Time Dilation, the Clock Paradox or Twin Paradox, and Relativity Theory

In the last section it was shown that special relativity is inherently a steady-state, or near-steady-state, theory. All precursor transient electromagnetic problems have to be analysed using quantum theory. Special relativity’s major assumption - that the velocity of light is constant in empty space - is unnecessary. The observed steady-state velocity of light in free space is always equal to c solely because the presence of the observer’s material detector makes it so. This last deduction arises from the correct application of the advanced potential solution of Maxwell’s equations.

Now let us consider the long-standing problem that exists in special relativity, namely time dilation. This problem is also often referred to as the “clock paradox” or the “twin paradox”. If an atomic clock is sent off on a long out-and-return journey, at a constant high velocity relative to the Earth (except for three short acceleration periods) then this travelling clock will record the passage of less time for the journey than an identical atomic clock which stays behind on the Earth. Likewise, a travelling identical twin, going on a similar out-and-return journey, will appear younger than the stay-at-home brother or sister when they are reunited. For brevity one often states that the travelling clock “goes slow”, but relativists prefer the more precise statement that the travelling clock “records the passage of less time for the given journey”.

This effect was demonstrated very clearly by Hafele and Keating. They organised an experiment where some atomic clocks were carried around the world in two commercial aircraft. By making use of the fact that the Earth’s spin velocity is about 1000mph at the equator, one aircraft travelled round the world at a net speed of about 1500mph, and the other at a net speed of about 500mph. The slowing of the atomic clocks in both of the aircraft, when compared with atomic clocks on the Earth’s surface, accorded with time dilation predictions. It is worth noting that appreciable gravitational corrections also had to be made because of the reduced gravitational field at the height of the aircraft. Clocks will run faster in a weaker gravitational field.

It seems difficult to explain this result using a true, symmetrical, relativity theory. Consider the case of the out-and-return journey experiment for identical twins. The Earth-bound twin sees the other twin travel out and back at a high velocity. But the travelling twin also sees the Earth-bound twin travel out and back at a high velocity! However, there is a major difference. The travelling twin experiences some short acceleration periods at the start, at the midway turn-around point, and at the end of his or her journey. The predicted time dilation is considered to arise as a direct result of the accelerations that have to be given to the clock, or twin, to achieve the two constant-velocity periods. But note that, because the acceleration periods are very short, the time dilation which occurs during the acceleration periods is neglected. All of the calculated time dilation occurs during the long constant-velocity periods when the clock is in an inertial frame. It is because of this particular analysis that many physicists still feel slightly uneasy about the clock paradox or twin paradox. It is therefore worthwhile to develop the problem of a travelling atomic clock a little further.

We have the situation where a standard atomic clock, travelling at a constant high velocity, is running more slowly than an identical clock which has been left behind on the Earth. But the Earth is moving relative to the background frame given by the most distant stars. Where can one find a standard atomic clock that runs at the fastest rate compared with all other identical clocks? It is then inevitable that there must be a unique inertial frame in the universe where a standard atomic clock runs at its fastest rate. This will be the frame located on the point in space where all matter and energy existed at the time of the Big Bang. The variations of all travelling clock readings must be referred to this standard inertial frame. Hence, there is an absolute inertial frame of reference in the Universe that establishes an absolute time standard. Few relativists like to admit to such an unambiguous statement. But the majority of relativists do agree that it is unlikely that the clock paradox or twin paradox would exist if the framework given by the most distant stars and galaxies disappeared.

Where does this leave us when considering special relativity? Time dilation comes out of Maxwell’s equations. Maxwell’s equations on their own, without using any assumptions from special relativity, predict that a time transformation occurs for electromagnetic waves travelling between moving inertial frames of reference. This deduction comes from Lorentz transformations, which are a direct consequence of Maxwell’s equations.

At first sight, one could logically restrict the application of the time transformation prediction of Lorentz transformations to electromagnetic waves travelling between inertial frames, and exclude any application to atomic clocks or twins. The analysis used to predict electromagnetic waves from Maxwell’s equations might suggest that this is all these equations, and Lorentz transformations, should predict. When only electromagnetic waves are being considered the acceleration histories of the frames are irrelevant because complete symmetry exists when there is an interchange of electromagnetic signals between inertial frames. It is the frequencies of the waves that change and the Lorentz time transformation predicts the relevant frequency shifts for both transverse and longitudinal motion of a moving source. This approach, of limiting Maxwell’s equations and special relativity to just electromagnetic waves, is also logical in terms of the assumptions made in special relativity. The time-assigning-function assumptions adopted by Einstein when establishing special relativity are based solely on a consideration of electromagnetic wave signals passing between inertial frames. On this restricted basis special relativity would be a true relativity theory. But if this restricted approach to special relativity were to be adopted then it would be necessary to introduce a completely new theory to explain the time dilation applicable to complete clocks and identical twins.

But there is a second choice. Is it possible that Maxwell’s equations contain within them a built-in knowledge of the existence of distant matter in the Universe? At first sight one might not imagine that a few equations, developed to deal with local electric and magnetic fields, could contain within them an inherent relationship between the local fields and distant matter in the Universe. But they do.

Free magnetic poles are not observed in nature and are not permitted in Maxwell’s equations. However, free electric charges are observed and their effects are included in Maxwell’s equations. It is important to appreciate that any analysis involving free electric charges requires careful interpretation. Mathematically, one may apply Maxwell’s equations to a single point charge situated in an otherwise empty universe. The predicted electric field of this point charge will extend to infinity. But the existence of such a single point charge is not physically possible. There must be distant matter in the universe in order to create the local single point charge. A second distributed charge, equal in magnitude and of opposite sign, will inherently be created on this distant background matter. If the single point charge is accelerated to a high velocity v, relative to the background matter frame, and is then oscillated, the resulting electromagnetic waves will be time transformed when compared with the radiation predicted for an identical hypothetical experiment where v = 0.

Maxwell’s equations contain more information than is immediately obvious provided they are always applied after the necessary material boundary conditions have been established. The need for a background frame of reference is a direct consequence of examining the physical aspects of Maxwell’s equations in more depth. The required background frame may seem to be more tenuous when related to an overall neutral body, such as an atomic clock, but a material background frame of reference must be there in principle. One might expect an atomic clock to go slow when travelling relative to the material background frame. Time dilation is then readily explained, and there is no “clock paradox” or “twin paradox”. Relative motion affects the analysis, as it is does in all Newtonian analysis, but there is no need for a “special” relativity theory.

Maxwell’s equations contain all of the information that is necessary to solve all steady-state, and near-steady-state, relativistic problems involving inertial frames and material boundaries. No further assumptions from special relativity, concerning the velocity of light in empty space and time-assigning functions, are necessary. In addition, Maxwell’s equations predict an absolute time frame when applied to the real world.

The mathematics of special relativity may still be used for all problems involving inertial frames, provided that any electromagnetic wave propagation problem is limited to the steady state or the near steady state. However, it is clear that this mathematics arises directly from Maxwell’s equations. One only needs the principle of relativity and Maxwell’s equations to fully establish the nature of relative motion at velocities approaching c.

It is finally worth noting that all time transformations in general relativity must have their origins in Maxwell’s equations.

I hope that this web site will stimulate interest in looking at the fundamentals of gravitation and relativity theory. Until we begin to understand the origins of gravity, and the true significance of advanced potential solutions and material boundary conditions, there seems to be little hope of achieving a link between quantum theory and gravitational theory.