RESEARCH
NOTES
ON
RELATIVITY
195
arise from
an
interaction between that
mass
and
the
remaining
masses
of
the
universe.
This
project
had
presumably
advanced little
by
4
July 1912,
the time of submission
of Einstein 1912h
(Doc. 8). Hoping
that
the
theory's equations
would remain invariant
under acceleration
and
rotational
transformations,
Einstein concludes
(pp.
1063-1064)
by
calling
on
his
colleagues to
search for
the
general
form
of
the
space-time
trans-
formation
equations
of
a
relativity
theory incorporating gravitation. By 16 August,
however,
Einstein could write from Zurich
to
Ludwig
Hopf
that unless
he
was com-
pletely mistaken,
he
had found the
most
general
equations.[14]
III
Einstein's
insight
into the
analogy
to
Gauss's
theory
of surfaces
was
just
a
first
step.
That
insight
had
now
to
be converted into
a
comprehensive
theory
that would
do
justice
to the
kinematics
and
dynamics
of
gravitational
and
nongravitational phenom-
ena.
The
development
of such
a
theory
was
furthered
immeasurably
when Grossmann
alerted Einstein
to
the absolute differential calculus of Ricci and Levi-Civita
1901,
which
proved
to
contain
precisely
the mathematical devices
needed to
complete
the
general theory
of
relativity.
Part
I
of
Doc.
10 begins
at
an
early stage
of
this
devel-
opment.
On
[pp.
1-2],
Einstein
uses a
nonstandard
uppercase
Guv
to
represent
the
metric
tensor,
before
reverting
to
the standard
guv
in
the remainder of the document.
The
elementary
nature
of the calculation
and
the
writing
out term
by
term
of
simple
sums
also
suggest
that Einstein
was
not
yet
familiar
with
the material.
As late
as
[p.
6],
there
is
no
indication of
techniques
characteristic of
the
absolute differential
calculus,
suggesting
that
Einstein
may
not
yet
have been alerted
to its
existence
by
Grossmann. Ricci
and
Levi-Civita,
for
example, distinguished
contravariant
and
covariant
systems.
The
distinction
does
not
appear
until
[p. 8].[15]
On
[p. 10] we
find
Ricci and
Levi-Civita's covariant differentiation
operation.
On
[p.
6]
we
also
find
direct evidence of Einstein's
awareness
of Gauss's
theory
of
surfaces. He
recapitulates
a
standard
result:[16]
The
trajectory
of
a
particle
free of forces but constrained
to
a
curved surface
is
a
geodesic
of
the
surface.
By [p. 10],
Einstein
has
developed
the
basic
law
governing
the
dynamics
of distributed
matter: the
vanishing
of
the
covariant
divergence
of
its
stress-energy
tensor.
By
the close of Part
I
Einstein has undertaken
calculations of considerable
sophistication.
His
confidence
has
grown by
[p.
25]
to
the
point
that
he is
prepared
to
attempt
a
classification of
tensors of various orders.
Einstein's
use
of the
term
"tensor" for other than second-rank
quantities is
an
innovation of Einstein and Grossmann
1913
(Doc. 13)
and
replaces
Ricci
and
Levi-
[14]See
Einstein to
Ludwig Hopf, 16 August
1912
(Vol.
5,
Doc.
416).
[15]Note
that the dramatic increase in
sophistication
of
[pp.
8-9]
and
possibly
also
[p. 7]
suggests
that these
pages may
have
been written much later than
those
that follow
in
the
notebook.
[16]The
result
is
developed
in
Grossmann's lecture
notes
mentioned above.
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