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\title{Coordinates Transformations and Masses involving Superluminal Velocities}
\author{Bernard R. Durney}
\date{2377 Route de Carc\'{e}s, F-83510 Lorgues, France durney@physics.arizona.edu}
\maketitle
\smallskip
\smallskip\textbf{Abstract}
Coordinate transformations (expressed in terms of four dimensionless parameters)
that leave invariant the metric of Special Relativity are considered. These parameters
must be functions of the dimensionless ratio {\it u/c} where {\it c} and
{\it u} are respectively the velocity of light, and the parametric velocity that
defines the transformation. For small values of {\it u}, this one is the velocity of the
moving frame of reference. Physically meaningful transformations are obtained by
imposing the condition that these parameters develop singularities at {\it u = c},
which in turn keep particles from crossing the light barrier. These transformations cover
velocities that are smaller than {\it c} (Sc) and larger than {\it c} (Lc) as well. For
superluminal velocities (associated with spacelike metrics), the relation between the square
of the metric and the square of the proper time differs from the one valid for Sc
velocities, associated with timelike metrics. As a consequence, the energy, of superluminal
particles (the time component of the four momentum) is as conventional as the one of
Sc-velocities. It is argued that the only component of the four-momentum accessible to us
is the enery, and that any attempt to define the remaining three-momentum component
(hypothesizing, e.g. an imaginay particle's mass) is in conflict with quantum mechanics. The
superluminal world is as logically consistent as the Sc one, and they must interact, at least
minimally through gravity.\\
\smallskip
\textbf{1. Transformations that conserve the metric of Special Relativity }
\smallskip
Consider the special relativity metric,
$$
ds^2 = - c^2 dt^2 + dx^2
$$
and the transformations of coordinates,
$$
x' = \alpha x - \beta c t, \mspace{15mu}t' = \gamma t - (\delta/c) x,
\eqno{(1)}
$$
where the quantities, $\alpha, \beta, \gamma $ and $\delta$ are dimensionless.
The equality,
$$
- c^2 dt^2 + dx^2 = - c^2 dt'^2 + dx'^2,
\eqno{(2)}
$$
leads immediately to the following relations,
$$
\gamma \delta =\alpha \beta,\mspace{15mu} \gamma^2 = 1 + \beta^2 , \mspace{15mu}
\delta^2 = \alpha^2 - 1
\eqno{(3a,3b,3c)}
$$
Therefore, it follows that,
$$ \alpha^2 \beta^2 = \gamma^2 \delta^2 =( 1 + \beta^2 )
(\alpha^2 - 1)\mspace{15mu} \longrightarrow \alpha^ 2 = 1 + \beta^2 = 1 + \delta^2,
\eqno{(4)}
$$
where the last equation follows from Eq.(3c). It is now straightforward to show with
the help of Eqs.(3), that the metric remains invariant if the equations below are fulfilled,
$$
\alpha^2 =1+\beta^2,\mspace{20mu}\gamma^2 = \alpha^2,\mspace{20mu} \delta^2 = \beta^2,
\mspace{20mu}\gamma \delta =\alpha \beta
\eqno{(5)}
$$
It is clear that, signs excluded, the knowledge of $\beta$ implies the knowledge of
all the remaining parameters defining the transformation. Concerning signs, $\gamma$
must positive, in both coordinate systems the flow of time must be in the same direction,
say, from the past to the future. Assume, e.g., that $\beta > 0,$ then it follows
from the last Eq. (5), that $\alpha,\delta$ must {\sl both} be positive or negative. These two
situations are associated with the direction of the moving particle's velocity.
The problem at hand contains two fundamental velocities, namely, the velocity of light, and
$u,$ the velocity that defines the transformation of coordinates. Because the parameters are
dimensionless, they must be functions of $u/c,$ (or of $c/u.)$ Consider the first case,
The choice of, e.g., $\beta = u/c,$ cannot lead to a physically meaningful
transformation because it doesn't provide a barrier for a particle to exceed the velocity of
light, as, e.g., $\beta = \gamma u/c$ does. In this case,
$$
\beta = \gamma u/c, \mspace{8mu} \alpha^2 = \gamma^2,\mspace{8mu} \gamma^2 = 1/(1 -(u/c)^2),\mspace{8mu} \delta^2 = \beta^2,\mspace{8mu}\alpha\beta
= \gamma \delta
\eqno{(6)}
$$
which is indeed the Lorentz transformation (concerning the Lorentz group, cf. Maggiore, 2005
p.16). If all the parameters are expressed in terms of $\delta,$ we obtain
$$
\alpha = \delta c/u ,\mspace{8mu} \beta^2 = \delta^2,\mspace{8mu} \gamma^2 =\alpha^2,
\mspace{8mu} \delta^2 = 1/((c/u)^2 -1),\mspace{8mu}\alpha\beta = \gamma \delta,
\eqno{(7)}
$$
which is another way of writing Eq.(6)
\smallskip
\textbf{2.Addition of Velocities} \\
\smallskip
The calculation of the addition of velocities involves, aside from $u$ defined above,
two more velocities, $v$ and $w,$ namely, the velocities of the particle in the primed
and original, unprimed frame of reference respectively. It follows from Eqs. (1) that,
$$
v = \frac {\alpha w - \beta c}{\gamma - \delta w/c},\mspace{30mu}
w = \frac {\gamma v + \beta c}{\alpha + \delta v/c} =
\frac { v + u}{1 + uv/c^2},
\eqno{(8)}
$$
where the last equation was obtained by replacing the values of the dimensionless
parameters as given by Eqs.(6) or Egs.(7). It is well known that for $|v|< c,$
also is $|w|< c.$ It is rarely quoted in the literature that the equivalent result
for superluminal velocities is also corroborated. It must be shown that $|v|> c$
implies $|w| >c,$ i.e., $(u + v)^2 > (1 + uv)^2,$ that is (setting $c =1$),
$$
\frac {u^2 + v^2}{1 + u^2 v^2 } > 1,\mspace{10mu} u^2 < 1,\mspace{10mu} v^2 > 1
\eqno{(9)}
$$
For velocities larger than $c,$ i.e v = $c(c/u),$ one would more conveniently use Eq.(7) to
calculate $w.$ It is found,
$ w= (v^2 + c^2)/2v =(v - c)^2/2v + c,$ always larger than $c.$
\smallskip
\smallskip
\textbf{3.Energy-Momentum }
\smallskip
The particle's proper time for a timelike and spacelike metric are respectively
defined by, (cf. Hartle (2003) Eq.(4.12)),
$$ d\tau^2 = - ds^2,{\mspace{20mu}} d\tau^2 = ds^2.
\eqno{(10)}
$$
and the components of the four velocity (for superluminal particles, namely,
$d\tau^2 = ds^2$) by,
$$
v^\alpha = \frac {dx^\alpha}{d\tau} = \frac{dt}{d\tau}(c,\mathbf{v}), {\mspace{10mu}} v^2 =c^2,
\mspace{10mu} \Bigl{(}\frac{dt}{d\tau}\Bigr{)}^2 = \frac{1}{(\mathbf{v}^2/c^2 - 1)}
\eqno{(11)}
$$
where the second equation follows from Eq.(10), and the last one, from the preceding
ones, (cf. Hartle, pages 82-89). In Eq.(11), $\mathbf{v} = \mathrm d\mathbf{x}/\mathrm d{t} $ is the three velocity. The four momentum is defined by,
$$
p = mv = \Bigl{(}m\frac{dt}{d\tau}c,m \frac{dt}{d\tau}\mathbf{v}\Bigr{)} = (E/c,\mathbf{p}),
\spa \mathbf{p} = \frac{m\mathbf{v}}{(\mathbf{v}^2/c^2 - 1)^{1/2}}
\eqno{(12)}
$$
where $\mathbf{p}$ is the three momentum, and $E$ is the energy. Because $p^2 =m^2 c^2$ it
follows that,
$$
E^2 = \mathbf{p}^2 c^2 - m^2 c^4 = \spa \frac{m^2\mathbf{v}^2}{(\mathbf{v}^2/c^2 - 1)} c^2
- m^2c^4 = \frac{m^2c^4}{(\mathbf{v}^2/c^2 - 1)},
\eqno{(13)}
$$
which should be compared with,
$$
E^2 = \mathbf{p}^2 c^2 + m^2 c^4 \spa = \frac{m^2c^4}{(1 - \mathbf{v}^2/c^2 )},
\eqno{(14)}
$$
for $\bv^2 < c^2.$
\smallskip
\textbf{4. Conclusions}
\smallskip
Recall that Lc and Sc stand for larger-, and smaller, than the velocity of light respectively.
It is clear that Eqs.(13) and (14) are both intuitively obvious: As a particle's velocity
approaches(from above or below) the velocity of light, it must gain energy that has to become
infinite for $\bv^2 = c^2 \mspace{15mu}$ in order to keep it from crossing the light barrier.
For $\bv^2 > c^2,$ the momentum decreases for increasing values of $|\bv|,$ and approaches
$mc\bv/|\bv|$ for large values of $|\bv|$ , which is indeed the smallest allowed momentum.
The contribution of this momentum to the energy squared at $|\bv| = \infty$ is $m^2c^4.$
For superluminal particles, the smallest energy cannot be other than zero, which accounts for
the $- m^2c^4$ term in Eq.(13). {\it It is clear, that not only Equation (14), but Equation (13)
as well, can be justified intuitively without any elaborated calculation}. The superluminal
world is indeed as logically consistent, as the smaller than $c$ one, and the Lorentz
transformation deals with both of them on an equal footing. If they do not interact, they could
both live peacefully, however they must interact, at least minimally, through gravity. In the
Lc-world, $m$ is a parameter that defines the energy of a particle (a "bunch of energy")
for a given value of $\bv.$ If we choose $|\bv| = \sqrt{2}c$ it is found that $E(|\bv| =
\sqrt{2}c)= m c^2.$ But there is nothing special about the $v= \sqrt{2}c$ velocity, whereas
there is indeed something special about $|\bv|=0$ in the Sc-world: it allows the value of $m$
to be determined. The special value in Eq.(13), namely, $|\bv|=\infty$ doesn't specify
$m$ because $E = 0.$ \\At the birth of the Universe it is reasonable to conjecture
that all particles compatible with the physical laws came into being. Encouraged by the
Lorentz transformation, assume then that this was the case with the Lc-particles. {\bf It is
often argued that these contradict the laws of physics. A more accurate statement would,
however, be that they are in disagreement with conventional field theories, which is not
surprising because it cannot be expected that theories aimed at explaining Sc-particles,
should remain relevant for Lc-paticles.} In fact even for ordinary particles, quantum field
theories are destined to be superseded in the future. In field theories, the interaction
between particles arouses from an interchange of {\it virtual} bosons, and the physics
behind this interchange is obscure. Contrast this theoretical approach with the one of G.R.
{\it Here, the interaction between the Earth and the Sun is not conveyed by virtual gravitons.
Instead, mass curves spacetime, a physical reality indeed.} The equivalent of G.R. for
particle physics, lies still in the future.\\
One of the fundamental principles of quantum mechanics is that information about physical
quantities cannot be assumed. Instead, it must be experimentally obtained. We cannot measure
the parameter $m(Lc)$, which must be imaginary for the Sc-energy equation
to be satisfied. {\it Following the same line of thought an Lc-member, who can neither
measure $m(Sc),$ would assign an imaginary mass to our particles.}
The presence of Lc-particles would manifest itself in our world as dark matter (or energy),
which is indeed the time component of the momentum vector, however {\it we cannot measure
the spatial momentum} and efforts of writing down the whole momentum vector cannot be
successful. On the positive side: the energy contains all the necessary information about
gravitational effects\\
Concerning the existence of Lc-particles, logical contradictions, however unlikely, would of
course be fatal. Nevertheless, to discard them, one shouldn't advocate requirements reminiscent
of a "causality principle": a "simultaneity principle" would wipe out Special Relativity.
Only one condition should be imposed on superluminal particles: to be compatible with General
Relativity. Optimism is justified. However, it is important to keep in mind, that this
constraint requires the existence of a dynamics for the Lc-particles.
\smallskip
\textbf{5. Acknowledgements}
\smallskip
I am grateful to Dr. Roger Clark for enlightening comments.
\smallskip
\smallskip \textbf{6. References}\\
\smallskip
Hartle, J.F.: {\it GRAVITY, An Introduction to Einstein's General Relativity}, Addison Wesley, California (2003)
\smallskip
Maggiore, M.: {\it A Modern Introduction to Quantum Field Theory}, Oxford University Press,
New York (2005)
\end{document}
\end{article}