Einstein and Beyond
by Jack Sarfatti
The reason that speeds of matter must be less than the speed of light in vacuum is because if that were not so, the future could influence the past. This would be a kind of "Destiny" model of physical reality. Most scientists today do not like this kind of teleology. For example, see the book, The Anthropic Cosmological Principle by John Barrow and Frank Tipler (Oxford). Nevertheless, Einstein's theory of special relativity still makes some sense if the missing mass of the universe is exotic and superluminal, therefore, perhaps, allowing traversable wormhole time travel star gates and faster-than-light warp drives for our star ships and alien UFOs in contradiction to Hawking's conjecture that time travel to the past is impossible.
OK, so just for fun as an exercise in conceptual art, here are the causality-violating super-boost equations in the elsewhere of the still frame-invariant light cone.
t' = [vt/c - x/c]/(v^2/c^2 - 1)^1/2
x' = [vx/c - ct ]/(v^2/c^2 - 1)^1/2
y' = y
z' = z
with invariant intervals
c^2t'^2 - x'^2 - y'^2 - x'^2 = c^2t^2 - x^2 - y^2 - x^2
Note that the "mass shell pole of the quantum field propagator" is
E^2 - px^2 - py^2 - pz^2 = -|m|^2
in this case when v > c. The mass shell is also invariant.
I have not worked out the second rank tensor transformations for the electromagnetic fields of charged tachyons. The abberration and Doppler effects are also of interest.
For length distortion, t = 0 in the lab frame, so
x' = vx/c/(v^2/c^2 - 1)^1/2
In the limit v -> infinity, x' = x so the transcendental frame replaces
the rest frame of the causal subluminal boost. As we approach the light
cone from the elsewhere the length of the rod stretches to an infinite
string along the direction of the boost.
For time distortion, x' = 0, if we use the same rule as in the
subluminal boost, though this is arguable. therefore,
vx/c - ct = 0
or
x = c^2t/v
t' = [vt/c - x/c ]/(v^2/c^2 - 1)^1/2
= [vt/c - c^2t/vc]/(v^2/c^2 - 1)^1/2
= t[v/c - c^2/vc]/(v^2/c^2 - 1)^1/2
= t[v^2/vc - c/v]/(v^2/c^2 - 1)^1/2
= t(c/v)[v^2/c^2 - 1]/(v^2/c^2 - 1)^1/2
Therefore,
t' = t(c/v)[v^2/c^2 - 1]^1/2
And again t' = t at infinite speed (i.e, zero energy and finite momentum),
but now we have a speeding up of time in the approach to the light cone
from its elsewhere outside it. Therefore, a highly energetic tachyon
decays faster than a low energy one and it is stretched into a string
in its direction of motion.
Consider the transformation of velocities of a particle for boost speed v > c along the x-axis.
dx' = [vdx/c - cdt ]/(v^2/c^2 - 1)^1/2
dt' = [vdt/c -dx/c]/(v^2/c^2 - 1)^1/2
Therefore, in the "longitudinal" direction of the boost:
dx'/dt' = (dx/dt) [v/c - cdt/dx]/[v/c - (1/c)dx/dt]
Note, that in the special case that dx/dt = c, dx'/dt' = c for all boosts v consistent with the frame-invariance of the speed of light in vacuum even though the particles can move faster-than-light with strong violation of causality.
A quantum field theory based on the extended Lorentz group could not use the Schwinger-Tomonaga microcausality axiom that spacelike separated local field operators commute.
The transverse speeds obey
dy'/dt' = (dy/dt)(v^2/c^2 - 1)^1/2 /[v/c - (1/c)dx/dt]
Consider the aberration of a light ray incident in the x-y plane at angle & to the y axis. Therefore,
dy/dt = c cos& and dx/dt = c sin&
It follows that
dx'/dt' = c sin&' = c sin& [v/c -1/sin&]/[v/c -sin&]
dy'/dt' = c cos&' = c cos&(v^2/c^2 - 1)^1/2 /[v/c -sin&]
Therefore
tan&' = tan& [v/c -1/sin&]/(v^2/c^2 - 1)^1/2
If the light ray is incident along the y axis, & = 0 and &' = 0. If the light ray is incident along the x axis, & = pi/2 and &' = pi/2. Suppose, & = pi/4, so tan& = 1, sin& = 1/2^1/2. Therefore
tan&' = [v/c - 2^1/2]/(v^2/c^2 - 1)^1/2
First take the infinite speed limit in which v/c -> infinity.
tan&' -> 1
This means zero aberration for the infinite boost which is the analog of the boost to the rest frame in familiar subluminal boost causal relativity.
Next take, the approach to the invariant light cone from its outside it in which v/c -> 1.
tan&' -> - infinity
This means that &' -> -pi/2. Take the positive angle as clockwise from the vertical y axis. Therefore the ray is bent away 90 degrees from the y axis in the counter-clockwise direction coming in from - infinity on the longitudinal x axis of the super-boost in this limit.
The Lorentz group boosts, both subluminal (v < c) and superluminal (v > c), can be represented by 4x4 matrices L< and L> respectively. Let the indices i,j ,i',j' range from ct, x, y, z respectively in that order. Use the standard names "gamma" and "beta" where beta = v/c in all cases. Gamma is 1/squareroot (1 - beta^2) for L<, and is 1/squareroot (beta^2 - 1) for L>.
Therefore, for Einstein's standard subluminal theory of special relativity that obeys the principle of retarded causality (i.e., causes before effects in all inertial frames),
L< = gamma -gamma beta 0 0
-gamma beta gamma 0 0
0 0 1 0
0 0 0 1
i.e., SUB-BOOST v < c
Note that the determinant of this matrix is:
gamma^2 - gamma^2 beta^2 = 1
In my new faster-than-light conjectured extension to Einstein's theory that strongly violates the classical principle of retarded causality and the quantum field axiom of microcausality:
L> = gamma beta -gamma 0 0
-gamma gamma beta 0 0
0 0 1 0
0 0 0 1
i.e., SUPER-BOOST v > c
Note that the determinant of this matrix is:
gamma^2 beta^2 - gamma^2 = 1
We are now ready to construct the transformation of the electromagnetic field of both subluminal and superluminal charged particles under the super-boost. Note, in ordinary relativity, one can treat the motion of a superluminal particle under a subluminal boost.
The electromagnetic field is an antisymmetric second rank tensor Fij = -Fji under the action of the Lorentz group in flat spacetime. The three non-vanishing mixed space-time components (Ftx, Fty, Ftz), form the electric field 3-vector and the three nonvanishing space-space "vortex" components (Fxy, Fxz, Fyz) form the magnetic field 3-pseudovector under the spacelike rotation subgroup of the full Lorentz group. Therefore, using summation convention, for either the sub or the super-boost,
F'i'j' = Li'i Lj'j Fij
Fxy = -Bz
Fyz = -Bx
Fzx = -By
Fxt = Ex
Fyt = Ey
Fzt = Ez
For example take the transformation that gives the final electric field component along the direction of the boost whether subluminal or superluminal. That is,
E'x' = F'x't' = Lx't (Lt'x Ftx + Lt'y Fty + Lt'z Ftz) + Lx'x (- Lt't Ftx + Lt'y Fxy + Lt'z Fxz) + Lx'y (- Lt't Fty - Lt'x Fxy + Lt'z Fyz) + Lx'z (- Lt't Ftz - Lt'x Fxz - Lt'y Fyz)
Recall that for the super-boost:
L> = gamma beta -gamma 0 0
-gamma gamma beta 0 0
0 0 1 0
0 0 0 1
Therefore,
Lt't = gamma beta, Lt'x = - gamma, Lx't = - gamma, Lx'x = gamma beta, Ly'y = 1, Lz'z = 1
and all other matrix elements are zero. Thus,
E'x' = F'x't' = - Ft'x' = (Lx'x Lt't - Lx't Lt'x) Fxt = ( gamma^2 beta^2 - gamma^2) Fxt = detL> Fxt = Fxt
Therefore, the electric field in the direction of the super-boost is invariant.
Next, take the transformation that gives the final electric field component Ey which is perpendicular to the direction of the boost whether subluminal or superluminal. That is,
F'y't' = E'y'
E'y' = F'y't' = Ly't (Lt'x Ftx + Lt'y Fty + Lt'z Ftz) + Ly'x (- Lt't Ftx + Lt'y Fxy + Lt'z Fxz)
+ Ly'y (- Lt't Fty - Lt'x Fxy + Lt'z Fyz) + Ly'z (- Lt't Ftz - Lt'x Fxz - Lt'y Fyz) = Ly'y (- Lt't Fty - Lt'x Fxy) = - gamma beta Ey - gamma Bz = -gamma ( beta Ey + Bz)
Note that the transformed transverse electric field component along the y-axis contains a contribution from the transverse magnetic field along the mutually perpendicular z-axis for the boost along the x-axis.
In the limit of infinite speed E'y' = Ey. Again this fiducial limit is for the superboost what the rest frame is for the subboost. There is a projective duality here between v and c^2/v which corresponds to de Broglie's quantum wave-particle duality.
The more interesting limit is the approach to the light cone. We see that the transverse electric field grows in strength without bound in this limit just as it does in the subluminal case. Therefore, the field pattern flattens into a transverse pancake outside the light cone as well as inside. My earlier guess that it squonched down into a string like flux tube appears to be wrong.