DOC. 24 PERIHELION MOTION OF MERCURY
113
IN
A
WORK RECENTLY PUBLISHED
in
these
reports,
I
set up
the
gravitational
field
equations
that
are
covar-
iant with
respect to arbitrary
transformations of determinant
1.
In
a
supplement I
showed that these
equations
are
generally
covariant if the contraction
of
the
energy
tensor of "matter"
vanishes,
and
I
demonstrated
that
no
important
considera-
tions
oppose
the introduction of this
hypothesis, through
which time and
space are
robbed of
the
last trace of
objective
reality.1
[1]
[2]
In the
present
work
I
find
an important
confirmation of
this
most
fundamental
theory
of
relativity,
showing
that it
explains
qualitatively
and
quantitatively
the secular rotation of
the
orbit of
Mercury
(in
the
sense
of the orbital motion
itself),
which
was
discovered
by
Leverrier and which
amounts to 45
sec
of
arc
per century.2 Furthermore, I
show that the
theory
has
as a
consequence
a
curvature
of
light
rays
due
to gravita-
tional
fields
twice
as strong
as was
indicated in
my
earlier
investigation.
[4]
The
Gravitational Field
From
my
last
two
communications
it follows
that the
gravitational
field
in
a vacuum
has
to
satisfy,
upon properly
choosing a
reference
frame,
the
equations
151
+1
W
=
0,
«
dx*
.0
(1)
where the
rjuv
are
defined
by
the
equations
47]
--i?4
dxv dx, dx,
(2)
Let
us
make,
moreover,
the
hypothesis
established
in
the last
communication,
that the contraction of the
energy
tensor
of
"matter"
always vanishes,
so
that,
in
addition,
the deter-
minantal
condition
is
imposed:
|guv|
=
-1.
(3)
A
point
mass,
the
sun,
is
located
at
the
origin
of the
co-
ordinate
system.
The
gravitational
field
this
point mass
produces can
be
calculated
from these
equations
by means
of
successive
approximations.
Nevertheless,
we
should consider that the
guv
are
still not
completely
determined
mathematically
by
equations
(1)
and
(3),
because these
equations are
covariant with
respect to
arbitrary
transformations of determinant
1.
Yet
we are
justified
in
assuming
that
all
these
solutions
can
be reduced
to
one
another
by
such
transformations
that
they are
distin-
guished
(by
the
given boundary conditions) formally
but
not,
however, physically,
from
one
another.
Consequently, I
am
satisfied for the time
being
with
deriving
here
a solution,
with-
out discussing
the
question
whether
the
solution
might
be
unique.
To
proceed,
let the
guv
be
given
in
the
0th
approximation
by
the
following
scheme
corresponding to
the
original theory
of
relativity:
-1 0 0
0
0
-1
0
0
0 0
-1
0
0 0 0 +1
(4)
[6]
or,
more
briefly,
9pv
=
Spa
9p4
=
g4p
= 0
g44
=
1
(4a)
Here
p
and
a
signify
the indices 1,2,3;
Spa
is
equal
to
1
or
0
if
p
=
a or p
#
r,
respectively.
I
assume
in what follows
that
the
guv
differ
from the
values
given
in
equation
(4a)
only by quantities
small
compared to
unity.
I
treat
this deviation
as a
small
quantity
of
first
order,
whereas functions of the nth
degree
in
these deviations
are
treated
as quantities
of the nth
order.
Equations
(1)
and
(3)
together
with
equation
(4a)
enable
us
to
calculate
by
succes-
sive approximations
the
gravitational field up to quantities
of
nth order
exactly.
The
approximation
given
in
equation
(4a)
forms the 0th
approximation.
The
solution has
the
following properties,
which determine
the coordinate
system:
1.
All
components are independent
of
x4.
2.
The solution
is
spatially symmetric
about the
origin
of
the
coordinate
system,
in the
sense
that
we encoun-
ter
the
same
solution
again
if
we
subject
it
to
a
linear
orthogonal spatial
transformation.
3.
The
equations
gp4
= g4p =
0
are
exactly
valid for
P
=
1,2,3.
4. The
guv
possess
the values
given
in
equation
(4a)
at
infinity.
first approximation
It
is easy
to
verify
that
to
quantities
of
first
order the
equations
(1)
and
(3)
are
satisfied for the
just-
named four conditions
by
the assumed
solution
(4b)
d2r
9pa
a
dxpdx,
gu
= 1
- -