194 DOC.
30
FOUNDATION OF GENERAL
RELATIVITY
may
take
on
any
values.
This
signifies
that
any
velocity
•
-
V(t):
dxo\2
/dXo\2
dxj \dx
may occur,
which
is less
than the
velocity
of
light
in
vacuo.
If
we
restrict
ourselves
to
the
case
which almost
exclusively
offers itself to
our experience,
of
v
being
small
as
compared
with
the
velocity
of
light,
this
denotes that
the
components
dx1
dx2
dx3
ds,
ds,
ds
are
to be
treated
as
small
quantities,
while
dx4/ds,
to
the
second
order
of
small
quantities,
is
equal
to
one.
(Second
point
of view
of approximation.)
Now
we
remark that from the
first
point
of view of
ap-
proximation
the
magnitudes
rTuv
are
all
small
magnitudes
of
at
least
the
first order. A
glance
at
(46)
thus
shows
that in
this
equation,
from
the
second
point
of view
of
approximation,
we
have to consider
only
terms
for which
u
=
v
=
4.
Re-
stricting
ourselves
to
terms of lowest order
we
first
obtain in
place
of
(46)
the
equations
d2XrT
-
dt2
144TIT
where
we
have
set
ds
=
dx4
=
dt;
or
with
restriction
to terms
which from
the
first
point
of
view
of
approximation
are
of
first
order:-
d?xr
dt2
=
[44,
r]
(t
=
1,
2,
3)
w
=
-
£44'
4-
If in
addition
we
suppose
the
gravitational
field
to be
a quasi-
static
field, by
confining
ourselves to
the
case
where
the
motion
of
the
matter
generating
the
gravitational
field is
but
slow
(in
comparison
with the
velocity
of
the
propagation
of
light),
we
may
neglect
on
the
right-hand
side
differentiations
with
respect
to the time in
comparison
with those with
re-
spect
to
the
space
co-ordinates,
so
that
we
have