Distance and the conventionality of simultaneity in special relativity

The conventionality of simultaneity within inertial frames is presented in a general formalism that clarifies the relationship of spatial measures to the choice of simultaneity. A number of claims that such measures undermine the conventional nature of simultaneity are presented and shown to be unfounded. In particular, a recent claim by Coleman and Korte [9] that such measures empirically establish a unique simultaneity relationship is shown to be in error. In addition, the general formalism enables the empirical status of simultaneity within an inertial frame to be clarified by presenting the choice of simultaneity as a gauge choice.1. Recent introductions to the literature have been given by Redhead [35], Ungar [47], Havas [21], and Vetharaniam and Stedman [48].2. The conventionalist position is by no means a uniform one, and in particular, it is worth noting an important distinction exemplified in the respective positions of Reichenbach and Grünbaum. For Reichenbach [37, p. 144f.] we have no empirical access to the one-way speed of light due to the nature of light as a first signal, and the conventionality comes from our absence ofknowledge about the one-way speed of light. For Grünbaum the one-way speed of light is actually objectively undetermined, and the physical attributes that sustain a speed in a given direction are non-existent. See, for example, [16, p. 87] and [17, p. 352]. Discussions of the differences between the positions of Reichenbach and Grünbaum may be found in [14] and [35]. Naturally, one may adhere to a position espoused by Reichenbach without the added ontological commitment of Grünbaum.3. Our is equivalent to (1 - 2), where is the symbol introduced by Reichenbach and customarily used in the discussions of the conventionality of simultaneity.4. An exposition of this argument may be found in the recent text by Lucas and Hodgson [28].5. Schrödinger [42, p. 78] has aptly labeled this quantity the distance of simultaneity.6. Examples of previous uses space-dependent synchrony parameters may be found in studies by Clifton [8], Havas [21], Anderson and Stedman [1], and Stedman [43; 44, § 2].7. This approach has been reviewed by Basri in [4] and [3].8. A number of faulty assessments of the empirical status of the conventionality of simultaneity may be similarly traced at least in part to overly simplistic assumptions on the nature of as Havas [21] and Clifton [8], for example, have had occasion to point out.9. See, for example, [1]. Kinematic formula relating other quantities in a treatment of STR without the standard convention on the one-way speed of light were first derived by Winnie [53].10. In comparison to other space dependent treatments of the synchrony parameter, ourh is analogous to defined by Clifton in Eq. (15) of [8], and equivalent to -f defined by Havas in Eq. (A1) of [21] and to defined in Eq. (6) of our earlier treatment in [1]. We take this opportunity to mention that the irrotational property ofh was inadvertently referred to as solenoidal in this work.11. Equation (26) is equivalent to Møller's expression in § 8.8 of [32] for the speed of light in terms of the metric components where our-h _i is equivalent to Møller's _i (g _i0)/ .12. Note as well, the expression of this operation in standard texts on STR by Rindler [38, pp. 27–28] and Mermin [30, p. 79] respectively: To measure the rod's length in any inertial frame in which it moves longitudinally, its end-point must be observed simultaneously... and, ...a measurement of the length of a moving meter stick involves determining how far apart the two ends areat the same time. In the same context of determining the length of moving rods, Mermin [30, p. 185] proposes that the sense of length entailing the concept of being determined at simultaneous times is inherent in the notion of rods: ...it is precisely the lines of constant time that determine whatA orB means by the stick. For the notion of the stick includes implicitly the assumption that all the points of matter making up the stick exist at the same moment.13. In many ways the claim that the special properties of proper lengths with Einstein synchronization undermines the conventionality of simultaneity is analogous to the claim that the correspondence of the slow-clock transport method of synchronization with that of Einstein synchronization provides an empirical determination of synchronization. The use of clock transport as a means for synchronization was discussed by Reichenbach [37, p. 133f], while the proposal that slow transport of clocks provides a unique form of synchronization was first argued for by Eddington [10]. Arguments that it undermines any significant sense of the conventionality in the one-way speed of light have been given by Ellis and Bowman [13] with responses by Grünbaum [19] and Salmon [41, 40].14. Coleman and Korte [9, pp. 423–425] claim their method is free from any assumptions on the one-way speed of light; however, they assume that is a constant 3-vector.15. Reichenbach explicitly mentioned in [36, § 43] that a condition equivalent to Eq. (13) is a sufficient condition for a constant roundtrip speed of light.16. The remarks of one of the referees have served to alert us to the need to emphasize both of these points.17. The manner in which gravity may be viewed as a gauge theory has been the subject of considerable discussion (see, for example, the discussion in [23] and [24]). We note that the manner in which we are takingh as a potential differs from the sense in which the Christoffel symbols as affine connections may be seen to play a role of gauge potentials in GTR.18. A discussion of the significance of Weyl's work and the importance of the round-trip measurements may be found in works by Yang [56] and Mills [31].19. In the context only of time orthogonal coordinates, an example of the fiber structure we are imposing on space and time may be found in [26, p. 71f]. Again we note that in a more general treatment, where the Christoffel symbols are considered as connections, the fiber structure instead consists of a bundle of linear frames of Riemannian spacetime (see, for example, the presentations in [46] and [23]).20. Our position is not unlike Göckeler and Schücker's [15, p. 75] claim that Einstein's particular choice of coordinates in GTR masks the general gauge structure of the theory.     

A modified Lorentz theory as a test theory of special relativity

A modified Lorentz theory (MLT) based on the generalized Galilean transformation has recently received attention. In the framework of MLT, some explicit formulas dealing with the one-way velocity of light, slow-clock transport and the Doppler effect are derived in this paper. Several typical experiments are analyzed on this basis. The results show that the empirical equivalence between MLT and special relativity is still maintained to second order terms. We confirm recent findings of other works that predict the MLT might be distinguished from special relativity at the third order by Doppler centrifuge experiments capable of a fractional frequency detection threshold of 10^–15.     

Antiparticles from special Relativity with ortho-chronous and antichronous Lorentz transformations

Special Relativity can be based on the whole proper group of both ortho- and antichronous Lorentz transformations, and a clear physical meaning can be given also to antichronous (i.e., nonorthochronous) Lorentz transformations. From the active point of view, the latter requires existence, for any particle, of its antiparticle within a purely relativistic, classical context. From the passive point of view, they give rise to frames dual to the ordinary ones, whose properties—here briefly discussed—are linked with the fact that in relativity it is impossible to teach another, far observer (by transmitting only instructions, and no physical objects) our own conventions about the choices right/left, matter/antimatter, and positive/negative time direction. Interesting considerations follow, in particular, by considering—as it is the case—theCPT operation as an actual (even if antichronous) Lorentz transformation.Work partially supported by FAPESP and CNPq (Brazil).     

Matrix formulation of special relativity in classical mechanics and electromagnetic theory

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed.     

Spatial measures in special relativity do not empirically determine simultaneity relations: A reply to Coleman and Korté

Coleman and Korté have restated and defended an earlier attempt to refute the traditional thesis of the conventionality of simultaneity within special relativity. Here we argue their attempt still fails and respond to criticisms of a paper in which we addressed the inadequacies of their earlier paper. The spatial criterion they use to argue for standard synchronization throughout an inertial frame is merely a definition and provides no demonstration that a unique distant simultaneity relation exists in nature.     

Superposition in quantum and relativity physics—An interaction interpretation of special relativity theory: Part III

With the interaction interpretation, the Lorentz transformation of a system arises with selection from a superposition of its states in an observation-interaction. Integration of momentum states of a mass over all possible velocities gives the rest-mass energy. Static electrical and magnetic fields are not found to form such a superposition and are to be taken as irreducible elements. The external superposition consists of those states that are reached only by change of state of motion, whereas the internal superposition contains all the states available to an observer in a single inertial coordinate system. The conjecture is advanced that states of superposition may only be those related by space-time transformations (Lorentz transformations plus space inversion and charge conjugation). The continuum of external and internal superpositions is examined for various masses, and an argument for the unity of the super-positions is presented.     

Modern classical physics: optics,fluids, plasmas,elasticity, relativity,and statistical physics

Semiconductor Physics. By K. Seeger. (Springer-Verlag, Wiens New York, 1973.) [Pp. 430.] DM 60. Scope: Textbook. Level: post-graduate.

The World of Walther Nernst:The Rise and Fall of German Science. K. Mendelssohn. (Macmillan, 1973.) [Pp. viii+191.] £4.95. Scope: Library. Level: General reader.

Cosmic Rays at Ground Level. A volume dedicated to George D. Rochester in the year of his retirement. Edited by A. W. Wolfendale. (Institute of Physics, London and Bristol, 1973.) [Pp. viii+ 233.] £12 (£6 to members).

Gravitation. By C. W. Misner, K. S. Thorne and J. A. Wheeler. (W. H. Freeman, 1973.) [Pp. xix+ 1279.] £19.20 cloth; £8.60 paper. Scope: Treatise/Textbook; Library. Level: Specialist/post-graduate.

General Theory of Relativity. By C. W. Kilmister. (Pergamon Press, 1973.) [Pp. ix+ 365.] Hard cover £3.50; Flexi £2.00. Scope: Textbook. Level: Post-graduate; Undergraduate.

Plasma Astrophysics. By S. A. Kaplan and V. N. Tsytovich, translator D. ter Haar. (Pergamon, 1973.) [Pp. xiii+ 301.] £9.60. Scope: Treatise/Textbook; Library. Level: Specialist/post-graduate.     

Lightlike simultaneity,comoving observers and distances in general relativity

We state a condition for an observer to be comoving with another observer in general relativity, based on the concept of lightlike simultaneity. Taking into account this condition, we study relative velocities, Doppler effect and light aberration. We obtain that comoving observers observe the same light ray with the same frequency and direction, and so gravitational redshift effect is a particular case of Doppler effect. We also define a distance between an observer and the events that it observes, called lightlike distance, obtaining geometrical properties. We show that lightlike distance is a particular case of radar distance in the Minkowski space-time and generalizes the proper radial distance in the Schwarzschild space-time. Finally, we show that lightlike distance gives us a new concept of distance in Robertson–Walker space-times, according to Hubble law.     

Cosmological relativity: A special relativity for cosmology

Under the assumption that Hubble's constant H₀ is constant in cosmic time, there is an analogy between the equation of propagation of light and that of expansion of the universe. Using this analogy, and assuming that the laws of physics are the same at all cosmic times, a new special relativity, a cosmological relativity, is developed. As a result, a transformation is obtained that relates physical quantities at different cosmic times. In a one-dimensional motion, the new transformation is given by

On Carmeli's exotic use of the Lorentz transformation and on the velocity composition approach to special relativity

As shown by Ramarkrishnan, the faithful mapping, in the sense of Lie groups, of the real line onto the finite segment–1<u<+1 is u=tanh A, from which follows the relativistic velocity composition law w=(u+v)/(1+uv) and the Lorentz-Poincaré transformation formulas. Composition of translations is merely one application of this. Carmeli has shown that composition of rotations is another one. There may be still others.     

基于同时的相对性对钟慢尺缩效应的再认识

同时的相对性、钟慢效应和尺缩效应是狭义相对论时空观的主要内容.鉴于同时性是时空测量的基础,本文从同时的相对性出发详述了对钟慢效应和尺缩效应的再认识:钟慢效应是运动时钟走时率变慢和校表问题的综合表现,其实质是同时的相对性在时间量度上的直接反映;尺缩效应的实质是同时的相对性在空间量度上的反映,也是不同观测者对同一客观事实的不同时空描述.     

Einstein's relativity and quantum physics

There are reasons to reject the idea that a field in empty space is a real physical entity. The nonexistence of the electromagnetic field and the gravitational field as physical entities leads to far-reaching consequences. The basic equations sufficient to constructclassical electrodynamics (the Maxwell equations and the Lorentz force equation) are obtained by combining quantum considerations with two premises: (a) there exists a subatomic particle, theemon, each concrete emon having a specific electric property described by aspacelike four-vector; (b) every concrete charged particle possesses a specific electric property described by atimelike four-vector. Some other points of interest are also discussed, in particular, ones related to Einstein's gravitational field as well as the action-at-a-distance versus local-action issue. Einstein's second postulate of special relativity is also shown to need some revision of principle.     

A note on Lorentz transformation properties of conserved charges

It is shown that in classical relativistic field theories there may exist global conserved charges which are not Lorentz-covariant quantities.     

Conventionalism in special relativity

Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions conerning light signal synchronization, as proposed, e.g., by Ellis and Bowman, are required in order to refute conventionalism in special relativity.     

Transformations in special relativity

By using the principle of relativity alone (no assumptions about signals or light) it is shown that a relativisitic group of linear transformations of a spacetime plane is, if infinite, either Galilean, Lorentzian or rotational. The largest such finite group is a Klein 4-group, generated by space-reversal and time-reversal. In the infinite case an invariant of the group, denotedc, appears. Whenc is real, nonzero, noninfinite, then the group is a Lorentz group andc is identified with the speed of light. Lorentz transformations are represented through an algebra ofiterants that provides a link among Clifford algebras, the Pauli algebra and Herman Bondi'sK-calculus.     

A note on classical and quantum unimodular gravity

We discuss unimodular gravity at a classical level, and in terms of its extension into the UV through an appropriate path integral representation. Classically, unimodular gravity is locally a gauge fixed version of general relativity (GR), and as such it yields identical dynamics and physical predictions. We clarify this and explain why there is no sense in which it can “bring a new perspective” to the cosmological constant problem. The quantum equivalence between unimodular gravity and GR is more of a subtle question, but we present an argument that suggests one can always maintain the equivalence up to arbitrarily high momenta. As a corollary to this, we argue, whenever inequivalence is seen at the quantum level, that just means we have defined two different quantum theories that happen to share a classical limit. We also present a number of alternative formulations for a covariant unimodular action, some of which have not appeared, to our knowledge, in the literature before.     

Finsler gauge transformations and general relativity

The theory of gauge transformations in Finsler space is applied to general relativity. It is seen that the transformations produce new metrics which correspond to the introduction of physical fields. The geodesic equation in the transformed space is equivalent to the equation of motion in the original space where the field is included by a force term. An example is given of a transformation and resulting metric in which the electromagnetic potential is related to parameters of the gauge transformation rather than to gauge potentials. This implies that the electromagnetic field corresponds to a connection instead of a curvature. Another example is given which shows how Weyl or conformal transformations are related to a class of the gauge transformations.     

The introduction of Superluminal Lorentz transformations: A revisitation

We revisit the introduction of the Superluminal Lorentz transformations which carry from bradyonic inertial frames to tachyonic inertial frames, i.e., which transform time-like objects into space-like objects, andvice versa. It has long been known that special relativity can be extended to Superluminal observers only by increasing the number of dimensions of the space-time or—which is in a sense equivalent—by releasing the reality condition (i.e., introducing also imaginary quantities). In the past we always adopted the latter procedure. Here we show the connection between that procedure and the former one. In other words, in order to clarify the physical meaning of the imaginary units entering the classical theory of tachyons, we have temporarily to call into play anauxiliary six-dimensional space-time M(3, 3); however, we are eventually able to go back to the four-dimensional Minkowski space-time. We revisit the introduction of the Superluminal Lorentz transformations also under another aspect. In fact, the generalized Lorentz transformations had been previously written down in a form suited only for the simple case of collinear boosts (e.g., they formed a group just for collinear boosts). We express now the Superluminal Lorentz transformations in a more general form, so that they constitute a group together with the ordinary—orthochronousand antichronous—Lorentz transformations, and reduce to the previous form in the case of collinear boosts. Our approach introduces either real or imaginary quantities, with exclusion of (generic) complex quantities. In the present context, a procedure—in two steps—for interpreting the imaginary quantities is put forth and discussed. In the case of a chain of generalized Lorentz transformations, such a procedure (when necessary) is to be applied only at the end of the chain. Finally, we justify why we call transformations also the Superluminal ones.     

Complex conformal rescalings and complex Lorentz transformations

Complex Lorentz transformations and complex conformal rescalings with independent conformal factors and are investigated in terms of elements of the group GL(2,C) G (2,C). It is shown how a general element of this group decomposes into a standard conformal rescaling (with =), a pure spin transformation, complex null rotations, and a complex boost-rotation. Of particular interest are the pure spin transformations that leave invariant the metric but transform the permutation spinors. It is these transformations that, when , are responsible for seemingly complicating the transformation law of the derivative operator and of spinors dependent thereon. It has been suggested that to avoid this complication one should allow the rescaled metric to have torsion. It is argued here that simplicity can be achieved even when the torsion-free condition is imposed.