The Speed of Light under the Generalized Transformations, Inertial Transformations, Everyday Clok Synhronization and the Lorentz- Einstein Transformations Bernhard Rothenstein Abstrat. Starting with Edwards synhrony parameter dependent transformation equations in two spae dimensions, transformation equations are derived for speeds. The obtained results are partiularized to the ase of the inertial (Tangherlini, Selleri) transformations and to the transformation equations that are the result of the everyday lok synhronization. We begin by extending Edwards [1] generalized transformations to two spae dimensions in order to derive the generalized transformation of the speed of light following Einstein s philosophy. Restriting the results, hoosing different values for the synhrony parameter present in the generalized transformations, we reover Selleri s inertial transformations [] and those whih are a result of the everyday lok synhronization proedure [3]. The purpose of our Note is to show that the results we present are a diret onsequene of the Lorentz-Einstein transformations (LET) whih are in turn a onsequene of the relativisti postulate and of the onstany of the one way speed of light. Many general transformations equations for the spae-time oordinates of the same event are proposed. A review of them is presented in [4]. Some of them work when in the stationary I inertial referene frame the loks are standard synhronized, following a synhronization proedure proposed by Einstein: A light flash is emitted by lok A in an inertial referene frame at a time t A (as read on lok A) and reeived by lok B in the same frame at time t B (as read on lok B). The two loks are defined to be standard synhronized if (t B -t A ) is equal to the distane between the loks. That is, the loks are synhronized if they measure the speed of a light signal traveling between them to be. [5]. In the moving referene frame, I, the loks are synhronized using a subluminal, one way signal, propagating in the positive diretion of the permanently overlapped OX(O X ) axes with speed + =/n; (where n>1 represents a synhrony parameter, defined by different authors in different ways). 1
With referene to the transformation equation of the spae oordinates, that expresses the spae oordinate of an event in I, (x ), as a funtion of its oordinates (x,t E ) in I, all the proposed general transformations [1], [], [6], [7], are synhrony parameter independent. Therefore, they oinide with the transformation proposed by Einstein s speial relativity theory, i.e., 1/ ( ) x = γ x t E (1) where 1 γ = represents the Lorentz fator. In order to extend the problem to two spae dimensions, we ould take into aount the invariane of distanes, measured perpendiular to the diretion of relative motion y = y () y = y whih is a diret onsequene of the relativisti postulate. Its proof does not involve light signals. [8] The study of the aberration of light effet [9] involves the angles θ and θ whih define the diretions in whih the same light signal propagates when we deteted them from I and I respetively. The trigonometri funtions whih define an angle being expressed as a quotient between two lengths measured in the same inertial referene frame, equations (1) and () tell us that the transformation equation whih relates the angles θ and θ should be synhrony parameter independent; hene it should be the same in all theories. Consider a signal that propagates in I along a diretion that makes an angle θ with the positive diretion of the OX axis. After a given time of propagation it generates the event E[x=rosθ;y=rsinθ;t E = r ]. (3) Performing the transformation of the spae oordinates of event (3) to I the results in r osθ = γ r osθ (4)
r sinθ = r sinθ (5) r = γ r osθ (6) and, therefore, the formulas that aount for the aberration of light effet are osθ osθ =. (7) osθ Solved for θ (7) leads to osθ + osθ =. (8) 1+ osθ Equation (8) is largely used in wave front relativity [10]. In referene to the transformation of the time oordinates different theories present synhrony parameter dependent transformation equations. In our notations that presented by Edwards [1] reads t = γ 1 (1 n) t E 1 n x + +. (9) The result is that if we define the speed of a photon measured in I using standard synhronized loks as =x/t E and the speed of the same photon measured in I using nonstandard synhronized loks as = r / t we obtain the following formula by whih they are related osθ =. (10) 1 + (1 n) n + osθ 3
Figure 1. The variation of 1 / / with θ for different values of / n= + We have shown that the inertial transformation equations proposed by Selleri [] are a partiular ase of Edwards transformation equations, orresponding to the ondition of absolute simultaneity. Speifially, it applies to the ondition that the equation that performs the transformation of the time oordinate (9) is spae oordinate independent. [10] i.e. 4
Figure. The variation of / 0 with the angle θ n= n + =0 (11) with whih (10) beomes osθ 1 + / =. (1) Expressing the right side of (11) as a funtion of θ via (8) it beomes n= 1 + / =. (13) 1+ osθ 5
In order to illustrate the anisotropy of the propagation of the synhronizing signal in I, in Figure 1 we present the variation of (, n= 1 + / / ) with θ for different values of /. Anisotropy fades away in the diretion of the O Y axis (θ =π/). Another non-standard loks synhronization proedure [3] is known as everyday loks synhronization and orresponds to a synhronizing signal that apparently propagates with an infinite speed i.e. orresponding to a synhrony parameter n=0. Under suh onditions the general transformation equation beomes n= 0 =. (14) osθ We present in Figure the variation of / n= 0 with θ. As we see it is / independent. Equations (13) and (14) are in agreement with Reihenbah s definition of the round trip speed of light [11] 1 1 = + + (15) where + orresponds to θ, orresponding to θ + π. Therefore, we have reovered that way all the results proposed in a three-spae dimensions approah reduing the problem to one of two spae dimensions one. [13] Conlusions The general transformation equations proposed by Edwards [1] ould be derived starting with the seond postulate stated as the onservation of the round trip speed of light but also transforming via the Lorentz-Einstein transformation the spae-time oordinates of the event generated by the synhronizing signals when they arrive at the loation of the lok to be synhronized. Partiular values of the synhrony parameter present in Edwards equation reover all the theories presented by different authors, (Tangherlini [6], Selleri [], Leubner, Aufinger and Krumm [3]). Knowing all that, we an sustain a onlusion presented by Rizzi, Ruggiero and Serafini [1] One orretly and expliitly phrased, the priniple of 6
SRT allow for a wide range of theories that differ from the standard SRT only for the differene in the hosen synhronization proedures, but are wholly equivalent to SRT prediting empirial fats. Referenes [1] W.F. Edwards, Speial relativity in anisotropi spae, Am. J. Phys. 31, 48 (1963) [] F. Selleri, Remarks on the transformation of spae and time, Apeiron, 4, 116-10 (1997) [3] C. Leubner, K. Aufinger, and P. Krumm, Elementary relativity with everyday synhronized loks, Eur.J.Phys. 13, 170-177 (199) [4] Yuan Zhong Zhang, Speial Relativity and its Experimental Foundations, (World Sientifi, Singapore 1997) [5] Thomas A. Moore, A Traveler s Guide to Spae-time: An Introdution to the Speial Theory of Relativity,(MGraw-Hill, In. New York 1995) p.30 [6] F.R Tangherlini, On energy-momentum tensor of gravitational field, Nuovo Cimento Suppl. 0, 351 (1961) [7] R. Mansouri and R.U. Sexl, A test theory of speial relativity. I: Simultaneity and lok synhronization, General. Relat.Gravit. 8, 497-513 (1977) [8] Thomas A. Moore, A Traveler s Guide to Spaetime: An Introdution to the Speial Theory of relativity, (MGraw-Hill,In. New York 1995) pp.65-68 [9] Thomas A. Moore, A traveler s Guide to Spaetime: An Introdution to the Speial Theory of Relativity (MGraw-Hill, In. New York 1995) pp.150-154 [10] William Moreau, Wave front relativity, Am.J.Phys. 6, 46 (1994) [11] H. Reihenbah, Axiomatization of the Theory of Relativity, (University of California Press 1969) [1] Guido Rizzi, Matteo Lua Ruggiero and Alessio Serafini, Synhronization gauges and the priniples of speial relativity, arxiv:grq/0409 105v 18 Ot 004 [13] Iyer Chandru, The speed of light under the IST and Lorentz transformations, arxiv0810 178v1 [physis.gen-ph] 7