D O C U M E N T 1 3 9 C A L C U L A T I O N S 2 4 3 139. Calculations on Decomposition of Riemann Curvature [Berlin, between 25 December 1925 and 9 January 1926][1] _____________________________________________________________ 12 23 23 12 13 22 22 13 1 1 1 1 _____________________________________________________________ _____________________________________________________________ Tμ ν 1 4 --ϕαβϕαβδμ - ν ϕμαϕνα¹ – © § · g = T4 4 1 2 -- - e2 – h2)δμ + ( ν e2 g + + Energiedichte. = = ϕ14 ∂x4 ∂ϕ1 ∂x1 ∂ϕ4 – eα = = ϕ23 ∂x3 ∂ϕ2 ∂x2 ∂ϕ3 – hα = = iklm) ( 1234 0 = 0 1223) ( 1 2 --[ - 12 23)–( , ( 34 14)] , 1 2 --[ - 12 23) , ( 14 43)] , ( + 1 2 --R13 - = = = 1 2 --G12 - 12 21) , ( 1 2 -- - [ 12 21) , ( 34 43) , ( – 1 2 --R11 - 1 4 --² - ¢ R22 R 2 --- –+ = = 1 2 --( - G11 G22) + 1 2 -- - 1331 – 2332 – 1 2 -- - 1441 – 2442 – 2112– 1 2 -- - – gilGkm gkmGil gimGkl – gklGim) – + ( Giklm kl Gim 1 2 - Gim Gim) + –--( –Gim = = = G ik, · lm Siklm + Gik lm , 1 2 --( - Rik lm , R ik,lm – ) = Sik lm , 2 3 -- - ϕik, ϕlm 1 2 --( - ϕilϕkm ϕimϕkl) – + = Sim Φim 1 2 -- - gimΦ Sim² ¢ – 2 3 -- - ϕiαϕm α² ¢ α – 1 2 -- - ϕiαϕm α k² ¢ – 1 4 -- - … – = = ϕiαϕm α k² ¢ – 1 4 -- - + = Gim Sim + 0. =