DOC. 32
INTEGRATION OF FIELD
EQUATIONS
201
Doc.
32
Approximative Integration
of
the
Field
Equations
of Gravitation
Session of the physicalmathematical class on June 22,
1916
by
A. Einstein
[p.
688]
For the treatment
of
the
special
(not basic) problems
in
gravitational
theory one can
be satisfied with
a
first
approximation
of
the
guv.
The
same reasons as
in the
special
theory
of
relativity
make it
advantageous
to
use
the
imaginary
time variable
x4 =
it.
By
"first
approximation"
we mean
that the
quantities
yuv,
defined
by
the
equation
guv
=
8uv
+
Yuv,
(1)
are
small
compared
to
1,
such that their
squares
and
products are negligible compared
with first
powers;
furthermore,
they
have
a
tensorial character under
linear,
orthogonal
transformations.
In
addition,
8uv
=
1
or
8uv
=
0
resp. depending upon
u =
v or u
#
v.
We
shall
show
that
these
yuv
can
be
calculated
in
a
manner analogous
to that
of
retarded
potentials
in
electrodynamics.
From this follows next that
gravitational
fields
propagate
at
the
speed
of
light. Subsequent
to this
general
solution
we
shall
investigate gravitational waves
and how
they originate.
It turned out that
my
suggested
choice
of
a
system
of reference with the condition
g
= guv/ =

1
is
not
advantageous
for
the calculation
of
fields in first
approximation.
A note in
a
letter
[1]
from the astronomer
De Sitter
alerted
me
to his
finding
that
a
choice
of
reference
system,
different from the
one
I
had
previously
given,1
leads
to
a
simpler expression
of
the
gravitational
field
of
a mass point
at rest.
I
therefore take the
generally
invariant field
equations as
a
basis in what follows.
§1.
Integration
of the
Approximated Equations
of
the Gravitational Field
[p.
689]
The field
equations
in their covariant form
are [3]
1Sitzungsber.
47
(1915), p.
833.
[2]