18
DOC.
2
COVARIANCE PROPERTIES
Published
in
Zeitschrift
für
Mathematik und
Physik
63
(1914):
215-225. Published
29
May
1914.
[1]Einstein
and
Grossmann 1913
(Vol. 4,
Doc.
13).
[2]Einstein
and
Grossmann 1914a.
[3]See
also
Vol.
4,
the editorial
note,
“Einstein
on
Gravitation and
Relativity:
The Collabo-
ration with Marcel
Grossmann,”
pp.
294-301,
for
a
discussion of Einstein’s
early
work
on
gen-
eral
relativity, including
this
paper.
[4]See
Einstein 1914d
(Vol. 4,
Doc.
26).
[5]In
a
letter
to
Besso of
ca.
10
March
1914
(Vol. 5,
Doc.
514)
Einstein claimed that
the
transformations for which the
gravitational
field
equations
were
covariant included
transfor-
mations
to
rotating systems.
In
the fall of
1915,
however, he realized that this
was
not
the
case
(see
Einstein
to
Erwin
Freundlich, 30
September
1915).
Einstein later listed this realization
as
one
of
the key insights
that led him
to modify
his
theory
of
gravitation
(see
Einstein
to
Arnold
Sommerfeld, 28
November
1915,
Einstein
to
H. A.
Lorentz,
1
January
1916,
and Einstein
to
Willem de
Sitter, 23 January
1917;
see
also the Introduction
to
this volume for
more
historical
background).
[6]The
right-hand
side of this
equation
should be
preceded by
a
minus
sign. "youTuv"
should be
"yuvTuo,"
both here and
in
eq.
(1).
[7]Einstein
and Grossmann 1913
(Vol. 4,
Doc.
13), pp.
23-27.
[8]Summation
over p
is
implied
in
eq.
(I).
These
equations
first
occur
in
Einstein 1913c
(Vol.
4,
Doc.
17).
[9]The
argument
that follows
is
known
as
the "hole
argument."
Earlier versions
are
given
in Einstein 1914d
(Vol.
4,
Doc.
26)
and Einstein 1914e
(Vol.
4,
Doc.
25).
See Norton
1984,
sec.
5,
Stachel
1989,
sec.
3,
and
Vol.
4,
the editorial
note,
“Einstein
on
Gravitation and Rela-
tivity:
The Collaboration with Marcel
Grossmann,”
pp.
297-298,
for
historical discussions.
See also Einstein 1914o
(Doc. 9), §§12-14,
for
a
later version of the
argument
and
a
revised
discussion of the covariance
properties
of the
theory.
[10]"yu"
should be
"yuv."
[11]"x"
should
be
"xv."
[12]See
Einstein 1914d
(Vol. 4,
Doc.
26).
[13]That
equation (III) implied
covariance under linear transformations
was
first discussed
in print in
Einstein
1913c
(Vol. 4,
Doc.
17), p.
1258. See
also the discussions
in
Einstein
to
H. A. Lorentz,
16 August
1913
[Vol. 5,
Doc.
470],
Einstein
to
Paul
Ehrenfest,
before
7
Novem-
ber
1913
[Vol. 5,
Doc.
481],
and Einstein
to
Paul
Ehrenfest,
second half of November
1913
[Vol. 5,
Doc.
484]).
[14]Paul
Bernays
(1888-1977)
was
Privatdozent
at
the
University
of Zurich.
He
was
also
the author
of
a
critique
of
special relativity (Bernays 1913).
[15]"2k" should be
"k."
[16]In
both
terms
one
"gTp"
should
be
a
"yTp."
[17]H
should be 2H.
[18]The
factor
-gyikgmndytn/dxk"
should be differentiated with
respect to
xr.
[19]The
symbol {ik, lm}
denotes the Riemann-Christoffel
tensor Rkiml. See
Einstein
and
Grossmann 1913
(Vol. 4,
Doc.
13),
pp.
35-36.