Relativity without Spacetime

Joseph K. Cosgrove

• Publisher: Palgrave Macmillan
• Year: 2018
• ISBN:
  9783319726311
• Topic: Philosophy
• Subtopic: Philosophy & Ethics in Science

In 1908, three years after Einstein first published his special theory of relativity, the mathematician Hermann Minkowski introduced his four-dimensional "spacetime" interpretation of the theory. Einstein initially dismissed Minkowski's theory, remarking that "since the mathematicians have invaded the theory of relativity I do not understand it myself anymore." Yet Minkowski's theory soon found wide acceptance among physicists, including eventually Einstein himself, whose conversion to Minkowski's way of thinking was engendered by the realization that he could profitably employ it for the formulation of his new theory of gravity. The validity of Minkowski's mathematical "merging" of space and time has rarely been questioned by either physicists or philosophers since Einstein incorporated it into his theory of gravity. Physicists often employ Minkowski spacetime with little regard to the whether it provides a true account of the physical world as opposed to a useful mathematical tool in the theory of relativity. Philosophers sometimes treat the philosophy of space and time as if it were a mere appendix to Minkowski's theory. In this critical study, Joseph Cosgrove subjects the concept of spacetime to a comprehensive examination and concludes that Einstein's initial assessment of Minkowksi was essentially correct.

© The Author(s) 2018

Joseph K. CosgroveRelativity without Spacetimehttps://doi.org/10.1007/978-3-319-72631-1_1

Begin Abstract

1. Introduction: A Critique of Minkowski Spacetime

Joseph K. Cosgrove¹

(1)

Philosophy, Providence College, Providence, RI, USA

An erratum to this publication is available online at https://​doi.​org/​10.​1007/​978-3-319-72631-1_​9

End Abstract

In 1908, three years after Einstein first published his special theory of relativity, the mathematician Hermann Minkowski introduced his four-dimensional “spacetime” interpretation of the theory. Einstein initially dismissed Minkowski’s theory as a “piece of mathematical trickery,” remarking wryly that “[s]ince the mathematicians have invaded the theory of relativity I do not understand it myself anymore.”¹ Yet Minkowski’s theory soon found wide acceptance among physicists, including eventually Einstein himself—his conversion to Minkowski engendered principally by the realization that he could profitably employ it for the formulation of his theory of gravity (the general theory of relativity). Thus in his popular book on the theory of relativity published in 1916, Einstein famously remarks that general relativity “would perhaps have gotten no farther than its long clothes” if it had not been for Minkowski’s innovation.²

The physical validity of Minkowski’s concept of merged spacetime has rarely been questioned by physicists or philosophers since Einstein incorporated it into his theory of gravity. This is strange in one sense, for there is no general agreement on the physical meaning of Minkowski’s theory. Indeed, one can only sympathize with Vesselin Petkov’s complaint that while physicists and philosophers of science routinely endorse and employ Minkowski’s four-dimensional formal apparatus, at the same time they habitually speak about the world as if it were really three-dimensional in the usual sense, giving back to themselves with one hand, as it were, what they have just taken away with the other.³ Physicists in fact often employ Minkowski spacetime with little regard to the whether it provides a true account of the physical world as opposed to a useful mathematical tool in the theory of relativity, while philosophers, for their part, sometimes treat the philosophy of space and time as if it were merely an appendix to Minkowski’s theory. The purpose of this book, then, is to subject the concept of spacetime to a much overdue critical examination, with a view toward a more physically intelligible interpretation of Einstein’s special and general theories of relativity. For I believe that Einstein’s initial assessment of Minkowski was essentially correct.

Anyone conversant with the theory of relativity knows, or at least thinks they know, that Minkowski “merged” space and time into a single four-dimensional geometrical continuum. However, the precise character of the intended merging of space and time is not always clear in the literature on spacetime. One often reads, for example, that from our present vantage point we can see that even Newton and Galileo employed a “four-dimensional spacetime continuum,” even if they treated time and space independently of one another. Einstein himself, for instance, suggests that the idea of a four-dimensional continuum is not something newly introduced by the theory of relativity, since classical mechanics also employed a four-dimensional continuum which, however, “falls naturally into a three-dimensional and one-dimensional (time), so that the four-dimensional point of view does not force itself upon one as necessary.”⁴

It would be helpful at the outset, then, to note briefly some of the senses in which it may be said that space and time are “unified” in a given theory of physics. The minimal notion of such unification is a mere “n-dimensional manifold” in which each point event can be associated with n numbers (coordinates). This amounts to saying that every event happens at a time and place. In this minimal sense, Newton’s physics indeed may be said to employ a four-dimensional spacetime manifold, something naturally suggested by diagrams in which time is symbolically represented by the length of a line in space (usually in Cartesian coordinates). But there is no single continuum of space and time in Newtonian physics, since space and time intervals are independent of one another, with no four-dimensional interval between events defined and therefore no metrical unification effected. While it is commonplace today to reconstruct Newtonian physics using the mathematical apparatus of differential geometry, as Michael Friedman notes, “We effect a relativistic unification of space and time only if we view space-time as a four-dimensional semi-Riemannian manifold.”⁵

Einstein’s 1905 special theory of relativity introduces a new kind of metrical entanglement of space and time. Here the measure of a time interval between two events in some inertial frame A depends, through the Lorentz transformation, on both the time and distance between those events as measured in an inertial frame B in motion relative to A, and likewise for a distance in A as a function of time and distance measured in B. As Einstein formulates it, the “necessity” of the four-dimensional point of view in special relativity, unlike the situation in Newtonian dynamics, lies in the “formal dependence between the way the space coordinates, on the one hand, and the temporal coordinates, on the other, have to enter into the natural laws.”⁶ By contrast, in pre-relativity mechanics, metrical intervals of time and distance are the same for all inertial reference frames.

Nevertheless, Einstein 1905 special relativity in no sense merges space and time into a single continuum. That is to say, even though in pre-Minkowski special relativity we regard the metrical properties of space and time as interdependent or entangled, space and time themselves remain distinct continua with no metrical unification per se. Thus, while in differential geometry , for instance, a metric continuum is defined by its distance function or line element (quadratic differential form), in Einstein 1905 there is no such distance function for time and space taken together. This point is easily obscured by the standard use in special relativity of Minkowski diagrams (which would be much more aptly termed “special relativity diagrams,” since they in no wise distinguish Minkowski’s theory conceptually from Einstein 1905), and the associated jargon of “world lines,” “light cone structure,” and so forth.

At the very least, then, it seems the term “spacetime” is associated with a number of distinct conceptions of the unification of space and time. But it is only with Minkowski’s introduction of the invariant “interval” or four-dimensional displacement vector that we encounter anything that could be properly termed the unification of space and time into a single continuum. By “spacetime” in this study, then, I shall always intend specifically “Minkowski spacetime” or the idea, set forth most famously by Minkowski in his Cologne lecture of 1908, that space and time as they physically exist are merged into a single continuum, geometrically determined by a four-dimensional line element analogous to the Pythagorean line element of standard differential geometry.⁷ To be sure, today we often speak of spacetime in a wider sense. However, the present terminological restriction is not at all arbitrary, for only with the advent of Minkowski’s theory do we find the essential condition for a metric continuum of space and time: a distance function. Clearly there is no such notion in Galileo or Newton, and therefore it is misleading to say that either of them employed a “four-dimensional continuum.”

It is unfortunate, in view of the very radicalness of Minkowski’s proposal, that Einstein’s 1905 special theory of relativity has come to be regarded as virtually synonymous with Minkowski’s 1908 theory, as if the latter simply elaborated what was already implicit in the former. Such an identification, which forecloses the possibility that one might wish to avail oneself of Einstein 1905 while abstaining from Minkowski 1908, reflects a kind of whiggish view of the history of relativity, according to which Minkowski’s theory represents the inevitable disclosure of the deep geometrical structure of Einstein’s 1905 theory, originally overlooked by Einstein himself. But the proposition is a dubious one. For even if Minkowski’s theory yields the same empirical results as Einstein 1905, the former nevertheless posits a new set of absolute geometrical objects that have no role whatsoever in the ontology of Einstein’s original theory—the absolutes of which are rather the laws of nature. Thus, unless we wish to insist that theories making the same empirical predictions are equi...

Table of contents

• Cover
• Front Matter
• 1. Introduction: A Critique of Minkowski Spacetime
• Part I. The Concept of Minkowski Spacetime
• Part II. Part II
• Part III. General Relativity Without Spacetime
• Part IV. Time Without Spacetime
• Erratum to: Relativity without Spacetime
• Back Matter