From: sarfatti@well.sf.ca.us (Jack Sarfatti)
Subject: Star Fleet Command Physics Notes 2
Organization: Whole Earth 'Lectronic Link
Date: Mon, 25 Jan 1993 06:56:42 GMT


Part 2  The Hinson Notes on coventional relativity with Sarfatti Commentary
on superluminal and transluminal matter causality-violating kinematics.

> Relativity and FTL Travel

>Outline:

I.      An Introduction to Special Relativity
        A.      Reasoning for its existence
        B.      Time dilation effects
        C.      Other effects on observers
        E.      Space-Time Diagrams
        D.      Experimental support for the theory
II.     The First Problem:  The Light Speed Barrier
        A.      Effects as one approaches the speed of light
        B.      Conceptual ideas around this problem
III.    The Second Problem:  FTL Implies The Violation of Causality
        A.      What is meant here by causality, and its importance
        B.      Why FTL travel of any kind implies violation of causality
        C.      A scenario as "proof"
IV.     A Way Around the Second Problem
        A.      Warped space as a special frame of reference
        B.      How this solves the causality problem
        C.      The relativity problem this produces
        D.      One way around that relativity problem
V.      Conclusion.

 >I. An Introduction to Special Relativity
The main goal of this introduction is to make relativity and its
consequences feasible to those who have not seen them before.  It should
also reinforce such ideas for those who are already somewhat familiar
with them.  This introduction will not completely follow the traditional
way in which relativity came about.  It will begin with a pre-Einstein
view of relativity.  It will then give some reasoning for why Einstein's
view is plausible.  This will lead to a discussion of some of the
consequences this theory has, odd as they may seem.  For future
reference, it will also introduce the reader to the basics of space-time
diagrams.  Finally, I want to mention some experimental evidence that
supports the theory.

>The idea of relativity was around in Newton's day, but it was
incomplete.  It involved transforming from one frame of reference to
another frame which is moving with respect to the first.  The
transformation was not completely correct, but it seemed so in the realm
of small speeds.  I give here an example of this to make it clear.

>Consider two observers, you and me, for example.  Lets say I am
on a train which passes you at 30 miles per hour.  I through a ball in
the direction the train is moving, and the ball moves at 10 mph in MY
point of view.  Now consider a mark on the train tracks.  You see the
ball initially moving along at the same speed I am moving (the speed of
the train).  Then I through the ball, and before I can reach the mark on
the track, the ball is able to reach it.  So to you, the ball is moving
even faster than I (and the train).  Obviously, it seems as if the speed
of the ball with respect to you is just the speed of the ball with
respect to me plus the speed of me with respect to you.   So, the speed
of the ball with respect to you = 10 mph + 30 mph = 40 mph.  This was
the first, simple idea for transforming velocities from one frame of
reference to another. In other words, this was part of the first concept
of relativity.

>Now I introduce you to an important postulate that leads to the
concept of relativity that we have today.  I believe it will seem quite
reasonable.  I state it as it appears in a physics book by Serway: "the
laws of physics are the same in every inertial frame of reference."
What it means is that if you observer any physical laws for a given
situation in your frame of reference, then an observer in a reference
frame moving with a constant velocity with respect to you should also
agree that those physical laws apply to that situation.

>As an example, consider the conservation of momentum.  Say that
there are two balls coming straight at one another.  They collide and go
off in opposite directions.  Conservation of momentum says that if you
add up the total momentum (mass times velocity) before the collision and
after the collision, that the two should be identical.  Now, let this
experiment be preformed on a train where the balls are moving along the
line of the train's motion.  An outside observer would say that the
initial and final velocities of the balls are one thing, while an
observer on the train would say they were something different.  However,
BOTH observers must agree that the total momentum is the same before and
after the collision.  We should be able to apply this to any physical
law.  If not, (i.e.  if physical laws were different for different
frames of reference) then we could change the laws of physics just by
traveling in a particular reference frame.

>A very interesting result occurs when you apply this postulate
to the laws of electrodynamics.  What one finds is that in order for the
laws of electrodynamics to be the same in all inertial reference frames,
it must be true that the speed of electromagnetic waves (such as light)
is the same for all inertial observers.  Simply stating that may not
make you think that there is anything that interesting about it, but it
has amazing consequences.  Consider letting a beam of light take the
place of the ball in the first example given in this introduction.  If
the train is moving at half the velocity of light, wouldn't you expect
the light beam (which is traveling at the speed of light with respect to
the train) to look as if it is traveling one and a half that speed with
respect to an outside observer?  Well this is not the case.  The old
ideas of relativity in Newton's day do not apply here.  What accounts
for this peculiarity is time dilation and length contraction.

>Here I give an example of how time dilation can help explain a
peculiarity that arises from the above concept.  Again we consider a
train, but let's give it a speed of 0.6 c (where c = the speed of light
which is 3E8 m/s).  An occupant of this train shines a beam of light so
that (to him) the beam goes straight up, hits a mirror at the top of the
train, and bounces back to the floor of the train where it is detected.
Now, in my point of view (outside of the train), that beam of light does
not travel straight up and straight down, but makes an up-side-down "V"
shape since the train is also moving.  Here is a diagram of what I see:


                         /|\
                        / | \
                       /  |  \
 light beam going up->/   |   \<-light beam on return trip
                     /    |    \
                    /     |     \
                   /      |      \
                  /       |       \
                 ---------|---------->trains motion (v = 0.6 c)

>Lets say that the trip up takes 10 seconds in my point of view.  The
distance the train travels during that time is:
        (0.6 * 3E8 m/s) * 10 s = 18E8 m.
The distance that the beam travels on the way up (the slanted line to
the left) must be
        3E8 m/s * 10s = 30E8 m.
Since the left side of the above figure is a right triangle, and we know
the length of two of the sides, we can now solve for the height of the
train:
        Height = [(30E8 m)^2 - (18E8 m)^2]^0.5  =  24E8 m
(It is a tall train, but this IS just a thought experiment).  Now we
consider the frame of reference of the traveler.  The light MUST travel
at 3E8 m/s for him also, and the height of the train doesn't change
because only lengths in the direction of motion are contracted.
Therefore, in his frame the light will reach the top of the train in
24E8 m /3E8 (m/s) = 8 seconds, and there you have it.  To me the event
takes 10 seconds, while according to him it must take only 8 seconds.  We
each measure time in different ways.

>To intensify this oddity, consider the fact that all inertial
frames are equivalent.  That is, from the traveler's point of view he is
the one who is sitting still, while I zip past him at 0.6 c.  So he will
think that it is MY clock that is running slowly.  This lends itself
over to what seem to be paradoxes which I will not get into here.  If
you have any questions on such things (such as the "twin paradox" --
which can be understood with special relativity, by the way)  feel free
to ask me about them, and I will do the best I can to answer you.

>As I mentioned above, length contraction is another consequence
of relativity.  Consider the same two travelers in our previous example,
and let each of them hold a meter stick horizontally (so that the length
of the stick is oriented in the direction of motion of the train).  To
the outside observer, the meter stick of the traveler on the train will
look as if it is shorter than a meter.  Similarly, the observer on the
train will think that the meter stick of the outside observer is the one
that is contracted.  The closer one gets to the speed of light with
respect to an observer, the shorter the stick will look to that
observer. The factor which determines the amount of length contraction
and time dilation is called gamma.

>Gamma is defined as (1 - v^2/c^2)^(-1/2).  For our train (for
which v = 0.6 c), gamma is 1.25.  Lengths will be contracted and time
dilated (as seen by the outside observer) by a factor of 1/gamma = 0.8,
which is what we demonstrated with the difference in measured time (8
seconds compared to 10 seconds). Gamma is obviously an important number
in relativity, and it will appear as we discuss other consequences of
the theory.

>Another consequence of relativity is a relationship between
mass, energy, and momentum.  By considering conservation of momentum and
energy as viewed from two frames of reference, one can find that the
following relationship must be true for an unbound particle:
        E^2  =  p^2 * c^2  +  m^2 * c^4
Where E is energy, m is mass, and p is relativistic momentum which is
defined as
        p  =  gamma * m * v     (gamma is defined above)
By manipulating the above equations, one can find another way to express
the total energy as
        E  =  gamma * m * c^2
Even when an object is at rest (gamma = 1) it still has an energy of
        E  =  m * c^2
Many of you have seen something like this stated in context with the
theory of relativity


* E^2  =  p^2 * c^2  +  m^2 * c^4

is the "mass shell" equation for slower-than-light (i.e., subluminal) real
particles that can be directly detected. It is a pole in the complex energy
plane for the particle propagator in relativistic quantum field theory.

Virtual particles are "off mass shell" and do not obey this equation in
conventional theory. Virtual particles are that part of the propagator not
due to the energy pole. The propagator is not only determined by the
position of the poles. It is also determined by the path or contour over
which the integral representing the propagator is computed. This is a
boundary condition and this is where causality makes its mark. The
principle of retarded causality (i.e. causes always before effects) is
defined by a certain path in the complex energy plane. It is, however, not
the path that Feynman uses in conventional quantum electrodynamics. Feynman
finds that in order to renormalize properly, to get finite answers, one
must use a contour that includes both retarded causality (i.e., past
cause/future effect) and "teleological" advanced causality (i.e., future
cause/past effect).

Faster-than-light (i.e. superluminal) particles (i.e. tachyons) moving in
real time (Lorentzian signature +++-) obey a different mass shell equation

E^2  =  p^2 * c^2  -  m^2 * c^4

Propagation require E and p real means that p > mc. The De-Broglie
probability waves of length h/p are shorter than the Compton wavelength
h/mc. The tachyon wave fronts move at v(wave) slower than light but the
mass-energy transport wave packet velocity v(particle) is faster than
light. This is just the opposite of an ordinary particle in which the wave
front moves faster than light but the mass-energy transport group speed is
slower than light. For both kinds of particles

v(wave) v(particle) = c^2

For an ordinary subluminal particle, increasing the energy E makes
v(particle) increase. In contrast, for a superluminal particle, increasing
E makes v(particle) decrease - like a smoke vortex ring or a "roton"
excitation in superfluid helium. Indeed, faster than light particles are
more string-like than point-like.

The gamma factor for the faster than light particle is

(v^2/c^2 - 1)^(-1/2)  with v = v(particle) so v/c > 1.

Superluminal particles grossly violate "causality" on the macroscopic scale
in Hinson's sense by which I mean "retarded causality". The question is do
they violate it in a consistent way or an inconsistent way? I suspect the
former is the case. If the latter is the case, then they cannot exist.

The string-like subnucleonic structure may mean that quarks are self-
trapped superluminal (or maybe transluminal) particles. This would
automatically explain the origin of the strong color force because color
was introduced to have the correct spin-statistics connection and
superluminal particles have the wrong spin-statistics connection (e.g. a
superluminal particle of spin 1/2 is a boson not a fermion.

Superluminal electrons or quarks in the free state would quickly radiate
photons in a Cerenkov cone speeding up to infinite speed at zero total
energy E but finite momentum p. This would explain why free quarks are not
seen. Condensed superluminal matter, if it could exist, would not obey the
Pauli exclusion principle and would not have the diverse and stable
organization of ordinary subluminal matter. Bound superluminal particles
constrained by a "bag" or by a force that increased with separation might
look like ordinary matter to an outside observer).

The ordinary subluminal Lorentz frame transformations describe both
subluminal and superluminal particle motions equally well and consistently.
Subluminal particles have a rest frame, superluminal particles do not. The
rest frame for a subluminal particle is defined by the particle's gamma =
E/mc^2 = 1 which means v(particle) = 0, E = mc^2, and p = 0.  Similarly,
the faster than light particle obeys the same equation for gamma. Now if
gamma = 1, v = sqrt2c. If v > sqrt2c , gamma is less than 1. In this region
we have string-like length expansion in the direction of motion and time
contraction. If, on the other hand,c < v < sqrt2c gamma is bigger than 1
like ordinary slower than light particles with length contraction and time
dilation.

The mass shell equation for transluminal particles moving in imaginary time
of quantum-gravity's Euclidean signature (++++) is

 E^2  =  -p^2 * c^2  +  m^2 * c^4

E and p real require p < mc which is the long wave limit which would be
most relevant to observational test. A transluminal particle moving locally
according to a Euclidean rather than Lorentzian metric signature would look
to our real time detectors like a new kind of particle with peculiar "dark
matter" kinematics and dynamics.

with gamma = (1 + v^2/c^2)^(-1/2) < 1 for all v.

Both the subluminal and superluminal particles in real time obey the
Einstein speed of light barrier. They are on opposite sides of the barrier.
Not so for transluminal particles which do not feel the barrier at all
since they are in a topologically distinct parallel universe connected to
ours by photons if we make the ansatz that a charged accelerating
transluminal particle emits photons in real time. But this may not be
correct. The question is neutral transluminal matter gravitate? How will
curvature in the Euclidean metric influence curvature in the Lorentz metric
to which it is connected by a Wick rotation. Will this explain the large
scale structure of the universe with its walls and voids?

Has Star Trek Command succeeded in converting among the subluminal,
superluminal and transluminal phases of matter at will? Note that a Star
Ship built of ordinary subluminal matter with subluminal life forms could
use a subluminal <---> transluminal matter converter to do two things.
First, transluminal matter ejected in a rocket exhaust at superluminal
speeds would be ultr-energy efficient enabling very heavy super-carrier
size craft to get close to the Einstein light barrier with small amounts of
fuel. Second, The transluminal matter is the exotic matter needed to
support stable traversable wormholes amplified out of the quantum foam for
warp drive.*

to be continued.