EINSTEIN ON THE
STATIC FIELD
123
When
Einstein
realized
in
1911
that
gravitational light
deflection should
be
exper-
imentally
observable for
light rays passing
near
the
sun,
he
took
up
the
problem
of
gravitation again
and
published
this
insight
in
Einstein
1911h
(Vol.
3,
Doc.
23).[6]
He
also
used
the
occasion
to
extend
his
theoretical discussion of
gravitation.
A
conse-
quence
of
the
equivalence principle
which he
had earlier mentioned
in
passing now
became of fundamental
importance.
Since Einstein
had
found that
the
coordinate time
measured
by
local
clocks
depends
on
the
gravitational potential
O, he had
concluded
that
the
velocity
of
light
c
should
also
depend
on
the
gravitational potential according
to
the relation[7]
c
=
c0 (1
+
O/c2).
This relation
was
not
only
crucial
for
deriving
the
bending
of
light rays
but
also
implied
that
the
principle
of
the
constancy
of
the
velocity
of
light
had
to
be
modified if
grav-
itational
fields
were
taken
into
account.[8]
Hence
the
assumption
of
a
variable
speed
of
light
became
the
basis
for
Einstein's
subsequent
papers on
the static
gravitational
field.
In
late
1911
and
early
1912
Einstein invested
great
effort
in
developing a
coherent
theory
of
the static
gravitational
field.
To
a
friend
he wrote:
"I
am
working
like
a
horse."[9]
Finally, on
24
February,
he
submitted
a
paper
to
Wilhelm
Wien, the
editor
of the
Annalen der
Physik,
entitled "The
Speed
of
Light
and the
Statics of
the
Grav-
itational
Field."[10] In this
paper
Einstein
gives
a
heuristic
argument
for
a
gravitational
field
equation
which
generalizes
the
Poisson
equation
of Newtonian mechanics.
Using
the
principle
of
equivalence
he first finds that the
speed
of
light
in
a
static
gravitational
field is
a
linear function of
the
spatial
coordinates.
By
studying
the
equations
of
motion
of
a mass
point,
he
then concludes
that the
speed
of
light plays
the
role of
the
gravi-
tational
potential.
On the
basis of these considerations Einstein
suggests
a
differential
equation
for
the
speed
of
light
c
in the
presence
of
matter
which
takes
the
place
of
Poisson's
equation:[11]
Ac
=
kcp.
[6]Einstein also
began corresponding
with
contemporary astronomers to encourage tests
of
the
experimental consequences
of his
analysis
of
gravitation;
see
in
particular
his
correspon-
dence with Erwin Freundlich
and W. H.
Julius from
this
period
in Vol. 5.
[7]See
Einstein
1911h
(Vol. 3,
Doc.
23),
p.
906;
see
also Einstein
1907j (Vol. 2,
Doc.
47),
p.
461.
[8]See
Einstein
1911h
(Vol. 3,
Doc.
23),
p.
906;
see
also Einstein
to
Jakob
Laub,
10 August
1911
(Vol. 5,
Doc.
275).
[9]"ich
arbeite
wie ein
Ross"
(Einstein
to Ludwig Hopf,
after
20
February
1912
[Vol. 5,
Doc.
364];
the
same
expression
is
used
in
Einstein
to
Heinrich
Zangger,
before
29
February
1912
[Vol. 5,
Doc.
366]).
[10]"Lichtgeschwindigkeit
und
Statik
des Gravitationsfeldes"(Einstein
1912c
[Doc. 3]).
[11]A
denotes
the
Laplacian operator,
p
the
density
of
matter, and
k
the
gravitational
constant.