102
DOC.
21
GENERAL RELATIVITY
Sim
=
sil
l/scxm
-
{im
p}{pl/l}.
(13b)
Under the constraint to
transformations with determinants
1,
not
only
(Gim)
is
a
tensor,
but
(Rim)
and
(Sim)
also have tensorial character. It follows indeed from
_
[il]
the fact that
\/-g
is
a
scalar,
and because
of
(6),
that
{il|l}
is
a
covariant four-
vector.
(Sim),
however, is,
due to
(29) l.c., nothing
other than the extension
of
this
four-vector,
which
means
it
is also
a
tensor.
From the tensorial character of
(Gim)
and
(Sim)
follows the
same
for
(Rim),
from
(13).
The tensor
(Rim)
is
of
utmost
importance
for the
theory
of
gravitation.
§2.
Notes
on
the Differential
Laws
of "Material"
Processes
1.
The energy-momentum theorem for matter (including electromagnetic
processes in
a
vacuum.
According
to the
general
considerations
of
the
previous paragraph, equation
(42a)
l.c. has
to
be
replaced by
E
sT/sXv
=
1
V-
tE
8
dg
Tß O/lV
rrt
V
äT
T
K.o’
(14)
where
Tva
is
an
ordinary
tensor,
Ka
an ordinary
four-vector
(not
a V-tensor,
V-vector,
resp.).
We have to
attach
a
remark to this
equation,
because it is
important
for the
following.
The
equations
of
conservation led
me
in the
past
to view the
quantities
1/2
Emgtm
sgmv/sX
as
the natural
expressions
of
the
components
of
the
gravitational
field, even
though
the formulas
of
the absolute differential calculus
seem
to
suggest
the
Christoffel
iva)
symbols
{vo|T}
instead,
as being
the
more
natural
quantities.
The former view
was
a
fateful
prejudice.
The
preference
for the
Christoffel
symbols justifies
itself
[p.
783] especially
because
of
the
symmetry
in their covariant indices
(here v
and
o)
and,
furthermore,
because
they occur
in the
fundamentally important equations
of
the
geodesic
line
(23b) l.c.;
and these latter
are-from
a physical point
of
view-the
equations
of
motion
of
a
material
point
in
a gravitational
field.
Equation (14)
cannot