A Raum with a View: Hermann Weyl and the Problem of Space

Abstract

A central issue in the philosophical debates over general relativity concerns the status of the metric field: should it be regarded as part of the background arena in which physical fields evolve, or as a physical field itself? In this paper, we approach this debate through its relationship to the so-called “Problem of Space”: the problem of determining which abstract, mathematical geometries are candidate descriptions of physical space. In particular, we explore the way that Hermann Weyl tackled the Problem of Space in the wake of general relativity, and argue that Weyl’s proposed solution reveals a “middle way” between bare-manifold and manifold-plus-metric accounts of spacetime.

1 Introduction

One of the central philosophical debates prompted by general relativity concerns the status of the metric field. A number of philosophers have argued that the metric field should no longer be regarded as part of the background arena in which physical fields evolve; it should be regarded as a physical field itself. Earman and Norton write, for example, that the metric tensor in general relativity ‘incorporates the gravitational field and thus, like other physical fields, carries energy and momentum’.¹ Indeed, they baldly claim that according to general relativity ‘geometric structures, such as the metric tensor, are clearly physical fields in spacetime’.² On such a view, spacetime itself—considered independently of matter—has no metrical properties, and the mathematical object that best represents spacetime is a bare topological manifold. As Rovelli puts the idea: ‘the metric/gravitational field has acquired most, if not all, the attributes that have characterised matter (as opposed to spacetime) from Descartes to Feynman... it is perhaps more appropriate to reserve the expression spacetime for the differential manifold, and to use the expression matter for everything which is dynamical, carries energy and so on; namely all the fields including the gravitational field’.³

Others, however, have strongly resisted this view, arguing that the paradigm spatio-temporal properties are precisely the metrical properties. Thus Maudlin has written:

…qua differentiable manifold, abstracting from the metrical (and affine) structure, space-time has none of the paradigm spatio-temporal properties. The light-cone structure is not defined; past and future cannot be distinguished; distance relations do not exist. Spatio-temporal structure is metrical structure.⁴

In resisting the adoption of a bare-manifold account of spacetime Maudlin has been joined by Hoefer, who has argued that the mere fact that the metric field appears to carry energy does not imply that it should be regarded as a physical field.⁵ More specifically, Hoefer has argued that the complexities arising from the fact that gravitational stress-energy is represented by a pseudo-tensor cannot be merely brushed aside, and that there are good reasons to be sceptical of any quick inferences based on the existence of gravitational energy in this context.⁶

This debate—between bare-manifold and manifold-plus-metric accounts of spacetime—touches on the long-standing philosophical problem of how mathematics represents the world. In the context of geometry, the most immediate aspect of this problem is the ‘Problem of Space’: the problem of determining which abstract geometrical structures are candidate physical geometries, i.e. candidate descriptions of physical space. It is only since the advent of non-Euclidean geometries that this problem has emerged, or could even be stated. For most of its history, of course, geometry was just Euclidean geometry, understood as the systematic description of spatial structure (‘the most ancient branch of physics’, as Einstein once put it).⁷ Hence it was only after the existence of non-Euclidean geometries was grudgingly accepted that it became possible to ask: which geometry actually describes space? By the end of the nineteenth century (as we summarise below) consensus formed around the following answer: the candidate physical geometries are the constant curvature geometries; the geometries in which congruence relations can represent the free mobility of rigid bodies.

However, this ‘classical’ solution to the Problem of Space was unequivocally undermined by general relativity—the theory of spacetime that employs precisely the kind of variably curved geometry that the philosophers of the nineteenth century thought they had ruled out. In this paper, we explore the new solution to the Problem of Space advanced by Hermann Weyl, drawing especially on the account Weyl gave of this problem in a series of lectures delivered in Barcelona in 1922.⁸ Weyl had an exceptionally nuanced understanding of the novel conception of spacetime implicit in general relativity, and our concrete goal in what follows is to show that an important insight made available by Weyl’s work is the unearthing of a ‘middle way’ between bare-manifold and manifold-plus-metric accounts of spacetime.

2 The Classical Solution

The classical solution to the Problem of Space is, in effect, the fruits of the cumulative effort of those who engaged with the problem in the second half of the nineteenth century. The formulation of the Problem of Space in this context is often referred to as the Helmholtz (or Helmholtz/Lie) space problem, but as Helmholtz himself pointed out it had already been treated substantially by Riemann. Helmholtz’s results were rigorously reworked and extended by Sophus Lie, who brought to bear the full power of his theory of continuous transformation groups.⁹ Poincaré also grappled with the Problem of Space, and although his philosophical stance differed significantly from Helmholtz’s, he clearly regarded the problem as more-or-less solved once Lie had put Helmholtz’s arguments on a sufficiently rigorous footing.¹⁰

The essential idea that emerged in this period was that the geometrical notion of congruence represented the possibility of the free mobility of rigid bodies. Treating such free mobility as a basic fact (and recognising the role that this fact seemed to play in the practice of measurement quite generally), a limited class of geometries could be specified as candidate descriptions of physical space. In particular, it was argued that only constant curvature geometries could represent the free mobility of rigid bodies because only these geometries have suitable congruence relations between geometric figures.

The close connection between congruence relations of geometrical figures and constant curvature was proved most rigorously by Lie, who considered the properties of the real, finite-dimensional transformation groups that correspond to classical congruence relations, but it is also easy to understand the basic point intuitively. Helmholtz discussed the example of the non-constant curvature of an egg-shaped surface, observing that a figure such as a triangle would have different internal angles if it were drawn near the pointed end of the egg, compared to if it were drawn near the base.¹¹ Similarly, two circles with the same radius would not in general have the same circumference. Thus sliding any such figure up or down the surface would not be possible unless it were flexible enough to change its dimensions as it moved. In the general case, in any space of varying curvature (with zero degrees of symmetry), a truly rigid figure—one that could not alter its dimensions without breaking—would not be able to be moved at all.

In this way, Helmholtz, and Poincaré after him, argued that geometries of non-constant curvature are not feasible candidates for describing physical space at all. Their reasoning depended on the premise that constructing a physical geometry depends essentially on the use of rigid bodies. According to Helmholtz, this is a generalisation of the requirement that material measuring instruments (rulers, compasses, and the like) must maintain their dimensions if they are to fulfil their function.¹² Helmholtz argued that if there were no rigid bodies, and hence no way of comparing spatial magnitudes, it would not be possible to construct any kind of physical geometry. Hence variably curved geometries, which lack the relevant congruence structure, cannot provide a useful description of physical space (for even if we lived in such a space, so the thought goes, we would not be able to construct a geometrical description of it). Thus Helmholtz declared: ‘all original spatial measurement depends on asserting congruence and therefore, the system of spatial measurement must presuppose the same conditions on which alone it is meaningful to assert congruence’.¹³ Poincaré, for his part, said of variably curved geometries that they could ‘never be anything but purely analytic, and they would not be susceptible to demonstrations analogous to those of Euclid’.¹⁴

This, then, was the classical solution to the Problem of Space: a geometrical structure could describe spatial structure only insofar as it could represent the free mobility of rigid bodies. Combined with Lie’s work, this postulate of free mobility implied a clear demarcation of candidate physical geometries from two directions. First, candidate physical geometries could only be constant curvature geometries. Second, the metric function must satisfy a generalised Pythagorean Theorem, i.e. the element of length must be given by the square root of a quadratic differential form.

However, the development of relativity—particularly general relativity—broke the back of this purported solution to the Problem of Space. One obvious change due to relativity was the new way in which space and time were welded together into spacetime, but the more immediately significant change was actually the threat to the notion of a rigid body. Already in special relativity it became clear that perfectly rigid bodies were simply incompatible with the theory.¹⁵ But it was only with the advent of general relativity that the classical solution to the Problem of Space was definitively undermined, for it is general relativity that employs precisely the variably curved geometrical structure that Helmholtz and Poincaré had exiled from the class of candidate physical geometries. As Scholz has put the matter, general relativity ‘posed, of course, a much greater challenge to the characterisation of the Problem of Space. Free mobility of finitely extended rigid figures became meaningless in the general case.’¹⁶

3 Weyl’s Problem of Space

In the wake of general relativity it is evident that the classical demarcation of candidate physical geometries is too restrictive—clearly, an adequate demarcation must include variably curved geometries too. This is the context in which Weyl sought to justify a new and broader conception of physical geometry. On Weyl’s view, the possibility of describing physical space in geometrical terms depends only on the possibility of infinitesimal comparisons of lengths and angles. This still allows for the construction of practically rigid bodies, so that, as long as circumstances are not too hostile, we can still survey the meso-scale structure of space in the way that Helmholtz and others envisaged. But Weyl’s solution also leaves room for the possibility of describing the geometry of a region of space encompassing such strongly varying gravitational fields that surveying it with rigid measuring instruments would be impossible.

An important upshot of Weyl’s approach to the Problem of Space is a distinction between the nature of the metric field, on the one hand, and the orientation of the metric field from point to point, on the other. The former is what determines the relative lengths of vectors at an arbitrary point, whilst the latter is what determines the relative lengths and angles of finitely separated vectors. Weyl uses this distinction to attribute the local metric properties to space itself, whilst attributing the non-local metric properties to the contingent distribution of matter and energy. In brief, rather than starting with the classical postulate of free mobility, Weyl starts with what he calls the ‘foundational fact of infinitesimal geometry’¹⁷—the idea that the notion of congruent transport (the transformation that preserves length) uniquely determines a notion of affine transport (the transformation that preserves parallelism). Weyl then proves that this foundational fact provides the basis for a new demarcation of physical geometries. According to Weyl, the candidate physical geometries are just the geometries with a Pythagorean-Riemannian ‘nature’; the geometries whose metrics have an infinitesimal Pythagorean form.

It seems that Weyl first became interested in the Problem of Space when he was called upon to edit Riemann’s 1868 Habilitationsvortrag for republication, for which he also provided a commentary.¹⁸ In the Habilitationsvortrag, Riemann gives the length of a line element as the square root of a quadratic form in the differentials, but remarks that this is merely the simplest case of a wider range of possibilities:

The next simplest case would probably comprise those manifolds in which the line element may be expressed by the fourth root of a differential expression of the fourth degree. To be sure, the investigation of this more general kind would not require any essentially different principles, but it would be rather time-consuming and would shed relatively little new light on the theory of space (especially as the results are not geometrically expressible); I therefore restrict myself to those manifolds where the line element is expressed by the quadratic root of a differential expression of the second degree.¹⁹

Weyl expresses Riemann’s observation as follows: if the interval at point P is expressed as a function of the differentials, ds = f _P(dx ₁, …, dx _n), then ‘f _P will be required to be a homogeneous function of the first degree, in the sense that upon multiplication of the arguments dx _i by a common real proportionality factor ρ, the function f _P is multiplied by |ρ|.’²⁰ An example of such a function is the familiar Pythagorean function:

$$\displaystyle \begin{aligned} \sqrt{(dx_1)^2 + (dx_2)^2 + \cdots + (dx_n)^2} \end{aligned} $$

(1)

Up to a choice of coordinates, any function given as a square root of some positive-definite quadratic form can be expressed in the form (1) (i.e. if $f_P^2$, at each point P, is a positive-definite quadratic form, then all the various f _P can be obtained from the function (1) by linear transformations of the variables). However, (1) is not the only homogeneous function of the first degree, and so the question arises: why use this function to define intervals, rather than any others? Riemann himself offered no satisfactory justification for why the expression for the square of the line element should be a quadratic form, and hence it was a signature achievement of the classical solution to the Problem of Space to show that, if every physical geometry must represent the free mobility of rigid bodies, every physical geometry must have a Pythagorean metric. But when general relativity undermined the classical solution to the Problem of Space, this justification for the Pythagorean form of the metric vanished with it.

Thus, if we follow Weyl and say that the nature of a (Weylian) metric is given by specifying the expression for the line element, then the problem at hand is to justify the Pythagorean nature in particular. Weyl’s solution to this problem begins from his generalisation of Riemannian geometry.²¹ In a Riemannian space, (M, g), where M is a manifold and g is a metric, consider two arbitrary tangent vectors, v ∈ T _pM and w ∈ T _qM, for finitely separated points p and q. Whether or not these two vectors are parallel is not, in general, determined absolutely, but only relative to the choice of a path connecting p and q. This becomes most apparent if we introduce the concept of an affine connection following Levi-Civita.²² An affine connection establishes the parallelism-facts amongst the vectors in infinitesimally separated tangent spaces; it is only in the special case of a flat affine connection, however, that these can be extended to establish absolute (i.e. path-independent) parallelism-facts between vectors in finitely separated tangent spaces.²³

For Weyl, Levi-Civita’s work presented a profound insight into the structure of Riemannian geometry. But he soon recognised that it pointed to a way in which Riemannian geometry was not as local as it could be. Although Riemannian geometry does away with an absolute notion of distant parallelism, it retains an absolute notion of distant length-comparison: there is always a definite answer to the question whether vectors v ∈ T _pM and w ∈ T _qM are the same length or not, even for finitely separated p and q. More generally, for any two tangent vectors, there is some fact of the matter about their lengths, and hence about their length-ratio.

In aiming for a truly local geometry, then, our first move should be to do away with the structure of Riemannian geometry which permits such comparisons. Consider a pair of conformally equivalent metrics on M: that is, metrics g _ab and $g^{\prime }_{ab}$ such that for some smooth, positive, nowhere-vanishing scalar field $\lambda : M \to \mathbb {R}$,

$$\displaystyle \begin{aligned} g^{\prime}_{ab} = \lambda g_{ab} \,.\end{aligned} $$

(2)

A conformal structure on M is an equivalence class of conformally equivalent metrics; a conformal manifold is a manifold equipped with a conformal structure.²⁴ As Weyl remarks, in a conformal geometry the inner product of two vectors (in the same tangent space) is ‘not absolute, but rather determined only up to an arbitrary non-zero proportionality factor’.²⁵

Conformal manifolds do not permit distant length-comparisons: we can only determine the length-ratio between two vectors if they are drawn from the same tangent space. But in the passage to conformal manifolds, we have removed more structure than we wished to. For the analogy to parallelism (in Riemannian geometry) to hold, we do not want determinations of length-ratio between distant vectors to be impossible: we just want them to be relative to a choice of path. Equivalently, we want there to be (absolute) facts about the length-ratios of infinitesimally separated vectors, just not about the length-ratios of finitely separated vectors.

We must therefore restore the kind of structure that will let us make such comparisons, i.e. something analogous to an affine connection but for lengths rather than directions: ‘a concept of transfer of the length-unit from a point P to its immediate neighbours’.²⁶ Let us refer to such a standard of length-transfer as a length connection.²⁷ By transferring an (arbitrarily chosen) length-unit from one point to another, we can compare the lengths of vectors in the tangent spaces at the two points, and the length connection operates in such a way that this length-ratio is independent of the choice of unit. Thus, in Weylian geometry, length-comparisons of distant vectors are once again possible but—in general—only relative to a path. As is the case for affine connections, things may work out such that the value of the integral is path-independent; in this case, we say that the length connection is flat, and the geometry is that of a Riemannian manifold up to an arbitrary global choice of length-scale. We will refer to a conformal structure together with a length connection as a Weylian metric, and a manifold equipped with a Weylian metric as a Weylian manifold.

Returning to our main theme, the groundwork for Weyl’s solution to the Problem of Space was laid already in his 1919 discussion:

Given the fundamental significance for the construction of geometry which, following recent investigations …, attaches to the basic affine concept [affine Grundbegriff] of the infinitesimal parallel transport of a vector, the question in particular arises, whether the manifolds of the Pythagorean class of spaces [i.e. those in which the line element can be given in the Pythagorean form (1)] are the only ones which permit the establishment of this concept, and which correspondingly possess not only a metric, but also an affine connection. The answer is most likely affirmative, but a proof has so far not been rendered.²⁸

As already noted, it is this idea which drives Weyl’s solution to the Problem of Space. More specifically, the key insight is that only the Pythagorean kind of spaces have the feature that they are associated with a unique concept of parallel transport. As is well-known, for a given Riemannian metric there is a unique symmetric affine connection compatible with it; compatible, that is, in the sense that parallel-transported vectors retain their length.²⁹ This result extends to Weylian manifolds: given a Weylian manifold, there is a unique compatible affine connection.³⁰

This fact—the uniqueness of the affine connection, given the metrical structure—was greatly striking to Weyl, and something he put great emphasis on:

And now we come to that fact, already anointed above as the foundational fact of infinitesimal geometry, which brings the construction of geometry to a wonderfully harmonious conclusion. In a metrical space, there is one and only one way to formulate the concept of parallel transport so that …this postulate is fulfilled: upon parallel transport of a vector, the interval determined by it should also remain unchanged. Thus, the principle of infinitesimal interval- or length-transfer, which underlies metrical geometry, automatically brings with it a principle of direction transfer; a metrical space naturally carries an affine connection.³¹

At the heart of Weyl’s solution to the Problem of Space is a demonstration that, among the much broader class of spaces obtained by allowing the line element to be an arbitrary homogeneous function of the differentials, only the Pythagorean spaces (i.e. the Weylian manifolds) will satisfy the following condition: ‘whatever quantitative configuration (within the scope of the nature of the metric) the metric field may have assumed, it invariably and uniquely determines the affine connection.’³²

4 Weyl’s Solution

We turn now to a reconstruction of (part of) Weyl’s argument. We follow the treatment given in Weyl (1923a): the text of a series of lectures on the Problem of Space which Weyl gave in the spring of 1922 in Barcelona and Madrid. There are some significant differences between the way in which Weyl carries out the argument here, compared to the way it is presented in his other work;³³ moreover, this text has not been translated, and so we hope that the discussion here can help bring these ideas to a wider audience.

Weyl reaches his solution via a group-theoretic analysis. Given an n-dimensional Weylian manifold, of whatever nature, let us say that a linear automorphism g of the tangent space at P is congruent if it preserves the interval: that is, if for any vector ξ at P, f _P(g(ξ)) = f _P(ξ). Since the composition of two congruent automorphisms will similarly be a congruent automorphism, as will the inverse of any congruent automorphism, the collection of all congruent automorphisms of P’s tangent space will form a group: let us refer to this group as the congruence group at P. For example, if f _P is the Pythagorean function (1), then the congruence group will be the orthogonal group O(n).³⁴

Note that in general, the congruence group is not sufficient to determine the nature of the metric. For example, the four-norm and the six-norm on $\mathbb {R}^2$, i.e. the functions

$$\displaystyle \begin{aligned} ||(x, y)||{}_4 := (x^4 + y^4)^{1/4} \end{aligned} $$

(3)

and

$$\displaystyle \begin{aligned} ||(x, y)||{}_6 := (x^6 + y^6)^{1/6} \end{aligned} $$

(4)

respectively, have the same congruence group: the group consisting of right-angle rotations and reflections about the x- and y-axes. ³⁵ Nevertheless, if the tangent space at P has O(n) as its congruence group, then the nature in question is Pythagorean (i.e. it is given by a positive-definite quadratic form) at P. So if Weyl can find conditions which guarantee that each tangent space has O(n) as its congruence group, then he will have succeeded in showing that only manifolds with a Pythagorean nature satisfy those conditions.³⁶

In the seventh of his Barcelona lectures, Weyl argues that if we postulate uniqueness of parallel transport, then the congruence group’s Lie algebra (i.e. the collection of infinitesimal congruent automorphisms)³⁷ must be of dimension n(n − 1)∕2, and must satisfy a certain kind of antisymmetry condition (stated in more detail below). In the eighth and final lecture, Weyl sketches a proof that these conditions entail that the congruence group is the orthogonal group O(n), and shows this by explicit calculation for the case n = 2; a complete proof (for the case of arbitrary dimensions) is provided in the appendices.³⁸

Weyl begins his argument by discussing the relationship between the metrical and the affine structure, i.e. between the transport of lengths and the transport of vectors. Recall that the length connection provides us with a unique way of transferring a length-unit from a point P to another point P _∗ in P’s immediate neighbourhood, and hence of determining whether vectors ξ at P and ξ _∗ at P _∗ are the same length. In light of this, let us follow Weyl by saying that a linear isomorphism $\xi \in T_P M \to \xi _* \in T_{P_*} M$ is a congruent transport just in case it preserves the lengths of vectors: i.e. if the length of ξ relative to a given length-unit at P is the same as the length of ξ _∗, relative to the transferral of that length-unit (using the length connection). Note that this is independent of the length-unit chosen at P.

At this stage we are restricting our attention to infinitesimal congruent transports: choosing some coordinate system around P, we are interested in those congruent transports ξ↦ξ _∗ for which

$$\displaystyle \begin{aligned} \xi^i_* = \xi^i + d \xi^i \end{aligned} $$

(5)

Since congruent transports are linear, it follows that $d \xi ^i = - \Lambda ^i_k \xi ^k$.³⁹ Let us say that the $\Lambda ^i_k$ are the coefficients of the congruent transport. Now, without yet specifying the dimensionality of the manifold, suppose that P _∗ is at the point (ε, 0, 0, …, 0). If we let $\Lambda ^i_{k1}$ be the coefficients of the congruent transport from P to P _∗, and $\Lambda ^i_{k2}$ be the coefficients of a congruent transport from P to the point (0, ε, 0, …, 0), then (for any $\alpha , \beta \in \mathbb {R}$) we can show that $\alpha \Lambda ^i_{k1} + \beta \Lambda ^i_{k2}$ are the coefficients of a congruent transport from P to the point at (αε, βε, 0, …, 0). Hence, once n congruent transports $\Lambda ^i_{kr}$ have been chosen, there is uniquely fixed a congruent transport to any point in P’s infinitesimal neighbourhood. As Weyl puts it:

…the formula,

$$\displaystyle \begin{aligned} d \xi^i = -\Lambda^i_{kr} \xi^k (dx)^r \end{aligned} $$

(6)

supplies a system of infinitesimal congruent transports to the totality of points P′ = (dx ¹, dx ², …, dx ⁿ) of the neighbourhood of P.⁴⁰

Weyl’s argument then runs as follows. Take a point P ₀ in our manifold M, and introduce some coordinate system around it. Take as given the congruence group G ₀ at P ₀ (but not the congruence group at any other point).⁴¹ Now let $\Lambda ^i_{kr}$ be an arbitrary collection of n ³ numbers. For every point P in the infinitesimal neighbourhood of P ₀, we can define a linear isomorphism from $T_{P_0} M$ to T _PM by (6). Since every vector determines a length-unit (the one according to which that vector is of unit length), we can use this to define a length-unit transfer from P ₀ to P, i.e. a length connection. Moreover, if we let the congruence group G at P be defined as the image of G ₀ under this linear isomorphism, then the isomorphism will be congruence-preserving, and hence Λ will represent a system of infinitesimal congruent transports.

Now that we have a length connection and congruence groups on the infinitesimal neighbourhood of P ₀, any infinitesimal congruent transport from P ₀ to P may be obtained from a given such transport by composition with some infinitesimal congruent transformation.⁴² Consequently, any system of infinitesimal congruent transports may be obtained from our original system (that encoded by the $\Lambda ^i_{jk}$) by specifying n such infinitesimal congruent transformations: one for each of the n linearly independent coordinate displacements dx ^r. If we let $A^i_{kr}$ be the infinitesimal congruent transformation associated with dx ^r, then the action of such a transformation on an arbitrary vector ξ at P is given (in terms of components) by:

$$\displaystyle \begin{aligned} \xi^i \mapsto \xi^i - A^i_{kr} \xi^k (dx)^r \end{aligned} $$

(7)

It is at this point that we impose the postulate mentioned above, ‘that among all these systems of infinitesimal congruent transports, a unique one is to be found which is simultaneously a possible system of parallel displacement’.⁴³ It follows that there is a unique array of $A^i_{kr}$ which will bring about such a system of parallel transport. Using the Christoffel symbols now ubiquitous in general relativity, a parallel transport can be represented by $\Gamma ^i_{jk}$, subject only to the symmetry requirement that $\Gamma ^i_{jk} = \Gamma ^i_{kj}$. We can then state Weyl’s postulate as follows: given any $\Lambda ^i_{jk}$, there is a unique system of parallel transport $\Gamma ^i_{jk}$ and a unique system of infinitesimal congruent transformations $A^i_{jk}$ such that

$$\displaystyle \begin{aligned} \Lambda^i_{jk} = \Gamma^i_{jk} - A^i_{jk} \end{aligned} $$

(8)

From this, Weyl proceeds to draw the following conclusions. First, if the dimensionality of the congruence group G (and hence, of its Lie algebra $\mathfrak {g}$) is N, then since every $\Lambda ^i_{jk}$ (with n ³ independent parameters) corresponds to a unique $\Gamma ^i_{jk}$ (n ²(n + 1)∕2 independent parameters) and $A^i_{jk}$ (nN independent parameters), then $n^3 = \frac {n^2(n+1)}{2} + nN$; that is,

$$\displaystyle \begin{aligned} N = \frac{n(n-1)}{2} \end{aligned} $$

(9)

Second, note that if $A^i_{jk} = A^i_{kj}$, then $\Lambda ^i_{jk}$ will represent a system of parallel transport, which must therefore be identical with that represented by $\Gamma ^i_{jk}$—from which it follows that $A^i_{jk} = 0$. So the family of the $A^i_{jk}$ has the feature that they are symmetric ($A^i_{jk} = A^i_{kj}$) only if they all vanish. These two conclusions are the conditions on the congruence group’s Lie algebra which—as discussed above—will lead us to the conclusion that the congruence group must be the orthogonal group (by an argument that we forbear from reconstructing here).

It bears emphasising how striking Weyl’s achievement here is. Not only has he shown how the Problem of Space may be reframed in the light of general relativity; he has also shown that a satisfactory solution may be arrived at by the requirement that one’s standard of length-comparison uniquely fixes one’s standard of direction-comparison. It should also be stressed that Weyl’s analysis is independent of his unorthodox geometrical background, insofar as Riemannian geometry is (up to an arbitrary choice of global scale) a special case of Weylian geometry.

In the wake of general relativity, it is evident that something fundamental has shifted in the implicit assumptions built into our practices of describing space geometrically. Weyl’s solution to the Problem of Space offers an insight into precisely this fundamental shift. For Weyl, the possibility of describing space in geometrical terms depends only on the possibility of an idealised observer at a point, ‘freely mobile’ in the sense of being free to rotate at that point and start moving in any direction. Although such an observer can compare the dimensions of (infinitesimal) bodies in her immediate vicinity—that is determined only by the nature of space itself—what she might go on to discover about the larger-scale structure of space as she explores larger regions of it is left maximally unconstrained.⁴⁴ Weyl sums up this new conception of space, implicit in general relativity as he understood it, with the following vivid metaphor:

Euclidean space may be compared to a crystal, built up of uniform unchangeable atoms in the regular and rigid unchangeable arrangement of a lattice; Riemannian space to a liquid, consisting of the same indiscernible unchangeable atoms, whose arrangement and orientation, however, are mobile and yielding to forces acting upon them.⁴⁵

5 The Status of the Metric Field

Let us return to a consideration of what Weyl’s work can contribute to the relatively recent debate over the status of the metric field. Recall that, on the one hand, because in general relativity the metrical field incorporates the gravitational field, some (including Earman, Norton, and Rovelli) have argued that the metric tensor should be regarded as representing a physical field, akin to the electromagnetic field. On such a view, it is the bare topological manifold, absent any metrical properties, that should be regarded as representing spacetime itself. On the other hand, others (including Maudlin and Hoefer) have argued that metrical structure remains at the heart of paradigmatic spatio-temporal structure.

In considering how Weyl himself might have responded to this debate, the following statement seems unequivocal:

…it is not correct to say that space or the world is in itself, prior to any material content, merely a formless continuous manifold in the sense of analysis situs; the nature of the metric is peculiar to it in itself, only the mutual orientation of the metrics at various points is contingent, a posteriori and dependent on the material content.⁴⁶

Thus, Weyl contrasts the nature of the metric—which, as we have seen, has its character fixed by the relationship between affine and metrical structure—with the orientation of the metric, encapsulating the remaining degrees of metrical freedom:

Thus one sees how the nature of the metric can be the same at every location, even while its quantitative determination, the—so to speak—mutual orientation of the metric at different points, is still very changeable and capable of continuously varying configurations. Thus, from this standpoint, the a prioristic essence of space (defined by the nature of the metric) …is divorced from the mutual orientation of the metric at the different points, which is a posteriori, i.e., contingent and naturally dependent on material content …⁴⁷

For present purposes, Weyl’s distinction between the a priori and a posteriori serves to indicate the metrical properties that he attributes to space itself as contrasted with the metrical properties he attributes to the particular distribution of matter and energy. The fact that Weyl explicitly states that empty space is not ‘merely a formless continuous manifold’ would seem to place him squarely against the view advanced by Earman, Norton, and Rovelli. But in fact Weyl’s analysis allows for a distinction that none of the more contemporary protagonists have in view whilst capturing motivations from both sides. On the one hand, there is the awkwardness of regarding the dynamical aspects of the metric as attributable to space itself; on the other hand, there is the fact that a bare manifold seems genuinely insufficient to represent anything we would recognise as space. But Weyl’s distinction between the nature and orientation of the metric field provides a way to retain the idea that space is intrinsically metrical without thereby being forced to attribute all the dynamical aspects of the metric field to space itself.⁴⁸ Weyl thus provides a “middle way” between bare-manifold and manifold-plus-metric accounts of spacetime, arguing that only the local metrical properties—properties which are independent of the variable distribution of matter and energy—are attributable to space itself.

One thing that emerges from the debate over the status of the metric field is that, in the wake of general relativity, we lack a principled means of identifying which mathematical structures represent features of space. By contrast, this was something that the figures of the nineteenth century had available to them: the classical solution to the Problem of Space provided a justification for why a particular collection of mathematical structures (the constant curvature geometries) could play this particular representational role. Once the classical solution became untenable, however, this justification went with it. Weyl’s new solution to the Problem of Space thus offers a new justification for why an enlarged collection of mathematical structures—differential manifolds equipped with an infinitesimal Pythagorean-Riemannian metric—are candidate descriptions of physical space.

Throughout this paper, we have been treating Weyl’s solution to the Problem of Space independently of his broader philosophical commitments. The fact that it is possible to do so points to the fundamental nature of the Problem itself. This is evident from the fact that, philosophical differences notwithstanding, there was broad agreement on the classical solution to the Problem of Space prior to general relativity. Poincaré, for example, could accept Helmholtz’s solution to the Problem of Space whilst disagreeing with Helmholtz’s claim that the value of the curvature of space would be determined by experiment. (For Poincaré, famously, the choice amongst constant curvature geometries was a matter of pure convention.) In a similar way, it is open to us to accept Weyl’s new solution to the Problem of Space (and the insight into the conception of space implicit in general relativity that Weyl offers) independently of Weyl’s own broader philosophical commitments.⁴⁹

Weyl’s argument seeks to show that reflection upon the concept of (physical) metrical structure—in particular, upon the required relationship between metrical structure and affine structure—provides a justification for the Pythagorean nature of the metric. This provides a different argument for regarding the metric as encoding spatial structure, beyond merely noting that certain ‘paradigmatically spatial’ properties depend upon it. In Weyl’s analysis the sine qua non of physical geometry is that it realises a concordance between parallelism and congruence, and so the physical geometries are those whose infinitesimal metrical structures uniquely determine affine structures over finite distances.⁵⁰ It is with this kind of insight in view that we urge that engaging with the Problem of Space remains important, not merely as providing a different answer to the question of what represents space, but rather as a means of shedding light on the question of what it is to represent space.