This page was revised (mainly by corrections to the discussion of tensors in general relativity) on 8 June 2007 and is a totally disorganised, rambling, informal supplement to (not a replacement of) the more concise proof paper at:

‘A theorem is only as good as the assumptions underlying it. … particularly in more speculative subject, like Quantum Gravity, it’s simply a mistake to think that greater rigour can substitute for physical input. The idea that somehow, by formulating things very precisely and proving rigourous theorems, correct physics will eventually emerge simply misconstrues the role of rigour in Physics.’

1. ‘There’s the issue of the theorem itself, and whether the assumptions that went into it are physically-justified.
2. ‘There’s the issue of a certain style of doing Physics which values proving theorems over other ways of arriving at physical knowledge.
3. ‘There’s the rhetorical use to which the (alleged) theorem is put, in arguing for or against some particular approach. In particular, there’s the unreflective notion that a theorem trumps any other sort of evidence.’

‘When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only the surface.’

There is already a frequency of oscillation in the photon before it hits the glass, and in the glass due to the sea of electrons interacting via Yang-Mills force-causing radiation. If the frequencies clash, the photon can be reflected or absorbed. If they don’t interfere, the photon goes through the glass. Some of the resonate frequencies of the electrons in the glass are determined by the exact thickness of the glass, just like the resonate frequencies of a guitar string are determined by the exact length of the guitar string. Hence the precise thickness of the glass controls some of the vibrations of all the electrons in it, including the surface electrons on the edges of the glass. Hence, the precise thickness of the glass determines the amplitude there is for a photon of given frequency to be absorbed or reflected by the front surface of the glass. It is indirect in so much as the resonance is set up by the thickness of the glass long before the photon even arrives (other possible oscillations, corresponding to a non-integer value of the glass thickness as measured in terms of the number of wavelengths which fit into that thickness, are killed off by interference, just as a guitar string doesn’t resonate well at non-natural frequencies).

What has happened is obvious: the electrons have set up a equilibrium oscillatory state dependent upon the total thickness before the photon arrives. There is nothing to this: consider how a musical instrument works, or even just a simple tuning fork or solitary guitar string. The only resonate vibrations are those which contain an integer number of wavelengths. This is why metal bars of different lengths resonate at different frequencies when struck. Changing the length of the bar slightly, completely alters its resonance to a given wavelength! Similarly, the photon hitting the glass has a frequency itself. The electrons in the glass as a whole are all interacting (they’re spinning and orbiting with centripetal accelerations which cause radiation emission, so all are exchanging energy all the time which is the force mechanism in Yang-Mills theory for electromagnetism), so they have a range of resonances that is controlled by the number of integer wavelengths which can fit into the thickness of the glass, just as the range of resonances of a guitar string are determined by the wavelengths which fit into the string length resonately (ie, without suffering destructive interference).

Hence, the thickness of the glass pre-determines the amplitude for a photon of given frequency to be either absorbed or reflected. The electrons at the glass surface are already oscillating with a range of resonate frequencies depending on the glass thickness, before the photon even arrives. Thus, the photon is reflected (if not absorbed) only from the front face, but it’s probability of being reflected is dependent on the total thickness of the glass. Feynman also writes:

‘when the space through which a photon moves becomes too small (such as the tiny holes in the screen) … we discover that … there are interferences created by the two holes, and so on. The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that … interference becomes very important.’

‘… the ‘inexorable laws of physics’ … were never really there … Newton could not predict the behaviour of three balls … In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’

So you could derive the Schroedinger from a corrected Maxwell ‘displacement current’ equation. This is just an example of what I mean by deriving the Schroedinger equation. Alternatively, a computer Monte Carlo simulation of electrons in orbit around a nucleus, being deflected by pair production in the Dirac sea, would provide a check on the mechanism behind the Schroedinger equation, so there is a second way to make progress

‘The picture [it is copyright, so get the book: see Fig. 7.1 on p.93 of Not Even Wrong] shows the SU(3) x SU(2) x U(1) transformation properties of the first three generations of fermions in the standard model (the other two generations behave the same way).

‘Under SU(3), the quarks are triplets and the leptons are invariant.

‘Under SU(2), the particles in the middle row are doublets (and are left-handed Weyl-spinors under Lorentz transformations), the other particles are invariant (and are right-handed Weyl-spinors under Lorentz transformations).

‘Under U(1), the transformation properties of each particle is given by its weak hypercharge Y.’

This makes it easier to understand: the QCD colour force of SU(3) controls triplets of particles (’quarks’), whereas SU(2) controls doublet’s of particles (’quarks’).

But the key thing is that the hypercharge Y is different for differently handed quarks of the same type: a right-handed downquark (electric charge -1/3) has a weak hypercharge of -2/3, while a left-handed downquark (same electric charge as the right-handed one, -1/3), has a different weak hypercharge: 1/3 instead of -2/3!

Clearly this weak hypercharge effect is what has been missing from my naive causal model (where observed long range quark electric charge is determined merely by the strength of vacuum polarization shielding of the electric charges closely confined). Energy is not merely being shared between the QCD SU(3) colour forces and the U(1) electromagnetic forces, but there is the energy present in the form of weak hypercharge forces which are determined by the SU(2) weak nuclear force group.

Let’s get the facts straight: from Woit’s discussion (unless I’m misunderstanding), the strong QCD force SU(3) only applies to triads of quarks, not to pairs of quarks (mesons).

The binding of pairs of quarks is by the weak force only (which would explain why they are so unstable, they’re only weakly bound and so more easily decay than triads which are strongly bound). The weak force also has effects on triads of quarks.

The weak hypercharge of a downquark in a meson containing 2 quarks is Y=1/3 compared to Y=-2/3 for a downquark in a baryon containing 3 quarks.

Hence the causal relationship holds true for mesons. Hypothetically, 3 right-handed electrons (each with weak hypercharge Y = -2) will become right-handed downquarks (each with hypercharge Y=-2/3) bought close together, because they then share the same vacuum polarization shield, which is 3 times stronger than that around a single electron, and so attenuates more of the electric field, reducing it from -1 per electron when widely separated to -1/3 when brought close together (forget the Pauli exclusion principle, for a moment!).

Now, in a meson, you only have 2 quarks, so you might think that from this model the downquark would have electric charge -1/2 and not -1/3, but that anomaly only exists when ignoring the weak hypercharge! For a downquark in a meson, the weak hypercharge is Y=1/3 instead of Y=-2/3 which the downquark has in a baryon (triad). The increased hypercharge (which is responsible physically to the weak force field that binds up a meson) offsets the electric charge anomaly. The handedness switch-over, in going from considering quarks in baryons to those in mesons, automatically compensates the electric charge, keeping it the same!

The details of how handedness is linked to weak hypercharge is found in the dynamics of Pauli’s exclusion principle: adjacent particles can’t have have a full set of the same quantum numbers like the same spin and charge. Instead, each particle has a unique set of quantum numbers. Bringing particles together and having them ‘live together’ in close proximity forces them to arrange themselves with suitable quantum numbers. The Pauli exclusion principle is simple in the case of atomic electrons: each electron has four quantum numbers, describing orbit configuration and intrinsic spin, and each adjacent electron has opposite spin to its neighbours. The spin alignment here can be understood very simply in terms of magnetism: it needs the least energy to have sign an alignment (hving similar spins would be an addition of magnetic moments, so that north poles would all be adjacent and south poles would all be adjacent, which requires more energy input than having adjacent magnets parallel with opposite poles nearest). In quarks, the situation regarding the Pauli exclusion principle mechanism is slightly more complex, because quarks can have similar spins if their colour charges are different (electrons don’t have colour charges, which are an emergent property of the strong fields which arise when two or three real fundamental particles are confined at close quarters).

Obviously there is a lot more detail to be filled in, but the main guiding principles are clear now: every fermion is indeed the same basic entity (whether quark or lepton), and the differences in observed properties stem to the vacuum properties such as the strength of vacuum polarization, etc. The fractional charges of quarks always arise due to the use of some electromagnetic energy to create other types of short range forces (the testable prediction of this model is the forecast that detailed calculations will show that perfect unification will arise on such energy conservation principles, without requiring the 1:1 boson to fermion ’supersymmetry’ hitherto postulated by string theorists). Hence, in this simple mechanism, the +2/3 charge of the upquark is due to a combination of strong vacuum polarization attenuation and hypercharge (the downquark we have been discussing is just the clearest case).

So regarding unification, we can get hard numbers out of this simple mechanism. We can see that the total gauge boson energy for all fields is conserved, so when one type of charge (electric charge, colour charge, or weak hypercharge) varies with collision energy or distance from nucleus, we can predict that the others will vary in such a way that the total charge gauge boson energy (which mediates the charge) remains constant. For example, we see reduced electric charge from a long range because some of that energy is attenuated by the vacuum and is being used for weak and (in the case of triads of quarks) colour charge fields. So as you get to ever higher energies (smaller distances from particle core) you will see all the forces equalizing naturally because there is less and less polarized vacuum between you and the real particle core which can attenuate the electromagnetic field. Hence, the observable strong charge couplings have less supply of energy (which comes from attenuation of the electromagnetic field), and start to decline. This causes asymptotic freedom of quarks because the decline in the strong nuclear coupling at very small distances is offset by the geometric inverse-square law over a limited range (the range of asymptotic freedom). This is what allows hadrons to have a much bigger size than the size of the tiny quarks they contain.

You can think of the strong force like the short-range forces due to normal sea-level air pressure: the air pressure of 14.7 psi or 101 kPa is big, so you can prove the short range attractive force of air pressure it by using a set of rubber ’suction cups’ strapped on your hands and knees to climb a smooth surface like a glass-fronted building (assuming the glass is strong enough!). This force has a range on the order of the mean free path of air molecules. At bigger distances, air pressure fills the gap, and the force disappears. The actual fall of course is statistical; instead of the short range attraction becoming suddenly zero at exactly one mean free path, it drops (in addition to geometric factors) exponentially by the factor exp{-ux} where u is the reciprocal of the mean free path and x is distance (in air of course there are weak attractive forces between molecules, Van der Waals forces, as well). Hence it is short ranged due to scatter of charged particles dispersing forces in all directions (unlike radiation):

‘… the Heisenberg formulae can be most naturally interpreted as statistical scatter relations, as I proposed [in the 1934 book The Logic of Scientific Discovery]. … There is, therefore, no reason whatever to accept either Heisenberg’s or Bohr’s subjectivist interpretation …’

• Sir Karl R. Popper, Objective Knowledge, Oxford University Press, 1979, p. 303.

(Note statistical scatter gives the energy form of Heisenberg’s equation, since the vacuum is full of gauge bosons carrying momentum like light, which above the IR cutoff start to exert vast pair-production loop pressure; this gives the foam vacuum.)

Dirac sea polarization (leading to charge renormalization) is only possible in volumes large enough to be likely to contain some discrete charges! The IR cutoff has a different explanation. It is required physically in quantum field theory to limit the range over which the vacuum charges of the Dirac sea are polarized, because if there were no limit, then the Dirac sea would be able to polarize sufficiently to completely eradicate the entire electric field of all electric charges. That this does not happen in nature shows that there is a physical mechanism in place which prevents polarization below the range of the IR cutoff, which is about 10^{-15} m from an electron, corresponding to something like 10^{20} volts/metre electric field strength.

Clearly, the Dirac sea is physically:

We only see ‘pair-production’ of Dirac sea charges becoming observable in creation-annihilation ‘loops’ (Feynman diagrams) when the electric field is in excess of about 10^{20} volts/metre. This very intense electric field, which occurs out to about 10^{-15} metres from a real (long-observable) electron charge core, is strong enough to overcome the binding energy of the Dirac sea: particle pairs then pop into visibility (rather like water boiling off at 100 C).

The spacing of the Dirac sea particles in the bound state below the IR cutoff is easily obtained. Take the energy-time form of Heisenberg’s uncertainty principle and put in the energy of an electron-positron pair and you find it can exist for ~10^{-21} second; the maximum possible range is therefore this time multiplied by c, ie ~10^{-12} metre.

The key thing to do would be to calculate the transmission of gamma rays in the vacuum. Since the maximum separation of charges is 10^{-12} m, the vacuum contains at least 10^{36} charges per cubic metre. If I can calculate that the range of gamma radiation in such a dense medium is 10^{-12} metre, I’ll have substantiated the mainstream picture. Normally you get two gamma rays when an electron and positron annhilate (the gamma rays go off in opposite directions), so the energy of each gamma ray is 0.511 MeV, and it is well known that the Compton effect (a scattering of gamma rays by electrons as if both are particles not waves) predominates for this energy. The mean free path for scatter of gamma ray energy by electrons and positrons depends essentially on the density of electrons (number of electrons and positrons per cubic metre of space). However, the data come from either the Klein-Nishita theory (an application of quantum mechanics to the Compton effect) or experiment, for situations where the binding energy of electrons to atoms or whatever is insignificant compared to the energy of the gamma ray. It is perfectly possible that the binding energy of the Dirac sea would mean that the usual radiation attenuation data are inapplicable!

Obviously the graining of the Dirac sea must be much smaller than 10^{-12} m because we have already said that it exists down to the UV cutoff (very high energy, ie, very small distances of closest approach). The amount of ‘energy’ in the Dirac sea is astronomical if you calculate the rest mass equivalent, but you can similarly produce stupid numbers for the energy of the earth’s atmosphere: the mean energy of an air molecule is around 500 m/s, and since the atmosphere is composed mainly of air molecules (with a relatively small amount of water and dust), we can get a ridiculous energy density of the air by multiplying the mass of air by 0.5*(500^2) to obtain its kinetic energy. Thus, 1 kg of air (with all the molecules going at a mean speed of 500 m/s) has an energy of 125,000 Joules. But this is not useful energy because it can’t be extracted: it is totally disorganised. The Dirac sea ‘energy’ is similarly massive but useless.

Let’s go right through the derivation of the Einstein-Hilbert field equation in a non-obfuscating way. To start with, the classical analogue of general relativity’s field equation is Poisson’s equation

div.²E = 4*Pi*Rho*G

The square of the divergence of E is just the Laplacian operator (well known in heat diffusion) acting on E and implies for radial symmetry (r = x = y = z) of a field:

div.²E

= 3*d²E/dr²

To derive Poisson’s equation in a simple way (not mathematically rigorous), observe that for non-relativistic situations

E = (1/2)mv² = MG/r

(Kinetic energy gained by a test particle falling to distance r from mass M is simply the gravitational potential energy gained at that distance by the fall!)

Now, observe for spherical geometry and uniform density (where density Rho = M/[(4/3)*Pi*r³]),

4*Pi*Rho*G = 3MG/r³ = 3[MG/r]/r²

So, since E = (1/2)mv² = MG/r,

4*Pi*Rho*G = 3[(1/2)mv²]/r² = (3/2)m(v/r)²

Here, the ratio v/r = dv/dr when translating to a differential equation, and as already shown div.²E = 3*d²E/dr² for radial symmetry, so

4*Pi*Rho*G = (3/2)m(dv/dr)² = div.²E

Hence proof of Poisson’s gravity field equation:

div.²E = 4*Pi*Rho*G.

To get this expressed as tensors you begin with a Ricci tensor R_uv for curvature (this is a shortened Riemann tensor).

R_uv = 4*Pi*G*T_uv,

where T_uv is the energy-momentum tensor which includes potential energy contributions due to pressures, but is analogous to the density term Rho in Poisson's equation. (The density of mass can be converted into energy density simply by using E=mc².)

However, this equation R_uv = 4*Pi*G*T_uv was found by Einstein to be a failure because the divergence of T_uv should be zero if energy is conserved. (A uniform energy density will have zero divergence, and T_uv is of course a density-type parameter. The energy potential of a gravitational field doesn't have zero divergence, because it diverges - falls off - with distance, but uniform density has zero divergence simply because it doesn't fall with distance!)

The only way Einstein could correct the equation (so that the divergence of T_uv is zero) was by replacing T_uv with T_uv - (1/2)(g_uv)T, where R is the trace of the Ricci tensor, and T is the trace of the energy-mass tensor.

R_uv = 4*Pi*G*[T_uv - (1/2)(g_uv)T]

which is equivalent to

R_uv - (1/2)Rg_uv = 8*Pi*G*T_uv

Which is the full general relativity field equation (ignoring the cosmological constant and dark energy, which is incompatible with any Yang-Mills quantum gravity because to use an over-simplified argument, the redshift of gravity-causing exchange radiation between receding masses over long ranges cuts off gravity, negating the need for dark energy to explain observations).

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):

The meaning of such a tensor is revealed by subscript notation, which identify the rank of tensor and its type of variance.

When you look at the mechanism for the physical contraction, you see that general relativity is consistent with FitzGerald's physical contraction, and I've shown this mathematically at my home page. Special relativity according even to Albert Einstein is superseded by general relativity, a fact that Lubos Motl may never grasp, he like other ‘string theorists’ calls everyone interested in Feynman’s objective approach to science a ‘science-hater’. To a string theorist, a lack of connection to physical fact is ‘science-loving’ while a healthy interest in supporting empirically checked work is ‘science-hating’. (String theorists borrowed this idea from KGB propaganda as explained by George Orwell as ‘doublethink’ in the novel 1984.) Because string theory agrees with special relativity, crackpots claim falsely that general relativity is based on the same basic principle of special relativity that is a lie because special relativity is distinct from general covariance that is the heart of general relativity:

‘... the law of the constancy of the velocity of light. But ... the general theory of relativity cannot retain this law. On the contrary, we arrived at the result according to this latter theory, the velocity of light must always depend on the coordinates when a gravitational field is present.’ - Albert Einstein, Relativity, The Special and General Theory, Henry Holt and Co., 1920, p111.

‘... the principle of the constancy of the velocity of light in vacuo must be modified, since we easily recognise that the path of a ray of light … must in general be curvilinear...’ - Albert Einstein, The Principle of Relativity, Dover, 1923, p114.

‘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). ...’ – Albert Einstein, ‘The Foundation of the General Theory of Relativity’, Annalen der Physik, v49, 1916.

‘According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Sidelights on Relativity, Dover, New York, 1952, p23.

‘The Michelson-Morley experiment has thus failed to detect our motion through the aether, because the effect looked for – the delay of one of the light waves – is exactly compensated by an automatic contraction of the matter forming the apparatus…. The great stumbing-block for a philosophy which denies absolute space is the experimental detection of absolute rotation.’ – Professor A.S. Eddington (who confirmed Einstein’s general theory of relativity in 1919), Space Time and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, Cambridge, 1921, pp. 20, 152.

The rank is denoted simply by the number of letters of subscript notation, so that X_a is a ‘rank 1’ tensor (a vector sum of first-order differentials, like net velocity or gradient over applicable dimensions), and X_ab 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, X_a) 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 ...x_n). [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 x_n.

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 = g_ct d(ct)² + g_x dx² + g_y dy² + g_z 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.)

G _ab^c = (1/2)g^cd [(dg_da/dx^b) + (dg_db/dx^a) + (dg_ab/dx^d)]

The Riemann curvature tensor is then represented by:

R^a_cbe = ( dG _bc^a /dx^e ) – ( dG _be^a /dx^c ) + (G _te^a G _bc^t ) – (G _tb^a G _ce^t ).

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.

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 hypothetical case of zero mass.

The components of the stress-energy tensor:

To get solutions, the source of gravity such as the energy of electromagnetic field, can in general relativity be treated as a 'perfect fluid' with no drag properties. Since the gravity source is conveyed by an intervening medium (the spacetime fabric, which we show to be dynamical Yang-Mills exchange radiation based), this medium when considered as an electromagnetic field, causes gravity by behaving as a perfect fluid.

Einstein’s method of obtaining the final answer involved trial and error and the equivalence principle between inertial and gravitational mass, but using Professor Roger Penrose’s approach, Einstein recognised that while this equation reduces to Newton’s law for low speeds, it is in error because it violates the principle of conservation of mass-energy, since a gravitational field has energy (i.e., ‘potential energy’) and vice-versa.

The average angle of the propagation of ray of light from the line to the centre of gravity of the sun during deflection is a right angle. When gravity deflects an object with rest mass that is moving perpendicularly to the gravitational field lines, it speeds up the object as well as deflecting its direction. But because light is already travelling at its maximum speed (light speed), it simply cannot be speeded up at all by falling. Therefore, that half of the gravitational potential energy that normally goes into speeding up an object with rest mass cannot do so in the case of light, and must go instead into causing additional directional change (downward acceleration). This is the mathematical physics reasoning for why light is deflected by precisely twice the amount suggested by Newton’s a = MG/r².

Light has momentum and exerts pressure, delivering energy. Continuous exchange of high-energy gauge bosons can only be detected as the normal forces and inertia they produce.

GENERAL RELATIVITY’S HEURISTIC PRESSURE-CONTRACTION EFFECT AND INERTIAL ACCELERATION-RESISTANCE CONTRACTION

Penrose’s Perimeter Institute lecture is interesting: ‘Are We Due for a New Revolution in Fundamental Physics?’ Penrose suggests quantum gravity will come from modifying quantum field theory to make it compatible with general relativity…I like the questions at the end where Penrose is asked about the ‘funnel’ spatial pictures of blackholes, and points out they’re misleading illustrations, since you’re really dealing with spacetime not a hole or distortion in 2 dimensions. The funnel picture really shows a 2-d surface distorted into 3 dimensions, where in reality you have a 3-dimensional surface distorted into 4 dimensional spacetime. In his essay on general relativity in the book ‘It Must Be Beautiful’, Penrose writes: ‘… when there is matter present in the vicinity of the deviating geodesics, the volume reduction is proportional to the total mass that is surrounded by the geodesics. This volume reduction is an average of the geodesic deviation in all directions … Thus, we need an appropriate entity that measures such curvature averages. Indeed, there is such an entity, referred to as the Ricci tensor …’ Feynman discussed this simply as a reduction in radial distance around a mass of (1/3)MG/c² = 1.5 mm for Earth. It’s such a shame that the physical basics of general relativity are not taught, and the whole thing gets abstruse. The curved space or 4-d spacetime description is needed to avoid Pi varying due to gravitational contraction of radial distances but not circumferences.

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.

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:

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 viscosity. Feynman was unable to proceed with the LeSage gravity and gave up on it in 1965.

The gravity force is the shielded inward reaction (by Newton’s 3rd law the outward force has an equal and opposite reaction):

F = (total outward force).(cross-sectional area of shield projected to radius R) / (total spherical area with radius R).

The cross-sectional area of shield projected to radius R is equal to the area of the fundamental particle (Pi multiplied by the square of the radius of the black hole of similar mass), multiplied by the ratio (R/r)² which is the inverse-square law for the geometry of the implosion. This (R/r)² ratio is very big for a falling apple! Because R is a fixed distance, as far as we are concerned here, the most significant variable the 1/r² factor, which we all know is the Newtonian inverse square law of gravity.

Illustration above: exchange force (gauge boson) radiation force cancels out (although there is compression equal to the contraction predicted by general relativity) in symmetrical situations outside the cone area since the net force sideways is the same in each direction unless there is a shielding mass intervening. Shielding is caused simply by the fact that nearby matter is not significantly receding, whereas distant matter is receding. Gravity is the net force introduced where a mass shadows you, namely in the double-cone areas shown above. In all other directions the symmetry cancels out and produces no net force. Hence gravity can be quantitatively predicted using only well established facts of quantum field theory, recession, etc. In the illustration above, only a ‘core’ of a fundamental particle (the shielding cross-section associated with the ‘Higgs-boson’ type mass-contributors in the standard model) does the shielding; the rest of the particle with its classical electron radius is generally much bigger but it doesn’t all contribute to the actual mass of the electron!

Gravity is not due to a surface compression but instead is mediated through the void between fundamental particles in atoms by exchange radiation which does not recognise macroscopic surfaces, but only interacts with the subnuclear particles associated with the elementary units of mass. The radial contraction of the earth's radius by gravity, as predicted by general relativity, is 1.5 mm. [This contraction of distance hasn't been measured directly, but the corresponding contraction or rather ‘dilation’ of time has been accurately measured by atomic clocks which have been carried to various altitudes (where gravity is weaker) in aircraft. Spacetime tells us that where distance is contracted, so is time.]

This contraction is not caused by a material pressure carried through the atoms of the earth, but is instead due to the gravity-causing exchange radiation of gravity which is carried through the void (nearly 100% of atomic volume is void). Hence the contraction is independent of the chemical nature of the earth. (Similarly, the contraction of moving bodies is caused by the same exchange radiation effect, and so is independent of the material's composition.)

The effective shielding radius of a black hole of mass M is equal to 2GM/c². A shield, like the planet earth, is composed of very small, sub-atomic particles. The very small shielding area per particle means that there will be an insignificant chance of the fundamental particles within the earth ‘overlapping’ one another by being directly behind each other.

The total shield area is therefore directly proportional to the total mass: the total shield area is equal to the area of shielding by 1 fundamental particle, multiplied by the total number of particles. (Newton showed that a spherically symmetrical arrangement of masses, say in the earth, by the inverse-square gravity law is similar to the gravity from the same mass located at the centre, because the mass within a shell depends on its area and the square of its radius.) The earth’s mass in the standard model is due to particles associated with up and down quarks: the Higgs field.

A local mass shields the force-carrying radiation exchange, because the distant masses in the universe have high speed recession, but the nearby mass is not receding significantly. By Newton’s 2nd law the outward force (according of a nearby mass which is not receding (in spacetime) from you is F = ma = m.dv/dt = mv/(x/c) = mcv/x = 0. Hence, by Newton’s 3rd law, the inward force of gauge bosons coming towards you from that mass is also zero; there is no action and so there is no reaction. As a result, the local mass shields you, so you get pushed towards it. This is why apples fall.

Shielding: since most of the mass of atoms is associated with the fields of gluons and virtual particles surrounding quarks, these are the gravity-affected parts of atoms, not the electrons or quarks themselves.

The mass of a nucleon is typically 938 MeV, compared to just 0.511 MeV for an electron and 3-5 MeV for one of the three quarks inside a neutron or a proton. Hence the actual charges of matter aren't associated with much of the mass of material. Almost all the mass comes from the massive mediators of the strong force fields between quarks in nucleons, and between nucleons in nuclei heavier than hydrogen. (In the well-tested and empirically validated Standard Model, charges like fermions don't have mass at all; the entire mass is provided by a vacuum 'Higgs field'. The exact nature of the such a field is not predicted, although some constraints on its range of properties are evident.)

The radiation is received by mass almost equally from all directions, coming from other masses in the universe; the radiation is in effect reflected back the way it came if there is symmetry that prevents the mass from being moved. The result is then a mere compression of the mass by the amount mathematically predicted by general relativity, i.e., the radial contraction is by the small distance MG/(3c²) = 1.5 mm for the contraction of the spacetime fabric by the mass in the Earth. Plotting the earth and the observable distant receding matter average distance circles (not to scale) the geometry of the mechanism becomes clear:

‘When one considers Maxwell’s equations for just the electromagnetic field, ignoring electrically charged particles, one finds that the equations have some peculiar extra symmetries besides the well-known gauge symmetry and space-time symmetries. The extra symmetry comes about because one can interchange the roles of the electric and magnetic fields in the equations without changing their form. The electric and magnetic fields in the equations are said to be dual to each other, and this symmetry is called a duality symmetry. Once electric charges are put back in to get the full theory of electrodynamics, the duality symmetry is ruined. In 1931 Dirac realised that to recover the duality in the full theory, one needs to introduce magnetically charged particles with peculiar properties. These are called magnetic monopoles and can be thought of as topologically non-trivial configurations of the electromagnetic field, in which the electromagnetic field becomes infinitely large at a point. Whereas electric charges are weakly coupled to the electromagnetic field with a coupling strength given by the fine structure constant alpha = 1/137, the duality symmetry inverts this number, demanding that the coupling of the magnetic charge to the electromagnetic field be strong with strength 1/alpha = 137. [This applies to the magnetic dipole Dirac calculated for the electron, assuming it to be a Poynting wave where E = c*B and E is shielded by vacuum polarization by a factor of 1/alpha = 137.]

‘If magnetic monopoles exist, this strong [magnetic] coupling to the electromagnetic field would make them easy to detect. All experiments that have looked for them have turned up nothing…’ - P. Woit, Not Even Wrong, Jonathan Cape, London, 2006, pp. 138-9. [Emphasis added.]

The Pauli exclusion principle normally makes the magnetic moments of all electrons undetectable on a macroscopic scale (apart from magnets made from iron, etc.): the magnetic moments usually cancel out because adjacent electrons always pair with opposite spins! If there are magnetic monopoles in the Dirac sea, there will be as many ‘north polar’ monopoles as ’south polar’ monopoles around, so we can expect not to see them because they are so strongly bound!

(1) required by the Dirac equation’s negative energy solution, and which is

(2) experimentally demonstrated by antimatter.

‘As long as the leadership of the particle theory community refuses to face up to what has happened and continues to train young theorists to work on a failed project, there is little likelihood of new ideas finding fertile ground in which to grow. Without a dramatic change in the way theorists choose what topics to address, they will continue to be as unproductive as they have been for two decades, waiting for some new experimental result finally to arrive.’

John Horgan’s 1996 excellent book The End of Science, which Woit argues is the future of physics if people don’t keep to explaining what is known (rather than speculating about unification at energy higher than can ever be seen, speculating about parallel universes, extradimensions, and other non-empirical drivel), states:

‘A few diehards dedicated to truth rather than practicality will practice physics in a nonempirical, ironic mode, plumbing the magical realm of superstrings and other esoterica and fretting about the meaning of quantum mechanics. The conferences of these ironic physicists, whose disputes cannot be experimentally resolved, will become more and more like those of that bastion of literary criticism, the Modern Language Association.’

L. Green, "Engineering versus pseudo-science", Electronics World, vol. 110, number 1820, August 2004, pp52-3:

‘… controversy is easily defused by a good experiment. When such unpleasantness is encountered, both warring factions should seek a resolution in terms of definitive experiments, rather than continued personal mudslinging. This is the difference beween scientific subjects, such as engineering, and non-scientific subjects such as art. Nobody will ever be able to devise an uglyometer to quantify the artistic merits of a painting, for example.’ (If string theorists did this, string theory would be dead, because my mechanism published in Oct 96 E.W. and Feb. 97 Science World, predicts the current cosmological results which were discovered about two years later by Perlmutter.)

This comment from Dr Love is extremely depressing; we all know today’s physics is a religion. I found out after emailed exchanges with, I believe, Dr John Gribbin, the author of numerous crackpot books like ‘The Jupiter Effect’ (claiming Los Angeles would be destroyed by an earthquake in 1982), and quantum books trying to prove Lennon’s claim ‘nothing is real’. After explaining the facts to Gribbin, he then emailed me a question something like (I have archives of emails by the way, so could check the exact wording if required): ‘you don’t seriously expect me to believe that or write about it?’

This elevation of a mere possibility to a truth, and then the use of this truth to convince oneself one has the correct theory, is a rather large extrapolation.’

A fruitful natural philosophy has a double scale or ladder ascendant and descendant; ascending from experiments to axioms and descending from axioms to the invention of new experiments. - Novum Organum.

This would allow LQG to be built as a bridge between path integrals and general relativity. I wish Smolin or Woit would pursue this.

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.)

- Feynman, QED, Penguin, 1990, page 54.

That's wave particle duality explained. The path integrals don't mean that the photon goes on all possible paths but as Feynman says, only a "small core of nearby space".

The double-slit interference experiment is very simple: the photon has a transverse spatal extent. If that overlaps two slits, then the photon gets diffracted by both slits, displaying interference. This is obfuscated by people claiming that the photon goes everywhere, which is not what Feynman says. It doesn't take every path: most of the energy is transferred along the classical path, and is near that.

Similarly, you find people saying that QFT says that the vacuum is full of loops of annihilation-creation. When you check what QFT says, it actually says that those loops are limited to the region between the IR and UV cutoff. If loops existed everywhere in spacetime, ie below the IR cutoff or beyond 1 fm, then the whole vacuum would be polarized enough to cancel out all real charges. If loops existed beyond the UV cutoff, ie to zero distance from a particle, then the loops would have infinite energy and momenta and the effects of those loops on the field would be infinite, again causing problems.

So the vacuum simply isn't full of loops (they only extend out to 1 fm around particles). Hence no dark energy mechanism.

Mainstream string theory or M-theory (due to Witten, 1995) theory is the 10 dimensional superstring / 11 dimensional supergravity unification which can't predict anything potentially checkable. It says that there are 10 dimensions of particle physics predicting 10^500 or so different Standard Models (because particle properties can take many values due to the many parameters of size and shape for the complex 6-dimensional Calabi-Yau manifold, which compactifies 6 of the 10 dimensions to give 4-d spacetime), each in a parallel universe! M-theory says that 10-dimensional superstring theory is a (mem)brane on 11-dimensional hyperspace of supergravity, like a 2-dimensional flat credit card containing a 3-dimensional hologram or 3 dimensional space containing ‘curvature’ due to time dimension(s). Despite all the ad hoc speculation, M-theory can’t give any checkable physics!

Unobservable extra dimensions curled up into imaginary Planck scale Calabi-Yau manifold strings, and there is postulated 1:1 boson:fermion supersymmetric partners for all Standard Model particles, to achieve ever-unobservable unification at the Planck scale. Watch how string theory dances around to impress the public without giving any real physics! It cannot ever go away because it is not a falsifiable theory. So after being ridiculed and dismissed, it always survives and come back again to sneer at alternatives which are checkable!

However, the extra dimensional speculation on general relativity, reinforced by confirmation of general relativity in various experimental tests, has led to a hardening of orthodoxy in favour of the real existence of extra dimensions. Although general relativity is 3 + 1 dimensional, the extra dimension being treated as a resultant (time), the Kaluza-Klein theory adds still another (fifth) dimension which is gives a way of combining electromagnetism and gravitation qualitatively (it makes no checkable predictions) through general relativity. The extra dimension was supposed to be rolled up into a small loop that constitutes a particle of matter. Vibrations of the loop or closed string allow it to represent different energy states, each corresponding to the different fundamental particles. There is no checkable prediction from this theory, not even the size of the loop, which is postulated to be Planck size due to Planck's fame. Planck's length - which he based on arbitrary dimensional analysis - is far bigger (G^1/2h^1/2c^-3/2 ~ 10^-35 m) than the black hole radius of an electron (2GM/c² = 1.3 x 10^-57 m) and so it is highly suspect whether the dimensional analysis numerology of the Planck size belies any real physics. The rest-mass energies of particles cannot be predicted from string theory. Later, the ad hoc suggestion was made that the Calabi-Yau six dimensional manifold be included in the string theory, leading to 10/11 dimensional superstrings/supergravity (unified by ideas like Witten's M-theory and the holographic conjecture) with a 'landscape' of 10³⁵⁰ or so values of the quantum field theory vacuum energy ground state.

The correct theory of quantum gravity to describe general relativity, applied to cosmology, must discriminate between the big bang induced cosmic expansion and the contraction of the dimensions describing matter due to gravity. There are three expanding dimensions in the big bang cosmology and three dimensions for matter that are contracted by motion and by gravitation.

Yang-Mills quantum field theory is abstract yet suggests physical dynamics: exchange of gauge bosons causes forces. This is clearly displaced by familiar Feynman diagrams depicting fundamental force exchange radiations. Via the October 1996 issue of the journal Electronics World, a mechanism was made available in an eight pages long.

Neither the equations of quantum mechanics nor Alain Aspects experiments disprove causality proper or prove Copenhagen philosophy/politics/religion.

Dr Thomas Love has proved that the entanglement philosophy is just a statement of the mathematical discontinuity between the time-dependent and time-independent Schroedinger wave equations when a measurement is taken. There’s no evidence for metaphysical wave function collapse in either the authority of Niels Bohr, the Solvay Congress of 1927, or Alain Aspect’s determination that the polarization of photons emitted in opposite directions by an electron correlate when measured metres apart.

Copenhagen quantum mechanics is speculative. So don’t build it up as a pet religion. The uncertainty principle in the Dirac sea has a perfectly causal explanation: on small distance scales, particles get randomly accelerated/decelerated/deflected by the virtual particles of the spacetime vacuum. This is like Brownian motion. On large scales, the interactions cancel out. If so, then photon polarizations correlate not because of metaphysical "wavefunction entanglement" but because the uncertainty principle doesn’t apply to measurements on light speed bosons, and only to massive fermions which are still there after you actually detect them.

A loop is a rotational transformation in the vacuum. The loop physically the exchange of energy-delivering field radiation from one mass to another, and back to the first mass again. Like the exchange radiation in Yang-Mills (Standard Model) theories, but with the added restriction of the conservation (looping between masses) of the exchange radiation? Things accelerated by a gravity field are losing gravitational potential energy and gaining kinetic energy, so the exchange radiation carries energy. If the LQG spinfoam vacuum does describes a Yang-Mills energy exchange scheme, you can get solid checkable predictions by taking account of the effect of the expansion of the universe on these conserved gravity field mediators.

If you observe two supernovae at the same time, you can in fact determine which occurred first by simply noting from their redshifts how far they are from you in time and space, and hence how long after the big bang each occurred. Hence there is an absolute time scale. Special relativity as usually taught denies absolute chronology, which doesn’t work where you can place absolute chronology on events like supernovae. A better theory will clearly separate the treatment of the expanding big bang spacetime dimensions (which measure the volume of the vacuum), from the local contractable/time dilation-able dimensions used for matter like clocks & rulers. Matter is contracted (in spacetime) by motion and gravity. But the big bang’s spacetime continues expanding. Hence the mathematical treatment of the universe needs to clearly distinguish between the 3 perpetually expanding spacetime dimensions for the volume of the universe, and the 3 contractable dimensions used to describe matter. When Einstein and Hilbert built general relativity in November 1915, they simply didn’t know that the volume of the vacuum was perpetually expanding. People thought it was static.

Above: mechanism of attraction and repulsion in electromagnetism, and the capacitor summation of displacement current energy flowing between accelerating (spinning) charges as gauge bosons (by analogy to Prevost’s 1792 model of constant temperature as a radiation equilibrium). The net exchange is like two machine gunners firing bullets at each other; they recoil apart. The gauge bosons pushing them together are redshifted, like nearly spent bullets coming from a great distance, and are not enough to prevent repulsion. In the case of attraction, the same principle applies. The two opposite charges shield one another and get pushed together. Although each charge is radiating and receiving energy on the outer sides, the inward push is from redshifted gauge bosons, and the emission is not redshifted. The result is just like two people, standing back to back, firing machine guns. The recoil pushes them together, hence the attraction force.