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Relativity : the Special and General Theory

Chapter 9: VI The Theorem of the Addition of Velocities Employed in Classical Mechanics
By Albert Einstein
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Science - Physics

About This Book

The text introduces basic notions of space, time, and measurement and develops the special theory by examining simultaneity, coordinate systems, Lorentz transformations, and resulting effects such as time dilation and velocity addition, culminating in a four-dimensional spacetime view. It then formulates the general theory by presenting the equivalence of inertial and gravitational mass, a generalized relativity principle, and a geometric description of gravity using non-Euclidean spacetime and coordinate methods. Supplementary sections treat cosmological considerations, simple mathematical derivations, and empirical tests including the perihelion advance of Mercury, light deflection, and gravitational redshift.

VI
The Theorem of the Addition of Velocities Employed in Classical Mechanics

Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly or, in other words, with what velocity W does the man advance relative to the embankment during the process? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total he covers the distance relative to the embankment in the second considered. We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained; in other words, the law that we have just written down does not hold in reality. For the time being, however, we shall assume its correctness.