DOC. 4 THEORY
OF STATIC GRAVITATIONAL
FIELD
119
8A
=
dc
8x
+
^-8y
dc
+
&
j
^/t
=
-Jc
|
d(fjdx)
+
^Tj
_
J
coadr.
dx
dy
dy
dz
To calculate
6E,
we
assume
that
8
j
c
fa
I
=
s|4jgrad2\/c^T|
=
S|4jgrad2udr
=
8f
du(
du
, ,
dr
=
81
f8u
-^ds
-
JAmSmJr|.
The first
of
these
integrals (the
surface
integral
over
the
infinitely
distant
surface)
vanishes because with
increasing
radius
vector
R,
the
quantities
6u and
du/dn go
to
zero as 1/R
and
1/R2.
The
second
integral,
however,
can
be reformulated with the aid
of the field
equation (3b'),
so
that
one
gets
5
jgrad2
dr
=
-4kju8uadr
=
-2k
ogcdr.
With the
help
of this
equation
one
obtains
8E
=
J(c8a
+
o8c
-
o8c)dT
=
dA.
This
proves
that
1/2kgrad2
c/c
is indeed
to
be viewed
as
the
energy density
of the
gravitational
field.
(Received
on
23
March
1912)
Note
Added
in
Proof
It is worth
noting
that the
equations
of
motion of the material
point
in a
gravitational
field
dc
X•
d
c
dx
St
etc
dt
m
_
2
1
-
*
1
-
c2
\
assume
a very simple
form if
one
gives
them the form of
Lagrange's equations.
Namely,
if
one
sets
they
read
H
=
-m\Jc2
-
q2,
d (dH\ dH