PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 8 SEPTEMBER 15. 1922 Number 9 
THE MEANING OF ROTATION IN THE SPECIAL THEORY OF 
RELATIVITY 
By Phiup Frankun 
Department of Mathematics, Princeton University 
Communicated July 27, 1922 
The novelty of the special theory of relativity consists in its replacing 
the equations for translation given by the Newtonian theory by the 
Torentz transformations. It is natural to ask what new equations should 
be used to express a rotation in this theory. Most of the writers on the 
subject ( e. g., A. Einstein, Leipzig, Ann. Physik, 49, 1916, §3; M. V. Taue, 
Die Relativitdtstheorie, Vol. 2, 1921, p. 162) have assumed that the New- 
tonian equations for rotation may be used without change in the new 
theory. While this is permissible as a "first approximation" for points 
near the axis of rotation, it is not a satisfactory transformation for the 
whole of space, since it involves velocities greater than that of light for 
points sufficiently far from the axis. The object of this note is to present a 
definition of rotation which is free from this objection, and to deduce some 
of its properties. 
In the Newtonian theory the equations of transformation connecting 
the coordinates in a system K with those in a system K' moving uniformly 
with respect to K along the ^c-axis are : 
x' = X — vt, y' = T, = z,t' ^ t. (1) 
For a rotation about the 2;-axis, we have : 
x' = X cosco^ + y sinco/, y' = — x sinoo/ + y cosco/, z' = z,t' — t, (2) 
or if polar coordinates are introduced in the xy plane : 
e' = e - cot, r' = r,z' ^ z, t' = t. 
(2a) 
