RESEARCH
NOTES
ON
RELATIVITY
197
Should the
gravitation
tensor
have
this
form,
then the
gravitational
field
equations
would reduce
to
Poisson's
equation
of Newtonian
theory
in
the
static,
weak
field limit.
Finally,
in
Einstein and Grossmann
1913
(Doc.
13), §4,
the authors indicated that
static
gravitational
fields
were
expected
to
reduce
to
the
simple
form
implicit
in
Ein
stein's
1912
theory
of static
gravitational fields;
that
is,
static
fields
admit
a
line
element
ds2
=
c2
(x,
y,
z)
dt2

dx2

dy2

dz2. (2)
This
assumption
is
also evident
in
Parts
I
and
II,
as
well
as
Part
III,
of
Doc.
10.[22]
IV
In
practical
terms,
the methods of Parts
I
and
II
were
too
crude
to
generate
the
com
plicated expressions
that could
yield
a
generally
covariant
gravitation
tensor. A
search
for such
expressions
must
exploit
the Riemann
curvature tensor,
whose
importance
in
precisely
this
context
was
well known from standard theorems
in
the
theory
of
qua
dratic differential
forms.[23]
Part
III
begins
on [p.
27]
with
the introduction of the
Riemann
curvature tensor, next to
which Einstein has written Grossmann's
name.[24]
It
is
contracted
once
to
form the Ricci
tensor, the
natural candidate for
the
gravitation
tensor and
the
choice
of
Einstein's final
theory
in the sourcefree
case.
The
tensor
has
four secondderivative
terms,
three
more
than
allowed
by eq.
(1).
On the last line
Einstein
expresses
his
concern
that
these three
terms
vanish
in
the weak
field
case.
On
[pp.
2836],
Einstein
proceeds
to
form
complicated
expressions
from the Riemann
curvature tensor.
On
[p. 37]
the search takes
a new
direction. Einstein writes
out
the
now
standard
technique
for
reducing
the Ricci
tensor to
the
form
of
eq.
(1), so
that
the Newtonian limit
can
be
recovered. While the
general
field
equations
Einstein
sought
were
to be
generally covariant,
the traditional Newtonian
equations to be
recov
ered
in
the weak
field
limit would have
only
limited covariance.
In
extracting
the
Newtonian
limit,
Einstein had therefore
to
restrict the coordinate
systems
under
con
[22]See,
for
example,
[p. 2], [p. 12],
and
[p.
42].
[23]See
Ricci and LeviCivita
1901,
chap. III,
§2.
Grossmann summarizes the
result in Ein
stein and Grossmann
1913 (Doc.
13),
p.
35:
"From covariant
algebraic
and differential
oper
ations
on
the Riemann differential
tensor
and the discriminant
tensor
(§3,
formula
38
[fully
antisymmetric
LeviCivita
tensor])
one
obtains the
complete system
of differential
tensors
(therefore
also differential
invariants)
of
the
manifold"
("Durch
kovariante
algebraische
und
differentielle
Operationen
erhält
man aus
dem Riemannschen Differentialtensor und dem Dis
kriminantentensor
(§3,
Formel
38)
das
vollständige System
der Differentialtensoren
(also
auch
der
Differentialinvarianten)
der
Mannigfaltigkeit").
[24]Contemporary
correspondence
confirms Einstein's
satisfaction
with
his
progress
(Einstein
to
Erwin
Freundlich, 27
October
1912
[Vol. 5,
Doc.
420], and
Einstein
to
Paul
Ehrenfest,
20
24 December
1912
[Vol. 5,
Doc.
425]),
as
well
as
his
gratitude
for the assistance
provided
by
Grossmann
(Einstein
to
Arnold
Sommerfeld, 29
October
1912
[Vol. 5,
Doc.
421]).
For further
discussion of these
pages
of Doc.
10,
see
Norton
1984, sec.
4.