Abstract
It is commonly asserted that superluminal particle motion can enable backward time travel, but little has been written providing details. It is shown here that the simplest example of a “closed loop” event – a twin paradox scenario where a single spaceship both traveling out and returning back superluminally – does not result in that ship straightforwardly returning to its starting point before it left. However, a more complicated scenario – one where the superluminal ship first arrives at an intermediate destination moving subluminally – can result in backwards time travel. This intermediate step might seem physically inconsequential but is shown to break Lorentz-invariance and be oddly tied to the sudden creation of a pair of spacecraft, one of which remains and one of which annihilates with the original spacecraft.1 INTRODUCTION
Objects traveling faster than light are discouraged by popular convention in Einstein’s Special Theory of Relativity ((Einstein, 1905)), which provides a profound, comprehensive, and experimentally verified description of particle trajectories and kinematics at subluminal speeds. Nevertheless, the vast distance to neighboring stars has caused superluminal speeds to continue to be discussed in popular venues. ((StarTrek, 1964)) To be clear, this work is not advocating that faster than light speeds for material particles are possible. Rather, the present work takes superluminal particle speeds as a premise to show how closed-loop backward time travel arises in a specific simple scenario.
Physics literature has indicated for many years that superluminal speeds can correspond to backward time travel. ((Tolman, 1917)) Such claims are pervasive enough to have become common knowledge, as exemplified by a famous limerick published in 1923: “There was a young lady named Bright, Whose speed was far faster than light; She set out one day, In a relative way, And returned on the previous night”.((Buller, 1923))
The possibly of closed-loop time travel within the context of special relativity was later mentioned explicitly in 1927 by Reichenbach. ((Reichenbach, 1927)) A prominent discussion on the physics of particles moving superluminally within the realm of special relativity was given in 1962 by Bilaniuk Deshpande, and Sudarshan. ((Bilaniuk, 1962)) The term “tachyon” was first coined for faster than light particles by Feinberg ((Feinberg, 1967)) who also derived transformation equations for superluminal particles. Tachyonic speeds have been suggested multiple times in the physics literature to address different concerns, for example being convolved with quantum mechanics to create pervasive fields ((Feinberg, 1967)), and to explain consistent results between two separated detectors in quantum entanglement experiments. ((Einstein, 1935)), ((Bell, 1966))
The
reality of particle tachyons or any local faster-than-light communication
mechanism is controversial, at best. Accelerating any material particle from
below light speed to the speed of light leads to a divergence in the particle’s
energy, a physical impossibility. For ,
The Lorentz-FitzGerald Contraction ((Lorentz, 1892)), ((FitzGerald, 1889))
becomes imaginary, leading
to relative quantities like mass, distance, and time becoming ill-defined,
classically. Simple tachyonic wavefunctions in quantum mechanics either admit
only subluminal or non-localizable solutions. ((Chase, 1993)) Experimental
reports of particles moving faster than light have all been followed by
skeptical inquiries or subsequent retractions. ((Opera, 2011)),((Opera, 2012)) Were
tachyonic communications to enable communications backward in time, violations
in causality seem to result, a prominent example of which is the Tachyonic
Anti-telephone. ((Benford, 1970)) For
this reason superluminal communication and backward time travel are thought to
be impossible. Experimentally, a recent search of Internet databases for
“unknowable-at-the-time” information that might indicate the possibility of
backward time travel came up empty. ((Nemiroff and Wilson, 2014))
Conversely, the existence of superluminal speeds for phase velocities and illumination fronts that do not carry mass or information are well established. ((Griffiths, 1994)) The ability of superluminal illumination fronts to show pair creation and annihilation events was mentioned by Cavaliere et al.((Cavaliere et al., 1971)) and analyzed in detail by Nemiroff ((Nemiroff, 2015)) and Zhong & Nemiroff((Zhong and Nemiroff, 2016)).
The possibility that a material object could undergo a real pair creation and subsequent annihilation event was mentioned in 1962 by Putnam ((Putnam, 1962)) including the possibility of pair events with regard to backward time travel. However, Putnam’s treatment was conceptual, gave no mathematical details, and the concept of a closed loop was not considered. In 2005 Mermin ((Mermin, 2005(@)) noted such behavior for an object moving subluminally but with an intermittent period of superluminal motion, reporting that such pair events would only be evident in some inertial frames. Mermin also never considered a closed loop event.
There appears to be no detailed treatment, however, showing how superluminal speeds lead to “closed-loop” backward time travel: a material observer returning to a previously occupied location at an earlier time. Treatments generally stop after showing that faster than light objects can be seen to create negative time intervals for relatively subluminal inertial observers. ((Tolman,1917)) To fill this void, the standard velocity addition formula of special relativity is here applied in the superluminal domain to show that closed-loop backward time travel can, in specific circumstances, be recovered – but perhaps in a surprising way. This scenario can also be considered a didactic and conceptual extension of the famous “twin paradox” ((Einstein, 1905)) to superluminal speeds.
2 OUT AND BACK AGAIN
The scenario explored here is extremely simple: an object goes out and comes back again. The return trip is important to ensure a “closed-loop”. Therefore, the scenario described can be thought of as an extension of the famous twin paradox to superluminal speeds. For the sake of clarity, to promote interest, and to place distances on scales where temporal effects correspond with common human time scales, the initial launching location will be called “Earth”, which can also be thought of as representing the twin that stays at home. The ballistic projectile will be referred to as a “spaceship”, which can also be thought of as representing the twin the travels away and then comes back. The turnaround location will be referred to as a “planet”. Furthermore, an example where the distance scale is on the order of light-years will be described concurrently.
The
following conventions are observed. In general, unless stated otherwise, all
times and distances will be given in the inertial frame of the Earth and from
the location of Earth. All relative motion for the spaceship will take place in
the line connecting the Earth to the planet, here defined as the axis.
All velocities are assumed constant. The planet is assumed moving away from the
Earth at the subluminal speed
as
measured in the Earth’s inertial frame. Times are given by the variable
,
and the standard time when the spaceship is scripted to leave Earth is set
to
. At this time
the spaceship leaves Earth from a location called the Launch Pad, and aims to
return to an Earth location called the Landing Pad. The distance on Earth
between the Launch Pad and the Landing Pad is considered negligible. Velocities
away from the Earth are considered positively valued, and velocities toward the
Earth are considered negatively valued. The outbound velocity of the ship
relative to the Earth is given by
,
the return velocity of the ship relative to the planet is given by
, and the return velocity of the ship relative to
the Earth is designated
.
Keeping both spaceship speeds at magnitude
is
a useful didactic simplification that demonstrates the logic of a much larger
set of event sequences when the outgoing and incoming spaceship speeds are
decoupled.
At
time , the planet is
designated to be a distance
and
moving at positive velocity
away
from the Earth, with respect to the Earth. Therefore, at time
the
distance between Earth and the planet is simply
|
(1) |
Similarly,
at time , the
spacecraft leaves Earth from the Launch Pad. After launch and before reaching
the planet, the distance between Earth and the spaceship is
|
(2) |
The x-coordinate will usually describe the spaceship and so, when it does, no subscript will be appended.
The
spacecraft reaches the planet at the time when . Combining the above two equations
shows that the amount of time it takes for the spacecraft to reach the planet
is
|
(3) |
The distance to both the spacecraft and the planet at this time is
|
(4) |
The spaceship turns around at the planet. For this calculation, the turnaround is considered instantaneous but the important point is that the turnaround duration is small compared to other time scales involved. After turnaround, the velocity of the spaceship relative to the Earth is
|
(5) |
where is
measured relative to the Earth but
is
measured relative to the planet. Only in Eq. (5) is
negative
as it describes the spaceship returning back to Earth – in all other
equations
is
to be considered positive as it refers to the speed of the ship leaving Earth.
Note that Eq. (5) is the standard equation of velocity
addition in special relativity and one that has been consistently invoked even
when superliminal speeds are assumed. ((Mermin, 2005(@)), ((Hill and Cox, 2012))Because of
the centrality of this equation to physics, it is retained here in its classic
form.
After the spaceship leaves the planet, its distance from the Earth is given by
|
(6) |
The
scenario is defined so that once the spaceship returns to Earth, it lands on
the Landing Pad and stays there. The spacecraft can only move between the Earth
and the planet – it does not go past the Earth to negative values.
The time it takes for the spaceship to return back to the Earth from the planet is
|
(7) |
The
negative sign leading this equation is necessary to make the amount of return
time positive when is
negative. Writing
in
terms of (positively valued)
,
,
and
yields
|
(8) |
The total time that the ship takes for this trip is
|
(9) |
where,
again, all interior speeds are defined as being positively valued and . Were the planet stationary with respect to
Earth, parameterized by
then
which
agrees with the non-relativistic classical limit, no matter the (positive)
value of
.
It is also
of interest to track how long it takes light signals to go from the spaceship
to Earth, as measured on Earth. The time after launch that an Earth observer
sees the spaceship at position will
be labeled
. Since
light moves at
in
any frame, then Earth observers will see the outbound spaceship at
position
at time
|
(10) |
where the first term is the time it takes for the spacecraft to reach the given position, and the second term is the time it takes for light to go from this position back to Earth. The time that Earth observers will see the spacecraft reach the planet is
|
(11) |
During the
spaceship’s return back to Earth, Earth observers will see the spaceship
at such
that
|
(12) |
The first
term is the time it takes for the spacecraft to reach the planet, the second
term is the time it takes for the spacecraft to go from the planet to
intermediate position , and
the third term is the time it takes for light to reach Earth from
position
.
A series
of threshold values
occur, which will be reviewed here in terms of increasing magnitude.
2.1 THRESHOLD SPEED:
The first
threshold speed explored is .
Below this speed, the spaceship is moving too slowly to reach the planet.
When
, the spaceship
has the same outward speed as the planet and only reaches the planet after an
infinite time has passed. This is shown by the denominators going to zero in
Eqs. (3, 8, and 9). Earth observers will see both the
spaceship and the planet moving away in tandem forever.
2.2 SPACESHIP SPEEDS
When , and when both the spaceship and the
planet are moving much less than
,
then Earth observers see the spaceship move out to the planet and return back
to Earth in a normal fashion that is expected classically.
When generally, then
. This results from
the magnitude of the spacecraft’s speed coming back to Earth,
,
being less than the magnitude of the spacecraft’s speed going out to the
planet,
,
even though the distance traveled by the spacecraft is the same in both
cases:
. In this
speed range, the occurrence of events in the Earth’s inertial frame proceeds as
expected in non-relativistic classical physics. As tracked from the Earth, the
spaceship simply goes out to the planet, turns around, and returns.
For
clarity, a series of specific numerical examples are given, with values echoed
in Table 1, and world lines depicted in the
Minkowski spacetime diagrams of Figures 1 and 2. In all of these examples, the planet
starts at distance light
years from Earth when
,
and the planet’s speed away from the Earth is
. For the speed range being investigated in
this sub-section, the spaceship has speed
. Then by Eq. (4) the spaceship reaches the planet when
both are
light
years from Earth. The duration of this outbound leg is given by Eq. (3) as
years. For clarity, the
speed of the ship’s return is computed from Eq. (5) as
, to four significant digits. The duration
of the ship’s return back to Earth is given by Eqs. (7, 8) as
years. The total time
that the spaceship is away is the addition of the “out” time and the “back”
time, which from Eq. (9) is
years. The world line of
this ship’s trip is given by the dashed line in Figure 1.
Due to the
finite speed of light, Earth observers perceive the spacecraft as arriving at
the planet only after the equations indicate that it has already started back
toward Earth. The closer is
to
,
the closer the spacecraft is to the Earth, as defined by Eq. (11), when Earth observers see the
spaceship arrive at the planet.
In the
concurrent example of ,
the spaceship reaches the planet at time
years, but light from this event does
not reach Earth until the time given by Eq. (11) –
years, well after the spaceship has
actually left the planet. However, the ship is first seen on Earth to arrive
back at Earth when it actually arrives back on Earth – 54.69 years after it
left.
Figure
1: A spacetime diagram is shown for the journeys taken by the subluminal ships
described in the text. Time is plotted against distance traveled, both measured
in the frame of the earth. Earth remains at
. The world line of the planet continually
receding from the earth at
,
in the earth frame, is depicted by the dark dotted line on the right. A space
ship traveling out at
in the
earth frame is shown by the light dotted line. This ship never reaches the
planet. A ship traveling out at
in
the earth frame, reaching the planet, and returning to earth at
in the planet’s frame is shown by the
dashed line. A ship similarly traveling at
is
shown by the solid line.
2.3 THRESHOLD SPEED:
The next
threshold value for the spaceships outbound speed explored is . Generally when
, however, then not only is the spaceships
speed
relative
to Earth on the way out, but it is
relative
to the planet on
the way out, it is
relative
to Earth on the way back, and it is
relative to the planet on the way back. Here
: the spaceship takes
the same amount of time to reach the planet as it does to return.
In the
example, the only quantity changed is the speed of
the spaceship. At
, the spaceship
catches up to the planet at
light years at time
years after launch. The speed of
return is
. The time it
takes for the ship to return is
years, meaning the total
time for the trip, as measured on Earth, is
years.
What Earth
observers see, in general, is quite different from the classical
non-relativistic cases. Neglecting redshifting effects, the spaceship would
appear to travel out to the planet normally, but would arrive back on Earth at
the same time that the spacecraft appears to arrive at the planet. In fact,
light from the entire journey back to Earth would arrive at the same time the
spacecraft itself arrived back on Earth. This is because as perceived from Earth,
the spacecraft, returning at speed ,
rides alongside all of the photons it releases toward Earth on the way back.
In the
specific example, the spaceship
is seen to arrive back on Earth when
, which Eq. (12) gives as
22.22 years. The solid
lines in Figures 1 and 2 depict the world lines of this
ship.
2.4 SPACESHIP SPEEDS
To explore
the question as to how faster-than-light motion can lead to backward time
travel, it is now supposed that superluminal speeds are possible for material
spaceships. In general, as greater spacecraft speeds are
considered in the range
,
the magnitude of the speed of the spacecraft’s return
increases
without bound. Here, in general,
.
In the
specific example, the spaceship speed is now taken to be . The spaceship catches up with the planet
at
light
years at time
years
after launch. The return speed is
so
that it takes the spaceship
years to get back to
Earth. Earth receives the spaceship back after 3.082 years away.
What Earth observers see, in general, is perhaps surprising, as tracked by Eqs. (10) and (12). First the spacecraft appears to leave for the planet as normal. Next, however, two additional images of the spacecraft appear on Earth on the Landing Pad, one of which stays on the Landing Pad, while the other image immediately appears to leave for the planet. The underlying reason for these strange apparitions is that spacecraft itself returns to Earth before two images of the spacecraft return to Earth. Therefore, after the spacecraft returns, the Earth observer sees not only the returned spacecraft, but an image of the spacecraft on the way out, and an image of the spacecraft on the way back, all simultaneously.
In the
specific example, an image of the spacecraft
is seen moving toward the planet and arriving, as determined by Eq. (11) at
years, even though the
spacecraft itself arrived back on Earth earlier – after only 3.082 years. The
light dotted line in Figure 2 describes the world line for this
trip’s journey.
It is
particularly illuminating to consider what is visible from Earth five years
after the spacecraft left the Launch Pad, after the actual spacecraft has
arrived back on Earth but before the spacecraft appears to have arrived at the
planet. First, assuming the returned spacecraft has remained on the Landing
Pad, there is the image of the returned spacecraft remaining on the Landing
Pad. Next, an image of the spacecraft going out to the planet is visible back
on Earth. Focusing on this image of the outbound craft, one can solve Eq.(10) for to find that
light years from Earth when this
image of the outbound ship arrives back on Earth. Last, a third image – an
image of the spacecraft on its return back from the planet – is simultaneously
visible back on Earth. At the five-year mark, one can solve Eq. (12) to find that
light years.
Therefore, Earth sees this third image as the spaceship is returning to Earth,
but still 2.135 light years distant.
To recap, five years after leaving the Launch Pad, three images of the spacecraft are visible on Earth. One image is emitted by the spacecraft as it remains sitting on the Landing Pad after its return, another image is emitted by the spacecraft on its way to the planet, and a third image is emitted by the spacecraft on its way back from the planet.
It is
further illuminating to check on the spacecraft images still visible on
Earth eight years
after it left the Launch Pad, three years after the previous check. At eight
years, assuming the spaceship remains on the Landing Pad, an image of the
returned spaceship remains visible on the Landing Pad. Solving Eq. (10) for now shows the “outbound image” of the
ship at location 6.667 light years from Earth, further out than it was before.
The image of the returning spaceship shows from Eq. (12) the ship at a distance 5.477 light
years from Earth. This might appear odd as the return image arriving at Earth
eight years out shows the ship as further away from Earth – not closer – than
the image of the returning craft that arrived after five years. It therefore
appears that this return spaceship is moving backward in time, as seen from
Earth.
For clarity, to recap again, eight years after leaving the Launch Pad, three images of the spacecraft remain visible from Earth: one image on the Landing Pad, one image on the way out, and one image on the way back. Both the outbound image and the return image show the ship appearing further away after year eight than at year five.
Right at
the time the spaceship returns to Earth, the number of spaceship images visible
on Earth jumps from one to three. Before this, Eq. (12) shows that both the image of the
spaceship on the Landing Pad and an image of the spaceship returning to the
Landing Pad have yet to reach Earth. Therefore for in
this interval, the spaceship reaching the Landing Pad marks an image pair
creation event.
Similarly,
when the spacecraft reaches the planet, both the outbound and the return images
of this event arrive back at Earth simultaneously, as can be seen from Eqs. (10) and (12). Because no further images of the
spacecraft going out or returning exist, these images then both disappear,
leaving only the spacecraft image on the Landing Pad. This disappearance is an
image pair annihilation event. These image pair events are conceptually similar
to spot pair events seen for non-material illumination fronts moving
superluminally. ((Nemiroff, 2015)) Image
events are entirely perceptual – the actual location of the spaceship is given at
any time by Eqs. (2) and (6). Observers at other vantage points – or
in other inertial frames – may see things differently, including possible
different relative timings of image pair creation and annihilation events.
Also, in this velocity range, since the location of the spacecraft in the Earth frame is unique, it is clear that
only one spaceship ever exists at any given time.
2.5 THRESHOLD SPEED:
At the
threshold speed , the
speed of return
of
the spacecraft to Earth diverges as the denominator of Eq. (5) goes to zero. Although slight
variations of
below
and above this threshold speed will yield
divergences
to positive or negative infinity, the negative infinity realization will be
discussed. At formally infinite return speed
,
the time it takes for the spacecraft to return to Earth is
. Therefore, as soon as the
spaceship reaches the planet it arrives back on Earth. To be clear, what
returns to Earth immediately is not only an image of the spaceship as perceived
on Earth, but the actual physical spaceship itself.
In a
specific example, the spaceship speed is now taken to be . This spaceship catches up with the
planet at about
light
years at time
years
after launch. With zero return time, Earth receives the spaceship back after
years away. The dashed line
in Figure 2 describes the world line for this
trip’s journey.
In
general, the Earth-bound observer sees the same series of events as perceived
when ,
they just happen a bit more compact in time. First, the spaceship is seen
leaving. Next, a pair of spaceships appears on Earth on the Landing Pad. One
ship from this image pair immediately leaves for the planet, while the other
spaceship – and its image – remain on Earth. Both outbound spaceship images
appear to reach the planet at the same time, and both then disappear from view.
In the
specific example when ,
the Earth observer first sees the spaceship leave the Launch Pad at
. At
years,
the Earth observer suddenly sees two images appear on the Landing Pad, one of
which immediately takes off – time reversed – toward the planet, while the
other image stays put. At
years,
the Earth observer sees both the outbound and return spacecraft images reach
the planet, and both disappear.
2.6 SPACESHIP SPEEDS
For
spaceship velocities , the
return velocity
,
in general, becomes formally positive. Since the spacecraft never moves toward
the Earth in this scenario, how can it return to Earth? Although such a
conundrum may seem like an end to a physically reasonable scenario, a
physically consistent sequence of events does exist that is compatible with the
formalism. This sequence is as follows. A spaceship leaves the Launch Pad on
Earth for the planet. At a later time, as measured on Earth, a physical pair of
spaceships materializes on the Landing Pad on Earth. One of these spaceships
immediately goes off to the planet, while the other spaceship remains in place
on the Landing Pad. Eventually both the spaceship that initially left the
Launch Pad and the spaceship that later left the Landing Pad arrive at the
planet simultaneously and dematerialize into nothing.
There is a
fundamental difference between this sequence of events and the events when . When the spacecraft has speeds in
this range, a real pair creation event occurs on the Landing Pad, and a real
pair annihilation event occurs at the planet. These are not images, but are
consistent with the location(s) of the actual spacecraft(s) to any observer in
the inertial frame of Earth, as computed by Eqs. (2) and (6). Interpreting these equations as
describing physical spaceship pair events is a natural extension of the image pair events
that occurs at lower speeds. Although observers in Earth’s inertial frame that
are located off the Earth may perceive events and sequences of events
differently, they all must use the actual locations of the spacecraft as
computed in the inertial frame of the Earth as the basis for what they see.
In a
specific example, the spaceship speed is now taken to be . The spaceship takes off from the Launch Pad
at
. Eq. (3) gives the outbound time as
years and Eq. (8) gives the return time as
years, so that the total
time the spaceship is away from Earth is
years. Therefore the next
thing that happens is that a pair of spaceships materialize on the Landing Pad
at
years.
One spacecraft stays on the Landing Pad. The speed of the spaceship that leaves
the Landing Pad is from Eq. (5)
. Therefore, even though this ship left
later, it is just the right amount faster to arrive at the planet at the same
time as the spacecraft that left the Launch Pad. Both spacecraft catch up to
the planet when it is, from Eq. (4),
light years distant. This occurs
at is
years.
At this time, both outbound spacecraft merge and dematerialize.
A
potential point of confusion is that the equation for the location of the
spacecraft that left the Landing Pad, Eq. (6), formally returns a negative valued
location for the spaceship when years.
It is claimed here that such locations are outside of the described scenario
and so do not occur. The two spaceships that materialize at
years on the Landing Pad do not have
previous positions described in this scenario. The situation is similar to the
equation for the spaceship that left the Launch Pad, Eq. (2). This equation also does not indicate
that the outbound spaceship that leaves the Launch Pad occupied negative valued
locations before
, as such
positions are outside the described scenario and do not occur.
Surprisingly,
perhaps, what Earth observers see, in general, is not conceptually different from
events perceived when the spacecraft has as described in the previous
two sections. Still the first event witnessed is the launching of the
spacecraft from the Launch Pad. Next, Earth observers see a pair of spacecraft
appear on the Landing Pad, one of which stays put and the other goes off to the
planet. Last, the observers see both spacecraft arrive at the planet at the
same time and disappear.
In the
specific example, a spaceship is seen from Earth to leave the Launch Pad
at . Suddenly,
at
years,
a pair of spaceships appear on the Landing Pad, one of which stays there, and
the other leaves for the planet. The spaceship that leaves the Landing Pad
appears time reversed and faster than the spaceship that left Launch Pad. The
images of the spacecraft arriving at the planet, described by Eq. (10) and Eq. (12), arrive back on Earth at
years, where the images appear to merge
and disappear.
Figure
2: A spacetime diagram is shown for the journeys taken by the superluminal ships
described in the text. Time is plotted against distance traveled, both measured
in the frame of the earth. Earth remains at
. The world line of the planet continually
receding from the earth at
,
in the earth frame, is depicted by the dark dotted line on the right. A ship
traveling out at
to the
planet is shown by the solid line. Spaceships traveling out from the earth at
speeds of (
,
,
,
AND
) are
depicted by the (dotted, dashed, dot-dashed, triple dot-dashed) world lines
respectively. World lines of similar type that connect back to earth at times
other than
are
formally described as returning to earth at the designated speed in the frame
of the planet. Are formally described as returning to earth at the designated
speed in the frame of the planet. However, as detailed in the paper, world
lines that “return” to earth at
are
actually time-reversed and described by the formalism as heading out from the
earth.. Therefore, these world lines better describe the trip of a second ship
that left earth before the first ship, and arrived at the planet simultaneously
with the first ship. All of these world lines depict actual spaceship
positions and not the images of the ships as seen back on earth.
2.7 THRESHOLD SPEED:
The
spacecraft speed is a threshold
value because here
, meaning that
and that the total time it
takes the spacecraft to go out to the planet and return back to Earth is zero.
This speed is one of two formal solutions to setting
, as given in Eq. (9), equal to zero. The other formal
solution,
always results
in a
and so
is discarded because it describes a scenario where the spaceship never reaches
the planet.
The general scenario has a spacecraft traveling in this speed range leave from the Launch Pad toward the planet. At the same time, a pair of spacecraft together materialize on the Landing Pad, one of which also immediately leaves for the planet, while the other spacecraft remains in place. The two simultaneously launched spacecraft both approach the planet, arrive simultaneously, and then de-materialize together at the planet.
In a
specific example, the spacecraft is assigned the speed . The return speed is computed from Eq. (5) to be, also,
. Therefore, at
, one spacecraft leaves the Launch Pad, while
another spacecraft leaves the Landing Pad. A third spacecraft remains in place
on the Landing Pad. Both outgoing spacecraft reach the planet at
years when the planet is
light years from Earth. At this
time, both spacecraft merge and dematerialize.
What is perceived as happening for spacecraft in this general speed range is qualitatively the same as what happens. Earth observers see a spacecraft image launch for the planet from the Launch Pad and, simultaneously, a second spacecraft image leave for the planet from the Landing Pad. The image of the other member of the spacecraft pair that appears on the Landing Pad stays put. Images of both outgoing spacecraft are seen to approach the planet, arrive at the planet at the same time, and disappear together when they reach the planet.
In the
specific example of ,
Earth observers see an image of the spacecraft leave from the Launch Pad – and
another image leave from the Landing Pad – at
. Images of both spaceships are seen to arrive at
the planet at
years,
whereafter the images merge and disappear.
2.8 SPACESHIP SPEEDS
Cases
where faster-than-light motion leads to closed-loop backward time travel can
finally be explained as a logical extension of previously discussed results. In
cases of increasing where
, in general, the
“return back” speed
is
not only positive – and so indicating motion away from the Earth – but
decreasing – and so indicating slower motion toward the planet. The result is
that the spacecraft “returns back” to the Landing Pad before it leaves
from the Launch Pad. Surprisingly, this scenario does not give the
straightforward out-and-back sequence of events commonly assumed for
superluminal time-travel. On the contrary, this scenario relies on
pair-creation and pair-annihilation events.
The first scenario event for spacecraft speeds in this general speed range is that two spacecraft appear on the Landing Pad, one of which immediately sets off for the planet. The next event, as described in the Earth frame, is that the initial spacecraft takes off from the Launch Pad and heads out toward the planet. Both the spacecraft that left from the Landing Pad and the spaceship that left from the Launch Pad reach the planet at the same time and de-materialize.
In the
specific example of ,
Eq. (5) now yields a “return back” speed
of
, a
positive value that describes movement away from the Earth. Note also
that
, so
that
describes
less rapid outward motion. Further, Eq. (3) shows
years, while Eq. (8) shows
years, so that their sum
has
years,
meaning that the spaceship “returns” 0.3367 years before it left. The
dot-dashed line in Figure 2 describes the world line for this
trip’s journey.
Eq. (2) describes the spaceship that left from
the Launch Pad at . Eq. (4) shows that this
spacecraft catches up with the planet
at
light
years from Earth. The time this spaceship catches up to the planet is at
years.
Eq. (6) describes the spaceship that left from
the Landing Pad at years.
This spacecraft also catches up to the planet at
years. Times greater than
years yield distance values in Eq. (6) larger than that of the planet, but
these are considered unphysical because the scenario gives the boundary
condition that the spacecraft turns around at the planet. Similarly, times less
than
years
yield negative distance values in Eq. (6), but these are also considered
unphysical because the scenario states a boundary condition that the spacecraft
travels only between Earth and the planet. One might argue that all times
before
are similarly
unphysical because the scenario dictated that the spaceship launched at
, but no such temporal boundary condition was
placed on the time of return.
It is
educational to query the locations of the outbound and “return back” spaceships
during their journeys to see how they progress. At time
years, Eq. (6) indicates that the return spaceship has
a location of
light
years, meaning the “return back” spaceship is still on Earth. At time
years, this equation indicates that the
return spaceship has a location
light years from Earth in the
direction of the planet. At time
years,
Eq. (2) indicates that
light years, meaning
that the “outbound” spaceship is still on Earth, while Eq. (6) indicates that
light years from Earth. At
time
years
the equations hold that
light years,
while
light years.
Next, at
years,
the equations hold that
light years,
while
light years.
Finally, at
years,
both
)
and
yield
10.03 light years.
What is seen on Earth in general for speeds in this range is again qualitatively similar to what physically happens. First, two spacecraft images are seen to appear on the Landing Pad, one of which is seen to launch immediately for the planet, while the other appears to stay put. Next, an image of the spacecraft on the Launch Pad launches for the planet. Both the image of the spacecraft that left from the Landing Pad and the image of the spacecraft that left from the Launch Pad appear to reach the planet at the same time. These two images merge and disappear.
In the
specific example where ,
Earth sees the spaceships that materialized on the Landing Pad at
years with no time delay because this
Landing Pad is on Earth. The spaceship that stays on the Landing Pad is always
seen on Earth to be on the Landing Pad with no time delay, from
years onward. The spaceship that left
the Landing Pad for the planet at
is
seen with increasing time delay due to the travel time of the light between the
spaceship and Earth. At
years,
the “return back” spaceship is at
light years from Earth but
because of light travel time, is seen when it was only at
light years away. At
years, a spaceship is seen to take off from
the Launch Pad, while the spaceship that left the Landing Pad while actually
at
light
years distant, appears as it did when at
light years distant. At
years
= 6.000 light years,
but due to light travel time, this “outbound” spaceship appears as it did when
at
light
years. Similarly, at
years,
the “return back” spaceship is at
light years
distant, but due to light travel time appears as it did when at
light
years. Images of both spaceships arriving at the planet are received back on
Earth at
years.
At this time, both images merge and disappear.
Even before the outbound spaceship leaves from the Launch Pad, astronauts on the spaceship that appeared and remained on the Landing Pad may come out, recount their journey, and even watch the subsequent launch of their spacecraft on the nearby Launch Pad. A physical conundrum occurs, for example, if these astronauts go over to the Launch Pad and successfully interfere with the initial spacecraft launch. This would create a causal paradox that may reveal any time travel to the past to be unphysical. ((Hossenfelder, 2012)) Alternatively, however, the presented scenario may proceed but the disruption event may be disallowed by the Novikov Chronology Protection Conjecture ((Novikov, 1992)) – or a similarly acting physical principle. Then, try as they might, the astronauts could find that they just cannot disrupt the launch. ((Lloyd, 2011)) In a different alternative, such disruption actions may be allowed were the universe to break into a sufficiently defined multiverse, with the disruption just occurring in a different branch of the multiverse((Everett, 1957)) than the one where the spacecraft initially launched. Then, life for the disrupting astronauts would continue on normally even after they disrupted the launch, even though they could remember this launch. It is not the purpose of this work, however, to review or debate causal paradoxes created by backward time travel, but rather to show how some superluminal speeds do not lead to closed-loop backward time-travel, while other speeds do, but by incorporating non-intuitive pair creation and annihilation events.
2.9 WHEN DIVERGES
Perhaps
counter-intuitively, an infinite amount of backward time travel does not result
when diverges.
In the general case, as
approaches
infinity, the time it takes for the spacecraft to reach the planet,
, approaches zero. However, Eq. (8) shows that the time it takes for the
spacecraft to return to Earth,
, does not approach negative infinity but rather
. The reason for
this is that, in Eq. (5),
only
approaches
as
diverges,
which is always superluminal and never zero. Therefore, the faster the planet
is moving away from the Earth, and the further the planet is initially from the
Earth, the further back in time the returning spaceship may appear on the
Landing Pad.
In the
specific example, diverging leads
to a maximum backward time travel according to Eq. (8) of
year. The return
speed
approaches
10 c. Therefore, in this scenario, the earliest a spaceship pair could appear
on the Landing Pad would be
year,
one year before the outbound spaceship leaves the Launch Pad. After this,
events would unfold qualitatively as described in the last section. The triple
dot-dashed line in Figure 2 describes the world line for this
trip’s journey.
3 PLANET-FREE SCENARIOS
One might
consider that the pair creation and annihilation arguments above only arise
because of the “trick” of involving a planet that has a non-zero and positive
speed .
In this view the planet’s speed, along with the relativistic speed addition
formula, act as a spurious door to mathematical possibilities that are
physically absurd. As evidence, one might take an example where a spaceship
leaves with a speed
relative
to the Earth and then returns at a speed
,
again relative to the Earth. The arbitrary turnaround location can be
labeled
. Then, in
the Earth frame, the time it takes for the ship to reach the turnaround
location would be
,
and the time it takes for the ship to return from the turnaround location would
be the same:
.
When
, a pair of
virtual images of the ship will again be seen, for a while, on Earth. However,
there are no velocities
and
,
subluminal or superluminal, where either
or
is negative, and so no
and
values
exist that create
.
Therefore, in this scenario the spaceship will never arrive back on Earth
before it left. Does this counter-example disprove the presented analysis?
No. The scenario of the previous paragraph does not create a situation where an object returns to the same location at an earlier time – a closed-loop backward time travel event. Therefore this scenario does not address the main query posed by this work – how faster-than-light travel enables backward time travel.
Scenarios
do exist, however, where superluminal travel creates closed-loop backward time
travel events, but where no intermediary planet is involved. Such scenarios,
which some might consider simpler, have the spaceship just go out at one speed,
turn around at an arbitrary location, and return at another speed. So long as
the return speed is generated and hence specified relative to the outbound
speed, then the relativistic velocity addition formula Eq. (5) may be used, and the same types of
results arise. Note that is only presumed that Eq. (5) is valid when one or both speeds and
are
superluminal – this presumption has never been verified.
A simple
planet-free scenario is as follows. A spaceship leaves Earth at speed .
At an arbitrary turnaround location, the ship changes its velocity by
, toward the Earth, relative to its outward motion.
For non-relativistic speeds, this turnaround would result in the ship heading
toward Earth at speed
.
For relativistic and superluminal speeds, however, the relativistic velocity
addition formula must be used, resulting in more complicated behavior.
Specifically, in this scenario, it is straightforward to show following the
above logic that
, so
that closed-loop backward time travel events occur for all spaceship speeds
of
. As before,
tracking spacecraft and image locations show that pair events also may occur.
This brings up the question: why does it matter against what the spaceship’s relative return velocity is measured – shouldn’t the physics be the same? Coordinate invariance – called general covariance, and inertial frame invariance – called Lorentz invariance – should make the physics the same no matter which coordinates are used for tracking and no matter which inertial frames are used for comparison. Specifically, in this case, closed-loop backward time travel should not depend on whether the spaceship’s return velocity is specified relative to the Earth, or a planet, or the spaceship’s previous velocity, or anything else. The reality of what happens should be same regardless.
The key
symmetry-breaking point is that the standard special relativistic velocity
transformation, Eq. (5), is not confined to be Lorentz
invariant when both subluminal and superluminal speeds are input.
Mathematically, the reason is that the denominator of the velocity addition
formula goes through a singularity at , a singularity that cannot be reversed by a
simple coordinate or inertial frame transformation. Physically, turning around
relative to a different object may change the scenario – a different physical
process may be described.
4 DISCUSSION AND CONCLUSIONS
An
analysis has been given showing how faster-than-light travel can result in
closed-loop backward time travel. The analysis focused on an extremely simple
scenario – an object going out and coming back – effectively extending the twin
paradox scenario to superluminal speeds. Further, only a single relativistic
formula was used – that for velocity addition. A surprising result is that, in
this scenario, backward time travel appears only when the turnaround location
is moving away from the launch location, and, further, is bound to the creation
and annihilation of object pairs. The underlying mathematical reason is that
the negative time duration for the return trip needed to create closed-loop backward time
travel is tied to spaceship motion away from the launch site, not toward it, as shown in
Eq. (7). This behavior neatly describes an
(Earth-observed) spaceship moving out toward the planet on the “return back”
leg of the trip in addition to the (Earth observed) spaceship moving out toward
the planet on the initial “outward” leg of the trip. One knows that the
spaceship does return, and so the second member of the pair-created spaceships
remains on the Landing Pad. Note that when superluminal spaceship speeds are
invoked, the spaceship always travels at superluminal speeds relative to the
Earth, and never accelerates through .
It is tempting to explain away these results as meaningless because the relativistic velocity addition formula, Eq. (5) was applied to a regime where it might not hold: where one speed is superluminal. However, the validity of this formula in the superluminal regime should be testable in a conventional physics lab where illumination fronts or sweeping spots move superluminally, in contrast to a detector that moves subluminally. Nemiroff, Zhong, and Borysow ((2016)) Further, to our knowledge, no other relativistic velocity addition formula has even been published.
Although not defined in the above equations, it is consistent to conjecture that the superluminal spaceship has negative energy. ((Chase, 1993)) This may be pleasing from an energy conservation standpoint because both pair creation and annihilation events always involve a single positive energy and a single negative energy spaceship – never two positive energy or two negative energy spaceships. Therefore, neither the creation nor annihilation of a spaceship pair, by themselves, demand that new energy be created or destroyed.
It is not clear how “real” the negative energy spaceships are to observers in inertial frames other than the Earth, including frames moving superluminally. The negative energy ships are surely real in the Earth frame in the sense that they give those observers positions from which spaceship images can emerge. However, these negative energy ships may not exist in some other reference frames, which appears to raise some unexplored paradoxes. Also unresolved presently is whether observer in a superluminal positive-energy ship that left the Launch Pad would be able to see a negative-energy ship that left from the Landing Pad. Since it is not in the scope of the above work to analyze what happens in inertial frames other than the Earth, then, unfortunately, this and other intriguing questions will remain, for now, unanswered.
Finally, this analysis may give some unexpected insight to physical scenarios that seem to depend on superluminal behavior. For example, implied non-locality in quantum entanglement typically posits some sort of limited superluminal connection between entangled particles, although one that does not allow for explicit superluminal communication. To the best of our knowledge, never has such supposed superluminal connection been tied through the special relativity addition formula to pair events. Perhaps one reason for this is that so few seem to know about it. Yet, as implied here – it may well be expected for observers in some reference frames.
Acknowledgements.
The authors thank Qi
Zhong, Teresa Wilson, and Chad Brisbois, for helpful conversations. Electronic address: nemiroff@mtu.edu
Electronic address: dmrussel@mtu.edu
REFERENCES
· Einstein ((1905)) A. Einstein, “Zur Elektrodynamik bewegter Korper”, Annalen der Physik 322, 891–921 (1905).
· StarTrek ((1964)) An example is the “warp drive” used in Star Trek, Desilu Studios (1964).
· Tolman ((1917)) R. C. Tolman, “The theory of the relativity of motion”, Berkeley: University of California press, (1917).
· Buller ((1923)) A. H. R. Buller, “Relativity”, Punch, or The London Charivari 165, 591 (1923).
· Reichenbach ((1927)) Hans Reichenbach (original: 1927), “The Philosophy of Space & Time”, translated by Maria Reichenbach and John Freund, New York: Dover publications, (1958).
· Bilaniuk ((1962)) O. M. P Bilaniuk, V. K. Deshpande, and E. C. G. Sudarshan, “Meta Relativity”, American Journal of Physics 30, 718-723 (1962).
· Feinberg ((1967)) G. Feinberg, “Possibility of Faster-Than-Light Particles”, Phys. Rev 159, 1089–1105 (1967).
· Einstein ((1935)) A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”, Physical Review 47, 777-780 (1935).
· Bell ((1966)) J. S. Bell, “On the Problem of Hidden Variables in Quantum Mechanics”, Reviews of Modern Physics 38, 447-452 (1966).
· Lorentz ((1892)) H. A. Lorentz, “The Relative Motion of the Earth and the Aether”, Zittingsverlag Akad. V. Wet. 1, 74–79 (1892).
· FitzGerald ((1889)) G. F. FitzGerald, “The Ether and the Earth’s Atmosphere”, Science 13, 390 (1889).
· Chase ((1993)) S. I. Chase, “Usenet Physics FAQ: Do tachyons exist?”,http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html (1993).
· Opera ((2011)) The Opera Collaboration et al., “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, http://arxiv.org/abs/1109.4897v1 (2011).
· Opera ((2012)) The Opera Collaboration et al., “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam”, http://arxiv.org/abs/1109.4897 (2012).
· Benford ((1970)) G. A. Benford, D. L. Book, and W. A. Newcomb, “The Tachyonic Antitelephone”, Physical Review D 2, 263-265 (1970).
· Nemiroff and Wilson ((2014)) R. J. Nemiroff & T. Wilson, “Searching the Internet for evidence of time travelers”, Winnower, June 2014, http://arxiv.org/abs/1312.7128 (2014).
· Griffiths ((1994)) See, for example, D. J. Griffiths, “Introduction to Quantum Mechanics” (First Edition), (1994).
· Cavaliere et al. ((1971)) A. Cavaliere, P. Morrison, & L. Sartori, “Rapidly Changing Radio Images”, Science, 173, 525 (1971).
· Nemiroff ((2015)) R. J. Nemiroff, “Superluminal Pair Events in Astronomical Settings: Sweeping Beams”, Publications of the Astronomical Society of Australia, 32, 1 (2015).
· Zhong and Nemiroff ((2016)) Q. Zhong & R. J. Nemiroff, research in progress (2016).
· Putnam ((1962)) H. Putnam, “It Ain’t Necessarily So”, Journal of Philosophy, 59 (October 11), 665-668 (1962).
· Mermin ((2005(@)) N. David Mermin, “It’s About Time, Understanding Einstein’s Relativity”, Princeton U. Press, Princeton (2005).
· Hill and Cox ((2012)) James M Hill & Barry J. Cox, “Einstein’s special relativity beyond the speed of light”, Proc. R. Soc. A, 3 October (2012).
· Hossenfelder ((2012)) For a good recent discussion, see, for example, S. Hossenfelder, “Quantum Superpositions of the Speed of Light”, Foundations of Physics, 42, 11, 1452-1468
· Novikov ((1992)) I. Novikov, “Time machine and self-consistent evolution in problems with self-interaction”, Physical Review D 45, 1989-1994 (1992).
· Lloyd ((2011)) See, for example, the experimental results of S. Lloyd, “Closed Timelike Curves via Postselection: Theory and Experimental Test of Consistency”, Physical Review Letters, 106, 040403 (2011).
· Everett ((1957)) H. Everett, “Relative State Formulation Of Quantum Mechanics”, Reviews of Modern Physics 29, 454-462 (1957).
· Nemiroff, Zhong, and Borysow ((2016)) An attempt to test this experimentally has begun by R. J. Nemiroff, Q. Zhong, and J. Borysow (2016).
|
|
|
|
|
|
( |
( |
(light years) |
(years) |
(years) |
(years) |
0.1 |
– |
|
|
– |
– |
0.5 |
-0.4210 |
12.50 |
25.00 |
29.69 |
54.68 |
1.0 |
-1.000 |
11.11 |
11.11 |
11.11 |
22.22 |
5.0 |
-9.800 |
10.20 |
2.041 |
1.041 |
3.082 |
10.0 |
|
10.10 |
1.010 |
0.000 |
1.010 |
15.0 |
29.80 |
10.07 |
0.6711 |
-0.3378 |
0.3333 |
19.95 |
19.95 |
10.05 |
0.5038 |
-0.5038 |
0.0000 |
30.0 |
14.95 |
10.03 |
0.3344 |
-0.6711 |
-0.3367 |
|
10.00 |
10.00 |
0.000 |
-1.000 |
-1.000 |
TABLE 1: Time of travel
parameters for the scenario given where a spaceship travels out to a planet at
constant speed ,
relative to the earth, and returns back at constant speed
,
relative to the planet. In this example the planet has initial distance from
the earth of
light years and constant speed
away
from the earth.
Showing 6 Reviews
-
0Perhaps the most detailed review, most welcomed, was the second journal's third referee's report (J2R3). Beyond educational clarity, J2R3 raised 6 potential scientific objections. We will go through them, with our responses, in order. As usual, no quotes are used and all indications of the referee's criticism as paraphrased in a way we feel is fair. It is noted that the referee took into account -- quite fairly -- educational aspects of the paper in addition to its scientific aspects.1. Why choose -v as the return velocity. Isn't this arbitrary?Our reply to 1: The value of v was chosen to be both the outbound and inbound speed of the spaceship for simplicity. The referee is correct that the inbound and outbound speeds could have been decoupled. However, it seemed to us that all of the physics could be explored within the context of that relatively simple scenario. In deference to the referee, this point is now mentioned in the manuscript. Even so, the Section labelled "Planet Free Scenarios" gives another example where the outbound and inbound speeds are coupled differently.2. The discussion after Eq. 5 (the relativistic velocity addition equation), was confusing.Our reply to 2: Yes, we labored over this point. We chose to include Equation (5) as presented because this velocity addition equation is best known to students and teachers as written. In fact, in its present form, the velocity addition equation is practically an established icon of special relativity, and so leveraging this icon we felt bolstered the manuscript and connected it more directly to common textbook discussions. Still, in deference to the referee, we have now clarified the text after Eq. (5).3. The phrase in the paper regarding the spaceship moving backward in time is confusing.Our reply to 3: We have now removed this sentence from the manuscript. In fact, the entire paragraph that contains this sentence has now been removed as it was deemed too speculative and distracted from the direct logic of the spaceship trajectories derived in the manuscript.4. How do the superluminal ships start and stop? There does not seem to be a way. This concern is noted as the greatest of all of J2R3's criticisms.Our reply to 4: This objection is valid for all objects moving superluminally, and is mentioned in the Introduction. It is specifically noted there that accelerating massive objects from below to above c requires infinite energy and so is likely impossible. However, the manuscript follows many previously published discussions that take superluminal motion as a premise -- just so as to see what happens. Also, many previously published works (by others) consider that superluminal motion may not involve the acceleration of an object across the c boundary but rather employ tachyons that always move faster than light, and cannot slow down below the speed of light. In writing this manuscript we decided to follow these previously established conventions but not discuss them in detail as that was not the crux of the manuscript. However, at the advice of the referee, we have now made added more text making this point clear.5. What are the superluminal to subluminal transformation formulae? They were not given and are not obvious.Our reply to 5: Many other published papers have delineated the likely transformations between sub- and super-relativistic frames. This goal of this manuscript was not to elucidate these already established transformations, but rather to describe a closed loop time superluminal event as simply as possible. To create this simplicity, we chose to a single reference frame. The description of events in other reference frames is surely possible, but to do so in our work would necessitate a much longer and possibly less clear manuscript. However, at the advice of the referee, we now point out explicitly that Feinberg (1967) describes such transformations.6. Towards the end of the paper, a separate situation is brought up where a spaceship changes its velocity by -2v relative to its initial velocity. This could mean that both velocities of Eq. 5, the equation of relativistic velocity addition, are superluminal. It is not clear that Eq. 5 would be valid in this case.Our reply to 6: This is an interesting point. In general, we believe that the standard relativistic addition formula should remain valid even when both speeds are superluminal because this formula covers all other cases, and we have no reason to believe that it would be invalid in this case. To the best of our knowledge, no other formula has been suggested in the refereed literature. If the referee has another formula in mind, we would be grateful to be given a reference to it. To comply with this referee's concern, we now mention in the manuscript that this formula is only hypothesized to be correct when both speeds are superluminal.You must be logged in to comment
-
0
The second journal's second referee, J2R2, had no criticisms and said that the paper deserved to be published.
-
0The main scientific criticism of the second journal's first referee (J2R1) was that the paper was elementary and derived results that one would expect. Referee J2R1 further claimed that the supposedly novel idea of pair creation of objects was unoriginal and occurred in a book by Reichenbach titled "The Philosophy of Space and Time" published initially in 1920.We ordered this book from Amazon and looked through it pretty thoroughly, but located only one passage that appeared to be relevant. On pages 141-142, the concept of a closed loop time travel is introduced, but pair creation and annihilation events are not mentioned. Also the twin paradox does not seem to be mentioned in the entire book. We therefore added the interesting Reichenbach reference to the paper, but since the specific criticism did not pan out, did not consider referee J2R1's criticism reason for rejection.
-
0The first journal's second referee (J1R2) gives as their main point of criticism that the well known velocity addition formula for relativistic velocities, w = (u + v) / ( 1 + u*v/c^2), (referred to as Eq. 5) does not hold when either u or v is greater than c. Referee J1R2 thus advised the paper be rejected.Now this is an interesting point! True, we do not know for sure that superluminal velocities add in this fashion. However, we are unaware of any other postulated formula for superluminal velocity additions. The citations this referee gave lead only to papers that discuss relativistic velocity additions in three dimensions when both velocities are subluminal. Truly, this leads to very complex mathematics, surely interesting in their own right, but not, it seems to us, proof that Eq. (5) is incorrect for parallel velocities when one or both speeds are superluminal. A literature search has not yielded an alternative to Eq. (5).It might be of interest to note that Eq. (5) might even be testable. It is well known that illumination fronts such as shadows can move superluminally (see, for example, Nemiroff, Zhong, and Lilleskov 2016). Even though these rapid fronts carry no mass, they might be used as one of the velocities to falsify Eq. (5).
-
0This paper has been submitted and rejected by two mainstream physics journals. Over the next few weeks, we (the authors) will attempt to summarize most key points of the reviewers and give brief rebuttals. We have decided not to name the journals nor directly quote the reviewers.For the first journal, the first referee (denoted "J1R1" for brevity) noted that to their knowledge, the paper was technically correct, in that the algebra was not found to contain mistakes. However, referee J1R1 questioned whether the derived sequence of images merited the conclusion of backwards time travel, therefore questioning whether the inference of backward time travel was a phony assertion.In response to J1R1, we agree that the algebra is correct, but feel that the paper's main purpose is to detail just how backwards time travel results from superluminal motion. We do not see another interpretation of the algebra provided, and referee J1R1 did not give any details of another interpretation. Also, we went to great lengths to detail how the images visible to the Earth observer are not the complete picture, a point that was not addressed by referee J1R1.
-
0This paper appeared in original form on the arXiv at the URL http://arxiv.org/abs/1505.07489. Also, this paper was briefly discussed on the Backreaction blog here: http://backreaction.blogspot.com/2015/06/does-faster-than-light-travel-lead-to.html . Insight can be gained by reading that blog post and many of the resulting comments.Additionally, Russell wrote up a brief invited report on this paper for PBS's Nature of Reality blog here: http://www.pbs.org/wgbh/nova/blogs/physics/2015/08/can-you-really-go-back-in-time-by-breaking-the-sp... . There is significant discussion following that post, some of which is insightful. Because of these reports and others, the arXiv version of this paper has a current (2016 June) Altmetric score of 172.
License
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