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Email: Nigel B. Cook


ASKING THE RIGHT QUESTIONS TO OBTAIN SCIENTIFIC ANSWERS

Nigel B. Cook
Alternate email:
nigelbryancook@hotmail.com

INTRODUCTION

‘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.

‘… the innovator has for enemies all those who have done well under the old conditions, and lukewarm defenders in those who may do well under the new. This coolness arises partly from fear of the opponents, who have the laws on their side, and partly from the incredulity of men, who do not readily believe in new things until they have had a long experience of them. Thus it happens that whenever those who are hostile have the opportunity to attack they do it like partisans, whilst the others defend lukewarmly…’ - http://www.constitution.org/mac/prince06.htm

‘(1). The idea is nonsense. (2). Somebody thought of it before you did. (3). We believed it all the time.’ - Professor R.A. Lyttleton's summary of inexcusable censorship (quoted by Sir Fred Hoyle in ‘Home is Where the Wind Blows’ Oxford University Press, 1997, p154).

Because of Drs Susskind and Witten, the media has let string theory go on without asking for definite testable predictions. I don’t think the layman public takes much notice of ‘theory’ it can’t understand. There are three types of not-yet-falsified theory:

1. Experimentally confirmed but mathematically abstract and possibly incomplete (Standard Model, relativity, quantum mechanics, etc.)

2. Not experimentally confirmed but popularised with best selling books, but possibly testable (Hawking radiation, gravity waves, etc)

3. Untestable/not falsifiable (over-hyped string theory’s vague landscape ‘predicting’ 10500 vacua, 10/11 dimensions, vague suggestions of superpartners without predicting their energy to show if they can be potentially checked or not, ‘prediction’ of unobservable gravitons without any testable predictions of gravity)

Back in 1996, ‘popular physics’ authors were flooding the media with hype about backward time travel, 10 dimensional strings, parallel universes and Kaku flying saucer speculation, and were obviously lying that such unpopular non-testable guesses were science.

1. Feynman diagrams, the physics behind the maths of quantum field theory, show that forces arise from the exchange of gauge bosons (coming from distances at light speed, hence coming from times in the past).

2. The big bang mass has an increasing speed, in our observable spacetime, from 0 toward speed of light c with times past of 0 toward 15 billion years (or distances of 0 to 15 billion light-years), giving outward force by Newton’s 2nd empirically based law: F = ma = m.dv/dt = m(c - 0) / (age of universe) = mcH, where H is Hubble’s constant (based on v = HR, where R is distance).

3. Newton’s 3rd empirically based law suggests equal inward implosion force, carried by gauge bosons, which shielded by mass, proves gravity and electromagnetism to within 1.65% (proof below). This mechanism also predicts particle masses and other observables, and eliminates most of the unobserved ‘dark matter’ speculation and the need for a cosmological constant / dark energy (the latest data suggest that the ‘cosmological constant’ and dark energy epicycle would need to vary with time!

These are all existing accepted facts; the Feynman diagrams are widely accepted, as is the spacetime, the big bang, Newton’s laws of motion. The result, that apples fall at the measured acceleration, is apparently ‘only a personal pet theory that should be suppressed from arXiv.org and ignored’. Drs Lee Smolin and Peter Woit could sit under an apple tree to verify that existing ‘string theory’ gravity is ‘speculative gibberish’: it is an effort to destroy science using untestable hocus pocus ‘string theory’!

Update: Lee Smolin has now kindly acknowledged the possibility of using this type of argument (that quantum field theory gauge boson exchange process predicts magnetic moments and Lamb shift, so an attempt to unify the spacetime fabric with Feynman path integrals is an empirically defendable physical reality, unlike ‘string theory’ speculation). This applies for some kind of spin foam vacuum in loop quantum gravity, as mentioned on Peter Woit’s blog. Smolin is committed to the very difficult mathematical approach, but was decent enough say:

Nigel Says: January 14th, 2006 at 2:18 pm

Some kind of loop quantum gravity is going to be the right theory, since it is a spin foam vacuum. People at present are obsessed with the particles that string theory deals with, to the exclusion of the force mediating vacuum. Once prejudices are overcome, proper funding of LQG should produce results.

Lee Smolin Says:
January 14th, 2006 at 4:41 pm

.. Thanks also to Nigel for those supporting comments. Of course more support will lead to more results, but I would stress that I don’t care nearly as much that LQG gets more support as that young people are rewarded for taking the risk to develop new ideas and proposals. To go from a situation where a young person’s career was tied to string theory to one in which it was tied to LQG would not be good enough. Instead, what is needed overall is that support for young scientists is not tied to their loyalty to particular research programs set out by we older people decades ago, but rather is on the basis only of the quality of their own ideas and work as well as their intellectual independence. If young people were in a situation where they knew they were to be supported based on their ability to invent and develop new ideas, and were discounted for working on older ideas, then they would themselves choose the most promising ideas and directions. I suspect that science has slowed down these last three decades partly as a result of a reduced level of intellectual and creative independence available to young people.

Thanks,
Lee

Sadly then, Dr Lubos Motl, string ‘theorist’ and assistant professor at harvard, tried to ridicule this aproach by the false claim that Dirac’s quantum field theory disproves a spacetme babric, as it is allegedly a unification of special relativity (which denies spacetime fabric) and quantum mechanics. Motl tried to ridicule me with this, although I had already explained the reason to him! "An important part of all totalitarian systems is an efficient propaganda machine. ... to protect the 'official opinion' as the only opinion that one is effectively allowed to have." - STRING THEORIST Dr Lubos Motl: http://motls.blogspot.com/2006/01/power-of-propaganda.html Here is a summary of the reasons why Dirac’s unification is only for the maths of special relativity, not the principle of no-fabric, and in fact Dirac was an electrical engineer before becoming a theoretical physicist, and later wrote:

‘… 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.

Thankfully, Peter Woit has retained so far a comment on the discussion post for loop quantum gravity which points out that Motl is wrong:

http://www.math.columbia.edu/~woit/wordpress/?p=330

anonymous Says:
January 21st, 2006 at 1:19 pm

Lumos has a long list of publications about speculation on unobservables. So I guess he’s well qualified to make vacuous assertions. What I’d like to see debated is the fact that the spin foam vacuum is modelling physical processes KNOWN to exist, as even the string theorists authors of http://arxiv.org/abs/hep-th/0601129 admit, p14:

‘… it is thus perhaps best to view spin foam models … as a novel way of defining a (regularised) path integral in quantum gravity. Even without a clear-cut link to the canonical spin network quantisation programme, it is conceivable that spin foam models can be constructed which possess a proper semi-classical limit in which the relation to classical gravitational physics becomes clear. For this reason, it has even been suggested that spin foam models may provide a possible ‘way out’ if the difficulties with the conventional Hamiltonian approach should really prove insurmountable.’

Strangely, the ‘critics’ are ignoring the consensus on where LQG is a useful approach, and just trying to ridicule it. In a recent post on his blog, for example, Motl states that special relativity should come from LQG. Surely Motl knows that GR deals better with the situation than SR, which is a restricted theory that is not even able to deal with the spacetime fabric (SR implicitly assumes NO spacetime fabric curvature, to avoid acceleration!).

When asked, Motl responds by saying Dirac’s equation in QFT is a unification of SR and QM. What Motl doesn’t grasp is that the ‘SR’ EQUATIONS are the same in GR as in SR, but the background is totally different:

‘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.

What a pity Motl can’t understand the distinction and its implications.

(See also http://nigelcook0.tripod.com/ and scroll down; http://lqg.blogspot.com/, and http://electrogravity.blogspot.com/)

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.

QUANTUM LOOP GRAVITY: SPIN FOAM VACUUM

The fabric of spacetime is a sea in which boson radiations spend part of their time converted into a perfect fluid of matter-antimatter.

‘In 1986, Abhay Ashtekar reformulated Einstein’s field equations of general relativity using what have come to be known as Ashtekar variables, a particular flavor of Einstein-Cartan theory with a complex connection. He was able to quantize gravity using gauge field theory. In the Ashtekar formulation, the fundamental objects are a rule for parallel transport (technically, a connection) and a coordinate frame (called a vierbein) at each point. Because the Ashtekar formulation was background-independent, it was possible to use Wilson loops as the basis for a nonperturbative quantization of gravity. Explicit (spatial) diffeomorphism invariance of the vacuum state plays an essential role in the regularization of the Wilson loop states. Around 1990, Carlo Rovelli and Lee Smolin obtained an explicit basis of states of quantum geometry, which turned out to be labelled by Penrose’s spin networks.’ - Wikipedia.

In the October 1996 issue letters page of Electronics World the basic mechanism was first released, with further notices placed in the June 1999 and January 2001 issues. Two articles in the August 2002 and April 2003 issues, were followed by letters in various issues. In 2004, the result r = rlocal e3 was obtained using the mass continuity equation of hydrodynamics and the Hubble law, allowing for the higher density of the earlier time big bang universe with increasing distance (divergence in spacetime or redshift of gauge bosons, prevents the increase in effective observable density from going to infinity with increasing distance/time past!). In 2005, a radiation pressure-based calculation was added and many consequences were worked out. The first approach worked on is the ‘alternative proof’ below, the fluid spacetime fabric: the fabric of spacetime described by the Feynman path integrals can be usefully modelled by the ‘spin foam vacuum’ of ‘loop quantum gravity’.

The observed supernova dimming was predicted via the Oct 96 Electronics World magazine, ahead of discovery by Perlmutter, et al. The omitted mechanism (above) from general relativity does away with ‘dark energy’ by showing that gravity generated by the mechanism of expansion does now slow down the recession. In addition, it proves that the ‘critical density’ obtained by general relativity ignoring the gravity mechanism above is too high by a factor of half the cube of mathematical constant e, in other words a factor of 10. The prediction was not published in PRL, Nature, CQG, etc., because of bigotry toward ‘alternatives’ to vacuous string theory:


http://www.math.columbia.edu/~woit/wordpress/?p=215#comment-4082:

Nigel Says:
July 7th, 2005 at 7:15 pm
Editor of Physical Review Letters says

Sent: 02/01/03 17:47
Subject: Your_manuscript LZ8276 Cook
{MECHANISM OF GRAVITY}
Physical Review Letters does not, in general, publish papers on alternatives to currently accepted theories…. Yours sincerely, Stanley G. Brown, Editor, Physical Review Letters

Now, why has this nice genuine guy still not published his personally endorsed proof of what is a ‘currently accepted’ prediction for the strength of gravity? Will he ever do so?

‘String theory has the remarkable property of predicting gravity’: false claim by Edward Witten in the April 1996 issue of Physics Today, repudiated by Roger Penrose on page 896 of his book The Road to Reality, 2004: ‘in addition to the dimensionality issue, the string theory approach is (so far, in almost all respects) restricted to being merely a perturbation theory’. String theory does not predict the strength constant of gravity, G! However, the Physical Review Letters editor still ‘believes in’ Edward Witten and Physics Today.

http://www.math.columbia.edu/~woit/wordpress/?p=215#comment-4081:
Peter Woit Says:
July 7th, 2005 at 7:27 pm

I’m tempted to delete the previous comment, but am leaving it since I think that, if accurate, it is interesting to see that the editor of PRL is resorting to an indefensible argument in dealing with nonsense submitted to him (although the "…" may hide a more defensible argument). Please discuss this with the author of this comment on his weblog, not here. I’ll be deleting any further comments about this.

‘(1). The idea is nonsense. (2). Somebody thought of it before you did. (3). We believed it all the time.’ - Professor R.A. Lyttleton's summary of inexcusable censorship (quoted by Sir Fred Hoyle in ‘Home is Where the Wind Blows’ Oxford University Press, 1997, p154).


http://www.math.columbia.edu/~woit/wordpress/?p=215#comment-4080:
Alejandro Rivero Says:
July 8th, 2005 at 6:34 am

currently accepted

is not different of the typical forms to request funds in some project, where you are basically asked what are you to discover, and when. I call this part of science, very botanic-wise, the "classification" side. The (also botanic) counterpart, "exploration", is always more problematic. Smolin article … was about this, wasn’t it?

*********************

The media is too chicken to report it, to save attacks from ‘string theorists’ who can’t even convince their own wives of their propaganda. It is independent proof of Catt’s experiment evidence that charges are gravitationally trapped energy. Fundamental particles have a black hole (not Planck) sized shielding area, predicting gravity. The mechanism for gravity is proved by a second calculation that doesn’t require shielding area (below).

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%.

The proofs below show that the local density (i.e., density at 15,000,000,000 years after origin) of the universe is: r (local) = 3H2/(4pe3 G). The mechanism also shows that because gravity is an inward push as reaction to surrounding expansion, there is asymmetry at great distances and thus no gravitational retardation of the expansion (predicted via October 1996 issue of Electronics World, before experimental confirmation by Perlmutter using automated CCD observations of distant supernovae). Because there is no slowing down due to the mechanism, the application of general relativity to cosmology is modified slightly, and the radius of the universe is R = ct = c/H, where H is Hubble constant. The observable recession velocity in spacetime is a = dv/dt = c/t = Hc.

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, see calculations later on), 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 heuristic explanation of this 137 anomaly is just the shielding factor by the polarised vacuum


‘All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs. The virtual fermions with charges opposite to the bare charge will be, on average, closer to the bare charge than those virtual particles of like sign. Thus, at large distances, we observe a reduced bare charge due to this screening effect.’ – I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424.

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
p 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 idea predicts that a particle core with n fundamental particles (n=1 for leptons, n = 2 for mesons, and obviously n=3 for baryons) coupling to N virtual vacuum particles (N is an integer) will have an associative inertial mass of Higgs bosons 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. We test this below against data for mass of muon and all ‘long-lived’ hadrons.

The problem is that people are used to looking to abstruse theory due to the success of QFT in some areas, and looking at the data is out of fashion. If you look at history of chemistry there were particle masses of atoms and it took school teachers like Dalton and a Russian to work out periodicity, because the bigwigs were obsessed with vortex atom maths, the ‘string theory’ of that age. Eventually, the obscure school teachers won out over the mathematicians, because the vortex atom (or string theory equivalent) did nothing, but empirical analysis did stuff. It was eventually explained theoretically!

There was a crude empirical equation for lepton masses by A.O. Barut, PRL, v. 42 (1979), p. 1251. We can extend the basic idea to hadrons. 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. This idea predicts that a particle core with n fundamental particles (n=1 for leptons, n = 2 for mesons, and obviously n=3 for baryons) coupling to N virtual vacuum particles (N is an integer) will have an associative inertial mass of Higgs bosons of: (0.511 Mev).(137)n(N + 1)/2 = 35n(N +1) Mev.

Accuracy tested against data for mass of muon and all ‘long-lived’ hadrons:

LEPTON (n=1)

Muon (N=2): 105 Mev (105.66 Mev measured), 0.6% error!

HADRONS

Mesons (contain n=2 quarks):

Pions (N=1): 140 Mev (139.57 and 134.96 actual), 0.3% and 3.7% errors!
Kaons (N=6): 490 Mev (493.67 and 497.67 actual), 0.7% and 1.6% errors!
Eta (N=7): 560 Mev (548.8 actual), 2% error!

Baryons (contain n=3 quarks):

Nucleons (N=8): 945 Mev (938.28 and 939.57 actual), 0.7% and 0.6% errors!
Lambda (N=10): 1155 Mev (1115.60 actual), 3.5% error!
Sigmas (N=10): 1155 Mev (1189.36, 1192.46, and 1197.34 actual), 3.0%, 3.2% and 3.7% errors!
Xi (N=12): 1365 Mev (1314.9 and 1321.3 actual), 3.8% and 3.3% errors!
Omega (N=15): 1680 Mev (1672.5 actual), 0.4% error!

The mechanism is that the charge of the bare electron core is 137 times the Coulomb (polarisation-shielded) value, so vacuum interactions of bare cores of fundamental particles attract 137 times as much virtual mass from the vacuum, increasing the inertia similarly. It is absurd that these close fits, with only a few percent deviation, are random chance, and this can be shown by statistical testing using random numbers as the null hypothesis. So there is empirical evidence that this heuristic interpretation is on the right lines, whereas the ‘renormalisation’ is bogus: http://www.cgoakley.demon.co.uk/qft/

Masses of Mesons:

Pions = 1.99 (charged), 1.93 (neutral)

Kaons = 7.05 (charged), 7.11 (neutral)

Eta = 7.84

Masses of Baryons:

Nucleons = 13.4

Lambda = 15.9

Sigmas = 17.0 (positive and neutral), 17.1 (negative)

Xi = 18.8 (neutral), 18.9 (negative)

Omega = 23.9

The masses above for all the major long-lived hadrons are in units of (electron mass)x137. A statistical Chi-squared correlation test against random numbers as the null hypothesis, indeed gives positive statistical evidence that they are close to integers. The mechanism is that the charge of the bare electron core is 137 times the Coulomb (polarisation-shielded) value, so vacuum interactions of bare cores of fundamental particles attract 137 times as much virtual mass from the vacuum, increasing the inertia that much too. Leptons and nucleons are the things most people focus on, and are not integers when the masses are in units of (electron mass)x137. The muon is about 1.5 units on this scale but this can be explained by a coupling of the core (mass 1) with a virtual particle, just as the electron couples increasing its magnetic moment to 1 + 1/(2.Pi.137). The mass increase of the muon is 1 + 1/2 because the Pi is due to spin and the 137 shielding factor doesn’t apply to bare cores in proximity.

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.

GENERAL RELATIVITY’S HEURISTICALLY EXPLAINED 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 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…

PROOF BY RADIATION PRESSURE: There is strong evidence from electromagnetic theory that every fundamental particle has black-hole cross-sectional shield area for the fluid analogy of general relativity.

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.

From the illustration above, the total outward force of the big bang,

(total outward force) = ma = (mass of universe).(Hubble acceleration, a = Hc, see detailed discussion and proof further on below),

while 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 (p multiplied by the square of the radius of the black hole of similar mass), multiplied by the (R/r)2 which is the inverse-square law for the geometry of the implosion. The total spherical area with radius R is simply four times p, multiplied by the square of R. Inserting simple Hubble law results c = RH and R/c = 1/H give us F = (4/3)p r G2M2/(Hr)2. We then set this equal to F=Ma and solve, getting G = (3/4)H2/(p r ). When the effect of the higher density in the universe at the great distance R is included, this becomes

G = (3/4)H2/(p r (local) e3).

Feynman discuss the LeSage gravity idea in ‘Character of Physical Law’ 1965 BBC lectures, with a diagram showing that if there is a pressure in space, shielding masses will create a net push. ‘If your paper isn’t read, they are ignorant of it. It isn’t even a put-down, just a fact.’ – my comment on Motl’s blog. The next comment was from Peter Woit: ‘in terms of experimentally checkable predictions, no one has made any especially significant ones since the standard model came together in 1973 with asymptotic freedom.’ Woit has seen the censorship problem! Via the October 1996 Electronics World letters, this mechanism – which Dr Philip Campbell of Nature had said he was ‘not able’ to publish – correctly predicted that the universe would not be gravitationally decelerating. This was confirmed two years later experimentally by the discovery of Perlmutter, which Nature did publish, although it omitted to say that it had been predicted. The standard approach of science-censors is ‘hear no evil, see no evil and speak no evil’, pretending these facts are illucid but not saying which facts they cannot grasp, preferring crackpot string theory which is a fraud from start to finish.

PROOF BY THE SPACETIME FOAM FABRIC:

Apples fall because of gauge boson shielding by nuclear atoms (mainly void space). The same pressure causes the general relativity contraction term.

STEP 1: Pressure is force/area. By geometry (illustrated here), the scaled area of shielding below you is equal to the area of space pressure above that is pushing you down. The shielded area of the sky is 100% if the shield mass is the mass of the universe, so: Ashielding = Ar M / Muniverse.

(1) Force, F = Pspace Ashielding = (Fspace /Ar).(ArM/Muniverse) = Fspace.M/Muniverse

Next (see step 2 below): introduce Fspace = mspace aH. Here, Hubble velocity variation in spacetime (v = HR) implies an acceleration equal to: aH = dv/dt = c/t = c/(1/H) = cH = RH2, while mspace = m(AR/Ar) = m(R/r)2, and the mass of the universe is its density, r , multiplied by its spherical volume, (4/3)p R3.

(2) F = Fspace.M/Muniverse = (mspace aH)M/Muniverse = m(R/r)2(RH2)M/ [(r 4pR3 /3)]

STEP 2: Air is flowing around you like a wave as you as you walk down a corridor (an equal volume goes in the other direction at the same speed, filling in the volume you are vacating as you move). It is not possible for the surrounding fluid to move in the same direction, or a void would form BEHIND and fluid pressure would continuously increase in FRONT until motion stopped. Therefore, an equal volume of the surrounding fluid moves in the opposite direction at the same speed, pemitting uniform motion to occur! Similarly, as fundamental particles move in space, a similar amount of mass-energy in the fabric of space (spin foam vacuum field) is displaced as a wave around the particles in the opposite direction, filling in the void volume being continuously vacated behind them. For the mass of the big bang, the mass-energy of Higgs/virtual particle field particles in the moving fabric of space is similar to the mass of the universe. As the big bang mass goes outward, the fabric of space goes inward around each fundamental particle, filling in the vacated volume. (This inward moving fabric of space exerts pressure, causing the force of gravity.)

‘Popular accounts, and even astronomers, talk about expanding space. But how is it possible for space … to expand? … ‘Good question,’ says [Steven] Weinberg. ‘The answer is: space does not expand. Cosmologists sometimes talk about expanding space – but they should know better.’ [Martin] Rees agrees wholeheartedly. ‘Expanding space is a very unhelpful concept’.’ – New Scientist, 17 April 1993, pp. 32-3.

The effective mass of the spacetime fabric moving inward which actually produces the gravity effect is equal to that which is exactly shielded by the mass (illustrated here). So mspace = m, but we also have to allow for the greater distance of the mass which is producing the gravity force by implosion. To take account of focussing due to the ‘implosion’ of space fabric pressure (see diagram) converging in to us in step 1 above (illustration above), we scale: mspace /m = AR / Ar. Hence: mspace = mAR / Ar = m(R/r)2. This is because nearby areas on which force acts to produce pressure are much smaller than the area of sky at the very great distances where the recession and density are high and produce the source of space pressure and thus gravity.

The big bang recession velocities vary from 0 to c with distance for observable times of 15,000 million years towards zero, so the matter of the universe has an effective outward acceleration of c divided by the age of the universe. This acceleration, a = c/t = cH = RH2 where H is the Hubble constant (in v = HR), is so small that its effects are generally undetectable. (Notice that if we could see and experience forces instantly, the universe would not show this acceleration. This acceleration is only real because we can’t see the universe at an age of 15 Gyr irrespective of distance. By Newton’s 2nd law, the actual outward force, when properly allowing for the varying effective density of the observed universe as a function of spacetime, is large and by Newton’s 3rd law it has an equal and opposite reaction, inward force which, where shielded, is gravity.)

(3) F = m(R/r)2(RH2)M/ [(r 4pR3 /3)] = (3/4)mMH2/(rpr2)

Next, for mass continuity, dr/dt = -Ñ.(rv) = -3rH. Hence, r = rlocal e3 (early visible universe has higher density, see below for a mathematical derivation). The reason for multiplying the local measured density of the universe up by a factor of about 20 (the number e3, the cube of the base of natural logarithms) is because it is the denser, more distant universe which contains most of the mass which is producing most of the inward pressure. Because we see further back in time with increasing distance, we see a more compressed age of the universe. Gravitational push comes to us at light speed, with the same velocity as the visible light that shows the stars. Therefore we have to take account of the higher density at earlier times. What counts is what we see, the spacetime in which distance is directly linked to time past, not the simplistic picture of a universe at constant density, because we can never see or experience gravity from such a thing due to the finite speed of light. The mass continuity equation dr/dt = -Ñ.(rv) is simple hydrodynamics based on Green’s theorem and allows the Hubble law (v = HR) to be inserted and solved. An earlier method of calculation for this the notes of CERN preprint EXT-2004-007, is to set up a formula for the density at any particular time past, so as to calculate red-shifted contributions to inward spacetime fabric pressure from a series of shells surrounding the observer. This is the same as the result r = rlocal e3.

(4) F = (3/4) mMH2/( p r2 r local e3 ) = mMG/r2, where G = (3/4) H2/( p r local e3 ) = 0.0119H2/r local = 6.7 x 10-11 Nm2kg-2, accurate to within 1.65% using reliable supernovae data reported in Physical Review Letters! If there were any other reason for gravity with similar accuracy, the strength of gravity would then be twice what we measure, so this is a firm testable prediction/confirmation that can be checked more delicately.

 

SYMBOLS

F = force = ma = PA

M = mass of Earth

m = mass of person or apple

P = force / area = F/A = ‘pressure’

A = surface area of a sphere, 4p times (radius squared)

r = distance from person to centre of mass of shield (Earth)

R = radius to big bang gravity source

H = Hubble constant = apparent speed v of galaxy clusters radially from us divided by their distance R when the light was emitted = v/R, hence v = HR = dR/dt, so dt = dR/(RH):

aH = dv/dt = [d(RH)]/[dR/(RH)] = RH.d(RH)/dR = RH2 = cH; a constant (Hubble saw light coming from fixed times past, not from stars at fixed distances).

r = density of universe (higher at great distances in space time, when the age was less and it was more compressed, dr/dt = -Ñ .(rv) = -3rH. So: r = r local e3, see be)

G = universal gravitational constant (previously impossible to predict from general relativity or string theory)

p = circumference divided by the diameter of a circle, approx. 3.14159265…

e = base of natural logarithms, approx. 2.718281828…

Mass continuity equation (for the galaxies in the space-time of the receding universe): dρ/dt + div.(ρv) = 0. Hence: dρ/dt = -div.(ρv). Now around us, dx = dy = dz = dr, where r is radius. Hence divergence (div) term is: -div.(ρv) = -3d(ρv)/dx. For spherical symmetry Hubble equation v = Hr. Hence dρ/dt = -div.(ρv) = -div.(ρHr) = -3d(ρHr)/dr= -3ρHdr/dr= -3ρH. So dρ/dt = -3ρH. Rearranging:-3Hdt = (1/ρ) dρ. Solving by integrating this gives say: -3Ht = (ln ρ1) – (ln ρ). Using the base of natural logarithms (e) to get rid of the ln’s: e-3Ht = density ratio. Because H = v/r = c/(radius of universe) = 1/(age of universe, t) = 1/t, we have: e-3Ht = (density ratio of current time to earlier, higher effective density) = e-3(1/t)t = e-3 = 1/20. All we are doing here is focussing on spacetime in which density rises back in time, but the outward motion or divergence of matter due to the Hubble expansion offsets this at great distances. So the effective density doesn’t become infinity, only e3 or 20 times the local density of the universe at the present time. The inward pressure of gauge bosons from greater distances initially rises because the density of the universe increases at earlier times, but then falls because of divergence, which causes energy reduction (like red-shift) of inward coming gauge bosons.

The physical content of GR is the OPPOSITE of SR:

‘… 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.

Notice that in SR, there is no mechanism for mass, but the Standard Model says the mass has a physical mechanism: the surrounding Higgs field. When you move a fundamental particle in the Higgs field, and approach light speed, the Higgs field has less and less time to flow out of the way, so it mires the particle more, increasing its mass. You can't move a particle at light speed, because the Higgs field would have ZERO time to flow out of the way (since Higgs bosons are limited to light speed themselves), so inertial mass would be infinite. The increase in mass due to a surrounding fluid is known in hydrodynamics:

‘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!

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 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…

‘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.

‘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.

‘All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs. The virtual fermions with charges opposite to the bare charge will be, on average, closer to the bare charge than those virtual particles of like sign. Thus, at large distances, we observe a reduced bare charge due to this screening effect.’ – I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424.

If the electron moves at speed v as a whole in a direction orthogonal (perpendicular) to the plane of the spin, then the c speed of spin will be reduced according to Pythagoras: v2 + x2 = c2 where x is the new spin speed. For v = 0 this gives x = c. What is interesting is that this model gives rise to the Lorentz-FitzGerald transformation naturally, because: x = c(1 - v2 / c2 )1/2 . Since all time is defined by motion, this (1 - v2 / c2 )1/2 factor of reduction of fundamental particle spin speed is therefore the time-dilation factor for the electron when moving at speed v.

Motl's quibbles about the metric of SR is just ignorance. The contraction is a physical effect as shown above, with length contraction in direction of motion, mass increase and time dilation having physical causes. The equivalence principle and the contraction physics of spacetime "curvature" are the advances of GR. GR is a replacement of the false SR which gives wrong answers for all real (curved) motions since it can't deal with acceleration: the TWINS PARADOX.

Strangely, the ‘critics’ are ignoring the consensus on where LQG is a useful approach, and just trying to ridicule it. In a recent post on his blog, for example, Motl states that special relativity should come from LQG. Surely Motl knows that GR deals better with the situation than SR, which is a restricted theory that is not even able to deal with the spacetime fabric (SR implicitly assumes NO spacetime fabric curvature, to avoid acceleration!).

When asked, Motl responds by saying Dirac’s equation in QFT is a unification of SR and QM. What Motl doesn’t grasp is that the ‘SR’ EQUATIONS are the same in GR as in SR, but the background is totally different:

‘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.

 

 

Phil Reed
Letters to the Editor
Electronics World

 30 June 2004

Dear Sir:
‘Electronic Universe’ article, Apr. 2003 EW, proves G = 3H2
/(4πρ). Here H is Hubble constant and ρ is density of universe responsible for causing gravity by reaction of Catt’s 377-ohm space to the big bang. Considering the density, it is highest at early times and thus density increases in the observable space-time trajectory, as we look further into the past with increasing distance.

But the increasing spread of matter with increasing distance partly offsets this increase, as proven when we put the observed Hubble equation (v = Hr) into the mass continuity equation and solve it. For spherical symmetry, dx = dy = dz = dr. Mass continuity implies: dρ/dt = -Ñ (ρv) = -Ñ .(ρHr) = 3d(ρHr)/dr = -3ρH. Solving dρ/dt = -3ρH by rearranging, integrating, then using exponentials to get rid of the natural logarithms (resulting from the integration) gives the increased density to be ρe3Ht, where e is Euler’s constant (2.718 ...). In the absence of gravitational retardation (i.e. with the cause of gravity as inward reaction of space to the outward big bang), H = 1/t when H = v/r = c/(radius of universe) = 1/t, where t is the age of the universe, so e3Ht = e3 and observed G = 3H2/(4πe3ρ).

Nugent, Physical Review Letters (v75 p394), cites decay of nickel-63 from supernovae, obtaining H = 50 km/sec/Mps (where 1 Mps = 3.086x1022 m). The density of visible matter at our local time has long been known to be 4x10-28 kg/m3. However, White and Fabian in the March 1995 Monthly Notices of the Royal Astronomical Society, using the Einstein Observatory satellite data, estimate that invisible gas increases this density by 15%.

Using these data, G = 3H2/(4πe3ρ) = 6.783x10-11 Nm2kg-2, 1.65% higher than the physical measurement for G of 6.673x10-11 Nm2kg-2. So current data predicts acceleration of just under 10 ms-2 at the Earth’s surface, compared to the observed value of about 9.8 ms-2. This proves Catt’s insistence on the reality of the 377-ohm fabric of space beyond any reasonable doubt.

Yours sincerely,
Nigel Cook

 

PROOF CHECK (LONG VERSION) Nigel Cook

Standard equation for mass continuity (conservation of mass in an expanding gas, etc):

dρ/dt + Ñ (ρv) = 0

Or

dρ/dt = -Ñ (ρv)

Where divergence term -Ñ .(ρv) = -[{d(ρv)x/dx} + {d(ρv)y/dy} + {d(ρv)z/dz}]

For spherical symmetry

dx = dy = dz = dr, where r is radius (note: this has nothing to do with the sum of the squares of the differential elements of distance, so the abusive anonymous ‘moderator’ on ‘Physics Forums’ who claimed this so, made vacuous personal sneers, and then banned all response, merely proved the ignorance of charlatans)

Hubble equation

v = Hr

Hence

dρ/dt = -Ñ (ρv) = -Ñ .(ρHr) = -[{d(ρHr)/dr} + {d(ρHr)/dr} + {d(ρHr)/dr}]

= -3d(ρHr)/dr

= -3ρHdr/dr

= -3ρH

So dρ/dt = -3ρH. Rearranging:

-3Hdt = ò (1/ρ) dρ. Integrating both sides:

-3Ht = (ln ρ1) – (ln ρ). Using the base of natural logarithms e to get rid of the ln’s:

e-3Ht = ρ1

Because H = v/r = c/(radius of universe) = 1/(age of universe, t) = 1/t:

e-3Ht = ρ1

= e-3(1/t)t = e-3

Therefore

ρ = ρ1e3 = 20.085537 ρ1.

So using the result in the April 2003 EW,

G = 3H2/(4πρ)

= 3H2/(4πe3ρ1)

where ρ1 is the local-time observed density of the universe. This is correct to within 1.7%.

It beats ‘quantum gravity’ speculations which give no prediction of G whatsoever, whereas general relativity actually uses the measured value of G as a constant to make calculations, rather than predicting it.