commentary, includes Janssen (2007), which presents new historical evidence on the origins of the hole argument. | |
31 | “The issues raised in this chapter [on “General Relativity and Substantivalism: A Very Holey Story”] have already touched off a lively debate. For a sampling of opinions, the reader is referred to the articles by J. Butterfield, T. Maudlin, J. Norton and J. Stachel in Fine and Leplin (1989) …” (Earman (1989), p. 219, note 28). Actually, Stachel’s contribution to this debate first appeared in Stachel (1993). | |
32 | Clearly, he meant to include time, too. When asked in 1921 “If matter were destroyed, then, what would happen to time and space?” Einstein replied “Then there would be no time and no space” (cited in Illy (2006), p. 222). | |
33 | The term “substantivalism” has largely replaced “absolutism” in recent philosophical discussions (see Section 5); it seems to have been introduced in Sklar (1974). | |
34 | For a brief historical review of these debates, see Stachel (2006b). | |
35 | This is the approach followed by most current textbooks. See, e.g., Hawking and Ellis (1973), Wald (1984), Stephani (2004), and Goenner (1996). | |
36 | See, e.g., Trautman (1970, 1980); Göckeler and Schücker (1987). Prugovečki (1992) even entitles his second chapter (pp. 30–65): “The Fiber Bundle Framework for Classical General Relativity.” For a modern treatment, see Fatibene and Francaviglia (2003); for a historical account, see Varadarajan (2003). | |
37 | I use this negative formulation because I believe Einstein’s vision is best represented by dynamic structural realism rather than by traditional forms of relativism (see Section 5). | |
38 | See Stachel and Iftime (2005). | |
39 | In this subsection, the nature of is not further specified. The number of its elements may be finite or infinite. And if infinite, the number may be denumerable or non-denumerable; and if non-denumerable, they may form a discretum or a continuum. | |
40 | An individual relation , element of a set , and point of a geometry , etc., will be denoted by italic symbols; and the corresponding sets of relations, elements and points will be denoted by boldface italic symbols: , , and , respectively. By so designating the points of a geometry, we are negating their homogeneity; so all meaningful geometric assertions must be invariant under any permutation of these names. Naming is just a particular case of coordinatization, discussed below. | |
41 | For the present, this notation will be used although it is usually reserved for sets with a finite number of elements. Later in this section, differentiable manifolds are discussed, for which is the symbol for the group that is analogous to . As in the case of , invariance under holds in the absence of any other geometrical structures on . A study of the subgroups of and their relation to each other is thus equivalent to a study of all possible geometries on and their relation to each other. | |
42 | Plane Euclidean geometry provides a simple example. The identity of a triangle, for example, is preserved under the group of automorphisms consisting of all translations and rotations of the points of a two-dimensional manifold homeomorphic to . | |
43 | The real numbers provide a simple example: Each real number is uniquely defined, and collectively they form an algebraic field under the operations of addition and multiplication. The identity elements for addition (“zero”) and multiplication (“one”) are uniquely fixed, and all rational numbers may be generated from them by iterating the operations of addition, multiplication and their inverses subtraction and division. Real numbers may then be defined by Dedekind cuts between sets of rational numbers. | |
44 | In the continuous case, complications arise in going from local coordinates patches to a global coordinatization. This is especially so for theories like general relativity, in which the global topology is not fixed in advance, but depends on the solution to the field equations. | |
45 | From the standpoint of category theory, “The sharp distinction between point transformations and coordinate transformations has disappeared: coordinate systems are simply local diffeomorphisms into ” (Nijenhuis, 1994). This is because category theory includes both algebras and geometries in the same category of sets. | |
46 | This is the most general type of relation, and includes all special cases. Any relation between a lesser number of elements of is equivalent to an that is always identically satisfied by all the remaining elements of the set. | |
47 | Of course, if all the relations were totally symmetric, i.e., invariant under all permutations of the relata, there would be no need to introduce permutations of the relations. But most geometries involve some order relations that are not symmetric. Suppose, for example, two points on a straight line obey the relation: . Then permuting the two points will violate the relation “”, so we must introduce the permuted relation “,” thus preserving the validity of the permuted order relation . | |
48 | For this subsection, Lawvere and Schanuel (1997) is particularly useful, especially Part II. | |
49 | The reader unfamiliar with the concepts of differentiable manifold, geometric object, and Lie group can consult Schouten (1954) or Isham (1999), for example. | |
50 | I adopt the Schouten (1954) kernel-index convention for symbols: Any addition to the symbol for a geometrical object placed to the left of the symbol denotes a different object of the same geometric type; while any addition placed to the right denotes the same object in a different coordinate system. Thus, if denotes a point of a differentiable manifold, denotes a different point of the same manifold, while denotes the same point in a different coordinate system. | |
51 | This account of the relation between various -structures is highly simplified mathematically. For a more complete account, based on jet prolongations of fiber bundles, see Sánchez-Rodríguez (2008) | |
52 | It is also called the stability group, the isotropy group and the little group (see Isham, 1999, p. 180). | |
53 | Note that are not vectorial component indices, but merely serve to enumerate the order of the sequence of vectors and of convectors. | |
54 | Of course, we could perform the opposite kind of dragging: Let and , but . This leads to the same equivalence classes of models of a covariant theory: It is only the relative dragging that matters. | |
55 | Useful references for this subsection are Fatibene and Francaviglia (2003), Michor (2008), Olver (1995) and Sharpe (1997). | |
56 | Note the difference in the meaning of : Kobayashi denotes by , what is denoted above by , the stabilizer group of what is denoted above by : the subgroup of defining the -geometry of . | |
57 | The full significance of this observation involves a discussion of sheaves of cross sections defined over a germ (see Stachel and Iftime, 2005). | |
58 | More carefully formulated: Some criteria must be given that define the maximal extension of a local solution of the field equations. For example, the Kruskal manifold is the maximal analytic extension of the Schwarzschild solution to the Einstein equations. For some results relevant to general relativity, see Isenberg and Marsden (1982). | |
59 | I shall not discuss the order of differentiability postulated, which varies with the theory being defined. In the mathematical literature, smoothness is often postulated, i.e., differentiability to all orders. But in physical theories defined by hyperbolic systems of partial differential equations, it is precisely the existence of non-smooth solutions that allows for the transmission of information. Indeed, the characteristic hypersurfaces of such a system may be defined as those hypersurfaces, along which discontinuities may propagate. | |
60 | Note that this is the coordinate-independent analogue of the traditional definition of the law of transformation of a geometric or natural object under a coordinate transformation. See, e.g., Stachel (1986). For a category-theoretical treatment of natural bundles, see the Appendix A. | |
61 | In order to treat theories involving more than one field, the object is not required to be irreducible. | |
62 | Again, we put aside the complications that arise when these objects are only defined locally and the global topology of the manifold is not defined a priori, but depends on the dynamics; i.e., on the solution to the field equations. | |
63 | “Indeed, every classical field theory can be regarded as taking place on some jet prolongation of some gauge natural (vector or affine) bundle associated with some principle bundle over some base manifold” (Matteucci, 2003, p. 115). | |
64 | It is possible to lift the fixed space-time structures from the base manifold into the fibers; but this does not change the essentially background-dependent nature of these theories (see Section 5). | |
65 | We shall not here discuss the important question of local versus global sections (see, e.g., Fatibene and Francaviglia, 2003). Unless the contrary is specifically stated, any mention of sections should be interpreted as a reference to local sections. For some further remarks on this question, see Section 6. | |
66 | This is clearly an equivalence relation (i.e., it is reflexive, symmetric and transitive), so it divides the set of all pseudo-metrics into equivalence classes. | |
67 | Again, here I sidestep the question of how one proceeds from the local to the global point of view. | |
68 | If is a vector field generating a one-parameter subgroup of such a symmetry group and is any geometric object involved in defining the space-time structure, then . | |
69 | For the benefit of those who did not read Section 4 and the Appendix B, this section reviews some concepts introduced there. | |
70 | “Substantivalism” is a neologism, not found it in any current dictionary of English or philosophy. Most dictionaries define “substantialism”: For example, the Oxford English Dictionary defines it as “The doctrine that there are substantial realities underlying phenomena.” It also defines “substantival” as “existing substantially.” | |
71 | “[T]he debate can be solved by advancing a tertium quid, a third option between classical substantivalism and relationism. Ths option, which I call structural spacetime realism, sides with the latter doctrine in defending the relational nature of spacetime, but argues with the former that spacetime exists, at least in part, independently of particular physical objects and events …” (Dorato, 2000, pp. 1607–1608). | |
72 | The first class will also include dynamical fields, certain derivatives of which will constitute the objects of the second class. For example, the electromagnetic potential vector would be in the first class, while the vectorial wave operator acting on it would produce a object in the second class. | |
73 | Note that the infinitesimal form of is where is a vector field generating a one-parameter subgroup of the symmetry group. | |
74 | Bergmann (1957) speaks of the “fundamentally trivial nature” of “weak covariance.” | |
75 | “The position that GTR is not ‘just another generally covariant theory’ was first clearly articulated in the philosophical literature by Stachel (1993)” (Earman, 2006, p. 464, note 5). | |
76 | I thank Tian Yu Cao for discussions of this point. | |
77 | Pooley speaks of “the objectless ontology of the ontic structural realist” (Pooley, 2006, p. 102; see Section 5.3). See also Stachel (2006a) for a critique of the concept of “relations without relata.” | |
78 | One distinguishes between a relation and the relata, the entities that it relates. | |
79 | In the former case one may speak of “things between relations,” in the latter more familiar case of “relations between things” (see Stachel, 2002). | |
80 | “For there are never two things in nature that are perfectly alike and in which it is impossible to find a difference that is internal, or founded on an intrinsic denomination.” (G.W. Leibniz, “The Monadology,” cited from Loemker, 1969, p. 643; our emphasis). See Henry (2006) for an extensive discussion of haecceity. | |
81 | Leibniz’s principle is often cited in discussions of the hole argument (see, e.g., Earman, 1989; Norton, 2011). | |
82 | Euclidean plane geometry provides a simple example. The relations characterizing any triangle are preserved under the automorphism group of the Euclidean plane, consisting of all translations and rotations. | |
83 | The real numbers provide a simple example, each element of which is uniquely defined. They form a field under the operations of addition and multiplication. | |
84 | They will of course be parallel to each other – parallelism in an affine-flat manifold is an equivalence relation. | |
85 | One might be tempted, by analogy with the special-relativistic case, to require that the curves be autoparallels of the inertia-gravitational connection (geodesics of the metric), but this is not necessary in theory, and in practice often not helpful: autoparallels in a gravitational field tend to form caustics. Einstein’s term for a similar construction was “mollusk of reference” (Einstein, 1952); but it was based on a coordinate system rather than a geometric construction. | |
86 | Of course, there is no need to confine evolution to this fibration. By the introduction of lapse functions and shift vectors, the evolution may be developed along any other foliation and fibration, respectively. But this does not change the circumstance that choice of an initial labeling of the world lines of the fibration with the associated proper times has destroyed diffeomorphism invariance in any but the trivial sense (see next footnote). The lapse and shift functions merely serve to situate points of space-time with respect to this initial individuating field. | |
87 | Again, the trivial identity still holds, of course. If we carry along all the fibrations with some diffeomorphism of the manifold, nothing has changed. | |
88 | Except in the trivial sense, diffeomorphism invariance is destroyed by the introduction of such a frame of reference. Often, an attempt is made to partially restore it by demanding invariance under all permutations of the labeling of the world-lines of the fibration. When a coordinate system is adapted to the fibration, this is described as three-dimensional diffeomorphism invariance or invariance under spatial diffeomorphisms, and the distinction is sometimes elided. “What are the quanta of the gravitational field? Or, since the gravitational field is the same entity as spacetime, what are the quanta of space?” (Rovelli, 2004, p. 18; see also Section 4.1, pp. 145–153). | |
89 | This is quite similar to Cartan’s method of generalizing a homogeneous Klein geometry to what is now called a Cartan geometry (see, e.g., Sharpe, 1997, “Preface” and Chapters 4 and 5). | |
90 | I have discussed a number of these issues at greater length in Stachel (2006a) and Stachel (2009). | |
91 | The isometry group is also referred to as the group of automorphisms or motions. | |
92 | For a review of the technique of finding group-invariant solutions to systems of partial differential equations, with applications to general relativity, see Anderson et al. (2000). | |
93 | More precisely, it is a Lie algebra over the field of real numbers. The Lie bracket of three generators is bilinear and satisfies the Jacobi identity, while the Lie bracket of a generator with itself vanishes. | |
94 | In category theory, the generalization from the category of manifolds to the category of sets can be carried out by the use of forgetful functors. | |
95 | Of course in a relativistic version of scattering theory, the number of particles of the same kind in the incoming and outgoing states need not be the same. But maximal permutability is still required of both the in- and out-states. | |
96 | That is, the group of automorphisms of the space-time manifold preserving the background structures. | |
97 | Under what conditions such a measurement process, which even ideally always occupies a four-dimensional region of space-time, can be modeled as occupying a lower-dimensional region is a delicate one. See, e.g., Micanek and Hartle (1996) for a discussion of non-relativistic quantum mechanics. | |
98 | One often refers to things as objects or subjects, depending on the role that they play in the theory. | |
99 | Note that this distinction is theory dependent. For, example, a property that is intrinsic at one level, may prove to be extrinsic at a deeper level. | |
100 | Rorty (1967) gives an excellent account of discussions about internal and external relations. |