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Tachyon

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Tachyons
There was a young lady named Bright,
Whose speed was far faster than light.
She went out one day,
In a relative way,
And returned the previous night!

-Reginald Buller

It is a well known fact that nothing can travel faster than the speed of light. At best, a massless particle travels at the speed of light. But is this really true? In 1962, Bilaniuk, Deshpande, and Sudarshan, Am. J. Phys. 30, 718 (1962), said "no". A very readable paper is Bilaniuk and Sudarshan, Phys. Today 22,43 (1969). I give here a brief overview.

Draw a graph, with momentum (p) on the x-axis, and energy (E) on the y-axis. Then draw the "light cone", two lines with the equations E = p. This divides our 1 dimensional space-time into two regions. Above and below are the "timelike" quadrants, and to the left and right are the "spacelike" quadrants.

Now the fundamental fact of relativity is that E2 - p2 = m2. (Let's take c=1 for the rest of the discussion.) For any non-zero value of m (mass), this is an hyperbola with branches in the timelike regions. It passes through the point (p,E) = (0,m), where the particle is at rest. Any particle with mass m is constrained to move on the upper branch of this hyperbola. (Otherwise, it is "off-shell", a term you hear in association with virtual particles - but that's another topic.) For massless particles, E2 = p2, and the particle moves on the light-cone.

These two cases are given the names tardyon (or bradyon in more modern usage) and luxon, for "slow particle" and "light particle". Tachyon is the name given to the supposed "fast particle" which would move with v>c. (Tachyons were first introduced into physics by Gerald Feinberg, in his seminal paper "On the possibility of faster-than-light particles" <Phys.Rev. v.159, pp.1089--1105 (1967)>).

Now another familiar relativistic equation is E = m*<1-(v/c)2>-1/2. Tachyons (if they exist) have v > c. This means that E is imaginary! Well, what if we take the rest mass m, and take it to be imaginary? Then E is negative real, and E2 - p2 = m2 < 0. Or, p2 - E2 = M2, where M is real. This is a hyperbola with branches in the spacelike region of spacetime. The energy and momentum of a tachyon must satisfy this relation.

You can now deduce many interesting properties of tachyons. For example, they accelerate (p goes up) if they lose energy (E goes down). Furthermore, a zero-energy tachyon is "transcendent," or infinitely fast. This has profound consequences. For example, let's say that there were electrically charged tachyons. Since they would move faster than the speed of light in the vacuum, they should produce Cherenkov radiation. This would lower their energy, causing them to accelerate more! In other words, charged tachyons would probably lead to a runaway reaction releasing an arbitrarily large amount of energy. This suggests that coming up with a sensible theory of anything except free (noninteracting) tachyons is likely to be difficult. Heuristically, the problem is that we can get spontaneous creation of tachyon-antitachyon pairs, then do a runaway reaction, making the vacuum unstable. To treat this precisely requires quantum field theory, which gets complicated. It is not easy to summarize results here. However, one reasonably modern reference is Tachyons, Monopoles, and Related Topics, E. Recami, ed. (North-Holland, Amsterdam, 1978).

However, tachyons are not entirely invisible. You can imagine that you might produce them in some exotic nuclear reaction. If they are charged, you could "see" them by detecting the Cherenkov light they produce as they speed away faster and faster. Such experiments have been done. So far, no tachyons have been found. Even neutral tachyons can scatter off normal matter with experimentally observable consequences. Again, no such tachyons have been found.

How about using tachyons to transmit information faster than the speed of light, in violation of Special Relativity? It's worth noting that when one considers the relativistic quantum mechanics of tachyons, the question of whether they "really" go faster than the speed of light becomes much more touchy! In this framework, tachyons are waves that satisfy a wave equation. Let's treat free tachyons of spin zero, for simplicity. We'll set c = 1 to keep things less messy. The wavefunction of a single such tachyon can be expected to satisfy the usual equation for spin-zero particles, the Klein-Gordon equation:

(BOX m2)phi = 0

where BOX is the D'Alembertian, which in 3 dimensions is just

BOX = (d/dt)2 - (d/dx)2 - (d/dy)2 - (d/dz)2.

The difference with tachyons is that m2 is negative, and m is imaginary.

To simplify the math a bit, let's work in 1 dimensions, with co-ordinates x and t, so that

BOX = (d/dt)2 - (d/dx)2

Everything we'll say generalizes to the real-world 3dimensional case. Now - regardless of m, any solution is a linear combination, or superposition, of solutions of the form

phi(t,x) = exp(-iEt ipx)

where E2 - p2 = m2. When m2 is negative there are two essentially different cases. Either |p| >= |E|, in which case E is real and we get solutions that look like waves whose crests move along at the rate |p|/|E| >= 1, i.e., no slower than the speed of light. Or |p| < |E|, in which case E is imaginary and we get solutions that look waves that amplify exponentially as time passes!

We can decide as we please whether or not we want to consider the second sort of solutions. They seem weird, but then the whole business is weird, after all.

1) If we do permit the second sort of solution, we can solve the Klein-Gordon equation with any reasonable initial data - that is, any reasonable values of phi and its first time derivative at t = 0. (For the precise definition of "reasonable," consult your local mathematician.) This is typical of wave equations. And, also typical of wave equations, we can prove the following thing: If the solution phi and its time derivative are zero outside the interval <-L,L> when t = 0, they will be zero outside the interval <-L-|t|, L|t|> at any time t. In other words, localized disturbances do not spread with speed faster than the speed of light! This seems to go against our notion that tachyons move faster than the speed of light, but it's a mathematical fact, known as "unit propagation velocity".

2) If we don't permit the second sort of solution, we can't solve the Klein-Gordon equation for all reasonable initial data, but only for initial data whose Fourier transforms vanish in the interval <-|m|,|m|>. By the Paley-Wiener theorem this has an odd consequence: it becomes impossible to solve the equation for initial data that vanish outside some interval <-L,L>! In other words, we can no longer "localize" our tachyon in any bounded region in the first place, so it becomes impossible to decide whether or not there is "unit propagation velocity" in the precise sense of part 1). Of course, the crests of the waves exp(-iEt ipx) move faster than the speed of light, but these waves were never localized in the first place!

The bottom line is that you can't use tachyons to send information faster than the speed of light from one place to another. Doing so would require creating a message encoded some way in a localized tachyon field, and sending it off at superluminal speed toward the intended receiver. But as we have seen you can't have it both ways - localized tachyon disturbances are subluminal and superluminal disturbances are nonlocal.
 
Dear Trott,

Thankyou for the post. The information you provided is most usefull to our current project of splitting energy. The idea is that if when we split light itself, we will be inducing a tachyon field by condensing light to the plancks length and then creating additional pryforce until the light which is condensed to the plancks length reaches it's maximum permiablility and tears in half. Since the light is condensed to a plancks length before tearing the light field, the field will also tear the plancks length at the region that the light field occupied just before being compressed to the plancks length. The reason I say>just prior to condensing to the plancks length< is because the uncertainty principle forbids us to know with certianty the location the actual location of the light field once it is focused to the plancks length because space is not quantized at regions smaller then the plancks length. Time is also not quantized at regions smaller then plancks length so when we split the field, the area that this tear will be manifested at must be larger then plancks length. Since the energy is conserved at plancks length still, there may still be a conection to the energy that bridges the imaginary location of the light field that is compressed to a plancks length and the point in time and position in space to which the light was localized just prior to transcending that barrier 10E-35 Meters.

If we tear the plancks constant by focusing a magnetic field(which consists of trons and light carrying charge) to the plancks length, then split the magnetic field creating tearing apart the light and plancks length that the light....occupies.... if that is the right word.....we believe that we will be pulling that negative real region of space-time and energy into this positive time universe thereby pulling tachyons from beyond centermass of the field from a negative superluminal region of space-time into this positive space-time universe. If we do this with a magnetic field won't we be introducing tachyons carrying a charge, in that the light field magnetic field is converted into charge carrying tachyons when the field is torn like this? Will the energy of the tachyons release this ever increasing radiation that you spoke of? Before answering please consider, because we here at Tap-Ten are currently working with Faraday Labs St. Petersburg Russia to develop a device to do just this. Is this a safe thing to attempt based on the math? One last thing...It can be mathematically proved that a mass that compressed beyond the zero point X, Y, and Z axis of the three dimensional shperical mass or field is to compress the mass to a static superluminal velocity: and therefore, back in time. Basically if you apply an infinite amount of kenetic energy to one side of a mass object you accelerate this object to the velocity of light. If you apply a greater then infinite amount of kenetic energy to one side of a mass, you accelerate that mass to a superluminal speed and therefore back in time. Now if you apply the infinite amount of kenetic energy to all sides of the sphere; instead of to just one side of the mass, you compress the mass to an infinitely small point. So my question is "what is the difference between applying an infinite amount of kenetic energy to one side of a mass to accelerate that mass to the velocity of light, and applying the same infinite quantity of kenetic energy to all sides of that mass to compress it to a singularity?". My answer is that there is no difference, after all , you are doing the same thing in both cases, applying infinite kenetic energy to that object. Therefore, to apply an greater then infinite kenetic energy to all sides of the object to compress the object beyond it's own center mass to a negative 3-d space is the same as accelerating the mass to the faster then light speed. Therefore, compression to a zero point is equal to acceleration to the velocity of light...and...compression of a mass to beyond zero point is equal to accelerating a mass to faster then light speed. So to pry split energy that is condensed to a plancks length is to pull that static superluminal negative 3-d spacelike region from beyond centermass through a zero point into the positive 3-d spacelike region and therefore exposing a negative energy or tachyon field into the positive universe in copious form. What do you think?

Inquisitively,

Edwin G. Schasteen
 
>So to pry split energy that is condensed to a plancks length is >to pull that static superluminal negative 3-d spacelike region >from beyond centermass through a zero point into the positive 3->d spacelike region and therefore exposing a negative energy or >tachyon field into the positive universe in copious form. What >do you think?

Oops! I accidently worded this last paragragh incorrectly. What I meant to say is....So to split energy that is condensed to a plancks length is to pull that static superluminal negative 3-d spacelike region through "the split zero point" into the positive 3-d spacelike region therefore, exposing electrically charged negative energy, aKa charged tachyon energy, into our positive space-time universe in copious form.

Trott does this sound accurate? I imagine that since the scientists that you mentioned were conducting experiments to see if they could witness evidence ofthe existance of tachyons is good enough evidence that the information they had about the subject suggested that it was safe enough to attempt. So I imagine that this would be a safe endeavor. What do you think though, based on your knowlege?

Inquisitively,

Edwin G. Schasteen
Inquisitively,

Edwin G. Schasteen
 
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