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, and to illustrate a flaw in pulsar timing algorithms.

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:

, (1)

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

(2)

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:

(3)

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 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 motional or kinetic energy. In the reference frame of the now moving clock, 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:

(4)

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

(5)

I have demonstrated in another paper that Newton’s mv²/2 is a low speed approximation, and that the final result of (4) and (5) is the correct solution, and not an approximation as indicated. 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 that 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, that 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.

We must consider what effect is to be seen on a clock constructed and calibrated in a non-inertial reference frame. To gain some insight, we look again at the clock in the gravitational field. If we construct a clock in free space, then lower this clock into a gravitational field, it will slow down due to the difference in energy between the clock and its surroundings. If we then construct a new clock at the same point in that field, the new clock will have the same rate as the clock brought in from the outside. Any energy delta experienced by the clock while it is being calibrated thus has the same effect as imparting that energy change after the clock is constructed. If we now take both clocks to a point outside the gravitational field, each clock will increase its rate by the same amount. We would then expect that, in the case of the Mossebauer rotor, even if we constructed the clock from scratch on the edge of the rotor while it was moving, the clock would still be slower than a stationary lab clock.

This effect has actually been tested by Hafele and Keating. Clocks constructed on the surface of the rotating earth run at a certain rate. If these clocks are flown eastbound, they slow down, but if they are flown westbound (at a velocity less than the rotational velocity of the earth), they actually speed up. This was demonstrated by Hafelle and Keating in their air-borne round-the-world clocks experiment. The reason this is so, is that circular motion is quite different from linear, inertial motion. In order to cause an object to move in a circle, we must continually supply energy, or accelerate, the object. Consider an atomic clock that is tuning itself with each "tick" of the clock. It takes a finite amount of time for the energy to be emitted and then absorbed, and during this time, energy is imparted to the system. Thus, even as we are constructing the clock, it is being calibrated such that it will run more slowly than if it were moving inertially. If this clock is now placed on a west-bound flying plane, its motion will become more inertial—we will be effectively removing the energy we have been adding due to circular motion. As a result, this clock will speed up with respect to an earth bound clock.

If the west-bound plane flies at a speed that allows it to circle the globe in exactly twenty-four hours, the clock will be running at its maximum rate. This is because the plane will effectively be stopped in space while the earth turns beneath it. The clock will no longer be experiencing rotational motion. The clock will also be running at the same rate as would be seen on a similar clock constructed in any inertial reference frame. Of course, in the above, we are ignoring or will have already compensated for, gravitational effects. If the plane increases it speed so that it now requires less than twenty-four hours to circle the globe, the clock will again begin slowing down. This is because the plane will no longer be "stationary" at a fixed point above the earth, but will begin orbiting the earth in the opposite direction. The higher the speed from this point on, the more the clock will slow. If the speed of the plane is such that it circles the globe in twelve hours, the west-bound clock will now have the same rate as a clock stationary on the earth.

We can calculate how much more slowly a clock constructed in circular motion will run than a similar clock constructed in an inertial reference frame. Since all inertial reference frames in a gravity-free environment are equivalent, we will take the time recorded on a clock constructed in such a frame to be metaphysical time. Such clocks will have the highest rate of all similarly constructed clocks in any other environment.

First we must write the expression for the work down on an object to confine it to motion in a circle. The incremental work done on the object, and thus the incremental energy imparted to the system at each instant of time is given by:

(6)

We can put this expression into a more useful form by noting that F = mdv/dt, and ds/dt = v, so that:

(7)

We may farther simplify (7) by noting the following relation:

(8)

We recognize that the quantity v² is constant with respect to time. Substituting (8) into (7) yields the final expression for the energy continuously imparted on the circularly moving object, as compared to an object at rest undergoing no acceleration:

(9)

A clock constructed in the lab frame, and then placed on the edge of a rotor, will slow down due to the acquired energy. When removed from the rotor and placed again in the lab frame, the clock will speed up to its original, inertial rate. Similarly, a clock constructed on the moving rotor would run at the same lower rate as the lab frame clock placed on the rotor. When removed from the rotor, both clocks will speed up by the same amount, reflecting the more inertial, lower energy environment. In the case of the west-bound flying clock, we are taking a clock that has been constructed on a rotor (the rotating earth), and then moved to a more inertial frame. Thus the clock speeds up as compared to an earth bound clock. We allow the west-bound clock to circumnavigates the globe until it is again at the same location on the earth as the earthbound clock. The west-bound clock will have accumulated more time than the earthbound clock. Being in a more inertial state, the clock rate speeds up as it approaches metaphysical time.

Imagine that we were to place a clock on a rocket, and launch it into space at the point of earth’s perihelion in its orbit around the sun. We keep powerful rockets pushing on this clock so that it remains at this point in space as the earth pulls away in its year-long journey. From the earth’s reference frame, this clock will have been accelerated mightily, and will have attained a tremendous velocity (30 km/sec) with respect to the earth. After a year has elapsed, and the earth is approaching perihelion and the space clock once again, we read the elapsed time on the space clock and compare it to one of our earth-bound clocks. We find that, even though the space clock was accelerated to a high velocity as measured with respect to us, it has recorded more time since we’ve been gone, not less as would be expected of a clock placed in motion.

Even after we adjust the time recorded on the earthbound clock for the slowing due to the earth’s rotation and the presence of the clock in the earth’s gravitational field, we still have accounted for less time than recorded on the space clock. The reason is that a clock stationary at the point of earth’s perihelion is in a more inertial reference frame than a similar clock in orbit around the sun. Thus this clock runs closer to metaphysical time, and, therefore, runs faster than the earthbound clock. The same would be true if we could set a clock in space and wait for the entire Milky Way to complete one revolution. As the solar system drags back around after millions of years, the space clock will have recorded much more time than a clock kept on the earth for the duration. Again, we have neglected or pre-corrected for all gravitational effects. The space clock, set free of the rotational grasp and effects of the Milky Way, will have been one step closer to fully inertial, metaphysical time.

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:

(10)

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].

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. 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 tosses the clock it received back to its original train. We will assume that each of the clocks that 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 that 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 at first glance, 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 that 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 Alice’s clock into his frame of reference, so that in his reference frame 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. This would confirm his suspicion that Alice’s clocks were initially calibrated to run fast. 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 that 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 that 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 that 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, that 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.

We are able to translate from the time read on a given atomic clock, or array of atomic clocks, on earth to the time of a hypothetical clock in a perfectly circular orbit and uniform gravitational field around the sun. This time is referred to as TDB, Barycentric Dynamical Time. This time represents a fixed offset from the time of a hypothetical clock located at the solar barycenter, which can simply be accounted for in an appropriate definition of a second. Thus we can interpret TDB as the time measured at the solar barycenter, and consider the solar barycenter to represent an inertial frame, at least for periods of tens to hundreds of years.

As pulsar timing data is accumulated, observatory readings obtained are transformed to TDB, thus we can plot the data as it would appear if the earth were an inertial system located at the solar barycenter. When this transformation is performed, one finds that millisecond pulsar data is very stable, with residual timing errors on the order of microseconds appearing basically randomly (white) around the zero point. The stability of these plots implies that the solar barycenter is an inertial system as viewed from the rest frame of the pulsar.

The transformation from TDB to that of a clock based at the solar barycenter is based on a trick devised by Hafele and Keating in their time-lagged clocks experiment. They were flying clocks around the earth and needed a stationary, inertial clock against which to compare their results. The clocks in the planes would be moving in a non-inertial manner with respect to any clock on the surface of the earth, and of course, the earth clock itself would be moving non-inertially, thus the test could not accurately be considered a test of SRT. What Hafele and Keating did was to transform the time of a clock on the surface of the earth to that of a hypothetical clock located at the center of the earth. Since the real clock would be moving at a fairly uniform speed of about 1000 km/hr with respect to this hypothetical clock, the time on that clock could be represented as a fixed offset from the real clock, or be absorbed in the definition of a second. As a result of this transformation, it was possible to pretend that the earth mounted clock, the west-bound flying clock and the east-bound flying clock were all traveling more or less inertially with respect to the earth centered clock.

As was explained in an earlier section, this approach works well, not because of the success of SRT when applied to this situation, but because clocks on a rotor run slower than an inertial clock by a factor proportional to the rotaional kinetic energy of their motion.

One can extrapolate these results to a certain degree without introducing error. For example, one can imagine a clock some distance away from the earth, at a point in space fixed with respect to the earth’s center. This clock will remain stationary with respect to the hypothetical earth-centered clock, and thus the slowing of the traveling clocks with respect to this clock would be identical to that with respect to the earth-centered clock. Thus, if the space clock were to send out periodic (say millisecond) pulses to be received by the traveling clocks, we could transform the observed times-of-arrival (TOA) to earth center time (ECT), and we would find that the stability of the transformed pulses is quite accurate. This transformation would, of course, require calculations of path length changes, gravitational field effects, and others, as is done in the case of pulsars, but the approach is still straightforward. It is only because the space clock is stationary with respect to, and therefore in the same reference frame as, the hypothetical earth-centered clock that this method works. If the space clock has its own proper motion, then the velocity (due to the rotation of) the earth is no longer a constant. At times it adds linearly. Twelve hours later it subtracts linearly, and at all other times it changes by the Pythagorean theorem as a function of the cosine of the angle of the velocity vectors. There is no way to model the Earth's rotational motion as inertial with respect to a moving space clock.

While one can, under the tenets of special relativity, transform quite effortlessly between any number of inertial frames, there is the requirement that the frames are indeed inertial. Transforming pulsar timing measurements to TDB works well, provided that the pulsar is not moving with respect to the sun. And of course, the chances of this being the case with any pulsar are certainly zero. Thus, there will be an annual periodic residual in pulsar timing measurements due to the error in correcting only to TDB. The magnitude of this error will vary based on the absolute magnitude of the velocity of the pulsar to the solar barycenter.

Special relativity states that time-dilation is due only to relative velocities, not imparted kinetic or rotational energies. But the trick of transforming to solar barycenter works only if clocks slow for the reasons presented in this paper. When we consider only the energy differentials between experienced by various clocks, whether in linear or circular motion, the time difference between the earth and sun mounted clocks remains constant. Further, the pulsar’s proper motion is basically inertial over long periods of time. Since clocks created in inertial systems all keep metaphysical time, there is no timing difference due to the pulsar’s proper motion with respect to the sun, unless that motion is changing rapidly.

In special relativity, we should be able to consider directly the time-dilation between the pulsar and the earth, due directly to the relative motion between the earth and the pulsar. This can, and should, be done without transforming through the solar barycenter frame, a frame in which no actual measurements or observations are made.

Consider the simple case of a pulsar moving radially away along a line joining the sun and the pulsar. It is a simple matter to show, by transforming the origin of the pulsar reference frame momentarily to the location of the sun, that any arbitrary motion of the pulsar can be modeled in this manner, except for the definition of the location of the Earth at which theta equals zero. We define theta as the angle that the Earth's velocity vector makes with the line joining the sun and pulsar. Figure 2 illustrates a pulsar moving away from the sun with a velocity v_x, while the velocity of the earth is given by:

(11)

The relativistic formula for the Doppler shift in pulse arrival time due to an arbitrary velocity is given by:

(12)

(13)

For a moving pulsar, v_r and v depend on the location of Earth in its orbit, and thus the angle theta, as follows:

(14)

Thus we can write the general equation for the Doppler shift in pulse arrival times as:

(15)

(16)

(17)

Taking the ratio of (17) to (16) yields:

(18)

This is, of course, what we expect to see, and this is the formula used in practice. The time dilation offset is constant throughout the orbit, so the entire Doppler shift in pulse arrival times is due to equation (18). The same analysis in the case of equation (15) yields the following:

(19)

(20)

(21)

(22)

It would appear that residuals on the order of one microsecond per pulse could be accumulated over long observing runs, and thus be detected directly. For example, for a proper motion of about 1,000 km/sec, six months worth of accumulated data would result in an excess (or deficit) of about 50 pulses for a one millisecond pulsar. This would be true if we knew anything firm about the pulsar itself. However, all we have is least-squares fitting of our observed data to a model of the pulsar. The pulsar’s characteristics are assumed to match those of whatever model we derive which minimizes the errors we see in our observed data. An error of plus or minus fifty pulses over the course of one year’s worth of millisecond periods is insignificant by current capabilities, and, even if it weren’t, the pulsar is modeled such as to absolutely minimize any apparent residuals in the data. Additionally, obtaining long run, consistent observations of a specific pulsar is not practical. Viewing time is expensive and rare, and, even if available, a given pulsar does not remain in view of a given observatory twenty four hours per day. It is even more rare to find pulsar’s whose signal strength is great enough to discern individual pulses in the first place. Thus, the direct accumulation of excess or deficit pulses is not the most likely method for detecting the presence or absence of these residuals.

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 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 but that the mechanisms for decay have been skewed by a factor allowing a longer average life 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 slow down because time itself slows down. No "mechanical" description is provided as to why the clocks slow, and, if it were, 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 constructs or calibrates a clock, her reference frame becomes the 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.