The Equivalence Principle, Uniformly Accelerated Reference Frames, and the Uniform Gravitational Field

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Embry-Riddle Aeronautical University Publications 3-2010 The Equivalence Principle, Uniformly Accelerated Reference Frames, and the Uniform Gravitational Field Gerardo Muñoz California State University, Fresno Preston Jones University of Louisiana at Monroe, [email protected] Follow this and additional works at: https://commons.erau.edu/publication Part of the Cosmology, Relativity, and Gravity Commons Scholarly Commons Citation Muñoz, G., & Jones, P. (2010). The Equivalence Principle, Uniformly Accelerated Reference Frames, and the Uniform Gravitational Field. American Journal of Physics, 78(4). https://doi.org/10.1119/1.3272719 This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. The equivalence principle, uniformly accelerated reference frames, and the uniform gravitational field Gerardo Muñoz∗ Department of Physics, California State University, Fresno, Fresno, CA 93740-8031 Preston Jones† Department of Mathematics and Physics, University of Louisiana, Monroe, LA 71209-0575 (Dated: March 17, 2010) The relationship between uniformly accelerated reference frames in flat spacetime and the uniform gravitational field is examined in a relativistic context. It is shown that, contrary to previous statements in the pages of this journal, equivalence does not break down in this context. No restrictions to Newtonian approximations or small enclosures are necessary. I. INTRODUCTION It would not be very controversial to state that there is universal agreement on the meaning and implementation of the equivalence principle within a Newtonian framework. As a review of the extensive literature on the subject will easily confirm, however, the situation changes drastically in the relativistic domain. Most authors correctly point out that, in general, the equivalence principle only applies locally,[1–6] but there are several others who deny its validity even at the local level.[7–9] The disagreement shows up not only in discussions of the foundations of general relativity, but also in applications such as the radiation from accelerated charges[10, 11], Unruh radiation[12–14], and uniform gravitational fields[15–17], to name just a few. Our aim in this paper is to present a compelling demonstration that a relativistic uniformly accelerated reference frame in flat spacetime is equivalent to a uniform gravitational field, as one would naively expect from the nonrelativistic version of the equivalence principle. It might be argued that this issue was settled by Rohrlich[15] in a definitive manner a long time ago, but a paper by Desloge[16] in this journal revisited the issue with the conclusion that “in a uniform gravitational field the space-time is as expected curved.” Because of its obvious importance, we believe that a careful reexamination of the problem is necessary. Most of our discussion will be easier to follow if the reader keeps a few simple facts about relativistic metric theories of gravity in mind. F1. Imposing a-priori meanings on the coordinates in a metric theory of gravity is to be done at one’s own peril. It is not uncommon to find that our preconceived notions fail in certain cases, or that our demands overconstrain the system. F2. Acceleration – either in the nonrelativistic form d2x/dt2 or in the special-relativistic form d2xµ/dτ 2 – does not have an absolute (geometric) meaning in a metric theory of gravity. In other words, unlike the 4-velocity dxµ/dτ, the object d2xµ/dτ 2 is not a 4-vector under general coordinate transformations, but rather an intrinsically observer- dependent notion. F3. One should never expect the relativistic description of a system to satisfy nonrelativistic constraints. These facts have been phrased in terms of a metric theory of gravity rather than general relativity in order to arXiv:1003.3022v1 [gr-qc] 15 Mar 2010 emphasize their independence from any particular set of dynamical equations. As the reader will notice, Sections II and III below hold in any metric theory of gravity; only when we get to Section IV will we require Einstein’s equations. II. INTUITIVE ATTEMPTS AT DEFINING A UNIFORM GRAVITATIONAL FIELD In Ref.[16], the metric (c = 1 hereafter) ds2 = α2(x) dt2 dx2 (1) − was motivated by the desire to represent a rigid frame in a gravitational field and a rigid frame uniformly accelerating in field-free space (the restriction to two-dimensions is of no consequence at this point; we shall take up the four- ∗Electronic address: [email protected] †Electronic address: [email protected] 2 dimensional problem in later sections). It is clear that, as intended, the x-coordinate has been given an a-priori meaning such that the length of an infinitesimal object equals its coordinate length dx, i.e., lengths measured with this coordinate system are given by their special-relativistic inertial values. Desloge defines the “local velocity” α−1dx/dt and “local acceleration” α−1d (α−1dx/dt)/dt and uses the latter together with the equation of motion of a free particle to identify α−1dα/dx as the proper acceleration of an observer at rest at x in the gravitational field. Taking the requirement that this acceleration be constant everywhere as a definition of the uniform gravitational field then leads to the conclusion that the Riemann tensor has nonvanishing components – hence his statement that “in a uniform gravitational field the space-time is as expected curved.” While Desloge’s “local acceleration” is a possible definition of acceleration, it is not mandated by general relativity or even special relativity. Moreover, it is an ill-motivated definition, since α dt is the time interval between two events occurring at a single spatial location (dx = 0), and hence bears no special significance for the motion of a freely-falling particle because in that case we are interested in events that necessarily refer to different spatial locations. The point is best illustrated by recalling that already in special relativity one defines the acceleration by resorting to the particle’s proper time rather than the proper time of the observer. The a-priori meaning of the spatial coordinate x (recall fact F1) together with the arbitrary assignment of a preferred status to the “local acceleration” (recall fact F2) collude to yield the erroneous conclusion that uniform gravitational fields must correspond to curved spacetimes. A few examples will help elucidate our claim. The geodesic equation of motion for an inertial observer is d2xµ dxν dxρ +Γµ =0 (2) dτ 2 νρ dτ dτ With the metric (1) we find the two equations d2t α′ dt dx +2 =0 (3) dτ 2 α dτ dτ 2 d2x dt + αα′ =0 (4) dτ 2 dτ Integrating equation (3) gives dt k = (5) dτ α2 with k > 0 an integration constant which may be fixed by imposing initial conditions. Replacing this expression in Eq. (4) yields d 1 dx α′ + =0 (6) dt α2 dt α which is equivalent to Desloge’s Eq. (6). The question before us now is how to define a uniform gravitational field based on the acceleration of a test particle. One possibility is to argue that, since Newtonian physics must be exact for vanishing velocities and weak fields, we should concentrate on particles dropped from rest at time t = 0 from a point x = 0 where α(0) = 1 (this normalization is possible for an arbitrary point because the field is uniform) and define the acceleration in a small neighborhood of that event as d2x/dt2. Equation (6) then implies d2x = αα′ (7) dt2 − in this small neighborhood. If we set the acceleration equal to the constant g we get the result − 2 α =1+2gx (8) As is well known, this is the correct nonrelativistic limit, but it should be clear from its derivation that this form of the metric cannot be expected to describe the field globally. Nevertheless, within its domain of validity, it represents a uniform gravitational field (in the intuitive sense that the acceleration equals g), and corresponds to a flat spacetime, − Rµνκρ = 0, at x = 0. A second possibility is to define the relevant acceleration as d (α−1dx/dt)/dt. It then follows from (6) that d 1 dx + α′ =0 (9) dt α dt 3 at the initial instant, provided again that the particle is released from rest. One would therefore naturally choose to set α′ = g and find as a consequence the Kottler-Møller[1, 19] metric ds2 = (1+ gx)2 dt2 dx2 (10) − The Riemann tensor associated with this metric vanishes everywhere. These two simple examples are perhaps sufficient to prove the point that defining the uniform gravitational field based on the naive “acceleration = g” condition is not satisfactory. One of the main obstacles to this approach is clearly fact F2, and most of the rest− of this paper will be dedicated to finding an unambiguous way of addressing this problem. Before we proceed in that direction however, let us briefly explain the technical reasons why Desloge’s “local acceleration” fails – as will almost every other definition – to give a flat spacetime if we start with the metric (1). As Rohrlich[15] and Tilbrook[18] have shown using different methods, a metric of the form ds2 = λ2(x) dt2 σ2(x) dx2 (11) − is equivalent to a flat spacetime only if 1 dλ(x) σ(x)= (12) g dx with g an integration constant. Adopting a metric of the form (1) means choosing σ = 1, which immediately turns λ(x) into α(x)=1+gx, and the metric (11) into the Kottler-Møller metric (10). In other words, once we adopt coordinates such that the metric takes the form (1), we can either impose a condition on a given definition of acceleration (in which case we would find a spacetime which is, in general, not flat), or we can impose the condition that the spacetime should be flat (and find that the only definition of initial acceleration that works is d (α−1dx/dt)/dt).

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