6 6 8 D O C U M E N T 4 1 7 O N G E N E R A L R E L A T I V I T Y [10]Similarly, Weyl had criticized Eddington’s approach for (i) not providing field equations for the components of the connection, (ii) believing that a generalized theory might solve the problem of the “unknown electron forces,” and (iii) failing to provide a conceptual foundation for the equality of iner- tial and gravitational mass see Weyl 1921b. In Appendix 4 to Weyl 1923, Weyl also criticized that an affine approach is incapable of giving an account of conformal, i.e., light-cone, structure. [11]Einstein’s remark here picks up on a theme that Ehrenfest had raised earlier and that Weyl was pursuing at the time. In 1917, Ehrenfest had raised the question as to how three-dimensional space is unique compared to spaces of other dimensions. Such uniqueness is given, for example, in that only in three (and two) dimensions circular orbits in a spherically symmetric force field are stable, and in that only in three dimensions the number n of components of the electric field is equal to the number of components of the magnetic field (see Ehrenfest 1917). After Weyl had made com- ments about the uniqueness of the four-dimensionality of space-time in his Weyl 1918a and 1919c, Ehrenfest published a German version of his earlier paper with an addendum, in which he posed the question of why the metric line element is given by a homogeneous quadratic form of the coordinates (Ehrenfest 1920). Then, Weyl had observed that in a variational formulation of his generalized rela- tivity theory the action for the electromagnetic field is an invariant only in a manifold of four dimensions. More generally, he had stated that in a general metric geometry a quantity that is formed from the metric field according to some rule, and which is a scalar density in a manifold of given dimension, will no longer be a scalar density in a manifold of different dimension. For details, see increasingly explicit comments in the different editions of Weyl 1919b, p. 261 Weyl 1921a, p. 259 Weyl 1923, p. 301 (Ehrenfest’s paper is cited by Weyl only in the fifth edition). Eddington referred to this observation by Weyl in a footnote to Eddington 1921a, p. 119. In Weyl 1922a and 1922b, as well as in a course of lectures that he gave in spring 1922 in Barcelona and Madrid, Weyl had investigated more deeply the general “problem of space,” of which the question of a justification for the unique- ness of the Pythagorean metric and the three-dimensionality of physical space are specific aspects. Einstein had studied Weyl 1922a in June 1922 (Doc. 219). [12]See note 25 below. [13]The parentheses are an allusion to the program of conceptualizing elementary particles as non- linear field configurations that would arise from suitably generalized field equations (see Doc. 219, note 4). [14]This appears to be the first unambiguous formulation of what later came to be known as the Palatini variational method. [15]From (3) it follows that , which in turn entails that . [16]Eddington 1921a, pp. 110, 113. [17]See Palatini 1919b. For Einstein’s familiarity with that paper, see Einstein to Attilio Palatini, 16 January 1920 (Vol. 9, Doc. 263) for a historical discussion, see Ferraris et al. 1982, Cattani 1993. [18]The very last index should be a μ. [19]For Einstein’s use of a semicolon to denote covariant derivatives, see Doc. 418, note 22. [20]Erroneous repetition of eq. (11) in the original. [21]In this equation, previous factors of 2 were erased. [22]With eqs. (11) and (14), we have (A) contracting (A) with yields (B) contracting (A) with , using (B), yields n n 1– ( ) 2 1 4 -- - fik f ik xd Γμα α 1 –g --------- -------------- –g ∂xm = ϕμν 0 gαβμ 1 2 --δμαgβσσ - 1 2 --δμβgασσ - gαβgστ μ δμαiβ δμβiα + + 0 = gαβ gαβgαβμ 1 3 --gμβgβσσ - 2 3 --iμ - + = δβμ
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