D O C . 3 1 C A L C U L A T I O N S 6 3 31. Calculations δR [Berlin, after 18 May 1923][1] AD. [30 164]. δR Rμνδgμν = κ 2 -- - gfστfστ κ 2 -- - ggσμgτνfμνfστ κ 2 -- - gαβ gσμgτμfμνfστ He = He δ κ g fμνfστgσμδ gτν κ ggσμgτνfστδfμν + = –gταgνβδgαβ κfσβfσαδgαβ –= +κfμνδfμν Hi κ β - –g –--gμνiμiν κ β - gμνiμiν –-- = = δHi κ β -- - iμiνδgμν 2-- κ β - iμδiμ = –gμαgνβδgαβ ∂δfμν ∂xν ------------- +-- κ β - iαiβδgαβ 2-- κ β∂xν - ∂iμ ------- -δfμν + = +-- κ§ β© - ∂iμ ∂xν ------- - ∂iν ∂xβ¹ --------· δfμν –fμν β© -- - ∂iμ ∂xν ------- - ∂xμ¹ --------·νi∂ = ∂fμν ---------- = ∂fμν ∂xν –---------- 1 β -- - = = [2] [3]
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