Derivation of the Lorentz Transformations
Gerardo Urrutia* 18 August 2013
*Facultad de Ciencias, Universidad Nacional Aut onoma de M exico, Mexico D.F. 04510, M exico *geursan@ciencias.unam.mx Abstract This article presents a simple way to derive the Lorentz transformations using calculations performed with basic algebra which are not separate from our physical intuition but assuming that time is not invariant, because it uses the postulate of Einstein that says: A beam has speed c in all inertial reference systems.
Introduction
The laws of physics must be the same in all inertial reference systems... A late 19th century electromagnetic theory reaches its highest peak with Maxwells equations, which can be veried experimentally which is very strong. But Maxwells equations do not preserve the structure to Galilean transformations. This was a problem because the electromagnetic theory could not relate to Newtonian mechanics, which was also very strong and important. Some mathematicians constructed linear transformations where remained invariant Maxwell equations (which is important for a law of physics), but there was no physical explanation for these transformations. After Einstein postulated the theory of relativity, the Lorentz transformations take physical explanation (which is developed throughout this paper). The study of the Lorentz transformations is important for modern physics, because modern phenomena and physical theories relate with particles and elds that propagate at very high velocities near the speed of light. Today every physical theory must remain invariant under these transformations, not to change its struc1
ture in dierent inertial frames
Lorentz Transformations
I want to remark that now the time is part of a very special coordinates are in Minkowski space, and we understand as x with = 0, 1, 2, 3 and in theory of relativity Greek indices are used to dene spacetime where xi con i = 1, 2, 3 its the spatial coordinates for example x, y, z . x0 = t its the time in covariant form and x0 = t the contravariant form. For a family treatment can work even in its classical form assuming that now the time is not universal as in Newtonian theory (or Foliation in Advanced Theory of Relativity). I say this to justify the metric structure that denes this Minkowski space (physical space-time) ds2 = dx2 + dy 2 + dz 2 c2 dt2 . We consider two frames O and O in relative motion with velocity v along their common xx direction. For a given point event, and observer in O will assign the space coordinates (x, y, z ) and the time t, while an observer in O will assign the space coordinates (x , y , z ) and the time t . We want to derive the set transformations that enable us to calculate the coordinates (x , y , z ) and the time t in terms
of the coordinates (x, y, z ) and the time t. Suppose that, at the instant the two origins of the frames (x + ct ) = A1 (x + ct) (9) coincide, a ash of ligth is produced at the origin. Each observer see the ligth propagating with the sa- whwew A can be determined by dividing Eq.(8) by me speed c in all directions. In other words, with Eq. (9) and using Eq. (7). After this process, we obrespect to his own frame, each observer must see the tain 1 = A2 (vt ct)/(vt + ct). Hence wave front as a sphere centered on the corresponding origin with radius c times the associated time. The (1 + v/c)1/2 equations of the wave fronts are (10) A= (1 v/c)1/2 x2 + y 2 + z 2 c2 t2 = 0 and x 2 + y 2 + z 2 c2 t 2 = 0 (1) The quantity A is shown to satisfy
so that we must satisfy
A + A1 = 2 where
and A A1 = 2v/c
(11)
x +y +z c t =x +y +z c t
2 2
2 2
(2) = 1 1 v 2 /c2 (12)
We assume that the coordinates transverse to the motion are invariant y =y (3) A = (1 + v/c) z =z Therefore Eq. (2) becomes x 2 c2 t 2 = x2 c2 t2 or equivalenty (x ct )(x + ct ) = (x ct)(x + ct) (6) (5) (4)
From combining Eqs. (11) we obtain
and A1 = (1 v/c)
(13)
We add Eqs. (8) and (9) and use Eqs. (13) to obtain
2x = (1 + v/c)(x ct) + (1 v/c)(x + ct) (14) After performing the specied algebra, Eq. (14) becomes x = (x vt) (15)
Now there is only ane relative speed v . Therefore, from the point of view of O the origin O must have We subtract Eqs. (8) and (9) and use Eqs. (13) to obtain coordinate x = vt, that is If x = 0 then x = vt (7)
2t = (1 + v/c)(x ct) (1 v/c)(x + ct) (16)
For the involved spacetime transformations we write which implies (x ct ) = A(x ct) (8) 2 t = (t vx/c2 ) (17)
Conclusions
y=y The equations (3), (4), (15) and (17) are the Lorentz transformations. Agrupandolas para la comodidad del lector, tenemos que x = (x vt) (18) z=z
t = (t vx /c2 ) In matrix form, the transformation A applied to a vector x (for Eqs. (18) with c = 1) in Minkowski space is v A = = 0 0 v 0 0 0 0 0 0 1 0 0 1
y =y z =z
t = (t vx/c ) For transformations of O relative to O just simply change the sign of the velocity (this physical sense) x = (x + vt ) (19)
(20)
And analogous to Eqs (19) changing the sign of the velocity v .
Referencias
[1] Wolfgang Rindler, Relativity, 2 ed, Oxford University Press, New York, 2006 [2] Bernard Schutz, A rst course in General Relativity, 2 ed, Cambridge University Press, UK, 2009 [3] Leonard Parker and Glenn M, Special Relativity and Diagonal Transformations, Am. J. Phys. 38, 218 (1970) [4] Leonard Parker, On the Lorentz Transformation, Am. J. Phys. 39, 223 (1971)
