Is Faster Than Light Travel or Communication Possible?

1. Cherenkov Effect

2. Third Party Observers

If a rocket A is travelling away from me at 0.6c in a Westerly direction, and another B is travelling away from me at 0.6c in an Easterly direction, then the total distance between A and B as seen in my frame of reference is increasing at 1.2c.  An apparent relative speed greater than c can be observed by a third person in this way.

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.

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 cannot 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 Newtonian 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).

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 v_ph from another velocity known as the group velocity v_gr which is given by,

   vgr = c / 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.

6. Superluminal Galaxies

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.

8. Speed of Gravity

9. EPR paradox

10. Virtual Photons

11. Quantum Tunnelling

Quantum Tunnelling is the quantum mechanical effect which permits a particle to escape through a barrier when it does not have enough energy to do so classically.  You can do a calculation of the time it takes a particle to tunnel through.  The answer you get can come out less than the time it takes light to cover the distance at speed c.  Does this provide a means of FTL communication?
ref:T. E. Hartman, J. Appl. Phys. 33, 3427 (1962).

The answer must surely be "No!" otherwise our understanding of QED is very suspect.  Yet a group of physicists have performed experiments which seem to suggest that FTL communication by quantum tunneling is possible.  They claim to have transmitted Mozart's 40th Symphony through a barrier 11.4cm wide at a speed of 4.7c.  Their interpretation is, of course, very controversial.  Most physicists say this is a quantum effect where no information can actually be passed at FTL speeds because of the Heisenberg uncertainty principle.  If the effect is real it is difficult to see why it should not be possible to transmit signals into the past by placing the apparatus in a fast moving frame of reference.
ref:
W. Heitmann and G. Nimtz, Phys Lett A196, 154 (1994);
A. Enders and G. Nimtz, Phys Rev E48, 632 (1993).

Terence Tao has pointed out that apparent FTL transmission of an audio signal over such a short distance is not very impressive.  The signal takes less than 0.4ns to travel the 11.4cm at light speed, but it is quite easy to anticipate an audio signal ahead of time by up to 1000ns simply by extrapolating the signal waveform.  Although this is not what is being done in the above experiments it does illustrate that they will have to use a much higher frequency random signal or transmit over much larger distances if they are to convincingly demonstrate FTL information transfer.

The likely conclusion is that there is no real FTL communication taking place and that the effect is another manifestation of the Heisenberg uncertainty principle.

12. Casimir Effect

13. Expansion of the Universe

According to the Hubble Law, two galaxies which are a distant D apart are moving away from each other at a speed HD where H is Hubble's constant.  In that case two galaxies which are a distance greater than c/H apart are moving away from each other faster than the speed of light.  This is quite correct.  The distance between two objects can be increasing faster than light because of the expansion of the universe.  However, it is meaningless to say that the universe is expanding faster than light because the rate of the expansion is measured by Hubble's constant alone which does not even have the units of speed.

As was mentioned above, in special relativity it is possible for two objects to be moving apart by speeds up to twice the speed of light as measured by an observer in a third frame of reference.  In general relativity even this limit can be surpassed but it will not then be possible to observe both objects at the same time.  Again, this is not real faster than light travel.  It will not help anyone to travel across the galaxy faster than light.  All that is happening is that the distance between two objects is increasing faster when taken in some cosmological reference frame.

14. The moon revolves round my head faster than light!

Stand up in a clear space and spin round.  It is not too difficult to turn at one revolution each two seconds.  Suppose the moon is on the horizon.  How fast is it spinning round your head?  It is about 385,000 km away so the answer is 1.21 million km/s, which is more than four times the speed of light!  It sounds ridiculous to say that the moon is going round your head when really it is you who is turning, but according to general relativity all co-ordinate systems are equally valid including revolving ones.  So isn't the moon going faster than the speed of light?  This is quite difficult to account for.

What it comes down to, is the fact that velocities in different places cannot be directly compared in general relativity.  Notice that the moon is not overtaking the light in its own locality.  The velocity of the moon can only be compared to the velocity relative to other objects in its own local inertial frame.  Indeed, the concept of velocity is not a very useful one in general relativity and this makes it difficult to define what "faster than light" means.  Even the statement that "the speed of light is constant" is open to interpretation in general relativity.  Einstein himself in his book "Relativity: the special and the general theory" said that the statement cannot claim unlimited validity (pg 76).  When there is no absolute definition of time and distance it is not so clear how speeds should be determined.

Nevertheless, the modern interpretation is that the speed of light is constant in general relativity and this statement is a tautology given that standard units of distance and time are related by the speed of light.  The moon is given to be moving slower than light because it remains within the future light cone propagating from its position at any instant.

Relativity arguments against FTL travel

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 cannot 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 cannot 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 cannot 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.

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.

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 Thorne 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.

Thorne 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 Thorne, Norton & co. (1994)
For an explanation of the multiverse see, "The Fabric of Reality" David Deutsch, Penguin Press.

21. Warp Drives

A warp drive would be a mechanism for warping space-time in such a way that an object could move faster than light.  Miguel Alcubierre made himself famous by working out a space-time geometry which describes such a warp drive.  The warp in space-time makes it possible for an object to go FTL while remaining on a time-like curve.  The main catch is the same one that may stop us making large wormholes.  To make it you would need exotic matter with negative energy density.  Even if such exotic matter can exist it is not clear how it could be deployed to make the warp drive work.
ref M. Alcubierre, Classical and Quantum Gravity, 11, L73-L77, (1994).

Conclusion

To begin with, it is rather difficult to define exactly what is really meant by FTL travel and FTL communication.  Many things such as shadows can go FTL but not in a useful way which can carry information.  There are several serious possibilities for real FTL which have been proposed in the scientific literature but there are technical difficulties.  The Heisenberg uncertainty principle tends to stop the use of apparent FTL quantum effects for sending information or matter.  In general relativity there are potential means of FTL travel but they may be impossible to make work.  It is thought to be highly unlikely that engineers will be building space-ships with FTL drives in the foreseeable future, if ever, but it is curious that theoretical physics as we presently understand it seems to leave the door open to the possibility.  FTL of the sort science fiction writers would like is almost certainly impossible.  For physicists the interesting question is "why is it impossible and what can we learn from it?".