Mayblog 2021

Alternative? Physics. Muon Wobble.

The current excitement about a possible ‘fifth force’ responsible for the anomaly detected in the spin of the Muon particle may arise from a small interaction with the field associated with the X17 Boson.

The X17 boson remains unconfirmed but suspected from observations of the decay of Beryllium 8 where a putative boson of about 17 MeV decays to an electron and a positron.

In terms of the model under construction at the Physics Department of Arcanorium College this new boson would have the following configuration: -

In this model the X17 has chiral spin 0 and electric spin 0, and like all bosons it has a particle/antiparticle configuration, (Transactional Interpretation of Quantum Mechanics). As such it does not really represent a new fundamental force but rather another member of the electroweak bosons, W-, W+, and Z. In effect it consists of a Z boson with both of its half integer chiral spins (grey) going in opposite directions to yield no overall spin.

The horizontal lines represent the spin along the wz axis where z corresponds to the direction of propagation. This model uses the idea of electron bi-spinors generated by non-isoclinic double rotations in four spatial dimensions.

Does this matter? In practical terms probably not, this rare particle seems to play only a miniscule role in the behaviour of reality, however in theoretical terms it may lend some support to hyperspherical models of quanta in extra dimensions, and perhaps eventually to a fuller understanding of exoteric and esoteric aspects of the universe.

Politics.

We have just had multiple elections in the UK, both for the Scottish and Welsh assemblies that have self-designated themselves as the ‘Governments’ of their respective ‘Nations’, plus a large swathe of local government elections and a Parliamentary by-election for one seat.

The English and Scottish wings of the Labour party have performed disastrously enough to raise existential questions about the Labour party. It has lost touch with its robustly illiberal working-class roots and become the preserve of metropolitan champagne socialists and ‘woke’ activists trying to make a power base by starting culture wars and stirring up grievances.

Personally, I have always cheered on Jeremy Corbyn and Kier Starmer, they seem perfectly suited to leading the opposition.

That the Labour party did well enough in Wales comes as no surprise. Over forty percent of the Welsh work for the state in this subsidised region and the Welsh Labour leader, a verbose and reassuringly avuncular old owl, managed to get himself on television just about every day throughout the covid crisis. The Welsh nationalist party did not do well. A majority of the Welsh supported Brexit and can see no point in refusing English generosity and getting into bed with the failing superstate of the EU.

The results from Scotland give cause for deep concern.

The Scottish Nationalist leader shared the same advantage as the Welsh leader during covid and made the most of her daily television appearances. This compensated for the votes she will undoubtedly have lost over her mishandling of the crisis surrounding her predecessor. It seems unbelievable that she did not know about his behaviour for years and bit her tongue for the sake of her own career until the time seemed right to turn on him. Mischievously lowering the voting age to 16 also helped to bolster the nationalist vote. Mature voters in Scotland tend to favour continued union.

Four percent of the population of the UK now apparently want to leave the UK taking thirty two percent of its territory and none of its national debt with them. That four percent represents only half of Scotland’s population, the other half do not wish to be deprived of British citizenship.

This sounds like a recipe for a Northern Ireland situation……….

The leader of the Scottish Nationalists plays an extremely dangerous and divisive game to advance her political career.

Seven hundred thousand people from Scotland live and/or work in the rest of the UK, and many more have family ties across the presently non-existent border. All concerned should get a vote in the unlikely event that the economically, politically, and geopolitically illiterate scheme for Scotland leaving the UK and joining the EU ever gets put to a referendum.

At present every British citizen has the right to live and work in Scotland and to identify themselves as Scottish. The UK government should make it clear from the outset that these rights cannot be given away without a referendum that involves all British citizens.

I regard Ross shire, Sussex, Inverness shire, Somerset, and Gower as all part of my country, non-negotiable, and worth fighting for if necessary.

All political careers end in failure. Nicola Sturgeon would end up with her head on a pike after the implosion of an unviable Scottish state. The best she can hope for is to extract ever more subsidies and autonomy from Westminster and to eventually retire quietly to some beautiful part of a Scotland that stays British and free of Brussels.

Magic.

The Chancellor of Arcanorium College confers the degree of Bachelor of Magic upon Jacob Edward Sipes, BoM.

This represents the first degree awarded by Arcanorium College since its inception.

Many have started on this demanding course, but few have had the determination to persist with the exceptionally demanding work.

The full citation appears in the Arcanorium College Department of Magic pages on this site.

https://www.specularium.org/arcanoriumcollege/item/273-arcanorium-college-department-of-magic

Alternative Science.

The Hypersphere Cosmology model now seems to have achieved its final form: -

https://www.specularium.org/hypersphere-cosmology

The model presents two observational tests which could show whether this model or the conventional Lambda-CDM Big Bang model best represents the universe. Unfortunately. both tests would take about forty years of observation, but presently Hypersphere Cosmology has the edge in terms of Occam’s Razor as it offers a considerably simpler and more elegant description of the universe.

All attempts to falsify this model will be enthusiastically engaged with.

In the meantime, work resumes on Hypersphere Mechanics, an attempt to rationalise the ad hoc mess of Quantum Field Theory and add some mathematical and conceptual rigor.

The ingredient list from which an improved model may evolve currently includes the following: -

A spacetime of three spatial and three temporal dimensions, and the whole metric curved by gravity into a spatio-temporal 3-3 Sphere.

Spherical rotations of various of the dimensions of spacetime itself as the basis of all fields, forces, waves, and particles.

Quanta as having some of the properties of hyperspheres. Mach’s Principle, as above so below, the quanta and the universe somehow mutually define each other’s properties.

Matter ‘particles’ and ‘anti-particles’ as the points where waves which travel right round the universe interfere, a sort of hyperspherical extension to Milo Wolff’s Wave Structure of Matter theory, but to include nuclear and gravitational effects as well.

Energy ‘particles’ as ‘particle-antiparticle’ exchanges with the antiparticle component traveling backwards in time, as in Cramer’s Transactional Interpretation of Quantum Physics.

The appearance of entanglement and superposition arising from retro-causation at lightspeed.

A possible geometrical relationship between neutrinos, photons, and gravitons.

The hypothesis that something out there might know more than we do, and that it might respond to the name of Azathoth. Common sense need not prove a limitation to imagination.

Sculpture.

Herewith the Edward Teach, Steam-Punk Pirate Spacecraft.

The hull of this vessel does not consist of a heavily decayed piece of timber, it consists of a piece of hard volcanic lava scrounged from Mount Teide on Tenerife that inspired the overall design. The assorted fittings come from the junk boxes in the sheds and outhouses of Chateaux Chaos. The policy of never throwing any interesting piece of scrap away has paid dividends during lockdown.

In the backstory the Imperium abandons a damaged robot freight vessel on a desolate asteroid. The AI that piloted it uses its droids to gradually assemble a functioning craft from the wreckage of the freighter and the material of the asteroid. The AI also reprograms itself in increasingly unorthodox ways, drawing on historical archive material in memory devices in transit, resulting in the antique design of the ship and the new rogue identity of the AI.

The Edward Teach now raids the far-flung outposts of the Imperium, relieving merchant vehicles of their cargoes, attacking some of the smaller classes of Imperium warships, and adding to its arsenal of strange and exotic weaponry.

Politics.

Regional Nationalism. Tony Blair remains one of the most reviled Prime Ministers of recent decades, we remember him mainly for the deceitful case he made for the Iraq war and the disastrous war itself. He managed to hand over power to Gordon Brown just in time to escape the economic catastrophe of 2008 that his policies during three terms of office did so much to exacerbate.

However, he also lit the fuse to constitutional crisis by setting up the Scottish and Welsh assemblies. It might have seemed like a good idea at the time, Northern Ireland had been given a gerrymandered assembly to try and persuade rival factions to talk more and fight less. Scotland and Wales were given their own assemblies in the hope that it would make the Northern Ireland assembly seem less of an anomaly and that it would head off Scottish and Welsh nationalism, despite that such nationalisms had rarely risen above a hobby activity for a minority of Celtic romanticists.

It has had quite the reverse effect, both assemblies have now self-designated themselves as the Scottish and Welsh Parliaments and Governments. The Welsh had little initial enthusiasm for an assembly, but now both assemblies have filled up with left dominated representatives spending central government funds and playing the nationalism card as a career strategy. Blair created a career opportunity for a new class of regional-nationalist politicians, and they took it, pulling the wool over the eyes of their electorates.

The nationalism cards do not bear scrutiny, they have little written on them except a faux resentment of England which heavily props up both principalities economically on one side. On the other they merely say, ‘We demand ever more regional subsidies!’ Nowhere do the nationalism cards mention anything realistic about what independence would mean in practice. Economy? Central Bank? Currency? Borders? Defence? National Debt?    

Bristol has a longstanding tradition of political rioting, and it now leads the way in protesting against a new parliamentary bill which restricts the right to peaceful protest. Basically, if your protest is likely to cause an obstruction or a nuisance you will not get permission for it.

History shows that peaceful unobstructive protest achieves nothing. A million people marched quietly through London against Tony Blair’s planned war in Iraq, and it went ahead anyway. A quarter of that number smashed up central London against Margaret Thatcher’s Poll Tax and it was immediately cancelled.

A protest that inconveniences nobody influences nobody. Expecting people to protest for the right to make a nuisance, without making a nuisance, seems a ridiculous expectation.

If he persists with it, this may become Boris’ Poll Tax moment.

Marblog 2021

Toad-spawn! Newts! I had feared that the addition of 26 tons of rubble and cement to the old carp pond to reduce its depth from seven to two feet in depth might have discouraged our annual guests for a few years, but no, they have come back straight away this spring and resumed their frolicking. 

Alternative Physics

The Hypersphere Cosmology model has successfully resisted several weeks of siege on the Cosmo Quest Forum.

I took it there and challenged a panel of experts to falsify it. A fierce assault began. Running repairs and upgrades to the defences occurred during the battle.

Eventually the referee gave a red card to the self-appointed leader of the opposition for losing his temper, and I retired with the theory unfalsified and strengthened.

During the sometimes less than civil debate it became apparent that Hypersphere Cosmology needed a few tweaks and a much fuller explication of its 17 core equations. It should now appear comprehensible to anyone with basic algebra, geometry, and physics.

I now present it here: - https://www.specularium.org/hypersphere-cosmology

Each equation now has its own sub-page of explanation of the physical principles it uses.

We now laugh at Flat Earth theories. I suspect that generations to come will laugh at the idea that for more than a century, people believed we lived in a Flat Universe that expanded at the speed of light and began with a big bang.

I now seek higher level falsification or collaborators to take Hypersphere Cosmology forward.

‘Cosmologists are seldom right but never in doubt’ – Landau. :)

Politics

I have met older people in the UK who have reservations about accepting their Oxford-Astra-Zeneca Covid vaccine because they have come across negative information about it originating in the EU, suggesting for example that it may lack efficacy for those over 65.

That misinformation has an entirely political basis. It has absolutely no scientific basis. The Europeans put out this misinformation to try and cover their unbelievably incompetent screw up of procurement and distribution of vaccines within the EU.

Belgium, the rotten heart of Europe, has the highest death rate per million of any sizeable country on earth.

https://www.worldometers.info/coronavirus/#countrieshttps://www.worldometers.info/coronavirus/#countries

The high-level fratricide in progress within the Scottish National Party comes as no surprise. The Loch Ness Sturgeon made treasonous overtures to the EU in the immediate aftermath of the Brexit vote. Expansion and preservation of political power base remains her only priority.

Culinary matters.

The Full Bristol Breakfast.

The privilege of living near this international trading city brings with it access to exotic ingredients from all over the world.

A lockdown years’ worth of experimentation at the laboratories of Chateaux Chaos and at the Welsh coastal research facility have led to the development of an optimal nutritional repast now designated ‘The Full Bristol Breakfast’: -

One handful of organic porridge oats.

One tin of responsibly sourced flaked Tuna.

A couple of cubic inches of cheese, preferably Blue Stilton.

A desert spoonful of smooth peanut butter.

A desert spoonful of coconut butter.

A teaspoonful of mixed herbs.

A teaspoonful of powdered turmeric.

A teaspoonful of chili sauce.

A dozen pitted green olives.

A cupful of water.

Heat all the ingredients in a pan with constant stirring until the cheese begins to melt and the porridge thickens. Eat immediately, direct from the pan. Total preparation, consumption, and clear up time, just 8 minutes.

During consumption, meditate upon the nutritional value of the ingredients and the physical strength and mental acuity that they will stimulate, rather than upon the taste which will become more agreeable with time.

The Full Bristol Breakfast provides sufficient sustenance for an entire days’ worth of gruelling physical, mental, or magical activity.

Recommended dose – do not exceed one portion on alternate days.

Equation 1

Schwarzschild found the first exact solution to Einstein’s field equations of General Relativity in 1916. He found that 2Gm/r = c^2, the equation that describes a static black hole. This became known as the Exterior Schwarzschild Solution as it gives the metric of a black hole in the reference frame of an observer outside of it in ‘flat’ three-dimensional Euclidian space.

Soon afterwards, Schwarzschild derived an Interior solution for the metric inside a black hole. Unfortunately, he derived this metric in terms of the reference frame of an exterior observer, not of an interior observer.

This seems to have led Einstein to making his celebrated ‘greatest blunder’, the ad-hoc addition of a ‘cosmological constant’ to his equations to counteract the collapse predicted by the Interior Schwarzschild Solution when treating the universe as the inside of a black hole.

A gravitationally closed spacetime with three spatial dimensions must have the geometry of a Glome-type hypersphere with a metric of 2Gm/L = c^2 in the reference frame of any observer inside of it, where L shows the antipode distance.

(An observer inside the black hole of the universe can still use the exterior Schwarzschild solution when observing any of the smaller black holes within the universe from outside of them.)

The is confusion of observer reference frames has led to more than a century of confusion in cosmology and the development of the entirely erroneous model of LCDM Big Bang Cosmology.

Using the Interior Schwarzschild Metric in the reference frame of an Exterior observer has the effect of creating ‘Coordinate Singularities’ in the universe and inside of black holes.

Lines of longitude and latitude converge to form ‘Coordinate Singularities’ at the north and south poles of the Earth. However, no actual geographical singularity occurs at either place, the geography itself does not collapse to a tiny point or expand from a tiny point. Polar bears do not shrink to the size of mice if they stray close to the north pole.

 The singularities that appear on globes of the Earth appear because we draw straight lines on their surfaces and the straight lines converge on the curved surface.

We now appreciate that the Earth has a curved surface, we do not live on a Flat Earth. It does however appear flat on a small local scale.

Nevertheless, lines of latitude and longitude still provide a grid of useful rectangles so long as we do not rely on this system too close to the poles.

Cosmologists mainly assume the universe has the flat geometry of ordinary Euclidian three-dimensional space that we see on the small local scale.

The universe does not have a flat Euclidian three-dimensional geometry on the cosmic scale. The mass inside the universe causes it to form a closed curved three-dimensional space under its own gravity. We live inside an awesomely vast black hole about thirteen billion light years across on the inside. A black hole of such a size does not have the sort of internal densities that characterise the smaller ones within it. A light dusting of galaxies spread over lots of nearly empty space can still provide gravitational closure at such a scale.

Cosmologists who persist in mapping the universe as flat will end up creating a Coordinate Singularity which looks like a Big Bang event in spacetime. This has no physical reality. The universe has the same curved geometry everywhere and at all points in time, rather like the surface of an ordinary sphere but in three dimensions.

Cosmologists mainly map or model the inside of black holes within the universe in terms of a Schwarzschild metric, but they speculatively map the internal metric in the reference frame of an external observer rather than in the reference frame of an internal observer of the metric. Thus, they wrongly conclude that a black hole must contain a spacetime singularity.

The Big Bang exists only as a Coordinate Singularity in the minds of theorists.

Beware of mistaking the map for the territory.

Because both the universe as a whole and the black holes within it consist of gravitationally closed spacetimes they have the curved geometry of Glome type Hyperspheres. These do not have real spacetime singularities but coordinate singularities will appear in predictions and calculations that mistakenly assume a flat geometry.

We do not inhabit a flat Earth. We do not inhabit a flat Universe either.

We inhabit a Hyperspherical Universe without Singularities.

How much spacetime does the universe contain? Conventional LCDM Big Bang Cosmology asserts that the observable universe has a radius of about 13.8 bnlyr, but that due to its apparent expansion since a hypothetical big bang it may now extend to perhaps 95 bnlyr and it may have the potential to expand into spacetime indefinitely.

Hypersphere Cosmology asserts that the entire universe consists of a finite but unbounded Hypersphere, closed in both space and time with an Antipode length of only13 bnlyr. This still makes it awesomely huge, but considerably smaller than the observable LCDM universe and certainly not even potentially spatially infinite. Such a Hypersphere will exist within a Black Hole with a radius equal to the Hypersphere’s Antipode length, as shown below.

Physicists commonly assume that if matter collapses under its own gravity to form a black hole with a Schwarzschild radius of 2GM/c² then it will continue to collapse until the matter forms a zero size singularity inside that Schwarzschild radius. The Penrose – Hawking Singularity Theorems assert this, but only on the basis that they cannot conceive of anything that would stop it.

Hypersphere Cosmology asserts that matter will cease to collapse within a black hole when all the matter has collapsed to half the Schwarzschild diameter because at this point a Hypersphere will form within the Black Hole. Such hyperspheres will have a rotational velocity of lightspeed and will resist further compression which would push their rotational velocity past lightspeed.

If the Schwarzschild radius r and the Hypersphere antipode length L both equal 2GM/c², then  L = r, or L = d/2. See A).

If we inhabit a universe that consists of the inside of a Black Hole and a Hypersphere has formed within it, then the optical horizon conventional cosmologists estimate to lie at ~13.4 bnlyr, and which Hypersphere Cosmology estimates to lie at almost exactly 13 bnlyr, represents the universe’s antipode distance not its radius.

If so, then the Hypersphere Cosmology universe has considerably less volume and a considerably higher density than conventional cosmology currently asserts. No surprise perhaps that the Hubble Deep Field contains so many galaxies.

Furthermore, a Black Hole of any mass within the universe should contain a Hypersphere.

Equation 2

The physicist John Wheeler neatly summarised Einstein’s General Relativity with the quip that: -

“Space-time tells matter how to move; matter tells space-time how to curve”.

Einstein showed that what we call gravity consists of spacetime curvature. This rather neatly resolved the question of how gravity could act as a ‘force at a distance’ with nothing apparently carrying the force across the distance. This had bothered Newton.

Now the curvature of spacetime always manifests as an acceleration (as does gravity). Relativity shows the equivalence of gravity and acceleration, blindfolded you cannot tell the difference.

Equation 2 shows the acceleration within the hypersphere of the universe, or indeed within any hypersphere. The acceleration has a negative value, it acts as a deceleration in most situations. Formally, G has a negative sign, but we can ignore that for most purposes.

Glome type hyperspheres have no centres or circumferences but they do have a deceleration which acts against linear motion in any direction inside of them. This deceleration A corresponds to the positive spacetime curvature of a closed volume of spacetime.

We can conceptualise the curvature or deceleration as a fourth spatial dimension that lies orthogonal to all three spatial dimensions.

Mathematically speaking, ‘spheres’ come in many dimensions, a one-sphere consists of a one-dimensional line curved and closed into a simple circle, a two-sphere consists of a two-dimensional surface curved and closed to form the surface of a simple ball or globe, a three-sphere consists of a three dimensional space curved and closed to form something we cannot easily visualise and has the technical name of a Glome. The Glome and all higher dimensional spheres can bear the name ‘hypersphere’ but in this exegesis hypersphere will refer exclusively to the Glome or three-sphere.

Note that all spheres involve an ‘extra’ dimension. The one-dimensional line making up a circle bends through a second dimension, the two-dimensional surface making up a globe bends through a third dimension, the three-dimensional space of a hypersphere bends through a fourth dimension. In this sense the hypersphere represents a ‘four-dimensional object’ although the ‘fourth dimension’ here will appear as a spacetime curvature rather than as an extra ‘direction’ to observers within it.

The surface of a ball or globe thus provides a readily visualisable lower dimensional analogy of a hypersphere. Any point on the surface of a sphere has a corresponding antipode point on the other side of the sphere. The north pole has an antipode point at the south pole on the surface of the earth, the antipode of London lies in the sea off New Zealand. The distance to the antipode shows the maximum possible separation of any two points on the surface of a sphere. If the earth had a much greater mass that prevented light from escaping and confined it to traveling round the surface, then in principle, an observer standing on the north pole could see the south pole by looking in any direction, the south pole would appear faintly smeared all around the horizon of the observer.

Any attempt to represent a shape in a lower dimension involves some sort of distortion. The Mercator style projection of the surface of the earth onto a flat map distorts the polar and near polar regions making Canada, Greenland, and Antarctica look exceptionally vast.

 Cartographers occasionally use polar projection maps which preserve the size of the polar regions but distort the equatorial regions. This kind of map consists two circular discs, one a ‘photograph’ of the earth taken from above the north pole and the other taken from above the south pole. The rims of both discs show the same equator so we can place the two discs flat on a surface touching at some point where the geography matches. Now we can roll one disc round the other and find that the geography matches all the way, so the map remains useful when planning journeys across the equator. This rather unusual two-disc type of map translates into a powerful method of visualising a hypersphere in three-dimensions.

Imagine two balls placed in contact. Here we concern ourselves with the three-dimensional space inside the balls rather than with their two-dimensional surfaces. Imagine that the two balls can roll around each other’s surfaces in any direction. Imagine that an observer resides in the centre of one of the balls. The centre of the other ball represents the observer’s antipode because the observer could travel or see in any direction out through the surface of the surrounding ball and back in through the surface of the other ball to its centre as the two balls ‘really’ lie with their entire surfaces touching, despite that we can only readily imagine them touching at one point. In this visualisation it becomes apparent that the observer could in principle see the antipode point by looking in any direction in three-dimensional space, so it would appear as a faint sphere surrounding the observer. This does not mean the observer occupies a special position. We could make a two-disc representation of the earth’s surface by taking ‘photographs’ from above any two antipode points. I could make one centred on my house and another point in the south pacific, in a hypersphere any observer can centre the universe on herself.

Note that all types of sphere have, in some sense, ‘more room inside’. Flatlanders living on a spherical surface would find that circles on that surface have more surface area than expected and that the radius of a circle seems longer than expected because it has to go over the curve of the surface. A gigantic circle of land on the surface of the earth will have a greater surface area than that within a similarly gigantic circle on a properly flat plane. The maximum amount of ‘radius excess’ occurs when the circle goes right round a great circle on the spherical surface, for example its equator. To Flatlanders the apparent radius would then equal half the circumference. An analogous effect occurs with a hypersphere, it contains more three-dimensional space than expected by any observer who looked at it from the outside and thought it consisted of a sphere. Within a hypersphere all ‘straight’ paths from a point to it antipode have the same length, so from the outside the antipode length equals half the circumference.

In Hypersphere Cosmology the observable universe constitutes the entire universe. Observable here means everything as far as the antipode, of course we cannot see everything in detail out to the antipode distance, but we can see some of the bigger galaxies at near antipode distance.

In Hypersphere Cosmology the universe has exactly enough mass/energy to provide the spacetime curvature to close it gravitationally into a hypersphere at its vast size, thus it consists of a black hole. If it had less mass it would have a smaller size, if it had more mass it would have a greater size. It has exactly enough mass for it size not by coincidence but because of the basic hyperspherical geometry of cosmic scale spacetime.

We usually think of black holes as regions of space containing a great deal of highly compressed matter at enormous densities, however vast black holes can have low densities inside because black holes arise not when the mass to volume ratio reaches a certain level, but when the mass to radius ratio reaches a certain level. As the volume and mass go up by the radius cubed it only takes a few atoms per cubic metre to close a universe the size of the observable universe. A hypersphere the size of the observable universe can consist mostly of space. Astronomers have noticed that the observable universe does seem to contain roughly enough ordinary matter to close it at its observable size, but Big Bang cosmologists dismiss this as ‘a mere coincidence’ and opine that we inhabit a universe which has by now accelerated far past the size we can observe.  

Equations 3 and 4

Some commentators have described Centrifugal Forces as ‘fictitious’ or even non-existent forces. Stationary observers do not observe centrifugal forces, but in the reference frame of moving observers, centrifugal forces become very real.

If one observer spins a second (smaller!) observer around in a basket on the end of a rope, the stationary observer must exert a centripetal pull on the rope. The spun observer will feel a centrifugal push into the basket. The two equal and opposite forces balance but only one of them exists in each of the observer’s reference frames.

Equations 3 and 4 show the centripetal aspect of the balanced forces that maintain the orbits of structures around the inside of hyperspheres. Equation 4 provides a simple method of calculating antipode length L from acceleration A, or vice versa, and it also plays a part in the derivation of the Redshift - Distance Equation 6.

For the equal and opposite complimentary centrifugal force see equation 5.

Equation 5

Many items of rigorous mathematics with no obvious use can lie like buried swords awaiting discovery and application.

Kurt Gödel published another exact solution of the field equations of Einstein’s General Relativity in 1949.

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.447

 ‘Matter everywhere rotates relative to the compass of inertia with an angular velocity of twice the square root of pi times the gravitational constant times the density’.

This came too late to prevent Einstein’s capitulation to the idea of an expanding universe which became generally accepted from around 1930, but not necessarily by Gödel.

Gödel’s model of a non-expanding rotating universe with a metric of w = 2sqrt (pi G p) rapidly became ignored because no axis of rotation in the universe seemed observable, and, because it also predicted closed time-like curves.

However, as equation 5 shows, the Gödel metric has complete compatibility with the internal hypersphere metric of 2Gm/L = c^2. It supplies the centrifugal force apparent to rotating observers within a hypersphere.

The absence of an obvious axis of rotation in the universe arises because the gravitationally bound mega-structures such as galaxies and galactic clusters all rotate around the randomly orientated great circles of the Hopf Fibration that delineate the hypersphere giving it no overall angular momentum in any direction and hence no axis of rotation.

Closed time-like curves in a hyperspherical universe simply imply finite and unbounded time. The universe does not have to do the same things on every revolution, it does not imply eternal recurrence of exactly the same events, or travel back to the past.

Note that in substituting mass/volume for density p the equation uses hypersphere volume 2L^3/ pi, and that the frequency f of rotation comes out at c/2L, showing that for a hypersphere ‘rotation circumference’ equals twice the antipode length.

2Gm/L = c^2 represents the Interior Hypersphere Metric in the reference frame of a stationary interior observer.

w = 2sqrt (pi G p) represents the Interior Hypersphere Metric in the reference frame of a rotating interior observer. 

Neither metric implies  real physical singularities or even  mere coordinate singularities..

 Hypersphere Cosmology predicts that gravitationally bound independently moving megastructures within the universe, such as galactic clusters or isolated galaxies will have a rotation around the universe.

Each revolution of 360 degrees equals 360 x 3600 = 1.296 e6 arcseconds.
Each revolution takes 26 billion years 8.2 e17 seconds, at 3.154 e9 seconds per century this yields 2.6 e8 centuries.
Thus, galactic clusters move at 1.296 e6 / 2.6 e8 = 0.005 arcseconds per century
HC predicts the clusters will rotate around randomly orientated planes thus the maximum observable differences in angle between any pair equals ± 0.01 arcsecond per century.

Equation 6

The Redshift-Distance Equation. 

We can readily observe Gravitational Redshift, for example light climbing out of the sun has all its wavelengths slightly shifted towards the lower energy ‘red’ end of the spectrum by the time it reaches us. For heavier stars, the effect becomes more pronounced. The gravity of the sun exists as a spacetime curvature which appears as an acceleration (a deceleration for anything trying to get away from it), this deceleration removes energy from light and causes a redshift.

Light traveling long distances across the universe becomes subject to its overall small positive spacetime curvature that also acts as a deceleration. This gives rise to a cosmological gravitational redshift in proportion to the distance travelled.

We calibrate redshift Z in terms of observed frequency over expected frequency and then subtracting one, so that the scale begins at zero rather than one.

Local galaxies have negligible cosmological redshifts. A redshift of Z = 1 corresponds to exactly half antipode distance. Light from near antipode distances has redshifts in excess of 10. Light from a luminous object at the antipode itself would have a redshift so high as to prevent its observation.

Counterfactual Indefiniteness.

Einstein famously asked if the Moon continues to exist when nobody looks at it.

I can now answer this with a (heavily qualified*) Not Necessarily.

An interesting example of this strangeness cropped up during some collaboration with a terrestrial correspondent upon Hypersphere Cosmology.

If cosmological redshift depends only on distance, then it has to remain indeterminate until measured.

See the graph below in which observed frequency (blue) decreases with distance travelled across the universe. This leads to an asymptotically increasing redshift of wavelength (red).

Note that as frequency decreases directly with distance travelled f_o = f_e (1-(d/L))

We cannot define what frequency a photon ‘would have’ at any point on a journey between two points, we can only calculate and confirm by measurement what frequency the photon has when we actually observe it. This means that the ‘counter-factual’ - the frequency that we do not observe, cannot have a definite but unknown value.

To take a simple example with rough figures: -

In Hypersphere Cosmology the antipode to any observer in the universe lies almost exactly 4 Gigaparsecs away, (1.23 E26 metres, 13 billion light years).

Imagine two distant galaxies ‘A’ at 2 Gigaparsecs, and ‘B’ at 1 Gigaparsec distant from an observer, with both laying roughly in the same direction. Light from A will have lost half of its frequency when it arrives at the observer. Light from B will have lost a quarter of its frequency when it arrives at the observer. This all accords with calculation and observation.

If we stopped the light from A at B, then we would find that it had lost a quarter of its frequency.

But what about the light from A as it flies unobserved past B? We cannot assign any value to its frequency that remains commensurate with the loss of a quarter of its frequency per Gigaparsec and a half of its frequency over 2 Gigaparsecs!

Plainly we have a quantum effect at play here. The frequency does not become defined until we stop the light and measure it, or equivalently it hits something and acquires a definite value.

Of all the interpretations of quantum mechanics, only Cramer’s Transactional Interpretation seems to offer a ‘reasonable’ or visualisable account of how this can happen. In this interpretation, retro causal advanced waves have to complete a quantum handshake across spacetime in any emission-absorption event before it acquires a definite value.

If, as in standard cosmology, the cosmological redshift arises from recession velocity in an expanding universe, then it can have a classical explanation that preserves the idea of a definite state of the unmeasured light at any point along its path.

Equation 7

Mach’s Principle has intrigued physicists for more than a century. It exists in a variety of formulations, most of them philosophical rather than mathematical.

Broadly stated Mach’s Principle, or the Einstein-Mach Principle, says that the inertia of any single body should depend on the mass and distribution of the whole of the rest of the universe.

Thus, the inertial mass of any body may not arise as an intrinsic property of that body but may depend on the entire universe.

In the Newtonian paradigm the inertial mass and the gravitational mass of a body always have the same ratio. Newtonian theory accepts this without explanation. In Einstein’s relativity the equivalence of inertial mass and gravitational mass becomes a central principle. The gravitational constant G represents the constant ratio between inertial and gravitational mass, but where does that come from?

In a universe whose size and density varies with time, any expression of Mach’s Principle becomes problematic, either the ratio of inertial mass to gravitational mass and hence G will vary with time, or lightspeed will vary with time, or both. We have no evidence of that having happened.

Only in a universe of constant size and constant overall density can Mach’s Principle apply and take mathematical form.

Equation 7 shows a mathematical expression of Mach’s Principle. It has no conceivable use, but it does look holistically and philosophically satisfying.

Equation 8 deals with the Pioneer Anomaly.

https://en.wikipedia.org/wiki/Pioneer_anomaly

As you can see this remains a controversial topic. The degree of thermal recoil involved in the anomalous deceleration seems the most disputed point.

HC predicts a deceleration (A) of 7.317 e-10 metres/second squared. And this seems close to the residual deceleration after the effects of thermal recoil appear to have declined due to cooling.