THE MASS-ENERGY RELATION
Let us also recall in this context that from as early as 1910, beginning
with Waldemar von Ignatowsky followed by Philipp Frank and Her-
mann Rothe, physicists and mathematicians realized that the (structure
of the) Lorentz transformations, and hence of the relativistic velocity
addition theorem as well, can be derived without invoking the light
postulate or any other reference to electromagnetic phenomena merely
by using general principles, such as the principle of relativity or the
isotropy and homogeneity of space.
31
Of course, such group-theoretical
derivations can involve only a limiting velocity α in lieu of c. The price
to be paid for not invoking the light postulate or any other equivalent
assumption is as Wolfgang Pauli phrased it: “Nothing can, naturally, be
said about the sign, magnitude and physical meaning of α.Fromthe
group-theoretical assumption it is only possible to derive the general
form of the transformation formulae, but not their physical content.”
32
The fact that for α =∞these equations degenerate into the Galilean
transformations of Newtonian physics and the mass-energy relation
E = mα
2
becomes meaningless can be interpreted as an indication that
this relation is an exclusively relativistic result. Conversely, it can also
be said that the mass-energy relation E = mc
2
or the velocity-dependent
equation of inertial mass can replace the second postulate in the logical
construction of the special theory of relativity.
33
As long as α remains
finite, its indeterminacy affects the numerical relation between mass and
energy but not the conceptual content of this relation.
The preceding derivations of the mass-energy relation belong to class
(I) in the classification described earlier. The first derivation belonging to
class (II) is Einstein’s 1906 second derivation. Like his first, it is based on
31
For bibliographical references up to 1964 see H. Arzeli
`
es, Relativistic Kinematics
(Oxford: Pergamon, 1966), pp. 80–82. Important more recent group-theoretical derivations
of (generalized) Lorentz transformations are: G. S
¨
ussmann, “Begr
¨
undung der Lorentz-
Gruppe allein mit Symmetrie- und Relativit
¨
atsannahman,” Zeitschrift f
¨
ur Naturforschung
24a, 495–498 (1969); V. Gorini and A. Zecca, “Isotropy of Space,” Journal of Mathematical
Physics 11, 2226–2230 (1970); A. R. Lee and T. M. Kalotas, “Lorentz Transformations from
the First Postulate,” American Journal of Physics 43, 434–437 (1975); J.-M. Levy-Leblond,
“One More Derivation of the Lorentz Transformation,” American Journal of Physics 44,
271–277 (1976).
32
W. Pauli, The Theory of Relativity (New York: Pergamon, 1958), p. 11.
33
For more details and a simple group-theoretical derivation of the (general) Lorentz
transformations see M. Jammer, “Some Foundational Problems in the Special Theory of
Relativity,” in G. Toraldo di Francia, ed., Problems in the Foundations of Physics, Proceedings
of the International School of Physics ‘Enrico Fermi’, Course LXXII (Amsterdam: North-
Holland, 1979), pp. 202–236.
77