DOC.
3
STATICS OF GRAVITATIONAL FIELD 99
that the field
is
determined
by
c.
According
to
(5)
and the
equivalence principle,
in
the
gravitational
field
corresponding
to
the acceleration field considered
in
§1,
the
equation
(5a)
Ac
=
+
^
+
&£_
= 0
dx2
dy2
dz2
is
satisfied,
and this makes it natural for
us
to
assume
that
we
have
to
view this
equation
as
valid in
every
mass-free static
gravitational
field.4
In
any
case,
this is the
simplest equation
that
is
compatible
with
(5).
It
is
easy
to
establish the
presumably
valid
equation
that
corresponds
to
Poisson's
equation.
For
it follows
immediately
from the
meaning
of
c
that
c
is
determined
only
up
to
a
constant
factor
that
depends
on
the constitution of the clock with which
one
measures
t at
the
origin
of
K.
Hence the
equation
corresponding
to
Poisson's
equation
must be
homogeneous
in
c.
The
simplest equation
of this kind
is
the linear
equation
(5b)
Ac
=
kcp,
where
k
denotes the
(universal) gravitational constant,
and
p
the matter
density.
The
latter
must
be defined in such
a
way
that it
is
already
given
by
the distribution
of
the
mass, i.e.,
that for
given
matter in
the
spatial
element it is
independent
of
c.
This
we
achieve
by setting
the
mass
of
a
cubic
centimeter of
water
equal
to
1,
regardless
of
the
gravitational
potential
in which
it is located;
p
is
then the ratio of the
mass
contained in
one
cubic centimeter
to this unit.
Now
we
seek
to
obtain the law
of
motion of
a
material
point
in
a
static
gravitational
field. To this end
we
seek the law of motion of
a
freely
moving
material
point
in the acceleration field
considered in
§1.
In the
system
Z this law
of
motion
is
E
= A1T
+
B1,
n
= A2T
+
B2,
C
=
A3T
+
B3,
where
A
and
B
are
constants.
By
virtue
of
(4),
these
equations
transform into the
following equations
valid for
sufficiently
small
t:
x
=
A1ct
+ B1
-
ac-2t2,
y
=
A2ct +
B2,
z
=
A3ct
+
B3.
By
differentiating
once
and then
once
again,
and then
setting
t
=
0, one
obtains from
4A
soon
to be
published paper
will show that
equation (5a)
and
(5b)
cannot
yet
be
exactly right.
However,
they
will be
provisionally
used in the
present paper.
[11]
[13]
[12]
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Extracted Text (may have errors)


DOC.
3
STATICS OF GRAVITATIONAL FIELD 99
that the field
is
determined
by
c.
According
to
(5)
and the
equivalence principle,
in
the
gravitational
field
corresponding
to
the acceleration field considered
in
§1,
the
equation
(5a)
Ac
=
+
^
+
&£_
= 0
dx2
dy2
dz2
is
satisfied,
and this makes it natural for
us
to
assume
that
we
have
to
view this
equation
as
valid in
every
mass-free static
gravitational
field.4
In
any
case,
this is the
simplest equation
that
is
compatible
with
(5).
It
is
easy
to
establish the
presumably
valid
equation
that
corresponds
to
Poisson's
equation.
For
it follows
immediately
from the
meaning
of
c
that
c
is
determined
only
up
to
a
constant
factor
that
depends
on
the constitution of the clock with which
one
measures
t at
the
origin
of
K.
Hence the
equation
corresponding
to
Poisson's
equation
must be
homogeneous
in
c.
The
simplest equation
of this kind
is
the linear
equation
(5b)
Ac
=
kcp,
where
k
denotes the
(universal) gravitational constant,
and
p
the matter
density.
The
latter
must
be defined in such
a
way
that it
is
already
given
by
the distribution
of
the
mass, i.e.,
that for
given
matter in
the
spatial
element it is
independent
of
c.
This
we
achieve
by setting
the
mass
of
a
cubic
centimeter of
water
equal
to
1,
regardless
of
the
gravitational
potential
in which
it is located;
p
is
then the ratio of the
mass
contained in
one
cubic centimeter
to this unit.
Now
we
seek
to
obtain the law
of
motion of
a
material
point
in
a
static
gravitational
field. To this end
we
seek the law of motion of
a
freely
moving
material
point
in the acceleration field
considered in
§1.
In the
system
Z this law
of
motion
is
E
= A1T
+
B1,
n
= A2T
+
B2,
C
=
A3T
+
B3,
where
A
and
B
are
constants.
By
virtue
of
(4),
these
equations
transform into the
following equations
valid for
sufficiently
small
t:
x
=
A1ct
+ B1
-
ac-2t2,
y
=
A2ct +
B2,
z
=
A3ct
+
B3.
By
differentiating
once
and then
once
again,
and then
setting
t
=
0, one
obtains from
4A
soon
to be
published paper
will show that
equation (5a)
and
(5b)
cannot
yet
be
exactly right.
However,
they
will be
provisionally
used in the
present paper.
[11]
[13]
[12]

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