Mach’s Principle Number 4
We continue our notes on Mach’s Principle (MP) and its various interpretations. Today we will discuss a fourth interpretation of MP of our list of 21, namely;
Mach Principle 4: “As a mechanical interaction of masses, in particular through as interactive Machian ‘kinetic energy’”.
That particular interpretation is not only interesting by itself, but also for its apparent historical origin. The above-mentioned ‘Machian kinetic energy’ is explained in the article by John D. Norton on pages 32-33 of [Barb95], from which my brief exposition is here based.
The idea emerges as a Machian hypothesis by its own essentially because it arises from a quote by Einstein in a 1913 paper, where he discussed “the hypothesis of the relativity of inertia” and cites an “ingenious pamphlet of the Viennese mathematician W. Hofmann”.
Essentially, Hofmann (1904) argued that the standard kinetic energy of a body of mass () was unsatisfactory, due to an intrinsic asymmetry: the kinetic energy of with respect to would not be the same as that of with respect to . This consideration would be a necessary outcome of the assumption of the relativity of inertia.
Therefore, in order to have a symmetric law for the kinetic energy, Hofmann proposed that
where is a constant and an undetermined function. The idea was: since the mass measured of a given particle should arise as a contribution of all masses in the Universe according to that law, then by an integration of all masses, that formula would have to converge to the usual formula for the kinetic energy.
Further work along that line was carried out by H. Reissner in 1914-15, but he cited Mach for the modified kinetic energy law, and Hofmann was not mentioned.
(Well, I was unsuccessful to find any further online references for W. Hofmann, either a bio or link to his “pamphlet”…)
For historical details, curiosities and references, the reader is referred to Norton’s article.