See also the sci.physics FAQ part 4:
  ftp://rtfm.mit.edu/pub/usenet-by-hierarchy/sci/physics/
  sci.physics_Frequently_Asked_Questions_(4_4)]

Tachyons are theoretical particles that always travel faster than
light.  Tachy meaning "swift."
 
There is a formula that relates mass to speed in the special theory
of relativity:
 
            m = m0 / SQR(1 - v^2/c^2)
 
where  m = energy divided by c^2 (sometimes called "relativistic mass")
      m0 = rest mass
       v = velocity of mass relative to you
       c = velocity of light (constant in all frames of reference)
 
So, as you see an object moving faster and faster, its mass
increases.  A simple experiment with electrons in a vacuum tube can
convince you that mass increases in this way.  So you get something
like:
 
                     v/c m/m0
 
                    0.0 1.000
                     .2 1.021
                     .4 1.091
                     .6 1.250
                     .8 1.667
                     .9 2.294
                     .95 3.203
                     .99 7.089
                     .995 10.013
                     .999 22.366
                    1.000 infinity
 
This led Einstein and others to conclude that it was impossible for
any material object to travel at or beyond the speed of light.
Because as you increase speed mass increases.  With increased mass,
there's a requirement for increased energy to accelerate the mass.
In the end, an infinite amount of energy is needed to move any object
*at* the speed of light.  Nothing would move you faster than the
speed of light, according to this type of analysis.
 
But, some researchers noted that light has no trouble moving at the
speed of light.  Furthermore, objects with mass have no trouble
converting to light.  Light has no trouble converting to objects with
mass.  So, you have tardyons and photons.  Tardy meaning slow.  These
classes of objects can easily be converted into one another.
 
Now, in terms of the equation given above, if you start out with
*any* mass you are constrained to moving less than the speed of
light.  If you start out with zero mass, you stay at zero mass.  This
describes the situation with respect to photons.  You have zero over
zero, and end up with zero....

But, what if you started out faster than the speed of light?  Then
the equation above would give you an imaginary mass, since v^2 / c^2
would be greater than 1 and that would be subtracted from 1 to
produce a negative number.  Then you'd take the square root of the
negative number and end up with an imaginary number.  So, normal
matter moving faster than the speed of light ends up with imaginary
mass, whatever that may be.
 
Imaginary mass travelling faster than the speed of light would show
up as regular mass to an observer at rest.
 
                         v/c m/m0 (m/m0)*i
 
                        infinity 0+0.000i 0.000
                         1,000 0-0.001i 0.001
                           100 0-0.010i 0.010
                            10 0-0.101i 0.101
                             8 0-0.126i 0.126
                             6 0-0.169i 0.169
                             4 0-0.258i 0.258
                             2 0-0.577i 0.577
                             1.5 0-1.118i 1.118
                             1.1 0-2.182i 2.182
                             1.05 0-3.123i 3.123
                             1.01 0-7.053i 7.053
                             1.000 0-inf*i infinity

So, if there was such a thing as imaginary mass, it would look like
normal mass but it would always travel *faster* than c, the speed of
light.  When it lost energy it would move faster.  When it gained
energy it would move slower.  So, in addition to tardyons and
photons, there might exist tachyons.
 
        Description Tardyon Photon Tachyon
 
        Gain energy  faster      c  slower
        Lose energy  slower      c  faster
        Zero energy    rest      c  infinity
    Infinite energy       c      c       c
 
Now, do tachyons exist?

If tachyons exist they can easily be detected by the presence of
Cerenkov radiation in a vacuum.  Cerenkov radiation is radiation
emitted when a charged particle travels through a medium at a speed
greater than the velocity of light in the medium.  This occurs when
the refractive index of the medium is high.
 
Cerenkov radiation is like the bow wave of a boat, or the shock wave
of a supersonic airplane.  Photons bunch up in front of the tachyon
and they're radiated away at an angle determined by the speed of the
tachyon.
 
Cerenkov detectors are useful in atomic physics for determining the
speed of particles moving through a medium.  Light slows as it passes
through a medium.  That's what's responsible for optical effects.
There's nothing mysterious about Cerenkov radiation in a medium.  So,
folks know how to make an operate Cerenkov detectors because they're
a useful speedometer when you're working with subatomic particles
 
Now, there have been a few studies looking for Cerenkov radiation in
a vacuum.  This would indicated the reality of tachyons.  Cerenkov
radiation has never been detected in vacuum.  So, most people believe
that tachyons don't exist.