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).
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: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 extremal) in a non-flat space-time with as its line element. By explicitly introducing the flat Minkowski pseudo-metric , this 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 , 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:
- determine the line element, representing the chrono-geometry of space-time;
- 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 differential 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 inertio-gravitational 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 differential 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 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.
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 effect 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 inertio-gravitational 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 , and a given four-dimensional coordinate system by a single letter . Suppose is a solution to a set of field equations. If both the coordinates and components of the metric tensor are subject to the transformation12
But if the field equations are generally covariant, Einstein noted, it follows that must also be a solution. He emphasized clearly that and represent two mathematically distinct solutions in the same coordinate system.13
Now consider a “hole” – a bounded, closed region of space-time, in which all non-gravitational sources of the field – represented by the stress-energy tensor – vanish; and suppose the field and all its sources 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 that reduce to the identity outside of and on the boundary of , together with all their derivatives up to any finite order; yet which differ from the identity inside . Such coordinate transformations will leave unchanged, and the resulting will still equal outside of and on the boundary of ; but will differ from inside .
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.
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 does differ mathematically from inside , they are merely different representations of the same physical solution. Properly-specified conditions outside the hole will suffice to specify a unique physical solution inside the hole.
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 , two different solutions and exist [see Section 2.3]. To imagine two solutions simultaneously on the same manifold has no meaning, and indeed the system 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:
- If two metrics in their respective different coordinate systems differ only in that one is the carry-along of the other, then physically there is no distinction between them.
- 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.
- In the absence of a metric tensor field, a coordinate system on a differentiable 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 coordinate-dependent 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 diffeomorphisms 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:
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 in such a way that for every point-event there is a corresponding system of values of the variables . To two coincident point-events there corresponds one system of values of the variables , i.e., coincidence is characterized by the identity of the co-ordinates. If, in place of the variables , we introduce functions of them, , 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 differing usages of the term “general covariance” – one involves the field equations, the other does not. This difference 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).
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 difference 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.
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 space-like 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 to these equations off 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).