DOC.
45
247
A
and
some
time thereafter
an
opposite
one
in
B. The two
impulses
compen-
sate each
other
so
that
they
do
not
modify
the
motion
of the rod.
The
case
looks
even
more
odd
if
we
ask about the
energy
at
a
time
when
the
impulse
in
A
had already ended
while that in
B
had not
yet
begun.
The
impulse
in
A
had
transferred
work
to
the rod (since the rod
was
in
motion); hence
the
energy
of
the rod
had to
increase
by
this
work. Yet
no
change
has occurred
either
in
the
velocity
of the
rod
or
in
any
other related
quantity
on
which
the
energy
function
might
be
made
to
depend. Thus
there
appears
to be
a
violation of the
energy
principle.
This difficulty
has
a
very
simple
solution
in principle.
By
implicitly
assuming
that
we can
completely
determine
the
momentary
state
of the
rod
by
the forces
acting
on
the
rod and
by
the rod's
velocity at
that
moment,
we
assume
that
an
increase in the
body's
velocity
is
produced
instantaneously
by
a
velocity-producing
force
acting
somewhere
on
the
body,
i.e.,
that the
spreading
of the force exerted
on one
point
of the
body
over
the
whole
body
does not require
time.
As
we are
going to
show,
such
an
assumption
is
not
compatible
with the
principle of
relativity.
We are
therefore
obviously
forced
to
postulate
in
our
case
that the effect
of
the
impulse
in
A
is
associated with
a change
of
state
of
unknown
quality in
the
body,
which
spreads throughout
it with finite
velocity and
produces
an
acceleration
of
the
body
in
a
short time unless
this effect is
compensated
by
the
effects
of
some
other
forces acting
upon
the
body
within that time.
Hence,
if
relativistic
electrodynamics
is correct,
we
are
still far
from
having
a
dynamics
of the
parallel
translation
of
the rigid
body.
We
will
now
show
that
not only
the
assumption
of
an
instantaneous
spread
of
some
effect,
but also,
more
generally,
any
assumption of
the
spreading
of
an
effect with
a
velocity greater
than the
velocity
of
light is
incompatible
with the
theory
of relativity.
Consider
a
material
strip
extending along
the x-axis of
a
coordinate
system
(x,y,z),
relative
to
which
a
certain effect shall
propagate
with
velocity
W,
and
let there
be at
x
=
0
(point
A)
as
well
as
at
x
=
+l
(point
B)
an
observer
who
is
at rest relative
to
the coordinate
system
(x,y,z).
By
means
of the above
effect,
the observer
in
A
sends
a
signal to
the observer in
B
through
the
material
strip,
which
is
not at rest but is
moving
in the
negative
x-direction with the
velocity
v
(
V).
It follows
[8]
[9]