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.  

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.