D O C U M E N T 3 0 M A Y 1 9 2 3 3 5
For the determinant , one obtains
where
Incl. the 2nd order terms, one hence obtains, if I now leave the factor out again
Maxwell’s energy tensor!
For you obtain the expression as before, where only takes the place of ;
I leave out the additional terms containing the current for now. If I then form out of
the equation : , I then obtain the law
Now I cannot avoid the following: (1) Maxwell’s energy tensor does indeed appear
here; but the equation only applies if it is weaker than the cosmological term .
And this appears to me to lie in the original assumption that always only an elec-
tromagnetic energy tensor of cosmological smallness can occur. (2) the sign is the
wrong one. According to the old theory it should read
thus
(Hence, either I get the cosmological term with the wrong sign or I get gravitational
repulsion instead of attraction.)
If in the symmetric part of one takes into account the additional terms con-
taining the current, then I find in addition to the usual Riemannian expression of
from the ’s the following additional terms— , means , where the
three-indices symbols are formed of course out of —:
γik ϕiαϕkα
–=
r riκ =
r g 1 ε2l … + + ( ), = l
1
2
--ϕikϕik.
- =
giα fα k g® δik
1
2
--lδik - ϕiαϕ kα –
© ¹
§ ·
+
¯ ¿
¾
­ ½
=
­ ° ° ® ° ° ¯
f
ik gϕik
=
Γkl α skl gkl
Rik gik = Rik sik – Riαsαk ⋅ =
Rik giαfαk g® δik
1
2
--lδik - ϕiαϕkα¹ –
©
§ ·
+
¯ ¿
¾.
­ ½
= =
δik
Rik
1
2
--Rδik - – g{ δik Maxw.
Energ.tensor};
+ =
–R 4 g and =
Rik g( δik – Maxw.) + =
Rkl
Rkl skl γi
iα·
α
© ¹
§
sik