296
EINSTEIN
IN
COLLABORATION WITH GROSSMANN
"Entwurf"
theory
was
rather
limited.
At
some point
in his
search for
a
theory
of
gravitation
Einstein turned
to Grossmann,
who drew
his attention
to
the work of
Riemann,
Christoffel,
Ricci,
and Levi-Civita
on
the
absolute differential
calculus.[10]
Grossmann
systematically
expounded
the
generalized tensor
calculus he and Einstein
had extracted from earlier mathematical studies
in
the second
part
of Einstein
and Grossmann
1913
(Doc. 13),
the
only
part
for which
he
took
direct
responsibil-
ity.[11]
During
their work
on
the "Entwurf"
paper,
Grossmann
helped
Einstein
in
his search
for
a gravitational
tensor,
and
he
apparently suggested
the
fourth-rank
Riemann
tensor
as a
starting
point.[12]
An obvious candidate for
a
gravitational
tensor
was
the Ricci
tensor,
constructed from
the Riemann
tensor;
but
at
some
point
in
their research Ein-
stein and Grossmann
came
to
reject it.[13] They
abandoned
general
covariance
and set
out to
derive
a
differential
equation
for the
gravitational field, taking
tensors
that
are
only
covariant
with
respect
to
linear transformations
as
their
starting point.
In the
derivation,
which
is
presented
in
§5
of Einstein and Grossmann 1913
(Doc. 13), they
use
the
requirements
of
covariance under
general
linear
transformations,
of
energy-
momentum conservation,
and of
recovering
the
correct
Newtonian limit
to find
the
field
equations.[14]
[10]For
evidence
of
Grossmann's role in
solving
the mathematical
problems
Einstein
was
facing,
see
Einstein
to
Erwin
Freundlich, 27
October
1912
(Vol.
5,
Doc.
420),
and Einstein
to
Arnold Sommerfeld,
29
October 1912
(Vol.
5,
Doc.
421).
For
an
account
of
the collaboration
between Einstein and Grossmann
as
well
as
of their
personal relationship,
see
Pais
1982, pp.
210-225. The
generalized
definition of the
tensor
concept as
it
appears
in
Einstein and Gross-
mann
1913 (Doc.
13)
merges
two
traditions that
up
to
that time had remained
essentially
separated:
the
study
of differential forms
(absolute
differential
calculus)
and
vector
calculus.
For
a
historical
discussion,
see
Klein,
F.
1927,
pp.
43-45;
Reich
1994,
sec.
5.3; and
Norton
1992a,
pp.
302-310.
In
the
vector analysis
tradition,
tensors
were
defined
more
narrowly
as
what
is
now
described
as
second-rank,
symmetric
tensors
(see, e.g., Sommerfeld 1910a,
p.
767),
whereas
in
Ricci and Levi-Civita
1901
objects
such
as
the
ones
defined
by
Einstein and Gross-
mann
were not
referred
to
as
"tensors" but
as
"covariant
or
contravariant
systems" ("des
systemes
covariants
ou
des
systemes contravariants,"
Ricci and Levi-Civita
1901,
p.
131).
See
also the editorial
note,
"Einstein's Research Notes
on a
Generalized
Theory
of
Relativity,"
pp.
195-196,
for further discussion.
[11]See
also Grossmann
1913
for
a
brief
account
of the
generalized vector calculus,
essen-
tially
similar
to
the
presentation
given
in
Einstein and Grossmann 1913
(Doc. 13).
On
pp.
291-
292 of his
paper
Grossmann
emphasizes
the
necessity
of
an
approach
to vector analysis
from
the
perspective
of the
theory
of
invariants,
as
opposed
to
the
understanding
of
vector
concepts
as
it is
customary
in
physics.
[12]See Einstein's research
notes
on a
generalized
theory
of
gravitation (Doc. 10),
in
particular
[p.
27]
and
[p.
44],
where Grossmann's
name occurs
in
connection with the Riemann and the
Ricci
tensor,
respectively.
These
tensors
would
play a key
role
in
Einstein's later
development
of the
general theory
of
relativity.
[13]See the editorial
note,
"Einstein's Research Notes
on
a
Generalized
Theory
of
Relativity," pp.
197-199,
for
more
details.
[14]Contrary
to
what Einstein and Grossmann
claim,
their
procedure
does
not
lead
to
unique
field
equations;
see
Norton
1984,
sec.
4,
for
a
discussion.
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