6 9 0 D O C U M E N T 4 1 8 C A L C U L A T I O N S I N D I A R Y at an angle, and in an unknown hand, and . [3]Einstein begins a calculation of , in which he starts from the covariant divergence of the covariant field tensor and raises the indices. [4]The result is related to but not identical with Einstein’s observation in his diary that in Weyl’s theory would produce the “useless” result see entry of 27 Octo- ber. Einstein’s use of the expression “Rang” is unusual he probably meant what Weyl referred to as “weight,” i.e., homogeneity with respect to a gauge transformation see the definition in Weyl 1921a, pp. 115–116. As Weyl remarked there, the contravariant tensor field is of weight 0, and multiplication with produces a tensor density whose weight is increased by n/2 (n = 4 for space-time theories). [5]At this point, Einstein specializes to the case of Weyl’s theory see, e.g., Eddington 1921a, pp. 109–110. [6]Five lines in the top right corner of this page were written in ink. [7]To be consistent with the usual conventions, should be . [8]The lines at the bottom of the page were written in ink. [9]Between [p. 2] and [p. 3] one sheet of paper was cut out from the bound notebook. Starting with this page, the remainder of the calculations in this part of the diary were written in ink. [10]At this point, Einstein introduces the electromagnetic vector potential as an independent variable, indicating that these entries were probably written after he found fault with his draft manu- script of 9 January see also note 21. [11]Einstein here implicitly introduces the notation . [12]Einstein’s false start in beginning with the word “crystal” may indicate that he was aware of the applicability of differential geometric concepts in solid state physics. [13]The Einstein tensor as the differential operator of a gravitational field equation is obtained in [eq. 5] by doing the variation of with respect to the metric components , as indi- cated in the first term of [eq. 4]. This variation is denoted here as . The six terms written in the boxed space underneath [eq. 5] [eq. 6], were obtained by doing the variation of with respect to , as indicated in [eq. 4]. This variation is denoted as and calculated in the bottom right half of the page. There, Einstein first wrote down the variational terms and in terms of expressions involving Kronecker deltas, which he then multiplied by and , respectively, and added both expressions to obtain [eq. 6]. [14]The first two lines on this page take up the result of [eq. 6] from the preceding page, divide it ko dt , t it , ------------- - 5|+ ϕμν ν ∂ϕμν –g ∂xν ------------------------ 0 = ϕμαϕα gμν λgμν → ϕμν –g hx hy ϕμ R –gRikgik ≡ Riκ 1 2 --giκR - – R giκ δ1 –g ∂gστ ∂xμ --------------------- - –g ∂gσν ∂xν ----------------------δμτ – –g( giσΓiτμ gσκΓκμ τ giκΓiσκδμ τ – gστΓμβ) β – + + R Γiακ δ2R ∂Riκ ν μ ∂Γστ, ---------------- ∂Riκ ∂Γσνμ ----------- - –g ∂giκ ∂xν -------------------- -– giκ –g