DOC. 30 FOUNDATION OF GENERAL RELATIVITY
193
Still
the
question
is not
without
a
formal interest,
whether
with
a correspondingly generalized
definition of
the
energy-
components
of
gravitational
field
and
matter,
even
without
specializing
the
system of co-ordinates,
it
is
possible
to
formu-
late
laws
of
conservation in the form
of
equation (56),
and
field
equations
of
gravitation
of
the
same
nature
as (52)
or
(52a),
in such
a manner
that
on
the left
we
have
a
divergence
(in
the
ordinary
sense),
and
on
the
right
the
sum
of
the
energy-components of
matter
and
gravitation.
I
have
found
that
in
both
cases
this
is
actually
so.
But I
do
not think
that the communication
of
my
somewhat
extensive
reflexions
on
this
subject
would
be
worth
while,
because
after all
they
do not
give
us
anything
that
is
materially
new.
E
§
21.
Newton's Theory
as a
First
Approximation
As
has
already
been
mentioned
more
than
once,
the
special
theory
of
relativity
as a
special case
of
the
general
theory
is
characterized
by
the
guv
having
the constant
values
(4).
From
what
has
already
been
said,
this
means complete
neglect
of
the
effects of
gravitation.
We arrive at
a
closer
approximation
to
reality
by
considering
the
case
where the
guv
differ
from the
values of
(4) by quantities
which
are
small
compared
with
1,
and
neglecting
small
quantities
of second
and
higher
order. (First
point
of view of
approximation.)
It
is
further
to
be
assumed
that in the
space-time
territory
under
consideration the
guv
at
spatial infinity,
with
a
suitable
choice of
co-ordinates,
tend toward the
values
(4);
i.e.
we are
considering gravitational
fields
which
may
be
regarded
as
generated
exclusively by
matter
in the
finite
region.
It
might
be
thought
that these
approximations
must
lead
us
to
Newton's
theory.
But
to
that
end
we
still need to
ap-
proximate
the fundamental
equations
from
a
second
point
of
view.
We
give
our
attention
to the motion
of
a
material
point
in accordance
with the
equations
(16).
In
the
case
of
the
special
theory
of
relativity
the
components
[32]
dx1
dx2 dx3
ds' ds'
ds