DOCUMENT 382
SEPTEMBER
1917
523
expression
for the
gravitational energy-momentum
pseudo-tensor
density.
In this
document,
a
roman
character
is used for
this
quantity.
[11]The
equation
referred
to
is:
^ggFvj ~
-(^o
+
to)
(Einstein
1916o
[Vol.
6,
Doc.
41]).
[12]In Nordstrom
1918a,
p.
1095
(p.
1078 in the
English translation),
the author writes:
“From
a
private correspondence
with
Einstein
I
learned that he has
proved
that these two
vectors
are really
identical,
at least when the
system
of
coordinates is thus chosen that
V-g
= 1
•”
[13]According
to
a
result derived for so-called
complete
static
systems
in Laue
1911a,
pp.
540-541,
the terms with
a
=
1,2,3
in
eq.
(8)
below vanish.
[14]See
note
10
above.
[15]See
note
11
above.
[16]See
Nordstrom
1918a,
sec.
3.
[17]Einstein
1916g (Vol.
6,
Doc.
32).
[18]At this
point
in the
original
text,
Nordstrom indicates
a
note
that he has
appended
at the foot
of
the
page:
“Als dies schon
geschrieben
war,
habe ich eine
Möglichkeit
gefunden
den
Widerspruch zu
erklären. Siehe Seite
5.”
See
“Fortsetzung,
28
Sept.”
below.
[19]The following
calculation
and the
(partial)
resolution
of
the
discrepancy
with Einstein’s
result
in
“Fortsetzung”
below,
can
be
found
in
sec.
2
of
Nordstrom
1918b,
submitted
to
the Amsterdam
Academy
in
January
1918
as a
sequel to Nordstrom 1918a.
[20]Droste 1916b,
eq.
(28).
[21]Eq.
(16)
below
gives
the
general
form
of
a
spherically
symmetric
metric in Cartesian coordi-
nates.
[22]The
expression
for
,.f-g
below follows from
eq.
(16)
by exploiting
the
spherical symmetry
of
the metric and
setting
x1
=
r,
x2 =
x3 =
0
(see
Nordstrom
1918a,
p.
1105
[p.
1087 in
the
English
translation]).
[23]Inserting
eq.
(17)
into
eq.
(16)
and
transforming
from
Cartesian to
spherical
coordinates, one
arrives at the metric
of
eq.
(15).
In Droste
1916b,
eq.
(18), v2
is the coefficient
of dû2
in
the
general
form
of
a spherically symmetric
metric in
spherical
coordinates.
[24]For
a
simple
proof
that the coordinates used in Einstein
1916g
(Vol.
6,
Doc.
32)
do
indeed
have
this
property, see
Einstein 1918a
(Vol.
7,
Doc.
1),
p.
159.
These
coordinates have
become
known
as
isotropic
coordinates.
[25]This example
of
how
gravitational
field
energy can
be transformed
away
is also
given
in Ein-
stein 1918a
(Vol.
7,
Doc.
1),
p. 159,
with
an acknowledgment
to
correspondence
with Nordstrom.
[26]In
Nordstrom
1918b,
sec.
2,
the author shows that
t4
vanishes
in
a
coordinate
system satisfying
V-g
=
1
and mentions without derivation that t4
does not vanish in
isotropic
coordinates. Nord-
strom writes that this makes Einstein’s choice
of
a
pseudo-tensor
for
representing gravitational ener-
gy-momentum
“somewhat less
sympathie”
and that it
supports
H. A. Lorentz’s alternative
choice of
a generally
covariant tensor
(see
Doc.
253,
note
3,
and Doc.
368,
note
6,
for
more
on
this
idea,
which
was championed by
Lorentz and Tullio
Levi-Civita).
[27]From
eq.
(11)
in Einstein
1916g (Vol.
6,
Doc.
32)
for
t^v
in
terms
of
Y'uv an expression
for
f44
results which is
equal
to
-2/k
times
t44
in
eq.
(28)
above
(in
Einstein’s
paper,
Ktuv
denotes what in
this
document
is called t,,,,). To
verify
this
result,
one
needs the relation
a
=
-kM/4^,
which is found
by
comparing
Einstein’s
approximate expression
for the metric in this
case (eq.
(14)
in
the
paper)
to
Droste’s exact
expression (see eqs. (16)
and
(24)
above).
[28]The
contradiction is,
in
fact,
due
to an
error
in
eq.
(11)
for
in Einstein
1916g
(Vol. 6,
Doc.
32),
which
was
corrected in
Einstein
1918a
(Vol.
7,
Doc.
1)
(see
especially
the footnote
on p.
157).
In the latter
paper,
the corrected
expression
for
t^v
is evaluated
explicitly
for the metric field
of
a
point
mass
in
isotropic
coordinates.
The result found
for
kí44
(Einstein
1918a
[Vol.
7,
Doc.
1], eq.
(13))
is
equal
to
the
expression
for
tf
in
eq. (28)
above
with
the
opposite sign.
Since
tßv
=
-tuv
in
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