234 DOC. 10

RESEARCH

NOTES

[67][Eq.

91]

is

the

fully

covariant form of the Riemann

curvature

tensor,

almost

exactly

as

given

in

Einstein and Grossmann

1913 (Doc.

13),

p.

35.

For

a

discussion of Einstein's collab-

oration with Marcel

Grossmann,

see

the editorial

notes,

"Einstein's Research Notes

on a

Gen-

eralized

Theory

of

Relativity"

and

"Einstein

on

Gravitation

and

Relativity:

The Collaboration

with Marcel Grossmann."

[68]Einstein

forms the

fully

covariant Ricci

tensor

by

contracting

once

and

proceeds to

compute

a

reduced form

([eq.

92])

for the

terms containing

first

derivatives

given

by summa-

tion

over

the Christoffel

symbols.

[69]These

three second-derivative

terms

from

the Ricci

tensor

"should vanish"

("sollte[n]

verschwinden")

to

leave

yklgim,

Kl

as

the

only

second-derivative

term in

the Ricci

tensor,

if

the Ricci

tensor

is

to satisfy

the condition

(1)

for

a

gravitation

tensor

discussed in

the editori-

al note,

"Einstein's Research Notes

on a

Generalized

Theory

of

Relativity."

Einstein

consid-

ers a

weak

field

metric with

components diag(1,1,1,1) to

zeroth order

so

that the

contracting

metric need

not appear explicitly

in

[eq.

93].

[p. 28]

/

d2Sil

\

p

=

I

yim\i

ö

im

imKl

V

dxKdx, dxKdxm

/

\

[70]

+

X

V-Jk/

i

m K

/

i i

K m

[eq. 94]

pGimK/ V

O

p

a

P /

fdSia

dg

mo

dgjm^\(dg

Kp

dg[p

[71]

+ +

p

oimKl

\

dx,n

dx dx

o /

V

dxl

dxK 3xp

dSKi

dsK,

~8

lC

dxm

°ma

dX;

im

dxa

dxK

^

,

^im ^im

dSim d1K,

Ki

d8Kt

+8ma

dx.

+y'm

dxa

)(g*P

dx./

S,pdxK

P

dx^

+

Jk

^'dx

}

171

/

O K

iv

p1*0

dy^ dy.

dig

G

aigc

'pm

'/p

dxm

dx,

PG

dx

O

dxp

V

_

J

Y

dypa

dx

a