304 DOC.
313
MARCH
1917
Three-dimensional
With
supernatural
masses.
Four-dimensional
Without
any
masses.
A
R2
3
R2
Coordinate
System
I
X1, X2,
x3,
et
:
x\
+
x\
+
x\ R2
xi, x2, x3,
x4
=
ict'
:
x\-\-x\+xl+x\
R2
944
1 gi4
=
0
9¡j.u
-
,
R2
-
(x2
xllxv
+
x2
+
x3)
9¡1V
^
[IV
Xy,XV
R2
-
(x2
+
x\
+
x\
+
x\)
ß
and
v
-
1, 2,
3,
4.
Coordinate
System
II
(Hyperspherical coordinates)
ds2
-
c2dt2
-
R2[dx2
+
sin2
x{d^2
+ sin2
ipdd2)\
-oo
t
+oo
0
d 2n
0 x,
^
?r
ds2
=
-
i?2[cL;2
+
sin2
eo{dx2
+
sin2 x(d02
+
sin2
ipdd2)}\
0
$
27T
0
a;,
x,
^
7t
Coordinate
System
III
(Cartesian)
is obtained from
II
through
a
“stereographic
projection”[7]
ds2
-
c2dr
-
dx2
+
dy2
+
dz2
[1
+
(x2
+
y2
+
z2)]2
2
-dx2
-
dy2
-
dz2 + c2dt2
ds1
=_
[l
+
4w(x2
+
y2
+
z2
-
c2t2)]
At
infinity:
guv
=
0 0 0 0
0
0 0 0
0
0 0 0
0 0
0
1
At
infinity:
guv =
0 0 0 0
0
0
0
0
0
0 0 0
0 0 0 0
Invariant under
all transforms.
with
t'
=
t.
Invariant
under
all transforms.
G44
=
0
Gii
=
2/R2gii
2
1, 2,
3 Gii
=
i
=
1,
2, 3,
4
To
find
the
relations between
À
and
R2,
I
have
the
field
equations
G%j
A-
n{Tij
^gijT).