SUPERLUMINAL VELOCITIES 59
conditions
on
the
magnetic
field.
After
formulating another,
more
general
definition
of
signal velocity,
Einstein abandons the
project
of
defining signal
velocities for
ab-
sorptive
media, pointing
out
that
an approach
using
Fourier
analysis might
be
neces-
sary.[16]
He
maintains, however,
on
the
general grounds
provided by
his
interpretation
of
Wiechert's
result,
that
dispersion theory,
because it
recognizes
no
forces that
are
not electromagnetic
in
origin, precludes superluminal signal propagation.
Wien
was
not
convinced
by
Einstein's
arguments[17]
and remained concerned with
the
problem
of
superluminal
velocities. Discussions with Sommerfeld
on
this
top-
ic[18]
in the
same summer
led the latter
to
a
systematic
study,
the results of which he
presented
at
the
meeting
of the Gesellschaft Deutscher Naturforscher und Arzte
in
Dresden in
September 1907.[19]
The
paper only gives
an
outline of the calculations
that
had
led Sommerfeld
to
conclude
that the
speed
of the front
of
a
wave-signal
can
never
exceed the
speed
of
light
in
vacuum;
full details
were
only given
several
years
later.[20]
In the discussion
following
the lecture Wien still showed himself
uncon-
vinced:
he
stated that the
physical meaning
of Sommerfeld's calculation
escaped
him. And
in
a
review
paper
on
radiation
theory
which
he
completed
in
November
1908
he
pointed
out
that the
problem
of what constitutes the
group velocity
in dis-
persive
media had
by
no means
yet
been
resolved.[21]
[1]The
letters from Wien
to
Einstein from this
period
are
not
available.
[2]See Heaviside
1889.
[3]See
Miller
1981, chap.
1,
for
a
review of
electron
theories,
and
pp.
118-119
for
more
on
the
early
debate
on
superluminal speeds.
[4]See
Sommerfeld 1904a,
1904b, 1904c, 1905.
[5]See
Wiechert
1905.
[6]The discussion,
which became rather
sharp
in
tone,
dealt with
a
variety
of
topics
concern-
ing
radiation
theory.
Wien's
point
of view
on
superluminal speeds
is
expressed
in Wien 1904a,
p.
637, in
response
to
Abraham's criticism in Abraham 1904b.
[7]"physikalisch wenig
wahrscheinlich"
(Wien
1906,
p. 35).
[8]The
nonphysicality
of such
speeds
is
demonstrated
through
several
examples:
the infinite
contraction of
objects moving at
the
speed
of
light
(p.
903),
the infinite
intensity
that
a
light-
source
would have for
an
observer
approaching
it
at
the
speed
of
light
(p.
912),
and the infinite
amount
of
energy
that would be needed
to
accelerate
an
electron
to
the
speed
of
light
(p.
920).
Einstein's conclusion
is
that
superluminal
velocities have "no
possibility
of existence"
("keine
Existenzmöglichkeit")
in
his
theory.
[9]See
Einstein 1907h
(Vol. 2,
Doc.
45),
pp.
381-382. Einstein
1907j (Vol. 2,
Doc.
47),
sub-
mitted
in
December of
1907, repeats
the
argument
without
significant
modification
(pp.
423-
424).
[10]"Wenn
dies Resultat
auch,
meiner
Meinung
nach,
rein
logisch genommen
keinen
Wider-
spruch
enthält,
so
widerstreitet
es
doch
so
unbedingt
dem Charakter
unserer
gesamten
Erfah-
rung,
daß durch dasselbe die
Unmöglichkeit
der Annahme
W V
zur
Genüge
erwiesen ist."
(Einstein
1907h
[Vol. 2,
Doc.
45],
p.
382).
V
is
the
speed
of
light, W
the
speed
with which the
signal
is
propagated.
[11]This
was
in
particular pointed out by
Max Laue
in
a
discussion of
the
propagation
of
ra-
diation
in
dispersive
and
absorptive
media
(Laue 1905).
Laue concluded that the
group
veloc-
ity is
the
velocity
with which
energy
is
transported
by
radiation,
except
in
highly absorptive
media,
where the
concept
of
group velocity
can no
longer
be used. The
history
of
the
concepts
of
group velocity
and
phase velocity goes
back
to
the work of
Rayleigh.
See,
e.g.,
Brillouin
1960, chap.
1,
for
a
brief historical review.
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