The Effects of Motion and Gravity on Clocks and an Examination of the Twin Paradox

Utilizing only the principles of equivalence and conservation of energy, the customary equations for the slowing of clocks due to motion are derived. It is shown that clocks slow proportionally to a Galilean transformation of the energy of the clock system from the initial rest frame of the clock to the new, moving inertial reference frame. Utilizing the same reasoning for the case of increasing gravitational potential, the customary equations for the slowing of clocks in a gravitational field are derived. This analysis, applied to the radiation continuum model of EM radiation, results in the correct equations for the time delay of a solar grazing light or radio signal. By considering the characteristic frequency absorbed or emitted by a hydrogen atom, it is demonstrated that only motion relative to the rest frame in which a clock is calibrated causes slowing. Thus if two observers initially in motion with respect to each other each construct identical clocks, at rest in their own inertial frames, the clocks will record identical time. If either clock is then placed in motion relative to the inertial frame in which it was calibrated, it will slow according to the energy considerations associated with this motion. A thought experiment involving riders on two trains exchanging and comparing the readings on atomic clocks explains the so-called "twin paradox" without resorting to either SRT or GRT. The derived equations are used to successfully analyze the Hafele-Keating traveling clocks experiment.

Introduction

In this paper we will demonstrate that the slowing of clocks placed in motion or lowered into a gravitational well can be explained utilizing only the principle of equivalence and conservation of energy. It is clear from this description that the effective slowing of these clocks has no effect on time itself, but only upon the instrumentation or processes by which we choose to measure time.

Clocks in a Gravitational Field

Realizing that frequencies shift to the blue (higher frequency) when brought into the presence of a gravitational field, it is natural to question what happens to a clock or any time-measured process in the same field. Let us begin by examining the case of a signal passed from a clock outside a gravitational field to one deep inside the field. We will use a highly stable oscillator as our clock, and let the frequency of the source represent the time unit of the clock. Thus if an observer located with the receiver is using an identical frequency source to time the received signal, he will obviously notice that the frequency generated by his clock is lower than that which he receives. From his point of view, the clock in the gravitational field must have slowed down. Similarly, for a signal going the other way, there will be a red shift in the received signal, and an observer with the clock outside the field will conclude that the clock inside the field is running slower than his own. If the two frequency sources are slowly brought together, either in the field or far removed from it, they will generate identical frequencies. What we have to decide is, did the clock inside the field slow down, or did the frequency decrease as it climbed out of the field. Clearly, only one of these effects occurred, otherwise the frequency would be shifted twice--once due to being generated at a lower frequency by a slower clock, and again by being gravitationally red-shifted as it climbed through the field. This is, of course, not what we observe.

Imagine a photon in free-fall entering a gravitational field. We know the frequency of this photon, as measured in the field with "local" clocks, will be increased or shifted to the blue. The photon, however, must be unaware of any change in its frequency, or it would become aware of its presence in the field--the principle of equivalence prohibits this.

Now, the photon, upon entering a gravitational field, or "well," acquires excess energy, much the way a ball dropped from a tower gains energy as it falls toward the ground. We can express the energy of a photon as mc² or as Planck's constant times c divided by the wavelength. Thus an increase in energy can be viewed as an increase in effective mass, or conversely as a shortening or "bunching up" of the various wavelength components. A visual representation of this is when smoothly flowing traffic suddenly comes upon slowing due to rubbernecking a stalled vehicle across the road. The traffic slows, and the cars which had a comfortable, even spacing now become bunched up as they pass through this area. Upon leaving the congestion, the cars once again resume their original, spread out configuration. Now, if a photon traveling at c with a particular wavelength finds itself bunched up so that the wavelength is now smaller, the photon will slow itself down such that, in its frame of reference, its frequency remains unchanged. This slowing of the photon has been verified to very high accuracy by Shapiro, et. al. Thus, in the photon's frame of reference, its frequency remains unchanged, as required by the principle of equivalence.

If the photon has managed to keep its frequency the same in its own frame of reference, then why does an observer situated deep within the gravitational field see a blue-shifted frequency? To see this, imagine a hydrogen atom far removed from the gravitational field. It will generate or absorb light or radio energy at a specific frequency, 1420 MHz. In fact, we use these emissions to define what 1420 MHz is. If any other source of producing energy at this frequency gives a value above or below this one, we assume the hydrogen atom to be correct, and the other source to be incorrect or shifted in frequency. Now, if we carry our hydrogen atom into the gravitational well, the resulting change in energy will cause the atom to generate and absorb a lower frequency than that which we saw outside the field. However, due to convention, we still use this lower frequency as a definition of 1420 MHz. Suppose another hydrogen atom generates a photon well outside the field, and that this photon then falls into the field, where we compare its frequency to one generated by our own hydrogen atom. What we find is that the photon has managed to keep its frequency unchanged, as measured locally by itself at each point along its journey, but upon reaching us, we have a new, lower definition of 1420 MHz. Thus when we compare the frequency of the photon to our definition, we find that it is higher--the frequency has been blue-shifted compared to an identical source inside the field. If we now build a clock based on the frequency required to tune these atoms, we will have to supply energy at this lower frequency to which the atom is now sensitive. Thus when we compare this frequency to one generated far removed from the field and sent to us, we will measure the incoming frequency as being blue shifted, even though to the photon itself it has not changed. Likewise, since our clock is tuned with a lower frequency, it will accumulate less time than a clock outside the field. When this clock is carried back outside the field after some duration, the elapsed time on this clock will read less than the elapsed time on a proper clock which never entered the field in the first place, even though the two clocks will be ticking synchronously once they are brought together.

We can express the energy of an atom compared to its surroundings as the difference between its mass energy and the energy of the gravitational field. Thus, at a distance R₁ from the gravitating mass, we have, to first order:

where we have effectively supplied part of the required excitation energy by lowering the floor.

If the atom required a photon of energy hn outside the gravitational field to become excited, it now requires a lower energy photon, given by:

Since the time unit of a clock based on this atom is defined by the frequency absorbed by the atom, we have that the elapsed time measured on the clock in the field is given to first order by:

Motion Effects

Suppose we have a cesium clock, calibrated, and stationary in our reference frame. If we now allow this clock to attain some velocity v with respect to our frame (the calibration rest frame of the clock), the total energy of the clock system will change, in a manner similar to that we saw when lowering the clock into a gravitational well. In our reference frame, we can express the total energy of the moving clock as the sum of its rest energy and its kinetic energy. To transform this energy to the reference frame of the moving clock, where it has no kinetic energy, we simply subtract the kinetic energy term from the energy we see in our frame.

Since any mass quantity and its associated kinetic energy would transform proportionally, we can derive an energy conversion formula by taking the ratio of the energy in the moving clock's reference frame to the energy measured in our reference frame. We know that this conversion takes the form of a proportionality constant due to the principle of equivalence. If such were not the case, then different energies would transform differently, and the moving clock could become aware of its constant velocity by measuring the differing degrees to which energy levels of some items change compared to their rest energies. A little algebra reveals that this proportionality ratio is equal to one minus one-half the velocity of the clock squared divided by c². Therefore, the energy required to cause the atoms in this clock which has been placed in motion to enter an excited state is reduced by the same ratio. One way to look at this is to say the cesium atoms in the frame of calibration required a certain frequency or energy to be applied to enter an excited state. By placing the entire mechanism in motion, we have already supplied a certain amount of that energy in the form of acquired kinetic energy. In the reference frame of the now moving clock, then, the atoms require less energy by the same amount as the applied kinetic energy. We could also state, as with the clock in the gravitational well, that the energy of the calibrated atomic clock compared with its surroundings equals the difference between the atom’s mass energy and the energy of the field, which now contains the imparted kinetic energy. In the gravitational case, we effectively supplied energy to the atom by lowering the floor. In this case, we have added kinetic energy to the system, so that the starting off point is already mv²/2 above the floor. If the rest energy of a cesium atom is E_o, and the required energy is E', we have:

Since the relation above applies to all energies by the principle of equivalence, we see that the frequency of the clock becomes:

Thus we see that an atomic clock which has been placed in motion is susceptible to a lower frequency, and thus accumulates less time, than a clock which remains stationary for any initial rest frame of reference we choose. Note that the stationary clock does not need to be actually present. Further, the actual nature of the presumed rest frame is not important. In other words, we do not need to presume the existence of any preferred, absolutely and universally stationary frame of reference. The energy required by any cesium clock atom that has been accelerated out of its calibration reference frame and is now moving uniformly with respect to that frame will be less than that required by a clock atom which remains stationary in our reference frame, as measured in our reference frame. What we must be careful to do then, when building and testing actual atomic clocks, which must be calibrated and synchronized, is to make certain that they are all calibrated and synchronized in the same reference frame prior to starting the test. This reference frame will then become the common rest frame for all clocks in any experiment we wish to perform.

It is extremely important to realize that time does not actually slow down due to this motion. Since cesium atoms of a given velocity require a specific frequency to reach the excited state, so atoms accelerated to a different velocity relative to the first require a different frequency, as measured in the reference frame of the first; shifted to the red according to the magnitude of the velocity by the factor g^-1. Since the frequency is lower, it takes more time for a fixed number of cycles to occur. With seconds in these clocks being defined as the length of time required for a specific number of cycles to occur, the moving clock slows down--more physical time is required for a given "second" to pass in the moving clock.

The Equivalence of Gravitational and Other Forms Of Energy

If the slowing of the moving clock is due to a change in kinetic energy, and the slowing of a clock in a gravitational field is due to a change in gravitational potential energy, we should be able to derive similar equations for each effect. In fact, gravitational potential energy can be converted to kinetic energy quite simply. If we hold a ball over the edge of the Harvard tower, it will posses a certain amount of gravitational potential energy as compared to an identical ball at rest on the ground twenty-two meters below. If the ball is now released, this potential energy will be converted to kinetic energy as it acquires speed on its trip toward the ground. At the instant before the ball strikes the ground, all of that excess gravitational potential energy will have been converted to kinetic energy. The amount of kinetic energy gained will exactly equal the amount of potential energy lost. That we can so easily shift from potential to kinetic energy further supports the equivalence of these expressions, and would lead us to believe that the expression for clock slowing due to gravitational potential is in fact identical to the expression for slowing due to acquired motion, with the expression for gravitational potential energy in the former replaced by the expression for kinetic energy in the latter. This is, in fact, the case.

Consider the case of the clock in a gravitational well. We were able to show that the time unit of the clock slowed by a factor proportional to the strength of the gravitational potential at that point in the field. In fact, the ratio of the slower time unit of the clock in the gravitational field to the "proper time" of a clock far removed from any gravitational fields is equal to the ratio of total energies experienced by an atom in these two situations. This ratio is independent of the actual mass or atoms considered, as it is the same for all mass energies. Thus the ratio of the inherent or rest energy of, say, a cesium atom plus the gravitational potential to the inherent energy of the cesium atom alone would be the same ratio we would see when comparing the rate of ticking of the two clocks. The reason this is so is that frequency and energy are linearly related, and, thus, a ratio of energies is equivalent to a ratio of frequencies. Since the time unit of the clocks is also linearly related to the frequency, then the ratio of energies is also equal to the ratio of time units.

In the case of a clock which has been brought from rest to some velocity, we showed that the degree of slowing was equal to the ratio of the inherent energy of the mass less the kinetic energy acquired to the inherent energy of the mass itself. Thus the expressions for gravitational slowing and slowing due to acquired motion are indeed equivalent. Beginning with the expression current energy over rest energy, we simply plug in the gravitational potential energy for the gravitational case, and plug in the kinetic energy for the case of a clock brought from rest to some velocity.

An interesting relation we can develop from previous equations is to determine what velocity a clock not in a gravitational well would need to achieve in order to experience the same slowing as a clock within a gravitational field. In this case, we have:

The interesting thing about the last relation is that the velocity required to obtain the same slowing of a clock as would the gravitational field is identical to the velocity required of an object within that field to exactly escape the confines of that field. This is exactly the result we would expect for our clocks if the two forms of energy are indeed equivalent.

We have thus far developed the usual equations for the slowing of clocks placed in motion and for clocks in a gravitational field. We have invoked only the principles of equivalence and of conservation of energy. Nowhere have we indicated that "time slows down." In fact, it is clearly apparent that time itself remains unaffected, and that simply changing the ratio of energy of the clock to that of its environment causes the slowing. This occurs whether we add energy to the clock system by accelerating it out of its calibration rest frame, or whether we effectively remove energy from its surroundings by lowering it into a gravitational energy well. The net effect is the same. When developed more fully and rigorously, the equations presented in this paper fully and accurately account for all so-called time dilation experiments to date, including the Shpiro time delay, the Hafele-Keating round-the-world clocks experiment [1], Bob Vessot's Scout-D Rocket experiment [2] and the Pound-Rebka gravitational red-shift measurements at Harvard [3].

Clocks Calibrated in Different IFRs

Suppose two independently moving experimenters, Alice and Bob, each construct and calibrate a clock in the respective inertial reference frames in which they reside, approaching at some fixed velocity. According to SRT, each would feel confident that their own clocks are correct, and that the other's is experiencing slowing due to motion. Assume that the clocks to be used are cesium clocks. Alice and Bob each construct and calibrate their clocks by an identical set of plans and procedures, thus the two clocks are identical except for the reference frames in which they reside. Alice uses her clock to send a signal to Bob, and Bob sends an equivalent signal to Alice. Now, since each observer knows the value of their velocity relative to the other, each can fully account for the motion induced Doppler shift, and therefore determine the effective rate of the other's clock. In this example, both Alice and Bob determine that the other's clock is ticking at exactly the same rate as their own! What happened to the slowing of clocks due to motion? The answer lies in the careful consideration of reference frames.

We have seen that clocks in motion slow down only when placed in motion relative to the rest frame in which they were constructed or calibrated. Obviously, a given cesium atom or collection of cesium atoms captured in any particular IFR will be susceptible to the same frequency, about 9 GHz.

Thus, any clock based on the cesium atom, having been constructed or calibrated in an IFR (not undergoing, or having undergone, acceleration or rotation) will keep the same time as an identical clock constructed in a different IFR. Since Alice's clock and Bob's clock each remain in their rest frame of calibration, each will record the passage of time accurately and synchronously. Recall that when a clock slows down due to being placed in motion, this has no actual effect on time itself, but only on the recording of time by that clock. The rate of the clock placed in motion becomes lower. Alice could argue that Bob's clock should be running slow, and that the reason it is not is that he has calibrated it improperly. Now we will see how the reference frames actually compare, and show why Alice cannot effectively support this argument. We will place Alice and Bob on two identically long trains on parallel tracks, heading toward each other at very high speeds, as is illustrated in the figure. Each observer is in the front of its respective train, and carries two identical, synchronized clocks. These clocks are constructed in such a manner that for every second which passes on the clock, one mark is made on a ticker tape. Thus the elapsed time of any experiment can be determined by simply counting the number of dots on the paper. As the trains pass each other, each observer tosses one of its clocks onto the passing train. When the last car of the passing train is along side the observer, each then stops the tape on the clock which was tossed to them and tosses it back to its original train. We will assume that each of the clocks which remained with Alice and Bob in their "stationary" frame of reference recorded a time of one-hundred seconds for the trains to pass each other. After walking to the back of their respective trains and checking the clocks which were placed in motion on the passing train and then returned, each finds that these clocks recorded only ninety seconds. Now that these moving clocks are back in their initial rest frames, it is found that they are each once again marking
the correct time.

Alice and Bob had each already concluded that the other's clock was initially running fast. This, reasons Bob, is why Alice recorded one-hundred seconds on her clock, while his clock, traveling with her, recorded only ninety seconds. Alice reasons the same way concerning the clock she gave to Bob. This seems fine, but an apparent paradox quickly arises if we follow things a little further, exposing the fallacy of this line of reasoning.

When Bob tossed his clock onto Alice's train, it slowed down due to that acquired motion, but then sped up again upon returning to Bob's frame of reference. The clock which Alice kept with her on her train, which is also in motion with respect to Bob, kept time at a faster rate than Bob's moving clock. If Bob were to bring this clock into his frame of reference, so that, as far as he is concerned, it is no longer moving, it seems that this clock should speed up by the same amount that his own clock did when brought to rest. However, we have already seen that Alice's clock, when brought into Bob's reference frame, slows down. How can this be--that one moving clock, when brought to rest, speeds up, while another slows down? The answer lies in the experimental setup--to obtain consistent results, all clocks in a given experiment must be calibrated in the same reference frame, which then becomes their common rest frame. In the above example, the clock which appeared to slow down when brought to rest was actually being placed in motion from its rest frame, thus it was slowing due to the change in energy associated with this acquired motion--the rest frame of this clock was not the same as the clock to which it was being compared.

Alice and Bob constructed and calibrated identical clocks by identical means in two different inertial frames of reference, and found them each to be marking proper time. The reason for this is that a cesium atom is susceptible to a specific frequency. In fact, the cesium atom is used to define that frequency, rather than the other way around. As these clocks are placed in motion, they slow down due to a change in state from the reference frame in which they were calibrated. In other words, contrary to the assertions of SRT, it is not enough simply to be in motion with respect to a given reference frame. The clock must actually have been placed in motion with respect to its rest frame of calibration. It is this non-inertial change in reference frames which causes the clock to slow. As the clock is being accelerated to its final velocity, with each "tick" the clock is traveling at a new velocity with respect to its rest frame. Thus, with each "tick" the clock is running slower than it was at the previous instant. Once the clock is no longer changing its velocity with respect to its rest frame, and is moving at a constant velocity with respect to that frame, it will not continue to slow down, but will run consistently at whatever low rate it has achieved. Thus, when Alice tosses her clock into Bob's reference frame, it slows down. Note that it is not simply an acceleration which causes a clock to slow. If such were the case, then clocks in a gravitational field or on a rotor would continually slow until they stopped, due to the constant acceleration. The only role the acceleration plays is in changing the magnitude of velocity, or speed, of the clock with respect to its rest frame. It is this relative change in energy with respect to the rest frame, caused by a change in speed, which causes the clock to run more slowly. Thus, a clock on the edge of a rotor will run at a constant slower rate than a clock at the center, despite the continual acceleration associated with circular motion. Only the acquired rotational kinetic energy has any effect on the rate of the rotor clock.

Now suppose that Alice decides to trick Bob. She knows that his clock will speed up when returned to his reference frame, so when he tosses the clock to her, she quickly recalibrates it to the same rate as her own clock and immediately tosses that clock to Bob rather than tossing one of her own. She thinks it will be funny when Bob gets this clock back, and finds that it speeds up upon returning to its rest frame, so as to record approximately one-hundred ten seconds for the trip rather than ninety. Unfortunately for Alice, this is not what happens. As soon as she recalibrated the clock, her reference frame became the clock's calibration rest frame. When she tossed this clock back to Bob's train, its rate slowed such that it would record only ninety seconds, just as if Alice had tossed one of her own clocks. As soon as Alice made an adjustment to the clock, she changed the nature of the experiment.

In the original experiment, Alice was a completely passive observer, thus she could observe the rate at which the clock was ticking without impacting its nature. As soon as she became non-passive, and made a change to the system by her observations, her intrusion caused a change in the experiment which could not be undone. Even if she had adjusted the clock back to the original slow rate at which she had found it, the clock's rate would still slow when tossed back to Bob's reference frame--he would find a clock ticking slow by twice the amount as would normally be expected due to the velocities involved. He would also know immediately that Alice had sabotaged the experiment by her prank.

This effect is not unique to saboteurs of atomic clocks. One of the basic assumptions in the field of quantum mechanics is that simply making an observation of a system alters the system. However, it would appear from the above example that the quantum theorists are not entirely correct. When Alice observed the system in a passive, or non-obtrusive, manner, she was able to extract information--in this case the rate of the clock--without altering the outcome of the experiment. However, when she made her observation in an obtrusive manner, changing the rate of the clock and then returning it to its original rate, the experiment was irrevocably altered. The reason the quantum physicists don't distinguish between obtrusive and non-obtrusive observations is that every observation they make is of the obtrusive form. This is due to the nature of the items they are measuring. Quantum physicists study photons and elementary particles, none of which can be individually discerned or measured by the naked eye. All observations in these experiments involve electromagnetic detection or perturbation of the items under consideration, which, by their nature, are obtrusive observations, and thus result in apparent paradoxes such as we see in double-slit photon experiments. Every time we make an obtrusive measurement in an experiment, that marks the start of a whole new experiment. In the case of Alice and the clock recalibration, that act ended the original experiment and began another, the results of which are easily predictable and consistent with other experiments of the same type.

Do we know that we can make a passive observation of a system without affecting it? The answer would appear to be yes, and it may be fairly simple to demonstrate. If we send a cesium clock on board the space shuttle, calibrated on earth, it will slow down due to the extreme velocity of the shuttle. Actually the effect is better demonstrated on a more linear trip, such as an Apollo moon mission, and the slowing of clocks on these missions has, in fact, been verified. If we now construct or perhaps simply recalibrate an identical clock on board the shuttle, using the exact procedures used on Earth, this clock should be found to be ticking slightly faster than the clock which was carried from and calibrated on Earth--the clock which was accelerated out of its rest frame.

The Twin Paradox

Consider the example commonly referred to as "The Twin Paradox." In this example, we have two twin brothers, each maintenance technicians, who have signed on to travel with two clocks A and B, and to maintain the clocks and the ships on which they respectively travel. Clock A is sent with its passenger on a long journey to the star Vega at a velocity approaching the speed of light. Clock B and its technician (much to his disappointment) stay on Earth. The experiment continues for over fifty years, when, one day, clock A and its passenger return. Before the earthbound technician opens the door to greet his brother, he notices that the readout for clock B indicates that only eight years have passed. The earthbound technician (who has in fifty years grown very jealous of his traveling brother), claims he is too old and weak to be able to release the door latch, and leaves his brother in the capsule, which sinks to the ocean floor.

Thus an important question remains unanswered. Was the returning brother 50 years older than when the experiment began, or only eight years older as the clock (perhaps) suggested? Since Congress blamed NASA for not rescuing the capsule, they refused to fund another such experiment, and so other means have been used to postulate an answer. One of these methods involves elementary particles called muons. These particles, when left stationary in [our] rest frame, have a half life of about 1.5 microseconds (ms). The process by which these particles (or any particles for that matter) decay is not entirely known, though the existence of the so-called "weak-force" is often postulated to explain this form of beta-decay. The weak-force is thought to break subatomic particles into electrons (beta-particles) and other minor particles. What is known is that if you increase the velocity of these muons (from our rest frame) to some high velocity, their half life will increase by the relativistic factor g.

Thus it would appear that the muons' life has been extended by the same factor that a clock traveling with them would have slowed. Based on this, one could conclude that one of the twin technicians was forty-two years younger than the other when he died. This certainly appears to be a vote in favor of all atomic processes slowing down, but not necessarily. Our only experience with measuring the lifetime of muons comes from measuring their decay at or near the surface of the Earth. One possibility may be that the mechanism of decay is external. Thus the properties which are abundant or well-tuned in our rest frame become scarce or improperly tuned as the particle's velocity is increased, in much the same manner as mass appears to increase with velocity when measured by purely electromagnetic means. In this case the means of measuring mass--the result of a magnetic force--has less and less effect on the particle as its speed increases relative to the field itself. A second possibility is that the cause of decay may be internal, requiring a transfer of forces inside the particle in a manner similar to the cesium clock (though not necessarily frequency tuning). Then, as with the clock, the decay of the moving muon will take longer than the decay of a stationary one, as the moving clock ticks more slowly than the stationary one. Of course a third possibility is that the mechanism of decay cannot be modeled completely in either manner. A neutron has a half-life of only about 620 seconds, yet when bound in the nucleus of an atom may exist for extremely long times without decay. In this case there is some mechanism that allows it to change its half-life other than by changing its velocity, though it seems safe to argue that the effect is still due to a change in energy.

When one considers any of the possibilities above regarding the life of a muon, it is at once apparent that time has not slowed down (as it was shown with the clocks), but that the mechanisms for decay have been skewed by a factor allowing a longer life (on average) for each particle. If this is the case, then atomic processes of all types may slow down in this manner, and it is not unlikely therefore that molecular ones do so also, though it is not guaranteed. If a neutron behaves differently in a bound state than when free, atomic processes in bound atoms and cells may occur differently than for free atoms as well. Hence we are left with uncertainty as to the apparent physical age of the returning twin.

Most importantly, however, it must be stressed again that time itself has not slowed down. Only the arbitrary units of measure with which we choose to mark time have slowed, whether atomic processes, frequency changes or molecular reactions. The distinction is important. In the relativistic model, clocks slowed down because time itself slowed down. No "mechanical" description could be provided as to why the clocks slowed, and, if it was, the effect would be additive. Quentin Smith has argued at length and quite successfully in Language and Time from a philosophical standpoint that "metaphysical time is the only time in the actual world and that it is the only time in any possible world in which there is time." [4]

Finally, we must consider the continual recalibration of the traveling twin's clocks. Recall how when Alice made an adjustment to Bob's clock, her reference frame became the new rest frame for that clock. This does not apply to the case of high speed muons, as their total life is too short for them to adjust to their new, high speed reference frames and recalibrate their clocks accordingly. But our traveling twin will be on the road for fifty years. His atoms will be continually absorbing and emitting photons, they will replace electrons brought with them for electrons native to the current environment. His cells will divide and re-divide, he will eat, drink and breathe. Each one of these actions takes him a little closer to being established in a new rest frame, the one representing his current velocity. Over the course of a fifty year trip, it is most likely that the traveler will age very close to fifty years--certainly more than the eight years predicted by SRT (actually, SRT alone cannot resolve the "paradox." Only by invoking GRT and the acceleration induced forces felt by the traveling twin is the question resolved in relativity theory). The process of continual recalibration will cause our twin to remain more or less in his continually revised rest frame for the entire trip, returning to earth as an old man, and making his brother very happy.

References:

[1] Hafele, J. C. and Keating, R. E., 1972, "Around-the-world atomic clock: predicted relativistic time gains," Science, vol. 177, pp 166-167; "Around-the-world atomic clock: measured relativistic time gains," Science, vol 177, pp 168-170

[2] Pound, R. V., Rebka, G. A., Jr. and Snider, J. L., 1965, "Effect of gravity on gamma radiation," Phys. Review, vol 140, B788-803

[3] Vessot, R. F. C. and Levine, M. W., et al, 1980, "Test of relativisic gravitation with a space-borne hydrogen maser," Physical Review Letters, Vol 45, 2081-4

[4] Smith, Quentin., Language and Time, Oxford University Press, New York, 1993, pg. 241

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