# Interpretations of Spacetime and the Principle of Relativity

Chapter
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 264)

## Abstract

Chapter 3 will introduce an interpretation of flat spacetime theories I call Primitive Motion Relationalism (PMR). According to this interpretation, motion should be thought of as a primitive entity, more fundamental than spatial points and temporal instants. Events are taken to be coincidences between motions; the identity of events depends on the identity of the underlying motions. The other main feature of this approach is that spacetime consists of a set of potential trajectories. The spacetime manifold and the metric defined on it should not be thought of as a field analogous to other material fields. Rather, spacetime determines the form of actual trajectories and relations between motions, in analogy with Aristotelian formal causes that determine the essence of a substance. One of the main advantages of PMR is that it explains the restricted Principle of Relativity (i.e., the equivalence between inertial reference frames), without presupposing the Principle of Relativity as a postulate.

## Keywords

Lorentz Transformation Causal Power Spacetime Symmetry Spacetime Structure Inertial Reference Frame
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 3 will introduce an interpretation of flat spacetime theories I call Primitive Motion Relationalism (PMR). According to this interpretation, motion should be thought of as a primitive entity, more fundamental than spatial points and temporal instants. Events are taken to be coincidences between motions; the identity of events depends on the identity of the underlying motions. The other main feature of this approach is that spacetime consists of a set of potential trajectories. The spacetime manifold and the metric defined on it should not be thought of as a field analogous to other material fields. Rather, spacetime determines the form of actual trajectories and relations between motions, in analogy with Aristotelian formal causes that determine the essence of a substance. One of the main advantages of PMR is that it explains the restricted Principle of Relativity (i.e., the equivalence between inertial reference frames), without presupposing the Principle of Relativity as a postulate.

To differentiate PMR from standard accounts of spacetime, this chapter delineates some of the dominant interpretations of spacetime. The chapter outlines three common approaches: the conventionalist, the geometric, and the dynamic interpretations of spacetime. While the conventionalist account is mostly out of favor today, the geometric interpretation is the accepted doctrine. Dynamic accounts constitute a minority view that wishes to undermine official doctrine. Each account of spacetime has important advantages, however each also carries some weaknesses and liabilities.

Primitive Motion Relationalism has affinities with the dynamic approach, as it does not suppose the independent existence of spacetime. Like dynamical relationalism, PMR does not separate between dynamic and kinematic aspects of physical knowledge. However, PMR also has affinities with the geometric approach, since it attempts to provide a unifying account for spacetime symmetries, while current versions of dynamical relationalism do not seek to do so.

I will restrict my attention to flat spacetimes, and will leave discussion of curved spacetime for future work. PMR argues that spacetime constitutes a range of possible trajectories, and is not itself an actualized entity. This possibilist conception of spacetime faces difficulties in the context of the General Theory of Relativity. If spacetime consists of a range of possible trajectories, it is difficult to conceive of spacetime as a contingent structure that is determined by how matter is actually distributed. While I do not think this problem is beyond resolution, I shall not consider it here and instead focus on the positive reasons for believing in PMR. The possibilist conception of spacetime and the fundamental nature of motion helps explain the restricted Principle of Relativity. Thus PMR has an important advantage over traditional approaches that either assume the Principle of Relativity as a postulate or provide an inadequate explanation for it. To demonstrate the benefits of adopting PMR, I devote this chapter to a brief and sketchy assessment of existing interpretations of spacetime. Section 2.1 will introduce the restricted Principle of Relativity and some recent discussions regarding its appropriate interpretation. I shall then consider how each of the three common approaches, i.e., the conventionalist (Section 2.2), geometric (Section 2.3) and dynamic (Section 2.4) interpretations of spacetime, accommodates the Principle of Relativity. I shall note what I think is the central weakness in each account, in this way preparing the way for evaluating the merits of PMR.

## 2.1 The Restricted Principle of Relativity

Einstein introduced the Principle of Relativity as a postulate of the theory of relativity. According to Einstein, the restricted Principle of Relativity, which is the equivalence between inertial reference frames moving with uniform rectilinear motions, is modeled after the “classical” Principle of Relativity, which is articulated in Corollary V to Proposition I in Newton’s Principia. The restricted Principle of Relativity is to be distinguished from the General Principle of Relativity, which according to Einstein amounts to the covariance of equations of motions under general coordinate transformations.1

Einstein argued that the restricted Principle of Relativity and the Light Postulate have an empirical basis. The laws describing Newtonian mechanics and electrodynamics do not include any property that makes reference to absolute rest. But in Einstein’s theory, the Principle of Relativity assumes the status of a postulate, which he explicates as follows:

Einstein’s Principle of Relativity. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion. (Einstein, 1952, p. 41)

According to Einstein’s Principle of Relativity, the laws are the same whether they are defined relative to one coordinate system, or to a coordinate system moving with uniform rectilinear motion relative to the first. A coordinate system assigns a 4-tuple x μ to any event that takes place, where $$\mu=0,1,2,3$$, and comes equipped with a set of measuring rods and clocks that are relatively at rest.2 His theory of relativity, he claims, is about the relations between these measuring devices and electromagnetic processes.

Einstein’s account of coordinate systems begs for further explanation. Clocks and rigid rods are macroscopic systems, and a clock that measures time at a particular infinitesimal point can only be a highly abstracted idealization. Nevertheless, the benefit of this idealization is that it provides Einstein with a conceptual tool for grounding his Principle of Relativity. One first imagines a coordinate system K consisting of a set of clocks and rigid rods that are relatively at rest. One then imagines another coordinate system K , whose clocks and rigid rods are relatively at rest, but move with uniform rectilinear motion relative to the clocks and rods at K. The origin in the coordinate system K , described with the 4-tuple x μ moves uniformly in a straight line in the coordinate system $$K=x^{\mu}$$. Thus $$x^{\mu}=\,<\alpha,0,0,0>$$ coincides with $$x'^{\mu}=\,<\beta,-v_1\beta,-v_2\beta,-v_3\beta>$$ (assuming that their origins coincide at $$x^0=x'^0=0$$). Once coordinate systems are given, the Principle of Relativity can be articulated. Any laws describing changes in states of a physical system in K will be the same in K .

The Principle of Relativity describes an isomorphism between laws articulated relative to different coordinate systems. However, Einstein also used the Principle of Relativity in deriving the Lorentz transformations, which are laws that transform between measurements performed relative to different coordinate systems. In the first step of his argument, Einstein derives generalized Lorentz transformations from the light Postulate, assuming that the translatory motion of K is in the x 1 direction3:
$$\begin{array}{rcl} x'^0 &=& \phi(v)\gamma\left(x^0-\frac{v^2}{c}x^1\right) \\ \nonumber x'^1 &=& \phi(v)\gamma\left(x^1-vx^0\right) \\ \nonumber x'^2 &=& \phi(v) x^2 \\ \nonumber x'^3 &=& \phi(v) x^3\end{array}$$
(2.1)
where $$\gamma=\left(1-\frac{v^2}{c^2}\right)^{-\frac{1}{2}}$$ and $$\phi(v)$$ is an unknown function of the relative velocities between the frames. To reduce (2.1) to the standard Lorentz transformations, Einstein looks at the transformation between K and K that moves with a velocity −v relative to K (so that K and K are at rest relative to each other). Let a rod of length 1 lie on the x 1 axis of system K . In the frame K, the length of this rod will appear contracted by a factor of $$\frac{\phi(v)}{\gamma}$$. In the frame K the same rod will appear contracted by a factor of $$\frac{\phi(-v)}{\gamma}$$. Thus, the rod in K will appear contracted in K by a factor of $$\frac{\phi(v)}{\phi(-v)}$$. Given the symmetry of the situation, Einstein argues, the transformations $$\Lambda:K \mapsto K'$$ and $$\Lambda':K' \mapsto K''$$ should look exactly the same, so one should conclude that
$$\phi(v)\phi(-v)=1$$
(2.2)
Einstein argues that since K and K are in fact at rest relative to each other, the transformations $$\Lambda :K \mapsto K'$$ and $$\Lambda':K' \mapsto K''$$ should also be considered as inverse transformations, so that $$\Lambda'(-v)=\Lambda^{-1}(v)$$. It therefore follows that $$\phi(v)=\phi(-v)=1$$, since otherwise the contraction parameter for the rods will depend on factors other than the relative velocities between frames. From this Einstein concludes that the generalized Lorentz transformations reduce to the restricted Lorentz transformations4:
$$\begin{array}{rcl} x'^0 &=&\gamma\left(x^0-\frac{v^2}{c}x^1\right)\\ x'^1 &=&\gamma\left(x^1-vx^0\right) \nonumber \\ x'^2 &=& x^2 \nonumber \\ x'^3 &=& x^3 \nonumber \end{array}$$
(2.3)

The notion is that the Lorentz transformations should form a group, and so the equality $$\Lambda(v)=\Lambda^{-1}(-v)$$ is referred to as the “Reciprocity Principle” by some commentators. But in asserting that the Lorentz transformations should conform to a group structure, Einstein in effect applies the Principle of Relativity to laws transforming between coordinate systems. As it is articulated, the Principle of Relativity describes an isomorphism between laws articulated relative to different frames; Einstein’s application of the principle, in deriving the Lorentz transformations, is to the coordinate systems themselves. It is at least logically possible to separate the laws articulated in each frame from the laws transforming between them. One could, for example, assume that the frame K is not the same as K.

Einstein uses the Principle of Relativity to justify the Reciprocity Principle, or the notion that rods and clocks are warped only as a result of the relative velocity boosts they experience. But it is not clear why the violation of the Reciprocity Principle is a violation of the Principle of Relativity. According to the Reciprocity Principle, it is supposed that if one has two sets of rods and clocks that are at rest relative to each other, they both provide the same “natural” units of length and time relative to clocks and rods at rest in another inertial reference frame. However, one could imagine, for example, rods and clocks made of different substances; one kind of substance would be appropriate for measuring the length in K, and one would be appropriate for measuring the length in K . Or, one could imagine different procedures by which clocks and rods in each frame were prepared. These two coordinate systems would have different units of length that are natural to them, even though they are at rest relative to each other. Because both K and K are coordinate systems, the fact that they are mutually at rest does not in itself violate the Principle of Relativity, since all one needs is for the laws of nature to be the same in K and K for the Principle of Relativity to hold. These considerations have led some to include the Reciprocity Principle as a separate axiom of relativity theory and to avoid appealing to the Principle of Relativity in justifying the restricted Lorentz transformations, given that the Principle of Relativity is only articulated for dynamic laws defined relative to frames (see, for example, Madarász et al., 2007).5

The application of the Principle of Relativity to the derivation of the specialized Lorentz transformations is logically independent of the isomorphism between laws articulated relative to different frames; one is a symmetry governing transformations between frames, the other is an isomorphism between laws articulated in different frames.

Einstein’s derivation makes it seem as if the kinematical results of relativity theory are directly derived from Einstein’s two postulates. When writing about the status of these postulates Einstein often compared them to the postulates of thermodynamics, suggesting that STR is a “principle theory” rather than a “constructive theory.”6 Constructive theories begin from a relatively simple formal scheme and construct from these elementary components the complex phenomena. For example, in the kinetic theory of gases, the macroscopic states of gases are constructed from the microscopic states of molecules and the laws governing these microscopic states. On the other hand, principle theories begin from empirically discovered principles, which describe general characteristics of natural phenomena. These principles give rise to criteria which the separate processes have to satisfy. Einstein provides thermodynamics as an example of a principle theory and places STR in the same category.

The distinction between principle and constructive theories is important in this context since it seems to provide an account of the kinematical results of STR. While one does not have a complete constructive account of composite structures such as clocks and rods in STR, Einstein argues, one can derive their behavior from broad phenomenological principles like the Light Principle and the Principle of Relativity. However, it is not clear, given that the Principle of Relativity itself seems to presuppose the existence of coordinate systems, how it applies to clocks and rods themselves. Either the presupposition is that an underlying dynamic account of clocks and rods is yet to be supplied, in which case the Principle of Relativity would be articulated as a symmetry of underlying dynamic laws that makes no reference to coordinate systems. Or, there is a constructive account of a different kind, perhaps involving a geometric account of manifolds, that should replace the phenomenological principle. In either case, the Principle Theory account of kinematic effects of relativity seems incomplete. The analogy to thermodynamic theory is illustrative. While it is legitimate to think of thermodynamic theory as a successful theory, the kinetic theory of gases does complete the thermodynamic theory and “grounds” it in fundamental facts.

One may feel as if a genuine constructive account is required for kinematics. For example, a possible explanation for length contraction may be that moving rods experience various forces due to their motion. These forces are velocity-dependent and they affect all rigid rods in the same way. Similarly, a moving clock will experience forces that make its parts move more slowly. The initial attempts to explain the Michelson-Morley experiment involved an interaction between the ether and charged bodies. Ether theorists such as Lorentz (1881, 1904) hypothesized that rigid bodies undergo some contraction after they are accelerated to some motion. While these proposals for a dynamic account of clocks and rods appear to violate the Principle of Relativity, it is not necessary that they do so. As Brown (2005, ) points out, one can hypothesize that these forces arise in proportion to the velocity of an object relative to an inertial reference frame, in this way keeping in tact the Principle of Relativity.

Einstein’s account of length contraction seems to supersede Lorentz’s theory and other ether theorists, since it doesn’t require an additional explanation over and above the axioms of relativity.7 The notion that dynamic forces explain the deformation of rigid rods and clocks is odd, given that they apply universally to all rigid rods and clocks, no matter what they are made of. But still, how does one justify the Reciprocity Principle if a constructive account of clocks and rods is not forthcoming?

So far I argued that it seems odd to subsume laws governing rods and clocks in different inertial reference frames under the same Principle of Relativity which governs dynamic laws. Another way to see the disparity between the two applications of the Principle of Relativity (the first to kinematic laws and the second to dynamic laws), is to consider the distinction between global and local versions of the Principle of Relativity. Einstein’s formulation of the Principle is articulated for “changes in the state of physical systems.” Einstein is implicitly referring to isolated systems, as the Principle of Relativity does not hold for a subsystem that interacts with another subsystem. However, it is logically possible for a universe to have indistinguishable dynamic models, but that the same symmetries do not apply locally to isolated subsystems of the universe.8

The global version of the Principle of Relativity is exemplified by Anderson (1967). Anderson articulates a precise formulation of the Principle of Relativity based on modern mathematical theories of differential geometry. According to Anderson’s approach, a physical theory consists of classes of kinematically and dynamically allowed models. Each model M consists of a manifold with spacetime structures and matter fields:
$$\left <M,O_1,\ldots,O_n\right>$$

Symmetry principles can now be introduced via the notion that various models are acceptable kinematic and dynamic models of the theory. Thus, the covariance of the theory is given via the equivalence between one model and another model where a diffeomorphism $$d:M\mapsto M$$ acts on the manifold, so that if $$\left <M,O_1, \ldots, O_n \right>$$ is a model of the theory and $$d*$$ is the drag on the objects of the theory, then so is $$\left <M,d*O_1,\ldots,d*O_n\right>$$. One can distinguish between spacetime symmetries and dynamic symmetries by considering whether any diffeomorphism of the manifold leaves the geometric structures invariant, so that the models $$\left <M, A_1,\ldots, A_n\right>$$ and $$\left <M, d*A_1,\ldots,d*A_n\right>$$ remain the same, i.e. $$d*A_i=A_i$$. The objects A i would then be the absolute (geometric) objects of the theory. A dynamic symmetry is a diffeomorphism which renders the shifted dynamic model acceptable, so that $$\left <M,A_1,\ldots,A_n,D_1,\ldots,D_n\right>$$ and $$\left <M, A_1,\ldots, A_n,d*D_1,\ldots,d*D_n\right>$$ are both dynamically acceptable.

Anderson’s account generates a natural connection between spacetime symmetries and the dynamic symmetries. On the one hand, one considers whether a certain transformation leaves a geometric object invariant, and then considers whether the same transformation renders acceptable the transformed dynamic objects. Assume that one defines frames on the manifold using a set of parallel time-like straight worldlines. Assume that there is a group of transformations, for example, the Lorentz transformations, for which each element of the group transforms one frame to another. One may think of the global Principle of Relativity as the notion that the group of transformations leaves the absolute objects of the theory invariant (in which case the group is a spacetime symmetry), or it leaves the dynamic objects of the theory invariant (in which case the group is a dynamic symmetry).

In this formulation of the Principle of Relativity, a direct connection appears to have been made between the spacetime symmetries and the dynamic symmetries. However, when the Principle of Relativity is formulated globally in this way, it is an inter-world principle describing the equivalence between various possible worlds. The global Principle of Relativity is not an inter-world symmetry between frames. Assume we have frames K 1 and K 2 in world w 1, and frames K 1 and K 2 in world w 2. If the worlds w 1 and w 2 are related through a kinematic symmetry, the frames K 1 and K 2 are mapped onto K 1 and K 2 via the spacetime symmetry. If the symmetry is also that of the dynamic objects of the theory, the results of experiments performed in K 1 are indistinguishable from experiments performed in K 1 . Moreover, experiments in K 2 are indistinguishable from experiments in K 2. But this does not imply an intra-world equivalence between K 1 and K 2 or $$K'_1$$ and K 2. The global Principle of Relativity does not imply the local Relativity Principle, which is an intra-world relation between frames (or the equivalence between laws applied to isolated systems).

Budden (1997) provides an example of a theory that many take as satisfying the global but not the local Principle of Relativity. Budden introduces a couple of theories with anisotropic spacetime that are analogous to the isotropic Minkowski spacetime. The theory includes the structures $$\left<\mathbb{R}^4,\lambda,N\right>$$, where $$\mathbb{R}^4$$ is the manifold, λ is the relation of lightlike connectibility, and N is a set of parallel lines in the spacetime, which amounts to a preferred frame. In one of Budden’s theories, inertial clocks do not obey the Lorentz dilation factor. Instead, the theory defines a temporal congruence relation ∼1, such that $$ab\sim_1ac$$, if and only if b and c lie in one of the null hyperplanes picked out by N. Thus, Budden’s theory picks a preferred direction in spacetime (i.e., a frame), and defines anisotropic congruence relations on spacetime intervals relative to this frame. The congruence relation does not define a unit of time, which implies that the clocks do not obey the Reciprocity Principle between frames. Budden’s theory enables one to define a dilation effect between the preferred frame and other frames:
$$D=D(1+v)^{-1}$$
(2.4)
where v is the relative velocities between the frames. In all other respects, the anisotropic theory satisfies the global symmetries of the theory of relativity, since a transformation from one possible world with a preferred frame to another will yield indistinguishable dynamic models. But clocks behave differently than in relativity, temporal measures sometimes dilating and sometimes contracting depending on the relative velocities between the preferred frame and the frame boosted relative to it.

Thus, Budden’s theory shows that one can have both a global kinematic and dynamic symmetry, e.g., Lorentz covariance, without having local kinematic symmetry. In Budden’s theory, dynamic laws relative to one inertial reference frame are isomorphic to dynamic laws of the frame which has actively been transformed under the Lorentz transformations. But time measurements in different frames do not obey the Lorentz transformations.

Recently Skow (2008) argued that Budden’s anisotropic theory fails to demonstrate that the global Principle of Relativity does not imply the local Principle of Relativity. According to Skow, the anisotropy of Budden’s spacetime implies that temporal processes in a non-preferred frame are correlated with remote temporal processes on the preferred frame. This somehow suggests that systems in the non-preferred frame that appear to be isolated are not genuinely isolated. Given that their evolution in time depends on its relation to the preferred frame, one cannot take them to be isolated. There must be some non-local interaction to account for Budden’s non-isotropic spacetime, perhaps mediated by spacetime itself, between systems residing in the non-preferred frame on the one hand and systems residing in the preferred frame on the other hand. Thus, what looks like a violation of the local Principle of Relativity, is not genuinely a violation of the principle, because in an anisotropic spacetime one cannot truly isolate a system from another system if a preferred direction of spacetime is given.

Perhaps what is at stake is a distinction that commentators fail to make, and that is the distinction between the failure of the Reciprocity Principle (that kinematic contraction and dilation effects only depend on the relation between frames, and not on the existence of a preferred frame), and the failure of the local Principle of Relativity (according to which the same dynamic laws apply to all isolated systems). Einstein argued that the Reciprocity Principle follows from the local Principle of Relativity. Given Einstein’s claim that the local Principle of Relativity implies the Reciprocity Principle, one is tempted to take Budden’s theory as a violation of the local Principle of Relativity. Skow is correct to argue that the failure of Reciprocity Principle does not imply the failure of the local Principle of Relativity. Skow’s argument amounts to the claim that when the Reciprocity Principle fails, the local Principle of Relativity is not shown to be false, because somehow an anisotropic spacetime suggests that no system can be genuinely isolated from another. And if no system can be isolated, one cannot argue that isolated systems in different frames disobey the local Principle of Relativity. But the correct response to Budden’s theory is that the Reciprocity Principle and the local Principle of Relativity are logically distinct, and the local Principle of Relativity could still hold despite the failure of the Reciprocity Principle.

To summarize the above discussion, despite Einstein’s claim, there is no direct logical connection between the restricted Principle of Relativity, and the Reciprocity Principle. While Einstein attempts to derive the relation $$\Lambda'(v)=\Lambda^{-1}(-v)$$ from the Principle of Relativity, he is not justified in doing so. In recent decades work has been done to demonstrate the gap between global, or purely geometric accounts of the Principle of Relativity, and the local Principle of Relativity, which asserts the dynamic equivalence between isolated systems. While this work is correct to point out the gap between global and local symmetries, much of it confuses the failure of the Reciprocity Principle with the failure of the local Principle of Relativity. What these arguments in fact point to is the distinction between the Reciprocity Principle and the Principle of Relativity.

## 2.2 Conventionalism

The conventionalist approach to spacetime was espoused by Einstein in his early work and by philosophers such as Poincaré (1905), Reichenbach (1927, 1969), Carnap (1937), Schlick (1920), and Grünbaum (1963).

One may detect a variety of conventionalists accounts, ranging from fairly modest claims about theories being underdetermined by the phenomena to radical claims about any axiomatic system amounting to an arbitrary linguistic construct. Poincaré, for example, famously believed that the nature of space gives rise to an underdetermined geometric structure. According to Poincaré, the choice between Euclidean, hyperbolic and spherical geometries is underdetermined by our measurements of spatial relations. The implication is that the axioms of geometry are neither empirical claims – since they are underdetermined by observations – nor are they necessary claims – since alternative axiomatizations of space can be given. According to Poincaré, there is a separate category for propositions that are neither empirical nor a priori or necessarily valid; these would be conventions. However, according to Poincaré the conventional nature of geometry does not extend to other branches of mathematics; the nature of arithmetics does not lend itself to a conventional choice about the axioms. Thus Poincaré’s conventionalism is fairly conservative in its scope.

Another version of conventionalism is neo-Kantian Conventionalism. According to this version, the interpretation of experience requires that our minds impose a rational form on intuitions. Kant believed that the rational form of intuition is necessary, but the neo-Kantian school allowed for those forms to change over time. In his earlier work, published in 1920 and entitled The Theory of Relativity and A priori Knowledge (1920), Reichenbach attempted to reconcile the insights of Kantian epistemology with the lessons of relativity theory. Reichenbach argued that some Kantian principles could be preserved in the light of the new theory, and could illuminate the epistemological nature of the theory; but other principles need to be revised. He argued, for example, that one could give a Kantian account of the relativistic principle that the speed of light provides an upper limit to all physical velocities. He claimed that there being an upper limit to the velocity of causal signals is a consequence of the principle of no action at a distance. The locality of causal action is itself a consequence of Kant’s a priori principle of permanence of substance (Reichenbach, 1920, p. 12). On the other hand, some of Kant’s claims about certain propositions being a priori must be given up in the light of relativity theory, including the absolute nature of time and the Euclidean character of space.

Reichenbach’s earlier work therefore attempts to salvage some of Kant’s claims. However, beyond accepting certain Kantian principles as being valid a priori, and rejecting others as invalid, Reichenbach also revised Kant’s notion of a priori itself. Kant equated a priori judgments with necessary truths. Reichenbach felt that the advent of relativity theory clearly demonstrated that a priori propositions are not necessary. Some principles, like the absolute nature of time, were held to be valid a priori in Newtonian physics, but were then taken to be false in the theory of relativity. However, Reichenbach argued, a certain interpretation of the a priori can be salvaged, as long as a priori judgments are not equated with necessary truths. But knowledge presupposes the cognitive coordination of individual terms of the theory with individual objects of experience. This coordination depends on the definition of the concept of object given by cognition. Without the concept of object, cognition would not be able to interpret intuitions, so that intuitions must have rational form. The philosophical problem is to identify those cognitive coordinations that are unique, so that no contradiction arises between theory and experience (1920, p. 47).

Reichenbach calls the principles of coordination that constitute the concept of the object Axioms of Coordination. Examples for Axioms of Coordination are the axioms of arithmetics with which the concept of a mathematical vector is defined. Without the mathematical theory of vectors, physical forces could not be conceptualized or identified in experience. Another Axiom of Coordination is the principle of genidentity, according to which the trajectory of a particle determines its identity. Euclidean geometry functions as an Axiom of Coordination in Newtonian physics, since it is not possible to identify in experience Newtonian physical objects without assuming Euclidean geometry as valid. But it is no longer an Axiom of Coordination in the General Theory of Relativity. Axioms of Connection, on the other hand, describe empirical connections between terms in the theory.

Axioms of Coordination have an a priori status in the theory, since without these axioms the theory is not able to receive empirical content. However, unlike Kant’s synthetic a priori judgments, these axioms are not necessarily valid for all theories. When new theories are formulated, different Axioms of Coordination are used to bridge between the formal system and experience. Friedman (1991, 1999) has coined the term “relativized a priori” to convey the epistemic status of Axioms of Coordination. The a priori nature of certain propositions arises from the “constitutive” role of principles that bridge between theory and experience. Nevertheless, over time, these principles can be revised. Thus, these principles are only a priori relative to a particular theoretical context. Euclidean geometry, according to Reichenbach, is constitutive in the context of classical mechanics, but only topology is constitutive in the context of the general theory of relativity. When a new principle of coordination is introduced, discarded principles of coordination can be shown to be approximations of newer principles, so that new theories can be shown to be an improvement over the older, less adequate theories.

What distinguishes a principle of coordination from any other empirical proposition, is that one can detect an element of arbitrariness in the principle. “The contribution of reason is not expressed by the fact that the system of coordination contains unchanging elements, but in the fact that arbitrary elements occur in the system” (1920, p. 89). Thus, the contribution of reason is made felt, so to speak, through the existence of conventional systems of representation; all adequate for the representation of experience, but none preferable. According to Reichenbach, the Principle of Relativity, since it allows for a conventional choice of inertial reference frames, demonstrates the contribution of reason to the object of understanding:

The theory of relativity teaches that the four space-time coordinates can be chosen arbitrarily, but that the ten metric function g μν may not be assumed arbitrarily; they have definite values for every choice of coordinates. Through this procedure, the subjective elements of knowledge are eliminated and its objective significance formulated independently of the special principles of coordinates. Just as the invariance with respect to transformations characterizes the objective nature of reality, the structure of reason expresses itself in the arbitrariness of admissible systems. (Reichenbach, 1920, p. 90)

Thus for Reichenbach, relativized a priori propositions introduce conventional, subjective elements to our theories. A convention is required because experience has to be understood via its rational form. Since this rational form is not necessary, one can bring to bear alternative rational forms to the same experience, introducing an element of arbitrariness to our representations.

But Reichenbach did not preserve his neo-Kantian view for long, and was influenced by Schlick to revise his account, thus forming a third kind of conventionalism of a positivist kind. Reichenbach slowly came to believe that Kant’s notion of synthetic a priori no longer holds if experience is to determine which principles of coordination are acceptable. If the theory of relativity is empirically superior to Newtonian physics, one can no longer treat the discarded principles of Newtonian physics, and the accepted principles of the Theory of Relativity, as equally valid. Thus, they could no longer be taken as a priori in good faith. The consequence is a retreat to a Humean-like fork: propositions must strictly be divided into analytic a priori, and synthetic a posteriori propositions. Reichenbach was also influenced by Schlick, who objected to the notion of the object of understanding being constituted by reason and to the neo-Kantian form of idealism. According to Schlick (1920), one may think of the totality of physical facts as objective claims that are distinguished from subjective experiences. Physics in fact describes mind-independent reality. Nevertheless, it is also possible to indicate the same set of facts by means of various systems of judgment (Schlick, 1920, p. 86). Thus, one can articulate various theories expressing the same set of facts, such that the choice between these theories is a matter of convention. While there is an indefinite number of conventional representations, one may use simplicity as a criterion for selecting between the various systems of judgments, but there is nothing in reality to tie us down to a specific representation as the correct one.

Influenced by Schlick’s positivist-conventionalism, Reichenbach (1927, 1969) revised his account of the a priori elements of theory. The cleavage between analytic a priori propositions and synthetic a posteriori claims demand that Reichenbach’s Axioms of Coordination be reevaluated. In a work published in 1924, entitled Axiomatization of the Theory of Relativity, Reichenbach introduced his new account of principles of coordination. He distinguished between two aspects of coordination. First, there are the linguistic conventions and definitions of the theory. In this, Reichenbach and Schlick were following Poincaré’s and Hilbert’s notions of implicit definition. According to Poincaré (1905, p. 92), for example, the Law of Inertia is not necessarily true. However, it is not an empirical claim, either. The notion that a force-free body will move with uniform rectilinear motion is partially determined by the meaning of “force” (and vice versa). One may associate a force with a change in position, or a change in acceleration; both would allow one to articulate alternative laws of inertia. Nothing in the observations will imposes Newton’s Law of Inertia. But once the Law of Inertia defines the force-free state of the particle as the uniform motion of an object, the Law of Inertia acquires the status of necessary truth within Newton’s axiomatic system. Similar views about the Law of Inertia were held by Reichenbach (1927, p. 116) and Hanson (1965).

However, the more significant transition in Reichenbach’s thinking is the revision of his account of coordination between theory and experience. Reichenbach is still arguing that for a mathematical theory to be applied to experience, individual terms of the theory must be coordinated with individuals of experience. However, now the coordination is not done from within the theory, and the concept of object is no longer determined by reason. Instead, the coordination between individual terms and individuals of experience is carried out via some experimental practice. The coordination of theory to experience is performed via what physicists do, not by the rational form of their intuitions. Thus Reichenbach is no longer speaking of Axioms of Coordination, but instead referring to them as coordinative definitions. In Reichenbach (1927), he articulates the notion of coordinative definition:

The mathematical definition is a conceptual definition, that is, it clarifies the meaning of a concept by means of other concepts. The physical definition takes the meaning of the concept for granted and coordinates to it a physical thing; it is a coordinative definition. Physical definitions, therefore, consist in the coordination of a mathematical definition to “a piece of reality”; one might call them real definitions. The concept of a unit of length is a mathematical one; it asserts that a certain particular interval is to serve as a [standard of] comparison for all other intervals. From this nothing can be inferred, however, as to which physical interval is to serve as the unit of length. The latter is first accomplished by the coordinative definition which designates the Paris standard meter as the unit of length. In this physical definition, the mathematical definition of the concept is presupposed. (Reichenbach, 1969, p. 8)

According to Reichenbach, complementing the linguistic conventions are definitions that coordinate between linguistic terms and individual objects of experience. In the case of physical geometry, concepts such as “unit of length” and “unit of time” have to be coordinated with physical objects such as measuring rods and clocks used to generate the relevant measures of length and duration. It is not that reason constructs for us the object of experience, rather, the objects are simply pointed out ostensibly. A coordinate system therefore consists of a set of physical measuring rods and clocks that are relatively at rest, and functions in our scientific practice as a coordinative definition. The implication of Reichenbach’s approach is that geometric axioms do not represent a single theory but a family of models. The symbols of the theory can be interpreted only after the coordinative definition is made (i.e., a coordinate system is selected) and the propositions of physical geometry gain physical (i.e., empirical) content.

Reichenbach’s transition from neo-Kantian to Positivist Conventionalism carries important epistemological consequences. In the neo-Kantian account, the Principle of Relativity marks the constitutive role of reason in constructing the concept of object. In the positivist account, the Principle of Relativity describes an isomorphism between the various models of the theory. In the positivist account, each model of the theory relies on a different coordinative definition. There is a crucial difference between an interpreted formal theory, and its interpretation after the individual terms are taken to represent specific objects. The formal theory is only a symbolic, syntactic structure, lacking any empirical or physical content. The interpreted theory has empirical or physical content in virtue of the objects to which the formal theory is referring. That is why logical positivists distinguish between formal and physical geometry. Formal geometry is a mathematical structure; the propositions of physical geometry are about real, physical objects.

The upshot of logical positivism is that two different interpretations of a formal theory include propositions that are not directly comparable. For example, one could conceive of Hilbert’s axiomatization of Euclidean geometry as a formal theory that is then given different interpretations. In one interpretation of the theory, the notion of a “point” could be taken to represent a point in physical space, and the notion of a “line” could be taken to represent a line in physical space. But the formal terms could be interpreted differently. For example, the notion of a “point” could be taken to represent a line in physical space, and the notion of a “line” could be taken to represent a point in physical space, leading to what is known as projective geometry. The consequence is that claims made within the context of one interpretation are not comparable or are not commensurable with claims made in the context of another interpretation, despite them being interpretations of a single theory. The physical content of these claims crucially depends on the coordinative definition, or on the interpretation of the formal system.

The logical positivists argue that Einstein’s restricted Principle of Relativity is a prime and central example of their epistemology. Einstein’s 1905 paper seems to follow conventionalist epistemology. The two postulates of the theory of relativity, the Principle of Relativity and the Light Postulate, provide the axioms of the theory. From these axioms one can deduce the theorems of relativity. In addition to the pure axiomatic system, the applied theory uses coordinate systems to bridge between the abstract conceptual system and physical experiences. Each inertial reference frame provides a particular interpretation of the theory of relativity. The frame coordinates between abstract concepts of “unit of length” and “unit of duration” and the physical objects used as standards of length and duration in experiments (i.e., rigid measuring rods and clocks).

According to the Principle of Relativity, whether a law is articulated relative to a coordinate system that is at rest, or whether it is articulated relative to a system that moves with uniform rectilinear motion relative to the first coordinate system, it assumes the exact same form. If one is reading into the Principle of Relativity a positivist epistemology, one is asserting the equivalence between different models of the theory. Each coordinate system is in effect a different interpretation of the formal system. If a certain coordinate system K is given, the term “unit of length” is coordinated with a unit of length marked on the measuring rods of K. One does not have to have the actual rods placed in the coordinate system, but at least in principle, for each unit of length in the relativistic geometry there should be a rod corresponding to it. Similarly, the term “unit of time” in the formal theory is coordinated with a period of a clock. The set of rods and clocks that are at relative rest comprise the coordinate system.

However, the logical positivist interpretation for the Principle of Relativity is untenable. Assume that one has at his disposal two coordinate systems K and K . One can choose to interpret the formal geometry by taking its terms to refer to rods and clocks in K, or one can take them to refer to rods and clocks in K . In each case, a different interpretation of the formal theory is provided. If the positivists take seriously the notion that physical geometry receives content only once the coordinative definition is made, then different coordinate systems in effect introduce alternative interpretations of the formal theory.

Given such a positivist gloss, the Principle of Relativity ceases to be a theoretical proposition. The principle asserts that different interpretations, stemming from different coordinate systems, produce the same dynamic laws. This type of equivalence is not a theoretical equivalence, because the different interpretations are not directly comparable. At most one can say that the different models are empirically equivalent, in that the same measurements will be predicted by each model. The upshot is that the Principle of Relativity is not itself an empirical claim within a particular model, but a meta-theoretical principle for constructing models. This implication of the positivist epistemology is not always emphasized or even recognized by positivist philosophers, since it is always stressed that the origin of the Principle of Relativity is empirical. Einstein argues that no physical theory ever makes reference to anything but relative velocities, which seems to provide the principle with great empirical support. But given the positivist reading, it may be that the Principle of Relativity received much empirical support, but the Principle itself is not a theoretical or a directly empirical claim. The Principle of Relativity asserts an isomorphism between different models of the theory, but is not itself part of the theory. Carnap (1937) seems to recognize the implication and treats the Principle of Relativity itself as a convention about the syntactical rules of the language of physics (see p. 328).

The problem with the positivist-conventionalist account of the Principle of Relativity is that the principle seems to be justified by empirical and theoretical considerations, but the principle itself cannot have any physical or empirical content. There seems to be an incongruity between the justification the Principle of Relativity receives from experiments and from theoretical considerations (these include, for example, the unification of the electric and magnetic fields), and the epistemic role the principle receives within the positivist account. If it is possible for the Principle of Relativity to receive empirical and theoretical justification, it seems odd to assert that the principle itself has no empirical or physical content. If the principle has no empirical and physical content, then all the justification it receives is irrelevant to its stipulated truth. If the Principle of Relativity is a mere convention, a meta-theoretical principle for the construction of models, then no justification for the principle is necessary.

The positivist-conventionalist seems untenable when nature of equivalence between models of the theory is considered. While the different models produce isomorphic structures, the equivalence between models is strictly empirical, i.e., there is no underlying theory that unifies the models into one overall interpreted theory. But if all models of the theory are isomorphic to one another, there is reason to suspect that they consist of different representations of the same world state. It would just be an enormous coincidence to have such a deep symmetry governing models which are based on conventional coordinative definitions. Friedman (1983, pp. 277–94) argues that the replacement of empirically equivalent models with equivalent representations is the process by which the vocabulary of a theory is revised in order to decrease the conventional and arbitrary elements of a theory. One may think of the Minkowski’s approach to spacetime as an attempt to do just that, i.e., to provide a four-dimensional geometric structure that unifies the various models of relativistic spacetime into one overall theory.

A similar development concerned the conventionalist interpretation of the Principle of Equivalence in the context of the General Theory of Relativity. At first, conventionalists viewed Einstein’s Principle of Equivalence as proof of the conventionalist thesis. According to Einstein, the equality of inertial and gravitational mass implies the equivalence between two descriptions; one is a frame of reference at rest with a uniform gravitational force, the other is a frame of reference accelerating uniformly but experiencing no gravitational force. Conventionalists argue that the choice between the two descriptions is conventional. However, as later interpreters have realized, the equivalence is only valid locally, and Einstein’s incorporation of the gravitational force into the spacetime metric does not allow the global elimination of gravitational effects. Thus, in Einstein’s theory there is no longer a conventional choice about the nature of gravitation, and the gravitational field could not be dispensed with throughout spacetime or be treated as an eliminable universal force.

While the conventionalist approach to spacetime is mostly out of favor today, the program for axiomatizing the theory of relativity is still in full force.9 According to this approach, one can gain insight to the foundations of a physical theory by reconstructing it within first order logic. Those who endorse this approach do not always insist on the conventionalist nature of the axioms of spacetime theory (by putting emphasis, for example, on the empirical origins of the Light Postulate). Neither do these modern revivals discuss the epistemological nature of different models of the theory. But they still view spacetime theory as a collection of empirically equivalent models. In Madarász et al. (2007), the project of constructing an first order logic axiomatization of relativity consists of constructing observable motions. A model of this theory includes a universe of bodies B and sets of four quantities Q describing the spatiotemporal location of bodies. But for the spatiotemporal locations to make sense, they must “belong” to a certain structure – the inertial system of an “observer,” which is a particular kind of body. Thus, each model of the theory includes a “worldview,” which is a six-place relation $$W(o,b,x,y,z,t)$$ stating that a body b has a spatial location $$x,t,z$$ at time t in o’s inertial reference frame. There are also special axioms fixing the motions of these observers. This approach to the axiomatization of relativity leads to various empirically equivalent models, depending on the “observer” or the inertial frame chosen as reference. The axiomatization of spacetime theory may be useful for gaining insight into the foundations of the theory.10

But such an axiomatization of spacetime remains an unsatisfying view of the nature of spacetime. As long as our axiomatization leads to empirically equivalent models, where each model is defined relative to a coordinate system or an “observer,” the Principle of Relativity itself becomes a non-theoretical principle about models – asserting the equivalence between them. It may be that spacetime theory is inherently limited, and that it is simply not possible to articulate a spacetime theory that eliminates observers or coordinates systems from the foundations of spacetime theory. However, if there exists an interpretation of spacetime that is able to unify the empirically equivalent models, the positivist-conventionalist interpretation of the restricted Principle of Relativity should be abandoned.11

Positivist conventionalism was abandoned in the second half of the twentieth century, as a result of many factors. Some include post-positivist critiques by Quine and Kuhn, other factors perhaps include sociological ones. In any case it is apparent that Einstein’s Principle of Relativity is not best understood by the positivist-conventionalist view. Some commentators, most notably Michael Friedman, recommend a return to the neo-Kantian views of early Reichenbach. But then it is not clear how one can retain the synthetic a priori nature of principles that are demonstrated to be empirically inadequate. Others attempt to revive the axiomatized approach to the theory of relativity, taking each observer to be constituting its own “worldview.” These commentators seem to be unaware of the epistemological weakness faced by the early conventionalists of the positivist school who took each inertial reference frame to be a different model of the formal theory.

## 2.3 The Geometric Approach to Spacetime

The conventionalist account of spacetime theory has by and large been superseded by the geometric approach to spacetime (see, e.g., Minkowski, 1952; Earman and Friedman, 1973; Nerlich, 1979; Mellor, 1980; Healey, 1995; Balashov and Janssen, 2003; Baker, 2005). One can trace the origins of this approach to Minkowski’s geometric formulation of the Special Theory of Relativity. The world, according to Minkowski, is a collection of worldpoints designated with a system of values $$x,y,z,t$$, where $$x,y,z$$ are spatial coordinates and t is a temporal coordinate (Minkowski, 1952, p. 76). A body’s motion through this world is described as a worldline; it is a set of worldpoints in which infinitesimal variations $$dx, dy, dz$$ dt. Minkowski erects an analogy between the transformation group G c that governs the spacetime structure of STR and the Euclidean group governing Euclidean space. The transformation group G c contains the Euclidean group of rotations and translations in the spatial dimensions (i.e., the transformations leaving $$x^2+y^2+z^2$$ invariant). But the group G c also includes velocity boosts that leave invariant the spacetime interval $$ds^2=c^2t^2-x^2-y^2-z^2$$. Minkowski argues that invariance in relation to G c is a general principle governing natural phenomena, and raises this property to the status of world-postulate.

Minkowksi concludes that the main lesson of relativity is that space and time can no longer be taken to exist independently:

We should then have in the world no longer space, but an infinite number of spaces, analogously as there are in three-dimensional space an infinite number of planes. Three dimensional geometry becomes a chapter in four-dimensional physics. Now you know why I said at the outset that space and time are to fade away into shadows, and only a world in itself will subsist. (Minkowski, 1952, p. 79)

Thus Minkowski believes that underlying the kinematics of relativity is an objective four-dimentional world with spacetime points and objective spacetime intervals defined between them. The Principle of Relativity is therefore not a phenomenological principle asserting an isomorphism between different models of the theory, but a geometric symmetry governing the underlying four-dimensional spacetime. Different inertial reference frames produce alternative representations of the same world-state.

Minkowski argues that his account of relativistic kinematics is more intelligible than that of Lorentz’s dynamic account of length contraction. Imagine a rod at rest in $$x, y, z, t$$. Such a rod looks like a band (see Fig. 2.1). The length of the rod PP is l. If one looks at a rod moving with uniform rectilinear motion relative to the original, its band will look slanted in the Minkowski diagram. The length of this rod is also l, but this length is measured relative to a rotated axes $$x', y, z, t'$$ so that $$Q'Q'=l$$. Since the length of this moving rod in $$x, y, z, t$$ is the cross-section along the axes of $$x, y, z, t$$, its length is $$l/\gamma$$ in this system. Fig. 2.1 Length contraction in Minkowski’s geometric approach

According to Minkowski, there is no dynamic account of length contraction and time dilation; these phenomena result from rods and clocks conforming to a four-dimensional underlying geometry. Each inertial reference frame, because of its orientation in spacetime, introduces different standards of length and duration, thereby producing different representations of the same spatiotemporal intervals. Moving rods appear contracted in the rest frame, since one is not taking the set of events that are considered simultaneous along the moving reference frame as defining a unit of length in the rest frame. A cross-section of non-simultaneous events produced by the moving rod is shorter when measured in the rest frame.

In Minkowski’s geometric approach, the lessons of relativity appear entirely different than Einstein’s conventionalist-leaning remarks. According to this new approach, space and time have to be fused together to form a four-dimensional structure. Understanding that space is only a substructure of spacetime

… is indispensable for the true understanding of the group G c , and when [this further step] has been taken, the word relativity-postulate for the requirement of an invariance with the group G c seems to me very feeble. Since the postulate comes to mean that only the four-dimensional world in space and time is given by the phenomena, but that the projection in space and in time may still be taken with a certain degree of freedom, I prefer to call it the postulate of the absolute world (or briefly, the world-postulate). (Minkowski, 1952, p. 83)

Thus in Minkowski’s account, the Principle of Relativity is not a phenomenological postulate of the theory, but a symmetry inherent in the geometric structure of a four-dimensional world. Rather than thinking of the Principle of Relativity as requiring an arbitrary coordinative definition, Minkowski thinks of it as a postulate of the absolute world, i.e., as a degree of freedom implicit in selecting arbitrary systems of reference used to construct representations of an objective world.

The view, which takes spacetime to describe an objective spacetime structure, has a philosophical advantage over conventionalist accounts. In the conventionalist account, different models of the theory are considered empirically equivalent. But in the geometric account, different representations of the spacetime are theoretically equivalent.

Minkowski’s view of relativity came to dominate relativity textbooks and philosophical explications of relativity theory. It also came to dominate Einstein’s own thinking while working on the General Theory of Relativity.

The trouble with the geometric approach is that it tempts us to think of spacetime as a material field that causally influences the behavior of bodies. Momentum and energy conservation laws assert that closed systems move along geodesics of the spacetime. However, what causes free particles and light rays to follow the geodesics of the spacetime? Since the path of a free particles is not caused by any other material body, the image of a four-dimensional manifold of events leads commentators to think of spacetime as the cause or the “origin” of kinematics. The view of spacetime as guiding inertial motions has its historical origins in Mach’s critique of Newton’s absolute space in The Science of Mechanics. Mach was concerned to show that Newton’s bucket experiment does not necessarily rule out every possible definition of relative motion as responsible for inertial effects (Mach, 1893, p. 300). But his other argument against Newton’s absolute space was that Newton could have posited the existence of a material medium, like the ether, which lies throughout space and directs bodies in their inertial motions (Mach, 1893, p. 282). Newton’s concept of absolute space is a metaphysical notion, precisely because it is a physical entity that has no causal influence on material bodies.

Einstein came to think of spacetime in much the same way as Mach proposes. According to Einstein, spacetime should be thought of as an entity that acts on bodies as this structure determines the inertial behavior of bodies. The problem with Newton’s mechanics and STR, according to Einstein, is that spacetime seems to act on matter without being acted upon in return. This “violation” of the principle of action equals reaction is corrected in The General Theory of Relativity where matter is said to be acting on spacetime by curving it. Thus, spacetime can be compared to an ether that causally influences the behavior of bodies:

The inertia-producing property of this ether [Newtonian spacetime], in accordance with classical mechanics, is precisely not to be influenced, either by the configuration of matter, or by anything else. For this reason we may call it “absolute”. That something real has to be conceived as the cause for the preference of an inertial system over a noninertial system is a fact that physicists have only come to understand in recent years … Also, following the special theory of relativity, the ether was absolute, because its influence on inertia and light propagation was thought to be independent of physical influences of any kind … The ether of the general theory of relativity differs from that of classical mechanics or the special theory of relativity respectively, insofar as it is not “absolute”, but is determined in its locally variable properties by ponderable matter. (Einstein, 1921, pp. 55–56)

Thus the difference between absolute and relative conceptions of spacetime, according to Einstein, is whether the spacetime structure is “mutable” or not.

Following Mach and Einstein, philosophers understood arguments for absolute space and time as an inference from inertial phenomena to the best explanation thereof. One first begins with inertial effects as observations in need of explanation. One next argues that no relative motion could provide a reasonable explanation for these effects. Then it is supposed that a spacetime background is needed relative to which inertial motions are defined. It is then argued, that without a spacetime structure, it is not possible to explain the motion of force-free particles.12 Finally, the argument concludes that because spacetime is indispensable for explaining physical phenomena, it is also real, and should actually be taken to have causal influence over physical bodies, in directing force-free particles and lights rays along the geodesics of the spacetime.

The reification of spacetime suggests that free particles could have failed to move along a geodesic. So the truth of the Law of Inertia is contingent and is analogous to any observed law of nature. Thus, the geometric interpretation of spacetime helps commentators replace the conventionalist account of the Law of Inertia with what appears like an empiricist’s account. Earman and Friedman (1973) argue that inertial reference frames are reducible to the independent structures of spacetime.13 If one has good reasons to think that the manifold and the spacetime structures are real, then one should treat the Law of Inertia as a directly verifiable prediction.

However, there are many drawbacks to reifying the spacetime structure and no benefits. Taking the spacetime itself as “explaining” the inertial behavior of bodies leads interpreters to say that the spacetime causally influences the behavior of isolated systems. Spacetime acts on bodies in that it “guides” them to move through geodesic lines. A useful way to think of these geodesic lines is to think of them as analogous to ruts that make the passage through spacetime easier. But this causal account of spacetime seems to stretch too far our common intuitions about explanation. One ought to feel uncomfortable when spatiotemporal points are attributed causal powers. For one, an entity which has causal powers seems to require some substance-like existence. It seems natural to think that causal powers must be inherent in some substratum. But spatiotemporal points by definition are not the kinds of things that persist. So in what sense do they have substance? Do we not need to think of an entity with substance as persisting? For these reasons, even Newton’s account of absolute space specifically precludes spatial points from being substance-like as Newton takes for granted that they are without causal efficacy.14

A typical example of an argument that reifies space is given by Nerlich:

Without the affine structure there is nothing to determine how the [free] particle trajectory should lie. It has no antennae to tell it where other objects are, even if there were other objects … It is because space-time has a certain shape that world lines lie as they do. (Nerlich, 1976, p. 264)

Nerlich’s imagery suggests that because spacetime has a certain shape, the trajectories receive a certain structure as a result. The implication that the shape of spacetime is explanatory in some way, i.e., it gives rise to the way in which free-particles, clocks and rods behave.15

However, the causal language describing the relation between spacetime and matter violates certain ingrained intuitions about causal explanations. Brown puts the objection as follows:

If free particles have no antennae, then they have no space-time feelers either. How are we to understand the coupling between the particles and the postulated geometrical space-time structure …? In what sense then is the postulation of absolute space-time doing more explanatory work than Molière’s famous dormative virtue in opium? (Brown, 2005, p. 24)

Brown’s worry is therefore with the cogency of taking spacetime to have causal powers analogous to the causal powers a physical substance or field may have. If the assumption – that spacetime has a causal efficacy – is cogent, one needs to consider whether the assumption provides a genuine explanation for inertial motion. To do so, one may analyze the causal relation between spacetime and matter with the help of counnterfactuals. Assume that the state ϕ of A causes the state θ of B. The causal relation between A and B implies the counterfactual statement “if A had not existed with property ϕ, then B would not have existed with property θ.” It is tempting to describe the relation between spacetime and matter as causal since it supports a similar counterfactual. If spacetime did not have certain properties (e.g., a pseudo-Riemannian structure, or a curvature), rigid rods and clocks would not behave as they do, and trajectories of free particles would not have been as they are.

One can parse Brown’s objection as follows. It is not always the case that such a counterfactual underwrites a causal relation. In the case where A having property ϕ states the condition for the possibility of B having property θ, the counterfactual does not describe causation between two independently existing things. For example, suppose we say that the Nobel Prize committee had awarded Einstein the Nobel Prize. One may say that a counterfactual claim is supported – had the Nobel Prize committee not given Einstein the title, he would not have been a Nobel Prize laureate. However, the act of giving the prize is not the cause of Einstein’s getting it. The act of giving the prize is part of the conditions for the possibility of earning the prize. The act of giving the prize and the event of earning it are one and the same; they are two different descriptions of the same thing. Similarly, that a counterfactual claim connects spacetime properties with material properties does not imply that an efficient causal relation is involved. Since spacetime defines the very distinction between trajectories of free particles and trajectories influenced by some dynamic force, the spacetime structure provides the condition for the possibility of rods, clocks and free particles having the properties that they have. Thus, the counterfactual does not underwrite a genuine efficient causal relation. There is no reason to suppose that spacetime causes free particles to move as they do in analogy to a billiard ball which causes another billiard ball to move. And since the counterfactual does not underwrite a genuine causal relation, the reification of spacetime is not a genuine explanation of inertial motions.

Now the response to such an argument may be that a geometric mode of explanation is non-causal, but may still be an independent means of explaining the behavior of bodies (Nerlich, 1979). But it then becomes a mystery why one ought to posit that the spacetime manifold and its metric should be thought of as a field existing independently and alongside matter fields. A geometric mode of explanation, at the end of the day, merely attempts to describe how bodies behave, and it is unnecessary to think that points exist independently of bodies and relations between them. Another reaction to the argument questioning the causal efficacy of spacetime might be that the mere existence of a counterfactual connecting properties of spacetime with properties of matter may underwrite the causal relation between spacetime and matter (Mellor, 1980). But if this line of argument is taken, then spacetime is explanatory in a way that an Aristotelian formal cause is explanatory of particular bodies. Assume that the explanation amounted to the counterfactual, “if spacetime M had not existed with property ϕ, then a free particle would not have existed with property θ.” Since the spacetime structure provides the means for describing θ, then spacetime is merely the articulation of the shape that a free particle has, in much the same way that a formal cause is merely a description of the form that a certain Aristotelian substance has. A square figure in Euclidean space has the form of a square, but in describing the square one is not giving an independently existing structure which is causally responsible for the existence of the square, one is describing the spatial form of the object.

It is worth noting another difficulty with the geometric approach. In the context of the General Theory of Relativity, the reification of the manifold gives rise to the Hole problem (Earman and Norton, 1987). The problem consists of there existing indistinguishable dynamic models of the theory, $$\left <M,g,T\right>$$ and $$\left <M,h*g,h*T\right>$$, where M is the spacetime manifold, g is the metric, T is the stress-energy tensor, and h is a diffeomorphism on the manifold. While indistinguishable from a dynamic perspective, these models differ in their mapping of empty regions of the manifold to the metrics g and h*g. This situation leads to indeterminism, since a manifold realist cannot predict which trajectory in the manifold will be realized by an object moving into the hole. The upshot seems to be that points in the manifold cannot retain their identity independently of the dynamic objects of the theory (Hoefer, 1996).

The geometric approach has a clear advantage over the conventionalist approach, in that the restricted Principle of Relativity can be viewed as stemming from a four-dimensional spacetime symmetry. The geometric approach was also instrumental in the construction of the General Theory of Relativity, and facilitated the merging of the gravitational field with the spacetime metric. However, there is a clear lacuna in the attempts to conceive of spacetime as another field lying alongside matter fields. The coupling between spacetime and matter becomes obscure if it is compared to the coupling between two material fields, and attempts to attribute the spacetime field efficient causal powers come close to being unintelligible. Thus, there is a steady stream of voices that attempt to argue against the official doctrine, which the geometric interpretation has become. Although it is difficult to overcome the appeal of the geometric approach, since Minkowski’s analogy between Euclidean and four-dimensional spacetime symmetries has strong intuitive appeal.

## 2.4 The Dynamic Approach to Spacetime

The difficulty in articulating the nature of the causal relation between spacetime and matter compels some to doubt the causal roles attributed to spacetime (see Stein, 1967; DiSalle, 1995; Brown, 2005; Brown and Pooley, 2006). Thus some commentators take the geometric interpretation to be untenable, and offer a program for reducing spacetime theory to dynamic laws. According to one articulation of dynamical relationalism (developed by Teller, 1987; Dieks, 2001a, b), one ought to think of spacetime as a collection of physical quantities actualized by material bodies, in a manner analogous to that of inherent properties such as mass or charge. The range of possible spacetime positions and velocities of each body is determined through the dynamical laws which characterize the theory. When a physical system actualizes a certain dynamic theory, it necessarily actualizes a set of possible trajectories and their relations. Thus, if the Hamiltonian of a system comprising of two particles is $$H=\frac{p^2_1}{2m}+\frac{p^2_2}{2m}-V(q_1,q_2)$$, the allowable coordinates and their variation would be those that satisfy the dynamics described by the Hamiltonian. The trajectories of the particles are physical possibilities that must conform to the dynamic laws. The spacetime symmetry, i.e., the equivalence between inertial reference frames, is determined by the symmetries of the dynamic laws.

However, there is a metaphysical difficulty with Teller’s and Dieks’ version of dynamical relationalism, which is made clear by the analogy they erect between spacetime quantities and other physical quantities. The plausibility of Teller’s dynamical relationalism turns on whether it is reasonable to treat spatiotemporal quantities as physical quantities. Is being at location x and then being at location x assimilable to the instantiation in the body of the physical quantity x, and then the instantiation of physical quantity x ? Presumably, whether a body possesses location x depends on the relation between this location and all the other spatiotemporal points. When one attributes a mass parameter to a body, one seems to be able to do so without any reference to other bodies. Thus, it is possible to conceive of a body that exists in empty space with a mass parameter, without assuming that any other body exists.16 That a body instantiates a certain mass parameter can be conceived independently of mass parameters actualized by other bodies. A spacetime position, in contradistinction, is necessarily relational. Treating the notion of spatiotemporal position as a monadic property therefore seems unintelligible.

Teller (1991) later clarified his account of spacetime positions as physical quantities, and reduced them to potential relations to actual bodies, endorsing a kind of liberal relationalism (to be distinguished from narrow relationalism, which reduces spacetime to actual relations between bodies). In this account, indistinguishable Leibnizian models of spatiotemporal relations are conceptualized as monadic physical quantities, whose values depend on the values of quantities instantiated by other bodies. Teller attempts to make this notion plausible by comparing spatiotemporal locations to the values of masses. The particular mass value a body instantiates is not entirely independent of mass values instantiated by other bodies. The particular mass value instantiated by the body would arbitrarily change if all masses and forces are scaled in the same proportions. However, it is not clear that this elaboration of dynamical relationalism renders it more plausible. It is not only that the value of the coordinate of a spatiotemporal point is determined in relation to other points. The identity of the point is also determined in relation to other points. There is some connection between the identity of objects and the identity of the points they occupy. Some physical models do not allow for two bodies to be in the same place. In these models, the location of the body identifies the body for us. The location of the body helps differentiate between this particle-mass over here from that particle-mass over there. One might explain the fact that two bodies cannot reside in the same place by positing a spatial exclusion principle, in analogy with Pauli’s Exclusion Principle, according to which two particles of spin 1/2 cannot be in the same state. According to the spatial exclusion principle, two particles cannot instantiate the same spatial quantity. However, it seems clear that the analogy between coordinates and physical quantities is stretched here. It is perfectly reasonable to take two bodies as having the same mass value, but mass values do not determine the identity of the body. However, the identity of spatiotemporal point is crucial for determining the identity of objects that occupy it, so that spatiotemporal positions do not seem to function like physical quantities. While the disanalogy between spacetime points and other physical quantities does not prove dynamical relationalism false, it does weaken the force of the analogy between a spatiotemporal point and other physical quantities.

Another version of dynamical relationalism is articulated in a stimulating book by Brown (2005). Brown argues that relativity theory was at first conceived by Einstein as a Principle Theory. Einstein compared the principles of STR to classical thermodynamics theory in that these principles are based on broad phenomenological principles. Thus, while kinematic effects of length contraction and time dilation are provided as consequences of the theory, these effects are merely described by the theory without any proper explanation. A fuller account of these kinematic effects would require a constructive theory that captures the behavior of macroscopic bodies such as clocks and rods. A constructive theory of spacetime would appeal to fundamental dynamic theories that explain the structure of composite material systems. According to this neo-Lorentzian strategy, spacetime effects are implicit in the dynamic theories explaining the fundamental structure of matter.

One of the difficulties with Brown’s strategy is that kinematical effects seem not to describe composite physical systems such as clocks and rods, but physical processes themselves. Thus, when one measures the half life of a decaying atom, its half life is dilated when it moves relative to the clocks and rods in the lab frame. But this dilation is a property of the decaying process, and a constructive account of clocks and rods would be irrelevant to the delay ascribed to the decaying process. Moreover, the dilation effect does not appear related to the specific dynamic laws governing the process of decay and the structure of the decaying atom, but a universal property of all processes taking place within time. All dynamic laws appear to have the general symmetries of relativistic spacetime, and so it seems as if one needs to find a unifying account for these effects that does not rely on the specific dynamic details. Relativistic effects are not a product of the structure of matter or the composite devices one uses for measuring time and length, but of time itself. This is the main reason the Lorentzian strategy for explaining relativistic effects seems beside the point, and that a direct account of spatiotemporal relations is needed independently of any dynamic theory explaining the structure of matter.

A more serious difficulty with both versions of dynamical relationalism is that they treat the Principle of Relativity and the symmetries of spacetime as brute, “accidental” features of the underlying dynamics. Brown readily admits as much:

In the dynamical approach to length contraction and time dilation that was outlined in the previous chapter, the Lorentz covariance of all the fundamental laws of physics is an unexplained, brute fact. This, in and of itself, does not count against the approach: all explanation must stop somewhere. What is required if the so-called space-time interpretation is to win out over this dynamical approach is that it offer a genuine explanation of universal Lorentz covariance. This is what is disputed. Talk of Lorentz covariance “reflecting the structure of space-time posted by the theory” and of “tracing the invariance to a common origin” needs to be fleshed out if we are to be given a genuine explanation here, something akin to the explanation of inertia in general relativity. Otherwise we simply have yet another analogue of Molière’s dormative virtue. (Brown, 2005, p. 143)

Brown therefore thinks that explanation ends exactly at the point where one finds an astounding symmetry governing all known dynamic laws. But the Principle of Relativity seems to beg an independent explanation since it would be a miraculous accident if it just happened that all dynamic laws are Lorentz-covariant. Why is it that future laws which are yet to be discovered are expected to have the property of being Lorentz-covariant? Either this expectation is unfounded or one has to find an explanation – a property that all dynamic laws share independently of their specific form.17 On the other hand, Brown is correct to doubt the substantivalist account of spacetime as giving a proper explanation. It is not clear why taking spacetime points to be real explains why the laws reflect the same symmetries as the underlying spacetime.18

## 2.5 Conclusion

The conventionalist account of spacetime dominated the philosophical literature initially. However, it soon gave way to the geometric interpretation of spacetime, which is philosophically superior. The geometric interpretation was inspired by Minkowski’s geometrization of Einstein’s theory and his description of relativistic effects as resulting from the structure of a four-dimensional spacetime manifold. If spacetime is taken to be a four-dimensional spacetime manifold, and various inertial reference frames are taken to be mere cross-sections of the same spacetime, then one has an intuitive grasp of what it means to have different representations of the same spatiotemporal structure, rather than empirically equivalent models. However, the geometric interpretation of spacetime, while providing an improvement over the conventionalist account, carries its own interpretive difficulties, since it compels commentators to “breathe life” into the shadowy spacetime and render it substance-like. Reifying spacetime leads commentators to the notion that the spacetime manifold is an independent physical entity, which results in dubious metaphors that credit the manifold with efficient causal powers. The efficient causal metaphors seem inappropriate, since they lend spacetime the appearance of a physical entity analogous to other material bodies. Spacetime provides the framework for interpreting causal relations, so it seems incongruous to take spacetime itself as an entity that has independent causal powers. Thus, while the geometric approach is dominant, there are critics of the geometric approach who argue that the kinematic effects of spacetime are implicit in the dynamic laws. However, it is not clear why the various dynamic laws conform to the same spacetime symmetries. It is difficult to see how various dynamic laws do not receive these symmetries from an underlying spacetime structure that is somehow “responsible” for these symmetries.

Einstein’s interpretation of spacetime shifted throughout his career, reflecting perhaps the appeal of each approach. His early work is couched in the conventionalist approach. His treatment of relations of simultaneity as a convention arising from the arbitrary choice of reference frame is explicated with the help of conventionalist epistemology. Initially, Einstein’s reaction to Minkowski’s geometrization of spacetime was not enthusiastic, but he later embraced Minkowki’s approach and made innovative use of a geometrized spacetime in the development of the General Theory of Relativity, incorporating the gravitational field into the curvature of spacetime. Finally, while he realized that spacetime was in fact treated by him as some kind of field or as some kind of ether, he also recognized the threat posed by the claim that clocks and rods are somehow determined by the geometry of spacetime, and flirted with a dynamic interpretation of spacetime.

An assessment of the three main interpretations of spactime suggests that none of these interpretations is fully satisfying. The difficulty with the conventionalist interpretation is that it permits a range of empirically equivalent models without seeking a theoretical framework for viewing these models as equivalent representations of the same world-state. The difficulty with the geometric account is that it lends commentators the impression that the spacetime manifold exists independently of material processes and that it causally interacts with material bodies. Finally, the difficulty with the dynamic interpretation is that it fails to recognize the unifying role of spacetime. In the next chapter, I introduce an alternative to these three approaches, that bears some similarity to both the geometric and the dynamic approaches, but is also distinct from both. The approach will attempt to provide a theoretical way for unifying the various inertial reference frames into one geometric theory, without assuming the independent existence of spacetime or an efficient causal relation between spacetime and matter.

## Footnotes

1. 1.

This interpretation of the General Principle of Relativity was later contested by many physicists and philosophers, since the General Principle of Relativity is violated by some theories of spacetime that are nevertheless invariant under general coordinate transformations. I shall not pursue this interpretive problem here.

2. 2.

A tacit assumption is that if rods in one coordinate system are boosted until they moved with uniform rectilinear motion relative to the first system, they would represent the same length in the new coordinate system. Similarly, the implicit assumption is that boosted clocks would also represent the same unit of time in the moving frame. See Brown (2005, p. 81) and footnote 41 for supporting quotations.

3. 3.

In the following I replace Einstein’s notation with the more readable modern notation.

4. 4.
Einstein justifies this conclusion as follows:
From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and −v are interchanged. Hence follows that $$\frac{l}{\phi(v)}=\frac{l}{\phi(-v)}$$ and
$$\phi(v)=\phi(-v)=1$$
5. 5.

Brown and Sypel (1995) argue that Einstein’s application of the Relativity Principle to derive the Lorentz transformations should not be surprising, since “the rods and clocks are themselves to be viewed not as primitive, structureless objects, but as solutions of the basic equations, treating clocks and rods as composite entities whose parts are governed by dynamic forces.” Thus given a dynamic approach to spacetime, Einstein’s application of the Principle of Relativity in this context is unproblematic. I shall consider dynamic approaches and their problems in Section 2.4.

6. 6.

See Brown (2005, ).

7. 7.

It took a while until the philosophical community came to grips with the status of the Lorentz-Fitzgerald contraction hypothesis (LCH). Popper (2003, p. 62) argued that the LCH is an ad-hoc hypothesis, since the prediction of Maxwell’s theory together with Newtonian mechanics regarding the motion through the ether was falsified by the Michelson-Morely experiment. The LCH was just introduced in order to avoid facing the falsification of accepted theories, and produced no new predictions. Grünbaum (1959) argued that the LCH in isolation was falsified by the Kennedy-Thorndike experiment. In this experimental setup, the interferometer used was similar to that of Michelson and Morley’s, only it had arms of different lengths. The difference in time between the two arms did not depend on the orientation of the interferometer. This shows that the LCH by itself is not sufficient to cohere with the data, and one needs to assume time dilation in addition to length contraction. See Grünbaum (1959), Evans (1969), and Erlichson (1971).

8. 8.

See Treder (1970, p. 86), Brown and Sypel (1995), and Budden (1997).

9. 9.

The program of axiomatizing spacetime theory has a long and interesting history. For a modern revitalization of the program see Andréka et al. (2006) and Madarász et al. (2007).

10. 10.

For example, the analysis of the logical foundations of relativity theory shows that it is possible to replace the Principle of Relativity with a weaker axiom (i.e., that observers agree on which events take place). The price for weakening the relativity postulate is that it is necessary to add a reciprocity relation between frames, i.e., time dilation and length contraction must be the same for any boosted frame relative to the frame at rest.

11. 11.

See Friedman (1983, chapter VII) for a similar complaint against conventionalism.

12. 12.

For the history of this reading of the argument, see Reichenbach (1927, pp. 210–18), Burtt (1954, pp. 244–55), Jammer (1994, p. 106), Lacey (1970), and Westfall (1971, p. 443).

13. 13.

A reference frame F is defined by a time-like vector field X, i.e., $$dtX\neq0$$. The trajectories of X could be interpreted as the worldlines of points in the spacetime. If there is a coordinate system $$\{x^\mu\}$$ in which the components of the affine connection vanish, or $$\Gamma^\gamma_{\mu\nu}=0$$, and the coordinate system is adaptable to F, i.e., the spatial coordinates $$x^{\alpha}, \alpha = 1, 2, 3$$ of the trajectories of X are constant, then F is an inertial reference frame (Earman and Friedman, 1973, section 3).

14. 14.

See Newton’s account of absolute space in the De Gravitatione, Newton (2004).

15. 15.
Even though Einstein described the relation between spacetime and matter as causal, he also thought that having two independent fields existing side by side is problematic. This is one of his motivations for searching for a unified field theory. In a lecture delivered at the Nobel Prize ceremony, he asserted the following:

The mind striving after unification of the theory cannot be satisfied that two fields should exist which, by their nature, are quite independent. A mathematically unified field theory is sought in which the gravitational field and the electromagnetic field are interpreted as only different components or manifestations of the same uniform field … The gravitational theory, considered in terms of mathematical formalism, i.e., Riemannian geometry, should be generalized so that it includes the laws of the electromagnetic field. (Einstein, 1923, p. 489)

16. 16.

In the following chapter I will argue for a different account of the mass parameter, which undermines the view that mass is an inherent property. However, I am using the popular (though false) understanding of mass to make a philosophical point.

17. 17.

See Balashov and Janssen (2003) for a similar argument against Craig’s neo-Lorentzian interpretation of relativity.

18. 18.

I should point out that Brown is not endorsing the adoption of an ether or an absolute frame of reference. Thus he is not attempting to resuscitate Lorentz’s specific strategy for explaining kinematic effects. Rather, he claims that dynamic laws should explain the kinematic effects of a set of rods and clocks moving relative to another system of clocks and rods. Brown argues that dynamics should explain kinematics, not that the Principle of Relativity is false.

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