Paradox of Lorentz force.
No solution. Right angle lever.
Special relativity "really" ends. (14/ 7/21)
(Fig.1) Light speed is always "c" in any inertial frames. → Relativity.
Special relativity (SR) is the physical theory dealing with spacetime transformation proposed in 1905 by Albert Einstein.
Theory of relativity is so mysterious and there are many objections to this theory.
As shown in Fig.1, in the special relativity, the speed of light is always "c" from the viewpoint of any observers moving at arbitrary speeds.
To satisfy this unrealistic condition, the time and space are contracted or dilated like "fantasy".
(Fig.2) Ball speed looks different depending on observer's velocity.
In Fig.2, a ball is moving at a velocity "V".
Accoding to ordinary classical mechanics, when the observer is moving at a speed "S" in the same direction, the speed of the ball looks " V - S ".
(Fig.3) Light speed is always "c", irrespective of the observer's speed ?
Accodding to special relativity, the speed of light is the same for all observers, irrespective of whether the observer is moving or stationary.
So the speed of light always looks "c" from the observer's viewpoints, both in Fig.3 right and left.
As I say later, if we assume the ether moving with the earth, we can naturally explain this constant speed "c".
(Fig.4) K' frame is moving at velocity "v" to the right along x axis.
At time t = 0 ( t' = 0 ), the origins of K and K' frames overlap, and the light is emitted from the light source at the origin.
K' frame is moving in the positive x direction at velocity "v" relative to K.
At time t = t ( t' = t' ), observers in both K and K' frames measure the position of lights.
According to special relativity, in both frames ( rest and moving ), the light speed must be constant "c", so the following equations are satisfied.
(Fig.5) In both frames, the light speeds are constant "c".
As you see Fig.4, x' coodinate is moving relative to x. ( x' = x - vt )
So within ordinary classical mechanics, it is impossible to satisfy both equations of Fig.5 at the same time.
To do so, we have to accept the strange idea that the time and space are contracted or dilated by observer's movement (= Lorentz transformation ).
(Fig.6) Light speed is always "c" in any frames under Lorentz transformation.
If the time and space coodinates transform like Fig.6 in K and K' frames, both two equations of Fig.5 hold.
But of course, changing basic idea about spacetime distorts other basic physical principles.
About the caluclation, see also this section.
(Fig.7) Fatal paradoxes → special relativity is wrong.
This Lorentz transformation causes serious paradoxes such as Electromagnetism and
Right angle lever.
Unfortunately, these paradoxes have NOT been solved.
When you see various solutions to these paradoxes, you find they are ad-hoc and based on wrong assumptions.
(Fig.8) Journals, universities, textbooks .. are hiding true paradoxes from students.
When you see various textbooks, journals, websites of universities, and other media, you notice Only "fake" paradoxes such as ladder paradox and twin paradox are shown.
True paradoxes such as Lorentz force and right angle lever cannot be seen at all.
( Or they are put in the shade, inconspicuous places. )
As far as they continue hiding these true paradoxes, students and ordinary people will be deceived and their precious future will be jeopardized.
(Fig.9) Ether was denied. → dark energy, matter, Higgs = ether revived !?
As shown on this site, it is said that special relativity denied luminiferous ether.
But instead, they needs many kinds of matters filling all space, such as dark matter, dark energy, Higgs, and virtual particles.
So, the kinds of ether-like matters are increased in special relativity !
If we accept only one kind of ether, we can naturally explain dark matter, uniform CMB, without depending on virtual particles violating relativity.
(Fig.10) Faster-than-light inflation = special relativity !?
Surprisingly, because special relativity denied aether, the current physics demands faster-than-light cosmic inflation.
To explain the uniformity of microwave background even in superluminal regions, space must have expanded with superluminal speed, physicists insist.
But as you feel, these explanations are completely inconsistent with the heart of special relativity, which claims any objects cannot exceed the light speed.
(Fig.11) Virtual photon = tachyon !? → special relativity is wrong.
In the relativsitic quantum field theory, virtual particles always appear in Feynman diagram.
As shown in this site, it is known that these virtual particles always violate special relativity.
But ordinary people do NOT know this important fact and they are deceived by various media and textbooks.
Special relativity insists Coulomb force is caused by exchange of these virtual photons.
For energy and momentum to be conserved, this virtual photon must be tachyon, as shown on this page.
In conclusion, special relativity contains self-contradiction from the beginning, which means special relativity is wrong.
(Fig.12) Influential Special relativity is used in various fields.
Special relativity was introduced by Einstein a very long time ago in 1905.
So this idea was used in various relativistic theories such as standard model, QED, black hole, expanding universe and string theory.
If special relativity is wrong. all these theories would break down.
This is the main reason why physicists aim at various fantasy such as 10-dimensional string theory and superluminal inflation even now, though they are clearly fantasy.
(Fig.13) 26 → 10 dimensional string theory is left as the ONLY unified theory.
As a result, as long as physicists believe special relativity, ignoring true paradoxes, 10-dimensional string theory is left as the only unified theory.
This means, the instant new students start to doubt "fantasy" string theory, they can NEVER advance in their physical careers. This is disaster.
(Fig.14) Mathematical trick = Change of variable. QED is wrong.
As shown on this page, the calculation of magnetic moment g-2 by quantum electrodynamics (= QED ) depends on wrong math tricks.
In Fig.14, "change of variables" ( k = l - β ) artificially changed the result itself.
This one-loop correction is ultraviolet finite.
In two-loop correction, even after ultraviolet (= UV ) divergence is renormalized, complicated infrared (= IR ) divergence is left.
As shown on this site (p.25-), the manipulation of infrared divergence is much more difficult than UV divergence in higher-order correction. ( There is NO universal rule in removing IR. )
As many as eight years after one-loop calculation, they introduced various artificial new concepts such as modified propagator, Yennie gauge, and contact term to adjust them to experimental values.
( See "Particles. Sources, and Fields" vol 3, Schwinger )
So the loop corrections of anomalous magnetic moment are NOT natural results, at all.
(Fig.15) QED cannot predict Lamb shift at all.
In Lamb shift calculation, they artificially divided calculation into high and low energy terms to avoid infrared divergence.
But high energy term contains virtual photons, and low energy terms contain real photon.
So basically it is unreasonable to recombine relativistic and nonrelativistic terms. ( See "advanced quantum mechanics", JJ. sakurai )
Furthermore, we have to depend on numerical ( NOT analytic ) calculations and virtual excited states to get proper Lamb shift values.
( See this site (p.23) and The Quantum Theory of Fields, vol 1 p.592, Weinberg. )
So Lamb shift by QED is NOT first-principle, either.
(Fig.16) Michelson-Morley experiment = "ether" moving with the earth ?
In Michelson-Morley experiment, they tried to observe the electromagnetic wave's interference changes.
( So the light speed "c" depends on the wave nature of the light. )
The earth is moving around the sun.
So if some medium (= ether ) fills the absolute space, "ether wind" is blowing in the opposite direction to the earth's movement, which would cause light interference changes.
But there were no interference changes in the various directions in this experiment.
Ordinary relativity textbooks often say this experiment denied the "ether" (= ether wind ).
(Fig.17) All charges are moving with earth !
If the ether is moving with the earth like the air and all other things, the speed of light is always constant "c", NOT causing "ether wind".
And all charged particles including our human bodies are moving with the earth, and it is natural to think the electric field E among them is moving with the earth.
The velocity of the electromagnetic waves completely depends on some medium such as air and water moving with the earth.
(Fig.18) Relativistic quantum field theory accepts "ether wind" !
When most people hear the word of "ether", they blindly reject the idea without thinking deeply.
But as I said above, the present quantum field theories such as QED and standard model use various kinds of "ether" like matter.
If you reject the "ether", you have to use infinite virtual particles to express the Coulomb and magnetic forces, which fill all space. These virtual particles do NOT obey special relativity (= Einstein formula ).
And other things such as dark matter, dark energy and Higgs field also fill all space.
But they try NOT to admit these "ether-like" things are ether, because they are "relativistic" quantum field theory.
And they are doing just "Shut up and calculate !" about the present particle physics.
( These contradictory ideas clearly obstruct the develop of science of our human beings. )
(Fig.19) Moving objects are contracted.
In Fig.19, the original length of stick at rest is " l0 ".
After it starts to move, it appears shorter ( l < l0 ) from stationary observer.
(Eq.1) Lorentz tansformation.
In Fig.20, there is a stick which rest length is l0.
This stick is moving with K' frame at speed "v" relative to K frame.
So in K' frame, this stick is at rest, and its both ends are always x'2 and x'1 in K' frame.
(Fig.20) Stick is moving with K' frame at velocity "v" relative to K.
At time "t", both ends of this stick are at x2 and x1 in K frame.
So the length of this stick in K frame is l = x2 - x1.
The length of the object means the value measured at the same time "t".
Using Lorentz transformation of Eq.1, x' coordinates ( in K' frame ) of both ends are
Considering the rest length of this stick is l0 = x'2 - x'1,
As shown in Eq.3, the moving object becomes contracted in the moving direction.
( The length as seen by K frame observer looks shorter than the original rest length l0. )
This is called "Lorentz contraction".
(Fig.21) Lorentz contraction is "real" ?
According to the special relativity, Lorentz-contracted length of the object is treated as "reality".
In Fig.21 the object with a rest length of l0 is moving at speed "v" with respect to K frame.
And a lot of cameras are put along the wall of K frame, they take a snapshot of the object at the same time of "t".
In these pictures, the object length is always "l", not "l0" according to the special relativity.
Though this measurement is so difficult that there has never been any direct test of length contraction, as shown on this site.
(Fig.22) Which is the true circumference of disc ?? → Paradox !
In Fig.22, the circumference of the round disc is 2πR ( R is radius ).
And this 2πR is just equal to N l0 in the stationary disc of Fig.22 left. ( 2πR = Nl0. )
When the disc starts to rotate, its circumference becomes Lorentz-contracted, because rotation means the movement in the tangential direction.
But the radius "R" is NOT Lorentz contracted, because there is no radial motion. ( R → R )
Considering each length l0 of this rotating disc is contracted, its circumference is NOT equal to 2πR. ( 2πR < Nl0 ? )
This is one of fatal paradoxes of special relativity. (= Ehrenfest paradox. )
Even if the rotating "rigid" disc is broken, each crack of it is also contracted. This is strange.
Unfortunately, these "real" paradoxes can NEVER be found in ordinary relativity textbooks. Why?
( Caution: observer A is in the stationary hole at the center of this disc, so centrifugal potential (= general relativity) has nothing to do with it. )
(Fig.23) "Lorentz-contracted" conveyor belt. ← Paradox !
The upper Ehrenfest paradox can be easily imagined using conveyor belt.
In Fig.23 left, a conveyor belt is not moving.
If you rotate this conveyor belt, this belt becomes Lorentz contracted according to the special relativity ( Fig.23 right ).
So the whole length of this belt becomes contracted !
This is clearly a fatal paradox, because inner square frame is not moving and keep the same length.
(Fig.24) Belt is contracted or not, depending on observer's motion.
As I said above, the moving object appears shorter from the perspective of a stationary observer.
Even when the observer starts to move relative to the belt, its length appears contracted.
(Fig.25) The relation between belt's length and observer's motion.
In Fig.25 left, conveyor belt is moving relative to the observer in K frame.
So in this case, moving conveyor belt is Lorentz-contracted.
When the observer starts to move at the same velocity as the belt, the belt looks stationary from that observer's viewpoint (= K', Fig.25 right ).
In K' frame, the length of belt at rest returns to the original, and becomes longer than that of K frame.
(Fig.26) Inner frame becomes shorter in K'.
On the other hand, the inner frame is at rest in K.
So while the observer is moving (= K' ), this inner frame becomes contracted.
Because from the viewpoint of K' observer, the stationary frame is moving in the opposite direction.
(Fig.27) Belt is "separated" from inner frame by observer's movement ??
From Fig.25 and Fig.26, these conveyor belts and inner frame become separated due to their different total length. This is very strange.
(Fig.28) Horizontal wires ( at detector ) become 5 → 3 by observer's movement ?
In Fig.28 left, the inner frame is at rest, and the belt is moving and contracted.
Suppose five parts of belts are contained in one side of the inner frame.
And each belt on the upper side extends a wire in the horizontal direction.
As a result, the detector ( which counts the number of wires ) always feels five wires in Fig.28 left case.
In Fig.28 right, from the viewpoint of the moving belt (= K' ), the belt returns to its rest length, and the inner frame is moving and contracted.
So only three belts are included in one upper side of the frame in K'. ( ← 5 = 3 )
In this case, only three horizontal wires are detected by the detector
This is clearly fatal paradox.
Because the number of wires which detector feels decreases from 5 to 3 only by observer's movement !
Continuous act ( such as Coulomb force and long wires ) is the weak point of special relativity.
(Fig.28') Each belt with long wire and point detector.
We supplement Fig.28 with Fig.28'.
Each belt is supposed to have long wire, which is always parallel to its belt.
As shown in Fig.28', point detector can detect this wire, when its belt exists Only on the upper side of the frame.
As a result, fatal paradox appears in the detected number of belts on the upper side ( 5 → 3 ).
This detector detects this wire (= belt ) continuously, when the belt is on the upper side.
So the logic of different times in defferent inertial frame cannot be used here, as shown in simultaneous problem.
Continuous act reveals true paradox.
See also Special relativity "really" ends.
(Fig.29) Moving pole and barn
Here one pole is moving and Lorentz-contracted with respect to K frame.
In K frame, barn is at rest, and the length of barn is the same as Lorentz-contracted pole.
So when we pull down the shutter at time "t", the pole can enter the barn completely.
(Fig.30) Moving pole and barn
In K' frame, the moving pole is at rest, and instead, barn is moving in the opposite direction.
So in K' frame, barn is Lorentz contracted instead of pole.
As a result, the pole cannot enter the barn only in K' ?
(= pole - barn paradox. )
(Fig.31) Clock ticks differently in K and K' frame.
This is what we call simultaneous problems of special relativity.
When the time in K frame is "t", the times at x=a and x=0 in K' frame are different.
So in K' frame, the time t' at x = a is earlier than that of x = 0.
(Fig.32) From moving pole, the front wall of barn is broken.
As a result, in K' frame, first, the pole collides with the front wall of the barn ( and the wall is broken ).
Next, the pole enters the barn, and then the shutter is down.
So also in K' frame, the pole can enter the barn, they insist.
But this pole-barn paradox cannot solve the basic problem contained in special relativity.
As a result, fatal paradoxes of conveyor belts and Ehrenfest happen.
(Fig.33) Right-angle lever rotates or not ?
In Fig.33 left, two forces acting on the lever are in equilibrium and the torque is zero with respect to the rest frame observer. So the lever is not rotating.
But in Fig.33 right, due to Lorentz contraction of one arm and the change of the force ( see this (p.232), this (p.675) ), this lever rotates just by simple observer's movement ! ( See this page. )
This right-angle lever paradox has NOT been solved, and shows special relativity is wrong.
( As far as I check various websites, the important transformation of ordinary force ( NOT artificial 4-force ) is rarely seen. )
(Fig.34) Observer's looking power moves the charged particle ??
This is clearly a paradox, and has NOT been solved.
By the simple observer's movement, the external charged particle is attracted to the electric wire (S'), though from the stationary observer (S) they are not attracted to the neutral wire.
Charge and currents are four-vector current, and transform like spacetime under Lorentz transformation, as shown on this site (p.3).
This direction is the weak point of the special relativity. See this page.
Because accordng to the special relativity, new strange charges are generated just by simple observer's movement !
(Fig.35) Electrons teleport to the lower wire → repulsive force ?
According to the four-current of special relstivity, minus charges of the upper wire teleport to the lower wire just by simple observer's movement !
In this case, fatal paradox appears as shown in Fig.34.
And as you notice, the state of Fig.35 is very unnatural.
Because it is natural that the strong repulsive force among electrons in the lower wire does NOT allow this teleportation.
As a result, special relativity, which contradicts the law of nature, is wrong.
(Fig.36) Two directions = weak point of relativity.
In conclusion, the fatal paradoxes of special relativity appear when we try to deal with two directions.
Two directions ( r and θ, x and y, circular wire ) are weak points of the special relativity.
And all people involved in physics education carry grave responsibility for teaching something related to relativity to students, hiding its fatal paradoxes.
( Includig QED, standard model, general relativity, and relativistic quantum chemistry. )
Because they damage the student's precious future by teaching paradoxical relativity.
(Fig.37) K and K' are moving in the opposite directions.
In Fig.37 left, K' is moving at v in the x direction with respect to K.
But from the viewpoint of moving observer K', K is moving in the minus x direction at v (= Fig.37 right ).
So, we just exchange their coodinates, and replace v by -v in ordinary transformation of
The coordinate in K' frame is ( t', x', y', z' ), and K frame is ( t, x, y, z ).
Inverse transformation of Eq.4 becomes
If you substitute Eq.5 into Eq.4, you can confirm Eq.5 is right.
(Eq.6) Time dilation of a moving clock K'.
Again, K' frame is moving at v in the x direction with respect to K frame.
Suprisingly, the clock fixed at x' = 0 of K' frame ticks more slowly than the observer at rest in K frame ! ( dt > dt', see this site. )
Substituting "x' = 0" into the first equation of Eq.5, you can get Eq.6.
(Eq.7) Time dilation of a moving clock K.
From the viewpoint of K' frame, K frame is moving in the opposite direction.
Conversely, the clock fixed at x = 0 of K frame ticks more slowly than K' frame. ( dt < dt' )
So, irrespective of K or K' frames, a moving clock ticks more slowly than stationary one.
(Fig.38) Which clock ticks more slowly ??
So according to the special relativity, the clock in the moving frame always ticks more slowly.
From K observer, K' is moving ( K' < K ). But from K' observer, K is moving ( K < K' ).
As a result, we cannot know which clock ticks more slowly.
This is clearly a paradox ( = twin paradox. ) and is one of mysterious phenomena in the relativistic world.
Ordinary relativity textbooks often say this twin paradox is not a real paradox, and is caused by our misunderstanding about the special relativity.
And they often say we have to use the general relativity to solve this paradox.
In fact, the famous GPS time shows twin paradox, which means GPS has nothing to do with relativity.
And transverse Doppler effect includes this twin paradox.
(Fig.39) Who ordered the muon ?
It is natural that we think the lifetime of the muon becomes longer due to its higher energy itself NOT by strange time dilation.
Unnecessary and unstable elementary particle muon is thought to be a combination of electron and high-energy de Broglie wave (= neutrino ).
Because the neutrino can penetrate the earth easily.
(Fig.40) Twin paradox in the two identical satellites.
According to the relativists, GPS satellite is moving faster than us, so the GPS clock ticks slower than ours by 7 μs per day.
And due to the weaker gravity, GPS clock ticks faster by 45 μs per day. As a result 45-7 = 38 μs faster.
You may notice, this GPS time calculation includes twin paradox, as I explained above.
Because from the veiwpoint of the moving satellite, our observers are moving faster.
And as shown on this page, when the two identical satellites are moving in the opposite directions on the same orbit around the earth, their clock times clearly show paradox.
We can prove these are clearly twin paradoxes using general relativity including the centrifugal potential.
So famous GPS satellite relativistic corrections which uses special relativity include twin paradox and are wrong.
Atomic clock of the satellite uses frequency of the electromagnetic waves.
( We are NOT looking at the hands of their clocks. )
It is natural that we think some medium in the air causes their frequency difference, rather than fictional gravitons.
(Fig.41) Special relativity ( Dirac equation ) is one of most influential theories.
In spite of those fatal paradoxes, the relativity survived for more than 100 years.
So we have to explain the reason why the relativity with fatal paradoxes survived.
( And we have to think about the reason why the ordinary relativity textbooks NEVER write about these important paradoxes. )
The main reason exists in the very important Dirac equation.
Before the computers become popular in 1980s - 1990s, the Bohr-Sommerfeld model could not deal with very compilicated three-body helium atoms.
In this situation, Dirac equation could combine spin and relativity properly.
And unrealistic quantum mechanics dislikes real de Broglie waves (= ether ), and loves strange parallel-worlds.
Of course, Dirac equation is just equal to Klein-Gordon equation, in which the relativity has survived.
(Fig.42) Fizeau experiment and aberration denied ether dragging ?
Fresnel's theory of aether dragging was confirmed by Fizeau experiment in 1851.
In this experiment, the moving water "partially" dragged the light (speed), as shown in Fig.42A.
Relativists often insist that this experiment denied the complete aether dragging theory.
But as you notice, their insistences neglect the power difference between the very "small" water and very "big" earth, and the relation between the medium motion and its refractive index change.
(Fig.43) Photon can NOT be slower in the water.
According to the relativity, when the particle with some mass is moving at the light speed c, its energy E becomes divergent to infinity ( Fig.43 upper ).
So the photon's mass must be zero.
If the photon's speed becomes slower to c/n in the water, its energy E becomes zero, because photon has no mass. ( Fig.43 lower. )
Of course, this is inconsistent with the photon's energy E = hν.
As a result, the relativity does NOT allow the photon to be slower than "c".
And the relativists often insist Fizeau experiment shows the Lorentz transformation.
But admitting "c/n" speed in the water means the light speed completely depends on the medium.
( "n" is the index of refraction for water = 1.33. )
(Fig.44) No refraction → telescope needs to be inclined.
And as shown in Fig.42B and 44, the explanations about aberration using ether drag are often misunderstood.
From the viewpoint of the moving earth, the emitted light from the star changes its direction like the rainfall.
In Fig.44, an apparent velocity of the light is a little larger in the outer space, and after it enters the "earth ether", its velocity becomes "c".
In this case, we need to consider the refraction.
When we calculate the angle of the aberration, we usually consider sinθ is equal to tanθ
This means we can neglect the second order effect of v/c. ( "v" is much smaller than "c" ).
So no refraction happens, and we need to incline the telescople for the light to reach the bottom of it.
(Fig.45) If ether "drags" light → no aberration ??
I found a wrong counterargument to ether dragging, as shown in Fig.45.
They insist, the light from some star is dragged by earth ether at the horizontal velocity of "v", the instant it enters earth ether.
As a result, from the viewpoint of the observer moving with the earth, he cannot see aberration like Fig.45 middle, so ether dragging is wrong, they insist.
Of course, this explanation is wrong. Can you find the fault of this ?
According to Snell's law, the refractive index (= n ) can be expressed as the ratio of velocities in two different media ( = v1/v2 ).
This is equal to the ratio of light angles ( v1/ v2 = sin θ1/ sin θ2 ).
If their objection is right, this ratio becomes infinity ( θ2 → 0 ), which is impossible.
So the right answer is, the light from the star is little bent (= Fig.44 ), and we can observe aberration in ether dragging theory naturally.
(Fig.46) Transverse Doppler effect.
It is said that "transverse" Doppler effect cannot be explained by nonrelativistic effect.
But is it really so ?
Due to time dilation of moving object (= light source ), its original light frequency becomes lower at the stationary observer, they insist.
(Fig.47) Transverse Doppler effect = time dilation !?
In Fig.46,47, the light source is moving to the right at the velocity v, and emits the light upward when it just passes the observer.
As I said above, the clock with moving object (= K' light source ) ticks slower than K frame clock.
(Eq.8) Original light frequency (= ν0 ).
During the time of dt', the number of emitted light (wavelengths) from the source is dn'.
So the light frequency in K' frame is ν0 = dn/dt'.
(Eq.9) Observed frequency (= ν' ) becomes lower than ν0.
While the time dt' has passed in K' frame, the time dt has passed in K frame. ( dt > dt' due to time dilation )
As a result, the observed light frequency in K frame becomes lower, because dt is bigger than dt' ( "dn" is the same ).
So they insist this transverse Doppler effect is caused by the time dilation.
But from the viewpoint of the light source, K frame observer is moving ( dt < dt' ) , so this result becomes opposite (= twin paradox ). This is strange.
As a result, transverse Doppler effect has nothing to do with time dilation and relativity.
(Fig.48) Transverse Doppler effect by classical mechanics.
Instead of using this "strange" time dilation, we can describe this effect by "classical" method.
As shown in Fig.48, due to the movement of the light source, the emitted electromagnetic wave (length) is elongated.
As a result, the observed light wavelength λ' is elongated like,
The frequency is inverse proportion to the wavelength.
So the observed light frequency (= ν') becomes
(Eq.11) Observed frequency (= ν' ) becomes lower than ν0 also in classical picture.
Eq.11 is equal to Eq.9.
So we can explain transverse Doppler effect by the classical electromagnetic wave.
As you notice, the result of Eq.9 and Eq.11 show intermediate frequency between the frequency of the light coming toward us and that of the light going away, because it's "transverse" direction.
If you insist that the transverse Doppler effect is due to purely relativistic effect, you need to use a photon particle, because the relativity is using virtual photons as Coulomb force.
But basically it is very strange that we think the "photon particle is oscillating" at the frequency ν0.
(Fig.49) Relativistic invariance = Math and infinity.
You may hear the relativity and quantum mechanics are incompatible with each other.
But in fact, the special relativity is the basis of all important relativistic theories such as QED, standard model and supersymmetry through Dirac equations.
Relativistic restriction is too strict, so all we can use is only very abstract Math, in which there are no concrete objects.
Furthermore, to keep Lorentz symmetry, relativistic wavefunction must include infinite kinds of momentum and energy, which diverge to infinity.
Dirac and Pauli disliked QED, which artificially removes infinity.
Pauli wrote a letter to Dirac. In this letter, he said he wanted to quit a physicist and might be a novelist, because he was dissapointed at the fact relativistic quantum field theories always diverge to infinity.
(Fig.50) Positron contradicts basic physical principle.
Positron is the antiparticle of the electron.
The positron has a opposite charge +e and the same mass as an electron.
According to the current quantum theory, a positron easily annihilates with electron, forming γ rays.
The important point is that the existence of these antiparticles are completely inconsistent with basic physical principles.
This means antiparticles are unreal.
(Fig.51) Positron is produced from photon. Why "nucleus" is necessary ?
According to this site and this site, when a high-energy electron comes near a nucleus, and is deflected in its path, it radiates photons.
These photons spontaneously turn into an electron-positron pair.
But if this explanation is right, the nucleus (= target ) is NOT necessary.
ONLY under the strong electromagnetic field, each electron can radiate high-energy lights !
Then why they must collide electrons with other nuclei to produce positron ?
The reason of this is very unreasonable.
(Fig.52) Momentum of incident light suddenly becomes zero ( p = 0 ) !?
This website says,
If the photon only just had enough energy to create the mass of the electron-positron pair then the electron and positron will be at rest.
This could violate the conservation of momentum since the photon has momentum ( p = E/c ) and the two resulting electrons have none ( p = 0 ) if they are stationary ( momentum = mass × velocity ).
This means that the pair production must take place near the nucleus since they can absorb the momentum of the original photon.
In fact, this explanation is self-contradictory. See also this site.
It is impossible for the nucleus to absorb ONLY momentum, without energy.
Because absorbing momentum means absorbing some kinetic energy.
(Fig.53) Only momentum is NOT conserved.
As shown in Fig.53, when the incident photon has the energy of just 2mc2, one pair of electron and positron can be created, according to Einstein relation.
But in this case, these resultant electron-positron do NOT have kinetic energy at all.
Because all energies of the photon are used for their mass energy.
Of course, the incident light has momentum ( p = E/c ).
But the resultant electron-positron has NO momentum ( p = 0 ). This is strange.
(Fig.54) Convenient nucleus can absorb "excessive" momentum ?
Of course, the total momentum ( and energy ) must be conserved.
To be consistent with this basic law, they think the nucleus near photon absorbs this excessive momentum.
As you may notice, this explanation is unreasonable.
(Fig.55) This nucleus has "momentum", but NOT kinetic energy !?
Strange to say, this nucleus has momentum, but NO kinetic energy !
In the realistic world, this is impossible.
This fact clearly shows pair production of positron-electron is impossible.
A positron is just a positive ion or proton ( or scattered electrons ), considering it always needs collision with nuclei.
As shown on this site (p.17-18) and this site, photon-photon collider just collide charged particles and virtual photons. So they don't use "real" light (= photon ) at all.
(Fig.56) Photon's ( light ) energy is just converted into a pair.
In this section, we explain why positron production is unrealistic.
In Fig.56, photon's energy is just 2mc2.
So a pair of electron and positron can be created, and they are at rest.
K observer is stationary, too.
(Fig.57) If photon has lower energy, this pair production is inhibited.
As a result, if the light source emits photon with slightly lower energy (= E - ε ) than Fig.56, a pair production does NOT happen.
Because the photon energy cannot reach 2mc2.
(Fig.58) When observer moves, new "kinetic energy" is given to position and electron ?
If the observer of Fig.56 starts to move (= K' frame ), both a positiron and an electron also start to move from the perspective of this K' observer.
So in K' frame, resultant positron and electron have more kinetic energies
in addition to the original rest mass energies.
Though the light sourse is emitting the same photon.
(Fig.59) When light sourse emit photon with lower energy ... ?
As shown in Fig.58, if resultant positron ( electron ) has excessive kinetic energy, the light sourse can make emitted photon's energy lower to produce a pair of positron and electron at rest.
But this fact clearly disagrees with Fig.57.
This means a pair production of antiparticle is inconsistent with special relativity.
According to Doppler effect, the phoron (= light ) energy increases in K' frame than K frame.
Because light frequency becomes higher.
This means the amount of photon's energy for a pair production depends on observer's movement !
This is strange.
(Fig.61) Kinetic energy depends on observer's motion !?
If the energy can be converted into rest mass energy of antiparticle, the observer's motion can control this production ?
Because object's velocity (= kinetic energy ) depends on oberver's motion.
Even if a positron cannot be produced in K frame, it can be generated in K' frame.
This paradox clearly shows the existence of antiparticle is very doubtful, as shown on this page.
(Fig.62) Distance between resultant position and electron is zero ... V = -∞ ?
When the photon's has the energy of just 2mc2, resultant positron and electron are at rest.
But in this case, the distance between them is almost zero, which means potential energy between them is minus infinity !
So the energy is NOT conserved ( 2mc2 → 2mc2 - ∞ ).
This is also a paradox.
(Fig.63) Even when we break mass into pieces, total mass remains the same.
The important point is that the concept of mass energy is unreal.
Even if you can break the mass of some particle (= m ) into ( infinitesimal ) pieces, total mass remains the same (= m ).
So the idea that mass can be changed into light with no mass is impossible.
Dirac equation is indispensable for QED, standard model and string theory.
Here we show that Dirac equation uses relativistic momentum and energy (= four vector momentum ).
And these four momentum causes fatal paradox of right angle lever paradox, which means all these relativistic quantum field theories are wrong.
In the relativistic theory, clocks tick differently depending on their frames.
So it is useful to introduce some new concept of time, which is invariant under Lorentz transformation.
This is called "proper time" ( τ ).
In Eq.12, "u(t)" is the object's velocity in K frame, an "t" is the time in K frame.
( In special relativity, the velocity becomes different in different frames. )
The proper time is the time which is measured in system of coordinates moving along with the object.
So this proper time does not change under Lorentz transformation (= Lorentz invariant ).
From Eq.12, the proper time is
Eq.13 is Lorentz invariant scalar ( see Fig.64 ).
We can treat this proper time as a "constant" under Lorentz transformation.
Considerting Fig.64 and Lorentz transformation, the special relativity defines four vector as
As I explain later, the relativistic mass of a moving object is heavier according to the special relativity.
So the relativistic momentum (= p ) can be expressed as
Eq.15 can be expressed as
From Eq.12-Eq.14 (= proper time), the denominator of Eq.16 can be considered as a constant.
So Eq.16 changes as x compoment of the four-vector under Lorentz transformation.
Replacing x by ct (= zero component of the four-vector, see Eq.14' ), Eq.16 becomes
From Eq.17, we can define the zero component of the four vector, as follows,
where E is the relativistic energy of
Here we use the notation, p0 = - p0.
As you see the derivation of Eq.17, these relativistic momentums transform like spacetime ( ct, x, y, z ) under Lorentz transformation.
So these are called "four-momentum".
As you see in Fig.64, and from Eq.15 and Eq.18, these relativistic momentum and energy satisfy the following important equation,
The important point is that Eq.21 does NOT change under Lorentz transformation (= scalar ).
In the quantum mechanics, the relativistic energy (E) and momentum (p) are replaced by the operators like
Using Eq.22, Eq.21 can be expressed as
where φ(x) is called Lorentz scalar field (= Klein- Gordon field ).
So Klein-Gordon (K-G) equation is just equal to special relativity.
For φ(x) to be scalar, this φ needs to include infinite kinds of momentums, which lead to ultraviolet divergence
As shown on this page, Dirac equation can be gotten from Klein-Gordon equation.
So Dirac equation itself is just equal to the special relativity.
Dirac (or K-G) equation uses the time and space coordinates ( = kx ) as four vector, which is based on the special relativity (Eq.24).
If the faster-than-light neutrino is correct, this means the Dirac equation and the special relativity are wrong.
If so, all other relativistic theories (= QED, standard model, string theory ) using Dirac equation, are all wrong.
See also this calculation.
In this section, we calculate constant light speed "c" under Lonretz transformation.
K' frame is moving at velocity "v" to the right along x axis relative to K frame.
(Ap.1) K' frame is moving to the right relative to K.
At time t = 0, the light is emitted from the source at the origin.
( At time t = 0, both origines in K and K' frames are in the same position. )
(Ap.2) Light speed is constant "c", in both K and K' frames.
After some time ( t in K frame, t' in K' frame ), both observers measure the positions of light in each frame.
Though only K' frame is moving, and K frame is at rest, the light speed must be the same "c" from the viewpoints of both these observers, according to strange special relativity.
(Ap.3) Light speed is always "c" irrespective of observer's movement.
As a result, special relativity requires that both two equations of Ap.3 hold at the same time.
To do so, the variables of time and space must transform like,
(Ap.4) Lorentz transformation.
According to Lorentz transformation of Ap.4, the moving clock ticks slower than one at rest !
And the length of any object in a moving frame appear contracted. See also this site.
This unnatural transformation causes serious paradoxes.
(Ap.5) Classical vs. Lorentz transformation.
As shown in Ap.5 upper equations, only x coordinate transforms like x' = x - vt, when K' observer is moving in the positive x direction, in classical mechanics.
In this case, two equations of Ap.3 cannot hold.
(Ap.6) Classical transformation.
But of course, in this classical mechanics, fatal paradoxes do NOT appear.
(Ap.7) Lorentz transformation is unrealistic.
When you substitute Ap.4 into the first equation of Ap.3, you find the second equation of Ap.3 also holds true.
This is called "Lorentz transformation".
But as you feel, it is very unrealistic to think the clock ticks slower only by object's motion.
The concept of "time" should be common irrespective of object's velocity.
(Ap.8) Michelson-Morley experiment = "ether" moving with the earth ?
When you look around us, all objects including air are moving with the earth.
All these things consist of electrons and nuclei which attract each other by electric field.
So if we assume the ether moving with the earth, we can naturally explain constant light speed "c", without depending on strange Lorentz transformation.
(Ap.9) Velocities change from different frames.
K' frame is moving in the x direction at "v" with respect to K frame.
From the observers of K and K' frames, the velocity of the same one particle can be expressed as
where u(t) is the velocity from K, and u'(t') is that from K'.
As shown in Ap.10, the velocity of the same one particle looks different in different frames.
Because the time and space change in different ways in different frames.
This velocity u(t) is different from v of β in Lorentz transformation (Ap.11).
"v" of β means the velocity of K' frame itself ( not particle ), which is constant in the special relativity.
The particle's coordinate ( x, y, z ) is the functions of the time t (or t' ), because its position is changing with time t.
So the particle's position ( x, y, z ) is NOT independent from the time t.
For example, the time transformation of Ap.11 can be expressed as
According to AP.11 (and Ap.12), the velocity in the x direction u'x(t') from K' can be replaced as
As I said above, inverse Lorentz tansformation is
From Ap.14, we have
Substituting Ap.15 into Ap.13,
Solving Ap.16, we obtain
Using Ap.11 and Ap.15, the y component of velocity becomes
Substituting Ap.17 into Ap.18 and solving it, we obtain
In the same way, the solution of z velocity is
Next we differentiate Ap.17 with respect to t.
From Ap.11, we get
Dividing Ap.21 by Ap.22,
Ap.23 means the acceleration from the viewpoint of K' inertial frame.
Here we choose the moment when the particle's velocity ux is just equal to "v" of K' frame's velocity.
So the particle is moving in K frame. But it is at rest in K' frame.
In K', the particle is at rest, so it must satisfy Newtonian mechanics using Newtonian force F', as follows,
Newtonian mechanics of Ap.25 is the basis of the relativistic momentum.
Substituting Ap.24 into Ap.23, we get
Here we use the relation of
From Ap.25-Ap.27, we get the relativistic momentum ( = px ).
To get the relativistic momentum, we have to define as follows,
The definition of Ap.29 (= transformation of Newtonian force ) is very important.
( x component of Newtonian force does NOT change under Lorentz transformation. )
But this definition is "artificial" to get the relativistic momentum.
We suppose the Newtonian force Fx doesn't change under Lorentz transformation in Ap.28.
But in other directions, we need to change the forces to define the proper relativistic mass.
( So the relativistic momentum is not a "natural" result gotten from the special relativity. )
In K' inertial frame, the acceleration in y direction is
Differentiating Ap.19 with respect to t,
Inserting Ap.31 and Ap.22 into Ap.30,
In Ap.32 we suppose only Fy exists.
And uy is zero here .
The particle is moving in the x direction in K frame, and the force Fy tries to accelerate the particle in the y direction.
K' is moving in the x direction at the velocity ux. Substituting Ap.24 into Ap.32,
Using the definition of relativistic momentum of Ap.28 and Ap.33,
As the particle is at rest in K' frame, we can use the usual classical relation of ( F' = m0a' ) in K' frame.
To match Ap.34 with Ap.28 (= relativistic momentum ), we have to change the force Fy, as follows,
This definition is inconsistent with Ap.29.
(Ap.36) Summary in Newtonian forces and relativistic mass.
The important point is that these artificial definitions cause fatal "right-angle lever paradox (Trouton-Noble experiment paradox)".
(Ap.37) Right-angle lever paradox.
Here is a right angle lever with a stationary point and two arms.
This lever is moving with K' frame, and is in equilibrium in K' frame.
( In K' frame, two arm's length and forces are equal, so its torque is zero. )
(Ap.38) Rotate only in K frame ?
But in K frame, the lever is moving in the x direction, so
one of the lever's arms is Lorentz contracted, and only Fy is weakened.
So the torque is NOT zero only in K frame, and the lever rotates only in K frame just by simple observer's movement !
(Ap.39) Which is true ?
As shown on this page, this fatal paradox has not been solved, and shows relativistic theories such as QED, standard model and string theory are all wrong.
Because all these theories depend on the relativistic momentum and energy through Dirac equation.
Next we try another method to get the relativistic momentum and energy.
We define " four-vector velocity " (= ω ) , as follows,
where the proper time (= τ ) is Lorentz invariant, so Eq.51 change as four-vector such as (t, x, y, z) under Lorentz transformation.
From here, we use the next notation of
i = 1, 2, 3 components of four-vectors are
where Eq.12 is used.
And ω0 becomes
where we use x0 = ct as shown on this page.
where g (or η ) means Minkowski metric tensor of
Using μ = 0, 1, 2, and 3, Lorentz transformation of Eq.2 can be expressed as
where Loretnz matrix is
As a result,
So we can get the relation of
Here we define new equations of
Using Eq.53 and Eq.54, we can get
When we use the same variable twice (like μ μ ), this means the sum of 0-3 components.
We differentiate Eq.62 by the proper time ( τ ), and divide it by 2,
where Eq.61 is used.
We introduce the new variable ωμ(τ) and fμ(τ) irrespective of reference frame.
So the next relation is naturally satisfied.
As shown in Eq.51, ωμ(τ) change as four-vector. So if fμ(τ) also change as four-vector,
We can get Eq.64.
So fμ(τ) of Eq.61 also change as four-vector like ωμ(τ).
We call these fμ(τ) " four-vector force ".
Here we define f i (τ) (i = 1, 2, 3 ) as
where F means Newtonian force.
This definition of Eq.66 is "artificial" to get the relativistic momentum.
And substituting Eq.53, Eq.54 and Eq.66 into Eq.64,
From Eq.67, the solution of 0 component ( = f 0 (τ) ) becomes
Using Eq.12, Eq.61 can be expressed as
Substituting Eq.53 and Eq.66 into Eq.69, we can get the equation of motion of
So we get the relativistic momentum ( = p ), as follows,
And substituting Eq.54 and Eq.68 into Eq.61,
So we get the relativistic energy ( = E ), as follows,
These results are the same as the upper section.
As I said, the definition of Eq.66, which represents the relation between Newtonian force F and the four-force, is artificial to get the relativistic momentum.
And this causes right-angle lever paradox, which rotation shows the weak point of the relativity.
According to Maxwell's equation, the current density (= J ) and charge density (= ρe ) satisfy
Of cource, the total charge is conserved irrespective of reference frame,
So the equations of Eq.74 must be Lorentz-invariant.
( See this page. )
But this definition of Eq.74 causes the violation of the total charge in the electric wire, as shown on this page !
This means the special relativity (= Lorentz transformation ) itself is wrong.
So we have to admit "ether" which is moving with the earth. (In this case, we don't need to use the relativistic restriction, because Lorentz transformation itself doesn't exist. )
Eq.74 can be expressed using four vector ( xμ ), as follows,
ct can be replaced by x0
And if we define the new four vector (= four current density ), as follows,
Eq.75 can be expressed as
Using Lorentz transfomation of Eq.56, the differentiation by xμ is
Substituting Eq.78 into Eq.77, we have
As a retult, Eq.77 (= Eq.74 ) proves to be Lorentz invariant.
2014/5/16 updated. Feel free to link to this site.