DOC. 24 PERIHELION MOTION OF MERCURY

113

IN

A

WORK RECENTLY PUBLISHED

in

these

reports,

I

set up

the

gravitational

field

equations

that

are

covar-

iant with

respect to arbitrary

transformations of determinant

1.

In

a

supplement I

showed that these

equations

are

generally

covariant if the contraction

of

the

energy

tensor of "matter"

vanishes,

and

I

demonstrated

that

no

important

considera-

tions

oppose

the introduction of this

hypothesis, through

which time and

space are

robbed of

the

last trace of

objective

reality.1

[1]

[2]

In the

present

work

I

find

an important

confirmation of

this

most

fundamental

theory

of

relativity,

showing

that it

explains

qualitatively

and

quantitatively

the secular rotation of

the

orbit of

Mercury

(in

the

sense

of the orbital motion

itself),

which

was

discovered

by

Leverrier and which

amounts to 45

sec

of

arc

per century.2 Furthermore, I

show that the

theory

has

as a

consequence

a

curvature

of

light

rays

due

to gravita-

tional

fields

twice

as strong

as was

indicated in

my

earlier

investigation.

[4]

The

Gravitational Field

From

my

last

two

communications

it follows

that the

gravitational

field

in

a vacuum

has

to

satisfy,

upon properly

choosing a

reference

frame,

the

equations

151

+1

W

=

0,

«

dx*

.0

(1)

where the

rjuv

are

defined

by

the

equations

47]

--i?4

dxv dx, dx,

(2)

Let

us

make,

moreover,

the

hypothesis

established

in

the last

communication,

that the contraction of the

energy

tensor

of

"matter"

always vanishes,

so

that,

in

addition,

the deter-

minantal

condition

is

imposed:

|guv|

=

-1.

(3)

A

point

mass,

the

sun,

is

located

at

the

origin

of the

co-

ordinate

system.

The

gravitational

field

this

point mass

produces can

be

calculated

from these

equations

by means

of

successive

approximations.

Nevertheless,

we

should consider that the

guv

are

still not

completely

determined

mathematically

by

equations

(1)

and

(3),

because these

equations are

covariant with

respect to

arbitrary

transformations of determinant

1.

Yet

we are

justified

in

assuming

that

all

these

solutions

can

be reduced

to

one

another

by

such

transformations

that

they are

distin-

guished

(by

the

given boundary conditions) formally

but

not,

however, physically,

from

one

another.

Consequently, I

am

satisfied for the time

being

with

deriving

here

a solution,

with-

out discussing

the

question

whether

the

solution

might

be

unique.

To

proceed,

let the

guv

be

given

in

the

0th

approximation

by

the

following

scheme

corresponding to

the

original theory

of

relativity:

-1 0 0

0

0

-1

0

0

0 0

-1

0

0 0 0 +1

(4)

[6]

or,

more

briefly,

9pv

=

Spa

9p4

=

g4p

= 0

g44

=

1

(4a)

Here

p

and

a

signify

the indices 1,2,3;

Spa

is

equal

to

1

or

0

if

p

=

a or p

#

r,

respectively.

I

assume

in what follows

that

the

guv

differ

from the

values

given

in

equation

(4a)

only by quantities

small

compared to

unity.

I

treat

this deviation

as a

small

quantity

of

first

order,

whereas functions of the nth

degree

in

these deviations

are

treated

as quantities

of the nth

order.

Equations

(1)

and

(3)

together

with

equation

(4a)

enable

us

to

calculate

by

succes-

sive approximations

the

gravitational field up to quantities

of

nth order

exactly.

The

approximation

given

in

equation

(4a)

forms the 0th

approximation.

The

solution has

the

following properties,

which determine

the coordinate

system:

1.

All

components are independent

of

x4.

2.

The solution

is

spatially symmetric

about the

origin

of

the

coordinate

system,

in the

sense

that

we encoun-

ter

the

same

solution

again

if

we

subject

it

to

a

linear

orthogonal spatial

transformation.

3.

The

equations

gp4

= g4p =

0

are

exactly

valid for

P

=

1,2,3.

4. The

guv

possess

the values

given

in

equation

(4a)

at

infinity.

first approximation

It

is easy

to

verify

that

to

quantities

of

first

order the

equations

(1)

and

(3)

are

satisfied for the

just-

named four conditions

by

the assumed

solution

(4b)

d2r

9pa

a

dxpdx,

gu

= 1

- -