Twin Paradox in Relativity
Explained and Diagrammed
in Absolute Terms
with a twin paradox animation
by Roger Luebeck
Cite the book, Relativity Trail
Luebeck, R. Relativity Trail. Mpls: L B Writ Publishing, (2008)
Cite this web page
Luebeck, R. (2010, Sept 10). Twin Paradox in Relativity.
Don't look elsewhere for literature that resembles what you read on this page. This document and the book Relativity Trail stem solely from the author's own contemplations and derivations. Relativity Trail is completetly consistent with, and subsumes, Einstein's treatment.
At the end of the kinematical section of his paper on special relativity, Einstein noted a "peculiar consequence" stemming from his analysis - that of a time differential (i.e. time-keeping differential) which would occur between reunited clocks.
Over the course of his remaining fifty years, Einstein offered no explanation for that time-keeping differential (which came to be known as the twin paradox).
Einstein's treatment was limited to assuming (thus assigning) symmetrical measures across inertial frames, such that his two postulates of measure would be satisfied.
Einstein's paper was published in 1905. As the years passed, a point of view took hold among most commentators that these symmetrical measured effects of relativity were confirmation that uniform motion is purely relative, and that there is therefore no meaning to be attached to absolute uniform motion, and therefore of course, to actual differences in clock rates, etc. In other words, they like to say that there "is no truth of the matter" regarding actual clock rates (or speed or length or mass).
The actual time-keeping differential between reunited clocks, a physical reality which necessarily favors one party over the other, does not fit with that interpretation.
As noted even by John A. Wheeler (a promoter of the purely relative approach), special relativity developed in absolute terms is completely consistent with Einstein's purely relative development, specifically including symmetrical measures across inertial frames and the consistent measure of light speed in all inertial frames. Wheeler notes that there is no physical experiment which can distinguish the absolute treatment from Einstein's purely relative treatment. [Wheeler, J., Taylor, E. (1992). Spacetime Physics, second edition. W. H. Freeman: New York, p. 88.]
As noted above, the actual time-keeping differential between reunited clocks, a physical reality which necessarily favors one party over the other, does not fit with a purely relative interpretation.
Yet, countless commentators on special relativity, including the eminent John Wheeler (the author of the above-mentioned spacetime "bible"), have claimed to explain the time-keeping differential without acknowledging an actual difference in clock rates.
Every one of those explanations falls into one of two categories, both incorrect:
1. The "inertial force" or "acceleration" explanation. For starters, neither force, nor identically acceleration, is allowed in special relativity, which addresses purely uniform linear motion. This is seen in Einstein's paper, as well as in all subsequent derivations of the Lorentz transformations. The time-keeping differential is deduced through purely linear uniform motion considerations. (The correct paradigm for linear motion studies is the transfer of clock information across inertial frames; and of course no force could possibly affect the simple act of starting a clock as an inbound astronaut passes an outbound astronaut, causing hundreds of years to disappear in a twinkling.) 2. The "lines of simultaneity" space-time explanation. This is identical to the "space-time diagram", "kink in space-time", "jump in time", "misperception", and "lattice of clocks" explanations. They are all one and the same explanation. These commentators are not aware that the construct known as space-time is dependent on Einstein's clock synchronization, a clock synchronization which is not required to deduce the results of special relativity, and which vacates, or neutralizes, the universal frame of reference. Similarly, Einstein's "relativity of simultaneity" vacates the universal frame of reference. It has only to do with pseudo-simultaneity, as in perceived simultaneity. All the results of special relativity can be explained and diagrammed independent of Einstein's clock synchronization against the backdrop of the universe in an absolute sense. ----------------------------------------
Journalists, and even physicists, routinely speak of one clock running slower than another, as though it is (correctly) an actuality. However, as we've pointed out, virtually every single physicist will at some point claim that one clock does not actually run slower than the other, rather that each clock's reality is as valid as the other's (due to the symmetrical measures across inertial frames), and then attempt to explain the time-keeping differential by way of pseudo-simultaneity argumentation. These authors cloak their ignorance with meaningless verbiage. Not surprisingly, this leads to a great variety of explanations with a great variety of meaningless verbiage.
Space-time is a useful calculation tool, yet it is a construct – entirely dependent for its existence on Einstein's particular method of clock synchronization. The best one can do with a space-time argument in the context of the twins paradox is to note that a traveler, upon his turn-around, will observe a jump in the reading of the clock time of the stay-at-home using the "lattice of clocks" method dictated by Einstein's particular clock synchronization.
But of course no such jump in anyone's aging (or identically on anyone's clock) actually occurs.
In contrast with the space-time argument:
When an outbound traveler transfers his clock reading to an inbound traveler, the incremental increase of the time differential between the travelers and the stay-at-home is obvious – as the participants can plainly see upon comparing notes later on. (Consider multiple iterations of this scenario, with ever longer round-trip journeys.)
Similarly, when a stay-at-home "A" and a traveler "B" check the status of each other's clock with the regular sending of radio signals, they note an incrementally increasing time-keeping differential, with the asymmetry being first noted with the signals emitted by B at B's turn-around point. (Although the assymetry is first noted by both parties beginning with the signals emitted by B at B's turn-around point, the difference in clock time between the two parties builds continuously throughout the journey. See the book, Relativity Trail, for the complete analysis.)
[Luebeck, R. (2008). Relativity Trail, L B Writ
Publishing. Mpls, pp. 12-14]
Space-time is a construct, dependent on an optional clock synchronization method, and generates a false jump in time reading.
Space itself is not a construct, nor is time-keeping. And a real-world experiment reveals a real-world incremental increase in the time-keeping differential.
Why should we expect anything different? The stay-at-home twin does not age several years in a twinkling just because some distant inbound astronaut starts his watch as he passes an outbound astronaut.
It is precisely one's motion with respect to the universe – and to the same end, one's speed as a percentage of the speed of light – that dictates that party's actual clock rate, resulting in the actual time-keeping differential which is seen upon reuniting with the other party.
Any attempt to explain an actual time differential while forbading "the actual" is doomed to fail in a purely logical sense; in a purely mathematical sense. Einstein's clock synchronization, and therefore also space-time, in fact vacates the actual as it vacates any sort of universal frame of reference.
Einstein's clock synchronization method (tB - tA = t'A - tB) dictates equal time passage for a ray of light, whereby spatially separated clocks of the same inertial frame record equal time passage for a light ray moving in either direction through the particular frame, even though the particular frame has non-zero speed relative to light speed. It does so by necessarily disregarding the physical nature of those clocks, and by disregarding the actual constancy of the speed of light.
Thus that equation neutralizes "the actual" – the physical reference frame of the universe. Such clock synchronization leads directly to the "lattice of clocks" methodology which is in lock step with space-time diagrams. It has the "jump in time" built in for any situation involving a change of inertial frame.
It is in a space-time diagram that we find the infamous sudden shift of a line of simultaneity. The further away (or the higher the overall speed) the inbound and outbound astronauts are from the stay-at-home when the "sudden turn-around" occurs, the greater the magnitude of the "jump in time" as dictated by the "lattice of clocks". It is the so called "misperception" explanation of space-time diagrams.
Space-time, being dependent on Einstein's clock synchronization, is simply a geometrical construct. Those who wrongly think of space-time as a physical reality will write, when attempting to explain the time-keeping differential, that we “travel” through space-time or that we take “different paths” through space-time; and this despite the fact that they describe the time differential as a sudden jump due to a sudden shift of pseudo-simultaneity.
Whether they see such “travel” as traveling through time or traveling through time-keeping is hard to discern. In either case, it's not rational.
We can travel through space. But to say that we can travel through time in a pliable manner would imply ungluing ourselves from the march of time, in the sense of the "march of history". The most one can realistically say about our relationship to (historical) time is that we are carried along with time in an unpliable manner in accordance with time. It is our travel through space alone that is actually pliable.
Time-keeping (mechanical clocks or biological aging) on the other hand is pliable. Our aging will slow when we are in motion relative to the totality of the universe (meaning that our aging is virtually always slower than what it could maximally be).
Note that the purely relative approach incorporates a hopelessly circular definition of inertial frame:
In Spacetime & Electromagnetism, Lucas and Hodgson, using the space-time paradigm, wrestle with the twin paradox for fifteen pages, and claim no resolution.
In a footnote on page 73, they write: "Is it fair to give the Earth-bound twin the vertical world line? Does not that beg the question in his favour? Why not draw another diagram with his world line set at an angle to the vertical, and his lines of simultaneity correspondingly inclined (but at a contrary angle) to the horizontal?" [J.R. Lucas & P.E. Hodgson, Spacetime & Electromagnetism, p 70-84.]
That alternative diagramming of the situation is identical to the consideration that one might just as well consider that it is the Earth, along with the entire cosmos, that changes inertial frames. That, in fact, is something we hear often from commentators on relativity, as they try to make their case for "no truth of the matter". Of course, it simply makes the twin paradox unresolvable, as either party can then lay equal claim to being the party that ages the least. This is demonstration that one must consider the entirety of the cosmos, the imparter of inertial properties, to be the judge of the matter regarding actual motion.
As another example of space-time madness, consider that the renowned physicist, John A. Wheeler, in his book Spacetime Physics, made two "space-time attempts" to resolve the twins paradox and failed each time.
After claiming to have solved it on page 131 of his book, he writes on page 170 that he will “finally! solve it”. But he remains trapped in the universe of pseudo-simultaneity and the lattice of clocks, and ends up mocking his own failed attempt.
He never stood a chance:
He failed to have an outgoing astronaut transfer his clock reading to an incoming astronaut, which would have plainly revealed the true incremental increase of the time differential.
Instead, he was constrained by Einstein's clock synchronization and the lattice of clocks methodology, which simply assigns an incoming astronaut a time reading for the earth clock that represents the entire ultimate time differential; all in one fell swoop.
Thus Wheeler has his astronaut proclaim – “as I turned around, a whole bunch of Earth clock ticks went from my future to my past. This accounts for the larger number of total ticks on the Earth clock.”
Wheeler continues: “The astronaut renounces her profession and becomes a stand-up comedian.”
Wheeler has at that point twice failed to resolve the matter.
In a footnote, he refers his readers to an old journal article as perhaps a way to place a stamp of legitimacy on the nonsense.
The article is "The Clock Paradox", American Journal of Physics, Volume 31, (1963). See page 59 of that article. Even though its author, Edward Lowry, in his informal and incorrect verbiage, feels compelled to claim “acceleration” for the returning twin, Lowry in fact, in his diagram and further discussion, specifies an instantaneous turnaround “at the event B”, and of course cannot provide any rational explanation for the ultimate time-keeping differential. Lowry instead attributes the time differential to changes in simultaneity as perceived by the parties involved.
It is simply the sort of nonsense that Wheeler had just engaged in, being devoid of clock reading transfer and devoid of acknowledgment of actual differences in clock rates.
Wheeler writes in that same book:
“Does something about a clock really change when it moves? Absolutely not!”
This is the very same John A. Wheeler who is pedestalized by the physics community. Clocks, including biological clocks, change fundamentally upon experiencing a change in motion.
Regarding this time-keeping differential, we'd like to make an even simpler argument, whereby we consider a gent A orbiting the earth twice for each orbit completed by a gent B at the same altitude, and then have them compare their recorded clock times each time A passes by B. But in practice that's not possible since orbital altitude dictates orbital speed.
However, why not let gent A orbit at a radius less than that of gent B's orbit and have them simply compare their recorded clock times once back together on Earth. Since, in this case, the gents would also have clock rate differences due to their differing accelerations and differing gravitational field strength, one would need to allow for that.
Such studies have, in effect, been carried out, and the kinematical (special relativity) portion of the time differential can be separated out from the total time differential. We find that the kinematical effect slows the clock of the faster moving gent A (in his lower orbit) more than it does that of gent B's clock.
What this example accomplishes is to eliminate any sort of turn-around or transfer of clock information across inertial frames, clearly illustrating the incremental nature of clock slowing due to what are clearly actual differences in speed.
The actual difference in recorded time due to what is clearly an actual difference in speed precludes any speculation about suddenly shifting "lines of simultaneity" or "coordinate hyperplanes", and you can use such an example to niftily refute arguments made by anyone who claims that differing speeds are merely relative.
The time registered on a person's clock is dependent on the combination of their speed relative to light speed and distance covered in absolute terms. The time contraction formula is not linear. Thus, the party that changes inertial frames will be the party whose clock registers the least time over the course of a "round trip". This is seen with clarity when everything is charted out in absolute terms, as we do in chapter one of Relativity Trail.
The indisputable actual time-keeping fluctuation, in conjunction with actual states of uniform linear motion and actual length contraction, produces all the symmetrical measured effects of special relativity as we know them in Einstein's treatment. Such actualities imply an absolute frame of reference, albeit an experimentally indiscernible one.
One need not, and in fact should not, consider the existance of an immutable aether. Nor should one consider the concept of "absolute space". Rather, the totality of the cosmos, the very thing that imparts inertial properties to objects, serves as the absolute frame of reference. In other words, we need to consider a system at rest with respect to the totality of the universe. (i.e. - a system at rest with respect to the barycenter of the universe.) In fact, general relativity provides us with such a heuristic model of continuity.
And to the same end, the actual constancy of the speed of light provides us with an absolute frame of reference.
It is meaningless to ascribe linear motion to the whole of the universe itself, considering that the universe is the baseline by which both accelerated motion and uniform linear motion acquire meaning. It is fundamentally Machian.
Although the twins paradox stemming from uniform motion is strictly an effect of special relativity, where no inertial force is involved, consider that A. P. French writes on page 150 of Special Relativity: “Note, though, that we are appealing to the reality of A's acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twins paradox exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large.”
In Mach's Principle, an object is affected by a change in motion relative to the matter of the universe at large. But such an effect cannot occur unless the object is in a relationship with the matter of the universe at large regarding its initial state of motion to begin with.
Remember, an effect due to a change in motion is not simply an "either or" effect, rather it is an effect of degree based on "degree of change". No effect of degree stemming from a degree of change can occur unless there is an effect based both on an initial state of motion and final state of motion. Put another way, a change in motion in the context of the universe is not meaningful without an initial and final state of motion in the context of the universe; and both those states are necessarily uniform states of motion, necessarily different from each other in an actual sense.
Mach himself regarded the matter of the universe at large to be an actuality, and the effect on the object to be actual. We can't have it both ways. If the matter of the universe at large is a reality which has an actual relationship with an object concerning a change in motion, then so too is it a reality which has an actual relationship with an object concerning motion itself, i.e. – the initial and final state of motion, related to each other by the act of change in inertial motion.
And again, our observations (measures) are something fundamentally different from the underlying reality – a reality which generates our observations (measures), which can occur only at light speed.
In special relativity, Einstein was able, in a pseudo manner, to make an absolute frame of reference superfluous by postulating only the constant measured speed of light. In general relativity, he made, in a pseudo manner, Mach's Principle (somewhat) superfluous by again holding to only a postulated measured speed of light.
The notion that there is no absolute frame of reference or actual differences in clock rates has caused widespread wonderment and confusion among physicists, who regularly state that the symmetrical measured effects of relativity, along with the "relativity of simultaneity", are confounding, deeply mysterious, and impossible to understand.
Two brief examples:
Physicist Banesh Hoffmann writes in Einstein, Creator and Rebel p. 76-78:
“Though [relativity of simultaneity] may be shocking, we have to learn to live with it. ... the logical consequences [of Einstein's postulates] are often such as to outrage common sense.”
No shock or outrage to our common sense is to be found in the context of the universal frame of reference, and there is nothing we need to “learn to live with”. We would be shocked only to learn that there is no time-keeping differential between reunited clocks, which of course is one of those logical consequences to which Hoffmann refers.
Without such concept of an absolute frame of reference, one is left with a clock paradox of one's own making, which cannot be resolved in any rational manner.
One is also left with no means to account for
the symmetrical assessments across inertial
frames as we know them in relativity.
The renowned physicist John A. Wheeler, writes in Spacetime Physics that relativity of simultaneity is “strange” and “difficult to understand”. Yet, Einstein's definition of simultaneity can be easily diagrammed against the rest frame of the universe, as can his clock synchronization method.
Once again, Einstein's definition of what constitutes a synchronization of two spatially separated clocks of the same inertial frame amounts to a disregard of an analytical incorporation of an absolute frame of reference.
In keeping with this, he utilizes inertial frames to which he arbitrarily assigns the status of "stationary" and "moving". His treatment does not address the question of which clock is actually running slower or faster over any interval of the analysis, nor, identically, the question of which entity's measuring rod is actually shorter. Symmetrical assessments across inertial frames are assumed, without any hope of diagramming the process. Over the course of Einstein's derivation, certain measures must simply be assigned to the entities involved for the sake of satisfying Einstein's postulates of measure.
In the end Einstein concludes, much to his surprise, that there is a time differential between reunited clocks; but with the absolute frame of reference neutralized by his methods, he cannot explain the missing time.
Why is it called a paradox? As we noted at the beginning of this introduction, Einstein's relativity is almost universally treated as though it precludes any actual difference of length and clock rates between different inertial frames. This leads immediately to the state of mind that "there is no truth of the matter" concerning inertial frames. That in turn, creates a seemingly paradoxical situation: Two reunited clocks do show an actual difference in recorded time, as though there must have been a "truth of the matter" regarding their clock rates as they moved uniformly.
But Einstein's treatment does not preclude such actual differences of clock rates. In fact, his postulates demand it, as he should have noted at the conclusion of his derivation. By extension of logic, the famous experiments performed around the turn of the century which drove Einstein's postulates also demanded it, if carried to their logical conclusion.
At the conclusion of Einstein's kinematical section, where he noticed the “peculiar” time-keeping differential between reunited clocks, he should have realized that his methods were obscuring the reality underlying the symmetrical measures across inertial frames.
Einstein's confusion regarding a universal (absolute) frame of reference versus purely relative frames of reference is plainly evidenced in his 1905 paper on special relativity, "On the Electrodynamics of Moving Bodies", submitted to Annalen der Physik.
In his initial wording, his second postulate states that “light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”
With the word "definite", Einstein implies that light has an absolute (actual) speed in reality. But he doesn't explicitly state that there is a physically defined universal reference frame against which light has this definite velocity.
Three pages later, when he restates this postulate, he uses a new wording which changes the meaning considerably:
“Any ray of light moves in the "stationary system" of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.”
Here he replaces "definite" with "determined" and uses quotes around stationary system. Thus he is already hedging on what he means by this postulate. To whatever extent he favors this second wording, he abandons the absolute character of his postulate as initially worded, indicating he is already preparing (with an eye on the results he anticipates) to abandon the very reference frame which could have brought clarity to his treatment.
Einstein spent the rest of his life avoiding the dilemma which he never resolved in his own mind.
That does not mean that Einstein's special relativity is not valuable. Einstein's treatment succeeds in predicting observations of electrodynamical phenomena.
When presenting relativity from the God's eye view, it is obvious that there is no clock paradox. From the God's eye view, light is regarded as having constant speed in an actual, not merely measured, sense.
Thus Einstein's second postulate is so replaced. Or we might say we are restoring Einstein's initial wording of his second postulate.
An actual difference in clock rates follows immediately from this postulate of the absolute nature of light, provided of course, that one assumes that photons are the maker of every relationship (specifically here, the regulators of atomic functioning; but also the carriers of force information and our means for perceiving events).
The consideration of photons being massless particles, along with the consideration that mass and energy are interchangable, serves as the basis for postulating that light has an absolute speed and is also the limiting speed, with the photon having the property of existing in the form of pure energy. The preceding properties of photons and matter were actually known prior to Einstein's theory. Einstein himself introduced the notion of light existing in the form of a quanta of energy.
Einstein's first postulate, which itself pertains only to measures of properties, is the Galilean Principle of Relativity extended to electromagnetism. As seen in the Michelson-Morley experiment, as well as in Galilean mechanical demonstrations, the Principle of Relativity is inextricably bound with synchronization of motion along different axes.
Einstein's first postulate is replaced, in the Gods' eye view, with the notion that there is actual synchronization at the base of our physical structures, for the sake of their stability.
The Machian notion of a particle's relationship to totality, along with absolute light speed, combine with the need for atomic synchronicity to explain actual length contraction of rigid bodies. This is because photons (or virtual photon events) are considered to be the fundamental agent of communication within atoms, maintaining the organized structure of the atom. It is the equivalent of the Michelson-Morley paradigm, but on the atomic scale. Implicit here, is that particles have both a translatory relationship with the universe, and a rotational/orientational relationship with their translatory path, in the Machian sense.
When light rays and the motions of objects, along with their photon clocks, are charted out against the absolute reference frame of the God's eye view, all the symmetrical measured effects of special relativity fall into place, including of course, consistently measured light speed in all directions, in all inertial frames.
There is no clock paradox from such a vantage point. It is seen with clarity why it is that the clock whose change of inertial frame brings the clocks back together is the one that records the lesser time over the course of the round trip.
Again, no aether need be considered in such a treatment, rather a structure which is made up of an evolving set of relationships between all elements of the universe, in keeping with the nature of general relativity.
Light is the maker of every relationship in this structure. The photon is the agent of communication of forces and of positioning information (virtual photon event). This structure is obviously not fixed, rather evolving. The sum total of the structure, identically with its point or points (as in the inflation case) of departure is regarded as being at rest.
Although the universe is expanding, special relativity easily holds true at any conceivable scale of distance at which we might test the theory. (It's hard to imagine it failing at any scale of distance.)
The description of this structure most simply incorporates the notion of a point of departure (the Big Bang) within the spatial dimensions of a Euclidean universe. No meaning can be attached to any movement of such a point of departure. But in the context of the notion of a universe of inflation, Euclidean or non-Euclidean, the description is fundamentally the same. In this latter case, the origin of the universe lies outside its spatial dimensions, and the membrane of space expands due to inflation along that other dimension. There are then any number of points of departure for motion with respect to the membrane of space. In this latter case, it is obvious there would be no single center point of the universe within its own spatial dimensions. But even in the former simple case, such an actual center point would be experimentally undetectable. This is detailed in Relativity Trail.
The relativity of Relativity Trail is fundamentally different from the relativity of Lorentz:
1. Lorentz embraced a fixed structure of space – the aether, with its problematic one-way autonomy. 2. Lorentz did not define time-keeping (clock functioning). (Nor did Einstein.) 3. Lorentz provided no clear basis for why clock rates should slow, or why the aether wind should cause rods to contract the needed amount to satisfy the Principle of Relativity. 4. Lorentz provided no diagrams of the measuring processes. (Nor could Einstein.)
The illustration above and the two-part animation below illustrate two basic possibilities for completing a round trip between two parties. They produce identical time differentials, demonstrating the impossibility of anyone determining their true motion status relative to the universe.
i.e. - none of the parties involved can assume anything about their actual state of motion; thus they cannot know whether they are participating in scenario 1 or scenario 2.
For maximum clarity, the trips themselves involve just a light second or two in distance, and all parties make use of photon clocks which are a whopping 1/2 light second in width. The photon clocks tell the story of time contraction.
The horizontal white lines represent the travelers. They each carry a photon clock. The fact that time differential is identical in both scenarios confirms that one cannot experimentally detect ones true motion status with respect to the universe.
In part one of the twin paradox animation below, an astronaut and a space station occupant start their clocks as the astronaut passes by. A second, incoming, astronaut starts his clock as he passes by the first astronaut. The incoming astronaut and the space station occupant stop their clocks as the incoming astronaut passes by the space station.
In part two of the twins paradox animation below, an astronaut and a space station occupant start their clocks as the astronaut passes by. A second, also outbound, astronaut starts his clock as he passes by the space station. This second astronaut chases down the first astronaut. As the first astronaut is caught, both astronauts stop their clocks.
In scenario 2, one needs to abandon the strict labels of "space station", "traveler one" and "traveler two", even though the video labels them as such for simplicity. In scenario two, the "space station" might regard itself as "traveler one" who meets "traveler two" coming from what the "space station" regards as the opposite direction of travel. The parties involved cannot know which scenario they are involved in.
The time contraction formula is easily obtained from the above diagrams. For instance, in the first diagram (animation), the clock at rest with the universe ticks off one cycle while the traveling clock ticks off 0.8 of a cycle. A simple application of the Pythagorean Theorem yields the following formula:
t' = t * sqr rt of ( 1 - V^2 )
where t' is the time recorded by the traveler, t is universal time (full clock rate, since at rest with the universe), and v is the speed of the traveler.
Keep in mind that the traveler, moving at 0.6 light second per second of universal time, went a distance, in absolute terms, of 0.6 light second.
A photon went the same distance in the station's clock as it did in the traveler's clock, namely, 1 light second.
0.8 = 1 * sqr rt of ( 1 - 0.6^2)
= 1 * sqr rt of ( 1 - 0.36)
= 1 * sqr rt of ( 0.64 )
(There is no need to use c (light speed) in the equation, since we are using units of light seconds. Light travels one light second in one second.)
See the book Relativity Trail for more details, including an analysis of how all parties involved assess each other's clock speed and lengths in symmetrical fashion, as well as detailed derivations of length contraction, the Lorentz transformations and e = mc^2.
Relativity Trail, with 192 pages, 65 diagrams and 75 illustrations, will provide you with complete detailed algebraic derivations of all the kinematical effects of special relativity. Everything is charted out in absolute terms against a system at rest with respect to the totality of the universe for perfect clarity as well as soundness of theoretical basis. It is the totality of the universe that imparts the inertial properties of clock rates and lengths which generate the effects of relativity. This is explained in detail in Relativity Trail.
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