The Hole Argument and Some Physical and Philosophical Implication Pt 1

The Hole Argument and Some Physical and Philosophical Implications Abstract This is a historical-critical study of the hole argument, concentrating on the interface between historical, philosophical and physical issues. Although it includes a review of its history, its primary aim is a discussion of the contemporary implications of the hole argument for physical theories based on dynamical, background-independent space-time structures. The historical review includes Einstein’s formulations of the hole argument, Kretschmann’s critique, as well as Hilbert’s reformulation and Darmois’ formulation of the general-relativistic Cauchy problem. The 1970s saw a revival of interest in the hole argument, growing out of attempts to answer the question: Why did three years elapse between Einstein’s adoption of the metric tensor to represent the gravitational field and his adoption of the Einstein field equations? The main part presents some modern mathematical versions of the hole argument, including both coordinate-dependent and coordinate-independent definitions of covariance and general covariance; and the fiber bundle formulation of both natural and gauge natural theories. By abstraction from continuity and di↵erentiability, these formulations can be extended from di↵erentiable manifolds to any set; and the concepts of permutability and general permutability applied to theories based on relations between the elements of a set, such as elementary particle theories. We are closing with an overview of current discussions of philosophical and physical implications of the hole argument. 1 Introduction “Hole argument” is the English translation of the German phrase “Lochbetrachtung,” used by Albert Einstein to describe his argument against the possibility of generally-covariant equations for the gravitational field, developed in late 1913 and accepted until late 1915. Einstein realized the desirability of general covariance, and showed that it was easily implemented for the rest of physics; but the hole argument purported to show why it could not be demanded of the gravitational field equations he was trying to formulate for the metric tensor.1 This article is a historical-critical study, in Ernst Mach’s sense.2 It includes a review of the literature on the hole argument that concentrates on the interface between historical, philosophical and physical approaches. Although recounting the history of the hole argument, the primary purpose is to discuss its contemporary significance – in both physics and philosophy – for the study of space-time structures. Like Mach, while presenting various other viewpoints, I have not hesitated to advocate my own. In physics, I believe the main lesson of the hole argument is that any future fundamental theory, such as some version of “quantum gravity,” should be background independent, with basic elements obeying the principle of maximal permutability. In the philosophy of space-time, this leads me to advocate a “third way” that I call dynamic structural realism, which di↵ers from both the traditional absolutist and relationalist positions. 1.1 Why should we care? One of the most crucial developments in theoretical physics was the move from theories dependent on fixed, non-dynamical background space-time structures to background-independent theories, in which the space-time structures themselves are dynamical entities. This move began in 1915 when Einstein stated the case against his earlier hole argument. Even today, many physicists and philosophers do not fully understand the significance of this development, let alone accept it in practice. So it is of more than historical interest for physicists and philosophers of science to understand what initially motivated this move, as well as the later developments stemming from it. Einstein’s starting point was the search for a generalization of the special theory that would include gravitation. He quickly realized that the equivalence principle compelled the abandonment of the privileged role of inertial (i.e., non-accelerated) frames of reference, and started to investigate the widest class of accelerated frames that would be physically acceptable. His first impulse was to allow all possible frames of reference; since he identified frames of reference and coordinate systems, this choice corresponds mathematically to a generally-covariant theory. But he soon developed an argument – the hole argument – purporting to show that generally-covariant equations for the metric tensor are incompatible with his concept of causality for the gravitational field. The argument hinged on his tacit assumption that the points of space-time are inherently individuated, quite apart from the nature of the metric tensor field at these points. Only two years later, after other reasons compelled him to reconsider general covariance, did Einstein finally recognize the way out of his dilemma: One must assume that, in an empty region of space-time, the points have no inherent individuating properties – nor indeed are there any spatio-temporal relations between them – that do not depend on the presence of some metric tensor field. Thus, general relativity became the first fully dynamical, background-independent space-time theory. Without some knowledge of this historical background, it is difficult to fully appreciate 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). either the modern significance of the hole argument, or the compelling physical motives for the requirement of background independence. 1.2 Summary: Where we are headed Einstein’s starting point in his search for a theory of gravitation was the theory we now call special relativity. From a contemporary viewpoint, its most important feature is that it has two fixed, kinematical space-time structures – the chrono-geometry embodied in the Minkowski metric tensor field and the inertial field embodied in the associated flat affine connection – both of which are invariant under the ten-parameter Lie group now called the Poincaré or inhomogenous Lorentz group.3 In Minkowski space-time, all dynamical theories must be based on geometric objects that form a representation (or more generally, a realization) of this group. There is a preferred class of spatial frames of reference in Minkowski space-time, the inertial frames. Einstein had shown how to define a class of physically preferred coordinate systems for each inertial frame of reference; in particular, he defined a clock synchronization procedure that provides a preferred global time for each frame. This enabled him to show how the principle of relativity of all inertial frames could be reconciled with the universal properties of light propagation in vacuum. The lesson he drew was the need to find a physical interpretation of the coordinates associated with an inertial frame of reference – a lesson that had to be painfully unlearned in his search for a generalized theory of relativity. In large part, the history of the hole argument is the story of that unlearning process. The end result was the formulation of the general theory of relativity, the first background-independent physical theory turning all space-time structures into dynamical fields. This was such a revolutionary break with all previous physical theories, in which spacetime structures constitute a fixed, non-dynamical background, that its ultimate significance is still debated by physicists.4 Understanding the hole argument in both its historical and contemporary aspects can help to clarify the issues at stake in this debate. The basic issue can be stated as follows: Given a physical theory, when should an equivalence class of mathematically distinct models of the theory be identified as corresponding to single, unique physical model? The hole argument shows that, for any theory defined by a set of generally-covariant field equations, the only way to make physical sense of the theory is to assume that the entire equivalence class of di↵eomorphicallyrelated solutions to the field equations represent a single physical solution. As will be seen later, mathematically this result can best be stated in the language of natural bundles. But a similar result holds for the even broader class of all gauge-invariant field theories, notably Yang–Mills theories: an equivalence class of gauge-related models of any such theory must be physically identified. Mathematically, broadening the question in this way requires the language of gauge-natural bundles. General relativity itself may also be treated by the use of gauge-natural bundle techniques: its similarities to and di↵erences from gauge theories of the Yang–Mills type will also be discussed. This move to natural and gauge-natural formulations of field theories also has important implications for the philosophy of space and time. The old conflict between absolute and relational interpretations of space and then space-time has been renewed on this new ground. But I shall argue that this reformulation of the question suggests a third position, around which a consensus is forming. This position has been given various names, but I prefer dynamic structural realism. 3 This the semi-direct product of the homogeneous Lorentz group SO(3, 1) and translation group T4 . 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. 4 1.3 Outline of the article Sections 2.1 – 2.5 recount the developments leading up to Einstein’s adoption of the hole argument against general covariance in 1913, how it misled him for over two years, the reasons for his rejection of it in late 1915, and its replacement by the point-coincidence argument for general covariance. Section 2.6 discusses Kretschmann’s 1917 critique of the concept of general covariance and Einstein’s 1918 reply; decades later this debate led Komar to propose the use of what are now called Kretschmann–Komar coordinates as a way of resolving the hole argument. Finally, Section 2.7 discusses Hilbert’s 1917 reformulation of the hole argument: He replaced the four-dimensional hole in space-time with a space-like hypersurface, on which he posed an initial value problem for the field equations; this was the first step in a series of developments culminating a decade later in a fully satisfactory formulation by Darmois of the general-relativistic Cauchy problem. Section 3 discusses the revival of interest in the hole argument in the 1970s, which grew out of an attempt to answer a historical question: Why did three years (1912 – 1915) elapse between Einstein’s adoption of the metric tensor to represent the gravitational field and his adoption of what are now called the Einstein equations for this field? Some highlights of this discussion are recalled, from the post-World War II revival of interest in general relativity up to the present. Section 4 presents a modern version of the hole argument in general relativity, and its generalization from metric theories of gravitation to gauge-natural field theories. By abstraction from continuity and di↵erentiability, a formulation general enough to include theories with discrete fundamental elements is obtained. The concept of general covariance of a field theory is similarly extended to general permutability, a concept wide enough to include theories based on relations between the elements of any set. Sections 5 and 6 focus on current discussions of philosophical and physical implications of the hole argument, respectively; no attempt is made to rigidly separate issues that overlap both areas. Section 5 discusses such issues as: the range of applicability of the hole argument, the correct mathematical definition of general covariance and its physical significance, the controversy between relationalists and substantivalists in discussions of space-time structures. The arguments of Earman, Pooley and Stachel are reviewed, and their convergence on a third alternative, which I call dynamic structural realism, is stressed. Section 6 discusses such issues as partially background-independent theories, including miniand midi-solutions to the Einstein field equations; the reformulation of general relativity as a gauge natural theory; and some implications of the hole argument for attempts to formulate a quantum theory of gravity. 2 Early History 2.1 From the special theory to the search for a theory of gravity Einstein attributed his success in formulating the special theory in 1905 in no small measure to his insistence on defining coordinate systems that allowed him to attach physical significance to spatial and temporal coordinate intervals: The theory to be developed – like every other electrodynamics – is based on the kinematics of rigid bodies, since the propositions of any such theory concern relations between rigid bodies (coordinate systems), clocks and electromagnetic processes. Not taking this into account insufficiently is the root of the difficulties, with which the electrodynamics of moving bodies currently has to contend (Einstein, 1905, my translation). His subsequent attempt to include gravitation in his theory focused on the equality of gravitational and inertial mass, and led him to adopt the equivalence principle: Inertia and gravitation are “wesensgleich” (the same in essence), and must be represented by a single inertio-gravitational field.5 The distinction between the two is not absolute (i.e., frame independent), but depends on the frame of reference adopted. In particular, he noted that a linearly accelerated (rigid) frame of reference in a space-time without a gravitational field is physically equivalent to an inertial frame of reference, in which there is a uniform, constant gravitational field: both result in equal acceleration of bodies moving relative to their respective frames. He concluded that, in order to include gravitation, one must go beyond the special theory, with its privileged role for inertial frames, and look for a generalized (“verallgemeinerte”) theory of relativity. In the simplest case, linearly accelerated frames in Minkowski space, the usual time coordinate loses its direct physical significance; and in uniformly rotating frames, a global time cannot even be defined. In the latter case, the spatial coordinates also lose their direct significance: the measured spatial geometry is no longer flat. I soon saw that, according to the point of view about non-linear transformations required by the equivalence principle, the simple physical interpretation of the coordinates had to be abandoned. . . . This recognition tormented me a great deal because for a long time I was not able to see just what are coordinates actually supposed to mean in physics? (Einstein 1933, translation from Stachel, 2007, p. 86). The equivalence principle made it not only probable that the laws of nature must be invariant with respect to a more general group of transformations than the Lorentz group (extension of the principle of relativity), but also that this extension would lead to a more profound theory of the gravitational field. That this idea was correct in principle I never doubted in the least. But the difficulties in carrying it out seemed almost insuperable. First of all, elementary arguments showed that the transition to a wider group of transformations is incompatible with a direct physical interpretation of the space-time coordinates, which had paved the way for the special theory of relativity. Further, at the outset it was not clear how the enlarged group was to be chosen (Einstein, 1956, my translation). 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). 2.2 From the equivalence principle to the metric tensor Einstein first attempted to develop a theory of the gravitational field produced by a static source, still based on the idea of a scalar gravitational potential. His earlier work had led him to consider non-flat spaces; this work led him to consider non-flat space-times: He found that his equation of motion for a test particle in a static field can be derived from a variational principle: Z 1/2 [c(x, y, z)]2 dt2 [dx2 + dy 2 + dz 2 ] = 0, (1) where he interpreted c(x, y, z) as a spatially-variable speed of light.6 Already familiar with Minkowski’s four-dimensional formulation of the special theory, he realized that this variational principle could be interpreted as the equation for a geodesic (i.e, an 1/2 as its line extremal) in a non-flat space-time with ds = [c(x, y, z)]2 dt2 [dx2 + dy 2 + dz 2 ] element. By explicitly introducing the flat Minkowski pseudo-metric ⌘µ⌫ , this ds can be rewritten in an arbitrary coordinate system. Einstein then made a big leap: He generalized the geodesic equation using a non-flat Riemannian pseudo-metric gµ⌫ , and assumed that it would still describe the path of a test particle in an arbitrary non-static gravitational field. The gravitational theory he was seeking must be based on such a non-flat metric, which should both: a) determine the line element, ds2 = gµ⌫ dxµ dx⌫ representing the chrono-geometry of spacetime; b) serve as the potentials for the inertio-gravitational field. While a student at the Swiss Federal Polytechnic, Einstein had learned about Gauss’ theory of non-flat surfaces, and realized he needed a four-dimensional generalization. His old classmate and new colleague, the mathematician Marcel Grossmann, told Einstein about Riemann’s generalization of Gauss’ theory and about the tensor calculus (“absolute di↵erential calculus”), developed by Ricci and Levi-Civita to facilitate calculations in an arbitrary coordinate system. Still identifying a coordinate system with a physical frame of reference, his goal of extending the principle of the relativity led Einstein to investigate the widest possible group of coordinate transformations. With Grossman’s help, he succeeded in formulating the influence of the inertiogravitational field on the rest of physics by putting these equations into a “generally covariant” form. The one exception was the gravitational field equations, the problem to which they now turned. “General covariance” then meant covariance under arbitrary coordinate transformations,7 so they turned to coordinate-invariant tensors formed from the metric. The concepts of covariant derivative and Riemann tensor were based on the theory of di↵erential invariants, and lacked a simple geometrical interpretation.8 Nevertheless, Einstein seriously considered the Ricci tensor, the only second rank contraction of the Riemann tensor, for use in the gravitational field equations. He tried, in linear approximation, setting it equal to the stress-energy-momentum tensor of the sources 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 di↵erentiable manifold and the distinction between di↵eomorphisms of the manifold and coordinate transformations in and between regions of Rn (the n-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. of the gravitational field; and even realized that, in order to obtain consistency with the vanishing divergence of the source tensor, the Ricci tensor would have to be modified by a trace term.9 However, after coming so close to the final form of field equations of GR, he retreated. His earlier work on static fields led him to conclude that, in adapted coordinates, the spatial part of the metric tensor must remain flat [see Eq. (1)], which is easily shown to be incompatible with field equations based on the Ricci tensor. So, as he later put it, he abandoned these equations “with a heavy heart,” and began to search for non-generally-covariant field equations. 2.3 From the metric tensor to the hole argument Einstein soon developed a meta-argument against a generally covariant set of field equations for the metric tensor. Why did he formulate this argument in terms of a hole – a finite region of space-time devoid of all non-gravitational sources?10 It was probably the influence of Mach’s ideas. One of Einstein’s main motivations in the search for a generalized theory of relativity was his interpretation of Mach’s critique of Newtonian mechanics. According to Mach, space is not absolute: It does not have any inherent properties of its own, and its apparent influence on the motion of a body – as manifested in the law of inertia, for example – must result from an interaction between the moving body and all the rest of the matter in the universe. Mach suggested that the inertial behavior of matter in the region of empty space around us is the e↵ect of all the matter in the universe that surrounds this region, or “hole.” When Einstein adopted the metric tensor as the representation of the potentials of the inertiogravitational field, he interpreted Mach’s idea as the requirement that metric field in such a “hole” be entirely determined by its sources – that is by the stress-energy tensor of the surrounding matter.11 The hole argument purports to show that, if the field equations are generally covariant, this requirement cannot be satisfied: Even if the field and all its sources outside of and on the boundary of the hole are fully specified, such equations cannot determine a unique field in the interior of the hole. Conclusion: the gravitational field equations cannot be generally covariant. We present the argument here in Einstein’s original coordinate-based formulation (see Section 4 for a modern, coordinate-free form). Let the metric tensor be symbolized by the single letter G, and a given four-dimensional coordinate system by a single letter x. Suppose G(x) is a solution to a set of field equations. If both the coordinates and components of the metric tensor are subject to the transformation x ! x0 = f (x) (2) from coordinates x to another coordinate system x0 , then G0 (x0 ) represents the same solution in the new coordinate system. This presents no problem – Einstein is quite clear on this point.12 But if the field equations are generally covariant, Einstein noted, it follows that G0 (x) must also be a solution. He emphasized clearly that G(x) and G0 (x) represent two mathematically distinct 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 G-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 (Tµ⌫ ), this therefore entails that the G-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). solutions in the same coordinate system.13 Now consider a “hole” H – a bounded, closed region of space-time, in which all non-gravitational sources of the field – represented by the stress-energy tensor Tµ⌫ – vanish; and suppose the field G(x) and all its sources Tµ⌫ are specified everywhere outside the hole and on its boundary, together with any finite number of normal derivatives of field on the boundary. This still does not suffice to determine a unique mathematical solution inside the hole. For there are coordinate transformations x ! x0 that reduce to the identity outside of and on the boundary of H, together with all their derivatives up to any finite order; yet which di↵er from the identity inside H. Such coordinate transformations will leave Tµ⌫ unchanged, and the resulting G0 (x) will still equal G(x) outside of and on the boundary of H; but G(x) will di↵er from G0 (x) inside H. In short, if the field equations are generally covariant, then specification of the gravitational field together with its sources outside of and on the boundary of such a hole do not suffice to determine the field inside. Einstein concluded that generally-covariant field equations could not be used to describe the metric tensor field, and began a search for non-covariant field equations. Now the question became: If Lorentz invariance is too little (equivalence principle) and general covariance is too much (hole argument), what is the widest possible group (if it is a group!) of coordinate transformations, under which one can demand the invariance of such equations? The hole argument is not valid for the inhomogeneous Lorentz (Poincaré) group; and Einstein concluded that the invariance group of the field equations should be extended only up to – but not including – the invariance group, for which the hole argument becomes valid in coordinates adapted to this group. 2.4 From the hole argument back to general covariance Because of problems unrelated to the hole argument, in mid-1915 Einstein abandoned the search for a non-covariant theory of gravitation and returned to the Riemann tensor. After several intermediate steps, by November of that year he adopted the set of generally-covariant equations now known as the Einstein equations. His successful explanation of the anomalous perihelion advance of Mercury convinced him – and many others – of the profound significance of the resulting theory, known today as general relativity. What about the hole argument? Einstein realized that, to avoid it, he had only to drop one of the premises that he had tacitly adopted: the assumption that the points of space-time in the hole are individuated independently of the metric field. If that assumption is dropped, it follows that when the metric is dragged-along by a coordinate transformation, all the individuating properties and relations of the points of space-time are dragged along too.14 While G0 (x) does di↵er mathematically from G(x) inside H, they are merely di↵erent representations of the same physical solution. Properly-specified conditions outside the hole will suffice to specify a unique physical solution inside the hole. 13 To avoid future confusion, let me emphasize that the terminology introduced in Section 4 di↵ers from that of Einstein, and indeed that of many contemporary accounts. In this review, the requirement that, if G(x) is a solution to the field equations, then G0 (x) must also be a solution is called covariance. General covariance is the requirement that G(x) and G0 (x) represent one and the same physical solution. Another di↵erence 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. 2.5 The point coincidence argument In order to better illustrate the flaw in the hole argument, Einstein developed a counter-argument, the point-coincidence argument.15 There are actually two versions of this argument, which have been called “the private” and “the public” one.16 First I shall cite the private version. In letters to friends, Einstein explained why the argument no longer applies to general relativity: Everything in the hole argument was correct up to the final conclusion. It has no physical content if, with respect to the same coordinate system K, two di↵erent solutions G(x) and G0 (x) exist [see Section 2.3]. To imagine two solutions simultaneously on the same manifold has no meaning, and indeed the system K has no physical reality. The hole argument is replaced by the following consideration. Nothing is physically real but the totality of space-time point coincidences. If, for example, all physical events were to be built up from the motions of material points alone, then the meetings of these points, i.e., the points of intersection of their world lines, would be the only real things, i.e., observable in principle. These points of intersection naturally are preserved during all [coordinate] transformations (and no new ones occur) if only certain uniqueness conditions are observed. It is therefore most natural to demand of the laws that they determine no more than the totality of space-time coincidences. From what has been said, this is already attained through the use of generally covariant equations (letter to Michele Besso, 3 January 1916, in Schulmann et al., 1998, p. 235, translation cited from Stachel, 1989, p. 86). Einstein’s argument consists of three points; in modern language, they are: 1. If two metrics in their respective di↵erent coordinate systems di↵er only in that one is the carry-along of the other, then physically there is no distinction between them. 2. Generally covariant equations have the property that, given a solution, any carry-along of that solution in the same coordinate system is also a solution to these equations. 3. In the absence of a metric tensor field, a coordinate system on a di↵erentiable manifold has no intrinsic significance.17 Note that points 1) and 2) were included in the original hole argument (see Section 2.3). Point 3) is the crucial new element. It follows from the three points that the entire equivalence class of carry-alongs of a given solution in the same coordinate system corresponds to one physical gravitational field. Thus, the hole argument fails. As will be seen in Section 4, point 1) constitutes a coordinate-dependent version, applied to the metric tensor, of what I call the basic or trivial identity; point 2) constitutes the coordinatedependent version, applied to the metric tensor, of my definition of covariant theories. I would apply the term generally covariant to the conclusion that an entire equivalence class of carry alongs corresponds to one physical solution. The coordinate-independent versions of all three concepts are obtained by substituting basis vectors for coordinates and di↵eomorphisms for coordinate transformations. Einstein’s 1916 review paper presents the public version of the argument to justify the requirement that any physical theory be invariant under all coordinate transformations: 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. Our space-time verifications invariably amount to a determination of space-time coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurements are nothing but verifications of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock-dial, and observed point-events happening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences. We allot to the universe four space-time variables x1 , x2 , x3 , x4 in such a way that for every point-event there is a corresponding system of values of the variables x1 . . . x4 . To two coincident pointevents there corresponds one system of values of the variables x1 . . . x4 , i.e., coincidence is characterized by the identity of the co-ordinates. If, in place of the variables x1 . . . x4 , we introduce functions of them, x01 , x02 , x03 , x04 , as a new system of co-ordinates, so that the systems of values are made to correspond to one another without ambiguity, the equality of all four co-ordinates in the new system will also serve as an expression for the space-time coincidence of the two point-events. As all our physical experience can be ultimately reduced to such coincidences, there is no immediate reason for preferring certain systems of coordinates to others, that is to say, we arrive at the requirement of general covariance (Einstein, 1916, pp. 776–777, reprinted in Kox et al., 1996, pp. 291– 292). In this version of the argument, there is no mention of dynamical equations or even of fields. Indeed, he proceeds to illustrate it with a version of the trivial identity applied to a system of particles, rather than fields, the model being any set of particle world lines, without any requirement that they satisfy equations of motion. Einstein also mentions “the requirement of general covariance;” but here it amounts basically to point 3) together with a generalization of point 1) to any objects used in a physical theory, whether or not they obey any field equations. It is essentially a coordinate-dependent version of the basic identity, extended from metrics to all geometric object fields of a certain type (see Section 4.2). We see here the origins of two di↵ering usages of the term “general covariance” – one involves the field equations, the other does not. This di↵erence has occasioned some confusion in recent discussions of the hole argument. (see Section 5.2 for an example of their conflation). My use of the term always involves the field equations. In later years, Einstein explicitly rejected any positivistic interpretation of the point coincidence argument. He wrote to Schlick: Generally considered, your presentation of the [point coincidence] argument does not correspond with my conception of it since I find your entire conception too positivistic, so to speak. Physics indeed provides relations between sense experiences, but only indirectly. For me, its essence is by no means exhaustively characterized by this assertion. Physics is an attempt at the conceptual construction of a model of the real world, as well as of its lawful structure. Indeed it must represent exactly the empirical relations between the sense experiences that are accessible to us; but only in this way is it linked to the latter (Einstein to Moritz Schlick, 28 November 1930; cited from Engler and Renn, 2013, p. 18, my translation). 2.6 From general covariance to Kretschmann’s critique In 1915, even before Einstein completed the general theory of relativity, Erich Kretschmann (1915a,b) had undertaken an investigation that led him to a version of the trivial identity. Kretschmann (1917) uses Einstein’s public point coincidence argument to conclude that any theory could be put into a form satisfying Einstein’s principle of general covariance. Einstein (1918) concedes the point, but argued, not very successfully,18 that an added criterion of simplicity gives the principle a heuristic significance. Evidently, he was not himself clear on the di↵erence between his two arguments: While the public point coincidence argument does not provide a criterion for singling out theories, the criterion of general covariance in the private argument does.19 Of greater future significance was Kretschmann’s suggestion: use four invariants of the Riemann tensor to fix a unique coordinate system (an individuating field in my terminology). Section III.1 of Kretschmann (1917) discusses the use of the principal directions of the Riemann tensor to fix the coordinate directions;20 and Section III.2 proposes the use of four mutually-independent invariants of the Riemann tensor and metric as the space-time coordinates.21 Apparently unaware of Kretschmann (1917), Arthur Komar (1958) also suggested the use of four invariants of the Riemann tensor as coordinates. In subsequent discussions of the problem of true observables in general relativity, they are often referred to as Kretschmann–Komar coordinates. Stachel (1989, 1993) noted their use as a way of individuating the points of space-time, and they have subsequently figured in many discussions of the hole argument. Kretschmann (1917) notes that: “This system of [principal] directions . . . naturally may be indeterminate or otherwise degenerate;” and that the four invariants may be used as coordinates “only by postulating that in no finite four-dimensional region are [they] mutually dependent.” Section 6.1 discusses the treatment of such cases, in which the symmetry or isometry group of an equivalence class of metrics is non-trivial. 2.7 The Cauchy problem for the Einstein equations: from Hilbert to Lichnerowicz As we have seen, in 1913 Einstein formulated his argument against generally covariant equations in terms of the non-uniqueness of the field in a “hole” in space-time. David Hilbert, the renowned mathematician, became interested in the problem of a unified gravitational-electromagnetic theory and followed Einstein in arguing against generally covariant field equations. Instead of a hole, however, he formulated the argument in a mathematically more sophisticated way, using a spacelike hypersurface.22 He showed that there is no well-posed Cauchy problem for generally covariant equations; i.e., no finite set of initial values on such a hypersurface can determine a unique solution 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. to these equations o↵ the initial hypersurface.23 After Einstein returned to generally covariant field equations, Hilbert dropped this argument against them, and Hilbert (1917) is the first discussion of the Cauchy problem in general relativity; but the analysis is far from complete.24 It was not until 1927 that Georges Darmois gave a reasonably complete treatment.25 His discussion included the role of null hypersurfaces as characteristics, the use of the first and second fundamental forms on a space-like hypersurface as initial data, and the division of the ten field equations into four constraints on the initial data and six evolution equations. Most post World-War II discussions of the Cauchy problem in general relativity are based on the work of André Lichnerowicz,26 but he acknowledges his debt to Darmois: In 1926 in Belgium, Darmois gave a course of four lectures on “the equations of Einsteinian gravitation” in the presence of De Donder. The monograph version (Darmois, 1927) . . . became my bedside reading. In this book . . . is the first rigorous analysis of the hyperbolic nature of the Einstein equations, i.e., the foundation of the relativistic theory of gravitation as a theory of wave propagation, With profound understanding, the splitting of the Einstein equations relative to the Cauchy problem into two sets is clearly discussed: one treats the initial conditions, and the other deals with time evolution (Lichnerowicz, 1992, p. 104). Many current discussions of the non-uniqueness problem in general relativity are formulated in terms of the Cauchy problem rather than the original hole argument (see, e.g., Belot and Earman, 2001; Rickles, 2005; Lusanna and Pauri, 2006). 23 For details, with references to Hilbert’s lectures and papers, see Renn and Stachel (2007). 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). 24