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 and independent of the source velocity - 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”. Although relativists declare that there is no paradox they still get into trouble when some aspects of Mach’s Principle are raised. Einstein showed that there is a difficulty in defining both inertia and an inertial frame of reference. At a really fundamental level one reaches a circular argument. Special relativity can only state that a particular frame is an inertial frame if this frame is found to be the one in which a standard clock indicates the largest passage of proper time between two events. Hence, one can only establish the result that one is trying to deduce by making an observation!
But let us look at the problem of the clock paradox and the twin paradox at a simpler level. There is no need to get involved with the epistemological difficulties within special relativity to solve this problem.
If an atomic clock is sent off on a long out-and-return journey, at a constant very high velocity relative to the Earth (except for three short acceleration periods at the start, at the turn-round point, and at the end) 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 in the following discussion I will state that the travelling clock “goes slow”, but the more precise statement is that a travelling clock “records the passage of less time for the given journey”.
This effect was demonstrated very clearly by Hafele and Keating in 1971. They organized 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.
Following this early experiment, further confirmation is given by the fact that all of the clocks used in satellites to provide satellite navigation systems are adjusted so as to allow for the time dilation of the moving clock.
It seems difficult to explain this aspect of time dilation using what appears to be a symmetrical theory like special relativity. 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 will also see 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 prediction, agreed by all relativists, that if a standard atomic clock is accelerated, and it then travels at a constant very high velocity relative to the Earth, it will run more slowly than an identical clock which has been left behind on the Earth. But the Earth is moving relative to the Sun, the Sun is moving relative to our galaxy, and our galaxy is moving relative to the background frame given by the most distant galaxies. All of these bodies have had a previous acceleration history.
A clear question arises. When considering only relative motion effects, where could one locate a standard atomic clock so that we may be certain that it will run at the fastest rate compared with all other identical clocks? Based on our present knowledge, this location will be in the inertial frame that is stationary against the overall background given by the most distant observable matter in the universe. The slowing of all moving clock readings must be referred to this standard preferred inertial frame. Hence, there is a unique preferred inertial frame of reference in our 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 galaxies disappeared. The reason that most relativists are prepared to go this far is because they accept Mach’s Principle. Mach’s Principle states that the whole set of inertial frames is determined by distant matter. If most relativists are prepared to go this far it seems strange that they will not accept an absolute time scale, based on a specific, unique, preferred inertial frame which is directly related to the totality of all of the matter in our universe. This is the only matter that could interact with any standard clock, even if other universes exist.
If most relativists agree that the clock paradox only arises because of the presence of the most distant matter in our universe, is there any other way that they can be persuaded of an even greater significance of this distant matter? It may be possible to convince relativists by going back to the fundamentals of Maxwell’s equation and special relativity. Special relativity is inherently subservient to Maxwell’s equations, and any new information coming out of Maxwell’s equations may be relevant.
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, and are an essential requirement if Maxwell’s equations are to apply in all inertial frames.
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 form of the analysis used to predict electromagnetic waves from Maxwell’s equations might suggest that this is all that 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. We have shown earlier that 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. These assumptions are not valid for pulse signalling where precursor transients are relevant, because quantum theory is then necessary. 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. Hence the mathematics needs to be interpreted carefully. Mach’s Principle also raises objections to the concept of having a single point charge, or a single transmitting antenna, situated in an otherwise empty universe.
For the case of the “single” point charge 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 prior application of these material boundary conditions is particularly important whenever one is considering the reception of electromagnetic energy, or the creation of electric fields in space.
The presence of a material receiving antenna (or optical detector) requires that the observed velocity of an electromagnetic wave in free space must be equal to c. This result arises from the correct application of the advanced potential solution. The mathematics of special relativity may then be developed without making the usual assumptions of special relativity. A theory with the minimum of assumptions is always more desirable.
The concept of radiating energy from a single transmitting antenna into an otherwise empty universe is a dubious concept. However, it is even clearer that the concept of having a single point charge, producing an electric field in an otherwise empty universe, is a totally flawed concept.
The need for a material background frame of reference is a direct consequence of examining the nature of the required boundary conditions for Maxwell’s equations in more depth. The required material 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 then expect an atomic clock to go slow when travelling relative to the material background frame. Time dilation is then readily explained in terms of motion relative to this specific background frame. There is no “clock paradox” or “twin paradox”.
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 (the mechanical steady state) provided that any electromagnetic wave propagation problem is limited to the electromagnetic steady state or the near steady state. However, it is clear that this mathematics arises directly from Maxwell’s equations. One only needs to assume the principle of relativity (that states that Maxwell’s equations should apply in all inertial frames) and Maxwell’s equations to fully establish the effects of relative motion at velocities approaching c.
It is worth noting that all time transformations in general relativity, which are produced by motion, must have their origins in Maxwell's equations. How does this deduction affect general relativity? As an example, consider the problem of predicting the orbits of the planets in the solar system. General relativity is unable to give a complete solution for any rotational problem, but can only give relativistic corrections to the Newtonian solution. The Newtonian solution for planetary orbits appears to be inadequate because, using Galilean geometry, it predicts that gravitational forces must act instantaneously over very large distances (the action-at-a-distance requirement) if angular momentum is to be conserved. However, special relativity suggests that it should be impossible to transmit this gravitational information faster than the velocity of light c. As was mentioned earlier, general relativity solves this problem by introducing an intermediate step in the analysis. Initially, the presence of all of the matter in the solar system warps local space-time, and this distortion of space-time travels over the region at a velocity equal to c. When the steady state has been reached the warped space-time then appears to act on the individual planets instantaneously and permits the predictions of the relativistic corrections to their Newtonian orbits.
General relativity is combining three distinct effects. Relativistic space distortion is primarily a local effect produced by the presence of local matter in the solar system. It is related to the coordinate system of the solar system. Relativistic space distortion arises from the assumption of the equivalence principle in general relativity. However, relativistic time dilation is made up from two components. One component is due to the gravitation field at a given point and is related to the frame of reference of the solar system. But there is also relativistic time dilation due to motion, which is produced by the motion of a planet relative to the coordinate system associated with the background of distant matter in the Universe. This second component is a consequence of the assumptions of Maxwell's equations.
It is usually considered that gravitational effects can always be transformed away (or materialized from nowhere) in general relativity by a suitable change of the coordinate system and geometry. Although Machian effects may arise in general relativity, it has been assumed that they can always be transformed away. This is why the present interpretation of general relativity is unable to incorporate the stronger form of Mach's Principle, as is illustrated by its inability to explain the classic two fluid stars problem. However, Maxwell's equations require the presence of substantial amounts of distant matter to produce the component of time dilation arising from motion in general relativity. It is Maxwell's equations that require general relativity to be grounded in the background material of the Universe.