*Reference: **Einstein’s 1920 Book*

*This paper presents Part 1, Chapter 13 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.*

.

## Theorem of the Addition of Velocities. The Experiment of Fizeau

Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment.

*Velocities in the
material domain are small compared with the velocity of light because of high
inertia. High velocities exist in the radiation domain only where the inertia
is much lower. Velocities closer to the velocity of light cannot be expected in
the material domain.*

In Section 6 we derived the
theorem of the addition of velocities in one direction in the form which also
results from the hypotheses of classical mechanics. This theorem can also be
deduced readily from the Galilei transformation (Section 11). In place of the
man walking inside the carriage, we introduce a point moving relatively to the
co-ordinate system ** K’**in
accordance with the equation

*x’ **= wt’.*

By means of the first
and fourth equations of the Galilei transformation we can express ** x’**and

**in terms of**

*t’***and**

*x***, and we then obtain**

*t**x **= **(v **+ **w**)t.*

This equation
expresses nothing else than the law of motion of the point with reference to
the system ** K**(of the
man with reference to the embankment). We denote this velocity by the symbol

**, and we then obtain, as in Section 6,**

*W**W **= **v **+ *** w**………………………………….

*(A)*But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation

*x’ **= **wt’*

we must then express ** x’ **and

**in terms of**

*t’***and**

*x***, making use of the first and fourth equations of the**

*t**Lorentz transformation*. Instead of the equation (A) we then obtain the equation

*Equation (B) is valid
only when c is infinite or very large compared to the material velocity v that
is being added to a radiation velocity w.*

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity *w*. How quickly does it travel in the direction of the arrow in the tube ** T**(see the accompanying diagram, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity

**?**

*v*In accordance with
the principle of relativity we shall certainly have to take for granted that the
propagation of light always takes place with the same velocity *w** with respect to the liquid*,
whether the latter is in motion with reference to other bodies or not. The
velocity of light relative to the liquid and the velocity of the latter
relative to the tube are thus known, and we require the velocity of light
relative to the tube.

It is clear that we have the problem of Section 6 again before us. The tube plays the part of the railway embankment or of the co-ordinate system ** K**, the liquid plays the part of the carriage or of the co-ordinate system

**, and finally, the light plays the part of the man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by**

*K’***, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment**

*W*^{1}decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow

*v*on the propagation of light is represented by formula (B) to within one per cent.

Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of Maxwell-Lorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of each other, on which electrodynamics was built.

*According to the principle of relativity the general laws are consistent with changes in uniform motion and inertia. This applies to the laws of electrodynamics too. Addition of velocities is valid when the inertia of the systems is of the same order of magnitude (both in the material domain). Lorentz transformation is valid when the inertia of the systems is many orders of magnitude apart (as in material and field domains). *

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## FINAL COMMENTS

**According to the principle of relativity the general laws are
consistent with changes in uniform motion and inertia. This applies to the laws
of electrodynamics too. **

**Addition of velocities is valid when the inertia of the two systems
is of the same order of magnitude (both in the material domain). Lorentz
transformation is valid when the inertia of the systems is many orders of
magnitude apart (as in material and field domains). Neither of the equations
may apply when the inertia of both systems is in the field domain. This is
because the inertia associated with the velocity of light is not exactly zero.**

**For equations to work flawlessly a relationship between velocity
and inertia needs to be derived for material domain as well as for field
domain.**

.