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General Relativity: Does it Prove the Cause and Strength of Gravity?

Nigel B. Cook
Alternate email:
nigelbryancook@hotmail.com
[Note: See
http://members.lycos.co.uk/nigelbryancook/ for more information.]


ABOVE: the two vital physical processes implied by general relativity: contraction and gravitation. Lee Smolin and others have shows how you can connect quantum field theory with general relativity, a process of summing the graphs for all vacuum interactions over spatial topography, Feynman’s sum-over-histories or path-integrals. The result discredits the idea that general relativity is simply special relativity with an extension. As Einstein stated, special relativity fails to deal with absolute motions and must give way to general covariance in general relativity.

The facts: gravity is the force of Feynman diagram gauge bosons coming from distances/times in the past. The Standard Model, the quantum field theory of electromagnetic and nuclear interactions which has made numerous well-checked predictions, forces arise by the exchange of gauge bosons. This is well known from the pictorial ‘Feynman diagrams’ of quantum field theory. Gravitation, as illustrated by this mechanism and proved below, is just this exchange process. Gauge bosons hit the mass and bounce back, like a reflection. This causes the contraction term of general relativity, a physical contraction of radius around a mass: (1/3)MG/c2 = 1.5 mm for Earth. Mass (which by the well-checked equivalence principle of general relativity is identical for inertial and gravitational forces), arises not from the fundamental core particles of matter themselves, but by a miring effect of the spacetime fabric, the ‘Higgs bosons’. Forces are exchanges of gauge bosons: the pressure causes the cosmic expansion. The big bang observable in spacetime has speed from 0 to c with times past of 0 toward 15 billion years, giving outward force of F = ma = m.(variation in speeds from 0 to c)/(variation in times from 0 to age of universe) ~ 7 x 1043 Newtons. Newton’s 3rd law gives equal inward force, carried by gauge bosons, which are shielded by matter. Spacetime, Newton’s laws, tensor calculus and geometry are all well accepted, so why isn’t this mechanism?

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.’

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). This formula, ds2 = å (dx2) = dx2 + dy2 + dz2d(ct)2, is known as the ‘Riemann metric’. 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. [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.]

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 dimensionless ‘Riemann metric’ (named the ‘metric tensor’ by Einstein in 1916):

g = ds2 = gm n dx-m dx-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.

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 some ‘string theorists’ do not grasp:

‘... 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 tensor 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). 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’ gibberish 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:

g = ds2 = gm n dx-m dx-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.)

Let’s examine the developments Einstein introduced to accomplish general relativity, which aims to equate 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. One crucial point to be made is the following:

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).

The point above is made clear by Professor Lee Smolin on page 42 of the USA edition of his 1996 book, ‘The Trouble with Physics.’

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:

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.

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. For understanding the physics, the Ricci tensor generally depends on g in the manner: Rm n = c2(dg /dx-m )(dg /dx-n ). Then the trace R = c2d2 g/ds2. In each case the resulting dimensions are (acceleration/distance) = (time)-2, assuming we can treat the tensors as real numbers (which, as Heaviside showed, is often possible for operators).

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.

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 Tm n is the mass-energy tensor, Tm n = r um un . 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:

 

 

Illustration credit: http://en.wikipedia.org/wiki/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.

According to most statements of Newton’s second law and universal gravitation law, F = ma = mMG/r2, but a serious flaw here is that F = ma is not an accurate statement because during acceleration the mass m varies with the speed (mass increases dramatically at relativistic velocities, i.e., velocities approaching c). A more accurate version of Newton's second law is therefore his original formulation, F = dp/dt where p is momentum (for low velocities only, p ~ mv). Even for the low velocity case where p ~ mv, this law expands by the product law in calculus to F = dp/dt ~ d(mv)/dt = (m.dv/dt) + (v.dm/dt). For the situation where m is a variable (relativistic velocities), the gravity law will therefore be complicated than Newton's universal gravitational law (F = mMG/r2). The Poisson equation for the Newtonian potential is Ñ2 F = 4p rG, where r is density. The Laplacian operator Ñ2 signifies the sum of second-order differentials of F; because there are three terms they add up (in spherical symmetry) to give 3a/r, where a is the gravitational acceleration along radius r. To convert Ñ2 F = 4p rG into the Einstein field equation requires replacing the mass density r by the energy-momentum tensor Tm n , so that field energy and pressure energy are included along with the energy equivalent of the mass density, and also replacing Ñ2 F by a rank-2 tensor.

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.

General relativity is an energy accountancy package, but you need physical intuition to use it. This reason is more of an accounting trick than a classical explanation. As Penrose points out, Newton’s law as expressed in tensor form with E=m c2 is fairly similar to Einstein’s field equation: Rm n = 4p GTm n /c2. Einstein’s result is: –½gm n R + Rm v = 8p GTm n /c2. The fundamental difference is due to the inclusion of the contraction term, –½gm n R, which doubles the value of the other side of the equality.

In an article by Penrose in the book It Must Be Beautiful Penrose explains the tensors of general relativity physically:

‘… 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, constructed from [the big Riemann tensor] R_abcd. Its collection of components is usually written R_ab. There is also an overall average single quantity R, referred to as the scalar curvature.’

Einstein’s field equation states that the Ricci tensor, minus half the product of the metric tensor and the scalar curvature, is equal to
8p GTm n /c2, where Tm n is the mass-energy tensor which is basically the energy per unit volume (this is not so simple when you include relativistic effects and pressures). The key physical insight is the volume reduction, which can only be mechanistically explained as a result of the pressure of the spacetime fabric.

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 (- gm n dx-m dx-n )1/2, while the proper time is ò (gm n dx-m dx-n )1/2. Notice that the ratio of proper length to proper time is always c.

Now, –½gm n R + Rm v = 8p GTm n /c2, is usually shortened to the vague and therefore unscientific and meaningless ‘Einstein equation,’ G = 8p T. Teachers who claim that the ‘conciseness’ and ‘beautiful simplicity’ of ‘G = 8p T’ is a ‘hallmark of brilliance’ are therefore obfuscating. A year later, in his paper ‘Cosmological Considerations on the General Theory of Relativity’, Einstein force-fitted it to the assumed static universe of 1916 by inventing a new cosmic ‘epicycle,’ the cosmological constant, to make gravity weaken faster than the inverse square law, become zero at a distance equal to the average separation distance of galaxies, and to become repulsive at greater distances. In fact, as later proved, such an epicycle, apart from being merely wild speculation lacking a causal mechanism, would be unstable and collapse into one lump. Einstein finally admitted that it was ‘the biggest blunder’ of his life.

There is a whole industry devoted toG = 8p T’ which is stated as meaning ‘curvature of space = mass-energy’ in an attempt to try to obfuscate so as to cover up the fact that Einstein had no mechanism of gravitation. In fact of course, Einstein admitted in 1920 in his inaugural lecture at Leyden that the deep meaning of general relativity is that in order to account for acceleration you need to dump the baggage associated with special relativity, and go back to having what he called an ‘ether’, or a continuum/fabric of spacetime. Something which doesn’t exist can hardly be curved.

The Ricci tensor is in fact a shortened form of a big Riemann rank 4 tensor (the expansions and properties of which are capable of putting anyone off science). To be precise, Rm v = Rmavb g-a-b , while R = Rm v g-m-v . No matter how many times people ‘hype’ up gibberish with propaganda labels such as ‘beautifully simplicity,’ Einstein lacked a mechanism of gravity and fails to fit the big bang universe without force-fitting it using ad hoc ‘epicycles’. The original epicycle was the ‘cosmological constant’, L . This falsely was used to keep the universe stable: G + L gm n = 8p T. This sort of thing is, while admitted in 1929 to be an error by Einstein, still being postulated today, without any physical reasoning and with just ad hoc mathematical fiddling to justify it, to ‘explain’ why distant supernovae are not being slowed down by gravitation in the big bang. I predicted there was a small positive cosmological constant epicycle in 1996 (hence the value of the dark energy) by showing that there is no long range gravitational retardation of distant receding matter because that is a prediction of the gravity mechanism on this page, published via the October 1996 issue of Electronics World (letters page). Hence ‘dark energy’ is speculated as an invisible, unobserved epicycle to maintain ignorance. There is no ‘dark energy’ but you can calculate and predict the amount there would be from the fact the expansion of the universe isn’t slowing down: just accept the expansion goes as Hubble’s law with no gravitational retardation and when you normalise this with the mainstream cosmological model (which falsely assumes retardation) you ‘predict’ the ‘right’ values for a fictitious cosmological constant the fictitious dark energy.

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.

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

Light has momentum and exerts pressure, delivering energy. The pressure towards us due to the gauge bosons (force-causing radiation of quantum field theory), produces the contraction effect of general relativity and also gravity by pushing us from all directions equally, except where reduced by the shielding of the planet earth below us. Hence, the overriding push is that coming downwards from the stars above us, which is greater than the shielded effect coming up through the earth. This is the mechanism of the acceleration due to gravity. We are seeing the past with distance in the big bang! Gravity consists of gauge boson radiation, coming from the past just like light itself. The big bang causes outward acceleration in observable spacetime (variation in speed from 0 toward c per variation of times past from 0 toward 15,000,000,000 years), hence force by Newton’s empirical 2nd law, F = ma. The 3rd empirical law of Newton says there’s equal inward force, carried by gauge bosons that get shielded by mass, proving gravity to within 1.65%.

Hence, outward force of big bang: F = Ma = [(4/3) pR3 r (local) ].[Hc] = c4 / (e3 G) = 6.0266 x 1042 Newtons. Notice the permitted high accuracy, since the force is simply F = c4 / (e3 G), where c, e (a mathematical constant) and G are all well known. (The density and Hubble constant have cancelled out.) When you put this result for outward force into the geometry in the lower illustration above and allow for the effective outward force being e3 times stronger than the actual force (on account of the higher density of the earlier universe, since we are seeing – and being affected by – radiation from the past), you get F = Gm2 /r2 Newtons, if the shielding area is taken as the black hole area (radius 2Gm/c2 ). Why m2 ? Because all mass is created by the same fundamental particles, the ‘Higgs bosons’ of the standard model, which are the building blocks of all mass, inertial and gravitational!

The muon is 1.5 units on this scale but this is heuristically explained by a coupling of the core (mass 1) with a virtual particle, just as the electron couples increasing its magnetic moment to about 1 + 1/(2.Pi.137). The mass increase of a muon is 1 + 1/2 because Pi is due to spin and the 137 shielding factor doesn’t apply to bare particles cores in proximity, as it is due to the polarised vacuum veil at longer ranges. This is why unification of forces is approached with higher energy interactions, which penetrate the veil.

The mechanism is that the 137 number is the ratio between the strong nuclear and the electromagnetic force strength, which is a unification arising due to the polarisation of the vacuum around a fundamental particle core. Therefore, the Coulomb force near the core of the electron is the same as the strong nuclear force (137 times the observed Coulomb force), but 99.27% of the core force is shielded by the veil of polarised vacuum surrounding the core. Therefore, if the mass-causing Higgs bosons of the vacuum are outside the polarised veil, they couple weakly, giving a mass 137 times smaller (electron mass), and if they are inside the veil of polarised vacuum, they couple 137 times more strongly, giving higher mass particles like muons, quarks, etc (depending on the discrete number of Higgs bosons coupling to the particle core: the for all directly observable elementary particle masses (quarks are not directly observable, only as mesons and baryons) is (0.511 Mev).(137/2)n(N + 1) = 35n(N + 1) Mev. This predicts that a elementary particle core containing n fundamental particles (n=1 for leptons, n=2 for mesons, and n=4 for baryons) couples to integer N virtual vacuum particles (Higgs field particles?), and hence has an associative inertial/gravitational mass of:

(0.511 Mev).(137/2)n(N + 1) = 35n(N + 1) Mev,

where 0.511 Mev is the electron mass. Thus we get everything from this one mass plus integers 1,2,3 etc, with a mechanism. This is right for lepton/hadrons masses (quarks can never be exist individually, only as pairs/triplets). The polarised dielectric of the vacuum around the core of a fundamental particle (be that a string or a loop) which shields the core force. Unification physically occurs when you knock particles together so hard that they penetrate the polarised dielectric which is shielding most of the electric field from the core, so that the stronger electric field of the core is then involved in the reaction. QCD involves gluons not photons as the mediator, but the strength of the forces becomes equal if you smash the particles together with energy enough to break through the polarised spacetime fabric around a fundamental particle. Suppose the Standard Model QFT is right, and mass is due to Higgs field. In that case, the Higgs field particles associating with the core of a fundamental particle give rise to mass. The field which is responsible for associating the Higgs field particles with the mass can be inside or outside the polarised veil of dielectric, right? If the Higgs field particles are inside the polarised veil, the force between the fundamental particle and the mass creating field particle is very strong, say 137 times Coulomb's law. On the other hand, if the mass causing Higgs field particles are outside the polarised veil, the force is 137 times less than the strong force. This implies how the 137 factor gets in to the distribution of masses of leptons and hadrons.

To recap, the big bang has an outward force of 6.0266 x 1042 Newtons (by Newton’s 2nd law) that results in an equal inward force (by Newton’s 3rd law) which causes gravity as a shielded inward force, Higgs field or rather gauge boson pressure. This is based on standard heuristic quantum field theory (for the Feynman path integral approach), where forces are due not to empirical equations but to the exchange of gauge boson radiation. Where partially shielded by mass, the inward pressure causes gravity. Apples are pushed downwards towards the earth, a shield:

‘… the source of the gravitational field [gauge boson radiation] can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Bernard Schutz, General Relativity, Cambridge University Press, 1986, pp. 89-90.

LeSage in 1748 argued that there is some kind of pressure in space, and that masses shield one another from the space pressure, thus being pushed together by the unshielded space pressure on the opposite side. Feynman discussed LeSage in November 1964 lectures Character of Physical Law, and elsewhere explained that the major advance of general relativity, the contraction term, shortens the radius of every mass, like the effect of a pressure mechanism for gravity! He does not derive the equation, but we will do so below.

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. However, we have a solution…

In quantum gravity, the big error in physics is that Edwin Hubble in 1929 divided the Doppler shift determined recession speeds by the apparent distances to get his constant, v/R = H. In fact, the distances increase while the light and gravity effect are actually coming back to us. What he should have done is to represent it as a variation in speed with time past. The whole point about space-time is precisely that there is equivalence between seeing thing at larger distances, and seeing things further back in time. You cannot simply describe the Hubble effect as a variation in speed with distance, because time past is involved! Whereas H has units s-1 (1/age of universe), the directly observed Hubble ratio is equal to v/t = RH/(1/H) = RH2 (and therefore has units of ms-2, acceleration). In the big bang, the recession velocities from here outward vary from v = 0 towards v = c, and the corresponding times after the big bang vary from 15,000 million years (t = 1/H) towards zero time. Hence, the apparent acceleration as seen in space-time is

a = (variation in velocity)/(variation in time) = c / (1/H) = cH = 6 x 10-10 ms-2.

Although a small acceleration, a large mass of the universe is involved so the outward force (F = ma) is very large. The 3rd law of motion implies equal inward force like an implosion, which in LeSage gravity gives the right value for G, disproving the ‘critical density’ formula of general relativity by ½ e3 = 10 times. This disproves most speculative ‘dark matter’. Since gravity is the inward push caused by the graviton/Higgs field flowing around the moving fundamental particles to fill in the void left in their wake, there will only be a gravitational ‘pull’ (push) where there is a surrounding expansion. Where there is no surrounding expansion there is no gravitational retardation to slow matter down. This is in agreement with observations that there is no slowing down (a fictitious acceleration is usually postulated to explain the lack of slowing down of supernovae).

If volume of universe is (4/3)p R3 and expansion is R=ct, then density varies as t-3, and for a star at distance r, absolute time after big bang will be t – r/c (where t is our local time after big bang, about 15 Gyr), so the density of the universe at its absolute age corresponding to visible distance r, divided by the density locally at 15 Gyr, is [(t – r/c)/t]-3 = (1 – rc-1t-1)-3, which is the factor needed to multiply up the nearby density to give that at earlier times corresponding to large visible distances. This formula gives infinite density at the finite radius of the universe, whereas an infinite density only exists in a singularity; this requires some dismissal of special relativity, either by saying that the universe underwent a faster than c expansion at early times (Guth’s special relativity violating inflationary universe), or else by saying that the red-shifted radiation coming to us is actually travelling very slowly (this is more heretical than Guth’s conjecture). Setting this equal to density factor e3 we see that 1 - r/(ct) = 1/e. Hence r = 0.632ct. This means that the effective distance at which the gravity mechanism source lies is at 63.2 % of the radius of the universe, R = ct. At that distance, the density of the universe is 20 times the local density where we are, at a time of 15,000,000,000 years after the big bang. Therefore, the effective average distance of the gravity source is 9,500,000,000 light years away, or 5,500,000,000 years after the big bang.

Maxwell’s 1873 Treatise on Electricity and Magnetism, Articles 822-3: ‘The ... action of magnetism on polarised light [discovered by Faraday not Maxwell] leads ... to the conclusion that in a medium ... is something belonging to the mathematical class as an angular velocity ... This ... cannot be that of any portion of the medium of sensible dimensions rotating as a whole. We must therefore conceive the rotation to be that of very small portions of the medium, each rotating on its own axis [spin] ... The displacements of the medium, during the propagation of light, will produce a disturbance of the vortices ... We shall therefore assume that the variation of vortices caused by the displacement of the medium is subject to the same conditions which Helmholtz, in his great memoir on Vortex-motion [of 1858; sadly Lord Kelvin in 1867 without a fig leaf of empirical evidence falsely applied this vortex theory to atoms in his paper ‘On Vortex Atoms’, Phil. Mag., v4, creating a mathematical cult of vortex atoms just like the mathematical cult of string theory now; it created a vast amount of prejudice against ‘mere’ experimental evidence of radioactivity and chemistry that Rutherford and Bohr fought], has shewn to regulate the variation of the vortices [spin] of a perfect fluid.’

‘… the source of the gravitational field can be taken to be a perfect fluid…. A fluid is a continuum that ‘flows’... A perfect fluid is defined as one in which all antislipping forces are zero, and the only force between neighboring fluid elements is pressure.’ – Professor Bernard Schutz, General Relativity, Cambridge University Press, 1986, pp. 89-90.

‘In this chapter it is proposed to study the very interesting dynamical problem furnished by the motion of one or more solids in a frictionless liquid. The development of this subject is due mainly to Thomson and Tait [Natural Philosophy, Art. 320] and to Kirchhoff [‘Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit’, Crelle, lxxi. 237 (1869); Mechanik, c. xix]. … it appeared that the whole effect of the fluid might be represented by an addition to the inertia of the solid. The same result will be found to hold in general, provided we use the term ‘inertia’ in a somewhat extended sense.’ – Sir Horace Lamb, Hydrodynamics, Cambridge University Press, 6th ed., 1932, p. 160. (Hence, the gauge boson radiation of the gravitational field causes inertia. This is also explored in the works of Drs Rueda and Haisch: see http://arxiv.org/abs/physics/9802031 http://arxiv.org/abs/gr-qc/0209016 , http://www.calphysics.org/articles/newscientist.html and http://www.eurekalert.org/pub_releases/2005-08/ns-ijv081005.php .)

So the Feynman problem with virtual particles in the spacetime fabric retarding motion does indeed cause the FitzGerald-Lorentz contraction, just as they cause the radial gravitationally produced contraction of distances around any mass (equivalent to the effect of the pressure of space squeezing things and impeding accelerations). What Feynman thought may cause difficulties is really the mechanism of inertia!

‘Recapitulating, we may say that according to the general theory of relativity, space is endowed with physical qualities... According to the general theory of relativity space without ether is unthinkable.’ – Albert Einstein, Leyden University lecture on ‘Ether and Relativity’, 1920. (Einstein, A., Sidelights on Relativity, Dover, New York, 1952, pp. 15, 16, and 23.)

‘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), MA, MSc, FRS, Space Time and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, Cambridge, 1921, pp. 20, 152.

So the contraction of the Michelson-Morley instrument made it fail to detect absolute motion. This is why special relativity needs replacement with a causal general relativity:

‘… with the new theory of electrodynamics [vacuum filled with virtual particles] we are rather forced to have an aether.’ – Paul A. M. Dirac, ‘Is There an Aether?,’ Nature, v168, 1951, p906. (If you have a kid playing with magnets, how do you explain the pull and push forces felt through space? As ‘magic’?) See also Dirac’s paper in Proc. Roy. Soc. v.A209, 1951, p.291.

‘It has been supposed that empty space has no physical properties but only geometrical properties. No such empty space without physical properties has ever been observed, and the assumption that it can exist is without justification. It is convenient to ignore the physical properties of space when discussing its geometrical properties, but this ought not to have resulted in the belief in the possibility of the existence of empty space having only geometrical properties... It has specific inductive capacity and magnetic permeability.’ - Professor H.A. Wilson, FRS, Modern Physics, Blackie & Son Ltd, London, 4th ed., 1959, p. 361.

 

The latest version of the gravity mechanism proof suggested may be found at http://nige.wordpress.com/2007/05/25/quantum-gravity-mechanism-and-predictions/ and http://quantumfieldtheory.org/1.pdf, while older material is located at http://quantumfieldtheory.org/proof.htm.

Acknowledgement

The author wishes to thank Walter Babin for hosting this and several other papers on the General Science Journal for the last seven years.