EINSTEINBESSO
ON THE MERCURY PERIHELION
349
(i) (D
or
the
product
of
two
factors
y^v
or
g^v
and
nothing smaller,
one
arrives at
(i)
~
()
~
(D
(2)
1
(0)
d 0) d
y^v (0)
(0) 9
Ynx
9
YvP
A
Yfxv
+
/
v
/ a
V
8
/
Yaß
§rp
~ ~
VVTfYaß
dx«
dxß
dxa
dxß
t
(1)
a
(1)
a
(,)
a
(1)
1
(0) (0)
°a
Srp °
Yrp
,1i
(0) (0) ° Srp
O
yrp
=
"
Ya^Yßv
+
~
Y^vYaß
,
(2)...
2
^
dxa öXr
4 LJ
dxa
dXßp
a
ßr
p
u p
a
ßz
p
"
00
where
g is
detg to
nth
order.
When
the
solutions for
gMV,
and
yflv
to first
order
are
substituted
in
eq. (2), one
obtains
equations
for
y^v
(see
[p.
3],
[eqs.
2022]).
These
equations can
be
solved
using
the
fact that
the field
of
the
sun
will be
spherically
symmetric.
This
means
that
the
components
of
the
metric
field in the
chosen Cartesian
coordinate
system
can
be
expressed
in terms
of
x, y,
z,
and
three
as yet
unknown
functions, N(r), T(r),
and O(r), where
r
=
(x2
+
y2
+
z2)1/2 (see
[p. 3], [eq.
17],
[p.
5]
and
[p. 7]).
The
same
is true
for
terms in the
power
series
expansion
of
the field.
The
(2)
equations
for
y/xu
now
become
equations
for
N(r),
T(r),
and
£(r).
They
are
solved
(see
[p.
6],
[eqs.
3639])
and the
resulting expression
for
y2Jv
is
added
to
y°Jv
+
yjv
to
give
the field
yßV
to
second
order.
Inverting
yßV,
one
obtains
g^v
to second order
(see
[p.
6],
[eq.
40]
and
[eq. 42]).
In
order
to find the
perihelion motion,
only
g44
is
needed
to second
order
(as
is
noted
explicitly
on [p.
16]).[28]
On
[p. 2],
the
44component
of
the
analog
of
eq. (2)
for
the
field equations in the
form

D^v =
K(t^v
+
T^v)
is
used
to find
g44.
When
a
trivial
sign
error
in this
calculation
is corrected, the
result
(see
[p.
2], [eq. 16])
is
the
same as
that of
the
more
laborious calculation
on
[pp.
37]
outlined
above.[29]
lb. Precession
of the
perihelion
of
an
orbit
in the
field of
a
static
sun
to
second
order
([pp.
815],
[pp.
2530],
[p.
33],
[p.
40]).
The
equations governing
the
motion
of
a point
mass m
in
a
metric
field
are
the
EulerLagrange
equations
for
the
action
S
=
f
H dt
(with the Lagrangian
H
=
m
ds/dt),
and the
corresponding
expression
for
the
Hamiltonian E. These
equations
are
copied directly
from the
"Entwurf"
paper
([p.
8],
[eqs. 4850]).[30]
For
a mass
point moving
slowly
in the weak field
of
a
static
sun,
the
EulerLagrange equations
become
(see
[p.
8],
[eq. 54])[31]
M •
\
d
x
\
1
dg44 x
ds
'I
*
/.
(3)
dt
\
I 2
dr
r
dt
[28]See
[p.
16],
note 78.
[29]See
[p.
2],
notes
68.
[30]See
Einstein and Grossmann
1913
(Doc. 13), pp.
68.
[31]In
the
manuscript,
no
vector
notation
is
used
at
this
point.