Colonel Corso's Time Traveling Saucers
A really crude toy model for
Imagine that m is the mass of the flying saucer with the Q-generator switched off.
1 + Q ~ 0 is the "warp bubble"
M = m(1 + Q)^1/2
is the effective mass seen by "us" looking from outside the warp bubble.
shows the warp bubble
Clearly this spacecraft can turn around and go back in time relative to us as is mentioned in Colonel Corso's
"The Day After Roswell"
Professor Milton Rothman speaks:
"My question is: how much energy does it take to generate a stream of tachyons? To provide a reasonable amount of thrust, the tachyon beam must have a certain amount of momentum. The relativistic relation between momentum and energy is surely the same for tachyons as it is for other particles." The late Milton Rothman of Skeptical Inquirer
No it's not Professor Rothman who we invoke in a seance. ;-)
When p =|M|c, E = 0 for a tachyon of imaginary mass M
And the mass-energy of the spaceship approaches infinity as the ship approaches the speed of light. So from where do we get this high energy efficiency? (Besides, nobody has seen a tachyon yet.)
Our Dearly Departed Debunker obviously did not know that the mass shell condition changes
E^2 = (pc)^2 + (Mc^2)^2 ordinary particle c = 1
E^2 = (pc)^2 - (|M|c^2)^2 tachyon
Therefore the energy of the tachyon E limits to zero as the momentum p limits to |M|c, which was my point - tiny energy at finite momentum.
It is well known that the special relativistic Bohm quantum potential for the Klein-Gordon equation bends the timelike world line of (in this case spin 0 massive particle) into a spacelike one outside its local light cone. Indeed you can make a closed timelike curve for time travel to the past . Ch 12 The Quantum Theory of Motion by Peter Holland, Cambridge University Press 1993
That is M = m(1 + Q)^1/2
Q ~ (h/mc)^2|Psi|^-1 D'Alembertian|Psi|
m = classical rest mass.
M is the quantum "dressed" rest mass.
Therefore, when Q < -1, M is imaginary, i.e. tachyon.
Note that relativistic quantum potential
Q ~ (Compton Wave Length)^2(Curvature)|Psi|^-1
The effective local space-time metric surrounding the particle in the stationary approximation is
g00 ~ 1 + Q
M^2 ~ g00m^2
1 + Q = 0 is a kind of micro-event horizon where g00 = 0
(Curvature) ~ |Psi|^uu
This can be negative.
Also the particle of quantum mass M runs around the core of the topological defect (here a vortex ring where |Psi| is tiny hence Q is large even if the gradients of the gradients of |Psi| are shallow as long as they give a negative effective local curvature trapping the particle.
Note that for a vortex ring's group speed
v ~ 1/E
Note the bigger the energy the larger the radius of the vortex ring.
Fomally similarly for a low energy tachyon - increasing energy of tachyon lowers its speed to the light cone.
A soft zero energy tachyon moves infinitely fast with a spatial periodicity lambda = h/|M|c, note that we can make |M| small.
(Note that we can even imagine for the height of the cosmic landscape at some point.
/\zpf^-1/2 = h/|M|c
1//\zpf = (h/|M|c)^2 = (h/mc)^2(1 + Q)^-1 = (1/g00)(h/mc)^2
if m is the Planck mass etc.
/\zpfLp^2 = 1 + Q
Q is the Bohm macro-quantum potential for the giant vacuum ODLRO order parameter of the pocket universes.)
Back to the relatively mundane
v = dE/dp
c = 1
E^2 = p^2 - |M|^2
2EdE = 2pdp
dE/dp = p/E = (E^2 + |M|^2)^1/2/E ~ |M|/E > vacuum speed of light
when E << |M|
in region p ~ |M|
Note also that since
E^2 = p^2 - |M|^2
In order for E to be real, (I put c back in)
p > |M|c
i.e. deBroglie wavelength h/p < Compton wavelength h/|M|c
Now normally this will excite virtual particle-antiparticle pairs into real pairs.
Not for on-shell tachyons however because the available energy E is too small.
h/|M|c = (h/mc)|(1 + Q)|^-1 >> (h/mc) indeed macroscopic
if we tune Q to keep 1 + Q ~ 0 on the "event horizon"
This is a kind of "particle" hidden variable analog to the Higgs field mechanism. In field theory with ODLRO pushing M imaginary triggers a vacuum phase transition e.g. "inflation".
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein