Energy conservation in the Standard Model and Unification of Forces

Update (15 August 2007):

Recent email exchanges with Guy Grantham, David Tombe, Jonathan vos Post and others have led me to reconsider the presentation of ideas delivered in this blog in posts such as this and this.  If you immerse yourself deeply in a particular problem, it can be difficult for others to grasp quick explanations you give.  I’ll leave both those posts and add here some introductory material.

The Standard Model U(1)xSU(2)xSU(3) is defective in including U(1) for electromagnetism which has only one charge and one gauge boson (reasons explained below and in previous posts).  U(1)xSU(2) requires a Higgs field to break the symmetry by giving the weak bosons of SU(2) mass at low energy, which limits their range.  The Higgs field cannot make falsifiable predictions and has not been verified.  In addition there are no falsifiable predictions from attempts to unify the U(1)xSU(2)xSU(3) standard model with gravitation.  Instead of U(1)xSU(2)xSU(3), consider the simplicity of SU(2)xSU(3), where SU(2) is an entire electroweak-gravity unification which can be understood by the SU(2) weak symmetry in the standard model as modelling weak isospin (a handed charge), weak hypercharge/electric charge, and gravitational charge (mass) all in one, by utilising 3 weak gauge bosons which exist in both massive and massless forms.  The massless forms of the W_+ and W_- gauge bosons mediate positive and negative electric force fields of electromagnetism, while the massless version of the Z_0 mediates gravity.  The massive forms of these three gauge bosons are the same as those in the existing weak isospin theory.  The mechanism by which mass is acquired by weak gauge bosons limits their interactions to one handedness.  The simplicity of having a massless spin-1 gauge boson for gravity is demonstrated by the following mechanism:

Galaxy recession velocity: v = dR/dt = HR.  Acceleration: a = dv/dt = d(HR)/dt = H.dR/dt = Hv = H(HR) = RH2 so: 0 < a < 6*10-10 ms-2Outward force: F = ma. Newton’s 3rd law predicts equal inward force: non-receding nearby masses don’t give any reaction force, so they cause an asymmetry, gravity. It predicts particle physics and cosmology. In 1996 it predicted the lack of deceleration at large redshifts.

Above, v = HR is the Hubble law (1929) based on empirical observations; see the great page discrediting lies about “tired light” which has no empirical evidence whatsoever unlike redshift due to the recession of light sources: http://www.astro.ucla.edu/~wright/tiredlit.htm.

Now remember that when we look to bigger distances R, we’re looking back in time (sun is seen as it was 8.3 minutes ago, next star is seen as 4 years ago, cosmic background radiation – the most distant light source – is redshifted fireball radiation coming from 14,700,000,000 years ago).

Hence, the Hubble parameter H = v/R can be written better in terms of v/time, v is recession velocity and time is the time in the past past when the light was emitted, because we only know the recession velocity v as a function of time past, not as a function of distance R irrespective of time (while the starlight was in transit to us for many years, the star’s velocity will have changed!).  Notice that the constant v/time is an acceleration.  Specifically:

a = dv/dt = d(HR)/dt = H*dR/dt = Hv = H(HR) = (H^2)R

This acceleration is very real.  It’s something like 6*10^{-10} ms^{-2} at the biggest distances, which is small compared to the accelerations we know, but is massive in terms of the amount of mass in the universe which is accelerating outward!

Newton says (2nd law): F = ma.  The mass of the observable universe is approximately 10^80 hydrogen atoms or about 10^53 kg (with a factor of ten error, depending mainly on what assumptions people make about whether dark matter exists, etc.).  This means that the outward force is on the order of F = (10^53)*(6*10^-10) = 6*10^43 Newtons (approximately).

That’s a big outward force, but what’s more exciting is that Newton’s 3rd law of motion says that there should be an equal inward reaction force:

F_outward = -F_inward

That means that pushing inward on every point there is 6*10^43 Newtons or whatever!  The only stuff it can be in the known facts of QFT and GR is graviton type gauge boson radiation.  Now imagine there is a shield called the Earth below your feet: each electron and quark has a cross-sectional area equal to its tiny black hole size, radius 2GM/c^2, which reflects gauge bosons (fermions are gravitationally trapped Heaviside energy currents, black holes).  What is the effect you calculate should result from the asymmetry in this 6*10^43 N shielding by the Earth?  Turns out, it predicts 10 ms^{-2} acceleration towards the earth at the surface!

Regarding the earlier post https://nige.wordpress.com/2007/06/20/the-mathematical-errors-in-the-standard-model-of-particle-physics/, fig. 4 shows  radiation is the propagation of energy, which occurs in two modes:

(1) massless, electrically “neutral” (as a whole, obviously within the spatial dimensions of the photon the electric field is not neutral) radiation propagates because it goes with an electric field (electric “charge”) which is varies in strength transversely in such a way that the magnetic fields cancel out, so there is no prevention of propagation due to infinite magnetic self-inductance.  That can go either with an oscillation longitudinally (like AC electricity, photons, gamma rays) or not (like DC electricity and as radiation these are gravity-causing gauge bosons).

(2) massless, electrically charged radiation propagates only when there is equal going from charge A to charge B and from charge B to charge A; this exchange makes the magnetic field curls cancel, allowing propagation.

A small sized pulse of oscillatory energy is a “real photon”.  If you have radiation trapped by gravity in a small loop, the opposite travelling radiation on each side of the the loop will cancel the magnetic field curl of the other, permitting it to exist.  This is how you get a fermion core, such as the core of the electron.

The E-field lines in the photon show where exchange radiation energy is going: the “displacement current” effect in a real photon is due to exchange radiation (virtual photons) because the field strengths involved are typically way lower than Schwinger’s threshold for pair production to occur: hence there is no actual “displacement current” of virtual vacuum charges in weak fields, just radiation effects.

To recap:

a) – there are two possibilities, neutral photons which can oscillate longitudinally as well as transversely (light photons, gamma rays, etc) and neutral photons which don’t oscillate longitudinally, and so only vary in strength transversely (flat-topped waves like DC, long TEM waves, or gravity-causing exchange radiation).  If such radiations are trapped by gravity into a small loop, you get a “static” charge of small spatial size, radius 2GM/c^2.  This looks like a “particle”.

b) – as fig4 of my reference explains, continuous emission of flat-topped transverse-only radiation (with no variation of field strengths longitudinally, unlike Maxwell’s picture of a light wave) is gauge boson radiation, which can be charged or uncharged.  If charged, it causes the electric fields in space between charges.  If uncharged, it causes gravity.

What is curious is the question of whether they are ignoring it 99% due to bigotry and 1% due to ignorance, or 100% due to bigotry.  If the former case is true, than I can potentially help matters by continuing efforts to overcome ignorance.  If the latter is true, it’s hopeless.  Another question, should I be concerned at all that work is ignored?  In other words, is the aim just to find out facts, or to find out and then publicise (market) the facts?  The marketing of ideas is quite a different area of work from coming up with them; in particular, it must address the facts you have to the needs of potential readers of the paper, which are possibly quite different in direction from most of the things you might actually prefer to write about.

On the other hand, the basic principle of modern marketing is having a great productString “theory” is an example of an successful marketing effort, although it should be criticised on the basis that “it” is 10^500 models, and therefore the best version of the theory is not even wrong.  I don’t however think that string “theory” is a good example of modern marketing.  In string “theory”, the public who get most excited are those people who turn to science as an alternative belief system to religion; religion loses credibility to them, and they believe in string instead.  The High Priests at the top of the string community like Witten believe in string as the “best” option just as the ignorant disciples at the grass roots of that community.  String “theory” is a great example of a financially successful modern religion involving several branches of mathematics.  It’s not a good marketing success in the scientific sense, because it’s not delivering the type of science that is now most needed.  Marketing cigarettes is not a good modern marketing strategy because it’s not giving people what they need: marketing an addiction to expensive products which cause an increased risk of lung and throat cancer is a poor business success.  String theory is similar.  It’s addictive, it destroys objectivity and replaces it by non-falsifiable faiths which lead the poor believer into a fantasy land of subjectivity; it’s based on ignorant speculations, not entirely upon solid empirical factual foundations.  Proper marketing of science should succeed at some point when the factual foundations can be made clear to all, and when bigotry towards “alternative” ideas has been dispelled.  God knows how long that will be, or how much effort it will take.

(End of update)

The electromagnetic gauge bosons: electricity (Heaviside energy or TEM – transverse electromagnetic – wave) energy waves are directly observable ‘gauge bosons’, because the light velocity force field which mediates electromagnetic interactions and accelerates electrons in conductors is the same thing as the gauge boson

I have added some updates to this post and want to make one point lucid at the start.  The normal mechanism for energy transfer in electricity is due to the electromagnetic force field which propagates at the velocity of light.  This was unknown to Maxwell, who wrote in his Treatise, 3rd ed., 1873, Article 574: ‘… there is, as yet, no experimental evidence to shew whether the electric current… velocity is great or small as measured in feet per second.’  The first evidence emerged in two years later in 1875, when Heaviside measured it experimentally, using logic signals (Morse) in the undersea telegraph cable between Newcastle and Denmark: electromagnetic energy goes at light speed.  Now Maxwell knew that the physical constants of electromagnetism yield light speed, but he didn’t think that electricity – which he associated with matter in conductors – would travel at the same velocity as radiation.  This is why in the same Treatise, Article 769, he wrote: ‘… we may define the ratio of the electric units to be a velocity… this velocity is about 300,000 kilometres per second.’  Maxwell was completely prejudiced (or ‘confident’, if you prefer that term) that, when a 300,000 km/s velocity comes out of electric and magnetic measurements using currents in wires to produce electric fields and electromagnets, the theory is demonstrating Faraday’s conjecture of 1846 that light is oscillations of electric and magnetic field lines.  He ‘knew’ that the speed was evidence of light because that speed was already known in one other context: the velocity of light.

Problem: unknown to Maxwell, the very stuff he was measuring these electric and magnetic constants with, electricity, has the unfortunate property of going at light velocity.  Or to be more precise: in an given medium (glass, plastic, air, vacuum, etc.), the velocity of light and of electricity is identical!  This is exactly analogous to Yukawa’s prediction of the field meson, hailed as evidence that the muon was Yukawa’s particle,  that the muon stopped the protons in the nucleus from exploding.  Just as in the case of the muon-pion (meson) muddle up, the photon has been muddled up with the electromagnetic gauge boson by Maxwell (who didn’t know Yang-Mills gauge theory, the speed of electricity, the fact that electrons are accelerated by a field composed entirely of gauge bosons, etc.).

Contrary to Maxwell’s conception of electricity, the electron drift in a wire involves only conduction band electrons which have a mass of about 0.5% the mass of the wire and which move at typically 0.001 m/s, so it doesn’t have a high energy density: the kinetic energy that the electrons have is 0.5mv^2.  So wire can carry about 0.5*0.005*0.001^2 = 0.000000025 Joules per kilogram via ‘electric current’ which flows at 1 mm/s, which is best emphasised by writing it out long hand.  There is no significant energy there.  That’s trivial energy.  You don’t need to be Einstein to point out that it’s a popular lie that this electric current has anything to do with the delivery of the electric energy we use!

Gauge bosons – composing the electromagnetic force field which moves at 300,000 km/s in vacuum or nearly that in air - are what we use as ‘electricity’.  This is the TEM wave or the Heaviside energy current, not electric drift current.  Gauge bosons form a field comprising of transverse electromagnetic radiation which induces forces on charges, accelerating the electrons in the conductor by delivering energy!  There is everything to be learned about quantum field theory by carefully studying this one macroscopic example of gauge boson radiation, which is on a scale we can cheaply and easily do experiments with.  It’s interesting that in the mid-1960s, a chip cross-talk engineer tried charging up a simple capacitor (with air as the dielectric between the conductors), while measuring what occurred using high speed sampling oscilloscopes which were not available to Maxwell or Heaviside:

‘a) Energy current [gauge bosons] can only enter a capacitor at the speed of light.

b) Once inside, there is no mechanism for the energy current to slow down below the speed of light.

c) The steady electrostatically charged capacitor is indistinguishable from the reciprocating, dynamic model.

d) The dynamic model is necessary to explain the new feature to be explained, the charging and discharging of a capacitor, and serves all the purposes previously served by the steady, static model.’

The funny thing is that nobody has ever observed a ‘charge’, which is simply the name given to what is presumed (without any evidence) to be the cause of a ‘static field’.  Problem is, Yang-Mills theory makes it clear that the cause of a ‘static field’ is moving exchange radiation, gauge bosons.  There is no theory of a static charge which works: the classical model of the electron (which assumes it to be static) gives a radius far bigger than observed in electron collision experiments, so it is wrong.  What people do is:

1. they observe an electric field.

2. they lie and claim that the observed electric field is fundamentally ‘static’, ignoring the fact it is composed of moving gauge bosons (i.e., ignoring Yang-Mills theory and the experiments quoted above).

3. they lie further and claim that the imaginary ‘static’ electric field proves the existence of ‘static’ charge, despite there being no way to actually probe the Planck scale or black hole scale for an electron.

4. they lie still further by claiming that, despite endless lying and total fantasy, they are scientists.

5. they won’t ever admit that they’re pseudoscientists (believing in a false religion which is based on lazyness, ignorance and lies).  To argue with such people is like playing a game with a cheat: if you do manage to win they won’t concede defeat, they will keep arguing endlessly.  You can’t win if you allow them to play the game by their bogus rules.

(For more on deducing the nature of electromagnetic gauge bosons using accurate observations of TEM waves in logic pulses, see figures 2, 3 and 4 of: https://nige.wordpress.com/2007/06/20/the-mathematical-errors-in-the-standard-model-of-particle-physics/.)

It’s exactly the way Aristarchus’ solar system was falsely dismissed: if people are prejudiced, they will interpret what they see in a lying way.  For example they will interpret the appearance of the sun as a result of ‘sunrise’ resulting from motion of the sun, instead of allowing the possibility that the effect they see is just due to the daily rotation of the planet they are living on.  It’s no use pointing out the facts to such people: they are too busy doing ‘useful string theory’ to listen, life is too short (because they waste time), and so on.  It’s a waste of time to argue with such prejudiced bigots too much.  If you explain the facts in a series of proved stages with them, they will simply have managed to forget the earlier stages by the end, and so the discussion is still inconclusive because it goes round in endless circles of repetition.  They will bring up ‘objections’ which have already been answered in earlier stages, because they are genuinely biased against learning anything new unless it comes printed in Nature or some other peer-reviewed journal which they consider the final judgement.  Eventually they might admit that they are confused, but they remain confident that there is some error in the facts, although they don’t know what it is!  With that mentality, no progress is ever possible, because any advance can be ignored that way.  Some evidence of attempted discussions are vital for the record to prove the existence of the problem and the sort of abusive which rewards stating experimental facts, but this fruitless effort to fight prejudice should not interfere with (or discourage) further progress in science:

Maxwell was wrong because he showed E field strength to vary with longititudinal propagation direction in the photon, instead of showing E to vary transversely: he drew a longitudinal wave and only one of the three dimensions on the diagram is a physical dimension, since the other two dimensions are used in the diagram to show relative electric and magnetic field strengths along the propagation direction, not to show spatial dimensions.  Even if you claim that the E and B field strengths are in transverse directions, they are still not varying transversely in Maxwell’s diagram, they are constant transversely and only varying as a function of longitudinal (propagation) direction.  So it’s still a longitudinal wave being passed off as a transverse wave by obfuscation.

Maxwell's folly

Above: the false Maxwellian light wave (illustration credit: Wikipedia).  This illustration of an electromagnetic wave was included in the 3rd edition of Maxwell’s Treatise, published in 1873.  Notice that there is no transverse variation at all, just a longitudinal variation.  Maxwell did not label his diagram like this one, and was probably confusing the sine wave lines (which are amplitudes, not field lines) with field lines of the sort Faraday suggested were the basis of light (Faraday’s 1846 paper, Thoughts on Ray Vibrations).  The only variation in the amplitude of the electric field strength which Maxwell shows is one with respect to longitudinal direction, like a longitudinal sound wave, where the intensity (pressure) varies as a function of the direction of propagation, not transversely:

 longitudinal wave, not transverse wave

Above: sound wave with longitudinal oscillation similarly to Maxwell’s longitudinal light wave.   Although the sine wave line looks as if it is a ‘transverse’ wiggle, don’t be deceived: it’s a graph with only one distance axis, the other (vertical in this case) axis is not height of wave but is pressure along the distance dimension.  So Maxwell fooled himself by mistaking the axes of his graph which are field strengths for spatial dimensions, which they aren’t (any more than this graph of sound wave pressure is a transverse wave!).

The waving lines are not electric and magnetic field lines, moving transversely.  They are not field lines at all, they are merely lines marking the intensity of the field.

If you draw a line of intensity and B and intensity of E varying as a function of one axis, then what you are plotting is a longitudinal wave.  People mistake this for field lines vibrating transversely!

To show a transverse wave, you have to have E and B varying as a function of transverse distance, not as a function of longitudinal distance.  Hence, you have to change the propagation direction 90 degrees, in these diagrams.

The two perpendicular sine waves labelled E- and B-fields are usually mistaken for E and B field lines oscillating in transverse z and y dimensions; but they are not.  They are lines representing the intensity of the fields, not the field lines.  The strength of a field is represented not by wiggling of field lines, but instead by the number density of field lines (just as in pressure maps: where the larger the number imaginary isopressure lines which occur in a unit area, the higher the rate of change of pressure being represented, so the faster the winds).

In order to correct Maxwell’s diagram to reality, you need to rotate it by 90 degrees so that it is a transverse wave, propagating in a direction at right angles the directions along which the field strengths vary, i.e., the positive and negative fields should propagate side by side with one another, rather than one behind the other as the Faraday/Maxwell false model shows.

All of this depends on the nature of the electromagnetic photon.  In path integrals:

‘Light … ‘smells’ the neighboring paths around it, and uses a small core of nearby space. (In the same way, a mirror has to have enough size to reflect normally: if the mirror is too small for the core of nearby paths, the light scatters in many directions, no matter where you put the mirror.)’ – R. P. Feynman, QED, Penguin, 1990, page 54.

The electromagnetic field energy exists everywhere in the vacuum, but it’s only observable if we can detect an asymmetry: magnetic fields are asymmetries in the usually balanced curls which cancel each other, positive electric fields are asymmetries in the normal balance of positive and negative field energy, etc.

Now, when you get near actual matter, the field intensity from any ‘static charge’ (trapped dynamic field) associated with the matter affects the light because it adds to one of the field components in it.

So sending a photon (electric and magnetic fields) through the electric and magnetic fields of matter gives a smooth effect on large scales (slowing down, refraction) that you can calculate easily from path integrals (see Feynman’s QED book) if your photon travels through electric fields weaker than 1.3*10^18 v/m.  If however, the photon is of short wavelength, it is less likely to be scattered even by a strong electric field, and will then penetrate much closer to a fermion, where it may then encounters a pair of virtual fermions created by pair production in the vacuum by electric fields exceeding 1.3*10^18 v/m (Schwinger’s threshold).  So gamma rays undergo different interactions from the usual scattering.  If you want to know why light bends on the presence of gravitating mass, this post  (figures 1, 2, 3 and table 1) deals with mass and gravity while this post (figures 2, 3, 4) explains gauge bosons and photons and how they are physically related to each other.

Massive electrically neutral bosons, like ‘Higgs’ bosons in some ways, provide mass.  However it is not done in the way of the usual symmetry breaking of U(1)xSU(2) into U(1) at low energy; instead the correct symmetry group is SU(2) and its 3 gauge bosons exist in massless forms at low energy which cause electromagnetism and gravity, and one handedness of them gets supplied with mass at high energy, when they are massive weak gauge bosons).  This has a rest mass of 91 GeV, which gives rise to the massive weak bosons at high energy.  At low energy, these neutrally charged, massive bosons cause smooth deflection of light by gravity because their electromagnetic fields (they contain electromagnetic fields, like a ‘neutral’ photon) interact with passing photons.   The more energy the passing photon has, the more strongly it interacts.  The ‘Higgs’ interacts with both electromagnetic and gravitational field; neutral and charged gauge bosons.  This is why it gives rise to mass the way we observe mass.  This makes a great many checkable predictions: it predicts the masses of all observable fermions.

Gauge boson mechanism for gravity and electromagnetism from SU(2) with three massless gauge bosons; the massive versions of the same gauge bosons provide the weak force.

Above: Gauge boson mechanism for gravity and electromagnetism from SU(2) with three massless gauge bosons; the massive versions of the same gauge bosons provide the weak force.  For a larger version (easier to read the type), please refer to Figs. 2, 3, and 4 in the earlier post here.

Energy Conservation in the Standard Model

Renormalization is explained by the running couplings, the varying relative charge for the same type of force in collisions at different energies, i.e., different distances between particles (the higher the collision energy, the closer the particles approach before being stopped and deflected by the Coulomb repulsion).

As explained in various previous posts, e.g. here, the relative strength of electromagnetic interactions increases by 7% as the collision energy increases from about 0.5 MeV to about 80 GeV, but the strong force behaves differently, falling as energy increases (except for a peaking effect at relatively low energy), as though it is being powered by vacuum effects caused by energy shielded from the electromagnetic force due to the radial polarization of virtual fermions, created by pair production in the Dirac sea above the IR cutoff (see Fig. 1).

Fig. 1 - How the renormalization group running couplings for the strong and electromagnetic forces are related.  The weak force is not shown; it is similar to the electromagnetic forces except that it is mediated by massive gauge bosons which give it a short range and a weak strength.

Fig. 1 – How the renormalization group running couplings for the strong and electromagnetic forces are related.  The weak force is not shown; it is similar to the electromagnetic forces except that it is mediated by massive gauge bosons which give it a short range and a weak strength.  Notice that at high energy the electromagnetic running coupling (relative electric charge) increases until it reaches the grain size of the vacuum (called the ‘UV’ cutoff energy, which isn’t shown here), while the strong coupling falls towards zero.  The mechanism is that the energy, sapped from the electromagnetic field by the radially-polarized virtual fermion pairs in the vacuum, goes into short ranged virtual particles and powers the strong force.  Calculating this energy is easy from electromagnetic theory, and allows the short-range nuclear force couplings to be calculated by subtraction.  All you have to do is keep accurate accounts for what energy is being used for.

Fig. 1 ignores the speculative theory of supersymmetry which is based on the false guess that, at very high energy, all force couplings have the same numerical value; to be specific, the minimal supersymmetric standard model – the one which contains 125 parameters instead of just the 19 in the standard model – makes all force couplings coincide at alpha = 0.04, near the Planck scale.  Although this extreme prediction can’t be tested, quite a lot is known about the strong force at lower and intermediate energies from nuclear physics and also from various particle experiments and observations of very high energy cosmic ray interactions with matter, so, in the book Not Even Wrong (UK edition), Dr Woit explains on page 177 that – using the measured weak and electromagnetic forces – supersymmetry predicts the strong force incorrectly high by 10-15%, when the experimental data is accurate to a standard deviation of about 3%.

This error is caused by the assumption that the strong coupling is similar to weak and electromagnetic couplings near the Planck scale; instead, it is much smaller, so the extrapolation of the supersymmetry unification calculation to lower energies yields a falsely high prediction of the strong force coupling constant.  Supersymmetry is totally wrong.  It is not science because nobody knows the values of all the 125 parameters the theory needs.  It can’t predict the masses of the supersymmetric partners it requires (an unobserved supersymmetric bosonic partner for every observed fermion and a supersymmetric fermion partner for every observed boson in the universe), so it makes no falsifiable prediction at all (except from the false prediction mentioned by Dr Woit).  It has no mechanism.  There is no evidence to support supersymmetry, there is no mechanism to support supersymmetry, there is no science in supersymmetry.  The idea it is based on, that all force couplings have a similar value near the Planck scale, is groundless ‘grand unified theory’ hot air, lacking any physics.

It’s interesting how the potential energy of the various (strong, weak, electromagnetic) fields varies quantitatively as a function of distance from particle cores (not just as a function of collision energy).

The principle of conservation of energy then makes predictions for the variations of different standard model charges with distance.

I.e., the strong (QCD) force peaks at a particular distance.

At longer distances, it falls exponentially because of the limited range of the massive pions which mediate it.

At much shorter distances (where it is mediated by gluons) it also decreases.

How does energy conservation apply to such ‘running couplings’ or variations in effective charge with energy and distance?

This whole way of thinking objectively and physically is ignored completely by the standard model QFT. As the electric force increases at shorter distance, the QCD force falls off; the total potential energy is constant; the polarized vacuum creates QCD by shielding electric force. This physical mechanism makes falsifiable predictions about how forces behave at high energy, so it can be checked experimentally.

Fig. 2 - Mainstream theory (M-theory) groupthink which contradicts empirical data, which lacks mechanisms, and which doesn't accomplish any falsifiable predictions because the 6 curled up extra dimensions in the Calabi-Yau manifold have unknown sizes and shapes since they are supposed to be rolled up into an unobservably small volume.  Without knowing the parameters for these 6 extra dimensions, there is no particular M-theory, just 10^500 possibilities, which means 10^500 theories all with different physics.  This is so vague it is non-falsifiable at present and so at present it is not even wrong, according to Woit.  Stringers should therefore shut up about string theory and give other alternatives which do make falsifiable predictions some chance to make themselves heard over the stringy hype noise level.

Fig. 2 – Qualitative (not to scale) representation of Mainstream theory (M-theory) groupthink which contradicts empirical data, which lacks mechanisms, and which doesn’t accomplish any falsifiable predictions because the 6 curled up extra dimensions in the Calabi-Yau manifold have unknown sizes and shapes since they are supposed to be rolled up into an unobservably small volume.  Without knowing the parameters for these 6 extra dimensions, there is no particular M-theory, just 10^500 possibilities, which means 10^500 theories all with different physics.  This is so vague it is non-falsifiable at present and so at present it is not even wrong, according to Woit.  Stringers should therefore shut up about string theory and give other alternatives which do make falsifiable predictions some chance to make themselves heard over the stringy hype noise level.  Again, the weak force is not shown in the diagram: it is generally fairly similar to electromagnetism although it has a different strength and a limited range.  Notice that in the diagram, both gravity and electromagnetism have a constant residual charge at low energies (i.e., mass as gravitational charge, and electric charge).  The strength of the gravity charge at low energy is about 10^40 times weaker than electromagnetism, so gravity is exaggerated in this diagram, just to show it exists at low energy.

Comparing Fig. 1 to Fig. 2, and you see the difference between hard physical reality with experimental evidence to back it up (see last half-dozen blog posts, for evidence), and Platonic fantasy.  Fig. 2, M-theory, is really in the Boscovich tradition: an attempt to unify all forces by mathematical means without a paradigm shift or any increase in physical understanding of the forces.

Now that quantum gravity is reasonably well sorted out, in my spare time I will write a paper on the Standard Model and its correct unification with gravity (see previous six posts on this blog for the basic evidence).  The main challenge now is to produce complete quantitative calculations for force unification by this mechanism, making additional predictions to those already made (and confirmed), and comparing the results to experimental data already available.  One interesting mathematical innovation worth mentioning, which may be helpful in dealing with relationships between quarks and leptons at very high energy, is category theory, which deals with transformations.  Kea (Marni Sheppeard) is an expert on this.

Update (22 July): Not Even Wrong has news that the SU(2) isospin charge of the weak force exhibits confinement properties:

‘There’s a potentially important new paper on the arXiv from Terry Tomboulis, entitled Confinement for all values of the coupling in four-dimensional SU(2) gauge theory. Tomboulis claims to prove that SU(2) lattice gauge theory has confining behavior (area law fall off of Wilson loops at large distances) for all values of the coupling at the scale of the cutoff, no matter how small. This conjectured behavior is something that quite a few people tried to prove during the late seventies and eighties, without success. Tomboulis is one of the few people who has kept seriously working on the problem, and it looks like he may have finally gotten there. The method he is using goes back to work of ‘t Hooft in the late 1970s, and involves considering the ratio of the partition function with an external flux in the center of SU(2) and the partition function with no such flux. For a recent review article about this whole line of thinking by Jeff Greensite, see here. For shorter, less technical articles by Tomboulis about earlier results in the program he has been pursuing, see here, here, and here.’

It is well known that the SU(3) strong force has confinement properties (confining quarks in hadrons), but the fact that SU(2) may do this job is interesting, and makes the relationship between SU(2) and SU(3) clearer.  There are several recent posts on this blog about SU(2).

One successful way to introduce gravity to the standard model is to change the interpretation of the standard model by using a version of SU(2) which produces electromagnetism, weak forces and gravity.  This can occur if the 3 gauge bosons can acquire mass according to handedness, or not acquire mass.  Those that acquire mass from a suitable mass-giving field become the 3 weak force gauge bosons, with weak strength and limited range.  Those that don’t acquire mass are 3 massless gauge bosons; one uncharged (photon-like) and two charge radiations which cannot propagate by themselves since they are massless and have infinite self-inductance.  These two charged radiations are gauge boson exchange radiation, mediating positive and negative fields respectively.  They can only propagate in two opposite directions at the same time (i.e., continuing exchange of radiation between charges), so that the magnetic field curls get cancelled out, as explained by Figures 2 and 4 of an earlier post, here.  The massless uncharged photon is the spin-1 (push and shove) graviton, predicting gravity.

The SU(3) force is similar in a strange way to this model for SU(2) as an electro-weak-gravity.  SU(3) has both direct mediation of strong forces via colour charged gluon exchange radiation (this binds quarks into hadrons), and indirect (longer range) mediation of strong forces via mesons like pions (this binds hadrons into nuclei against the repulsive Coulomb force that exists between protons).

So you get two types of forces created by SU(3): small-scale, directly-mediated, gluon forces and indirectly-mediated, longer range meson forces.  Similarly, for SU(2) we have two types of forces: three massive gauge bosons mediating short-ranged weak nuclear interactions, and three massless versions mediating long-range (inverse square law) electromagnetism and gravitation.  There is a nice economy here, ‘two force systems for the price of one’ in both SU(2) and SU(3) symmetry groups.  (Quite the opposite of string theory, where you get zero real physics in return for infinite mathematical complexity.)

Update (24 July): Dr Peter Orland has made a comment to this post about confinement.  Confinement under a short ranged force such as the colour force in SU(3) is represented by the increase in colour charge as quarks move apart: this means that they are confined because the attractive charges increase as they move apart, slowing them down (higher energies in Figs 1 and 2 correspond to smaller distances between quarks).  Over a range of distances between quarks, this variation in effective charges means that the charge variation with distance offsets the inverse square law (all forces, including the short ranged strong force, is proportional to the product of the coupling constants or charges involved in the interaction, divided by the square of the distance, although unlike the long range gravity and electromagnetism force laws, the coupling constant or relative charge is not constant but varies with distances and falls exponentially to zero at long ranges in short-ranged nuclear interactions).  This allows quarks asymptotic freedom to move about within a certain volume.  If quarks stray too far, the attractive strong force predominates, and the quarks are pulled back, and confined.  There is also the effect that the vast amount of energy you need to knock a quark out of a hadron exceeds the amount of energy needed to create a new quark-antiquark pair, so instead of getting a free quark isolated, you end up creating a new hadron instead.  That’s why quarks can’t be isolated.  Here are some more details on the physics for electromagnetism and for SU(3):

‘The Landau pole behavior of QED is a consequence of screening by virtual charged particle-antiparticle pairs, such as electron-positron pairs, in the vacuum. In the vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases.

‘In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves carry color charge, and in a different manner. Roughly speaking, each gluon carries both a color charge and an anti-color charge. The net effect of polarization of virtual gluons in the vacuum is not to screen the field, but to augment it and affect its color. This is sometimes called antiscreening. Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so the contribution of this effect would be to weaken the effective charge with decreasing distance.

‘Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors, as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark flavors.’ – http://www.answers.com/topic/asymptotic-freedom?cat=technology

As far as SU(2) confinement is concerned, a meson contains a quark-antiquark pair, and this is due to SU(2) isospin.  The evidence in the previous half dozen posts is that SU(2) gauge bosons without mass are electromagnetism and gravity, replacing U(1).  The atom is an example of the lack of electromagnetic confinement: the electron can be isolated simply because the energy needed for pair production of leptons is lower than the binding energy of an electron to an atom.  This is because the electric charge doesn’t increase as the electron-proton distance increases.  (For quarks, the opposite is the case: the pair production energy for quark-antiquark pairs is lower than the energy needed for a quark to escape from a hadron.  Hence, the fact that quarks are confined and can’t ever be isolated in nature is a purely quantitative result due to the increasing charge with distance and fact that the quark binding energy is bigger than the quark pair production energy.  The fact that electrons can escape from atoms individually is just due to the lower binding energy of electrons in atoms, and the fact that the attractive electromagnetic force between electrons and protons falls instead of increasing as distance increases.)  Gravity also comes out of this SU(2) with massless gauge bosons.  Gravity tends to confine masses into lumps because it is always attractive.

Copy of a comment intended for Not Even Wrong blog, which unfortunately contained a typing error and was deleted:

It appeared in the March 1 1974 issue with the title Black Hole Explosions?. Taylor’s paper (with P.C.W. Davies as co-author) arguing that Hawking was wrong appeared a few months later as Do Black Holes Really Explode?

This idea that black holes must evaporate if they are real simply because they are radiating, is flawed: air molecules in my room are all radiating energy, but they aren’t getting cooler: they are merely exchanging energy. There’s an equilibrium.

Moving to Hawking’s heuristic mechanism of radiation emission, he writes that pair production near the event horizon sometimes leads to one particle of the pair falling into the black hole, while the other one escapes and becomes a real particle. If on average as many fermions as antifermions escape in this manner, they annihilate into gamma rays outside the black hole.

Schwinger’s threshold electric field for pair production is 1.3*10^18 volts/metre. So at least that electric field strength must exist at the event horizon, before black holes emit any Hawking radiation! (This is the electric field strength at 33 fm from an electron.) Hence, in order to radiate by Hawking’s suggested mechanism, black holes must carry enough electric charge so make the eelectric field at the event horizon radius, R = 2GM/c^2, exceed 1.3*10^18 v/m.

Schwinger’s critical threshold for pair production is E_c = (m^2)*(c^3)/(e*h-bar) = 1.3*10^18 volts/metre. Source: equation 359 in http://arxiv.org/abs/quant-ph/0608140 or equation 8.20 in http://arxiv.org/abs/hep-th/0510040

Now the electric field strength from an electron is given by Coulomb’s law with F = E*q = qQ/(4*Pi*Permittivity*R^2), so

E = Q/(4*Pi*Permittivity*R^2) v/m.

Setting this equal to Schwinger’s threshold for pair-production, (m^2)*(c^3)/(e*h-bar) = Q/(4*Pi*Permittivity*R^2). Hence, the maximum radius out to which fermion-antifermion pair production and annihilation can occur is

R = [(Qe*h-bar)/{4*Pi*Permittivity*(m^2)*(c^3)}]^{1/2}.

Where Q is black hole’s electric charge, and e is electronic charge, and m is electron’s mass. Set this R equal to the event horizon radius 2GM/c^2, and you find the condition that must be satisfied for Hawking radiation to be emitted from any black hole:

Q > 16*Pi*Permittivity*[(mMG)^2]/(c*e*h-bar)

where M is black hole mass. So the amount of electric charge a black hole must possess before it can radiate (according to Hawking’s mechanism) is proportional to the square of the mass of the black hole. This is quite a serious problem for big black holes and frankly I don’t see how they can ever radiate anything at all.

On the other hand, it’s interesting to look at fundamental particles in terms of black holes (Yang-Mills force-mediating exchange radiation may be Hawking radiation in an equilibrium).

When you calculate the force of gauge bosons emerging from an electron as a black hole (the radiating power is given by the Stefan-Boltzmann radiation law, dependent on the black hole radiating temperature which is given by Hawking’s formula), you find it correlates to the electromagnetic force, allowing quantitative predictions to be made. See https://nige.wordpress.com/2007/05/25/quantum-gravity-mechanism-and-predictions/#comment-1997 for example.

You also find that because the electron is charged negative, it doesn’t quite follow Hawking’s heuristic mechanism. Hawking, considering uncharged black holes, says that either of the fermion-antifermion pair is equally likey to fall into the black hole. However, if the black hole is charged (as it must be in the case of an electron), the black hole charge influences which particular charge in the pair of virtual particles is likely to fall into the black hole, and which is likely to escape. Consequently, you find that virtual positrons fall into the electron black hole, so an electron (as a black hole) behaves as a source of negatively charged exchange radiation. Any positive charged black hole similarly behaves as a source of positive charged exchange radiation.

These charged gauge boson radiations of electromagnetism are predicted by an SU(2) electromagnetic mechanism, see Figures 2, 3 and 4 of https://nige.wordpress.com/2007/06/20/the-mathematical-errors-in-the-standard-model-of-particle-physics/

For quantum gravity mechanism and the force strengths, particle masses, and other predictions resulting, please see https://nige.wordpress.com/about/

Update (25 July): Because I’ve been busy preparing for major exams, I didn’t read the paper Dr Woit linked to about SU(2) confinement at the time, and mentioned the news because it seemed relevant.  Now I have read the relevant paper, http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.2179v1.pdf, it’s nowhere as physically interesting as expected, although remarkably it does clearly describe the physical problem in quantum field theory which it addresses mathematically, on page 2:

‘The origin of the difficulty is clear. It is the multi-scale nature of the problem: passage from a short distance ordered regime, where weak coupling perturbation theory is applicable, to a long distance strongly coupled disordered regime, where confinement and other collective phenomena emerge. Systems involving such dramatic change in physical behavior over different scales are hard to treat. Hydrodynamic turbulence, involving passage from laminar to turbulent flow, is another well-known example, which, in fact, shares some striking qualitative features with the confining QCD vacuum.’

Page 3 mentions the relationship of the new approach to other symmetry groups:

‘Only the case of gauge group SU(2) is considered explicitly here. The same development, however, can be applied to other groups, and, most particularly, to SU(3) which exhibits identical behavior under the approximate decimations.’

It is unfortunately applied to bogus stringy ideas dating from the late 1960s, which will mislead students just as epicycles did for centuries; the analogy of forces to elastic bands under tension is only useful for forces that increase with distance, instead of falling with increasing distance.  See page 34:

‘It is worth remarking again that in an approach based on RG [renormalization group] decimations the fact that the only parameter in the theory is a physical scale emerges in a natural way. Picking a number of decimations can be related to fixing the string tension. That this can be done only after flowing into the strong coupling regime reflects the fact that this dynamically generated scale is an ‘IR effect’. The coupling g(a) is completely determined in its dependence on a once the string tension is fixed. In particular, g(a) → as a → 0. Note that this implies that there is no physically meaningful or unambiguous way of non-perturbatively viewing the short distance regime independently of the long distance regime. Computation of all physical observable quantities in the theory must then give a multiple of the string tension or a pure number. In the absence of other interactions, this scale provides the unit of length; there are in fact no free parameters.’

The string analogy is irrelevant as stated; the fact that a force can be considered a string tension is neither here nor there (analogies abound in physics; just because an analogy exists, it does not mean that it is physically real since there could be a better [more predictive, falsifiable] analogy out there, and in fact there is).  It’s interesting if the author’s claim, of getting the whole theory from merely the scale and dispensing with all other parameters, is correct.  If that is the case, it makes things simpler.

It would be nice in future for real experts to check the content of new papers they report on, determining whether they are actually correct or not.  Otherwise, what these people tend to do is comment on new arxiv papers without actually committing themselves to saying the paper is right or wrong.  That’s a good way to be fashionable, but it’s not really that scientific.  What ends up occurring, is that groupthink and consensus emerge, not scientific fact.  Papers get mentioned not because the person mentioning them has really checked them and found them correct, but because they ‘look interesting’ or the author ‘has been working a long time’ on the topic, or some other scientifically-irrelevant chatter.  It reminds me of the peer-review process.  A new innovator of a totally radical approach to a subject doesn’t – by definition – have any ‘peers’ who are up to speed on the new idea.  Peer-review even at the best of times doesn’t involve experiments being replicated and calculations checked; the peer-reviewer is more likely to endorse the paper if he or she ‘respects’ the author and finds the paper ‘interesting’.  If the peer-reviewer who gets 50 manuscripts a week checked radical ideas in each paper, it would they would never take (at least) several weeks of non-stop, full-time work to carefully evaluate and check in detail all the results in each paper, and the peer-review system would become clogged and break down.  So instead, trust is placed in the author.  This trust is based on the author being either well known to ‘peers’ or else being affiliated with a trustable institution.  (It is for this reason that peer-review in many cases is anti-science.  It’s the old boys club principle; the mutual back-slapping tea party, which is very elitist and excellent at applying groupthink-based censorship criteria to heretical new developments that would adversely affect the status of the more senile members, and it’s ingenious at rewarding orthodoxy and conformity, while not caring about actual physical facts quite so much as the social side of conferences, the networking of contacts, the getting potential peer-reviewers on the ‘right side’ by explaining ideas over a few beers, bribes, corruption, etc.  This sounds scientific in one sense, but it’s not what science is all about; it’s missing the whole point.  This reminds one of Lord Cardigan’s charge of the light brigade in 1854, where cavalry charged against overwhelming odds and the French Marshal Pierre Bosquet commented: C’est magnifique, mais ce n’est pas la guerre.  The thing is, if it had been filmed, it would have looked like war.  The idea that a successful war is one where there isn’t too much carnage along the way, is just one idea or fashion about what ‘war’ is supposed to be about.  Getting off topic a bit, people get used to seeing a lot of blood in war films, and if reality doesn’t match celluloid, then the army gets a ticking off for taking things too easy, and approaching the enemy too cautiously!  They’re being paid to risk their lives today, whereas yesterday they were being paid to save lives and preserve liberty.  That’s exactly the sort of subtle change in public perception of what people’s jobs are, that creeps up on society in politics and the media.  It makes you sick.  Groupthink is very fickle!  There’s no science involved in all these political and social areas, where definitions are arbitrary and so can change at whim.  Moving back to the analogy of science orthodoxy: getting a paper hyped up in the news is not necessarily the same thing as actually doing science, it’s actually almost irrelevant.  Publication is only relevant to science if other people are actually in a position to read tha paper and switch their own ideas and research areas in that direction, if needed.  If those who could help are busy socialising and riding a bandwaggon with their peer-reviewers, and that kind of thing, then nothing can possibly happen.)

Update (27 July 2007):

Physical position of electric and magnetic fields in photons and gauge bosons

Light is an example of a massless boson. There is an error in Maxwell’s model of the photon: he draws it with the variation of electric field (and magnetic field) occurring as a function of distance along the longitudinal axis, say the x axis.

Maxwell uses the z, and y axes to represent not distances but magnetic and electric field STRENGTHS.

These field strengths are drawn to vary as a function of one spatial dimension only, the propagation direction.

Hence, he has drawn a pencil of light, with zero thickness and with no indication of any transverse waving.

What you get occurring is that people look at it and think the waving E-field line is a physical outline of the photon, and that the y axis is not electric field strength, but is distance in the y-direction.

In other words, they think it is a three dimensional diagram, when in fact it is one dimensional (x-axis is the only dimension; the other two axes are field strengths varying solely as a function of distance along the x-axis).

I explained this to Catt, but he wasn’t listening, and I don’t think others listen either.

The excellent thing is that you can correct the error in Maxwell’s model to get a real transverse wave, and then you find that it doesn’t need to oscillate at all in the longitudinal direction in order to propagate!

This is because the variation in E-field strength and B-field strength actually occurs at right angles to the propagation direction (which is the opposite of what Maxwell’s picture shows when plotting these field strengths as a variation along the longitudinal axis or propagation direction of light, not the transverse direction!).

This is useful for discriminating between a longitudinally oscillating real photon, and a virtual boson which has no longitudinal oscillation, just a transverse wavelength, and can be endlessly long in the direction of propagation in order to allow smooth transfer of force by virtual boson exchange.  The virtual boson doesn’t oscillate charges it encounters like a photon; it merely transfers energy and momentum p = E/c (if absorbed in the interaction without re-emission) or p = 2E/c (if absorbed and then re-emitted with opposite direction; i.e., if reflected back the way it came).  This discriminates gravitons (virtual photons) from real photons.  The gauge bosons of electromagnetism are distinct because they are charged; positive charged exchange radiation is possible because it is going in both directions between two protons (mediating a positive electric field in the vacuum) and so it is travelling through itself in two directions at once (see figures 2, 3 and 4 here).  Similarly for negative gauge bosons being mediated between two electrons.  The similar charges get knocked apart by the exchange, just as two people firing guns at one another tend to recoil apart (both from actually firing bullets and from being hit by them).  For attractive forces, shielding from the inward force due to the reaction to the outward force of the big bang (accelerating mass gives an outward force by Newton’s 2nd law, which by Newton’s 3rd law is accompanied by an inward reaction force, which turns out to be carried by gauge boson radiation) creates attraction, just as in the case of the gravitation mechanism, although you need to allow for the fact that the path integral of charged gauge bosons will allow a multiplication of force across the universe because some special paths will encounter (by chance) alternating positive and negative charges (like a series of charged capacitor plates with vacuum dielectric between them) making the effective potential multiply up by a large factor, while the majority of likely paths which encounter positive and negative charges at random and so will behave like a series of charged capacitors randomly orientated in series (think about a battery back; if you put a large number of batteries into it randomly, i.e., without getting them all the same way around, on average the voltage will be be cancelled out to give zero output).  Hence, only the special path works.  The path integral geometry shows that the special path is a zig-zag like a drunkard’s walk between alternating positive and negative charges across the universe.  This is not as efficient (for creating a net force in a line) as a straight line series of alternatively positive and negatively charged capacitor plates, but it does multiply up the force a lot.  The resulting force is equal to that of gravity times 10^40.

So this mechanism makes checkable, falsifiable predictions for force strengths, particle masses, cosmological stuff (the biggest falsifiable prediction was made and published in 1996, years before the observational confirmation of that prediction, which showed that the universe is indeed not decelerating; this is actually due to a lack of gravitational mechanism at great distances because of the geometry of shielding mechanism and or you can consider the weaking of gravity due to the redshift of gravitons exchanged between receding masses over vast distances in the expanding universe; include this quantum gravity model as a correction to general relativity, and then ‘evolving dark energy’ and its small positive cosmological constant become as redundant, misleading, unnecessary, unfalsifiable, pseudoscientific, hogwash as would be the inclusion of caloric in modern heat theory).

Maxwell’s drawing of a light photon in his final 1873 3rd edition of A Treatise on Electricity and Magnetism is actually a longitudinal wave because the two variables (E and B) are varying solely as a function of propagation direction x, not as functions of transverse directions y and z which aren’t represented in the diagram (which uses y and z to represent field strengths along x, instead of directions y and z in real space).

The full description of the electromagnetic gauge boson and its relationship to a photon can be found in figures 2, 3 and 4 of:

https://nige.wordpress.com/2007/06/20/the-mathematical-errors-in-the-standard-model-of-particle-physics/

Further update (27 July): Charge experiment: charge up anything with electricity. You can do that by sending in a light-velocity logic step: the energy flows in at light velocity.  It then has no mechanism to slow down below light velocity.

Thus, static electricity is ‘composed’ in a sense of energy going at light speed in all directions, in an equilibrium of currents.  (Remember that in electricity, the gauge bosons of the electromagnetic field carry the energy, and the drift of electrons carries trivial kinetic energy because the electrons on go at a snails pace and have small masses; the kinetic energy of the electrons is half their mass multiplied by the square of their net velocity.)  The magnetic field curls cancel out because there is always as much energy going in direction y as in direction -y.  So only electric field (the only thing about charge that is observable) is experienced as a result.

The question is why the discoverer doesn’t forcefully state this, and why he doesn’t extend it properly by considering the smallest possible unit charge, the electron (which itself must be trapped light velocity energy, and we can then do a lot).  Instead, like Kepler, he bogs down his few vital laws in loads of false trivia (Kepler was also an astrologer and had planets being held in their orbits by magnetism; his books are full of pseudoscience and Newton’s chief biographer pointed out that Newton’s genius was in part being able to wade through Kepler’s vast output of nonsense and pick out the useful laws, ignoring the rest).  One of the claims popularly made against that discoverer is the of dismissing ‘charge’ as a fundamental entity.

Actually, everyone who claims to have observed charge has just observed a field, say an electric field, not the core of the electron itself.  So ‘charge’ has never been directly observed and there is no evidence that it is not composed of trapped field bosons.  In fact, charges can be created in the vacuum within strong or extremely high frequency fields by pair production, given just gamma rays which have an energy exceeding the rest masses of the charges.  Particles have spin and one known way to allow for the known quantum numbers such as spin properties consistently is to get such a model of a particle as a ‘loop’ or ‘string’ (not M-theory superstring which has extra spatial dimensions) to create charge as a permanent field by trapping radiation in a small loop or similar, that would then be indistinguishable from ‘charge’.  Scientifically, the word ‘charge’ is only defensible so far as it has been observed.  The electric charge has not been directly observed (the electric field is observed) since it exists on a scale so small it is unobservable (Planck scale or black hole scale, which is still smaller), so people observe the electric field and decide whether that implies a charge or not.  There are electric fields in radio and other waves.  Therefore, when you observe such a field, it does not automatically prove what the cause of the field is.  When you name ‘charge’ you don’t know, directly, anything about what it is that you are naming.  The word charge is vague: it is used to denote any appearance of an electric field associated with apparently static electricity, but there is no reason why a ‘charge’ shouldn’t just be a trapped non-static field.  On the contrary, there is plenty of evidence for it; it unifies matter and radiation.

Update (16 August 2007):

ENERGY DENSITY IN ELECTRIC, GRAVITATIONAL, & NUCLEAR FORCE FIELDS

Here’s a demonstration of how to calculate the energy density of various fields for working out how energy is conserved when short range nuclear forces are created from electromagnetic force which is shielded by the polarized particles of the disrupted fabric of the vacuum at very high energy (very close to a particle). 

Energy density in an electric field is easy to calculate in electromagnetism because you can charge up a capacitor to a constant potential or voltage v (two parallel flat metal plates with an “x” metre gap such as vacuum between then, the vacuum being called the dielectric of free space) and there is then a constant electric field of v/x volts/metre between the plates.  Knowing how much electrical energy you put into the capacitor to charge it up, allows you to relate the electric field strength v/x to the energy per unit volume in the field (i.e., the energy used to charge the capacitor, divided by the product of the gap between the plates “x” and the area of the plates).

Coulomb’s law for electric charges q and Q is:

F = qQ/(4*Pi*permittivity*r^2)

the strength of an electric field v/x (I’m not using E for electric field here or it will be too confusing; I’m using E only for energy) from charge Q is given by

F = (v/x)q

Hence

Electric field strength, v/x = F/q

= Q/(4*Pi*permittivity*r^2).

Now from the analysis of a capacitor, the energy density of an electric field is

E/V = 0.5*[permittivity]*(v/x)^2

where V here is unit volume and has nothing to do with voltage v (reference: see http://hyperphysics.phy-astr.gsu.edu/hbase/electric/engfie.html )
So the energy density of electric fields is (substituting the previous expression for electric field strength v/x around charge Q into the last formula):
 
E/V = 0.5*[permittivity]*[Q/(4*Pi*permittivity*r^2)]^2
 
= (1/32)*Q/[(Pi^2)*permittivity*r^4]
 
Hence, the energy density of a field varies as the 1/r^4.
 
Now, we have the energy density of an electric field – which derives from the Coulomb force which is an inverse-square law rather Newton’s in some respects, can we use the analogy between Newton’s law and Coulomb’s law to derive the energy density of a gravitational field?
 
If we assume for quarks or electrons or whatever that Newton’s law is just something like 10^40 times weaker than Coulomb’s electric force law, then presumably the energy density of the gravitational field will be simply the value we calculated from Coulomb’s law, divided by 10^40:
 
E/V ~ (1/32)*Q/[(10^40)*(Pi^2)*permittivity*r^4]
 
This equation allows you to calculate approximately the energy density of the field around a unit mass like a fermion.  It’s clear that the energy density varies very rapidly with distance from the middle.  This is why I don’t see how you are going to get a constant energy density for space from an inverse square law: the Joules of field energy per cubic metre fall off rapidly with increasing distance.  You would have to find a way to average the energy density by integrating the total energy as a function of radius.
 
Notice that the classical electron radius is based on this approach for the energy density of Coulomb’s law.  You integrate the energy density over space from a small inner radius out to infinite radius, and you set the result equal to the known electron rest-mass energy E = mc^2 where m is electron mass.  The maths then tells you the value of the inner radius you need to start the calculation (if you took the inner radius to be zero, you would get a wrong answer, infinity).  The calculated inner radius is 2.818 fm, see http://en.wikipedia.org/wiki/Classical_electron_radius
 
In previous posts’ comment sections, it is pointed out that within such a radius the vacuum energy is disorganised with pair production spontaneously creating short-lived pairs of particles which then annihilate in collisions (with average time scales as indicated by Heisenberg’s statistical impact uncertainty formula).  Because this energy at short ranges is so disorganised, it has high entropy and cannot be extracted, so it’s not useful energy.  Because it can’t be extracted, that short ranged chaotic field energy is not included in the equation E=mc^2.  The discrepancy between the classical electron radius and the known shorter ranged physics in quantum field theory is due to the assumption that the releasable energy E=mc^2 is the total energy, when it is in fact just that portion of the total energy which is sufficiently organised that it can be converted into gamma rays or whatever when matter and antimatter annihilate; the rest of the energy remains unobservable as gauge bosons with high entropy, going in all directions at once between all charges.

Such disorganised energy doesn’t contribute to organised energy any more than you can extract the energy of air molecules hitting you continually.  Air molecules at room temperature and pressure have a mean speed of 500 m/s, so the 1.2 kg of air in just one cubic metre at sea level contains air with kinetic energy of 0.5*1.2*500^2 = 150,000 Joules.  But this energy is almost totally useless to you because it is disorganised.  It isn’t useful energy, because you can’t use it to do work.  You can’t use the energy of air molecules to power your laptop or light.  It will mix gases slowly (by diffusion), but that’s about it.  This is why “energy” is such an abused term in the media.  Just because you technically have an immense amount of energy, all energy is not the same thing and what matters in practice is how easy it is to extract it.  (The ocean contains 9 million tons of gold, so if you believe that extracting something is always economical then you can just go down to the seaside with your distillation set and get rich: dissolved in ocean water you’ll have access to 180 times the total amount of gold ever mined on land throughout the whole of history.  Obviously, this is useless advice since 3.5% of sea water is a mixture of salts, and you would have to separate the gold from the salts.  It’s just more expensive to get gold that way than to go into a shop and buy gold!  Germany did extensive research on this after WWI, when its leading chemists seriously considered the extraction of gold from sea water as a means to pay war reparations to France.  When the efforts to accomplish it failed, hyperinflation resulted and the dissent led to fascism and WWII.)

The crackpots who write fancyful things promising the use of “zero-point energy” to cure disease and power spaceships, before they even have a clue as to how much such energy there is or whether it can be extracted, are really anti-science: because they are just trying to impose a religious belief system ahead of the facts.  (It’s a situation like the charlatan string theorists who celebrate and hype their “discovery” of quantum gravity in lucrative articles and books before they have even identified a speculative theory from their landscape of 10^500 variants.)

Update (17 August 2007):

I should obviously work on the quantification of force unification indicated by Fig. 1 above.  My idea is to calculate quantitatively the way the electric charge increases in apparent strength above the IR cutoff (collision energy of 0.5 MeV per electron), due to electrons approaching closely and seeing less polarized vacuum shielding the core charge (see Figure 26.10 of the paperback edition of Penrose’s book, Road to Reality).  The significance of the IR cutoff energy is that it corresponds to the outer radius at which the electric field strength of a fermion is strong enough to cause pair-production the the vacuum.  This threshold electric field strength required for pair production was first calculated by Julian Schwinger and his formula can be found as equation 359 in Freeman Dyson’s lecture notes on Advanced Quantum Mechanics http://arxiv.org/abs/quant-ph/0608140 and as equation 8.20 in Luis Alvarez-Gaume and Miguel A. Vazquez-Mozo, Introductory Lectures on Quantum Field Theory, http://arxiv.org/abs/hep-th/0510040.

It is easy to translate the IR or UV cutoff energy into the actual distance from a fermion of unit charge, at least approximately.  For example, if you assume Coulomb scattering occurs, two electrons of IR cutoff energy (0.511 Mev each) will approach each other until the potential energy of Coulomb repulsion has negated all of their kinetic energy and their velocities have dropped to zero.  At that time, they are at the distance of closest approach, and thereafter will begin accelerating apart.  (Obviously such a simple calculation ignores inelastic scatter effects like the release of x-ray radiation accompanying deceleration of charge.)   From this comment:

‘The kinetic energy is converted into electrostatic potential energy as the particles are slowed by the electric field. Eventually, the particles stop approaching (just before they rebound) and at that instant the entire kinetic energy has been converted into electrostatic potential energy of E = (charge^2)/(4*Pi*Permittivity*R), where R is the distance of closest approach.

‘This concept enables you to relate the energy of the particle collisions to the distance they are approaching. For E = 1 MeV, R = 1.44 x 10^-15 m (this assumes one moving electron of 1 MeV hits a non-moving electron, or that two 0.5 MeV electrons collide head-on).

‘But just thinking in terms of distance from a particle, you see unification very differently to the usual picture. For example, experiments in 1997 (published by Levine et al. in PRL v.78, 1997, no.3, p.424) showed that the observable electric charge is 7% higher at 92 GeV than at low energies like 0.5 MeV. Allowing for the increased charge due to reduced polarization caused shielding, the 92 GeV electrons approach within 1.8 x 10^-20 m. (Assuming purely Coulomb scatter.)

‘Extending this to the assumed unification energy of 10^16 GeV, the distance of approach is down to 1.6 x 10^-34 m, and the Planck scale is ten times smaller.

‘If you replot graphs like http://www.aip.org/png/html/keith.htm (or Fig 66 of Lisa Randall’s Warped Passages) as force strength versus distance form particle core, you have to treat leptons and quarks differently.

‘You know that vacuum polarization is shielding the core particle’s electric charge, so that electromagnetic interaction strength rises as you approach unification energy, while strong nuclear forces fall.

‘Considering what happens to the electromagnetic field energy that is shielded by vacuum polarization, is it simply converted into the short ranged weak and strong nuclear forces? Problem: leptons don’t undergo strong nuclear interactions, whereas quarks do. The answer to this is that quarks are so close together in hadrons that they share the same vacuum polarization shield, which is therefore stronger than in leptons, creating vacuum energies that allow QCD. If you consider 3 electron charges very close together so that they all share the same polarized vacuum zone, the polarized vacuum will be 3 times stronger, so the shielded charge of each seen from a great distance may be 1/3 of the electron’s charge (a downquark).’  (Obviously, weak isospin charge and weak hypercharge make things more complex, masking this simple mechanism in general, see for example my post here for details, as well as other recent posts on this blog, i.e., last 6-7 posts.)

Carl Brannen and Tony Smith have kindly made some interesting comments about the possibility of preons in quarks which seem to me to explain vital aspects of the SU(3) colour charge (strong force) which is discussed on Kea’s blog Arcadian Functor here.  Anyway, my idea is to use the logarithmic correction law for energy-dependence on charge between IR and UV cutoffs, such as equation 7.13 or 7.17 (the summation in that equation is for the fact that at higher energies you get pair production of heavier charges contributing, i.e., above 0.511 MeV/particle you get electron and positron pair production, above 105 MeV/particle you get also muon and anti-muon pair production, and you get pairs of hadrons as well as leptons at the higher energies), on pages 70-71 of http://arxiv.org/PS_cache/hep-th/pdf/0510/0510040v2.pdf.  (Notice the footnote on page 71 thanking Lubos Motl, the result of an email he sent the authors after I asked Lubos on his blog why the calculation for the electric charge of an electron is wrong according to the earlier version of that paper: see lying equation 7.17 on page 70 of the original version of the paper which is still also held on arxiv as http://arxiv.org/PS_cache/hep-th/pdf/0510/0510040v1.pdf.  This original version of the equation falsely claims that the electron’s charge increases from the relative value of 1/137 at low energies (i.e., all energies below and up to the IR cutoff of 0.511 MeV) to 1/128 at an energy of 92 GeV just as a result of electron and positron pair production.  But when you put the numbers into that original version of the equation, you get the wrong answer.  This puzzled me, because I’m used to textbook calculations being checked and workable.  Clearly the authors hadn’t actually put the numbers in and done the calculation!  So it turned out, because between 0.511 MeV and 92 GeV there are loads of other vacuum creation-annihilation loops of leptons and hadrons other than merely electron and positron pair production.  The total effect is that at all distances beyond the IR cutoff, i.e., a radius of 1.44 fm, the electron’s charge is the normal value in the textbook, e = 1.60*10^-19 Coulombs (which is 1/137 in dimensionless quantum field theory charge units, see this post for a discussion of why).  As you go to an energy of 92 GeV i.e., as you get approximately 92,000/0.511 or 180,000 times closer to the electron than you are at 1.44 fm, you find that the electron’s charge apparently increases by 7% to 1.71*10^-19 Coulomb (or 1/128 in QFT dimensionless charge units).  This 7% increase was experimentally verified as shown by Levine, Koltick, et al., in a good paper published PRL in 1997.  The physical reason for this “increase” is that as you get closer to the electron core, there is less shielding (i.e. less polarized pair production particles) in the space between you are the core of the electron.  It’s like climbing above the clouds in an aircraft: the sunlight doesn’t increase because you are closer to the sun, but because there is less condensed water vapour between you and the sun.  For an electron, the cloud of polarised pair production charges extends out to the IR cutoff or something like a radius of 1.44 fm.  However, this exact number is controversial because it differs somewhat from the radius corresponding to Schwinger’s threshold electric field strength for pair production, which is 1.3*10^18 volts/metre, and this field strength occurs out to a radius of r = [e/(2m)]*[(h-bar)/(Pi*Permittivity*c^3)]^{1/2} = 3.2953 * 10^{-14} metre = 32.953 fm from the middle of an electron.  This radius also differs from the classical electron radius of 2.818 fm already discussed, so there are some issues over the precise value of the IR cutoff you should take; should it correspond to a distance from an electron of 1.44 fm, 32.95 fm, or 2.82 fm?  Renormalization does not answer this question, because the physics is not sensitive to the precise energy of the IR cutoff when calculating the magnetic moment of leptons or the Lamb shift (some of the few things that can be accurately calculated from QFT).  There are two cutoffs, one at low energy (hence “IR” meaning infrared, which is the name given to the cutoff in visible light at the low energy end of the visible spectrum) and one at high energy (hence “UV” meaning ultraviolet, which is the cutoff in visible light at the high energy end of the visible spectrum).  The UV cutoff seems to occur simply because once you get within a certain extremely small distance of the core of a charge, there is physically not enough room for pair production and polarization of those charges to occur in that tiny space between you and the core, so there is no way physically that the mathematical logarithmic equation for the running coupling can continue to apply.

People who think that this is strange are in awe of the mathematics (like a religious belief in the equations), and can’t see that a continuously variable equation which is accurate for describing large numbers (statistically many polarized pairs of particles) will of necessity break down in physical validity when it is applied to distances so small that the distances are shorter than the average physical size of a single, radially-polarized pair or particles resulting from pair production.  [It's a bit like the problem of miniaturising computers: if you take Moore's empirical law, the number of transistors on a chip doubles every two years (or whatever).  Now it is obvious that this law is theoretically defective because you can't make a transistor the size of a single atom, so there is an obvious physical limit on how far Moore's empirical law can be applied.  The very idea that there should not be a UV cutoff is equally absurd, because it suggests that the vacuum has no quantum grain size and that scaling extends indefinitely.  Of course it doesn't.  Everyone with practical experience in physics knows that all observed physical laws are liable to break down at some extreme limit due to factors which are of no consequence well away from that limit, but which become inportant as that limit is approached.  This is the whole way in that physics progresses.  You look for new physics by trying to work out why laws break down at extreme limits, not by simply calling the break-down an "embarrassing or heretical anomaly" and censoring out all investigations into it.  However, in practice the breakdowns of physics at the boundaries between classical and quantum physics have been dealt with in this way by people like Niels Bohr at the Solvay Congress of 1927.  The reason for this was probably pressure due to Einstein who tried to dismiss or ridicule quantum field theory.  Einstein tried to obtain a classical continuum unified field theory and failed.  However, this story is usually told in a prejudiced way and it is clear that Einstein was right with regards to the Bohr's and Heisenberg's "Copenhagen Interpretation" of quantum mechanics being completely speculative, religious junk.  There is no evidence for the "Copenhagen Interpretation", and Feynman's advice on the matter is dismissive: "shut up and calculate".  (Feynman specifically debunks the copenhagen interpretation in footnote 3 to chapter 2 of his book, QED: "I would like to put the uncertainty principle in its historical place ... If you get rid of all the old-fashioned ideas and instead use the ideas that I'm explaining in these lectures - adding arrows for all the ways an event can happen - there is no need for an uncertainty principle."  Probably this is Feynman's neat revenge for Bohr's ignorant dismissal of Feynman's path integrals back at the 1948 Pocono conference, where Bohr angrily ridiculed Feynman: "Bohr … said: “… one could not talk about the trajectory of an electron in the atom, because it was something not observable.” … Bohr thought that I didn’t know the uncertainty principle … it didn’t make me angry, it just made me realize that … [ they ] … didn’t know what I was talking about, and it was hopeless to try to explain it further. I gave up, I simply gave up …” – Feynman, quoted by Tony Smith.  Facts emerge in physics from factual numerical evidence, not from the speculative consensus of experts or the mere authority of big-mouthed dictatorial, ignorant leader-figures like Niels Bohr.]

Renormalization works by taking the relative low-energy charge (i.e., the charge observed in all normal laboratory and daily life physics up to collisions of 0.511 Mev, which are correspond to relativistic beta particle collisions) to be lower than the bare core charge of an electron by a factor which makes the theory give make useful predictions.  This does not indicate the absolute values of cutoffs accurately, only the relative low energy charge of the electron which is about 1/137.  Mass has to be adjusted in exactly the same way as electric charge to make QED predict things correctly.  (This suggests that mass physically is associated with particles by a mechanism utilising the electric field, because mass itself can’t be shielded in the same way was electric charge can; the field corresponding to mass as charge is of course gravity, and you cannot shield gravity by polarized pairs of masses because masses don’t polarize; they instead all fall the same way in a gravitational field.)

Obviously, as Lubos pointed out, you have to include contributions from all the pair production particles to calculate the total screening of the electric charge to a particular energy, if that energy is high enough to include the possibility of pair production of species of particles other than just electron and positrons.

What’s pretty obvious from this fact, before doing any calculations at all, is that the ‘curve’ for relative electric charge as it increases is not completely smooth, but instead it should have changes in gradient at points corresponding the the energy for the onset of pair production of each new spacetime loop (i.e., a ‘loop’ of pair production virtual fermions being created from gauge bosons and then annihilating back into gauge bosons, and repeating the cycle in and endless ‘loop’ which is easily seen when this cycle is shown on a Feynman diagram).  So as Fig. 1 above shows, there is a change in running coupling gradient at the IR cutoff energy (1.022 MeV) because the charge is constant with respect to energy below the IR cutoff, but at the IR cutoff it starts to increase (as a weak function of energy).  Similarly, above the muon-antimuon creation energy (211.2 MeV) the gradient of the total electric running coupling as a function of energy should increase slightly.

It’s really weird that nobody at all has ever – it seems – bothered to work out and publicise graphs showing how the running couplings (relative charges) for different standard model forces (electromagnetic, weak, strong) vary as a function of distance.  I’ve been intending to do these calculations by computer myself and publish the results here.  One thing I want to do when I run the calculations is to integrate the energy density of each field over volume to get total energy present in each field at each energy, and hence calculate directly whether the rate of decrease in the strong charge can be quantitatively correlated to the rate of increase in electromagnetic charge (see Fig. 1) as you get closer to the core of a particle.  I have delayed doing these detailed calculations because I’m busy with other matters of personal importance, and those calculations will take several days of full-time effort to set up, debug and analyse.

All that people seem to have done is to plot these charges as functions of collision energy, which is somewhat abstract.  If you produce a graph accurately showing how these charges vary as a function of distance from the middle of the particle, you will be able to start to address quantitatively the reasons why the short range strong charge gets weaker as you get closer to the particle core, while the electromagnetic charge gets stronger over the same range: as explained in several previous (recent) posts, the answer to this is probably that electromagnetism is powering the strong force.  The energy of the electromagnetic gauge bosons that get shielded by polarized pairs of fermions, gets converted into the strong force.  It’s easiest to see how this occurs when you consider that at high energy the electromagnetic field produces virtual particles like pions, which cause an attractive nuclear force which stops the repulsive electric force between protons from blowing apart the nuclei of all atoms with atomic numbers of two or more: the energy used to create those pions is electromagnetic energy.  The strong nuclear force in terms of colour charge is extremely interesting.  Here are some recent comments about it via links and comments on Arcadian Functor:

“… I think that linear superposition is a principle that should go all the way down. For example, the proton is not a uud, but instead is a linear combination uud+udu+duu. This assumption makes the generations show up naturally because when you combine three distinct preons, you naturally end up with three orthogonal linear combinations, hence exactly three generations. (This is why the structure of the excitations of the uds spin-3/2 baryons can be an exact analogue to the generation structure of the charged fermions.) …” – Carl Brannen

“In my model,
you can represent the 8 Octonion basis elements as triples of binary 0 and 1,
with the 0 and 1 being like preons, as follows:

1 = 000 = neutrino
i = 100 = red up quark
j = 010 = blue up quark
k = 001 = green up quark
E = 111 = electron
I = 011 = red down quark
J = 101 = blue down quark
K = 110 = green down quark

“As is evident from the list, the color (red, blue, green) comes from the position of the singleton ( 0 or 1 ) in the given binary triple.

“Then the generation structure comes as in my previous comment, and as I said there, the combinatorics gives the correct quark constituent masses. Details of the combinatoric calculations are on my web site.” - Tony Smith (website referred to is here).

“Since my view is that “… the color (red, blue, green) comes from the position of the singleton ( 0 or 1 ) in the given binary triple …[such as]… I agree that color emerges from “… the geometry that confined particles assume in close proximity. …” – Tony Smith.

More on this here.  If this is correct, then the SU(3) symmetry of the strong interaction (3 colour charges and (3^2)-1 = 8 gluon force-mediating gauge bosons) changes in interpretation because the 3 represents 3 preons in each quark which are ‘coloured’, and the geometry of how they align in a hadron gives rise to the effective colour charge, rather like the geometric alignment of electron spins in each sub-shell of an atom (where as Pauli’s exclusion principle states, one electron is spin-up while the other has an opposite spin state relative to the first, i.e., spin-down, so the intrinsic magnetism due to electron spins normally cancels out completely in most kinds of atom).  This kind of automatic alignment on small scales probably explains why quarks acquire the effective ‘colour charges’ (strong charges) they have.  It also, as indicated by Carl Brannen’s idea, suggests why there are precisely three generations in the Standard Model (various indirect data indicate that there are only three generations; if there were more the added immense masses would have shown up as discrepancies between theory and certain kinds of existing measurements), i.e.,

Generation 1:

  • Leptons: electron and electron-neutrino
  • Quarks: Up and down

Generation 2:

  • Leptons: muon and muon-neutrino
  • Quarks: Strange and charm

Generation 3:

  • Leptons: Tau and tau-neutrino
  • Quarks: Top and bottom

Update (18 August 2007):

”… I believe what is needed is as much new mathematical ideas as new physical ones.’ – Dr Peter Woit.

The issue is where mathematical physicists should get such new ideas from, i.e, whether they should guess new ideas off the top of their head, or whether they should be experimentally guided by trying to rearrange the known solid facts (not the speculative interpretations of the facts, but just the raw facts as observed in natural data, e.g. it isn’t a fact that we seen the sun rise in the morning – it’s actually the earth’s rotation which brings the sun into our field of view - we just see a relative motion).  Sir Roger Penrose wrote:

‘In the present climate of fundamental research, it would appear to be much harder for individuals to make substantial progress than it had been in Einstein’s day. Teamwork, massive computer calculations, the pursuing of fashionable ideas – these are the activities that we tend to see in current research. Can we expect to see the needed fundamentally new perspectives coming out of such activities? This remains to be seen, but I am left somewhat doubtful about it. Perhaps if the new directions can be more experimentally driven, as was the case with quantum mechanics in the first third of the 20th century, then such a “many-person” approach might work.’

- Penrose, THE ROAD TO REALITY: A COMPREHENSIVE GUIDE TO THE LAWS OF THE UNIVERSE, Jonathan Cape, London, 2004, page 1026.

I first took a look at Penrose’s book in March 2005 and put some comments on the internet somewhere (I can’t find the page now).  At that time, the only chapter I found interesting was chapter 19, The classical fields of Maxwell and Einstein, dealing with Maxwell’s equations and Einstein’s field equation of general relativity.  The first part of the book was stuff I already knew from undergraduate studies, while the last part of the book was exactly the kind of non-calculating, non-physical speculative, abstruse, drivel that inspired my interest in physics in the first place (because if the mainstream think that such half-baked mathematical claptrap is impressive, maybe they’re a lot of ignorant mathematicians who don’t have real physical intuition for the mechanisms of nature).

The thing about mathematical physics is that the newer the branch, generally speaking, the more abstruse it is.  Maths is most impressive when: reducing apparent chaos in the data of the natural world to simplicity by finding hidden patterns, summarising vast amounts of data with a compact formula, understanding or at least investigating the possible quantitative  interconnections between different variables in a physical situation, and that kind of thing.

Maths is least impressive when it is used to create a landscape of 10^500 variants of string theory which describe ‘possible’ universes derived from different parameters for 6 compactified extra spatial dimensions in a postulated (unobserved) Calabi-Yau manifold of postulated (unobserved) Planck size believed (without any objective evidence) to constitute fundamental ‘stringy’ particles

The reason is that in string ‘theory’, the role of the mathematical model is exactly the opposite of the role of a mathematical model in successful areas of physics.  Mathematics is used to obfuscate physics in string ‘theory’, not to reduce chaos to simplicity, but to transform relative simplicity (observed data as summarised in the standard model and in general relativity) into chaos (I’m talking of chaos in the sense of ‘confusion’, not the Poincare chaos which results from the 3+ body problem and explains how the Schroedinger equation and uncertainty principle result from the random electron paths on small spatial distance scales in an atom, due to quantum field interferences, as Feynman explained with path integrals in his book QED).  String ‘theory’ is a perfect example of the abuse of mathematics for the purpose of making a quick buck by obfuscating physical reality behind a smokescreen of mathematical confusion, and conning the gullible.  It only works because very famous people (e.g. Witten, Hawking, and the paranormal-seeking Nobel Laureate Josephson) are strongly behind it, together with their respective fan clubs and popular ‘physics’-book authors.  Fortunately a few more honest people are against it, such as Penrose and Woit.  It’s  interesting that the people who speak up most loudly against it are those with alternative ideas (twistor theory for Penrose, representation theory for Woit, etc.).

Anyway, I want to update here my views on Penrose’s book Road to Reality.  It’s clearly written as the sort of book a young mathematician (with an interest in frontier physics) would like to have.  I find the length and style of the book repelling, Penrose should bring out a more portable two or three volume version, and I don’t like the mathematical style at all: he doesn’t forcefully argue at the beginning of each chapter why the reader should devote the time and energy to studying the abstruse material.

He is assuming that the reader is going to see it is written by Penrose, and take the whole thing on trust.  The first 150 pages (up to the end of chapter 8) are just his own (none too concise) treatment of a few of the topics on the A-level mathematics exam I took at school 17 years ago.  Chapter 4, on complex numbers, is well worth reading for its clarity but the reader will find much of the rest in any good textbook.  Chapter 9 on Fourier analysis is undergraduate physics material, so after page 150 he moves on from school maths to college material.  Now the problem start, because his treatment of Fourier analysis is lengthy and doesn’t convey the whole point that it is not just an abstract way to ‘decompose waves’ into a sum of sine and cosine waves: it’s vitally important for converting a graph of wave amplitude versus time into a graph of wave amplitude versus frequency.  In other words, it Fourier analysis is used to convert waveforms into corresponding frequency spectra!  This is vital in audio, radio and other situations where waveforms are produced as input and in order to analyse them, you need to calculate the frequency spectrum to show the signal strength as a function of frequency.  But what really gets me annoyed is where in section 9.6 Penrose claims that an Fourier series [sin A] + [(1/3) sin (3A)] + [(1/5) sin (5A)] + … gives rise to a sine wave.  Yes, if you take an infinite number of terms, it would in principle tend towards a square wave.  But in reality you can’t.  The problem with imagining that square waves are really just composed of sine waves is that you get problems in the real world with discontinuities.

For example, a square wave has an increase at the step from 0 to full amplitude A in a time of 0.  This means that the rate of change of amplitude at the step is dA/dt = A/0 = infinity.  Consider a square wave electric signal fed through a cable.  The rate of change of current will be zero for the flat-topped portion of the square wave (full amplitude) and it will be zero when the amplitude is zero, but in the transition (the vertical step) it will be infinity.  This would mean that the radio emission (which is proportional to the rate of change of current, which for the step is infinity) will be infinity.  If so, it would destroy the universe, which is just absurd.  In the real world, what people like Heaviside thought to be vertical steps in electric waveforms turn out to not be perfectly vertical.  The mathematical idea suggested by Fourier analysis that a ‘real world vertical step’ can be explained as an infinite series of sine wave terms, is very dangerous because it stops people thinking about physically real phenomena and turns them into mathematical philosophers who lose contact with the real problems of the real world, like correcting Heaviside’s error and designing computers that don’t crash due to cross-talk.

Chapter 10 (beginning on page 179) and thereafter are more advanced and introduce some physics (section 10.5 on the Cauchy-Riemann equations is very good), but again Penrose focusses on teaching mathematics, not teaching physics.  Any general maths about manifolds is mathematics, not physics, unless or until it is applied to a physical situation and shown to be a useful model of that physical situation.  There is so much mathematical machinery that the physics is largely covered up by the tools.  If Penrose was a driving school instructor and had the same philosophy, the students would never get into the car until the last day of the course; they would spend all their time leading up to that moment studying the machinery in the factory which produces the tools that are used to manufacture the car, they would study the blueprints for the car, they might take the engine to pieces and put it back together again, and on the last day of the course the students would be allowed to sit in the driving seat and turn the engine on for a moment before switching it off again.  Very exciting.  Fine.  Those students then know plenty about the technical details, but have they learned much about actually driving a car?  (Can they apply the mathematics from his book to practical situations in real physical world?)

Obviously that’s a bit unfair, since Penrose wasn’t intending to just write an improved physics textbook, he was setting out to write primarily about the mathematical tools and trusts that the reader will be able to figure out the relatively easy part (of applying those tools to the real world) practically unaided.  Chapter 11 on quaternions is something I’m actually against because I waded through a lot of drivel on quaternions in Maxwell’s treatise, which turned out to be clutter.  You don’t need quaternions to understand Maxwell’s equations or electromagnetism.  Mathematics is an enormous subject, full of abstruse, abstract areas each of which can easily take a lifetime to understand properly, and the one thing you don’t want to do is to hoard mathematical clutter that is of no use physically, if your interest is in physics.  Mathematics is an elite subject, and if mathematicians want to spend their lives on abstract stuff and can fund that research by teaching, or by getting grants, good luck to them.  However, the same chapter goes on to Clifford algebras (section 11.5) which have some relevance to physics.  Reading this, I find nothing remotely physical or interesting in it.  It’s a lot of abstract ‘mathematics’ of the drivel variety: the less useful an area of mathematics is, the more rigorously it is proved and presented (in order to ‘make up’ for the fact that it has no real uses).  The result is that the mathematics is less exciting to read than watching paint dry, none of it sticks in your mind because it’s just an exercise in following a lot of boring, physically meaningless, symbolism, and it just wastes your time.  Chapter 12 on n-dimensional manifolds is more readable but only because I’ve seen references in technical papers to some of the terms Penrose defines, like ‘configuration spaces’ (technically precise jargon for what in 3-dimensions is merely ‘a space … whose different points represent the different physical locations of the body…’).

Chapter 13 on symmetry groups is of course extremely interesting from my perspective, although it is lengthy and doesn’t really tell me anything new.  I want to learn more about SU(2) and SU(3) symmetry groups, but instead of that there is just a lot of abstract background stuff I don’t need (which is of course typical of mathematics).  Chapter 14 contains an interesting section, 14.4, showing the origin of the Bianchi identity (which is used to obtain Einstein’s field equation).  Chapter 15 starts by discussing the failed Kaluza-Klein extra spatial dimension ‘theory’ but goes on to a very interesting discussion of fibre bundles and the Mobius strip.  The next interesting thing is Figure 17.8 in chapter 17.  Chapter 19 is excellent, dealing with Maxwell’s equations and the general relativity field equation.  Chapter 20 is about lagrangians and hamiltonians and is very useful introductory material on that stuff.  Chapter 21 is a terribly standard introduction to quantum mechanics that looks at the equations without grasping where they break down, what is being represented physically by the hamiltonian (ignoring as is usual the interesting physical relationship between Maxwell’s equation for displacement current and the hamiltonian form of schroedinger’s time-dependent equation in quantum mechanics).  A sound wave equation for air pressure in a sound wave will break down and give false results when the amount of air being described is just a few (or a single) air molecules.  This doesn’t mean that the equation is ‘telling us’ that the position of air molecules become real and magically ‘result’ from the ‘collapse of the wavefunction in a sound wave equation’.

Similarly, for quantum mechanics, the average locations of electrons can be represented by a schroedinger equation because the electrons behave like a wave on small scales (being jostled around by the randomly located pair-production occurring around the electron spontaneously at short distances, causing deflections to the electron’s motion as Feynman explained in his discussion of path integrals for the path of the electron inside an atom, see his book QED).  The mainstream still ignores Feynman’s path integrals explanation for what occurs to the electron inside the atom, preferring mystery and obfuscation like some metaphysical ‘interpretation’ of quantum mechanics using parallel universes or other weirdness.  The mainstream leaders don’t want reality to be simple, so they always go for the weird ‘possibilities’ and dismiss the facts using arguments that are none other than pseudo-physics.  They think it makes physics more popular (actually A-level physics uptake in Britain has been falling at 4% per year since mainstream string theory hype by Hawking started).  (The people who buy popular physics books of the parallel universe variety are the same people who believe in UFOs, and not the people who are seriously interested in studying physics.  Making a subject more ‘popular’ amongst the feeble-minded and the deluded is not really a step forward.)

Chapter 24 deals with Dirac’s equation very quickly and simply, chapter 25 does the same for the standard model (unfortunately it doesn’t include anything useful or new about the symmetry groups involved), and chapter 26 does the same for quantum field theory (sadly containing next to no mathematics; Penrose gives loads of mathematical insights in physically irrelevant drivel chapters and then dries up in the interesting ones).  The remainder of the book is on standard basic cosmology and then speculations.

Update (19 Aug 07):

Penrose on the electron as a black hole (proved from calculations based entirely on empirical experimental facts here)

Quotation from Road to Reality, chapter 31, section 31.1, page 870:

“… the standard model is not free of infinities, being merely ‘renormalizable’ rather than a finite theory.  Renormalizability just allows certain calculations to be performed, giving finite answers to most questions of interest within the theory, but it does not provide us with any handle on certain of the most important parameters, such as the specific values of the mass or electric charge of particles described by the theory.  These would have come out as ‘infinity’ (or perhaps ‘zero’), were it not for the renormalization procedure itself, which evades these infinite scalings through a redefinition of terms, and allows finite answers for other quantities to be obtained.  Basically, one ‘gives up’ on mass and charge, whose values are just inserted into the theory as unexplained parameters; indeed, there are some 17 or more such parameters, including coupling constants of various kinds in addition to the mass values of the basic quarks and leptons, the Higgs particle, etc. that need to be specified [contrary to Penrose, the corrected standard model parameter values for coupling constants and masses can actually be calculated from causal mechanisms of the gauge boson physics, the dynamics by which forces arise, and in particular from the virtual particle polarization mechanism which explains the running couplings and which shows the physical nature of the mechanism by which mass is acquired by charge cores - the predictions are accurate - see here and subsequent posts and comments to those posts for updates].

“There are considerable mysteries surrounding the strange values that Nature’s actual particles have for their mass and charge.  For example, there is the unexplained ‘fine structure constant’ alpha, governing the strength of electromagnetic interactions … [a] function[...] of the energy of the particles involved in the interaction … [this is dealt with here].”

On page 832 (in section 30.5 of chapter 30), Penrose discusses the electron as black hole:

“… I cannot resist making a comparison with another observation, due originally to Brandon Carter which, in a different context, has a significant similarity to the argument just given, although it has never been presented as a ‘derivation’ of anything.  We recall that a stationary charge-free black hole is described by the two Kerr parameters m and a, where m is the hole’s mass and am its angular momentum (and where for convenience [i.e. without regard to the annoyance resulting for busy readers who don't have the time to keep converting into real, physical units] I choose units for which c = G = 1, such as Planck units…).  A generalization of the Kerr metric found by Ezra T. Newman (usually referred to as the Kerr-Newman metric) represents an electrically charged rotating stationary black hole.  We now have three parameters: m, a, and e.  The mass and angular momentum are as before, but there is now a total electric charge e.  There is also a magnetic moment M = ae, whose direction agrees with that of the angular momentum.  Carter noticed that the gyromagnetic ratio (twice the mass times magnetic moment divided by the charge times angular momentum, which for a charged black hole is 2m*(ae/e)*am = 2), being completely fixed for a black hole (i.e. independent of m, a, and e), actually takes precisely the value that Dirac originally predicted for the electron, namely 2 [Bohr magnetons] (where for the Dirac electron, the angular momentum is (1/2)*h-bar and the magnetic moment is (1/2)*h-bar*e/(mc), again giving a gyromagnetic ratio of 2, taking c = 1). …

“Can we regard this argument as providing a derivation of the electron’s gyromagnetic ratio, independently of Dirac’s original argument?  Certainly it does not, in any ordinary sense of the term ‘derivation’.  It would only apply if an electron could be regarded as being, in some sense a ‘black hole’. [It is!]  In fact, the actual values for the a, m, and e parameters, in a case of an electron, grossly violate an inequality (m^2) {is greater than or equal to, i.e. the symbol combining > and =} (a^2) + (e^2) that is necessary in order that the corresponding Kerr-Newman metric can represent a black hole.”

However, I recall reading a paper by Tony Smith related to this.  I can’t find the paper I need by Tony Smith’s just now, but there is some related discussion on his webpage here.  I’ll return to this question when I’ve found the right paper and looked into the details.

Update:

High energy unification

High energy unification is believed by the mainstream because it at higher energy, particles approach more closely in collisions than they do at lower energy. I.e., the extra kinetic energy allows charged fermions to overcome Coulomb repulsion more before being stopped and repelled.

Because they approach closely, there is physically less room for vacuum effects to occur between the two particles, so you would expect that vacuum loop effects (like polarization of virtual fermion pairs in the field, which has the effect of weakening the field) would be minimal. If the differences between different fundamental forces is caused by virtual particle loops in the vacuum, then at high enough energy there’s simply no room for a lot of vacuum loops to occur between the two interacting particles due to the closeness of their approach. So you would expect that charge equalization might occur.

In order for gravity to unify in this way, it has to become about 10^40 times stronger as you approach the Planck energy (or whatever the imaginary – unobserved – unification scale is believed to be).

The mainstream idea is that at high energies the energy of the gravitational field interacts with itself strongly, producing more and more gravitons, which explains (qualitatively, not quantitatively) the presumed vast multiplication in gravitational coupling constant at extremely high energy.

This seems to be the rationale for a belief in unification at high energy. The idea is that at low energy, the symmetry between all the different fundamental forces is broken because of the different ways that virtual particles in the vacuum are affected. However, such quantum gravity ideas don’t work. The perturbative expansion for quantum gravity would make it non-renormalizable because successive terms (for every more complex loop situations) wouldn’t get smaller fast enough for the infinite series of terms to give a finite answer. Renormalization is merely about cutting off calculations at a particular energy to make the calculations work (effectively, adjusting the mass and coupling constant for vacuum polarization effects); renormalization it is not about simply ignoring an infinite series of terms in the perturbative expansion to make the theory work.

Hence, the problem of divergence in quantum gravity requires the invention of a supersymmetric field with superpartners that can cancel out the loop divergence problem in the perturbative expansion for quantum gravity. There is so much invention in such theories (they don’t predict the energies of the superpartners, which can be fiddled to any higher-than-observable value that is required to make the theory ‘work’), and so little concrete factual basis, that the theory can ‘predict’ anything depending on your assumptions. So it is just a religious-style belief system, i.e. wishful-thinking, and is not checkable science. A theory which is so vague that it covers all eventualities is just not capable of making useful scientific predictions.

Another problem is that this way of thinking ignores the conservation of energy for the various force fields mediated by gauge bosons that are exchanged between charges. For example, the increase in electric charge (electromagnetic coupling) at energies between the IR and UV cutoffs, raises the question of what happens to the electric field energy which is lost due to vacuum polarization. Clearly, the electric energy shielded by the vacuum’s virtual fermions ends up in the vacuum at high energy (short distances).

This is exactly where the short-ranged nuclear force fields appear. So a mechanistic approach to what is occurring is that the energy lost from the electric field due to vacuum pair production and polarization at high energy, gets converted into short-ranged nuclear force fields. This is a checkable, falsifiable prediction because existing high-energy measurements have shown the rate at which the electric charge increases with collision energy, and the rate at which say the strong nuclear charge decreases with increasing collision energy. By calculating the energy density of each field as a function of distance from the centre of a hadron, we can find out if the shielding of electric charge is indeed powering the short-ranged nuclear forces.

We can calculate the absolute energy density (Joules per cubic metre) of any force field for which the relative coupling constant (or running couplings, for collision energies between between IR and UV cutoffs) is known, by the following method.

You can easily calculate the energy density (energy per unit volume) of an electric field: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/engfie.html

You can calculate the electric field at any distance from a charge very simply because electric field strength (volts/metre) is just equal to force divided by charge (see http://www.glenbrook.k12.il.us/GBSSCI/PHYS/CLASS/estatics/u8l4b.html), and you get charge from Coulomb’s law, giving the electric field strength as a function of distance from a charge, http://en.wikipedia.org/wiki/Electric_field#Coulomb.27s_law

This allows you to work out the energy density of the electric field. Because the gravity coupling constant is 10^40 times weaker than the electromagnetic coupling strength at low energies, the energy density of the gravitational field is equal to 10^{-40} of that of the electric field.

Similarly, we can calculate the energy densities of strong and weak nuclear forces from their relative running couplings. Although running couplings are almost always reported in terms of the coupling as a function of collision energy, we can convert collision energies into distances as follows. The kinetic energy is converted into electrostatic potential energy as the particles are slowed by the electric field.

Eventually, the particles stop approaching (just before they rebound) and at that instant the entire kinetic energy has been converted into electrostatic potential energy of

E = (charge^2)/(4*Pi*Permittivity*R),

where R is the distance of closest approach. This concept enables you to relate the energy of the particle collisions to the distance they are approaching. For E = 1 MeV, R = 1.44 x 10^{-15} m (this assumes one moving electron of 1 MeV hits a non-moving electron, or that two 0.5 MeV electrons collide head-on). (There are other types of scattering than the simple Coulomb scattering at higher energies.)

Metrics and gravitation

Fig. 1 - Newton's Principia, revised 2nd edition, 1713: Book 1, The Motion of Bodies, Section II: The Determination of Centripetal Forces, Proposition 1, Theorem 1.

Fig. 1 – Newton’s geometric proof that an impulsive pushing graviton mechanism is consistent with Kepler’s 3rd law of planetary motion, because equal areas will be swept out in equal times (the three triangles of equal area, SAB, SBC and SBD, all have an equal base of length SB, and they all have altitudes of equal length), together with a diagram we will use for a more modern analysis.  Newton’s geometric proof of centripetal acceleration, from his book Principia, applies to any elliptical orbit, not just circular orbits as Hooke’s easier inverse-square law derivation did.  (Newton didn’t include the graviton arrow, of course.)  By Pythagoras’ theorem x2 = r2 + v2t2, hence x = (r2 + v2t2)1/2. Inward motion, y = x – r = (r2 + v2t2)1/2r = r[(1 + v2t2/r2)1/2 - 1], which upon expanding with the binomial theorem to the first two terms, yields: y ~ r[(1 + (1/2)v2t2/r2) - 1] = (1/2)v2t2/r. Since this result is accurate for infidesimally small steps (the first two terms of the binomial become increasingly accurate as the steps get smaller, as does the approximation of treating the triangles as right-angled triangles so Pythagoras’ theorem can be used), we can accurately differentiate this result for y with respect to t to give the inward velocity, u = v2t/r. Inward acceleration is the derivative of u with respect to t, giving a = v2/r. This is the centripetal force formula which is required to obtain the inverse square law of gravity from Kepler’s third law: Hooke could only derive it for circular orbits, but Newton’s geometric derivation (above, using modern notation and algebra) applies to elliptical orbits as well.  This was the major selling point for the inverse square law of gravity in Newton’s Principia over Hooke’s argument.

See Newton’s Principia, Book I, The Motion of Bodies, Section II: Determination of Centripetal Forces, Proposition 1, Theorem 1:

‘The areas which revolving bodies describe by radii drawn to an immovable centre of force … are proportional to the times on which they are described.  For suppose the time to be divided into equal parts … suppose that a centripetal [inward directed] force acts at once with a great impulse [like a graviton], and, turning aside the body from the right line … in equal times, equal areas are described …  Now let the number of those triangles be augmented, and their breadth diminished in infinitum … QED.’

This result, in combination with Kepler’s third law, gives the inverse-square law of gravity, although Newton’s argument is using geometry plus hand-waving so it is actually far less rigorous than my rigorous algebraic version above.  Newton failed to employ calculus and the binomial theorem to make his proof more rigorous, because he was the inventor of them, and most readers wouldn’t be familiar with those methods.  (It doesn’t do to be so inventive as to both invent a new proof and also invent a new mathematics to use in making that proof, because readers will be completely unable to understand it without a large investment of time and effort; so Newton found that it payed to keep things simple and to use old-fashioned mathematical tools which were widely understood.)

Newton in addition worked out an ingeniously simple proof, again geometrically, to demonstrate that a solid sphere of uniform density (or radially symmetric density) has the same net gravity on the surface and at any distance, for all of its atoms in their three dimensional distribution, as would be the case if all the mass was concentrated in a point in the middle of the Earth. The proof for that is very simple: consider the sphere to be made up of a lot of concentric shells, each of small thickness. For any given shell, the geometry is such as that a person on the surface experiences small gravity effects from small quantities of mass nearby on the shell, while most of the mass of the shell is located at large distances.  The inverse square effect, which means that for equal quantities of mass, the most nearby mass creates the strongest gravitational field, is thereby offset by the actual locations of the masses: only small amounts are nearby, and most of the mass of the shell is at a great distance.   The overall effect is that the effective location for the entire mass of the shell is in the middle of the shell, which implies that the effective location of the mass of a solid sphere seen from a distance is in the middle of the sphere (if the density of each of the little shells, considered to be parts of the sphere, is uniform).

Feynman discusses the Newton proof in his November 1964 Cornell lecture on ‘The Law of Gravitation, an Example of Physical Law’, which was filmed for a BBC2 transmission in 1965 and can viewed on google video here (55 minutes).   Feynman in his second filmed November 1964 lecture, ‘The Relation of Mathematics to Physics’, also on google video (55 minutes), stated:

‘People are often unsatisfied without a mechanism, and I would like to describe one theory which has been invented of the type you might want, that this is a result of large numbers, and that’s why it’s mathematical.  Suppose in the world everywhere, there are flying through us at very high speed a lot of particles … we and the sun are practically transparent to them, but not quite transparent, so some hit.  … the number coming [from the sun's direction] towards the earth is less than the number coming from the other sides, because they meet an obstacle, the sun.  It is easy to see, after some mental effort, that the farther the sun is away, the less in proportion of the particles are being taken out of the possible directions in which particles can come.  So there is therefore an impulse towards the sun on the earth that is inversely as square of the distance, and is the result of large numbers of very simple operations, just hits one after the other.   And therefore, the strangeness of the mathematical operation will be very much reduced    the fundamental operation is very much simpler; this machine does the calculation, the particles bounce.  The only problem is, it doesn’t work.  …. If the earth is moving it is running into the particles …. so there is a sideways force on the sun would slow the earth up in the orbit and it would not have lasted for the four billions of years it has been going around the sun.  So that’s the end of that theory. …

‘It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’

The error Feynman makes here is that quantum field theory tells us that there are particles of exchange radiation mediating forces normally, without slowing down the planets: this exchange radiation causes the FitzGerald-Lorentz contraction and inertial resistance to accelerations (gravity has the same mechanism as inertial resistance, by Einstein’s equivalence principle in general relativity).  So the particles do have an effect, but only as a once-off resistance due to the compressive length change, not continuous drag.  Continuous drag requires a net power drain of energy to the surrounding medium, which can’t occur with gauge boson exchange radiation unless acceleration is involved, i.e., uniform motion doen’t involve acceleration of charges in such a way that there is a continuous loss of energy, so uniform motion doesn’t involve continuous drag in the sea of gauge boson exchange radiation which mediates forces!  The net energy loss or gain during acceleration occurs due to the acceleration of charges, and in the case of masses (gravitational charges), this effect is experienced by us all the time as inertia and momentum; the resistance to acceleration and to deceleration.  The physical manifestation of these energy changes occurs in the FitzGerald-Lorentz transformation; contractions of the matter in the length parallel to the direction of motion, accompanied by related relativistic effects on local time measurements and upon the momentum and thus inertial mass of the matter in motion.  This effect is due to the contraction of the earth in the direction of its motion.  Feynman misses this entirely.  The contraction of the earth’s radius by this mechanism of exchange radiation (gravitons) bouncing off the particles, gives rise to the empirically confirmed general relativity law due to conservation of mass-energy for a contracted volume of spacetime, as proved in an earlier post.  So it is two for the price of one: the mechanism predicts gravity but also forces you to accept that the Earth’s radius shrinks, which forces you to accept general relativity, as well.  Additionally, it predicts a lot of empirically confirmed facts about particle masses and cosmology, which are being better confirmed by experiments and observations as more experiments and observations are done.

As pointed out in a previous post giving solid checkable predictions for the strength of quantum gravity and observable cosmological quantities, etc., due to the equivalence of space and time, there are 6 effective dimensions: three expanding time-like dimensions and three contractable material dimensions. Whereas the universe as a whole is continuously expanding in size and age, gravitation contracts matter by a small amount locally, for example the Earth’s radius is contracted by the amount 1.5 mm as Feynman emphasized in his famous Lectures on Physics.  This physical contraction, due to exchange radiation pressure in the vacuum, is not only a contraction of matter as an effect due to gravity (gravitational mass), but it is also a contraction of moving matter (i.e., inertial mass) in the direction of motion (the Lorentz-FitzGerald contraction).

This contraction necessitates the correction which Einstein and Hilbert discovered in November 1915 to be required for the conservation of mass-energy in the tensor form of the field equation.  Hence, the contraction of matter from the physical mechanism of gravity automatically forces the incorporation of the vital correction of subtracting half product of the metric and the trace of the Ricci tensor, from the Ricci tensor of curvature.  This correction factor is the difference between Newton’s law of gravity merely expressed mathematically as 4 dimensional spacetime curvature with tensors and the full Einstein-Hilbert field equation; as explained on an earlier post, Newton’s law of gravitation when merely expressed in terms of 4-dimensional spacetime curvature gives the wrong deflection of starlight and so on.  It is absolutely essential to general relativity to have the correction factor for conservation of mass-energy which Newton’s law (however expressed in mathematics) ignores.  This correction factor doubles the amount of gravitational field curvature experienced by a particle going at light velocity, as compared to the amount of curvature that a low-velocity particle experiences.  The amazing thing about the gravitational mechanism is that it yields the full, complete form of general relativity in addition to making checkable predictions about quantum gravity effects and the strength of gravity (the effective gravitational coupling constant, G).  It has made falsifiable predictions about cosmology which have been spectacularly confirmed since first published in October 1996.  The first major confirmation came in 1998 and this was the lack of long-range gravitational deceleration in the universe.  It also resolves the flatness and horizon problems, and predicts observable particle masses and other force strengths, plus unifies gravity with the Standard Model. But perhaps the most amazing thing concerns our understanding of spacetime: the 3 dimensions describing contractable matter are often asymmetric, but the 3 dimensions describing the expanding spacetime universe around us look very symmetrical, i.e. isotropic. This is why the age of the universe as indicated by the Hubble parameter looks the same in all directions: if the expansion rate were different in different directions (i.e., if the expansion of the universe was not isotropic) then the age of the universe would appear different in different directions. This is not so. The expansion does appear isotropic, because those time-like dimensions are all expanding at a similar rate, regardless of the direction in which we look. So the effective number of dimensions is 4, not 6.  The three extra time-like dimensions are observed to be identical (the Hubble constant is isotropic), so they can all be most conveniently represented by one ‘effective’ time dimension.

Only one example of a very minor asymmetry in the graviton pressure from different directions, resulting from tiny asymmetries in the expansion rate and/or effective density of the universe in different directions, has been discovered and is called the Pioneer Anomaly, an otherwise unaccounted-for tiny acceleration in the general direction toward the sun (although the exact direction of the force cannot be precisely determined from the data) of (8.74 ± 1.33) × 10−10 m/s2 for long-range space probes, Pioneer-10 and Pioneer-11.  However these accelerations are very small, and to a very good approximation, the three time-like dimensions – corresponding to the age of the universe calculated from the Hubble expansion rates in three orthagonal spatial dimensions – are very similar.

Therefore, the full 6-dimensional theory (3 spatial and 3 time dimensions) gives the unification of fundamental forces; Riemann’s suggestion of summing dimensions using the Pythagorean sum ds2 = å (dx2) could obviously include time (if we live in a single velocity universe) because the product of velocity, c, and time, t, is a distance, so an additional term d(ct)2 can be included with the other dimensions dx2, dy2, and dz2. There is then the question as to whether the term d(ct)2 will be added or subtracted from the other dimensions. It is clearly negative, because it is, in the absence of acceleration, a simple resultant, i.e., dx2 + dy2 + dz2 = d(ct)2, which implies that d(ct)2 changes sign when passed across the equality sign to the other dimensions: ds2 = å (dx2) = dx2 + dy2 + dz2d(ct)2 = 0 (for the absence of acceleration, therefore ignoring gravity, and also ignoring the contraction/time-dilation in inertial motion); This formula, ds2 = å (dx2) = dx2 + dy2 + dz2d(ct)2, is known as the ‘Riemann metric’ of Minkowski spacetime. It is important to note that it is not the correct spacetime metric, which is precisely why Riemann did not discover general relativity back in 1854.

Professor Georg Riemann (1826-66) stated in his 10 June 1854 lecture at Gottingen University, On the hypotheses which lie at the foundations of geometry: ‘If the fixing of the location is referred to determinations of magnitudes, that is, if the location of a point in the n-dimensional manifold be expressed by n variable quantities x1, x2, x3, and so on to xn, then … ds = Ö [å (dx)2] … I will therefore term flat these manifolds in which the square of the line-element can be reduced to the sum of the squares … A decision upon these questions can be found only by starting from the structure of phenomena that has been approved in experience hitherto, for which Newton laid the foundation, and by modifying this structure gradually under the compulsion of facts which it cannot explain.’

[The algebraic Newtonian-equivalent (for weak fields) approximation in general relativity is the Schwarzschild metric, which, ds2 = (1 – 2GM/r)-1(dx2 + dy2 + dz2) – (1 – 2GM/r) d(ct)2. This only reduces to the special relativity metric for the impossible, unphysical, imaginary, and therefore totally bogus case of M = 0, i.e., the absence of gravitation. However this does not imply that general relativity proves the postulates of special relativity. For example, in general relativity the velocity of light changes as gravity deflects light, but special relativity denies this. Because the deflection in light, and hence velocity change, is an experimentally validated prediction of general relativity, that postulate in special relativity is inconsistent and in error. For this reason, it is misleading to begin teaching physics using special relativity.]

WARNING: I’ve made a change to the usual tensor notation below and, apart from the conventional notation in the Christoffel symbol and Riemann tensor, I am indicating covariant tensors by positive subscript and contravariant by negative subscript instead of using indices (superscript) notation for contravariant tensors. The reasons for doing this will be explained and are to make this post easier to read for those unfamiliar with tensors but familiar with ordinary indices (it doesn’t matter to those who are familiar with tensors, since they will know about covariant and contravariant tensors already).

Professor Gregorio Ricci-Curbastro (1853-1925) took up Riemann’s suggestion and wrote a 23-pages long article in 1892 on ‘absolute differential calculus’, developed to express differentials in such a way that they remain invariant after a change of co-ordinate system. In 1901, Ricci and Tullio Levi-Civita (1873-1941) wrote a 77-pages long paper on this, Methods of the Absolute Differential Calculus and Their Applications, which showed how to represent equations invariantly of any absolute co-ordinate system. This relied upon summations of matrices of differential vectors. Ricci expanded Riemann’s system of notation to allow the Pythagorean dimensions of space to be defined by a line element or ‘Riemann metric’ (named the ‘metric tensor’ by Einstein in 1916):

g = ds2 = gm n dx-mdx-n. The meaning of such a tensor is revealed by subscript notation, which identify the rank of tensor and its type of variance.

‘The special theory of relativity … does not extend to non-uniform motion … The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. Along this road we arrive at an extension of the postulate of relativity… The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is, are co-variant with respect to any substitutions whatever (generally co-variant). … We call four quantities Av the components of a covariant four-vector, if for any arbitrary choice of the contravariant four-vector Bv, the sum over v, å Av Bv = Invariant. The law of transformation of a covariant four-vector follows from this definition.’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

The rank is denoted simply by the number of letters of subscript notation, so that Xa is a ‘rank 1’ tensor (a vector sum of first-order differentials, like net velocity or gradient over applicable dimensions), and Xab is a ‘rank 2’ tensor (for second order differential vectors, like acceleration). A ‘rank 0’ tensor would be a scalar (a simple quantity without direction, such as the number of particles you are dealing with). A rank 0 tensor is defined by a single number (scalar), a rank 1 tensor is a vector which is described by four numbers representing components in three orthagonal directions and time, a rank 2 tensor is described by 4 x 4 = 16 numbers, which can be tabulated in a matrix. By definition, a covariant tensor (say, Xa) and a contra-variant tensor of the same variable (say, X-a) are distinguished by the way they transform when converting from one system of co-ordinates to another; a vector being defined as a rank 1 covariant tensor. Ricci used lower indices (subscript) to denote the matrix expansion of covariant tensors, and denoted a contra-variant tensor by superscript (for example xn). But even when bold print is used, this is still ambiguous with power notation, which of course means something completely different (the tensor xn = x1 + x2 + x3 +… xn, whereas for powers or indices xn = x1 x2 x3 …xn). [Another step towards ‘beautiful’ gibberish then occurs whenever a contra-variant tensor is raised to a power, resulting in, say (x2)2, which a logical mortal (who’s eyes do not catch the bold superscript) immediately ‘sees’ as x4,causing confusion.] We avoid the ‘beautiful’ notation by using negative subscript to represent contra-variant notation, thus x-n is here the contra-variant version of the covariant tensor xn. Einstein wrote in his original paper on the subject, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916: ‘Following Ricci and Levi-Civita, we denote the contravariant character by placing the index above, and the covariant by placing it below.’

This was fine for Einstein who had by that time been working with the theory of Ricci and Levi-Civita for five years, but does not have the clarity it could have. (A student who is used to indices from normal algebra finds the use of index notation for contravariant tensors absurd, and it is sensible to be as unambiguous as possible.) If we expand the metric tensor for m and n able to take values representing the four components of space-time (1, 2, 3 and 4 representing the ct, x, y, and z dimensions) we get the awfully long summation of the 16 terms added up like a 4-by-4 matrix (notice that according to Einstein’s summation convention, tensors with indices which appear twice are to be summed over):

g = ds2 = gmn dx-mdx-n  = å (gm n dx-m dx-n )= -(g11 dx-1 dx-1 + g21 dx-2 dx-1 + g31 dx-3 dx-1 + g41 dx-4 dx-1) + (-g12 dx-1 dx-2 + g22 dx-2 dx-2 + g32 dx-3 dx-2 + g42 dx-4 dx-2) + (-g13 dx-1 dx-3 + g23 dx-2 dx-3 + g33 dx-3 dx-3 + g43 dx-4 dx-3) + (-g14 dx-1 dx-4 + g24 dx-2 dx-4 + g34 dx-3 dx-4 + g44 dx-4 dx-4)

The first dimension has to be defined as negative since it represents the time component, ct. We can however simplify this result by collecting similar terms together and introducing the defined dimensions in terms of number notation, since the term dx-1 dx-1 = d(ct)2, while dx-2 dx-2 = dx2, dx-3 dx-3 = dy2, and so on. Therefore:

g = ds2 = gct d(ct)2 + gx dx2 + gy dy2 + gz dz2 + (a dozen trivial first order differential terms).

It is often asserted that Albert Einstein (1879-1955) was slow to apply tensors to relativity, resulting in the 10 years long delay between special relativity (1905) and general relativity (1915). In fact, you could more justly blame Ricci and Levi-Civita who wrote the long-winded paper about the invention of tensors (hyped under the name ‘absolute differential calculus’ at that time) and their applications to physical laws to make them invariant of absolute co-ordinate systems. If Ricci and Levi-Civita had been competent geniuses in mathematical physics in 1901, why did they not discover general relativity, instead of merely putting into print some new mathematical tools? Radical innovations on a frontier are difficult enough to impose on the world for psychological reasons, without this being done in a radical manner. So it is rare for a single group of people to have the stamina to both invent a new method, and to apply it successfully to a radically new problem. Sir Isaac Newton used geometry, not his invention of calculus, to describe gravity in his Principia, because an innovation expressed using new methods makes it too difficult for readers to grasp. It is necessary to use familiar language and terminology to explain radical ideas rapidly and successfully. Professor Morris Kline describes the situation after 1911, when Einstein began to search for more sophisticated mathematics to build gravitation into space-time geometry:

‘Up to this time Einstein had used only the simplest mathematical tools and had even been suspicious of the need for “higher mathematics”, which he thought was often introduced to dumbfound the reader. However, to make progress on his problem he discussed it in Prague with a colleague, the mathematician Georg Pick, who called his attention to the mathematical theory of Ricci and Levi-Civita. In Zurich Einstein found a friend, Marcel Grossmann (1878-1936), who helped him learn the theory; and with this as a basis, he succeeded in formulating the general theory of relativity.’ (M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1990, vol. 3, p. 1131.)

General relativity equates the mass-energy in space to the curvature of motion (acceleration) of an small test mass, called the geodesic path. Readers who want a good account of the full standard tensor manipulation should see the page by Dr John Baez or a good book by Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity.

Curvature is best illustrated by plotting a graph of distance versus time and when the line curves (as for an accelerating car) that curve is ‘curvature’. It’s the curved line on a space-time graph that marks acceleration, be that acceleration due to a force acting upon gravitational mass or inertial mass (the equivalence principle of general relativity means that gravitational mass = inertial mass).

This point is made very clearly by Professor Lee Smolin on page 42 of the USA edition of his 1996 book, ‘The Trouble with Physics.’ See Figure 1 in the post here.  Next, in order to mathematically understand the Riemann curvature tensor, you need to understand the operator (not a tensor) which is denoted by the Christoffel symbol (superscript here indicates contravariance):

G abc = (1/2)gcd [(dgda/dxb) + (dgdb/dxa) + (dgab/dxd)]

The Riemann curvature tensor is then represented by:

Racbe = ( dG bca /dxe ) – ( dG bea /dxc ) + (G tea G bct ) – (G tba G cet ).

If there is no curvature, spacetime is flat and things don’t accelerate. Notice that if there is any (fictional) ‘cosmological constant’ (a repulsive force between all masses, opposing gravity an increasing with the distance between the masses), it will only cancel out curvature at a particular distance, where gravity is cancelled out (within this distance there is curvature due to gravitation and at greater distances there will be curvature due to the dark energy that is responsible for the cosmological constant). The only way to have a completely flat spacetime is to have totally empty space, which of course doesn’t exist, in the universe we actually know.

To solve the field equation, use is made of the simple concepts of proper lengths and proper times. The proper length in spacetime is equal to cò (- gmn dx-m dx-n)1/2, while the proper time is ò (gm n dx-mdx-n)1/2.

Notice that the ratio of proper length to proper time is always c. The Ricci tensor is a Riemann tensor contracted in form by summing over a = b, so it is simpler than the Riemann tensor and is composed of 10 second-order differentials. General relativity deals with a change of co-ordinates by using Fitzgerald-Lorentz contraction factor, g = (1 – v2/c2)1/2. Karl Schwarzschild produced a simple solution to the Einstein field equation in 1916 which shows the effect of gravity on spacetime, which reduces to the line element of special relativity for the impossible, not-in-our-universe, case of zero mass.  Einstein at first built a representation of Isaac Newton’s gravity law a = MG/r2 (inward acceleration being defined as positive) in the form Rm n = 4p GTm n /c2, where Tmn is the mass-energy tensor, Tm n = r um un. ( This was incorrect since it did not include conservation of energy.) But if we consider just a single dimension for low velocities (g = 1), and remember E = mc2, then Tm n = T00 = r u2 = r (g c)2 = E/(volume). Thus, Tm n /c2 is the effective density of matter in space (the mass equivalent of the energy of electromagnetic fields). We ignore pressure, momentum, etc., here:

The components of the stress-energy tensor

Above: the components of the stress-energy tensor (image credit: Wikipedia).

The scalar term sum or “trace” of the stress-energy tensor is of course  the sum of the diagonal terms from the top left to the top right, hence the trace is just the sum of the terms with subscripts of 00, 11, 22, and 33 (i.e., energy-density and pressure terms).

The velocity needed to escape from the gravitational field of a mass (ignoring atmospheric drag), beginning at distance x from the centre of mass, by Newton’s law will be v = (2GM/x)1/2, so v2 = 2GM/x. The situation is symmetrical; ignoring atmospheric drag, the speed that a ball falls back and hits you is equal to the speed with which you threw it upwards (the conservation of energy). Therefore, the energy of mass in a gravitational field at radius x from the centre of mass is equivalent to the energy of an object falling there from an infinite distance, which by symmetry is equal to the energy of a mass travelling with escape velocity v. By Einstein’s principle of equivalence between inertial and gravitational mass, this gravitational acceleration field produces an identical effect to ordinary motion. Therefore, we can place the square of escape velocity (v2 = 2GM/x) into the Fitzgerald-Lorentz contraction, giving g = (1 – v2/c2)1/2 = [1 – 2GM/(xc2)]1/2.

However, there is an important difference between this gravitational transformation and the usual Fitzgerald-Lorentz transformation, since length is only contracted in one dimension with velocity, whereas length is contracted equally in 3 dimensions (in other words, radially outward in 3 dimensions, not sideways between radial lines!), with spherically symmetric gravity. Using the binomial expansion to the first two terms of each: Fitzgerald-Lorentz contraction effect: g = x/x0 = t/t0 = m0/m = (1 – v2/c2)1/2 = 1 – ½v2/c2 + … .  Gravitational contraction effect: g = x/x0 = t/t0 = m0/m = [1 – 2GM/(xc2)]1/2 = 1 – GM/(xc2) + …, where for spherical symmetry ( x = y = z = r), we have the contraction spread over three perpendicular dimensions not just one as is the case for the FitzGerald-Lorentz contraction: x/x0 + y/y0 + z/z0 = 3r/r0. Hence the radial contraction of space around a mass is r/r0 = 1 – GM/(xc2) = 1 – GM/[(3rc2]. Therefore, clocks slow down not only when moving at high velocity, but also in gravitational fields, and distance contracts in all directions toward the centre of a static mass. The variation in mass with location within a gravitational field shown in the equation above is due to variations in gravitational potential energy. The contraction of space is by (1/3) GM/c2. This physically relates the Schwarzschild solution of general relativity to the special relativity line element of spacetime.

This is the 1.5-mm contraction of earth’s radius Feynman obtains, as if there is pressure in space. An equivalent pressure effect causes the Lorentz-FitzGerald contraction of objects in the direction of their motion in space, similar to the wind pressure when moving in air, but without molecular viscosity (this is due to the Schwinger threshold for pair-production by an electric field: the vacuum only contains fermion-antifermion pairs out to a small distance from charges, and beyond that distance the weaker fields can’t cause pair-production – i.e., the energy is below the IR cutoff – so the vacuum contains just bosonic radiation without pair-production loops that can cause viscosity; for this reason the vacuum compresses macroscopic matter without slowing it down by drag). Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965.

More information can be found in the earlier posts here, here, here, here, here and here.