DOC. 14
EINSTEIN
AND
BESSO MANUSCRIPT
373
[p.
7]
(Einstein)
[27][P.
7]
is
the
verso
of
[p. 6].
[28]On
[p. 7], [eq.
17]
on [p. 3]
for
a
spherically symmetric
metric
in Cartesian coordinates
is
derived. The
same argument
can
be found
in
a
later
paper
by
Droste
(Droste 1914,
pp.
999-1000;
the
argument is
discussed
in
Earman and Janssen
1993,
pp.
144-145).
The
problem is to
find
the
metric
at
an
arbitrary point
P with coordinates
(x, y,
z) (see
the
figure
in the
upper
left
corner
of
[p.
7]).
Consider coordinates
(x', y',
z'),
related
to
the coordinates
(x, y,
z)
through
a
rotation
around the
origin,
such
that the
point
P
lies
on
the
x'-axis
(the
point
is
indicated
in
the
figure
but
is not labeled).
The
primed
coordinates of P
are
(r, 0, 0),
where
r
=
(x2
+
y2
+
z2)1/2.
In
the
primed
coordinates, the
metric
has
a very
simple
form.
All
off-diagonal components are
zero.
For
g'4i
and
g'i4
this
follows
from
the
static
character of
the metric;
for
g'ij
(i
# j)
it
follows
from
the
spherical symmetry
(see
Droste's
discussion). Furthermore, it
follows
from the
spherical
symmetry
that
g'22
= g'33.
[29][Eq.
43]
gives
the
spherically symmetric
metric
in
the
primed
coordinates
(see note
28).
In this
coordinate
system,
it is
easy to
evaluate the determinant of
the
metric. The determinant
is
just
R T2
p.
When p
is replaced
with p
and R with N
+
T
(see
[p.
3],
[eq.
18]),
this
expression
turns into
[eq. 19] on [p.
3].
[30][Eq. 43]
for the metric
in
primed
coordinates
is
now
transformed
to
the
metric
in
unprimed
coordinates.
To this end,
the transformation
law
[eq.
45]
from
guv
to
g'uv
(which
appears
to
be copied
from Einstein and Grossmann
1913
[Doc. 13],
p.
24)
is inverted to
[eq.
44].
The
transformation matrices
pik
and
jzki are
defined
as
(see
ibid.,
p.
24):
dx
i
dx't
Pik
T
J Hki
-
dxk
dxk
[31]With
the
help
of
[eq.
44],
g11
is
computed.
Since
the
transformation
from
the
primed
to
the
unprimed
coordinates
is
just
a
rotation
(see
note
28), one
can
use
x211 +
x212
+
x213
=
1
(see
[eq.
32]
on [p.
5]
and
note
23).
For
x11
one
has
x11
=
dx'/dx
=
dr/dx
=
x/r.
Using
these
relations,
one
arrives
at
[eq.
46].
[32]With
the
help
of
[eq.
44]
and the relation
x11x21
+
x12x22
+
x13x23
=
0
(see [eq. 32]
on [p.
5]
and
note 23),
g12
is
computed.
When N
is
substituted for
R
-
T
(as
was
done
in
[eq.
18] on [p.
3]),
[eq.
46]
and
[eq.
47]
turn
into the
corresponding components
of
the
metric
in
[eq.
17]
on
[p.
3].
[33]The
material
in
the lower
right
corner
is in
Besso's hand.