256 DOC. 10 RESEARCH NOTES

[p. 46]

[117]From

the

quantity

[eq.

169],

which

is

a

tensor

under unimodular

transformations, Ein-

stein

generates

another

tensor,

[eq.

183],

under

an even more

restricted

group

of

transforma-

tions.

It has

the form of condition

(1)

of

the

editorial

note,

"Einstein's Research Notes

on a

Generalized

Theory

of

Relativity,"

without

the

need

to

invoke

the

condition

y

=

0.

In its

place, he

further restricts transformations

by

the

condition that

«

transforms

as a

tensor.

This

quantity

also

plays

an

important

role

on [pp.

7-9].

[118][Eq.

178]

is

[eq.

169]

with the

Christoffel

symbols expressed

in

terms

of

by

means

of

the

identities

[eq.

175]

and

[eq.

177].

[Eq.

178] expands to

[eq. 180]

where

a^Kß^/Ka^/A.ß"

Since both

terms

on

the

right

are

tensors

under unimodular

transformations for which

^

is

a

tensor, the

left-hand

side is

a

tensor

as

well.

[119][Eq.

181],

the covariant

divergence 7™$

is

also

a

tensor

under the restricted

group,

as

is

[eq.

182],

which

is

an

expansion

of

[eq.

181]

with the

discarding

of

terms

in

tensorial under

the

restricted

group.

[Eq. 182]

can

be

further reduced

by

discarding

the final

term,

which

is

equal to

ypßd./p3logG/3.Xß

and

therefore

a

tensor

under

the

restricted

group.

Substraction of

[eq. 182]

from

[eq. 180]

gives the final

result

[eq. 183].

In

expanding

[eq.

178]

to

[eq. 180]

Einstein should have included

y

within

the

scope

of the

operator

in

the

first

two terms

of

[eq. 180],

as

is suggested

by

Einstein's

marking

of these

terms.

This

correction

requires

a

similar correction

to

[eq. 183],

which then does

turn

out to

have tensori-

al

properties

under

the

restricted

group. (An

additional

term

y

ft

can

be discarded since

it

proves to

be

tensorial under

the

restricted

group.)

Y,k

fdgj

K

dgKf

+ +

V

dxi

dXi dXK

/

It

dgu

iK

a*K

I

dxK

^

dx

*gi

kY a

[120]

v

dx,

)

Kixdxx

(VkKSiJ

[eq. 184]

Zurück transformiert

PipPkcPix

PipPkoP^x^ [eq.

185]

Ausführlich

^+PlJ^g"+pj£?gfK

[eq. 186]

x V"X V"X