3. Shadows and Light Spots
Think about how fast a shadow can move. If you
project a shadow of your finger using a nearby lamp
onto a far away wall and then wag your finger,
the shadow will move much faster
than your finger. If your finger moves parallel to the wall,
the speed will be multiplied
by a factor D/d where d is the distance
from the lamp to your finger and D is the
distance from the lamp to the wall. It can actually be
much faster than this if the wall is at some oblique angle.
If the wall is very far away the movement of the shadow
will be delayed because of the time it takes light to
get there but its speed is still amplified by the same
ratio. The speed of a shadow is therefore not restricted
to be less than the speed of light.
Others things which can go faster than the speed
of light include the spot of a laser which is pointed
at the surface of the moon. Given that the distance to
the moon is 385,000 km try working out the speed of the spot
if you wave the laser at a gentle speed. You might also like to
think about a wave arriving obliquely at a long
straight beach. How fast can the point at which the
wave is breaking travel along the beach?
This sort of thing can turn up in nature. For example
the beam of light from a pulsar can sweep across a dust
cloud. A bright explosion emits an expanding spherical
shell of light or other radiation. When it intersects a surface
it creates a circle of light which expands faster than light. A
natural example of this has been observed when an electromagnetic
pulse from a lightning flash hits an upper layer of the
atmosphere.
These are all examples of things which can go faster than
light, but which are not physical objects. It is not possible to
send information faster than light on a shadow or light spot so
FTL communication is not possible in this way. This
is not what we mean by faster than light travel although
it shows how difficult it is to define what we really
do mean by faster than light travel.
(see also Relativity FAQ
The Superluminal Scissors)
4. Rigid bodies
If you have a long rigid stick and you hit one end,
wouldn't the other end have to move immediately? Would
this not provide a means of FTL communication?
Well it would if there were such things
as perfectly rigid bodies. In practice the effect of
hitting one end of the stick propagates along it at
the speed of sound in the material which depends on
its elasticity and density. Relativity places an
absolute limit on material rigidity so that the
speed of sound in the material will not be
greater than c.
The same principle applies if you hold a long string or
rod vertically in a gravitational field and let go of the top end.
The point at which you let go will start to move immediately
but the lower end can not move until the effect has propagated
down the length at the speed of sound in the material.
It is difficult to formulate a general theory of
elastic materials in relativity, but the general principle
can be illustrated with Newtonain mechanics. The equation
for longitudinal motion in an ideal elastic body can
be derived from Hooke's law. In terms if the mass per unit
length p and the Young's modulus of elasticity
Y the longitudinal displacement X
satisfies a wave equation,
(see for example "Classical Mechanics" Herbert Goldstein)
d2X d2X
p --- - Y--- = 0
dt2 dx2
Plane wave solutions travel at the speed of sound s
where s2 = Y/p. This wave equation does
not allow any causal effect to propagate faster than s.
Relativity therefore imposes a limit on elasticity:
Y < pc2. In practice no known material
comes anywhere near this limit. Note that even if the
velocity of sound is near c the matter does not
necessarily move at relativistic speeds. But how can we
know that no material can possible exceed this limit?
The answer is that all materials are made of particles
whose interaction are governed by the standard model
of particle physics, and no influence faster than light
can propagate in that model (see below about quantum field
theory)
Although there is no such thing as a rigid body
there is such a thing as rigid body motion but this
is another example in the same category as the shadows
and light spots described above which do not give you
FTL communication. (see also the relativity FAQ articles
The Superluminal Scissors
and The Rigid Rotating Disk in
Relativity).
5. Phase velocity
Look at this wave equation:
d2u d2u
-- - c2 -- + w2 u = 0
dt2 dx2
This has solutions of the form:
u = A cos( ax - bt )
c2 a2 - b2 + w2 = 0
These solutions are sine waves propagating with a speed,
v = b/a = sqrt(c2 + (w/a)2)
But this is faster than light, so is this the equation
for a tachyon field? No it is the usual relativistic equation
for an ordinary massive scalar particle!
The paradox is resolved by distinguishing this velocity
which is known as the phase velocity vph
from another velocity known as the group velocity
vgr which is given by,
vgr = c2 / vph
If a wave solution has a frequency dispersion it
will take the form of a wave packet which travels at the
group velocity which is less than c. Only its
wave trains travel at the phase velocity. It is only
possible to send information with such a wave equation
at the group velocity so the phase velocity is yet another
example of a speed faster than light which cannot
carry a message.
7. Relativistic Rocket
A controller based on Earth is monitoring a
space-ship moving away at a speed 0.8c.
According to the theory of relativity he will
observe a time dilation affecting the clocks
on the ship and slowing them down by a factor
of 0.6, even after he has taken into account
the Doppler shift of signals coming from the
space-ship. If he works out the distance
moved by the ship divided by the time elapsed
as measured by the on-board clocks, he will
get an answer of 4/3 c. This means
that the occupants of the ship are traversing
the distances between stars at effective speeds
greater than the speed of light when measured with
their clocks. From the point
of view of the occupants, it is the distance
between the stars which is contracted by a factor
of 0.6 and they also agree that they are
covering the known distances between stars at
4/3 c.
This is a real effect which in principle
could be used by space travellers to cover
very large distances in their lifetimes. If they
accelerate at a constant acceleration equal to the
acceleration due to gravity on Earth, they
would not only have a perfect artificial
gravity on their ship, but would also be able
to cross the galaxy in only about 12 years of
their own proper time.
(see relativity FAQ What
are the Equations for the Relativistic Rocket?)
However, this is not true FTL travel. The effective
speed calculated used the distance in one reference
frame and the time in another. This is not the real
speed. Only the occupants of the ship benefit from
this effective speed. The controller will not see
them travelling large distances in his lifetime.
15. What Does Faster Than Light Mean?
The cases given so far just go to show how difficult
it is to pin down exactly what we mean by Faster Than Light
travel or communication. It does not mean things such
as shadows so what does it mean?
In relativity there is no such thing as absolute
velocity, just relative velocity, but there is a clear
distinction between the world lines which
are timelike, lightlike and spacelike. By "world line"
we mean a curve traced out in the 4 dimensions of
space-time which could be the history of a particle
or a point on a shadow. If the world line of something
is space-like then it is going faster than light.
So there is a clear meaning
of what is meant by a "faster than light"
velocity which excludes the case of third party
observers.
But what do we mean by an "object" if we
don't want to include shadows. We could agree to say it is any
thing which carries energy, charge, spin or information
or just that it must be made of atoms, but there are
technical problems in each case. In general relativity
energy can not be localised so we had better avoid using
that in our definition. Charge and spin can be localised but
not every object may have charge or spin. Information is
better but tricky to define and sending information faster
than light is really just FTL communication not FTL travel.
Another difficulty is knowing when an object seen at A
is the same as the one which was seen at B when we
claim that it has travelled there faster than light. Could
it not be a duplicate? It could even be argued that FTL
communication makes FTL travel possible because you can
send the blueprint for an object FTL as information then
reconstruct the object, though not everyone would accept
such teleportation as FTL travel.
The problems of specifying just what we mean by FTL are
more difficult in general relativity. A valid form of FTL travel
may mean distorting space-time (e.g. making a wormhole)
to get from A to B without going on a spacelike
curve locally. There is a distinction between going faster than light
locally and getting from A to B faster than
light globally. When a gravitational lens bends the light coming from
a distant galaxy, the light coming round the galaxy on one side
reaches us later than light which left at the same time and
went round the other side. We must avoid a definition of FTL travel
which says that a particle going from A to B
gets there before light which has made the same journey along
a lightlike geodesic. This makes it very difficult, perhaps impossible,
to define global FTL unambiguously
If you were expecting me to finish this section with a
precise definition of what is meant by FTL travel and FTL
communication I am afraid I have to disappoint you. The
above difficulties are insurmountable. Nonetheless, you
will probably recognise the real thing when confronted
with it now that I have given some examples of what
would not be FTL.
16. The infinite energy argument
When Einstein wrote down his postulates for
special relativity he did not include the
statement that you can not travel faster than light.
There is a misconception that it is possible to
derive it as a consequence of the postulates he
did give. Incidentally, it was Henri Poincare who
said "Perhaps we must construct a new mechanics,
... in which the speed of light would become an
impassable limit." That was in an address to
the International Congress of Arts and Science in
1904 before Einstein announced special relativity in
1905.
It is a consequence of relativity that the energy
of a particle of rest mass m moving with
speed v is given by
E = mc2/sqrt(1 - v2/c2)
As the speed approaches the speed of light the energy
approaches infinity. Hence is should be impossible to
accelerate an object with rest mass to the speed of light
and particles with zero rest mass must always go at exactly the
speed of light otherwise they would have no energy. This is
sometimes called the "light speed barrier" but
it is very different from the "sound speed barrier".
As an aircraft approaches the speed of sound it starts to
feel pressure waves which indicate that it is getting close.
With some more thrust it can pass through. As the light speed
barrier is approached (in a perfect vacuum) there is no such
effect according to relativity. Moving at 0.99999c
is just like standing still with everything rushing past you at
-0.99999c. Particles are routinely pushed to these
speeds in accelerators so the theory is well established.
Trying to get to the speed of light in this way is like trying
to get to the pot of gold at the end of a rainbow.
This explains why it is not possible to exceed the
speed of light by ordinary mechanical means,
however, it does not in itself rule out FTL travel.
It is really just one way in which things cannot be made to
go faster than light rather than a proof that there is no way
to do it.
Particles are known to decay instantly into other particles
which fly off at high speed. It is not necessary to
think in terms of the particles having been accelerated
so how could we say that they could not go faster than light?
What about the possibility of particles which might have
always been going faster than light and which might be used
to send information if they can be detected and deflected
without ever slowing down to less than the speed of light?
Even if such particles don't exist there may be ways of
moving matter from A to B, faster than
light can get there by the usual root, but without anything
having to go at a FTL speed locally.
17. Quantum Field Theory
All physical phenomena except gravity which have been
observed until recently are consistent with the standard
model of particle physics. The standard model is a
relativistic quantum field theory which incorporates the
nuclear and electromagnetic forces as well as all the observed
particles. In this theory, any pair of operators corresponding
to physical observables at space-time events which are separated
by a space like interval commute. In principle this means
that effects can not propagate faster than light in the
standard model and it can be regarded as the quantum field
theory equivalent of the infinite energy argument.
However, there is no completely rigorous proof of anything
in the quantum field theory of the standard model since nobody
has yet succeeded in showing that it is self consistent. Most
likely, it is not consistent. In any case, there is no
guarantee that there are not other undiscovered particles and
forces which do not obey the rule. Nor is there any generalisation
which takes gravity and general relativity into account.
Many physicists working on quantum gravity doubt that
such simplistic expressions of causality and locality will
be generalised. All told, there is no guarantee that light
speed will be meaningful as a speed limit in a more
complete theory of the future.
18. Grandfather Paradox
A better argument against FTL travel is the grandfather
paradox. In special relativity a particle moving FTL in
one frame of reference will be travelling back in time
in another. FTL travel or communication should therefore
also mean the possibility of travelling back in time or
sending messages into the past. If such time travel is
possible you would be able to go back in time and change
the course of history by killing your own grandfather.
This is a very strong argument but it leaves
open the possibility that we may be able to make limited
journeys at FTL speed which did not allow us to come back.
That is not very likely. Relativity implies that anything
which can be done in one reference frame should be possible
in another. Or it may be that time travel is possible and
causality breaks down in some consistent fashion when FTL
travel is achieved. That is not very likely either but
if we are discussing FTL we had better keep an open mind.
Conversely, If we could travel back in time we might also
claim the ability to travel FTL because we can go back in
time and then travel at a slow speed to arrive somewhere
before light got there by the usual route ( See
the relativity FAQ article on Time
Travel for more on this subject.)
Open Possibilities for FTL travel
In this last section I give a few of the
speculative but serious suggestions for possible
faster than light travel. These are not the kind
of thing which are usually included in the FAQ
because they raise more questions than answers.
They are included merely to make the point that
serious research is being done in this direction.
Only a brief introduction to each topic is
given. Further information can be found all over
the internet.
19. Tachyons
Tachyons are hypothetical particles which travel
faster than light locally. They must have imaginary
valued mass to be able to do so, but they have real
valued energy and momentum. Sometimes people imagine
that such FTL particles would be impossible to detect
but there is no reason to think so. The shadows and
spotlights suffice to show that there is no logic in
the suggestion because they can go FTL and still be
seen.
No tachyons have been definitely found and most
physicists would doubt their existence. There was
a claim that experiments to measure neutrino mass in
Tritium beta decay indicated that they were tachyonic.
It is very doubtful but not entirely ruled out. Tachyon
theories have problems because, apart from the
possibility of causality violations, they destabilise the
vacuum. It may be possible to get round
such difficulties but then we would not be able to
use tachyons for the kind of FTL communication that
we would like.
The truth is that most physicists consider tachyons
to be a sign of pathological behaviour in field theories,
and the interest in them among the wider public stems mostly
from the fact that they are used so often in science fiction.
(see the Physics FAQ article on
Tachyons)
20. Worm Holes
A famous proposition for global FTL travel is to
use wormholes. Wormholes are shortcuts through space-time
from one place in the universe to another which would
permit you to go from one end to the other in a shorter
time than it would take light passing by the usual route.
Wormholes are a feature of classical general relativity but to
create them you have to change the topology of space-time.
That might be possible in quantum gravity.
To keep a wormhole open, regions of negative energy
would be needed. Misner and Thorn have suggested using
the Casimir effect on a grand scale to generate the
negative energy while Visser has proposed a solution
involving cosmic strings. These are very speculative ideas
which may simply not be possible. Exotic matter with
negative energy may not exist in the form required.
Thorn has found that if wormholes can be created then
they can be used to construct closed timelike loops in
space-time which would imply the possibility of time
travel. It has been suggested that the multiverse
interpretation of quantum mechanics gets you out of
trouble by allowing time to evolve differently if
you succeed in going back to a previous time.
Hawking says that wormholes would simple be unstable
and therefore unusable. The subject remains a fertile area for
thought experiments which help clarify what is and what
is not possible according to known and suggested laws
on physics.
refs:
W. G. Morris and K. S. Thorne, American Journal of Physics 56, 395-412 (1988)
W. G. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Letters 61, 1446-9 (1988)
Matt Visser, Physical Review D39, 3182-4 (1989)
see also "Black Holes and Time Warps" Kip Thorn, Norton & co. (1994)
For an explanation of the multiverse see,
"The Fabric of Reality" David Deutsch, Penguin Press.
