A Development of the Relativity of Lorentz and Poincare
Lorentz derived the equations we refer to as the Lorentz transforms by a process of trial and error. His aim was to show how Maxwell's equations could be equally valid both in a reference frame at rest in the aether and in the laboratory which was supposed to be in motion through the aether.
Lorentz went on to show that according to Maxwell's equations, a system of charges in equilibrium such as that which ponderable matter is composed of would contract in the direction of motion by a factor . Fro this starting point, the rest of the Lorentz transforms can then be derived as well as the transformations for electric and magnetic fields.
Poincare proved that the Lorentz transforms together with rotations of space form a mathematical structure called a group. The outcome of this is that if two observers are moving through the aether at different velocities, the measurements of time and position of events that they both record can be related one to the other by Lorentz transforms once the co-ordinate axes have been adjusted so that both x axes are in line with their relative velocity and clocks adjusted so that clocks at the origins of the two systems both read zero at the moment the origins coincide.
We can see from the last of his 1906 lectures at Columbia University, New York that Lorentz was not aware of Poincare's work when he first read Einstein's 1905 paper. In those lectures he gives Einstein the credit for first realising that the Lorentz transforms were of universal application.
Sir Edmund Whittaker in his 1953 History of the Theories of the Aether and Electricity gives an account of what we would now call the Special Theory of Relativity under the chapter heading "The Relativity Theory of Poincare and Lorentz" giving one Einstein the credit only for the equation for the relativistic Doppler effect. While Whittaker is regarded as being "over the top" in this treatment, he makes the point well that Einstein was not the originator of the mathematics of Special Relativity.
With hind sight, we can identify two rival relativities in the years after 1905, one popular theory according to Einstein and the other half forgotten according to Lorentz and Poincare. The difference between the two is conceptual, not mathematical and they both share the same formulae and results. Conceptually, Lorentz Poincare relativity is based on Maxwell's assumption that the aether is the seat of all electromagnetic interactions. It seeks to explain why it has been impossible to detect the motion of the laboratory through the aether. Einstein does not deign to use the word aether, but refers to "the stationary system" and shows that the concept is superfluous.
We can best understand the difference by considering the Lorentz-Fitzgerald contraction. In LP relativity, the motion of matter through the aether results in the generation of a magnetic field by the motion of the electric field. The magnetic field is also in motion and it attempts to generate its own electric field. The result is a modification of the electric field of the moving charge. In a lump of solid matter held in shape by the equilibrium of the forces between all its positive and negatively charged components, the equations governing the equilibrium can be changed back into their original form by the substitution and it is this which accounts for an actual physical contraction in the direction of motion. This contraction then has an effect on clock rate and synchronisation such that if an object at rest in the aether were seen from our laboratory system, it would appear to be contracted.
If we look at the situation as Einstein described it, two observers in relative motion each try to set up a system of co-ordinates and clocks. Each knows that the speed of light measured on a path to a mirror and back is a constant value and so they use this method to synchronise their clocks assuming that the clock by the mirror should be synchronised on the basis that the light takes equal times for the there and back journeys. This results in disagreements about synchronisation which results in different perceptions of each other's system. An object at rest in the other observer's system appears contracted and a clock in his system appears to run slow.
It seems to me that the reason why Einstein's concept gained over that of Lorentz and Poincare was that there were so many problems with aether theory that it really did not give a satisfactory basis upon which to develop other theories. Indeed Whittaker goes to great lengths to show how Poincare and Lorentz had difficulty with the concept of the aether and might well have thought of their relativity in much the same way as Einstein.
In latter years, the spelling has changed, but in the world of alternative science, ether theorists of one type and another still abound. One of them referred to my "Stasis Theory" as "An ether theory without any ether", and that remains to date the best way of understanding it.
Stasis theory tries to explain why light travels at a constant speed and why and how electric and magnetic fields interact.
We start with the astronomical observation of a class of distant objects which show a periodic variation in brightness and frequency of the spectral lines of the light coming from them. We call these objects binary stars, however the phenomena is now also used to identify star and massive planet systems. The fact that we can observe these phenomena at all can only be explained by assuming that the light from them travels through each local region of space en route at the same speed. The velocity of the individual star from which it came affects only its frequency and not its speed. This is a statement about the equal speed of photons from different sources through the same region of space at the same time and in the same direction. Note that this is quite different from the statement that any attempt to measure the speed of light in a vacuum will produce the same numerical result. We make the assumption that each region of space has a local property of "background-ness" which is responsible for moderating the speed of the photons and ensuring that their velocities relative to that background are constant.
We next note that currents in wires generate magnetic fields. We assume that the magnetic field is generated by the relative motion of electric fields through one another and consider how this might work in the case of two, three of more charges in an otherwise empty universe. The simplest and most logical explanation is that the strength of the interaction within each charge's electric field is proportional to the velocity of that charge. That then begs the question as to what its velocity is measured with respect to. It seems reasonable to suppose that nearby charges will have a greater influence than those at a distance. We can take this into account by calculating a weighted average velocity with respect to the other charges. As to a weighting factor, field strength would seem to be a reasonable first guess.
These arguments lead to a formal definition
The vector is a local property at a point P of space with the dimensions of velocity measured with respect to an inertial frame. Each charge in the universe has velocity in that frame and is at a distance from P.
The motion of an individual charge in that reference frame generates a magnetic intensity and magnetic induction .
where points from the i th charge to the point P in space at which its magnetic intensity is experienced. is the contribution to the magnetic intensity at P by the i th charge. is the microscopic magnetic intensity which exists in the space between and surrounding charges. The symbol in the equation means 'would like to be equal to' and indicates that the magnetic induction also has other laws to obey such as continuity. These equations do not include the affects of varying magnetic fields and electromagnetic radiation.
In the case of photons of light and of radio waves, we know that Maxwell's equations describe a situation in which the motion of the electric field generates the magnetic field and the motion of the magnetic field in turn generates the electric field, the two sustaining each other. We now only have to make one more assumption and that is that the same background presence against which the velocity of the electric field works to generate the magnetic field is the same presence against which the velocity of the magnetic field generates an electric field.
Thus stasis theory offers us an alternative to the various aether and ether theories. If we calculate the stasis vector for our laboratory, we get the sum of two vectors, one of about 16 m/s pointing back along the earth's orbital path and the other of about m/s pointing to the west. For more details, see my paper on Stasis Field Theory.
One of the problems with Lorentz Poincare relativity is that the fundamental processes have to be causal. Einstein's relativity depends only the method of clock synchronisation. In his 1915 publication of his 1906 lectures, Lorentz details a theory accounting for the increase in mass with high velocity. A number of determinations for electrons emitted as beta radiation from radioactive material had been carried out. By 1915, a firm pattern had emerged in which the electron behaved according to classical principles of mechanics and electricity and magnetism as if it had two different masses opposing acceleration in the direction of motion and acceleration perpendicular to it. These two masses were called respectively longitudinal and transverse mass and were given by the formulae:
Lorentz gives a brief account of a derivation of his of this result in his 1906 lectures: however, in 1905, this pattern was not supported by the experimental results which favoured an alternative result by Abraham which Lorentz described in detail. By 1915 new experimental results favoured Lorentz's theory and he notes this in the publication of his lectures.
Lorentz's result depends on a combination of three factors. The strength of the magnetic field surrounding the charge due to its motion is enhanced by a factor ; the charge and its field suffer a Lorentz contraction and the energy content of the electric field of the charge somehow remains constant. This last factor is highly dubious. In my second paper on inertia as an electromagnetic phenomena, I show how Lorentz's result can be be justified by assuming that the equipotential surfaces of the electric field of the charge and its electric induction both suffer Lorentz contractions.
Let us first describe the electric field of a stationary charge. It has three descriptors in classical theory which can all be calculated at a point in space given by the position vector from the centre of the charge and are called potential, electric field strength and electric induction.
The electric induction is given on the assumption that the charge is surrounded by vacuum, but at the scale of electronic components, it behaves like a flux with continuity rules. This electric flux can only terminate at the surfaces of charges. Looking at a single charge, we see that the total amount of electric flux emerging from its surface is fixed. There is also a relationship between the electric field strength and the potential which can only be described mathematically.
These last two equations define an unbreakable relationship between and . To examine the immediate geometrical implication of this, we need to refer to a rigorous mathematical definition of the gradient of a potential function . First we must understand the potential is a function of position and has values everywhere in space. Second that we can draw equipotential surfaces through all the points in space with the same value of . The gradient at a point P is formally defined as being normal (perpendicular from any view) to the equipotential surface through P.
Now the electric field of a moving charge is derived in a number of different ways in the text books all giving the formula
and the texts describe this as a radial field. That means that always points along the radius vector . If we combine this with the fact that is always normal to the equipotential surfaces, then these surfaces must be spherical about the centre of the charge. This implies that any two must be everywhere the same distance apart and that implies that the magnitude of must be independent of direction contradicting the formula. The only explanation is that the electric field strength must be the sum of two sources.
Where is the electric field due to the charge and associated with the spherically symmetric potential and is the induced electric field due to the motion of the charge.
There are two significant experimental facts. Moving clocks run show and moving objects behave as if their mass has increased. In Lorentz's relativity, these are real causal effects while in Einstein's they are due to the observation of events occurring in one inertial frame by an observer in another. Let us perform a little thought experiment.
Let us imagine a long straight train track of the future in which trains run at very high speeds in an evacuated tube on maglift suspension. We measure out a section of track down which the train will run at constant speed allowing track for acceleration and deceleration at either end. At the ends of our measured track are two clocks which are synchronised with each other using the radar method. A receiver on the train is able to interrogate each clock as it passes.
On the train are a number of clocks. An Einstein light clock, a quarts crystal clock and several others each consisting of a rotating cylinder mounted on friction-less bearings with their axes pointing in various directions. These latter record the passage of time in the same way that one of the famous twins might count the revolutions of his craft against the background stars if it were set spinning to provide artificial gravity by centrifugal force. We will only consider things from the point of view of the observers on the train and consider the scenario in which all the clocks on the train keep time with each other. The observers discover that their clocks have run slow recording a shorter interval than that found by subtracting the two time signals from the track side clocks which they passed. Perplexed by this, they make the return journey discovering the same effect still occurs.
The observer of the light clock is quite happy and says she can explain the slowing in terms of the diagonal path of the light pulse as it moves backwards and forwards between the moving mirrors. A little effort with Pythagoras Theorem yields the result that the clock runs slow by a factor of and this indeed is the result which they have all observed. She has read physics at university and understands about these things. In terms of the factor , the clock has run slow by a factor . ( being greater than 1, that means that the period between ticks is times longer and the recorded period of time on the train is found by dividing the track-side period by .)
The other two observers are stubborn believers in aether theory and prefer Lorentz's explanation. One of them argues thus. One of my cylinders was mounted with its axis pointing in the direction of motion. None of the forces on it during acceleration could have had any effect on its rotation. Therefore its rotational kinetic energy remained unchanged by the acceleration. But its period of revolution did change and therefore something else had to change to slow it. Its radius remained the same maintaining its radius of gyration k, so that only leaves its mass.
The angular velocity is times the number of revolutions divided by the elapsed time which is smaller by a factor . For the rotational kinetic energy to remain constant, the mass must increase by a factor . That is to say that he can explain the slowing of the rate of revolution by a factor of by assuming that the mass somehow increases by a factor of
The observer of the quartz clock knows that the crystal oscillates with simple harmonic motion governed by the equation.
Here is the angular velocity and the SHM has a period . The term is derived from the modulus , (restoring force per unit displacement) divided by the mass.
The exact constant of proportionality depends on the type and geometry of the SHM system, but we have the same relationship between mass and and consequently time. So the observer of the quarts clock can also explain its slowing by assuming that mass increases by a factor .
Since in a thought experiment, we are not too limited by practical considerations, yet another time measuring device might be constructed in the form of a mass sliding along a smooth horizontal rod across the width of the carriage. During its acceleration, there is no component of force perpendicular to the direction of the train and the component of its kinetic energy in that direction remains constant. The situation is similar to that of the cylinder clocks and again the slowing of time can be explained by assuming that the mass increases by a factor of .
It is very important to realise that the two effects
are completely undetectable by the occupants of the train because they always cancel each other out.
If this line of thinking is correct, we can take a Lorentzian view and conclude that motion through the stationary system results in the generation of magnetic fields surrounding all elementary charged particles which are enhanced by a factor . This results in their inertial mass increasing by a factor of slowing all time dependent processes by a factor , and the two always combine to make any attempt to measure the effect from within the moving system impossible. We can alternatively view it from an Einsteinian perspective and say that it appears to us when viewing mechanical clocks at rest in the moving system S' that they have slowed by a factor of and that if we were assume a definition of mass within S' such that moduli of elasticity, etc. are the same in S' as they were before it was accelerated, it would appear that mass is increased by a factor of .
The observer of the light clock, schooled in modern relativity might bulk at this suggestion, but reference to Einstein's 1905 paper show that he first derived a factor of for the transverse mass. However, the following quotes from Einstein's 1905 paper amply demonstrate that there is a degree of arbitrariness in these definitions.
"Now if we call this force simply 'the force acting upon the electron,' * and maintain the equation -mass x acceleration = force- and if we also decide that the accelerations are to be measured in the stationary system K, we derive from the above equations"
"With a different definition of force and acceleration we should naturally obtain other values for the masses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously."
The footnote says "* The definition of force here given is not advantageous, as was first shown by M. Planck. It is more to the point to define force in such a way that the laws of momentum and energy assume the simplest form."
Our only guide is to maintain self consistency. The observers on the train are forced into discovering that all their clock ran slow by the same factor compared to track-side time. Everything appears to be the same when moving as it was when stationary. Things are the same size, the mechanical and electrical properties of materials should be the same. Only inertial mass can change, and that only if we assume a Lorentz type model for the electron.
It is interesting to note that we have had to use the principle of conservation of energy in these thought experiments. Had we used the conservation of momentum, we would not have obtained the correct result. This suggests that the conservation of momentum is not universally applicable lending comfort to the student of electrodynamics who discovers that magnetic fields exert forces on moving charges which are perpendicular to their direction of motion and preserve kinetic energy but not momentum.
We often think of time dilation as being very important. We might imagine that a clock in a satellite orbiting 7,000 miles per second or more runs slow. The fact of the matter is that the effect of gravity is far more dominant. The clocks on GPS satellites run fast by a factor of about 5 times the amount by which we might expect them to run slow.
Gravity, or to be more precise, gravitational potential slows the rate of time dependent processes. The so called atomic clocks do not set an absolute standard of time: they are affected by their height above sea level running slower if they are taken to a lower location and faster if taken to a higher one. Each clock must be "steered" to the correct rate after installation. If the gravitational potential at a point is defined as the work per unit mass which be done remove a mass from that point to an infinite distance, the time dilation factor is . (Note that as the work done in moving the mass away, is a negative quantity and is less than 1.)
An odd quirk of relativity is that Einstein derived the equations for the effect of gravity on time in a thought experiment using a gramophone turntable world in which centrifugal force acted as an artificial gravitational field. The clock slowing is due to the velocity of the clock on the rotating turntable with respect to the rest frame of the centre. This has an interesting consequence for a clock on a ship. Sea level is affected by centrifugal force so that as the ship moves from the high to low latitude, although its clock should run faster because it is moving further away from the centre of the earth, it should run slower by exactly the same amount because its velocity due to the rotation of the earth is increasing.
Einstein's wonderfully naive deduction of the result goes like this. The velocity of a point on the turntable is and so the time dilation is given by . The centrifugal force on a mass is and the equivalent of gravitational potential is given by
So if we write , and make use of the fact that , the time dilation factor is
Remembering that gravitational potential is by convention negative, that is to say that the time interval measured by the clock at gravitational potential is less than the time interval measured by the clock at zero gravitational potential because t
he former runs slower. This formula agrees with experimental observations: wonderful, or just a happy coincidence?
Due the polarisation of space, energy is stored in the electric fields of electrons, and the quarks form which neutrons protons are built . The brace theory of gravity assumes that the internal stress of the electric field must be braced against the fabric of space. This results in a minute distortion of the fabric of space reducing the amount of energy stored. If we consider a mass brought from great distance towards a planet, the distortion its constituent charges causes to space is insignificant, but it experiences the distortion caused by the electric fields of all of the constituent charges of the planet. As it gets closer to the planet, energy is released from its electric field. Half of this is transferred to the fabric of space and half is available as gravitational energy to do work in accelerating the mass and overcoming resistance to its motion. If the gravitational energy released as the charge moves a distance in the -x direction (towards the planet), an amount of gravitational energy is realised. This is equal to the an increase in energy stored in the distortion of the fabric of space and the source of this energy is a change in the energy content of the electric field of the charge by . The relationship between these quantities being expressed in the three equations.
We can represent this diagrammatically by greatly exaggerating the distortion.
For each charge, the diagram shows the fraction of energy lost from the electric field. The electric potential at the surface of each charge is reduced from its original to V due to the loss in energy, and this change is proportion to change in energy contents.
We could at this point get involved in a long argument about whether or not the total energy content of a charge is equal to the energy content of its electric field, and the validity of Einstien's equation . However, whatever conclusions we come to only effect the stiffness which we attribute to the fabric of space. As soon as we use those assumptions again in calculating the relationship between gravitational energy and energy content of the electric field, the same factors come into play and cancel out.
At this stage, we say that the the effect of bringing a charge into the gravitational field of a massive body is twofold. The total energy content of its electric field is reduced and the electric potential at its surface is decreased. We know that the effect is a slowing of time dependent process. In motion at high velocity, we saw a similar effect and attributed this to an increase in inertial mass resulting in a slowing of the angular velocity associated with SHM. To have the required effect on time dependent processes, we must increase mass or reduce the coupling forces. We can achieve the desired result by assuming that the inertial mass remains constant and that the coupling forces are reduced. This requires us to understand the process as follows.
The ability of the electric field of the charge to generate magnetic fields by virtue of its motion remains constant, but the property of the field by which it stores energy and generates electrostatic force is reduced.
If we now remember the relationship between the SHM angular velocity and the mass and elastic modulus of a vibrating system.
We see that a decrease in by the same fraction as an increase in m would give the same effect on the time dependency of the oscillations. Decreasing decreases increasing the period of oscillation and slowing down the time dependent process. If SHM describes the oscillation of the quartz crystal of a clock, the clock runs slow by a factor equal to the square root of the change in the modulus of elasticity of the crystal. Since this depends on the electrostatic forces between atoms, the effect is proportion to the change in the gradient of the the electric field potential of the constituent charges of matter. This is proportional to the change in potential at the surfaces of the charges and the time dilation factor is:
Which is the result predicted by Einstein in his General Theory of Relativity.
These two results are identical, but the reasoning behind the is quite different. While Einstein bases his theory on the belief that God designed the universe so that we poor creatures would all see nature behaving according to the same laws of physics regardless of our velocity of location in space, the derivation I have given is causal. Einstein views time as variable. I view time as an absolute and look for factors affecting the rate at which clocks attempt to measure the passage of time. To explain gravity, Einstein relies on mathematics so hideous that the standard texts omit the more demanding parts of his derivations. The brace theory on the other hand is simple elegance.
If we consider a photon to consist of energy stored in equal amounts its electric and magnetic fields, it then has a certain similarity with the Pure Charge Model of an electron or quark. The only difference lies in the geometry of the fields. We would expect the photon to have a gravitational mass proportional to its total energy and an inertial mass in some way related to the inertial mass of a charge. But since it lacks spherical symmetry, the same constants of proportionality may not apply. As a photon approaches a star, what happens is that the star's gravity acts on the photon doing work which increases its energy and consequent frequency. If the photon is on a path which will pass close to the star, then a component of the gravitation force will act in the transverse direction and accelerate the photon. We know that experimental results have indicated the correctness of of the formula:
where is the angle of deviation and d is distance of closest approach. We take this result and see what relationship between gravitational mass and inertial mass would be required to explain the bending of light by a gravitational field in purely Newtonian terms.
Consider the motion of the photon at some time after it has passed the point of closest approach. We impose Cartesian Co-ordinates with the origin at the point of closest approach, the x axis pointing along its path and the y axis towards the sun. It has travelled a distance and the direction of the sun makes an angle with the direction of motion at the closest point of approach. The gravitational pull of the sun therefore has a component in a perpendicular direction of
This force acts for a time causing an increase in the transverse velocity
Thus the component of transverse velocity acquired after passing the point of closest approach is
The angle of deviation is then given by
On substitution of we get the required result
This requires us to understand how the gravitational mass of the photon might be twice its transverse mass. According to the brace theory, both magnetic and electric fields generate an internal stress in the fabric of space. Distortion of the fabric of space effects magnetic fields in exactly the same way as it affects electric fields. The gravitational mass of a photon is thus determined by its total energy which is . On the other hand, inertial mass is associated with the generation of the magnetic field by the motion of the electric field. The inertial mass is therefore determined by .
Newton defined three types of mass according to the generation of inertial forces, the experience of gravitational forces and the generation of gravitational fields. For mater at speeds much less than the speed of light, these three types of mass are, as Newton said, identical. The behaviour of high speed particles shows that this is not universally true. It is but a small step to realise that the geometry of the electric fields plays a role in determining the mass. This requires a deeper understanding of the action of gravity on photons in which both the bending of the path of the photon and its changes of frequency are accounted for. The allocation of gravitational mass and longitudinal mass of and a transverse mass of half that value to the photon, together with the fact that its speed is constant can account for the observed gravitational effects on light without any need to invoke the theories of general relativity.