DOC. 9 FORMAL FOUNDATION OF RELATIVITY
81
m
qx
I-q2
=
-
m
v'l
-
q2
"I*
=
m
3
Jl-q2
i4 =
m
i
v'l
-
q2
(86a)
The choice
of
an imaginary
time coordinate
implies
here that the
energy
is
not
represented by
the
quantity
I4-as it is in (52)-but rather
by i
I4.
In the absence
of
external
forces,
one
gets-due to
(84a)-instead
of
(50b)
d(-Io)
dt
=
4E
2
dx
dhVT dx
v
-I)
dt
(87) [49]
Equations (85), (87),
(86)
replace
the
Newtonian
theory
in
first
approximation.
Newton's
theory as
an
approximation.
We arrive
at
this
approximation by
[p.
1083]
treating
the
velocity q as infinitesimally small,
and
by retaining
in the
equations only
those terms which contain the
components
of
q
in
the lowest
power.
In
place
of
(85)
one gets
the
equations
hov
=
0
(as long
as
not
v
=
o
=
4)
h44
=
-Kp0
(85a)
and in
place
of
(87) one
gets
d(m
q) m,
"V
= 2P
"« (87a)
From
(85a)
one
concludes
in
this
case
(with
suitable
boundary
conditions
at infinity)
that all
hov
vanish,
except
for
h44.
From
(87a)
one
concludes that
(
-h44
2)
plays
the
role of
the
gravitational potential. Calling
the latter
quantity
o,
one gets
the
equations
K •M
2P*
** "
*
.
-m grad
j.
dt
(88)
which
agrees
with
Newton's
theory,
provided
d2o
dt2
can
be
neglected compared
to