1 8 D O C . 1 3 C O M M E N T O N G E N E R A L R E L A T I V I T Y
.
(4)[5]
The quantities s and i are thereby dependent on the quantities r as follows: Let
be the normalized subdeterminants to the ’s, furthermore, let be the associ-
ated tensor density
, (5)
let be decomposed into the symmetric and antisymmetric components accord-
ing to the formula
(6)
The quantities s are derived from the quantities f according to the relations:
(7)
(8)
, (9)
the quantities i from the quantities # and f according to the relations
(10)
(11)
. (12)
By applying (4) in (2) one obtains the field equations
,
(13)[6]
which—broken down into their symmetric and antisymmetric components—yield
the field equations of gravitation and electromagnetism. means the Riemann
tensor of curvature, formed out of the ’s as the fundamental metric tensor.
Equations (13) can be brought into the form of Hamilton’s principle
, (14)[7]
where R is the scalar density of the Riemannian curvature belonging to the funda-
mental tensor and the variation is to be performed over the and ’s.
Equation (14) shows that—contrary to the view stated in the first paper—to each
solution there belongs another which distinguishes itself from the former only by
Γkl α
1
2
--sαβ¨ -
∂xl
∂skβ
∂xk
∂slβ
∂xβ¹
∂skl·
–+
©
¸
§
1
2
--skliα - –
1
6
--δk - αil
1
6
--δlαik - + + =
rkl
rkl rkl
rkl rkl r –=
[p. 77]
rkl
rkl #kl fkl. +=
sαλsβλ
δα
β
=
s skl =
#kl skl s –=
ik
∂xα
∂
fkα =
ik ik s –=
ik skαiα =
rkl Rkl
1
6
-- -
∂xl
∂ik
∂xk¹
∂il
–
©
¨ ¸
§ ·
ikil + +=
Rkl
skl
δ® 2 r –– R
1
6
--#αβiαiβ - –+ dτ
³
¯ ¿
¾
­ ½
0 =
sik #kl’s fkl