EINSTEIN-BESSO

ON THE MERCURY PERIHELION

345

the basic effect

(i.e.,

the

perihelion

motion

produced

by

the

sun

conceived of

as a

static

mass

distribution)

is

correct,

but

there

is

a

mistake of

a

factor

10

in the

number that

is

inserted for

the

mass

of

the

sun,

which

yields

a

mistake of

a

factor

100 in the final result.

Einstein writes

on [p.

28]:

"1821"

=

30'

independently

checked."[12]

The mistake

is

discovered

in the

following pages

(on [p.

30]

by

Einstein,

on [p. 35] by Besso),

but

there

is

no

clear

and

unambiguous

statement

of

the correct

result

in the

manuscript.

The so-called Scratch

Notebook[13]

contains

an

expression

for

the

perihelion

advance

of

Mercury

which

to

a

good approximation is equivalent to

the

expressions

given

in

this

manuscript.

In the

"Scratch Notebook"

the correct

numbers

are

inserted and

the

end result

is

given as

17".

More

important

than

the

actual numbers

is

Einstein

and

Besso's derivation of

the

expression

for

the

perihelion

motion

predicted

by

the

"Entwurf"

theory.

It turns out

that

the

method used in

1913

is virtually

identical

to the

method Einstein

used in his

November

1915

paper

on Mercury.[14]

This

may

help

to

explain

why

Einstein

was

able

to

write

this

paper

in such

a

short

time.[15]

The remainder of

this

editorial

note

is organized

as

follows.

In

sec.

II

a

brief outline

will be

given

of

the

three main derivations

in

the

manuscript,

with

further details

pro-

vided

in

footnotes

to the

transcription.

These three

derivations, all in the

context

of

the

"Entwurf"

theory, concern

the

motion of

perihelia

in the

metric

field

of

both

a

static

and

a rotating

sun

(see

[pp.

1-30] and

[pp.

32-35],

plus

some

material

on [p.

40]) and the

motion of

nodes

in

the field

of

a rotating

sun

(see

[pp.

45-49],

plus some

material

on [p.

31]

and

[pp.

41-42]).

These three

topics occupy

39

of

the 53

pages

of

the

manuscript.

The

remaining

14

pages

deal

with the

following

topics:

a

plan

for

various corrections

to the

analysis

in Newcomb 1895

on

the basis

of

the

"Entwurf"

theory

([p.

31]);

the

metric

field

inside

a

rotating

shell and the

relativity

of inertia

([pp.

36-38]);[16]

the

perihelion

motion

in

a

special

relativistic

gravitational theory

([p.

39]); an expression

for

the

period

of

a

Newtonian orbit

in terms

of

the

orbiting particle's

total

energy

([p.

40]);

the

"Entwurf"

field

equations

and

Minkowski

space-time

in

a rotating

coordinate

system

([pp.

41-42]);

the

"Entwurf"

field

equations

for

what

is

called

the

"Eulerian

case"

("Eulerscher Fall")

([pp.

43-44]); the

metric

field inside

a

rotating

ring

and the

[12]"...

unabhängig geprüft."

[13]Vol. 3, Appendix A,

[p.

61].

[14]See

Earman and Janssen

1993,

pp.

142-143,

pp.

156-157.

[15]See

David Hilbert

to

Einstein,

19

November

1915:

"...

congratulations

on

conquering

the

perihelion

motion. If

I

could calculate

as

fast

as you,

the

electron

would be

forced

to

surrender

in

the face of

my

equations

and the

hydrogen atom

would

have

to

present

an

excuse

for

the

fact

that

it

does

not

radiate"

("...

herzliche Gratulation

zu

der

Überwältigung

der

Perihelbewegung.

Wenn ich

so

rasch rechnen

könnte, wie

Sie,

müsste

bei

meinen

Gleichungen entsprechend

das

Elektron

kapituliren

und

zugleich

das

Wasserstoffatom seinen

Entschuldigungszettel aufzeigen,

warum es

nicht

strahlt").

[16]On

these

pages

Einstein calculates

the

Machian effects

he

described

in

letters

to

Mach

and

Lorentz of this

period:

the

gravitational

effects

at

the

center

of

a

spherical

mass

shell

when the

shell

either

rotates

uniformly

or

is

accelerated

uniformly

and

rectilinearly

(see

Einstein

to

Ernst

Mach, 25

June

1913

[Vol. 5,

Doc. 448], and

Einstein

to H. A. Lorentz,

14 August 1913

[Vol. 5,

Doc.

467]).