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