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
1
β©
-- -
∂iμ
∂xν
------- -
∂xμ¹
--------·νi∂

§
=
∂fμν
----------
=
∂fμν
∂xν
–----------
1
β
-- - = =
[2]
[3]
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