196 RESEARCH
NOTES
ON RELATIVITY
Civita's
terms
contravariant and covariant
"systems."[17]
Einstein also deviates from
Ricci and Levi-Civita's
notation,
where
covariant
systems
are
represented
by
lowered
indices and contravariant
systems
by
raised indices. Instead
he
follows
the
convention
of Einstein and Grossmann
1913
(Doc. 13).
All
indices
are
lowered;
covariant
quan-
tities
are
represented
by
Latin letters
(e.g.,
guv)
and the
corresponding
contravariant
quantities
are
represented
by
the
corresponding
Greek letters
(e.g.,
yuv).
Summation
over
repeated
indices
is not
routinely
assumed,
although
the
summation
symbol is
sometimes
suppressed.[18]
The
purpose
of
many
of
the
calculations of Parts
I
and II
remains unclear.
Repeated
themes
are
the
investigation
of coordinate
transformations,
including
infinitesimal
transformations and unimodular
transformations, and the
construction of
quantities by
differential
operations
that remain invariant under these transformations. In
this task,
Einstein
employs
the
four-dimensional
analogs
of Beltrami's
first
and second
operators
A
and
A2
which
are
given
by[19]
A4) =
7^4v4,v A((j),
4»)
=
7^4),A«*
A24
=
7^4».^
for
the
scalar
fields
Q
and
W.
In
several
places,
Einstein
constructs
quantities solely
from
the
metric
tensor and its derivatives.[20]
This
suggests
that
Einstein
was
seeking
gravitational
field
equations.
In
Einstein and Grossmann
1913
(Doc. 13), §5,
the
prob-
lem of
finding
these
field
equations
is
posed.
The
equations
were
to generalize
Pois-
son's
equation
Acp
=
4ukp
of Newtonian
gravitation
theory[21]
and
were
expected
to
have
the
form
K()
=
r
uv
a uv'
where
k
is
a
constant,
0uv
the
stress-energy tensor,
and the
gravitation
tensor,
Tuv,
a
second-rank covariant
tensor, is
constructed from
the
metric
tensor
and its
derivatives
up
to
second order.
It
became
clear, furthermore,
that
the
authors
expected
the
grav-
itation
tensor to
have
the
form
Edxr(dxb)
+
further
terms
that vanish
on
taking
the first
(1)
approximation.
[17]See
also the
editorial
note,
"Einstein
on
Gravitation
and
Relativity:
The Collaboration
with Marcel Grossmann,"
p.
296.
[18]In
this note
and the
notes to
individual
pages,
we
shall
follow the Einstein and Grossmann
convention.
Summation
over
repeated
indices
will,
however,
often
be
assumed for
compactness.
A
comma
will
represent
differentiation with
respect
to
a
coordinate.
A
semicolon will
represent
covariant
differentiation.
[19]For
a
discussion of Beltrami's
invariants,
see
Darboux
1887-1896, vol.
3,
pp.
193-217,
Wright
1908,
pp.
52-57,
and Bianchi
1896, chap.
2.
[20]See,
for
example,
[p. 3], [pp.
12-14],
[pp.
16-18],
[pp.
22-23],
and
[pp.
25-26].
[21]Q
is
the
gravitational potential,
p
the
mass
density,
and
k a
constant.
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