# Einstein

Relativity: The Special and General Theory

Albert Einstein: Relativity

Part I: The Special Theory of Relativity

The Lorentz Transformation

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

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,

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