162
DOC.
13
GENERALIZED THEORY OF RELATIVITY
d2y d2y/xv
1
a2y
\
+
+
_
2
V
dx(
dx
2
^3
dx-
dx
4
If the field
is
static and
only
guv
is
variable,
we
thus arrive
at
the
case
of the
Newtonian
theory
of
gravitation
if
we
take the
expression
obtained for the
quantity
ruv
up
to
a
constant.
Hence
one
might
think
that,
up
to
a
constant factor,
the
expression
(a)
must
already
be the
generalization
of Ap that
we
are
seeking.
But this would be
a
mistake;
for
alongside
this
expression,
in
a
generalization
of this kind there could also
appear
terms
that
are
themselves
tensors
and that vanish when
we
neglect
the kinds
of
terms
just
indicated.
This
always
occurs
when
two
first derivatives of the
guv
or
Yuv
are
multiplied by
each
other.
Thus,
for
example,
?)Saß
dyaß
E
aß
dx..
p
dx
V
is
a
covariant
tensor
of the second rank
(with
respect
to
linear
transformations);
it
becomes
infinitesimally
small
to
the second order if the
quantities g«ß
and
y«B
deviate from
constant
values
only
infinitesimally
to
the first order.
We
must
therefore
allow still other
terms in
Tuv,
in
addition
to
(a),
which
terms,
for
now, must
satisfy
only
the condition
that,
taken
together, they
must
possess
the character of
a
tensor
with
respect
to
linear transformations.
We make
use
of the
momentum-energy
law
to
find these
terms.
To make
myself
clear about the method
used,
I
will first
apply
it to
a
generally
known
example.
In electrostatics
-
dq/dxvp
is
the
vth
component
of the
momentum
transferred
to
V
the
matter
per
unit
volume,
if
p
denotes
the
electrostatic
potential
and
p
the electric
density.
We seek
a
differential
equation
for
p
of such kind that
the
law
of the
conservation of
momentum is
always
satisfied.
It is
well known that the
equation
E-
=
p
v dXv
solves the
problem.
The fact that the
momentum
law
is
satisfied follows from the
identity
E
a
dcp
dip
a
1 dip dip
E
32cp
__
ap.
P
2
dx dx
dxv
dx
dx..
\ P \
pj
V
p
dxp
dx..
/
Thus,
if the
momentum
law
is satisfied,
then
an
identity
of the
following
construction
must
exist
for
every
v:
On the
right
side,
-
dq/dxv
is
multiplied by
the left
side of the differential
equation;
on
the left side of the
identity
there
is
a sum
of the
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