THE NEW KINEMATICS 157querel Rays [i.e., /?-rays] and the Apparent Mass of the Electron' [K3]. Stimu-
lated by these investigations, Abraham soon thereafter produced the complete
answers for the electromagnetic energy (Ec[m) and the electromagnetic momentum
(pc\m) °f an electron considered as a hard sphere with charge e and radius a and
with uniform charge distribution (|3 = v/c, fi = 2e^2 /3ac^2 ):*As usual, we assume the electron to move in the x direction. Equations 7.33 and 7.34 were first
published in 1911 by von Laue [L10].At the 74th Naturforscherversammlung, held in Karlsbad in September 1902,
Kaufmann presented his latest experimental results [K4]. Immediately after him,
Abraham presented his theory [Al]. Kaufmann concluded that 'the dependence
[of E on v] is exactly represented by Abraham's formula.' Abraham said, 'It now
becomes necessary to base the dynamics of the electrons from the outset on elec-
tromagnetic considerations' (in 1903 he published his main detailed article on the
rigid electron [A2]). One sees what Lorentz meant in his Columbia lectures: if it
would have been true, if it could have been true, that the E-v relation were
experimentally exactly as given by Eq. 7.29, then two things would have been
known: the electron is a little rigid sphere and its mass is purely electromagnetic
in origin.
Such was the situation when in 1904 Lorentz proposed a new model: the elec-
tron at rest is again a little sphere, but it is subject to the FitzGerald-Lorentz
contraction [L9]. This model yields a velocity dependence different from Eqs. 7.29
and 7.30:where ju 0 = 3ju/4, ju, = 5/i/4, and n is as in Eqs. 7.29 and 7.30. Lorentz, aware
of Kaufmann's results and their agreement with Abraham's theory, remarked that
his equations ought to agree 'nearly as well ... if there is not to be a most serious
objection to the theory I have now proposed' and did some data-fitting which led
him to conclude that there was no cause for concern.
In order to understand Lorentz's equations (Eqs. 7.31 and 7.32) and Poincare's
subsequent proposal for a modification of these results, it is helpful to depart
briefly from the historic course of events and derive Lorentz's results from the
transformation properties of the electromagnetic energy momentum tensor density
T,,, [P13]. With the help of that quantity we can write (in the Minkowski
metric)*