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.