46 DOC.
9
FORMAL FOUNDATION OF RELATIVITY
the fact that the transformation
equations
of the
tensor
components are
linear
and
homogeneous.
This in
turn
causes
the
tensor
components
to
vanish relative
to
any
coordinate
system
if
they
vanish relative
to
just
one
such
system. Therefore,
once a
group
of
equations
of
a
physical system
has been
brought
into
a
form which shows
the
vanishing
of
all
components
of
a
tensor,
then this
equation system
is
independent
of
the
choice of
the
coordinate
system.
In order
to
establish such
equations
of
tensors,
one
has to know the laws
by
which
new
tensors
can
be formed from
given ones.
It
has
already
been discussed how this
can
be done in
an
algebraic manner.
We still
have to derive the laws
by
which
new
tensors
can
be formed from known
ones
through
differentiation. The laws of these differential
expressions
have
already
been
given by
Christoffel,
Ricci, and
Levi-Civita.
I
give
here
a
particularly simple
derivation for
this,
which
appears
to
be
new.
[p.
1047]
All differential
operations
on
tensors
can
be traced
to
the so-called "extension."
In the
case
of the
original theory
of
relativity,
i.e.,
the
case
where
only
linear
orthogonal
substitutions
are
admitted
as
"admissible,"
the law
can
be
phrased as
follows.
If
To1...o1
is
a
tensor
of
rank
l,
then
dto1...o1/dxl
is
a
tensor
of rank
l
+
1.
The
so-called
"divergence"
of
tensors follows
easily now
with
equation
(10)
of
§6
from
the
special
tensor
8vu,
which under restriction
to
linear
orthogonal
transforma-
tions-where
the distinction between covariant and contravariant is
void-is
to be
replaced by
the notation
8uv.
By means
of inner
multiplication
of the
tensor
8uv
with
the tensor
of
rank
l
+
1,
obtained
by "extension,"
we get
the
tensor
of
rank
l
- 1
r)T r)T
{7} T
=
ai-_ V
"i
"ai
a/S a,
This is the
divergence
of
the
tensor
To1...a1
formed relative to the index
al.
It
is
now
our
task
to
find the
generalization
of this
operation
in
case
substitutions
are
not
restricted
to
the
previous
conditions
(linearity-orthogonality).
Extension
of
a
covariant
tensor.
Let
(/)(x1...x4)
be
a
scalar and
S
a
given
curve
in
our
continuum.
The
"length
of
arc"
s
is measured from
a
point
P
on
S
in
a
distinct
direction,
as
has been
explained
in
§§1
and
8.
The values of the function
$
at
points
on
S in
our
continuum
can
then
also
be looked at
as a
function
of
s. Obviously,
the
[p.
1048] quantities d(f)/ds,
d2(f)/ds2, etc.,
are
then
also
scalars, i.e., quantities
that
are
defined
independently
of
the coordinate
system. However,
since
f
'
E
(25)
ds
ox"
ds
a*
and since from
every point
the
curves
S
can
be drawn in
any
direction,
the
quantities
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