130
DOC.
9
FORMAL
FOUNDATION
OF RELATIVITY
[32]See
Einstein
1916b
(Doc. 27)
for
an
alternative
treatment
of the
electromagnetic equa-
tions
in
which the
use
of dual six-vectors
is avoided.
[33]The
reference
is
to eq. (24)
on p.
1044.
[34]The argument presented
in
§12
is
known
as
the "hole
argument."
Earlier versions
are
given
in Einstein 1914d
(Vol. 4,
Doc.
26),
Einstein 1914e
(Vol. 4,
Doc.
25),
and Einstein and
Grossmann 1914b
(Doc.
2).
See
Norton
1984,
sec.
5,
Stachel
1989,
sec.
3,
and Vol.
4,
the edi-
torial
note,
"Einstein
on
Gravitation and
Relativity:
The Collaboration with Marcel Gross-
mann,"
pp. 297-298,
for historical discussions.
See
also Einstein's retraction of
the
conclusion
drawn
at
the end of the third
paragraph
below
in
Einstein
to
Paul
Ehrenfest, 26
December
1915.
[35]The
limitation
to
linear transformations
is
considered
in
Einstein and Grossmann
1913
(Vol. 4,
Doc.
13);
in
Einstein and Grossmann 1914b
(Doc.
2)
it is
shown that the Einstein-
Grossmann
theory
is
covariant under
a
wider
group
of transformations. The
argument
devel-
oped
in
this section elaborates
on
the latter
paper.
[36]In
the fall of
1915
Einstein realized that it
was a
mistake
to
assume
that H
is
invariant
under linear transformations
and
that the
use
of
eq.
(77)
on p.
1075 to fix
H
was
fallacious
as
well
(see
Einstein
to
H. A. Lorentz,
12
October
1915,
and Einstein
1915f
[Doc.
21],
p.
778;
see
also Einstein 19160
[Doc.
41],
eq.
(13),
from which
it follows that
linearity
of H alone
implies
condition
(77)).
He
later listed this
insight
as one
of the
key
steps
toward
revising
his
theory
of
gravitation (see
Einstein
to
Arnold
Sommerfeld, 28 November
1915,
and Einstein
to
H. A.
Lorentz,
1
January
1916).
See also Norton
1984,
secs. 6
and
7,
for
a
historical discussion.
[37]See
also Einstein's elaboration of the
argument
of the remainder of this section
in
Ein-
stein
to
H. A. Lorentz, 23 January 1915,
which
responds
to objections
raised
by
Lorentz
in
a
letter
to
Einstein,
between
1
and
23 January
1915.
[38]The
minus
sign
on
the
right-hand
side should
be
dropped;
a
summation
over \i
is
implied.
[39]The reasoning
in
§14
leading to
the
proof
of the
tensor
character of
smv on p.
1073
was
later criticized
by
Tullio Levi-Civita in
a
series of letters
to
Einstein.
In
his
first
reply to
Levi-
Civita,
Einstein called
this
proof
"the
most
important
proof
of the
theory,
achieved
at
the
cost
of
streams
of
perspiration"
("den wichtigsten,
mit Strömen
von
Schweiss erkauften Beweis der
Theorie." Einstein
to
Tullio
Levi-Civita,
5
March
1915).
See the
correspondence
between Ein-
stein and Levi-Civita from March
to May
1916 in Vol.
8
for
more
details;
see
also Cattani and
De Maria 1989 for
an
analysis.
In
a
letter
to
David Hilbert
of
30
March
1916,
Einstein hints
at
an error
discovered
by
Hil-
bert
in
this section
as
well and admits that his
assumption
of
commutativity
of the variation
5
and the derivative made
on p.
1073 is
erroneous
(see
also Norton
1984, note 62,
for
an
inter-
pretation
of Einstein's remark
to
Hilbert).
[40]"±"
should
be "V."
[41]"xa'"
in
the second
and
fourth
expressions
should
be
"x0,."
[42]Einstein later realized that
eq. (77) only expresses
the invariance of
H
under linear
trans-
formations and does
not fix
H
at
the form of
eq.
(78),
as
is
claimed here.
See
also
note
36.
[43]The
first
term
on
the
right-hand
side should
be
preceded
by a
minus
sign.
[44]A
minus
sign is missing
between the
two expressions
in
the
parentheses.
[45]The sign
of the second
term
in
the
parentheses
on
the
right-hand
side should
be
minus.
[46]The sign
of the third
term
on
the left-hand side should
be
minus.
[47]This problem
was
also raised
in
a
letter
to
Lorentz of
17 January
1916.
[48]In
1949
Kurt Gödel showed that
general relativity
admits solutions that include
closed
timelike
curves
(see
Gödel
1949).
See,
e.g.,
Geroch and
Horowitz
1979,
pp.
232-243,
for
a
discussion of
causality
in
such
spacetimes.
[49]"da" should
be
"d."
[50]"xa2"
should
be "dx2
."
[51]Both
the result for the redshift and the
one
for the
speed
of
light
in the
next equation
already
occur
in
Einstein's
first paper
on
gravitation,
Einstein
1911h
(Vol. 3,
Doc.
23),
where
they
were
directly
derived from the
equivalence principle.
See
also Einstein
1914k (Doc.
3),
note 4,
for
more on
the
attempts
to measure
these effects.