426 DOC.

420 DECEMBER

1917

principle

in

gravitation

theory.[2]

The

reason

I did

not

do

so

long ago

is

that

I

have been

busily making

calculations

using

the

theory,

in search of

something

new.

Some

little

things

have

come

out of

it, yet

it

seems

that

I

am

not

making any

more

progress

for

the

time

being

and would

like to

take

the

liberty

of

reporting

to

you

about

my

efforts.

Although

it

seems

probable to

me

that

you

have

long

been

acquainted

with these

sidetracks, among

the

lot connected

to

your theory,

and

have

come

farther

along

them

than

I,

it could nevertheless be

possible,

and

it would

please

me greatly,

if

you

were

to benefit from

one or

the other

in

some

way.

First

of

all,

I

would

like

to

correct

an error

in

my

earlier letter.[3]

I gave

an

example

there

of

a

space

of

nonvanishing

curvature in which

the

energy

tensor of

matter

was supposed

to vanish.

Subsequently

I

noticed

nonvanishing

components

in

a

hidden

corner

of

the

tensor, though.

1)

My

main efforts since

receipt

of

your

letter

have been

devoted toward

unify-

ing

the

electromagnetic

field

with

the

gravitational field.[4] I

have

not

yet

found

a

solution.

I

performed larger

calculations in

one

vein

by proceeding

from

an

asym-

metric fundamental

tensor

guv.

I would

like to

present

the

following approaches:

a)

Reduction into

a

symmetric

and

an

antisymmetric

part

guv

= suv

+

auv.

Deter-

minant:

g, or

subdeterminant

s,

divided

by

g

(or s)

itself:

guv, or

suv.

Yguagva

=

Evu

=

0

or

1,

likewise

gaugav

=

Evu,

by

contrast

gaugva

=

avu(

#

1);

guagav

=

ivu;

hence

guv

=

aavgau

=

iaugva; guv

=

aaugva

=

ivagau;

iva

.

aau

=

iau

.

avu

=

eva.

dguv

=

-gavgußdgaß;

dguv =

-gavgußdgaß;

logg/dt

=

guvdguv/dt

=

-guv

(:t

one

parameter:).

b)

If

Av is

a

vector,

then

ft»

=

+ A

'

~dx^

is a

tensor,

likewise

dAv

dA^ d(fu

dxa dxa dxa

Others

in

addition

to

this

through

multiplication by

guo, or

gov,

etc.

c)

From

the

64

derivatives of

guv

the

40

second derivatives of

the

xo,s

can

plainly

be

calculated

if 24

relations exist between

the

guv’s

and

g'u'v's.

These

relations

imply

that the

following

system

of quantities is

a

tensor: