DOC.
3
STATICS OF GRAVITATIONAL FIELD 97
£
=
X
+
at2
+
(2)
r
=
ß
+
yt
+
8t2
+
The coefficients
of
these
series,
which hold for
sufficiently
small
positive
and
negative
values
of
t,
shall be considered
provisionally
as
unknown functions of
x.
Limiting
ourselves
to
the
terms
written
down,
we
obtain
by
differentiation
(3)
=
(X'
+
a't2)dx
+
2atdt,
dr
= (j3'
+
y't
+ 8't2)dx + (y + 28t)dt.
We
imagine
that
the time
is
measured in the
system
E in such
a
way
that the
velocity
of
light
is
equal
to
1.
Then
we can
write the
equation
of
a
shell
expanding
from
an
arbitrary spacetime point
with the
velocity
of
light
in the
following
form,
if
we
restrict ourselves
to the
infinitesimally
small
region surrounding
the
spacetime
point:
dE2
+ dn2 + dc2
-
dr2
=
0.
In the
system
K,
the
same
shell
must
have the
equation
dx2 +
dy2
+ dz2
-
c2dt2
=
0.
The substitution
equations
(2)
must
be such that these
two
equations
be
equivalent.
Because
of
(1),
this
requires
the
identity
(4) d£2
-
dT2
=
dx2
-
c2dt2.
If
one
substitutes in the left side of this
equation
the
expressions
in
dx
and dt
obtained
through equation (3),
and if the coefficients of
dx2, dt2,
and dxdt
are
set
equal
to
each other
on
both
sides,
one
obtains the
equations
1
=
(X' +
cc't2)2-^'
+
Y7+
ö't2)2,
-
c2
=
4a2/2
-
(y + 2öt)2,
0
=
(X'+
a'tz)-2at-(ß'+ y't+
ö't2)^
+
261).
These
equations
are
identities in
t
up
to
powers
of
t
so
high
that the
terms
omitted
in
(2)
still have
no
effect, thus,
the first
equation
up
to
the
second,
and the second
and third
up
to the first
power
of
t.
This results in the
equations
l
=
V2ß'2, 0
=
P'y\
2Act'-Y'2-2ß'ö'
=
0,
-C
=
-Y2,
0=yö
0=
ß'Y, 0
=
2aA'-2ß'6-YY'.
Since
y
cannot vanish, it
follows from the first
equation
in the third line that
ß'
=
0.
Thus,
ß
is
a
constant
that
can
be
set
equal
to
zero
if
starting
times
are
chosen
appropriately. Further,
the coefficient
y
must
be
positive;
hence,
according
to
the first
equation
on
the second line
we
have
y
=
c.
According
to
the second
equation
of
the second
line,
we
have
Ö
=
0.
[8]
[9]