152
ELECTRODYNAMICS OF MOVING
BODIES
£2
+
v2
+
£2
=
R2.
Expressed
in
x, y, 2,
the
equation
of
this surface
at
time
t
=
0
is
x
y2
+ Z2
=
Ä2
+
1
-
v
V
A
rigid
body
that has
a
spherical
shape when
measured
in the
state
of
rest
thus
in
the
state
of
motion-observed
from
a
system
at
rest-has
the
shape
of
an
ellipsoid
of
revolution with
axes
7
R
1
-
v
,R,R
V
Thus,
while the
Y
and
Z
dimensions of the
sphere
(and
hence
also
of
every
rigid
body,
whatever
its
shape) do
not
appear
to
be
altered
by
motion,
the
X
dimension
appears
to be
contracted
in
the ratio
1 :
1
-
(v/v)2,
i.e., the greater the
value
of
v,
the greater
the contraction.
At
v
=
V,
all
moving
objects-observed
from
the
system
"at
rest"-shrink into
plane
structures. For
superluminal
velocities
our
considerations
become
meaning-
less;
we
shall
see
in
the
considerations that follow that
in
our
theory
the
velocity
of
light physically
plays
the part
of
infinitely
great
velocities.
It is clear that the
same
results
apply
for bodies
at rest
in
a
system
"at rest" that
are
observed
from
a
uniformly
moving
system.
We
further
imagine
that
one
of
the
clocks that is able
to
indicate
time
t when
at rest
relative
to
the
system
at rest
and
time
r
when
at rest
relative
to
the
system
in
motion,
is
placed
in the
origin
of
k
and
set such
that it indicates the
time
r.
What
is the
rate
of this clock
when
observed
from
the
system
at
rest?
The
quantities
x,
t,
and
r,
which
refer
to
the
position
of this
clock,
are
obviously
related
by
the
equations
and
v
T
=
t
-
Ji
X
1
1
V
V