In 1687, Newton published his laws of motion in Philosophiae Naturalis Principia Mathematica. The laws were three scientific laws, which could basically explain and predict the behaviour of moving bodies. Later on however, experiments with high velocity particles were found to give different results then those predicted by Newtons Laws. Newton seemed to be in trouble. The Kinetic energy was no longer being proportional to the velocity squared, but was heading to a much higher value. The more the particles velocity approached the speed of light c, the greater the discrepancy between the measured KE and Newton's predicted KE. At this point, Lorentz came up with the idea of a multiplying correction factor, g= 1/Ö(1-v2/c2).
This lead to the idea of relativistic mass, a mass equivalent to gmo, where mo is the rest mass. This was further developed by Einstein in his special relavity (SR) theory, which was more or less the public version of Lorentz & Maxwell's work. SR had introduced for the first time quite 'weird' effects like time dilation and increase in mass of a moving body. One of the strangest parts of special relativity as we know it today is the conclusion that two observers who are moving relative to one another, will get different measurements of the length of a particular object or the time that passes between two events. Consider two observers, each in a space-ship laboratory containing clocks and meter sticks. The space ships are moving relative to each other at a speed close to the speed of light. Using Einstein's theory, each observer will see the meter stick of the other as shorter than their own, by the same factor g. This is called length contraction. Each observer will see the clocks in the other laboratory as ticking more slowly than the clocks in his/her own, by a factor gamma. This is called time dilation. This is what special relativity predicts, and although experimental results seem to agree, everybody still feels that there is something wrong. Newton's laws became the result of SR equations for the condition g=1, and as long as the mathematical predictions were then in perfect agreement with experimental values, everyone was happy to accept the requirement for such weird effects to be part of nature, even though no logical explanation was ever found. Although this solved the discrepancy between theory and experiment, it degraded the scientific laws, as the correction factor could not be explained in terms of a physical model.
In an attempt to visualise a physical model, I transfered both Newton's law and experimental results onto a geometric diagram to better interprete Lorentz factor. The below diagram has been sketched following Newton's laws, experimental evidence and common sense.
A spherical particle of mass mo leaves the source to reach its destination, distance S apart in time t. Note that although a point particle (zero dimensional object) is still accepted in most physics textbooks, it is an impossibility and cannot be used to define a particle. Its mean translational velocity is equal to v = S/t. Experimental evidence shows that this translational velocity can be in the range zero to very close to c, the speed of light, so geometrically, v can be shown as the projected shadow of velocity c, which makes at angle of q with v. And since v<=c, than c must always be the hypotenuse of triangle c,v,a.
Also, from Newtonian mechanics, we know that the total KE is equal to the sum of the body's translational kinetic energy and its rotational energy, or angular kinetic energy:
Total KE= Translational KE + Rotational KE .... If Vreal is the resultant total velocity, then
VREAL2 = v2 + Vo2 .... where v and Vo are the translational and orbiting velocities at a point in time
This relation shows us that VREAL, v, Vo form a second right angled triangle, with VREAL being the hypothenuse. The translational kinetic energy of such a moving particle is equal to 1/2mv2, where v is the linear velocity of the sphere, that is equal to the straight line distance S between source and destination, divided by the time t taken to travel through the whole path. This equation holds very well for non-relativistic mechanics, but experiments involving particles travelling at relativistic speeds, show that KE does no longer obey the equation for translational velocity v=S/t, but shoots up to infinity for v approaching c. This implies that although we 'see' the particle leave the source, and reach its destination S metres apart in t seconds, its real resultant KE is somehow not equal to the calculated translational KE 1/2mv2 or 1/2m(S/t)2. How can this be? This is where Lorentz, and Einstein had their fatal mistake. They reasoned, ..well, if KE=1/2mv2 is not being followed, and v=S/t, then, the particle's mass must be changing. As you will see, they were wrong! The mass is not changing at all, it is the real path of the particle which can no longer be approximated as a straight line, especially when v approaches c. When one looks again at the relation for VREAL their mistake becomes obvious - they assumed a zero rotational KE, that is a null Vo. The object would in fact be rotating/spinning around the path connecting source to destination, a helical path being a good example. This is the key to understand relativity. One could easiely understand how motion about its own axis can actually change its path length, whilst still reaching its destination point. The distance from the source to the destination divided by the time taken is due the translational velocity path limited to c, but the actual helical path divided by the same time taken results in a much higher velocity, not limited to c. So at any point in time, the real velocity is in fact travelling at an angle to the linear velocity v, at a higher speed. We also know that velocity of light as seen by a particle is totally independent of its real velocity, so the velocity of the real path taken by the particle is always normal to the velocity of light. So we know that c is perpendicular to VREAL. Now we also know that as v tends to zero, angle q tends to 90 degrees, and VREAL and v become almost equal meaning that they tend to become parallel to each other. As v tends to c, angle q tends to zero, and VREAL and v will approach an angle of 90 degrees to each other, whilst VREAL will grow infinitely long. This means that the angle between v and VREAL is equal to 90 - q. Since triangle v,Vreal,Vo is a right angle triangle, and the angles between vectors c & v, and Vreal & Vo are equal, then triangles 'c,a,v' is a similar triangle to trianlge v, VREAL,Vo.
Consequences of the above description:
Lorentz and Einstein were wrong in their interpretation of experimental results.
The path travelled by a particle can only be approximated as a straight line either in calculus, or as the mean velocity tends to zero. So strictly speaking a particle travels in a straight line only at v=0, in other words, a particle CANNOT travel in a perfect straight line. Nor do electromagnetic waves travel in a straight line, they only spiral along a line.
For Newtons laws of motion to apply at relativistic speeds, the velocity taken into account must NOT be the mean velocity v=S/t but the real velocity VREAL along the real particles path. It's not Newton's law which need a correction factor, we only need to take into account an orbiting velocity which is ALWAYS greater than zero.
Although the particle cannot reach its destination before an other particle which could theoretically cross the path in a straight line at the speed of light, its REAL VELOCITY along its real path, can exceed by far the speed of light. Still, its information content in the direction of the 'imaginary' straight line path cannot travel faster than light. I refer to the straight line path as imaginary for the reason that nothing is really travelling in this path, but only spiralling around it.
The velocity of light as seen by the particle real path is totally independent of the particle's velocities.
Derivation of Lorentz Factor using Newtons Laws on a body having both linear and angular velocities
Now that we are armed with a better understanding of the actual velocity components of any moving particle, we can easiely derive Lorentz factor using the above diagram, by applying simple geometry!
VREAL2 = v2 + Vo2.... (1) by Pythagoras
VREAL/Vo = c/v .... from similar triangles
Vo = v*VREAL/c .... (2)
Substitiuting for Vo in equation 1
VREAL2 = v2 + (v2/c2)*VREAL2
VREAL2(1-v2/c2) = v2
VREAL = v/Ö(1-v2/c2)
VREAL = gv.... where g= 1/Ö(1-v2/c2)
This means that most mathematics derived by Einstein and Lorentz still hold true, but have a different meaning, a meaning which unlike time dilation and distance contraction, does make sense and can be easiely explained by a physical model of the particles' actual path of travel.
From the above, it follows that travelling in a straight line is something which nature abhors, and a perfect straight line travel occurs only at v=0, or in calculus as dS/dt tend to zero. Also, refering again to our relativity velocity vector diagram, the resultant real velocity VREAL is made up of two normal vectors, one of which is v, which points in the same direction joining the source to destination. So, we know, that VREAL is really the resultant of two velocity vectors v and Vo which are normal to each other. This might not make much sense until you follow the helical path diagram which shows how such path must look to satisfy all the above conditions.
A helical path is one example in which the resultant velocity is made up of two velocity vectors v & Vo which are always normal to each other at any point in time. This is a path in which the ratio of VREAL to v is eqaul to Lorentz factor, resulting in a kinetic energy value which goes to infinity as the mean velocity v tends to c, but where no distances shrink, no time dilates and no mass goes to infinity! At low mean velocity v, much less than c, Vo the orbital velocity of the spiral will be very small, the path will resemble much to a straight line, and VREAL will be almost eqaul to v and to S/t. In such a case Newton's laws will give the correct results even if the path is approximated as a straight line, and angular velocity assumed null. As velocity increases, the orbiting speed will also increase and VREAL will increase to superluminal helical velocities, but the magnitude of the mean linear velocity to its destination will be still less than c. Applying Newton's laws on a straight path will no longer yield the correct results, because the angular velocity is no longer negligible. So, the correction factor g is only required if one totally ignores the angular motion. Once angular velocity is put into the equation, the correct kinetic energy is obtained.
This model does not exclude superluminal speeds, however it still has the speed of light limit within its linear part, the velocity which we measure by measuring the time it takes for a particle to travel from source to destination. This also clearly explains why we do not see photons along their travel. We see photons at the radiating source, and at destination, but since they are superluminal during their helical journey, they are not visible along their path! Also, it is kind of silly to assume that when a photon is released, its total KE is only made up of translational KE and totally ignore its rotational KE. In fact from the above it is obvious that a mass with zero angular KE is not a mass at all.
From the particle section discussion, we know that matter (defined as having mass) is made up of standing waves. The below picture shows a simple form of matter made up of a helical standing wave. A pair of helical waves is all required to generate matter. All elementary particles should be of this form, yes, whether it is an electron, an atom, a quark or any other newly discovered particle it will be of this form. An electron is such example. The positron is exactly the same but goes backwards in time. All it means is that v,Vo and VREAL point to the opposite directions. The spin is simply v.
For the condition Vo/v= Ö2, or q=35.264 degrees, we get Vreal = cÖ2.
Applying Newton's law to find the total internal energy of a particle:
E= 1/2m [Ö(2)*c]2
E = mc2
So knowing about the helical path we can derive Einstein's equation directly from Newton's equation, not the other way round! More important is the fact that we can finally construct a physical model for ALL matter. Note that the above standing wave is made up of pure electromagnetic waves, and that the whole circular helix has the properties of matter. Now, if external energy (kinetic & rotational) is supplied to this helix, the whole structure will start moving in a helical path of greater dimensions. The grey entities moving around the bigger circular helix will thus be the original helices. The bigger helix will still have the properties of matter, but its standing wave will be home to a number up of smaller 'particles' which can be made to increase or decrease in quantity by kicking them with enough energy. This mechanism is the fundamental mechanism of nuclear theory. All smaller helices within one larger helix will have exactly the same properties and be similar in size and frequency. Each helix size will thus exist in a different heirarchy level, with the lowest level being the smallest helix that is made up of pure electromagnetic waves, that is with no internal circular helices. The relation between heirarchy levels is governed by the fine structure constant which actually defines the maximum speed limits for v and Vo at which lower stage helices can move around the main circular helix.
The figure above, is a much better scientific explanation of the origin of matter and what one would expect to get when bombarding matter in particle accelerators. It also solves the enigma of the point particles. Nobel laureate Paul Dirac, who developed much of the theory describing the quantum waves of the electron, was never satisfied with the point-particle electron because the Coulomb force required a mathematical correction termed renormalization. In 1937 he wrote, This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small not neglecting it because it is infinitely large and you do not want it! [P. A. M. Dirac, Nature, 174, 321, p 572 (1937)]. The below figures is basically what present books teach (not a joke).
So, we collide two cherries, and get pears, bananas, apples and all fruit varieties! Of course, with no physical model for matter, nuclear theory offers a lot of enigmas and surprises to scientists. Once we start sorting out matter on different heirarchy levels, everything starts to get more clear. For example, we can now say that all known atoms exist in the same heirarchy level. They differ only in the number of helical anti-nodes (protons), their v/Vo ratio (proton to neutron ratio), and the number of smaller helices moving around (electrons), but they are on the same hierarchy level. The fact that the number of protons is usually equal to the number of electrons indicates, that for a stable atom, each structure antinode can handle one lower hierarchy helix within it. Let's say these higher level structures are the cherries. Once we collide these two big helices into each other, some of the lower level(s) helices, get mechanically dislodged and since they are standing wave circular helices on their own, they will be detected as independent matter, say pears and bananas. So, why were pears and bananas not visible in the first place? Simply because their helical velocity is faster than the speed of light, and anything faster than the speed of light cannot be detected! The pears and bananas are no longer spiralling around the cherry structure at superluminal speeds, but have now been kicked off their orbit and travelling at a much lower velocity resulting after impact. If one splits the resulting pears, then apples may be detected and the process continues, with energy levels going up as lower heirarchy levels are approached. The process continues until plancks energy level is reached, at which point the resulting outcome of the bombardment will not be a standing wave (detected as matter) but pure travelling electromagnetic waves at Plancks frequency and energy, travelling at the speed of light.
Calculating speed limits for v and Vo.
ERydberg/2/EBohr = EBohr/ECompton = ECompton/Eclass = a= 1/137.036
From E=mc2, we can therefore get the relation in terms of masses:
MRydberg/2/MBohr = MBohr/MCompton = MCompton/MClassical = a= 1/137.036
This clearly shows that a is nothing but the mass or energy ratios of a circular helix standing wave to a similar helix of higher hierarchy level. If we take the lower hierarchy level helix as our 'stationary mass' Mo, we have:
MREAL = gMo.... where g= 1/Ö(1-v2/c2)
...but MREAL= 1/a Mo, which implies that for sequential hierarchy levels in matter, a = 1/g
a= 1/g = Ö(1-v2/c2)
Also, c a = Ö(c2-v2), that's why I have put c a in the relativity diagram on top of this page.
The fine structure angle q = ArcSin(a), so:
Fine structure angle q = ArcSin(1/136.036)= 0.418111 degrees. The real superluminal helical path velocity at which the internal hierarchy levels move within the structure is
VREAL= g v = v/a= 137.036v
VREAL= 137.036*0.999973374c= 137.032c
So, strictly speaking Einsteins equation E=mc2 is not exact since it assumes that the translational velocity v can reach c, whilst in fact it is limited to a maximum limit of 0.999973374c. So the exact equation for energy-mass equivalence is:
E= 0.999946748 mc2