DOC. 4 THEORY OF STATIC GRAVITATIONAL FIELD
109
from
which it
immediately
follows that
u
=
cb'
or
=
-
tJ
if
one
sets
t"
=
-
etc.
c
Further,
let
dc
denote the
change
that
c
experiences during an
infinitesimally
short
time
at
a
point
of
the
system K,
and
d's'
the
corresponding change
that
s'
experiences
at
the
momentarily coinciding point
of E
at
the
corresponding
time. At
the
start
of
the
infinitely
small time intervals
dt
and dx
let t
=
t
=
0;
at
that time
s
= S'.
However,
the
latter
equation
is
no
longer exactly
valid
at
the end
of dt
or
dx
for
two
reasons.
First,
because
at the end of
dx
the
system point
of K is
no
longer
coincident with that
of
the
system
E;
but this
can
be
ignored
because this
displace-
ment
is
infinitesimally
small of the second order.
Second,
during
the
infinitely
short
time in
question,
the
point
of
the
system
K attains
a
velocity gdx
in the direction
of
the
e-axis; thus,
in
order to obtain
s
at
the end of
dx,
one
has
to
refer the
electromagnetic
field
to
an
acceleration-free
system
that,
relative
to
E,
moves
in the
direction of the
positive
e-axis
with the
velocity gdx.
The
electromagnetic
field
transforms here in the familiar
way.
In
light
of the indicated
arguments one
obtains
ds =
d'&'
+
[g
$']dt,
[7]
or, taking
into
account
the last of
equations
(2a)
3®'
1
a®
lrA
Ä1
dr
~
c
dt
c®®1
But from
equations (2)
one
obtains
[8]
a
1
dc
c
c
dx"
hence,
because
c
is
independent
of
y
and
z,
9
=
-
gradc.
c
Thus,
one
finally gets
d&
_
Id®
C
(
d*
-
o
ar=
c
[gradc,£]
[9]
and in
a
wholly analogous way
4^=14^+[gradc-®].
If
one
also
takes into
account
now
that,
according
to
the rules
of
vector
calculus,
c
rot
4?
+
[grad c,
$]
=
rot
(c$),
and that the
analogous equation
holds for
rot
(c
©), one
obtains from
equations
(1),
if
one
takes into
account
the
results
of
the
previously
stated
arguments,
the
following
equations
referred
to
the
system K: