Faster Than Light
While Einstein held that it was not possible to alter a causal chain of events, there persists
the view by many that relativity theory (“RT”) implies that travel faster than light – if such
were possible – would allow for time travel, or at least for a signal to be sent from the
future to the past. In this paper, I will show how such a conclusion can be arrived at
through an improper application of mathematics. The example I will use is set out in a book
entitled About Time, Einstein’s Unfinished Revolution by Prof. Paul Davies published by
TOUCHSTONE (Simon & Schuster) 1996 paperback edition (copyright 1995). I want to make
clear that I do not know Prof. Davies and do not have any personal issue with him or his
work. I simply stumbled across his example of time travel when researching my own
(unpublished) work: Time, Space and Relativity Theory (December 2009) from which the
following paper is derived. I contacted Prof. Davies shortly after finishing my draft in order
to give him the opportunity to review the critique of his time travel argument (that follows
below) but an associate told me the professor was too busy to take on such a task and
therefore he has not read the critique that follows, to my knowledge, and certainly has not
offered any rebuttal to it. My only objective here is to show how the mathematics of RT can
be misapplied and lead an incorrect conclusions. I do not wish to impugn Prof. Davies in any
way.
Similarly, I will first discuss briefly another example in a book entitled Spacetime Physics by
E. F. Taylor and J. A. Wheeler published by W. H. Freeman & Co. New York in 2003. Again, I
do not know these authors and they have not read the critique of their example of
theoretical time travel that follows below.
In this paper, I will not consider speculation about the possibility of faster-than-light speeds.
However, it should be emphasized that RT asserts that it is not possible to accelerate
anything or anyone to the speed of light – let alone beyond it. Yet, as we have seen, there
continues to be speculation about super-light travel and this is sometimes attributed to RT.
Some have speculated there may be particles (tachyons) that exceed light speed, but our
technology is not yet able to observe them or to measure anything moving faster than light
(for example, see Davies, p. 80). If we allow for the possibility of a faster-than-light entity,
then it might be possible to send a signal at super-light speed. This by itself would not be
time travel as I would define it since humans would not be able to travel along with these
undiscovered particles. But we should consider whether the ability to send or receive signals
at speeds faster than light using such particles would allow for a possible change in a causal
sequence of events. Davies asserts that tachyons “could be used to send signals into the
past (ibid)”. Further, if such particles are found to exist that exceed light speed then it
would not be an undue speculation to suggest that one day humans might be able to
develop technology to emulate them. So we should consider both possibilities – travel by
humans and by signals at speeds greater than c.
Let us first consider super-light signals. Clearly, if such signals were ‘visible’ to humans,
they would have a major impact on perception and knowledge. But would they allow for
time travel? Let us take an example from Superman movies. Aliens are traveling at light
speed to attack and destroy the Earth. Superman receives a super-light signal warning of
the attack and is able to fly out into space and prevent the attack. This is prevention but not
change in a cause-and-event sequence that has already happened. What would Superman
be able to do if he was away on another planet and received a super-light signal that Earth
had already been attacked and destroyed? No matter how fast he could travel, it would be
too late to save Earth. Similarly, if Superman was standing beside a villain who fired a laser
weapon at Lois Lane, he could fly at super-light speed and outrun the laser beam to sweep
Lois out of the way before she was fatally struck by the beam. Alternatively, he could send
Lois a super-light signal warning her about the incoming laser beam that she had not yet
seen.
But what if Superman was elsewhere and the first thing he saw was Lois being
struck by the laser? Even if he could fly at any speed, how could he undo her fate? Clearly it
would be possible to have situations where one could see an effect occur before its cause
with super-light signals but, still, that would not allow the observer to undo the cause-andeffect sequence. Seeing something after it has happened is, by definition, not the same as
being there when or before it happens. Consider that c is only a speed. How would it change
anything with respect to cause-and-effect or the flow of time if the maximum speed of the
universe was greater? Would c plus 1% make a difference? All that would happen is that
things would go faster. Also, something might reach us before we could see it coming. That
would be different, certainly, but I do not see how it could reverse the order of past and
future events or allow for any change in causal sequences, or time travel.
While there seems to be no answer to these questions and a causal sequence remains
inviolate even with the possibility of speeds exceeding light, there are those writing about
RT who suggest that time travel is possible and causal sequences could be reversed if only
we could exceed light speed. An example is found in Taylor & Wheeler, pp. 108-09. Again, I
am using this example not to single out Taylor & Wheeler for criticism but instead because
their diagrams and explanation in the text are so well detailed. Their objective in this
exercise is to show "why no thing travels faster than light":
Why does no signal and no object travel faster than light in a vacuum? Because if
either signal or object did so, the entire network of cause and effect would be
destroyed, and science as we know it would not be possible. (Taylor & Wheeler, p.
109, emphasis added)
The only proof offered by Taylor & Wheeler is based solely on the Lorentz equations so this
is a perfect example of mathematicism (my term for cases where mathematical formulae
take on a life of their own and drive the argument rather than simply describing observed
natural phenomena). Their example involves Klingons destroying a Federation vessel
(traveling at 0.6 light speed) by firing a super-light missile that can travel at three times the
speed of light. Four years after the Federation visits the Klingons to negotiate a peace
treaty, the Klingons launch the missile. It takes the missile another year to reach the
Federation ship which has by then traveled 3 light-years away from the Klingons (0.6c x 5
years elapsed time). From the Klingons perspective nothing is odd here regarding cause and
effect. However, when the time and distance variables are converted using the Lorentz
equations to create a "spacetime diagram of the departing Federation negotiators", the
causal sequence involving the missile becomes reversed. The first event ‘seen’ by the
Federation negotiators is their ship being blown up and the final event they ‘see’ (after
already having been blown up) is the missile arriving back at the Klingon base.
The silliness in this diagram is obvious in several respects. In the 'first' event, the
Federation ship is not seen to launch the missile back toward the Klingons; instead it is seen
to be destroyed by the missile along with the missile itself, so this first event could not
possibly be the cause of the missile traveling backward toward the Klingons. Then the final
event, the arrival of the missile at the Klingons, does not involve impact and destruction of
the missile and the Klingons but, instead, the launch of the missile is seen. Both of these
events involve physical processes in nature and Taylor & Wheeler are not claiming that
these processes themselves have been - or even are seen to be - changed or reversed.
Further, the Klingons still see things in the normal causal sequence while the Federation
negotiators supposedly have the opposite perception. So the cause-and-effect sequence of
events is not changed, it is only seen in reverse order by the Federation negotiators, even
though the very first thing they see is themselves being blown up so it seems unlikely they
would ‘see’ anything after that. So this example is no proof whatsoever that “the entire
network of cause and effect would be destroyed” if a speed greater than light speed could
be achieved. All this scenario illustrates is that the missile could outrun the light beams
carrying the successive images of the missile, so the negotiators would not see the missile
coming until it had struck them.
But this example also illustrates mathematicism. The second diagram is created using the
Lorentz equations: "Substitute these values into equations (L-11) to reckon the rocket
coordinates….(Taylor & Wheeler, p. 109)" The (L-11) equations are found on page 103, and
the first equation Taylor & Wheeler show, which is for calculating the time of event 3,
involves as its first term the negative relative velocity between the Federation ship and the
Klingons, which is -0.6c. By inserting a negative term for relative velocity into a
mathematical formula, Taylor & Wheeler imply the possibility of negative time which, in
turn, suggests a reversal of time order so that an effect could happen before its cause. If t
= d / V and if the distance covered is positive but velocity is negative, we would have
negative time, as -t = d / -V. Again, we must ask if the RT equations are meant to assume
that the time and space variables are amounts (scalar only) or whether they are supposed
to have a qualitative aspect (direction).
The misleading impression that RT presents the possibility of time travel or faster-than-light
travel is due in large measure to mathematicism, especially the appearance of the term –t
in various equations. This can lead to the derivation of the term –v, the term in the example
we have just examined, but the insertion of –x for a negative distance measurement can
lead to the term –v as well. But now consider what –t implies for the speed of travel over a
given distance, d. At the maximum speed of travel, c, according to RT, the formula for light
velocity is c = d / t. If we could substitute in this formula for the original value of t any
value less than t, then we would be saying we could cross the distance d at a speed fasterthan-light. But the term –t appears to mean less-than-zero time. So not only is –t less than
t, it is 200% less! Therefore, the speed possible in the formula V = d / -t is much faster
than light speed, to the point where it is not clear that this expression has any real
meaning. In terms Percy Bridgman might use, it does not have any real physical content.
Nevertheless, you can see how the insertion by a human of a negative sign in front of the
term for time, t, can lead to all sorts of speculations about time travel, time reversal, and
faster-than-light travel.
A better example of a misleading discussion about RT is found in Davies, p. 234-36, where
he attempts to show that faster-than-light travel can reverse time order. To be fair, it is
most important to observe that, in the first example (which is almost identical to the
Klingon missile scenario in Taylor & Wheeler we have just examined) Davies does not
suggest a reversal of cause and effect – his discussion concerns only what would be seen by
an observer. (Davies also makes the useful distinction between time reversal, where “the
arrow of time itself is inverted, making time “run backwards””, so things happen in reverse
order (they are not just seen to happen that way), and “travel into the past”, which “leaves
the direction of time unchanged, but involves somehow visiting an earlier epoch. (Davies, p.
234, emphasis added)” I would point out that neither of these phrases (“time reversal” and
“travel into the past”) are well-defined and that in neither scenario does the observer simply
‘see’ an effect event before the related causal event. However, we shall see that in a
subsequent example of faster-than-light travel involving twins, he concludes that there is
the possibility that a causal sequence could be changed in a situation where there are
super-light signals. I believe this conclusion is wrong and his text is quite misleading, and
we will see that his conclusion is based on a crucial error in the way he has applied the RT
mathematics.
First, Davies proposes a tachyon gun that fires faster-than-light bullets at twice light speed.
What an observer would see would depend on their own motion but, if they move “in the
same direction as the bullet at 90 percent of the speed of light”, their observations would be
similar to those of the Federation negotiators in the previous example. First the target
would be seen to shatter and then “the bullet would appear to travel backwards from the
target into the barrel of the gun. (Davies, p. 235, emphasis added)” While this conclusion
derives from the equations of relativity, Davies does not provide them for this example. Yet
we can discover several things on further examination. First, take note that to ‘see’ anything
in any situation of this kind we rely on photons of light and not on the tachyons. Next,
consider what would be seen by an observer located at the same position as the gun, who
remains there. The explosion of the gun would be seen simultaneously. Then, at each point
along its path, the tachyon bullet would emit photons in all directions, presumably, but at
least in the direction of the observer because otherwise the ‘bullet’ would not be seen at all,
which is contrary to what Davies describes. At the end of its journey, the bullet shatters the
target and photons from that event are emitted back to the observer. So what we have is a
sequence of events with photons emitted from each one, in sequence, and these photons
are received at the eyes of the observer in the same sequence, so absolutely nothing
unusual happens. None of the photons emitted later can overtake any photon emitted
earlier in order to be seen out-of-order by the observer. All that happens is the series of
events would take half the time compared to a bullet that could travel at light speed. The
same number of photons would be received by the observer and in the same order, but in
half the amount of elapsed time. This time difference is due to the speed of the tachyon
bullet that allows the photon emission events to occur twice as fast – but they still occur at
the same positions as for a light-speed bullet or an even slower bullet.
But the foregoing is not what Davies describes. His observer is in motion, and this is where
the mathematics of the Minkowski space-time interval comes into play, just as it did in the
Taylor & Wheeler example above. Note that this scenario is only concerned with
appearances and there is no suggestion that the cause-and-effect sequence might be
changed because of super-light speeds. “For faster-than-light motion, the time sequence of
events E1E2 is no longer fixed but can appear reversed. (Davies, p. 235, emphasis added)”
The reason is that, for an observer located near the target, photons will arrive from the
bullet shattering the target before the photons from the firing of the gun will arrive. The
tachyon bullets outrun the latter photons due to their super-light speed. Imagine a triangle
with the causal event at A, the effect event at B and the observer at any point O. The
observer will always see A before B because light from A must travel only the A~O distance
while light from the event B must travel the B~O’ distance only after light has traveled the
A~B distance, and it is clear that the A~B~O’ distance will always be greater than the A~O
distance since the speed to cross all distances is c. However, in Davies’ tachyon gun
scenario, the A~B distance is covered in half the time due to the super-light speed of the
tachyon bullet. This allows an observer located near B to see B before A. But there is no
effect on causation: “…the tachyons appear to travel backwards in time relative to normal
physical processes. (Davies, p. 235, emphasis added)” Note that Davies here implies that
the process of a gun firing at a target defines the arrow of time and to see such a process in
reverse, as a movie played backward, is equivalent to apparent travel backward in time.
Yet, so far, such things have never been observed in nature.
However, Davies next proposes a twin scenario where one twin, Betty, departs at 10:00
A.M. and the other, Ann, remains on Earth, and then they send signals to each other at 4x
the speed of light. The Earth twin sends her signal at noon, her time, and the traveling twin
sends a reply immediately upon receiving the Earth signal. The result is that the reply from
the traveling twin arrives back at Earth at 11:07 1/2 Earth time, 52 1/2 minutes before the
signal was sent from Earth. This also is an example of an effect appearing before a cause,
but it is qualitatively different from the tachyon gun example because it suggests that the
causal event sequence could be changed. What is to stop the Earth twin from deciding not
to send the signal at noon? This is the standard paradox that always arises whenever
‘closed time loops’ are proposed.
The first point to consider in this example is how the proposed speed for the signals greater
than c would change the kinematics in a situation where, according to the Special Theory of
Relativity (“ST”), a causal sequence cannot be seen in reverse order let alone undergo a
change in the event sequence. All we are doing here is substituting for the value of velocity,
c, some higher value. But the mathematical equations themselves do not change and
neither do the mechanics they describe. Only the speed of signaling is greater, and this
should not affect the order of events in a causal sequence. Of course, the order in which we
‘see’ things can be different than the actual causal order of events. Because we see using
light, and light has speed c, travel at speeds greater than c can allow for effects to be seen
before causes. As just described in regard to the imagined triangle A~B~O, this would allow
for the A~B~O distance to be traveled in a shorter time than the A~O distance it would take
light to travel from the causal event A to the observer at O, because the A~B segment could
be crossed at a speed much greater than light speed if there were a super-light signal. Then
the observer at O could see B happen before seeing A even though A caused B by emitting
the super-light signal. But seeing the events in reverse order would not empower O to
change this causal event sequence.
In Davies’ twins scenario, the specific speed of the signals does not matter so long as it is
greater than the speed of the traveling twin so that the signal from Earth can overtake her.
The first signal links the first event (sending the Earth signal) to the second event (receipt
of the Earth signal by the traveling twin), and the second signal links the third event
(emitting the reply signal by the traveling twin) with the fourth event (reception of the reply
signal by the Earth twin). The order of these events does not change, in spite of Davies’
mathematics. The speed of the traveling twin relative to the other twin does not matter as
long as it is less than the speed of the signal from Earth. Events 2 and 3 take place at a
single position in space, which we shall call point A, that is the point 2 light-years away
reached by the traveling twin at the instant the signal from Earth arrives.
Davies carries on about the two different reference frames of the twins, their clocks, and the
time dilation factors, but it is all irrelevant. This is a one-clock, single observer situation
similar to the observer I have described above who remains at the site of the tachyon gun.
The Earth twin sends a signal out and gets a signal back, like sending a light beam to a
mirror and having it reflect back – it does not matter at what speed the light beam travels
to the mirror and back, it arrives back later than it was sent.
So how is Davies able to conclude, based on the mathematics of RT, that the return signal
will arrive before the outgoing signal is sent? This is another example of mathematicism –
Davies allows the mathematics to take over and drive his thought experiment. He applies
the relativity equations to show how things would look to the twins who are the observers in
two separate reference frames even though only the Earth twin’s observations matter in this
case. After all, the function of the Lorentz transformation is to convert the observations of
one reference frame into those of another. The same is true of the Minkowski equations for
calculating the interval. So Davies is using these equations to translate terms for time and
distance between the reference frames of the two twins. For example, he writes: “The
distance from Earth as measured in Betty’s reference frame will also be different.” (Note
that here, as in all other such examples by various authors, there is no suggestion of any
kind as to the method by which either twin would be able to “measure” the distance
between them.)
But whatever distances the twins might “measure” does not matter to the signals, and it is
the signals that are in motion. They travel over only one specific distance, and they define
that distance, not the twins. Assuming Earth’s position is relatively fixed, the first signal
travels from Earth to point A and the second signal travels from point A back to Earth and,
since these two points are the same for both trips, the Earth~A distance is equal to the
A~Earth distance as far as the signals are concerned. Further, the Earth~A journey happens
first and the A~Earth trip occurs immediately after: there is no overlap and no gap in
between. So the total distance traveled by the signals is one amount of distance that is
Earth~A~Earth. That distance can be determined by multiplying the time elapsed on the
single clock located at both the first and last events (the Earth clock) by the speed of the
signals, which Davies proposes to be 4x the speed of light. But his conclusion is that the
elapsed time is negative and, since the distance is clearly positive, no matter what the
amount of it, he would have to conclude that the speed of the signals is negative, which is
absurd and not the positive speed he has stipulated: positive distance, d, = -t times - v.
Davies asserts, and purports to illustrate, the following:
…the outgoing signal travels into the future, and the reply travels into the past. By a
suitable arrangement of speeds, the reply can get back to Earth before the original
signal is sent. (Davies, p. 235, emphasis added)
And he concludes:
…the return signal arrives on Earth 52 ½ minutes before Ann sent the original signal.
(Davies, p. 236)
The explanation for Davies’ error is that his calculations show not what the Earth twin would
see on her clock, which is what his conclusion above actually states, but, instead, what the
Earth clock would look like to the traveling twin! “…in Betty’s frame, it is Ann’s clock that is
dilated, by a factor of 0.6. The 1 7/8-hours’ total time as told on Betty’s clock therefore
translates into 1 7/8 x 0.6 = 1 1/8 hours on Ann’s clock, which therefore reads 11:07 ½
A.M. (Davies, p. 236)” But here he omits a most important qualifier at the end of the
sentence, which should read something like “as seen by Betty” since he is writing about
what is seen from Betty’s frame of reference. In fact, he never does get back to Ann’s
reference frame on Earth to calculate how she would observe the arrival of the reply signal,
yet his conclusion clearly seems to say that Ann sees the reply arrive before she sends the
original signal.
This is just so much playing with numbers that have been abstracted from the
(hypothetical) real world situation that has been proposed. Among the various unreal
aspects of this example is the puzzle about how the traveling twin light-years out in space
could measure the distance she had traveled or ‘see’ the Earth twin’s clock at the time the
reply signal arrives. What Davies really means to show is what the traveling twin would
calculate as the time showing on the Earth clock when the signal arrives. (The physics
seems to show she would be in error.) But this is only a nuance and should not detract from
the main point of this exercise which is that one can get tangled up in mathematics, make
errors, thereby arrive at serious misconceptions about physics, and attribute these
misconceptions to RT. Clearly, it is wrong to suggest to readers that, based on the ST
equations, a faster-than-light signal can travel back in time and potentially change a causal
sequence of events. “…perceiving something is a way of being causally affected by it. So, by
the causal analysis, what one perceives must always occur earlier than one’s perception of
it. Perceiving the future would be an instance of backwards causation…(Roger Le Poidevin,
Travels in Four Dimensions Oxford University Press 2003, p. 219)” So Davies cannot be
correct to say that “the reply travels into the past”, unless he means to assert the possibility
of reverse causation (which thesis I oppose and he has not proven). Of course, it is not
wrong to say that with super-light speeds we may be able to see an effect before seeing its
related cause if our position and state of motion are such that we are at a distance from the
causal event and closer to the effect event (as in the case of the tachyon gun), but this is
quite a different conclusion than the one made by Davies discussed here.
There remains a loose end from the foregoing: if Davies’ calculations based on the RT
equations are correct and the time showing on the Earth clock when the reply signal arrives
as determined by the traveling twin is as Davies describes even though this is not physically
possible and the Earth clock, showing proper time, will be different than the time calculated
by the traveling twin, must we conclude that Davies has made an error or that the there is a
flaw in the RT equations? Clearly something is wrong here because the equations are
supposed to translate the time co-ordinate for a given event in one frame into that same
co-ordinate as it would be measured in the other frame. Davies calculates that when the
reply signal arrives on Earth, the time will be 11:52 ½ A.M. in Betty’s frame but it will be
11:07 ½ in Ann’s frame on Earth. So, if the first time is what Betty actually sees on her
clock when the event occurs, the second time must be what Ann sees on her clock,
according to the equations, and not simply the time Betty sees on Ann’s clock (assuming
that were even possible). In other words, the equations translate between the observations
of two observers for the same event.
First, let us see where the twins are in agreement. They both agree on the departure time
of 10:00 A.M. They both agree, presumably, on the value of the constant c, the velocity of
Betty’s rocket of 0.8c, and the speed of the signals of 4 x c. So first we can say what
happens from the point of view of the signals and, since Ann is situated with her clock at the
point of emission of the first signal as well as the point of return of the reply signal, her time
is also the time for the signals and it is the ‘proper’ time. It is clear from the foregoing
assumptions that by the time Ann sends the first signal at 12:00, Betty has traveled 1.6
light-hours (2 hours elapsed time times 0.8c). At 4x light speed, the signal will cover this
distance in 0.4 hours. But Betty is still moving at 0.8c during the signal’s journey. It is
simple mathematics to determine that, at this speed, Betty will be overtaken by the signal
at point A at 12:30 Earth time since she will have traveled 2 light-hours by then (2.5 hours
elapsed time times 0.8c = 2 light-hours) and during the half hour between 12:00 and
12:30, the outgoing signal will have traveled the same distance (0.5 hours x 4c = 2 lighthours). If Betty transmits the return signal immediately, the signal has to retrace the exact
same distance traveled by the first signal (this assumes no motion by the Earth) and it does
so at the same speed. This takes another half hour. Therefore the reply signal arrives at
Earth at 13:00 as shown on Ann’s clock.
How then does Davies determine that Betty will calculate the arrival time showing on Ann’s
clock as 11:07 ½ A.M. and not 13:00 P.M.? He uses the time dilation formula. With 0.8c as
the term for relative velocity, the time dilation factor is 0.6 as calculated from the Lorentz
formula: square root (1- V2/c2) = square root (1 – 0.8x0.8 / 1) = square root (1 – 0.64) =
0.6. Therefore Davies asserts the 2.5 hours of elapsed time in the Earth reference frame is
only 1.5 hours for Betty’s frame. Therefore, at speed 0.8c, she has traveled only 1.2 lighthours and her clock shows a time of only 11:30 A.M. With an immediate reply, Davies
calculates that the return signal will take 22 ½ minutes to get back to Earth. Here Davies
makes a small error – he assumes the signal will also have to travel the distance that Betty
will cover as she moves past point A but that is not correct. The return signal only has to
retrace the A~Earth distance. In Betty’s frame, that is calculated to take 18 minutes. That is
because the distance Betty ‘measures’ from Earth to point A where she received the first
signal is 1.2 light-hours and at 4c the signal would take 18 minutes to travel this distance
(1.2 / 4 = 0.3 and 0.3 hours = 18 minutes). Since Betty receives the first signal at 11:30
A.M. her time (according to Davies), the return signal will arrive at Earth at 11:48 A.M. her
time (and not 11:52 ½ A.M. as calculated in error by Davies).
Already, we have a time that is earlier than the 12:00 time on Earth when the first signal
was emitted. However, Davies goes on to assert that because Betty sees Earth in motion
rather than herself, she applies the time dilation factor to the elapsed time on her clock,
which is 1.8 hours (1.5 hours + 0.3 hours = 1.8 hours, not the 1 7/8 hours calculated by
Davies). That means, as calculated by Betty, only 1.08 hours of total elapsed time would
show on the Earth clock meaning an arrival time for the reply signal of 11:04 4/5 A.M. (not
11:07 ½ A.M.). So this explains the mathematics, but it is still only what Betty calculates in
her frame of reference and not what Ann (or Betty) actually sees. The signal does not
actually arrive before it is sent. We can prove this using Davies’ own method and the same
set of numbers.
If Betty receives the first signal at 11:30 A.M. her time and calculates that the signal
traveled 1.2 light-hours (her distance measurement as asserted by Davies) and everyone
agrees that the signal travels at 4c, then it must have been emitted 18 minutes before
11:30 A.M. Betty’s time, or 11:12 A.M. That means an elapsed time since Betty’s departure
of 1.2 hours her time. Applying the same time dilation methodology described above that
was used by Betty to calculate the total elapsed time on Ann’s clock of 1.08 hours, we
derive an elapsed time between Betty’s departure and the sending of the first signal of 0.72
hours (1.2 x 0.6 = 0.72), which is 43.2 minutes. Therefore the first signal was sent not at
12:00 on the Earth clock but instead was sent at 10:43 1/5 A.M. Earth time, as calculated
by Betty using exactly the same calculation procedure Davies used to show the reply signal
arriving back on Earth at 11:04 4/5 A.M. By using the same procedure consistently in only
one frame, we have shown that the first signal was sent well before the reply signal
returned to Earth. As long as we compare the times of the events as seen or calculated
within the same frame of reference, the causal event sequence does not change for any of
the frames (Ann’s, Betty’s, or the signals), no matter what speed we assume for the signals.
But there is a more fundamental point to make about this exercise. Why does Betty
calculate an incorrect reading for Ann’s clock, which means the Lorentz transformation failed
to work properly here? The answer is that, like all the twins paradox scenarios, this is not a
typical ST situation with two inertial reference frames in relative motion. The twins start out
together in the same frame but then one of them is accelerated. This acceleration causes an
actual physical change in the run rate of Betty’s clock, as I believe the RT requires, and as
reflected in Davies’ assertion that Betty reads only 1.5 hours of elapsed time on her clock
between departure time and the arrival of the signal from Earth. So, while Davies has made
some mistakes in his calculations, he nevertheless has correctly assumed that RT requires a
physical change in Betty’s clock due to acceleration, meaning a slowing of its run-rate
relative to Ann’s clock, which change is real and not simply apparent (apparent here
referring to how Betty’s clock would ‘appear’ to Ann in her reference frame on Earth).
Furthermore, Davies is wrong to say that Betty can assume that she is the one at rest and
the Earth is the frame in motion because Betty can perceive her own motion due to the
acceleration she experienced in departing from Earth in her space ship.
As always, I welcome any comments or rebuttal.
C. Terrence Fisher
February 2017
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