Equation 9

Galaxies rotate but they do not rotate in accordance with our standard ideas about gravity.

A large discrepancy exists between the rotation curves for disc galaxies expected from a classical or relativistic consideration of their baryonic mass distributions, and the rotation curves observed. This discrepancy has led to the hypothesis that a mysterious ‘Dark Matter’ must make up the difference.

Gomel and Zimmerman show that an apparently ‘Non -Inertial System Component’ in the form of an angular velocity ‘w’, can account for this discrepancy.

v(r) = vi(r) + w(r) 

https://www.preprints.org/manuscript/201908.0046/v1

This paper asserts that the angular velocity w arises from a Gödelian rotation component of the galaxies where w = 2sqrt (pi G p)

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

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

Thus, v(r) = vi(r) = 2sqrt(pi G p)  r

Now because the wr component of galactic rotation dominates at the extremities of galactic discs, thus creating ‘flat’ rotation curves, and because the outer edges of all disc galaxies rotate with the same time period of almost exactly one billion years, irrespective of their varied sizes.

https://astronomy.com/news/2018/03/all-galaxies-rotate-once-every-billion-years

It follows that we can calculate w with high accuracy to w = 2 e-16 radians/second.

Calculation: distance/velocity = time

2 pi r / wr = t

2 pi / 2 e-16 = t

pi e16 = t = 3.142 e16 seconds. (A billion years equals 3.155 e16 seconds)

Thus, it seems far more likely that all disc galaxies have a Gödelian rotational component of precisely 2 e-16 radians/second rather than dark matter halos with sizes and masses precisely and mysteriously tailored for each galaxy’s distribution of ordinary baryonic matter to create the observed rotation curves.

(Aside - just a little aside I came across whilst reading science news daily site - a flurry of papers have come out about 'Self-Scattering Dark Matter', apparently it will have to have this amongst its other miraculous properties to account for some of its predicted distribution. Phlogiston ended up with a negative weight variety.)

Equation 10

Here we show the modification to Newtonian Dynamics expected in a positively curved spacetime that has a negative acceleration A. Force does not exactly equal mass times acceleration on the very large scale, we have to subtract the very small deceleration.

Substituting this effect into the equation for the acceleration due to gravity we find that it increases that acceleration and also the orbital velocity.

As hyperspheres, black holes have an orbital velocity of lightspeed. 

A black hole hypersphere within the hypersphere of the universe will tend to acquire extra orbital velocity from the spacetime curvature of the universe. The effect will increase with the size of the black hole.

Exactly what will happen when the material inside a black hole begins to push up against the lightspeed limit remains an open question, but it seems likely that the hole will begin to shed mass/energy in some form or other.

This predicted effect seems similar to the Hawking Radiation predicted to emerge from smaller black holes.

The effect may prove difficult to observe because large black holes usually remain obscured by dense clusters of stars and large fluxes of radiation from their accretion discs and infalling matter.

Equation 11

The Dirac Large Numbers Hypothesis suggests that a huge dimensionless number, or perhaps just a few huge dimensionless numbers, seem to characterise many ratios in the observed universe that relate the quantum scale to the cosmic scale.

https://en.wikipedia.org/wiki/Dirac_...ers_hypothesis

https://arxiv.org/ftp/physics/papers/0502/0502049.pdf

If the universe exists in a state of expansion, then the equality of all these ratios appears as an extraordinary coincidence at only this particular epoch of the universe and we just have the good luck to observe it now.

On the other hand, if the universe exists as a non-expanding Hypersphere, finite and unbounded in space and time, then the Large Numbers simply represent the permanent ratio between the smallest components of the universe as shown by the quantum scale and the whole universe.

The universe and the quantum realm have characteristic units of mass, length, time, energy and acceleration, the same equations govern both: -

m/l = c^2/G     t = c/l   e = mc^2   a = c^2/l

A single number underlies the ratios of the quantum scale to the cosmic scale, this number which we can call The Ubiquity Constant U has a value of the order of 10^60, or e60.

All the other ‘cosmic coincidences’ such as e20, e40, e60, e80, e120 all arise as multiples of it and reflect something fundamental about the universe at any point in time.

Amusingly the ‘vacuum catastrophe’ which arises if you mistakenly try to explain the non-existent cosmological constant in terms of the non-existent quantum vacuum energy, comes out by a factor of e120 wrong, the largest mistake cosmologists have ever admitted to.

The origin of U, the ratio of the quantum scale to the cosmological scale remains a mystery but at least it reduces all the mysterious large number coincidences to a single mystery.

Equations 12, 13, 14, and 15 show how the Ubiquity constant defines the size of quanta, the pixilation of the universe, the effective uncertainty, and the mass of stable particles in the universe.

Equation 12

The Bekenstein-Hawking conjecture arose to explain the behaviour of entropy in a black hole.

https://en.wikipedia.org/wiki/Black_hole_thermodynamics#Overview

It asserts that the entropy does not disappear but becomes preserved by an increase in the surface area of the hole in Planck units if matter falls in.

As the universe itself consists of a black hole in Hypersphere Cosmology then its surface area should represent the total amount of entropy that it contains.

Using the square of the antipode length L over the Planck area to approximate the entropy we obtain an entropy of the order of e120.

If the entropy of a system equates to its information content, then e120 bits of information must suffice for the e180 Planck volumes within the universe.

This implies only one bit of information per e60 Planck volumes or one for each e20 Planck length.

The universe thus has insufficient information to specify anything below this level of its effective pixilation.

See equations 13,14, and 15 for the consequences of this.

None of this supports the Holographic Universe hypothesis. The surface area defines the entropy/information content, but the information is not ‘encoded’ on the surface.

We can obtain a similar result by using the ratios of 3D surface hyper-areas. 

The Conservation of Entropy.

Hypersphere Cosmology asserts that the overall entropy of the universe remains constant.

The second law of thermodynamics states that the entropy of a closed system must either increase or remain constant, it can never decrease.
The sky is cold, the stars are hot, plainly the observable entropy remains fairly low, but many of the processes going on around us appear to increase entropy

Now as the limits of observation represent a temporal horizon rather than a temporal boundary, the matter and energy within it has had unbounded time during which it has plainly not achieved thermodynamic equilibrium and maximum entropy.

Thus the universe must have some ongoing processes which increase entropy and some which do the opposite, and between them maintain an overall constant entropy. Gravitation itself may supply the main entropy reversing mechanism.

The principle of never decreasing entropy, the second law of thermodynamics, grew out of the age of steam and observations of the flow of heat in gasses. It takes no account of the attractions between particles. Observe some gas with an uneven distribution of heat or pressure and it will soon homogenise or dissipate. However, a really vast amount of gas will compress itself and ignite as a star. I submit that it be considered as an entropy reversing effect.

If gravity worked repulsively it would fit the simple view of entropy nicely, matter and energy would dissipate. Interestingly the sign of G is formally negative, although we can usually ignore that, but in a sense, gravity reverses the (entropic) arrow of time.

'Entropy increases with time because we measure time in the direction in which entropy increases' - Stephen Hawking.

Equation 10 shows that black holes within the universe will not persist indefinitely.

Neutrons build up as stars burn and create heavier elements but heavier elements breakdown into neutron plasma inside neutron stars which eventually explode. Free neutrons in space decay back into hydrogen.

The recently discovered Giant Gas Halos around nearby galaxies suggest that all galaxies probably have these difficult to observe halos. Such halos, which also contain heavy element dust from exploded stars, may well act as the recycling yards for the stellar parts of the galaxies.

We have no means of determining the ages of black holes and some structures within the universe seem to have an age very close to, or in excess of, the temporal horizon.   

It does not seem unlikely that the closed spacetime of the Hyperspherical Universe can maintain a constant leel of entropy.    

Equation 13

The universe has an information deficit in the sense that it has only one bit of information/entropy for every e20 Planck lengths and this represents the universe’s pixilation or ‘grain size’. Equations 14 and 15 show what this means for quantum particles.

The pixilation level also has the effect of raising the effective Heisenberg uncertainty/indeterminacy level of the universe by twenty orders of magnitude. This does not make the universe excessively Chaotic in principle, but it does make it about as Chaotic as it appears in practice.    

Equation 14

The universe has an information deficit in the sense that it has only one bit of information/entropy for every e20 Planck lengths and this represents the universe’s pixilation or ‘grain size’. In practise we do not observe anything physically real at LESS than twenty orders of magnitude Above the Planck Length or Above the Planck Time.

See below: -

https://en.wikipedia.org/wiki/Orders_of_magnitude_(length)

https://en.wikipedia.org/wiki/Orders_of_magnitude_(time)

Equation 15

It may seem odd that whilst the ‘enhanced’ Planck length and Planck time set a lower limit for quantum particles, (see equation 14) whilst a similarly ‘reduced’ Planck mass sets an upper limit for the size of stable quanta.

However, the wavelength of a quantum is inversely proportional to its mass, heavier quanta have smaller not larger wavelengths.

Stable quanta cannot have a mass in excess of e-20 Planck mass or their wavelengths would fall below the pixilation level of the universe.

See below: -

https://en.wikipedia.org/wiki/Orders_of_magnitude_(mass)

Equation 16

A stereographic projection can show the effect of projecting an n-sphere into an n-dimensional space.

For the purposes of Hypersphere Cosmology, we require a projection of a 3-sphere into three-dimensional space as observers will measure light as having travelled to them in ‘straight’ lines.

The diagram shown projects a 1-sphere (a circle) into a 1-space (a line), but this will suffice as we only need to know the distortion of distance.

Imagine first the projection of a 2-sphere (say a globe) into a 2-space (a flat surface). Imagine that we construct a wire frame globe with lines of latitude and longitude made of wires. If we place the globe south pole down on a sheet of paper and then place a lightbulb just inside the smallest ring of latitude around the north pole. The light will throw a shadow of the latitude wires of the globe onto the paper. The latitude rings close to the south pole will appear as circles close to the south pole on the paper, but latitude rings further from the south pole will start to appear exponentially larger and further away from the south pole on the paper. In principle, the shadow from latitudes very close to the north pole will go off the edges of a sheet of paper of any size.

Now a Glome-type hypersphere has the property that its circumference equals twice its antipode length L, so the distance d equals the distance around the circumference from the observer at the origin O at which we seek to determine the projection.

Such stereographic projections project a circle into an ever more widely spaced series of points on a line. A sphere becomes a series of ever more widely spaced concentric circles on a plane, a Glome hypersphere becomes an ever more widely spaced series of concentric 3D spheres. The equation shows the same distance relationship in all the various dimensional scenarios.

By expressing distance d over antipode length L and converting to redshift Z using the Redshift-Distance Equation we can relate the distance distortion from apparent to actual distance caused by lensing directly to redshift.

This shows that the universe does not undergo an accelerating expansion and does not need Dark Energy. It also allows a precise determination of the antipode length of the universe by Equation 17.  

Equation 17

The Redshift-Distance and the Lensing equations 6 and 16 together yield a derivation for the Antipode length L for the universe.

This derivation depends only on measured redshifts and measured apparent distances derived from measured apparent magnitudes.

As shown in the extensive calculations here: -

https://www.specularium.org/hypersphere-cosmology/item/270-type-1a-supernovae-and-hypersphere-cosmology

A value of L close to 1.23 e metres emerges in every case, within the limits of observational inaccuracies in apparent magnitude observations. This corresponds to 13 billion light years.

The substitution of this value for L into equation 1 gives the exact value of the mass of the universe M at 8 e52 kilograms, eighty octillion metric tonnes.

The substitution of this value for L into equation 2 gives the exact value of A the spacetime curvature of the universe expressed as an acceleration of 7.317 e-10 metres per second squared.

The substitution of this value for L into equation 5 gives the exact angular velocity of megastructures around the universe as 0.005 arcseconds per century on the basis that they rotate around the great circles of a Hopf Fibration of the hypersphere with a circumference of twice the antipode length L.