D O C . 4 8 0 O N K A L U Z A ’ S T H E O R Y , P A R T 2 7 5 9 version of) Kaluza’s paper to the Prussian Academy for publication see Einstein to Theodor Kaluza, 14 and 29 May 1919 (Vol. 9, Docs. 40 and 48). Einstein wrote to Kaluza again on 14 October 1921 (Vol.12, Doc. 270) and suggested that he might present the paper to the Prussian Academy after all. Kaluza acknowledged that the difficulty remained unresolved in his answer of 24 October 1921 (Vol. 12, Doc. 281). Klein, O. 1926 and Einstein in the present document only escaped this difficulty by assuming (via the sharpened cylinder condition) that is a constant. With this assumption, the five-dimensional geodesic equation becomes equivalent to the Lorentz force law in four dimensions. For a comparison of the articles by Einstein, Mandel, Klein, and Fock, see Goenner 2004, sec. 6.3. In contrast to Reichenbach’s attempt (see the Introduction, sec. III, for further discussion) of repre- senting charged particles subject to electromagnetic fields by four-dimensional geodesics (of an af- fine connection custom-tailored for this purpose), the five-dimensional approach advocated in the present document allows for bodies with arbitrary specific charge to correspond to five-dimensional geodesics. [8]Kaluza 1921, pp. 968–969, derived the Einstein-Maxwell equations from the (implicit) assump- tion that the five-dimensional Ricci tensor vanishes for the special case of weak fields, i.e., using the additional assumption that the five-dimensional metric tensor deviates only slightly from flatness. In addition to obtaining the Einstein-Maxwell equations in four dimensions, Kaluza also obtained a fif- teenth field equation for the component of the five-dimensional metric, which corresponds to a new scalar field in four dimensions. For Einstein’s reasons for setting via introducing the sharpened cylinder condition, see note 3 above. [9]This is the five-dimensional equivalent of Einstein’s “ΓΓ−Lagrangian,” whose Euler-Lagrange equations in four dimensions are the Einstein field equations. It first appears in Einstein to Hendrik A. Lorentz, 19 January 1916 (Vol. 8, Doc. 184). For analysis of Einstein’s variational derivation of the Einstein field equations using this Lagrangian, see Janssen and Renn 2007, sec. 9. [10]The draft of this addendum [1 068] reads: “Herr H. Mandel macht mich darauf aufmerksam, dass die von mir hier mitgeteilten Ergebnisse nicht neu sind. ¢Insbesondere ist² Die ‘ -Transforma- tion’ schon von Klein (Z. f. Phys. 3. 37) verwendet worden. Noch mehr Berührungspunkte hat Fock’s Untersuchung (Z. f. Phys 39, 226, 1926) mit meinen Betrachtungen. Wenn ich meine Mitteilungen aufrecht erhalte, so ist es nur ¢die² in der Meinung, dass sie als Ergebnis selbstständiger Betrachtung ein gewisses Interesse haben können, und dass sie dazu beitragen können, das Interesse der Fachge- nossen auf die von Kaluza angebahnte Auffassung des Zusammenhanges von Gravitation und Elek- trizität zu lenken. Es verdient hervorgehoben zu werden dass sowohl Klein als auch Fock und Mandel selbstständig, d.h. ohne die Kenntnis von Kaluzas Arbeit auf die Heranziehung des fünfdimensiona- len Kontinuums zur formalen Vereinigung von Gravitation und Elektrizität gekommen sind.” (“Herr H. Mandel alerts me to the fact that the results communicated here are not new. The ‘‘ -Transfor- mation’’ has already been used by Klein (Z. f. Phys. 3. 37). There are even more similarities to Fock’s investigation (Z. f. Phys 39, 226, 1926). If I keep up my own communications, then I do so only because I am of the opinion that as a result of independent thought they might be of a certain interest, and they might contribute to directing the interest of the community to the approach connecting grav- itation and electricity pioneered by Kaluza. It deserves to be noted that Klein, Fock, and Mandel inde- pendently thought of using the five-dimensional continuum in order to achieve a formal unification of gravitation and electricity, i.e., they did so without knowledge of Kaluza’s work.”) The physicists named are Heinrich Mandel (Leningrad), Vladimir Fock (Leningrad), and Oskar Klein. The papers cited are Klein, O. 1926, Fock 1926, and Mandel 1926. For a comparison of the theories of Kaluza and Klein, see Goenner and Wuensch 2003 for a comparison of the respective contributions of Kaluza and Klein with those of Einstein, Mandel, and Fock, see Goenner 2004, sec. 6.3. γ00 γ00 γ00 1= x0 x0