4 7 6 D O C U M E N T 4 8 0 K A L U Z A ’ S T H E O R Y : P A R T 2 We can and shall set this constant equal to 1, which does not signify any restriction then dτ has the meaning of an arc length in R4 . The variation with respect to the coordinate yields,[6] if (12) and (13) are taken into account after the variation: , (14) with (15) This is exactly the equation that in the general theory of relativity has hitherto been regarded as the equation of motion of a charged mass point, whose ratio is given by –2A.[7] It should be noted that A is an invariant with respect to -transforma- tions. §(3). The Hamilton Function of the Field Equations Kaluza transferred the field equations (16) to R5 and showed that in this way one can arrive at field equations of gravitation and of the electromagnetic field, which to first approximation agree with those that the general theory of relativity had introduced in line with the semi-empirically ob- tained field equations of Maxwellian theory.[8] We shall show that Kaluza’s thought leads exactly to these equations, rather than merely at first approximation. In order to demonstrate this, one only needs to express the Hamiltonian function (in R5) in terms of and : . (16a)[9] The are to be those of R5 γ denotes the determinant of . From (9) it follows that . (17) This immediately follows if one multiplies the last column of (9) by , and sub- tracts it from the ath column. Furthermore, it is simple to verify that the can be represented in the form [p. 28] xs s 0) ≠ ( gmsxm ·· m n s xmxn · · 2Aφsnxn · + + 0 = φsn 1 2© --§ - ∂xn ∂φs ∂xs¹ ∂φn· – = ξ μ -- - x0 Rik 0= gnm φm H γγμν( Γμν α Γαβ β Γμα β Γνβ α – ) = Γμν α γμν γ g = φa γμν