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.