7 4 D O C U M E N T 1 2 O N K A L U Z A ’ S T H E O R Y
on the field concept. This criterion is a recurring theme in Einstein’s later work on unified field theo-
ries. For a historical discussion of the role of particle solutions in five-dimensional unified field the-
ories, see Dongen 2002 and 2010, chap. 6. For historical discussions of the role of particle solutions
in Einstein’s unified field theory program more generally, see Goenner 2004, especially pp. 7–10,
secs. 3.2 and 4.3.4; Goldstein and Ritter 2003, pp. 112–113. For the importance of the condition for
the solutions to be free of singularities, see Earman and Eisenstaedt 1999.
Einstein had earlier investigated spherically symmetric static solutions to purely gravitational
field equations, namely, when he aimed to derive Mercury’s perihelion (see in particular Einstein
1915h [Vol. 6, Doc. 24]). However, Einstein only gave an approximate solution, to which Schwarz-
schild 1916 found the rigorous equivalent. On p. 833 of Einstein 1915h, he lists four conditions that
the sought-for solution has to fulfill, and which partly mirror the conditions given in the present paper.
However, the fourth condition of Einstein 1915h, which in modern terms corresponds to demanding
asymptotic flatness, is not introduced as part of the list here. Instead, it is introduced at the very end
of the paper (p. 5). For a detailed historical discussion of the perihelion calculations, see the editorial
note, ‘‘The Einstein-Besso Manuscript on the Motion of the Perihelion of Mercury’’ (Vol. 4, pp. 344–
359), and also Earman and Janssen 1993.
The manuscript version has “ , und ” instead of only “ .”
After eq. (7), the manuscript version contains the following sentence, which was crossed out:
“Damit im Nullpunkt die Determinantengleichung
erfüllt sei, darf dort weder γ noch λ verschwinden.” For discussion, see the following note.
In the manuscript version, after “welche,” the following words were crossed out: “ohne Verlet-
zung der Stetigkeitsbedingung im Nullpunkt die.” Following the definition of “singularity-free” in
their footnote 1, this deleted passage, derived from the deleted passage noted in note 13, corresponds
to a different conclusion. As it stands, Einstein and Grommer conclude that no centrally symmetric
solutions exist; the deleted passage instead claims that no centrally symmetric nonsingular solutions
exist. At first sight, this looks like a weaker statement than the one they ended up with; however, the
nonsingularity of the solution would have stemmed from an independent argument.
g44 g45 g55 g44
+ 1– = =