George David Birkhoff (1884–1944)

Birkhoff bases his analysis on the assumption that distance is determined by the time a light flash needs to travel that distance. In modern relativity, this idea is ensconced in the Minkowski metric and in the availability of local inertial coordinates. Then follows a discussion of isotropic and aleotropic metrics These concepts are now expressed in terms of symmetries and Killing vectors. Next comes an analysis, based on the above assumption, of relativity in two dimensions, which does indeed capture many of the important concepts and is easy to read.

After covering four dimensional tensor analysis, Birkhoff discusses a classification of tensor equations which probably corresponds to the Petrov types. Then he proves four important results; only the last of which seems to be widely known. The first is that the equations of electromagnetism (Maxwell) are determined by the space time concept of general relativity. They could not take any other form. The next is that a charged particle of zero mass will accelerate to the speed of light in an electromagnetic field. This suggests that charge is a property of mass and not a seperate fundamental entity as commonly supposed. The third is that the radius coordinate used in the Schwarzschild solution is the only radius that will allow the paths of particles to be expressed in terms of a central force and hence easily related to Newtonian dynamics. Finally he shows that any spherically symmetric empty space solution of the Einstein field equations, must be static. This means no spherically symmetric gravitational waves. This seems to suggest that gravitational waves must be transverse. Some have criticized the details of this theorem, but most physicists respect it and there have been no counterexamples in 87 years.

My copy of this book was obtained on inter-library loan from the University of Santa Clara in California. I noticed that the last time the book was checked out was 2002. Before that it was1970, but in the 50's and early 60's there were many more check outs. In reading this book I was reminded of Edgar Allen Poe's phrase "a quaint and curious volume of ancient and forgotten lore". In terms of modern relativity theory, Birkhoff's book may well be ancient, quaint and curious, but it should not be forgotten.… (mais)