DOC.
9 FORMAL FOUNDATION OF RELATIVITY
55
l~gi.iz~~LVtv+fl
V
V 2,~
ax0
(42a)
In
keeping
the above-listed
names,
we
denote
the
components
Iva
of
according
to
the scheme
v
=
1
v
=
2 v
=
3
v
=
4
a
=
1
-Pxx
-Pxy -Pxz
-ix
a
=
2
-Pyx
-Pyy -Pyz
-iy
a
=
3
-Pzx
-Pzy
-Pzz
-iz
a
=
4
Sx Sy Sz n
(43)
[25]
and the
components
of
ÄCT
according
to the scheme
7 =
1 -fx
7
=
2
-fy
7
=
3
-fz
7 =
4
w
(44)
The
purely
covariant
(or
purely
contravariant)
tensor that is associated
to Iva
is
symmetric.
One understands
easily
that
equations (42a)
transform into
equations (42)
when the
quantities
gßV
take the
special
values
-1
0 0 0
0
-1 0 0
0 0 -1
0
0 0
0
1
(45)
Discussion of
(42a). First,
we
consider the
special
case
that there is
no
gravitational
field, i.e.,
that all the
g^v
are
to
be
seen
as
constants. The first
term
on
the
right-hand
side
of
(42a)
must then vanish.
Say
that
the
system
under consideration
extends
spatially,
i.e.,
with
respect
to
x1,
x2,
x3,
and is finite. The
integral
of
a
[p.
1057]
quantity
/,
when extended
over
the entire
system,
shall be denoted
by
f.
This
type
of
integration over
x1, x2,
x3
yields
from
(42a)