The Lorentz transformation

(1324) | |||

(1325) | |||

(1326) | |||

(1327) |

This transformation is tried and tested, and provides a very accurate description of our everyday experience. Nevertheless, it must be wrong! Consider a light wave which propagates along the -axis in with velocity . According to the Galilean transformation, the apparent speed of propagation in is , which violates the relativity principle. Can we construct a new transformation which makes the velocity of light invariant between different inertial frames, in accordance with the relativity principle, but reduces to the Galilean transformation at low velocities, in accordance with our everyday experience?

Consider an event , and a neighbouring event , whose coordinates
differ by , , , in , and by
, , , in . Suppose that at the event
a flash of light is emitted, and that is an event in which some
particle in space is illuminated by the flash. In accordance with the
laws of light-propagation, and the invariance of the velocity of
light between different inertial frames, an observer in will
find that

for . Any event near whose coordinates satisfy

(1330) |

(1331) |

where , , ,

where is a function of , , , and . [We can prove this by substituting into Eq. (1332) the following obvious zeros of the quadratic in Eq. (1328): , , , , , : and solving the resulting conditions on the coefficients.] Note that at is also independent of the choice of standard coordinates in and . Since the frames are Euclidian, the values of and relevant to and are independent of the choice of axes. Furthermore, the values of and are independent of the choice of the origins of time. Thus, without affecting the value of at , we can choose coordinates such that in both and . Since the orientations of the axes in and are, at present, arbitrary, and since inertial frames are isotropic, the relation of and relative to each other, to the event , and to the locus of possible events , is now completely symmetric. Thus, we can write

(1334) |

Equation (1335) implies that the transformation equations between primed and unprimed coordinates must be

The linearity of the transformation allows the coordinate axes in the
two frames to be orientated so as to give the *standard configuration*
mentioned earlier. Consider a fixed plane in with the equation
. In , this becomes (say)
,
which represents a moving plane unless
.
That is, unless the normal vector to the plane in , ,
is perpendicular to the vector
. All such planes
intersect in lines which are fixed in both and , and which are
parallel to the vector
in . These lines must correspond
to the direction of relative motion of the frames. By symmetry, two such
frames which are orthogonal in must also be orthogonal in . This
allows the choice of two common coordinate planes.

Under a linear transformation, the finite coordinate differences satisfy the
same transformation equations as the differentials. It follows from
Eq. (1335),
assuming that the events coincide in both frames,
that for any event with
coordinates in and
in , the following relation holds:

where is a constant. We can reverse the directions of the - and -axes in and , which has the effect of interchanging the roles of these frames. This procedure does not affect Eq. (1337), but by symmetry we also have

(1338) |

as in the Galilean transformation.

Equations (1336), (1339) and (1340) yield

where is a constant (possibly depending on ). It follows from the previous two equations that

where and are constants (possibly depending on ). Substituting Eqs. (1342) and (1343) into Eq. (1341), and comparing the coefficients of , , and , we obtain

(1344) | |||

(1345) |

We must choose the positive sign in order to ensure that as . Thus, collecting our results, the transformation between coordinates in and is given by

This is the famous

(1350) | |||

(1351) | |||

(1352) | |||

(1353) |

Not surprizingly, the inverse transformation is equivalent to a Lorentz transformation in which the velocity of the moving frame is along the -axis, instead of .