|Consider a thought experiment in which one atomic clock is placed at sea level and an identical second clock is accelerated along the earths surface into a sea level orbit at a velocity of 7905 m/s. Because of this increase in velocity, the orbiting clock would run at a slower rate than the clock at the earths surface. If this second clocks orbit were increased from sea level to a synchronous Comsat orbit of 6.61538 earth radii, it would be necessary to slow its orbital velocity to 3,076 m/s and reduce its escape velocity to 4,346 m/s. Both of these changes in velocity would cause the clock to run faster than it did in a low earth orbit and slightly faster than the clock at the earths surface.
Next, consider the case of a clock carried up a ladder from the Equator to the height of a synchronous Comsat orbit. Both the orbiting clock and the clock atop the ladder would have no relative velocity between them. They would have the same values for kinetic energy and gravitational potential energy and would keep the same time. The only difference between them is that it took far less energy to carry the clock up the latter than to put the other one in orbit. The orbiting clock must be accelerated into a low earth orbit and then decelerated into the much higher Comsat orbit.
Now consider a third clock that is in the same orbit but traveling in the opposite direction. This third clock is the same distance from the earths center as the other two clocks and its orbital velocity of 3,076 m/s is identical to the velocities of the other clocks. When they pass twice a day, the third clock is moving relative to the first two at a velocity of 6,152 m/s. Since the dynamics of all three clocks are identical then they must all keep the same time. How is this possible with so much relative motion between them? An absolute velocity of 6,152 m/s should cause a clock to increase the length of its measured time intervals to 1.00000000021. Then 6 hours later, when the orbiting clocks are on opposite sides of the earth, they are both moving in the same direction with no relative velocity between them. How can both clocks run at a constant rate when the relative velocity between them is constantly changing?
This paradox is particularly difficult for relativity enthusiasts to answer because any explanation directly contradicts the fundamental tenants of relativity theory. If they claim that the clocks constantly change their rates in relation to the relative velocity between them, then all motion is absolute and the very idea of relative motion must be discarded. If during each orbit, the clocks undergo a cycle of speeding up and then slowing down, then the clocks must be changing their rates in relation to, not each other, but rather to an absolute preferred inertial reference frame. Each clock must monitor its changes in velocity relative to the 2.7o CBR photon rest and then change its rate of ticking according to its vector of motion within this frame. This must be done without the clocks experiencing any measurable inertial acceleration. If, to the contrary, it is claimed that both clocks maintain the same constant rate then, the principle of relative motion is still violated. In order to put two identical clocks into opposite synchronous orbits, it is necessary to accelerate each to velocities of 3,076 m/s in opposite directions. If relative motion changes the rates of clocks, how is it possible for synchronous clocks to maintain identical rates when they are then accelerated to a relative velocity of 6,152 m/s?
A possible way to measure this effect would be to synchronize two identical clocks in Equador and then transport one clock halfway around the world to Borneo. These clocks could then be monitored for a number of years to see if their elapsed times remained the same despite their constant motion in opposite directions at the relative velocity of 896 m/s. If the clocks did loose their synchronicity over time it would show that it is relative motion and not absolute motion that changes the rate of clocks. However, relativity theory has no means for predicting which clock will slow down or speed up. If we expand this experiment from two clocks to a great many clocks located at equal distances along the equator, it would seem impossible to determine how the rates would change for each clock.
The most likely result of such an experiment would be that clocks at sea level maintain constant rates of elapsed time while at any point on the equator. There are two possible explanations that could explain this result. While present day atomic clocks are not accurate enough to measure such an effect, one possibility is that each clock speeds up and then slows down during each orbit as its velocity changes within the 2.7o CBR photon rest. In this case all clocks would be running at a different and changing rate during any interval of time but their average rate would be the same and they would record exactly the same number of time intervals for each rotation of the earth. The other possibility is that each clock maintains the same exact rate throughout the day and year after year.
Neither of these two possibilities is compatible with the principle of relative motion. The first requires the preferred reference frame of absolute rest and the second completely refutes the relativists claim that clocks in different states of motion must run at different rates. At this point in the paradox the relativity buff must drag out his old faithful four dimensional non-Euclidean geometry equations and try to show that space and time can be curved in a non-intuitive way that is not possible to visualize. When clocks are in the presence of such a curvature they can maintain constant rates regardless of their changing states of motion. It is explained that the equations can describe what is happening even though the rational mind is unable to visualize it. It is a metaphysical occurrence the dynamics of which can neither be measured nor rationalized. It can only be described with equations.
The changing rate of clocks caused by changes in both velocity and gravity are well documented and have been determined to a high degree of accuracy. Measuring the rates of orbiting atomic clocks is a highly developed science necessitated by the need to maintain the accuracy of the constellation of Global Positioning Satellites (GPS). The clocks in many different satellites must be designed to run as synchronously as possible with clocks on earth. This is difficult because a clocks rate is influenced by both the kinetic time dilation of its orbital velocity and the gravitational escape velocity at the orbits distance from the earths center. The clocks in the lowest orbits run the slowest because both their orbital velocities and escape velocities are greater than those of the higher orbits. Clocks speed up as they are placed in higher and higher orbits because they must be decelerated to both lower orbital velocities and lower escape velocities.
In order to synchronize the clocks in the GPS constellation, technicians must first synchronize the cesium clock to be put in orbit with a cesium clock on Earth. They must then calculate the rate at which the clock will run when it is placed in its desired orbit and then calibrate it to run faster or slower than the earth clock so that it will run at the same rate when in orbit. The reason the clocks rate changes is because the rate at which cesium atoms vibrate decreases when they are accelerated to higher velocities and increases when they are decelerated to lower velocities. The clocks recording mechanism must be calibrated to make up for the changed vibration rate of its cesium atoms when in orbit. The ideal orbit for GPS clocks is 2.694 Earth radii. At this orbit the sum of its orbital and escape velocities is the same as the sum of these velocities at the earths surface. Clocks placed in this orbit would be synchronous with clocks on the earths surface without the need of any special calibration. Higher orbiting clocks would run faster and clocks in lower orbits would run slower.
Since the process of gravitational expansion changes the absolute values of mass, space and time, gravitational motion, such as an orbit, occurs along a forth vector that is independent of the first three vectors of Euclidean space. When two clocks are accelerated into the same orbit in opposite directions, they remain in sync with one another because they both slow by the same amount due to the same increase in motion along the forth vector.