# Relativity: The Special and General Theory

The Lorentz Transformation

The results of the last three sections show that the apparent incompatibility of the law of

propagation of light with the principle of relativity (Section 7) has been derived by means

of a consideration which borrowed two unjustifiable hypotheses from classical

mechanics; these are as follows:

1. The time-interval (time) between two events is independent of the condition of

motion of the body of reference.

2. The space-interval (distance) between two points of a rigid body is independent of

the condition of motion of the body of reference.

If we drop these hypotheses, then the dilemma of Section 7 disappears, because the

theorem of the addition of velocities derived in Section 6 becomes invalid. The

possibility presents itself that the law of the propagation of light in vacuo may be

compatible with the principle of relativity, and the question arises: How have we to

modify the considerations of Section 6 in order to remove the apparent disagreement

between these two fundamental results of experience? This question leads to a general

one. In the discussion of Section 6 we have to do with places and times relative both to

the train and to the embankment. How are we to find the place and time of an event in

relation to the train, when we know the place and time of the event with respect to the

railway embankment ? Is there a thinkable answer to this question of such a nature that

the law of transmission of light in vacuo does not contradict the principle of relativity ? In

other words : Can we conceive of a relation between place and time of the individual

events relative to both reference-bodies, such that every ray of light possesses the

velocity of transmission c relative to the embankment and relative to the train ? This

question leads to a quite definite positive answer, and to a perfectly definite

transformation law for the space-time magnitudes of an event when changing over from

one body of reference to another.

Before we deal with this, we shall introduce the following incidental consideration. Up to

the present we have only considered events taking place along the embankment, which

had mathematically to assume the function of a straight line. In the manner indicated in

Section 2 we can imagine this reference-body supplemented laterally and in a vertical

direction by means of a framework of rods, so that an event which takes place anywhere

can be localised with reference to this framework. Similarly, we can imagine the train

travelling with the velocity v to be continued across the whole of space, so that every

event, no matter how far off it may be, could also be localised with respect to the second

framework. Without committing any fundamental error, we can disregard the fact that in

reality these frameworks would continually interfere with each other, owing to the

impenetrability of solid bodies. In every such framework we imagine three surfaces

perpendicular to each other marked out, and designated as " co-ordinate planes " (" co-

ordinate system "). A co-ordinate system K then corresponds to the embankment, and a

co-ordinate system K' to the train. An event, wherever it may have taken place, would be

fixed in space with respect to K by the three perpendiculars x, y, z on the co-ordinate

planes, and with regard to time by a time value t. Relative to K1, the same event would be

fixed in respect of space and time by corresponding values x1, y1, z1, t1, which of

course are not identical with x, y, z, t. It has already been set forth in detail how

these magnitudes are to be regarded as results of physical measurements.