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The Principle of Equivalence As a Criterion of Identity Ryan Samaroo Forthcoming in Synthese The Principle of Equivalence as a Criterion of Identity Ryan Samaroo Somerville College, Woodstock Road, Oxford, OX2 6HD, United Kingdom [email protected] Dedicated to the memory of William (Bill) Demopoulos Abstract In 1907 Einstein had the insight that bodies in free fall do not “feel” their own weight. This has been formalized in what is called “the principle of equivalence.” The principle motivated a critical analysis of the Newtonian and special-relativistic concepts of inertia, and it was indispensable to Einstein’s development of his theory of gravitation. A great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether it has any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. I argue that the principle should be understood as a kind of foundational principle known as a criterion of identity. This work extends and substantiates a recent account of the notion of a criterion of identity by William Demopoulos. Demopoulos argues that the notion can be employed more widely than in the foundations of arithmetic and that we see this in the development of physical theories, in particular space-time theories. This new account forms the basis of a general framework for applying a number of mathematical theories and for distinguishing between applied mathematical theories that are and are not empirically constrained. 1. Introduction “The principle of equivalence,” which Einstein originally used to refer to one particular statement, has come to refer to a number of interrelated principles in the theory of gravitation.1 1 Einstein’s first formulation of the principle can be found in his article “On the Relativity Principle and the Conclusions Drawn from It” (1907, p. 454). Other early accounts include his “On the Influence of Gravitation on the Propagation of Light” (1911, pp. 898-99), his review article “The Foundation of the General Theory of Relativity” (1916a, pp. 772-3), his note “On Friedrich Kottler’s Paper: ‘On Einstein’s Equivalence Hypothesis and Gravitation’” (1916b, p. 639), his popular exposition Relativity: The Special and the General Theory (1916c [2004], Chapter 20), The principle formalizes an insight into gravitation that Einstein had in 1907. This is the insight, roughly speaking, that objects in free fall do not “feel” their own weight. The principle motivated a critical analysis of the Newtonian and 1905 inertial frame concepts, and it was indispensable to Einstein’s argument for a new concept of inertial motion. But setting aside its role in Einstein’s argument, the principle is generally held to be one of the foundations of Einstein’s theory of gravitation, even if the sense in which it is a foundation is disputed. For these reasons, a great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether indeed it has any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. A methodological analysis must consider two basic questions: what kind of principle is the equivalence principle? What is its role in the conceptual framework of gravitation theory? I maintain that the existing answers are unsatisfactory and I offer new answers. I argue that the equivalence principle should be understood as expressing a kind of foundational principle known as a criterion of identity. The principle functions as a criterion for recognizing when the motions of different reference frames are the same motion; it has the consequence that the motion of an inertial frame is the same as the motion of locally freely falling frame. My new account illuminates the methodological role of the equivalence principle in the conceptual framework of gravitation theory and also our understanding of the application of the theory of pseudo-Riemannian manifolds in Einsteinian gravitation. Furthermore, this account of the role of the principle informs our understanding of Einstein’s analysis of the inertial frame concept, and so of the transition from the conceptual framework of special relativity to that of the general theory. This is a novel use of the notion of a criterion of identity, one that may be surprising even to those already acquainted with the literature in the philosophy of mathematics and the metaphysics of individuals. This study owes several things to the former and nothing to the latter. It builds on the recent account of the notion of a criterion of identity by Demopoulos in Logicism and its Philosophical Legacy (2013). Demopoulos argues that the notion of a criterion of identity and his Princeton Lectures (1922, Lecture III). So far as I know, the first time he uses the term “principle of equivalence” is in his reply to Kottler. Further remarks about Einstein’s insight of 1907, including his remark that it was “the happiest thought of my life,” can be found in his “Fundamentals and Methods of the Theory of Relativity” (1920). It is worth noting, however, that the principle would find a more precise formulation only much later in the general relativity literature. 2 can be employed more widely than in the foundations of arithmetic and that we can see this in the development of physical theories, in particular space-time theories. Demopoulos’ contribution is a penetrating analysis of the criterion of identity and this study aims to further extend and substantiate it. Although this employment of the notion of a criterion of identity in the foundations of space-time theories may seem at first surprising and even questionable, I hope to show that it is in fact a natural one and that in the case of the equivalence principle it allows us to recover the features of the gravitational interaction that the principle is generally held to establish. In §2 I will introduce the 1905 inertial frame concept. Since the equivalence principle motivates a critical analysis of this concept, I will present the concept in overview, beginning with its Newtonian and nineteenth-century antecedents. In §3 I will consider a number of principles that have been called “the equivalence principle,” and I will examine the relations between them. I will draw attention to one particular principle that, I will argue, fully captures Einstein’s insight of 1907. In §4 I will survey and evaluate some notable accounts of the principle. In §5 I will present Demopoulos’ account of the criterion of identity and his claim that other criteria of identity underlie the analysis and interpretation of a number of space-time theories. I will argue that understanding the equivalence principle as a criterion of identity illuminates its methodological role in the conceptual framework of gravitation theory. Last, in §6, I will examine a few implications of this account. I will show, in particular, that it isolates what is distinctive about Einstein’s contribution to our understanding of the gravitational interaction. 2. Background: The 1905 Inertial Frame Let us begin by getting clear on the concept of an inertial frame, specifically, Einstein’s 1905 concept that was the object of his 1907 analysis. It is instructive to introduce this concept by way of Newton’s. Newton’s laws of motion express empirical criteria for the application of the basic concepts of mechanics, namely force, mass, and inertial motion – and all those concepts that depend on them. Inertial motion is that state in which a body moves in uniform rectilinear motion unless acted upon by a force.2 Now associate with a body moving inertially a reference frame. In 2 It is worth noting that all three laws of motion are necessary to give an account of inertial motion – the first law is sufficient only for ideal point-particles. This is clear in Newton’s account of inertia. Hence, referring to the first law as “the principle of inertia” encourages the widespread view that it alone is enough. So far as I know, Newton himself did not use the term. I do not know at what point it appeared, but it can be found in Euler’s “Réflexions sur 3 the most general sense, a frame is a space. It is a “small” space in the sense that it is sufficiently local, homogeneous, and isotropic; furthermore, it is a space in which an accelerometer would detect no acceleration. We can give a geometrical description of bodies in the space among themselves using a coordinate system. But we can do more than just give a geometrical description: in any such space, the outcomes of mechanical experiments, calculated using the laws of motion, will be the same. (And the same outcomes would be calculated in any space in uniform rectilinear motion relative to it – this is the Galilei-Newton relativity principle.) This is the Newtonian concept of an inertial frame. While these frames are empirically indistinguishable, for Newton, they were not theoretically equivalent: Newton thought of them as moving with various velocities relative to what he called “absolute space,” even though those velocities cannot be known. Although Newton introduced this term to refer to the resting backdrop against we can talk about uniformly moving relative spaces, many of his contemporaries understood it to have certain ontological implications and criticized its introduction on those grounds. It was not until the nineteenth century that Newton’s theory was given its proper form, by the insight into its complete independence from the notion of absolute space in the work of Neumann (1870), Thomson (1884), Lange (1885), and others.3 The nineteenth-century inertial frame concept was the outcome of their work.
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