*Reference: **Einstein’s 1920 Book*

*This paper presents Part 1, Chapter 11 from the book **RELATIVITY: THE SPECIAL AND GENERAL
THEORY by A. EINSTEIN.
The contents are from the original publication of this book by Henry
Holt and Company, New York (1920).*

*The paragraphs of the original
material (in black) are accompanied by brief comments (in color) based on the present
understanding. Feedback on these comments is appreciated.*

*The heading below is linked to
the original materials.*

.

## 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:

- The time-interval (time) between two events is independent of the condition of motion of the body of reference.
- The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

*Einstein’s conclusions above are correct. These conclusions, however, do not take **into account** the vector of motion.*

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.

*Einstein
formulates the question, “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?” The formulation of the equation should be such that
it allows the velocity of light c to become infinite and still yield a result.*

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

**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’***K**by the three perpendiculars

**on the co-ordinate planes, and with regard to time by a time-value**

*x, y, z***. Relative to**

*t***,**

*K’**the same event*would be fixed in respect of space and time by corresponding values

**, 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.**

*x, y, z, t.**An event may be localized completely only in a framework of infinite inertia. Einstein is looking at two inertial frameworks in material domain that are relatively localized.*

Obviously our problem can be exactly formulated in the following manner. What are the values ** x’, y’, z’, t’** of an event with respect to

**, when the magnitudes**

*K’***of the same event with respect to**

*x, y, z, t,***are given? The relations must be so chosen that the law of the transmission of light**

*K**in vacuo*is satisfied for one and the same ray of light (and of course for every ray) with respect to

**and**

*K***. For the relative orientation in space of the co-ordinate systems indicated in the diagram (Fig. 2), this problem is solved by means of the equations:**

*K’*This
system of equations is known as the “Lorentz transformation.” ^{1}

^{1 }A simple derivation of the Lorentz transformation is given in
Appendix I.

*We also
know that velocity becomes zero when inertia is infinite, and velocity becomes
infinite when inertia is zero. That means the product of inertia and velocity is
a constant. Since inertia is directly proportional to frequency, we may say
that the product of frequency and velocity is a constant. This leads to the
interesting result:*

*λ f ^{2}= const. (the units of acceleration)*

If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:

This
system of equations is often termed the “Galilei transformation.” The Galilei
transformation can be obtained from the Lorentz transformation by substituting
an infinitely large value for the velocity of light ** c** in the latter transformation.

*Galilei
transformations are correct, and they should apply to light also if the inertia
of light is taken into account.*

Aided
by the following illustration, we can readily see that, in accordance with the
Lorentz transformation, the law of the transmission of light *in vacuo* is satisfied both for the
reference-body ** K** and for the reference-body

**. A light-signal is sent along the positive x-axis, and this light-stimulus advances in accordance with the equation**

*K’**x = ct,*

i.e. with the velocity ** c**. According to the equations of the Lorentz transformation, this simple relation between

**and**

*x***involves a relation between**

*t***and**

*x’***. In point of fact, if we substitute for**

*t’***the value**

*x***in the first and fourth equations of the Lorentz transformation, we obtain:**

*ct*from which, by division, the expression

*x’ = ct’*

immediately
follows. If referred to the system ** K’**, the propagation of light takes
place according to this equation. We thus see that the velocity of transmission
relative to the reference-body

**is also equal to**

*K’***. The same result is obtained for rays of light advancing in any other direction whatsoever. Of course this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.**

*c*.