DOC.
13
GENERALIZED THEORY OF RELATIVITY
161
Given this
state
of
affairs,
and in view of the old
theory
of
relativity,
it
seems
natural
to
assume
that
the
transformation group
we are
seeking
also includes
the
linear
transformations.
Hence
we
require
that Fuv be
a
tensor
with
respect
to
arbitrary
linear transformations.
Now
it is
easy
to
prove
(by
carrying
out
the
transformation)
the
following
theorems:
1.
If
®aß...x
is
a
contravariant
tensor
of rank
n
with
respect
to
linear transforma-
tions,
then
V
v
^
^
dx
fi U*V
is
a
contravariant
tensor
of rank
n
+
1
with
respect
to
linear transformations
(expansion).14
2.
If
©^
x
is
a
contravariant
tensor
of rank
n
with
respect
to
linear transforma-
tions,
then
x
dxx
is
a
contravariant
tensor
of rank
n
-
1
with
respect
to
linear transformations
(divergence).
If
one
carries
out
these
two
operations on a
tensor in succession,
one
obtains
a
tensor
of the
same
rank
as
the
original one (operation
A,
carried
out
on a
tensor).
For the fundamental
tensor
yuv
one
obtains
(a)
'
dy
if
C»
E-f-
aß
°Xa
™ß
,
One
can
also
see
from the
following argument
that
this
operator
is
related
to
the
Laplacian operator.
In
the
theory
of
relativity (absence
of
gravitational field)
one
would
have
to
set
g11
= g22 = g33 =
-1,
g44 =
c2,
guv
=
0,
for
u
#
V;
I
hence
y11
=
y22
= y33 =
-1,
Y44
= -7'
Yuv
=
0,
for
u
#
v.
c
If
a
gravitational
field
is
present
that is
sufficiently
weak,
i.e.,
if the
guv
and
yuv
differ
only infinitesimally
from the values
just given,
then
one
obtains instead of the
expression (a), neglecting
the second-order
terms,
14
Yuv
is the contravariant
tensor
reciprocal
to
guv
(Part
II,
§1).
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