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[Article] Uniformly accelerated observers in special relativity

[Article] Uniformly accelerated observers in special relativity

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INVESTIGACI ´ ON REVISTA MEXICANA DE F ´ ISICA 52 (1) 70–73 FEBRERO 2006 Uniformly accelerated observers in special relativity G.F. Torres del Castillo Departamento de F´ ısica Matem´atica, Instituto de Ciencias, Universidad Aut´onoma de Puebla, 72570 Puebla, Pue., M´exico C.I. P´erez S´anchez Facultad de Ciencias F´ ısico Matem´aticas, Universidad Aut´onoma de Puebla, Apartado postal 1152, 72001 Puebla, Pue., M´exico Recibido el 25 de octubre de 2005; aceptado el 9 de enero de 2006 The red shift for an electromagnetic wave measured by two observers in a uniformly accelerated frame, which, according to the equivalence principle, should correspond to a gravitational red shift, is calculated as well as the bending of light rays. Keywords: Special relativity; red shift; bending of light rays. Se calcula el corrimiento al rojo para una onda electromagn´etica medido por dos observadores en un sistema de referencia uniformemente acelerado, el cual, de acuerdo con el principio de equivalencia, debe corresponder a un corrimiento al rojo gravitacional, as´ ı como la desviaci´on de rayos de luz. Descriptores: Relatividad especial; corrimiento al rojo; desviaci´on de rayos de luz. PACS: 03.30.+p; 04.20.-q 1. Introduction Just like in Newtonian mechanics, in the special theory of rel- ativity the inertial frames of reference play an essential role (see, for example, Refs. 1,2) and the Lorentz transformations give the relationship between the space-time coordinates of events measured in two different inertial frames. However, in Newtonian mechanics, as well as in special relativity, one can make use of non-inertial reference frames. In fact, the equivalence principle states that, in the absence of gravita- tional fields, a reference frame that is linearly accelerated is locally identical to a reference frame at rest in a gravitational field. In particular, making use of the equivalence principle, it is possible to derive the existence of a gravitational red shift for electromagnetic waves and of a bending of the light rays by considering their propagation viewed from an accelerated frame in the absence of a gravitational field (see, for example, Refs. 3-5). References 3 and 4 contain computations of these effects based on Newtonian relations. The aim of this paper is to find, in the context of spe- cial relativity, the red shift for an electromagnetic wave, mea- sured by two observers in a uniformly accelerated frame sep- arated by a fixed distance, and the trajectory of a light ray. In Sec. 2 we derive the relationship between the space-time co- ordinates of events measured by an inertial frame and a ref- erence frame whose origin has a constant acceleration with respect to an instantaneously co-moving inertial frame (see also Refs. 6-8). This coordinate transformation is employed in Sec. 3 to find the exact red shift formula and the bending of light rays.
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