EINSTEIN
IN
COLLABORATION
WITH GROSSMANN
297
III
In
spite
of
his achievements,
Einstein
was
dissatisfied
with the lack of
general
covari-
ance,
which
he
later called "this
ugly
dark
spot,"
and
did
not
have much faith
in
the
new
theory.[15]
In
collaboration with Grossmann
he
continued
to
search for the
most
general
transformations under which the
gravitational
field
equations
of
the
"Entwurf"
theory
are
covariant,
but
by
mid-August
of
1913
he
was
still
unable
to
find
a
single
nonlinear transformation admitted
by
the field
equations.[16]
Around that time he
apparently gave up
these
attempts
because he found
two
argu-
ments
which convinced him that the restriction of
general
covariance
was a
necessity.
The
first
of these
arguments
follows from
a
consideration of conservation laws and
was
found
by
Einstein
on
15 August 1913.[17]
He had
suddenly
realized that the
conservation
law
for
matter
and the
gravitational
field
as
derived
in
Einstein and
Grossmann 1913
(Doc. 13), part I,
eq.
(19)
is
only
covariant under linear transfor-
mations because
it
involves
a
coordinate
divergence
and not
a
covariant
divergence.
The
implicit assumption
made here
is
that the
stress-energy
tensors
for
matter and
for
the
gravitational
field
are
both
generally
covariant. Einstein's conclusion
was
that
his
theory
could
only
be covariant under linear
transformations.[18] As he
later
pointed
out,
this
conclusion
was
incorrect. For
a theory
that
is not
generally
covariant,
the
stress-energy
tensor
need
not be
a
generally
covariant
tensor,
and
the
conservation
laws
do not
have
to be
covariant
only
under linear transformations
(see
Einstein and
Grossmann
1914b).
Einstein's second
argument against
general covariance,
probably
discovered
only
shortly
afterwards,
seems
to
imply
that the metrical
tensor
guv
cannot be
uniquely
determined
by generally
covariant
field
equations.
It has been
called
"the hole
argument."[19]
The "hole
argument"
has been
extensively
discussed
in the
secondary
literature because of
its
significance
for
reconstructing
Einstein's
early understanding
[15]See Einstein
to
H. A. Lorentz,
14 August
1913
(Vol. 5,
Doc. 467);
for Einstein's reference
to
"dieser hässliche dunkle
Fleck,"
see
Einstein to H.
A. Lorentz,
16 August
1913
(Vol. 5,
Doc.
470).
[16]See Einstein
to H. A. Lorentz,
14 August
1913
(Vol. 5,
Doc.
467).
[17]See
Einstein
to H. A. Lorentz,
16 August
1913
(Vol. 5,
Doc.
470).
See also
Einstein
to
Paul
Ehrenfest,
before
7
November
1913 (Vol.
5,
Doc.
481).
[18]This
argument
is
presented
in
detail in the lecture Einstein delivered
to the
meeting
of
the
Gesellschaft Deutscher
Naturforscher
und
Ärzte
in
Vienna
on
23
September
(see
Einstein
1913c
[Doc. 17], §6).
[19]"Lochbetrachtung"
(see
Einstein
to
Michele
Besso,
3 January
1916).
For
historical dis-
cussions of this
argument, see
Norton
1984,
sec.
5;
Norton
1987,
pp.
168-184; Kox
1988,
pp.
69,
72-73;
Stachel
1989, sec. 3;
and
Howard and Norton
1993.
The
argument
is
alluded
to in
Einstein 1913d
(Doc. 15);
Einstein
1914g (Doc. 16);
Einstein
1913c
(Doc. 17),
p.
1257,
fn.
2;
Einstein to
Ludwig Hopf,
2
November
1913
(Vol.
5,
Doc.
480);
and Einstein
to
Paul
Ehrenfest,
second half of November
1913
(Vol.
5,
Doc.
484).
It
was
first
published
in
January
1914,
in
a
supplement to the republication
of the "Entwurf"
paper
in the
Zeitschrift für
Mathematik und
Physik (Einstein
1914d
[Doc.
26]).
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