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 6 7 1 [p. 51v] gμτgνσ ∂ϕμν ∂xσ ----------- ϕανΓσμ∗ α – ϕμαΓσ∗α – ∂ϕμσ ∂xσ ----------- - ϕμν----------- ∂gνσ ∂xσ – ϕανgνσΓσμ∗α – +ϕμν[ gντΓτσ σ gαβΓαβ ν + ] ϕανgνσΓσμ∗α – ∂ϕμσ ∂xσ ----------- - ϕμσΓσα α – ϕμτgαβ αβ τ – + ∂ϕτσ ∂xσ ----------- ϕμν ∂gμτ ∂xσ ----------gνσ - ∂gνσ ∂xσ -----------gμτ + – ϕανgμτgνσΓσμ∗α – + 1 2 --ϕανgμτgνσgαλ - ∂gσλ ∂xμ ----------- ∂gμλ ∂xσ ---------- - ∂g ∂xλ -----------σμ –+ – +--ϕαν 1 2 - ∂gνα ∂xμ -----------gμτ ∂gατ ∂xσ ----------gνσ - ∂gντ ∂xλ ----------gαλ –+ Γστ' ρ Γab-------- n ∂xρ-'----------------- ∂xn ∂xa ∂xσ'∂xτ' ∂xb ∂2xn xσ' xτ' ∂∂ -------------------------- ∂xρ-' ∂xn + = +ϕασ-----------xσ∂ ∂gατ ∂ϕτσ ∂xσ ----------- ϕντ----------- ∂gνσ ∂xσ – ∂gνσ ∂xσ –ϕατgαν----------- ∂ ∂xτ ------- ∂xρ' ∂xn -------- ---------- ∂xn ∂xσ' . ∂2xρ' xn xτ∂xσ' ∂∂ --------------- ----------n ∂x += ∂ ∂xσ' --------- ' ∂x ∂xτ ---------ρ gαν gνλΓλσ σ gαβΓαβ ν + ∂ϕτσ ∂xσ ----------- ϕτλΓλσ σ gμτgνσgαβϕμα(gσβϕν gνβϕσ gσνϕβ) –+ –+ ϕτσ------------ –g ∂ ∂xσ ---------- 1 –g - gμτgασϕμαϕσ + (gμτgανϕμαϕν – –4gμτgαβϕμαϕβ ) ϕτνϕν ϕτσϕσ 4ϕτβϕβ) –+ ( – 2ϕταϕα –g ∂ϕτα ∂xα ---------------------- g – 2ϕταϕα ⋅ + Tensordichte vom Rang 0 = Γab n Anα-----------a ∂Aα- ∂xb = ∂xρ ∂xn' ---------------- - ∂xn' ∂xα - δρα∂xβ ∂ ------- - = ∂xρ ∂xn' -------- - ∂2xn' xα xβ ∂∂ ---------------- - ⋅ +------- ∂ ∂xβ - ∂xρ ∂xn' -------- - ∂xn' ∂xα -------- - 0= 0 [3] [4] [5] 0