114
DOC. 24 PERIHELION MOTION OF
MERCURY
The
g4p
(gp4)
are
determined
by
condition
3,
the
r
denotes the
quantity +
v/x12
+
x22
+
x32,
and
a
is
a
constant
determined
by
the
mass
of the
sun.
[7]
That condition
3
is fulfilled to
terms
of
first
order
we see
immediately.
More
simply,
the
field equations
(1)
are
also
fulfilled in
the first
approximation.
We
need
only
to
consider
that
upon neglect
of
quantities
of
second
and
higher
order, the
left
side of
equation
(1) can
be
permuted
successively
through
EadTauv/dxa
Ead/dxa[uva],
where
a
runs over
only
1-3.
As
we
perceive
from
equation
(4b),
my theory implies
that
in
the
case
of
a
resting mass,
the
components
g11
up
to
g33
are
in quantities
of
first
order
already
different from
0.
There-
fore, as we
shall
see
later,
no
disagreement
with Newton's law
arises
in
the first
approximation.
This
theory,
however,
pro-
duces
an
influence of the
gravitational field
on a
light ray
somewhat different from that
given
in my
earlier
work,
because the
velocity
of
light
is
determined
by
the
equation
[8]
Eguvdxudxv
=
0.
(5)
Upon
the
application
of
Huygen's
principle,
we
find from
equations
(5)
and
(4b),
after
a simple
calculation,
that
a light
ray passing
at
a
distance
A suffers
an
angular
deflection
of
magnitude
2a/A, while
the earlier calculation,
which was not
based
upon
the
hypothesis
£
Tuu
=
0,
had
produced
the value
a/A.
A light ray grazing
the surface of
the
sun
should
experi-
ence
a
deflection of
1.7
sec
of
arc
instead of
0.85
sec
of
arc.
In
contrast to
this
difference,
the result
concerning
the shift of
the
spectral
lines
by
the
gravitational potential,
which
was
confirmed
by
Mr. Freundlich
on
the
fixed
stars
(in
order of
magnitude),
remains
unaffected,
because this result
depends
only on
g44.
[9]
[10]
Since
we
have obtained the
guv
in
the
first approximation,
we can
also calculate the
components
Tauv
of the
gravitational
field to
the first
approximation.
From
equations
(2)
and
(4b)
we
have
[11]
Pp*=
-a
3
Xpx^x
(6a)
[12]
where
p, o, r
signify
any
one
of the indices
1,
2, 3,
and
a
x.
ru-a--5r,.
(6b)
where
a
signifies
the index
1,
2, or 3.
Those
components in
which the index
4 appears
once or
three times vanish.
second approximation
It
will
subsequently
be
seen
that
we
need
to
determine
only
three
components
Ta44
exactly to
quantities
of
the
second order
in
order
to be
able
to
determine
the orbits
of
the
planets
with the
appropriate degree
of
accuracy.
For
this
process,
the last
field
equation, together
with the
general
conditions
we
have
imposed on our
solution,
suffices.
The last
field
equation,
#dxa
+
1^
=
0,
ax
becomes
upon
consideration of
equation
(6b)
and
upon
neglect
of
quantities
of third and
higher
order
[13]
dn
zL.
?
dx,
2?
From this
we
deduce, upon considering equation
(6b)
and the
symmetry properties
of
our
solution,
Ta44
=
-
a/2xa/r3
(1
-
a/r).
(6c)
The
Motion
of
the Planets
The equation of motion of the point mass
in
the
gravita-
tional
field
yielded
by
the general theory of relativity reads
d2xv/ds2
=
EarTvar
dxa/ds
dsr/ds
(7)
From this equation
we
first deduce that
it
contains the
Newtonian equations of motion as a first approximation.
Of
course, if
the
motion of
the
planet takes place with
a
velocity
less
than the velocity
of
light, then
dx1, dx2, dx3
are smaller
than
dx4.
In consequence,
we
get a first approximation
in
which
we
consider on the right
side
only the term
a
=
r
=
4.
Upon considering equation
(6b), we
obtain
d2xv/ds2
=
Tv44
=
-
a/2xv/rv3(v
= 1,2,3)
(7a)
d2x4/ds2
=
0
These equations show that
we
can
set
s
=
x4
for the first
approximation. Then the
first
three equations are exactly the
Newtonian equations. If
we
introduce polar variables
in
the orbital plane, then, as
is well
known, the energy
law
and
the law of
areas yield
the equations
1/2u2
+ O =
A
(8)
r2do/ds
=
B