FASTER-THAN-LIGHT PARTICLES: A REVIEW OF ) (TACHYON CHARACTERISTICS
OCT B0 E A PUSCHER F49620-77-C-0023
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A RAND NOTE
FASTER-THAN-LIGHT PARTICLES: A REVIEW OF ) (TACHYON CHARACTERISTICS
Edward A. Puscher
October 1980
N-IS30-AF
Prepared For The United States Air Force
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Relativity Theory Physics
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UNCLASSIFIED
Documents an analytical prediction of some
of the characteristics which presently undiscovered faster-than-light sub-atomic
particles (called tachyons) must possess
if they are to exist without violating the
Theory of Special Relativity. A brief review of necessary concepts from the Special
Theory is included so that the reader might
more readily understand the reasoning as it
is developed. Necessary, but not all,
characteristics of tachyons are then identified and presented. Finally, an interesting potential relationship between
tachyons and anti-gravity is discussed.
UNCLASSIFIED
SOCURITY CLASSIPICATION Of THIS := as D Me
A RAND NOTE
FASTER-THAN-LIGHT PARTICLES: A REVIEW OF
TACHYON CHARACTERI STICS
Edward A. Puscher
October 1980
N-1530-AF
Prepared For The United States Air Force
A
Rand
SAWA MONICA, CA. 90406
APPROVED FOR PUILIC lELIMA14 DISINCTION UNLMITID
PREFACE
This Note was prepared to document the results of the first phase
of an analytical effort that explored the possibility that anti-gravity
might be a result of elementary (but as yet undiscovered) particles that
travel faster than light. This effort falls largely outside the normal
management confines of Rand's Project AIR FORCE, and is more aptly
described as "exploratory research." Although it is only in the early
stages of development, this information should be of special interest to
particle physicists.
Accession For
j ; s t i , c-_
i I' F:"' '
A .. ,._.
SUMARY
This Note documents an analytical prediction of some of the characteristics which presently undiscovered faster-than-light sub-atomic particles (called tachyons) must possess if they are to exist without
violating the Theory of Special Relativity. A brief review of necessary
concepts from the Special Theory is included so that the reader might
more readily understand the reasoning as it is developed. Necessary,
but not all, characteristics of tachyons are then identified and
presented. Finally, an interesting potential relationship between
tachyons and anti-gravity is discussed.
-viiCONTENTS
PREFACE ........................................................... iii
SUMMARY ........................................................... v
FIGURES ........................................................... ix
Section
1. INTRODUCTION ................................................... 1
II. REQUIRED CONCEPTS FROM THE SPECIAL THEORY OF RELATTVITY .... 3
III. CONSERVATION LAWS AND SYMMETRY ................................ 12
A. RELATIVISTIC DEPENDENCE OF ENERGY UPON MOMENTUM ...... 12
B. RELATIVISTIC DEPENDENCE OF ENERGY UPON VELOCITY ...... 14
C. DEPENDENCE OF ENERGY UPON MOMENTUM FOR TACHYONS ...... 18
D. DEPENDENCE OF ENERGY UPON COORDINATE SYSTEM SPEED
FOR TACHYONS ............................................ 19
IV. THE VALUE OF SPACE-TIME CONSIDERATIONS ....................... 22
V. TACHYON CHARACTERISTICS ....................................... 29
VI. FIRST CONSIDERATION OF AN ANTI-GRAVITY THEORY BASED ON
TACHYONS: NEWTON'S LAW ........................................ 31
VII. TENTATIVE CONCLUSIONS ......................................... 34
REFERENCES ............................................................ 37
-ixFIGURES
1. Origin of Mathematical Reasoning ................................ 4
2. Length Contraction and Time Dilatation--i ....................... 6
3. Mass ind Velocity Dependence upon Reference Systems ........... 7
4. Mass and Velocity Dependence upon Reference Systems,
Limiting Process ............................................... 9
5. Length Contraction and Time Dilatation--2 ....................... 9
6. Conservation of Relativistic Momentum and Energy .............. 12
7. Relativistic Dependence of Energy on Velocity--Equations ...... 15
8. Relativistic Dependence of Energy on Velocity--Graphics ....... 17
9. Plot of Energy-Momentum for Faster-than-Light Particles ....... 19
10. Dependence of Energy upon Coordinate System
Speed for Tachyons ............................................ 20
11. Visualization of Time ........................................... 22
12. Space-Time for 2 Observers in Relative Motion
along X Axis ................................................ 24
13. Dependence of Time upon Velocity of Particle and
Relative Velocity of Reference System S' WRT
System S--Event End Point above X' Axis ...................... 25
14. Dependence of Time upon Velocity of Particle and
Relative Velocity of Reference System S' WRT
System S--Event End Point below X' Axis ...................... 26
15. Plot of Energy-Momentum for Faster-than-Light Particles,
Time Reversal ............................................... 27
16. Tachyon Characteristics ......................................... 30
17. Newton's Law of Gravitation for Tachyons ....................... 32
I. INTRODUCTION
Several articles have recently been published regarding the possible existence of faster-than-light particles, which have been named
"tachyons" [1, 2, 3, 4]. Some researchers [2, 4] have attempted to
present a relativistic quantum mechanical theory of tachyons, primarily
with the point of view of attempting to predict how these elusive particles might be discovered in the laboratory. Considerable emphasis has
been placed on the elimination of philosophical problems associated with
causality (cause and effect) when particles that travel faster than
light are used in thought experiments. It is sufficient for this paper
simply to note that there are some good arguments [1, 2] that an
apparent violation of causality is not a compelling argument against the
existence of tachyons.
The intent of this paper is to enumerate some of the characteristics which tachyons must possess if they are not to violate Special
Relativity Theory. After showing a brief development of thinking which
culminates in a presentation of some of the necessary characteristics of
tachyons, an interesting (but not compelling) possible connection of
tachyons to a tentative concept of anti-gravity will be presented. Of
course, anti-gravity is a very interesting topic, but one which will be
discussed in the next paper on this study.
However, a desire to provide some new ideas for anti-gravity has
been the driving interest of this paper. Thus, some decisions are made
here which are necessary to identify particular characteristics of
tachyons which would be useful in a tentative theory of anti-gravity
-2-
(for example, tachyon energy must remain a real quantity or an effective
real repelling fcrce could not be produced.)
-3-
II. REQUIRED CONCEPTS FROM THE SPECIAL THEORY OF RELATIVITY
This section will provide the necessary concepts of the Special
Theory of Relativity which are needed to develop the ideas of the following sections [5, 6].
Near the end of the 19th century, very accurate experiments were
performed by Michelson and Morely which indicated that the speed of
light was the same in all inertial reference frames. This led Albert
Einstein to state his kinematic postulate of the Special Theory of Relativity: The speed of light in a vacuum has the same value relative to
all inertial reference frames and is independent of the relative velocity of the light source and the observer. This postulate can also be
used to derive the Lorentz equations, although these equations were
first derived by Lorentz while attempting to reconcile the MichelsonMorely experimental results with Newtonian mechanics and the laws of
electromagnetism, but without Einstein's postulate. Figure 1 contains a
description of the derivation of the Lorentz equations.
Suppose, while referring to Fig. I that both observer 1 and
observer 2 were originally somewhere to the left of the figure, while a
stationary man who holds a light source (which is turned off) is at
position x=O, x'=0. Both observer 1 and observer 2 are moving fast with
respect to each other, along the horizontal axis, with observer 2 moving
faster then observer 1, and slightly higher than observer I so that they
do not hit each other when passing. When observer 1, observer 2, and
the man holding the light source are exactly on top of each other, t=O,
t'=0, and the man instantaneously turns on the light. As observer 1 and
-4-
NOTE: The speed of light is a universal constant (by experiment)
Event 1(
xrct
Uniform r- C
Reaty V -r XXax Direction
Velocity x' 0, 01 02 •x Direction
between
2 observers
2 2 '2
From pyti,uyorean theorem, (r) (ct) -(x')
(r)2 (ct) 2 -(x) 2
But r=r' since that distance is i to relative motion of observers
and since c - c'
Then - (r) (rt)2 2 (ct)2- _ ) 2 (ct) 2-(x) and
, x-vt)
- - (V21]2 - y z - z
Fig. 1--Origin of mathematical reasoning
observer 2 pass the light (which travels perpendicular to the observer's
line of relative motion), they both see the light turn on. They both
see the light bit a reflector at Event 1 at some later time, but even
though they both see the same event, they disagree on the coordinates
necessary to describe the location of the event. Because observer 2 has
been going faster, he perceives that his position x' is greater than x
(the position of observer 1), and he believes that Event 1 is located at
a different angle from that which observer I believes.
-5-
Because light travels at the same speed regardless of the particular inertial reference frame chosen, it travels at the same speed for
both observers, so that in the figure rr'and cc'. After some algebra,
the equations given within the box in Fig. 1 are derived. The algebraic
solution method consists of assuming straight line, uniform, nonaccelerated motion, where x'=K(x-vt), t'=Mt-Nx, where K, M, and N are
constants to be found, and v is the relative velocity between observer 1
and observer 2.
There are some important observations which should be made about
the equations in the box of Fig. 1. The first is that there is a denominator which consists of a square root term. This means that mathematically, at least, the term could either be positive or negative and may
be correctly written with either a positive or a negative sign in front
of the radical. The second important observation is that the apparent
position of observer 1 depends on the relative speed of the other
observer with respect to himself as a reference, and vice versa for
observer 1. Similarly, the time that a fast moving observer would measure depends upon the time measured by another observer and the relative
velocities between observers. It should also be noted that spatial
directions y and z are considered to be perpendicular to directions t
and x so that there is no effect of relative motion in these directions. f
Figure 2 contains the transformation formulas which relate the
apparent length and the apparent time that would be measured by a moving
observer with respect to that which would be measured by am observer at
rest. Notice that each observer relates all experiences to his frame of
reference as a stationary frame and other frames as moving with respect
-6-
to his. Therefore, each observer would notice that the other one has
made incorrect measurements with respect to his own.
Figure 3 shows how mass varies as the velocity varies. Notice that
the denominator contains the same radical term and that when an object
is moving fast with respect to an observer, the mass Gf the object
appears greater than it does when at rest in the observer's system.
Again the velocity is the variable which controls the amount of
increase. We shall return to this equation again later.
V2
L =L C2
,henever one observer is moaving with respect to another, whether approaching
or separating, it appears to both observers that everything about the other has
shrunk in the direction of motion. Neither observer notices any effect in his own
system, however.
v2 t' t I -1 -C
Whenever two observers are moving at const velocity relative to each other, it
appears to each that the other's time processes are slowed down.
Fig. 2--Length contraction and time dilatation--l
-7-
(A) MASS INCREASE WITH VELOCITY
m
-2
,'Vhen an object is moving with respect to an observer, the mass of
the object appears greater then when at rest in the observer's system,
with the amount of increase depending on the relative velocity between
object and observer.
(B) VELOCITY TRANSFORMATION LAW
v' v-w (vw)c2
wv) or v' -c 2
( _2 c
2 - wv
,Vhen an object is moving with respect to two observers (s and s'), the
velocity of the object appears different than when at rest in each
observer's system, with the magnitude and direction of change depending
on the relative velocity between the object and s observer and that
between the observers.
Fig. 3--Mass and velocity dependence upon reference systems
The velocity transformation law does not have the radical term in
it (Fig. 3). Instead, the speed w, which is the speed at which the
Coordinate system S' moves along the x axis of coordinate system S, with
respect to the reference frame of S. is defined. The speed v continues
to be the speed along the x axis at which a particle is racing in the
system S. Then the velocity transformation law is derived by considering the Lorentz relationships for x' and t', realizing that v' is the
speed that a particle moves along the x' axis with respect to system S',
and then dividing x' by t'.
-8-
Now that the necessary relationships are available for us, the very
important question should be asked, "What is the maximum permissible
velocity?" Figure 4 again presents the relativistic mass relationship.
In 1905, Einstein used this equation to predict that the speed of light
was the maximum permissible speed. This can be easily seen, for if we
take the limit of the relativistic mass m' as v approaches c, we see
that the v/c term approaches 1, which means that the denominator
appr-aches zero. Of course, the zero in the denominator means that the
mass would have to be infinite if the velocity of a particle were
speeded up to equal that of light. This equation clearly predicts that
no particle which can exist at a speed slower than that of light could
ever be accelerated to a speed equal to that of light.
The equations presented in Fig. 5 illustrate that, as v approaches
c, the relativistic length approaches zero and that the time processes
also slow down and approach zero. This is even more astounding, however, when one remembers that the mass has simultaneously approached
infinite size. Again, the equations clearly indicate that accelerating
a particle to the speed of light is impossible.
But what if the particle were forced to travel at the speed of
light when it was produced? That is, what if the particle never existed
in a state where it travelled at a speed less than that of light, and
when it was produced it was produced while travelling at the speed of
light. Now one might say that there is no change--the relativistic mass
equation still predicts an infinite mass. But this is true only if the
rest mass (the numerator) is not zero. If the rest mass or, as more
-9-
(A) MASS INCREASE WITH VELOCITY
m_ I
m' mm= f Note: lir m' = infinitely large
c2
When an object is moving with respect to an observer, the mass of
the object appears greater then when at rest in the observer's system,
with the amount of increase depending on the relative velocity between
object and observer.
Fig. 4--Mass and velocity dependence upon reference systems,
limiting process
L ' L I- Note: lim L' =0 But Real
v-C
Whenever one observer is moving with respect to another, whether approaching
or separating, it appears to both observers that everything about the other has
shrunk in the direction of motion. Neither observer notices any effect in his own
system, however.
= t I- - 2 Note: lim t = 0 But Real:
V- C Time is relative
Whenever two observers are moving at const velocity relative to each other, it
appears to each that the oiher's time processes are slowed down.
Fig. 5--Length contraction and time dilatation--2
-10-
properly called, the "proper mass", and the denominator were both zero,
then the relativistic mass would simply be undefined.
In fact, this is precisely the case for the photon. It is a particle which travels at the speed of light--never any slower or faster. It
cannot exist at speeds slower than light or at speeds which are faster,
and therefore has a zero proper mass. The equation which appeared to
predict that the speed of light is the maximum speed has been shown to
be invalid for some particles!
Suppose that some new particles exist which have as yet never been
identified in a laboratory. The unique characteristic of these particles is that they exist only while travelling faster than the speed of
light. (This idea could be predicted by symmetry, since known subatomi, paticles exist which travel below and at the speed of light.
The only other regime is the one which is faster than light.) These postulated new particles are called "tachyons".
Another look at the equation in Fig. 4 will show that for tachyons,
(whose velocity is always greater than that of light i.e., v>c), since
v>c the denominator term becomes negative, so that the square root of a
negative number must be taken. Of course, this would result in an imaginary denominator--clearly an impossible situation. Or is it? Suppose, as was the case for photons, the numerator were adjusted to
account for the the imaginary denominator. If the numerator were made
imaginary also (i.e., if the proper mass were imaginary), the imaginaries would cancel out, leaving the relativistic mass to be real. Of
course, it is the relativistic mass which enters into reactions, so the
imaginary proper mass simply means that the tachyons could not exist at
rest in any reference frame travelling at speeds less than that of
light.
-12-
III. CONSERVATION LAWS AND SY.METRY
A. RELATIVISTIC DEPENDENCE OF ENERGY UPON MOMENTUM
Perhaps the most powerful of the conservation laws and symmetry
ideas is the conservation of relativistic energy and momentum. Figure 6
shows the pertinent three-dimensional equation. Without loss of generality, the coordinate axes can be chosen so that only the X direction
Equation: E
2 p 2 c2 - p2C 2 = m 2 c4 where E Energy
y P --Momentum
or E _ 2 2 2 m c 4 C Speed of fight in vacuum
x E
. ,1O c 2 Each point on curve
represents the E and P
E 1that a particle would X
2 2 11=have in a given frame
2 2c . 2 x of reference
E Luxons (E 2 P2c 2 =01)
E = -P. X- X
[ mcc -mc
I - / anti-particles ', i om a
Fig. 6--Conservation of relativistic momentum and energy
-13-
is important, as is shown in the second equation of the figure. From
the analytic geometry, this equation can be plotted as in the plot
shown. Note that since the equation consists of terms which are
squared, the plot must have two halves, and in fact, is the plot of a
hyperbola which breaks upward and downward. It is interesting to note
that only the top half of the hyperbola is considered to be acceptable
according to most physicists today, since we do not have a good perception of the physical interpretation of the negative energy which is
described by the bottom half of the hyperbola. Nevertheless, in 1928
Dirac predicted that there was an anti-particle for every particle of
physics. Soon thereafter the positron (the anti-particle of the electron) was discovered, thus lending credibility to the Dirac theory, and
allowing many physicists to believe that there was indeed an explanation
to the bottom half of the hyperbola. Notice the symmetry of the plot.
Each point of the curves represents the energy and the momentum that a
particle would have, as perceived in a given Lorentzian frame of reference, when undergoing some experiment. In other words, although the
experiment is the same, different observers would "see" different
amounts of energy and momentum during the experiment, with the amounts
of each based only upon the particular frame of reference being used
(but these parameters must, of course, be read from points on the
curves). The asymptotes for the curves are the lines which represent
particles of light (luxons).
The actual equations which were used to produce the sketch are also
shown on the left side of Fig. 6. It should be noted that the top curve
is simply the positive term taken from the square root of the denomina-
-14-
tor and that the bottom curve is a representation of the energy and
momentum associated with the negative sign in front of the radical term.
Notice that this plot shows that as positive energy increases, so
does momentum, and vice-versa. Of course, the momentum could be taken
as positive or negative, depending upon the direction of the coordinate
system chosen. The plot also shows that as negative energy increases in
a negative direction, so does momentum increase (in either a positive or
a negative direction, depending upon the direction of the coordinates
chosen.)
B. RELATIVISTIC DEPENDENCE OF ENERGY UPON VELOCITY
The equation which describes the relativistic relationship between
energy and velocity is given in Fig. 7. An assumption must now be made.
One can see by looking at the equation that whenever v becomes greater
than c (c=the speed of light), the energy would become imaginary. But
imaginary energy can have no meaning to people on earth and therefore
can have nothing worthwhile to do with any usable theory of antigravity. We must force the energy to remain real at all times. The
assumption is that since the speed of light is real, the relativistic
mass must be real or the energy could not be measured (or, perhaps more
correctly for those who believe that energy itself is an abstract term
and cannot be measured directly, that the capacity to do work could not
be inferred).
There are only three possibilities regarding the particle velocities:
-15-
+ m c2 m0 proper (rest) mass
Euto E m - 0 m= relativistic mass 112 v particle velocity in
this reference frame
Assumption E is real
Since c2 is real, : must be real or E could not be measured
Possibilities (a) v ' c E would be real
+ inc 2
(b) v -c E = - which is impossible unless 0
m- (rest mass) - 0. Therefore, rest mass of
p photon must be zero and experiment
verifies tnis.
.inmc2
Wc V > c E 0which is complex and could i[()2 11L2 not be measured unless m
c is imaginary 0
Note: Relativistic mass remains real
even though rest mass is imaginary
Fig. 7--Relativistic dependence of energy on velocity--equations
o The particle is travelling (with respect to a particular reference frame) at a speed less than that of light. This would
result directly in a real energy (as seen from the equation of
Fig. 7) and is the common case for everyday physics.
o The particle travels at a speed equal to that of light. In
this case, the rest mass must equal zero and we have the case
of the photon, whose zero rest mass has been verified by experiment. The energy, although undefined, is still real and
presents no obstacles to our thinking.
-16-
0 The particle travels at a speed greater than that of light. In
this case, the numerator, i.e., the proper mass, must become
imaginary in order to make the energy remain real (and measurable). Again note that the relativistic mass remains real and
presents no contradiction to verifiable physical experiments.
Figure 8 is a sketch of the equation shown on that page. Since
this equation has a radical term for the denominator, it also can be
written with both a positive and a negative sign in front of the right
hand side. Therefore, the equation will plot as upper and lower curves
which represent the positive energy and the negative energy portions.
Note that in these curves, the asymptote is again the speed of light.
The center part of the curves represents the normal case where particle
velocity is less than that of light. The tachyon portions of the curves
would be a plot of imaginary energy unless the ordinate were changed to
represent the normalized energy which is associated with the imaginary
proper mass. Notice how the curves as plotted in Fig. 8 provide both
left-right and top-bottom symmetry.
Figure 8 shows that for ordinary particles (tardyons), the energy
increases as the velocity of the particle increases, up to an infinite
energy associated with particle speeds equal to light. Notice that the
addition of even infinite amounts of energy could not accelerate the
particle to or past the speed of light. The plot also shows that for
the negative energy tardyon section, the subtraction of energy (or the
addition of negative energy) also would accelerate the particle, but
never to speeds equal to or greater than light.
-- -
-17-
2 2 2 4 m 0 V Equation E px c =m0 c But px =mv V x oI
so thatE1
E !
m0 2 2
c
Tachyon s
2.' Imaginary unless m0 is moo
S1 2 Velocity of
or particle in - reference
system s
-V c v c
Fig. 8--Relativistic dependence of energy on velocity--graphics
The case for the tachyons is different. Figure 8 demonstrates that
as energy is added to tachyons, the tachyon velocity would decrease.
Even infinite amounts of additional energy could not succeed in slowing
the particles down to the speed of light. Subtracting positive energy
is seen to increase the tachyon velocity, thus leading to the unfamiliar
possibility of infinite speeds with zero energy at the limit. Since the
curves are symmetric, the figure shows that as negative energy is
-18-
reduced, the tachyon velocity is also increased without limit. Although
these are strange characteristics when compared to our everyday experience, they are certainly not impossible to accept.
C. DEPENDENCE OF ENERGY UPON MOMENTUM FOR TACHYONS
Since tachyons must have imaginary proper masses, the imaginary
term must be written on the right side of the energy-momentum conservation equation of Fig. 6. But the square of the imaginary part simply
turns out to be the negative of a real term, as can be seen from the
equation in Fig. 9. The negative term forces the hyperbola to break
left and right, rather than up and down as it did in Fig. 6. Again,
this plot completes the symmetrical representatio,. of the conservation
equation.
However, a new complication appears to arise. The curves now can
be seen to cross the abscissa, which means that a single event could be
noticed by two different Lorentzian observers, differing only by the
relative speeds by which they travel past the event, and one observer
would detect a negative energy while the other observer could detect a
positive energy. Figure 6 shows that this never occurs with tardyohs.
Since our physical concept of "negative energy" is somewhat less than
"well based in experiment," this at first appears to be a major stumbling block to the existence of tachyons. I shall return to this point
later.
-19-
Note: E 2-p2 c2- (mo) 2 c
IIE c2but (m;) 2 = 2M I ml c2 bu =m
which is real, but negative
XLuxons'.
Fig. 9--Plot of energy-momentum for faster-than-light particles
D. DEPENDENCE OF ENERGY UPON COORDINATE SYSTEM SPEED FOR TACHYONS
With a considerable amount of algebra, the equation stated in Fig.
10 can be reduced to an easily plotted equation also given in Fig. 10.
Of course, this equation again provides a symmetric plot. The plot
shows that any event that had a positive energy would appear to decrease
rapidly in energy, and in fact, would pass through zero energy and enter
the negative energy region, with the rapidity of change being a function
of the relative speeds of the coordinate systems. It is important to
-20-
mc2
Equation: E
2 pxc 2 2 M 4 or E 2 where
or C )
E (c2_wvl V) = (v-w)c 2
S2 c
2 v' = tachyon velocity
- in frame s'
E w = relative speed of
2 Is' RT s frames
Assumption: E must remain real axes are aligned
%b v = tachyon velocity
% in frame s
%_________ w
O Crossover occurs
C2 li , at w : -
Fig. 10--Dependence of energy upon coordinate system speed for tachyons
note that on this plot, the relative velocitiv of the coordinate systems
always remains less than that of light, and the crossover occurs whenever the relative speed of the coordinate systems (w) times the velocity
of the particle as viewed from an observer in one of the coordinate systems (v) is equal to the square of the speed of light. In other words,
a fast traveller could view a tachyonic event as having negative energy,
while a slower observer could view the same event as having a positive
energy, or vice-versa. Again, the cross-over of energy from positive to
-21-
negative or from negative to positive which we first observed in Fig. 9
could occur.
-22-
IV. THE VALUE OF SPACE-TUIE CONSIDERATIONS
Before returning to the positive-to-negative energy transitions, I
shall introduce some necessary concepts of space-time. The left side of
Fig. 11 consists of my attempt to sketch the four-dimensional space of
space-time on a two-dimensional surface. All four axes are orthogonal
to each other. All light pulses which are emitted at the event which is
at the origin of the figure will travel along the outside surface of the
Ct Ct
Absolute future
Lgtcone- '
450 '-World line of all light
V pulses (for one observer)I
0x OR X
Absolute past
Fig. 11--Visualization of time
-23-
light cone, and the light cone always makes an angle of 45 degrees with
the time axis. Notice that the time axis is really a distance axis for
purposes of plotting space-time, and is equal to "ct." This definition
allows reasonable plots to be made. All tardyon particles which are
emitted at the origin of space-time coordinates must travel at speeds
slower than light, and must, therefore, have paths (or "world lines")
which are constrained to lie within the light cone. All world lines
which extend toward the positive time axis are travelling into the
future, and those which travel down would appear to travel into the
past.
The right side of Fig. 11 is a two dimensional representation of
the light cone and the space-time coordinates.
If another coordinate system, which travels along the X axis of the
first, is superimposed on the first frame, the plot would appear as in
Fig. 12. Notice that the second frame does not have orthogonal coordinates, that the skewness of it is a function only of its speed with
respect to the orthogonal frame, and that the relative velocities of the
frames remain less than light.
Suppose that a tachyon were emitted at the origin of the space-time
curve represented in Fig. 13. Further suppose that the tachyon were
sent out in a straight path along the X axis. Since the speed of a
tachyon is constrained to be greater than that of light, it must be constrained to exist only outside of the light cone. Since the light cone
can be represented by the same line to both observers, the only question
is, "Should the event which signifies the disappearance of the tachyon
be represented by a point as at "E" (which is above the X' axis), or
should it be represented by a point below the V' axis?"
-24-
Ct c t' Light cone
X
* ew-tan (WC
* Eqn of light cone in s coordinates is x = ct
* Eqn of light cone in s coordinates is x' = (ct)'
* Plot of light cone in both s and s systems is blue line
Fig. 12--Space-time for 2 observers in relative motion along X axis
Suppose that the event signifing tachyon "birth" was at the origin
in Figure 13 and that signifying "destruction" were at point E. Then,
according to the observer in the orthogonal frame, the particle would
appear to be emitted at the origin, travel along the X axis, and at some
later time ctA it would appear to be at position X According to the
faster observer who is also travelling along the X axis, at some different (from ctA) time ct'B, the particle would appear to be at position
-25-
(v-w) c2
(1) Equation v (v-w)
2
C -W V , ( w c
Integrating x'= fv'dt' gives '= 2v c t'
(2) Sketct ct ct'
Light cone
ctA -- E X
ci' - 'I °
XA
:3) Analysis *For tachyons, v c. This places E outside light cone.
0 v > w always since v is for tachyons and w < c for Lorentz transformation
Numerator is always positive
Fig. 13--Dependence of time upon velocity of particle and relative
velocity of reference system S' WRT system S--event
end point above X' axis
X' There is nothing strange about this, since, for each observer, the
particle appears to have been shot along his position axis, and at some
later time from the initial event it appears to have moved.
Now suppose that the event representing tachyon destruction
occurred below the X' axis as at point EA in Fig. 14. Further suppose
that initial conditions remain as were those for point E. Now the
observer who sits on his orthogonal frame would notice that at some time
which he calls zero his tachyon is emitted and at some later time it is
-26-
(1) Equation v (v-w) c 2
C -wv =(-~
Integrating x' =f v' t' gives x' c2w l (CV)
(2) Sketch ct
ct'
Light cone
X, B X1
ct,B.
(3) Analysis For tachyons, v > c. This places E or EA outside light cone.
v > w always since v is for tachyons and w < c for Lorentz transformation
Numerator is always positive
X' is negative if wv>c 2 unless Iet') is negative
2 Whenever wv>c , time reversal must occur
Note that E/ 1m5 c
2 becomes negative when w v> c
2
Fig. 14--Dependence of time upon velocity of particle and relative
velocity of reference system S' WRT system S--event
end point below X' axis
absorbed. Again, there is nothing strange about this. But the other
observer notices that at some time, which he calls zero', he sees the
emission of the tachyon and at some EARLIER time (ct' B) he sees that the
tachyon has been absorbed! For this observer, a strange thing has happened. He thinks that the tachyon has been absorbed before it was emitted (and that "causality" has been violated)!
What are the conditions under which this seemingly impossible
situation arises? Note from the equation in Fig. 14 that the denomina-
L-- I -
-27-
tor turns negative whenever the product of the relative speeds of the
frames and the velocity of the tachyon exceeds the square of the speed
of light.
By referring to Fig. 10, one can see that these are the same conditions under which the energy of a tachyon turns from positive to negative or from negative to positive. Figure 15 is another look at Fig. 9
which shows that time reversal also occurs whenever a particular
observer senses that the tachyonic energy turns negative.
Note: E2 2- c 2 (M (,)I c4
E but (m;)2 = _2
which is real, but negative
/acyons Px
TIME REVERSAL OCCURS FOR
NEGATIVE ENERGY TACHYONS
Fig. 15--Plot of energy-momentum for faster-than-light particles
Ai
-28-
This has led some researchers [1] to identify a "Reinterpretation
Principle": The proper interpretation of negative energy particles that
travel backward in time is that they can be considered to be positive
energy particles which travel forward in time. This reinterpretation
invalidates causality objections to the existence of faster-than-light
signals and permits construction of a consistent theory of tachionic
behavior. Although this "reinterpretation principle" seems at first
exposure to be a hastily conceived artifice for removing objections to
tachyon existence, further consideration demonstrates that the "artifice" may have considerable merit and should not be taken lightly.
4
Ai
-29-
V. TACHYON CHARACTERISTICS
The foregoing analysis has provided information concerning the
characteristics of these new particles, tachyons, if they really exist.
If they do exist, then tachyons must have at least those characteristics
(but not only those characteristics) which have been presented in this
paper. Because of their importance to the remaining research, these
characteristics are summarized in Fig. 16.
-30-
(1) Must have imaginary rest miss
(21 Always travel faster than light
(3) Could have either G or 0 relativistic mass (in')
(4) 2ill have real energy
(5i -nergy m~y be either G or (
i oI Energy could be either G or G and still have 0 momentum
or
Energy could be either ( or ( and still have (D momentum
1 fh'ILJ have im3ginary 'proper length''
' I MUSt have imaginary "proper lifetini '
(91 Addn-n 0 real energy would slow down the 0 energy tachyons
(V;i Adding (D real energy would slow down the C) energy tachyons
(11 J Nay have infinite speeds
(12, Infinitely fast tachyons have zero energy
(13) Infinitely fast tachyons still have momentum
(14i Real energy could be measured (either 6 or 0) by an observer in an s,
frame mving fast WRT s frame, but at w < c.
(151 There are many s' frames where an energy which appears ( to an observer
in the rest frame s would appear as 0 to the observer in s'.
o' Saame as (15) except for appears as 0 to s observer would appear ( to s'
observer.
(17 Time reversal will occur whenever (15) or ( 16) occurs. (whenever wv > c2
(181 lachyons which have (0 (or ( ) energy can also appear to SAME observer
to be going backward in time. (but not necessary)
(19i Objects made of tachyons would appear to have (0 length unless 0 '
is used.
(20) Objects made of tachyons would appear to have tim? reversal unless (D)
is used.
(211 Since entropy must increase, tachyons would appear to violate 2nd Law of
Thermo except for time reversal occurs at sam? time, ie, entropy decreases
are accompanied by time reversal. (natural occurrences tend to go from
higher ordered systems to lower ordered systems - low ordered systems have
low entropy).
Fig. 16--Tachyon characteristics
-31- VI. FIRST CONSIDERATION OF AN NEWTON'S ANTI-GRAVITY LAWTHEORY BASED ON TACHYONS:
Since Newtonfs Law has predicted the correct results of all nonrelativistic experiments which have been performed using "ordinary"
masses, it should be a good starting equation to help with anti-gravity
research. The purpose of the anti-gravity research would be to identify
an equation similar to Newton's, which would correctly predict measurable and therefore useful anti-g'avity forces. These anti-gravity
forces might then be produced and applied in our everyday lives.
Figure 17 provides Newton's Law and identifies all terms used.
Because the force F has been found to be an attractive force in all
experiments that have been done to date, and because the force F
represents the gravitational force (which has been defined as positive),
one would expect the anti-gravity force to show up as a negative force
in the same equation. But how does one make the force show up as negative?
For all ordinary gravitational phenomena, the masses have been
defined as positive, and since the distance between the masses is a
squared term, the denominator is always positive. The constant is simply a device which we use tu make the equation give results which agree
with experiments, and therefore has been defined as positive. If we
were to assign a negative value to the constant, we would predict a
negative force. However, the result would contradict all experiments
that have already been performed. The equation terms themselves, rather
than an arbitrary choice of sign for the constant, must dictate the sign
of the force.
-32-
~ m~ 1 rn2 m
F2 1 mss of object I
r m2 miss of object 2
r - distance between m1 and m2
G = constant
F = Force between m1 and m2
For anti-gravity, F must be negative
mf m
For msses composed ot tachyons, F = G 2
r
F remains real
Fig. 17--Newton's law of gravitetion for tachyonF
If the constant were always taken as positive, the force would
always be positive (attraction), provided the masses were either both
positive or both negative (composed of sometimes hypothesized antiparticles). So even the negative mass particles (if they should exist),
would not provide an anti-gravity (negative) force. If one mass were
positive and the other were negative (if negative masses exist), then a
negative or anti-gravitational force would be predicted by the equation.
-33-
However, this possibility seems to violate the results of experiments,
since all masses appear to be composed of (different) quantities of the
same particles, and therefore all macroscopic masses which undergo gravitational forces are expected to be of the same sign (either positive or
negative).
If imaginary terms were considered in Newton's equation, the distance (r) could be imaginary. Then the denominator would be negative
and we would achieve a negative force. However, this possibility would
contradict the desired goal--to find an expression which could be useful
for actual anti-gravity (e.g., to have a force of repulsion experienced
by two masses which are a specified real distance apart). If the gravitational constant were imaginary, the force experienced by the masses
would also be imaginary, providing an uninteresting result.
However, if the proper masses were imaginary (i.e., if the masses
consisted of tachyons), then the force would be a real, but negative,
force. An unchanged Newton's Law, using the premises that tachyons
actually exist and that some masses are composed of tachyons (under some
conditions), could then be used to predict anti-gravity without violating the results of any experiments that have already been accomplished.
-34-
VII. TENTATIVE CONCLUSIONS
The foregoing analysis is not meant to be convincing. I retain
considerable skepticism myself, particularly questioning the legitimacy
of utilizing parts of a particle theory (which is unproved) for investigating macroscopic phenomena and applying the concept of rest masses to
test a gravitational formnila which has been proved inapplicable for
relativistic velocities. I have not intended to provide a complete
rationale for saying that "tachyons" really exist, nor do I mean to
imply that the only theory that makes sense for anti-gravity is a
tachyon theory. Admittedly, much "arm waving" has been done. However, I did intend to point out that
a. Other researchers [1, 2, 31 have shown that the existence of
tachyons is not inconsistent with special relativity or relativistic quantum mechanics.
b. Other researchers [1, 3) have shown that the causality arguments against the possible existence of tachyons are not compelling.
c. Energy/momentum conservation and particularly symmetry arguments seem to predict that tachyons could exist.
d. If tachyons actually do exist, they must have at least the
characteristics listed in Fig. 16.
e. A connection between particles that travel faster than light
and anti-gravity could exist.
-35-
Much more work must be done to identify additional characteristics
of tachyons, particularly whether or not they should have electrical
charges and their expected spin. An investigation should be conducted
into the way a tachyon would react if placed into a force field. A
solid theoretical model of the relativistic quantum mechanical description of tachyons must be developed if tachyons are soon to be found
in the laboratory. But if the characteristics of tachyons are to be
used for postulating new ideas while simply tentatively assuming that
tachyons exist, perhaps enough characteristics are already known to
make legitimate guesses at the framework of suitable theories. This
will be the topic of the next paper.
-37-
REFERENCES
1. Bilaniuk, 0. M., and E. C. G. Sudarsham, "Particles Beyond the Light
Barrier," Physics Today,, May 1969, pp. 43-51.
2. Feinberg, G., "Possibility of Faster-Than-Light Particles," Physical
Review, Vol. 159, No. 5, 25 July 1967, pp. 1089-1105.
3. Terletskii, Y. P., Paradoksv Teorii Otnositelnosti, Nauka Press,
Moscow 1966; ENglish Translation: Paradoxes in the Theory of Relativity, Plenum Press, New York (1968).
4. Tanaka, S., Progr. Theoretical Physics, (Japan) 24, 177 (1960).
5. Einstein, A., Relativity, The Special and the General Theory, as
translated by R. W. Lawson, Crown Publishers, Inc., New York, 1956.
6. Skinner, R., Relativity, Blaisdell Publishing Co., Waltham, Massachusetts, 1969.
xmzm PAGE BL -NoT ni D
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