The Apophenion: A Chaos Magick Paradigm - Peter J. Carroll 2008

The Hyperspherical Universe
Appendix

Key to symbols.

G = Gravitational Constant.

M = Mass of Universe

m = Mass

c = Lightspeed

d = Density (Mass divided by volume)

A = Anderson Acceleration

a = Acceleration

= Orbital Velocity

or r, = Three radius of a sphere

= Four radius of a hypersphere

W = Angular velocity in radians per second

L = Antipode distance in a hypersphere, ()

l = length

I have a hunch that the universe runs on fairly simple algebra/geometry like 'force equals mass times acceleration', or 'energy equals mass times lightspeed squared'.

I suspect that really complex formulae do not apply to fundamental phenomena.

Part 1. The Vorticitating Hypersphere.

'Matter everywhere rotates relative to the compass of inertia with the angular velocity, (W), of twice the square root of pi times the gravitational constant times density'

-Kurt Gödel.

(Equation 1)

(Gödel derived this as a possible solution to Einstein's field equations).

Now substituting the mass of the universe M, and volume of a sphere, 4/3

for density, and then substituting (the formula for a photon sphere) into equation 1, and then simplifying, we obtain:

(Equation 2)

A Photon sphere consists of an object about which light approaching it tangentially would go into orbit. Equation 2 shows that the Gödel universe would have an orbital velocity of c, lightspeed, at its circumference, and a centrifugal acceleration of: - This balances a similar centripetal (gravitational) acceleration.

To give a hypersphere the properties of an orbital velocity of lightspeed means that

So working backwards and inserting the mass of the universe M, and hyperspherical 3-surface volume, , for density, and (the formula for a hypersphere with an orbital velocity of lightspeed), we recover: -

(Equation 3)

This shows the vorticitation of a hypersphere, in which the entire 3 dimensional surface rotates relative to the orthogonal curvature axis.

Such a structure has a centrifugal acceleration of: -

(Equation 4)

Part 2. The Size of the Universe.

A universe consisting of a hypersphere with , has the equation; -

(Equation 5)

And thus a centripetal (gravitational) acceleration of, to balance the centrifugal acceleration in equation 4.

Now if we equate the Anderson acceleration A,

(Measured at 8.74 x 10^-10 metres/second^2), with the centripetal/centrifugal accelerations in a vorticitating hyperspherical universe, then we can easily calculate L and M, and also the temporal horizon of the universe T, to yield the following values: -

M = 1.39 x 10⁵³ kilograms.

L = 1.03 x 10²⁶ metres, about 11 billion light years.

T = 3.34 x 10¹⁷ seconds, about 11 billion years.

Angular rotation = 0.006 arc-seconds per century.

Note that these figures have an uncertainty of about 15% arising from difficulties in precisely measuring the Anderson acceleration. The universe will actually look a little larger than L and T because of hyperspherical lensing.

Part 3. The Anderson Acceleration.

The centripetal/centrifugal effect of the Anderson acceleration in a vorticitating hypersphere gives rise to an omni-directional resistance to linear motion and an omni-directional boost to any kind of gravitational orbital motion.

As , light from antipode distance becomes redshifted to oblivion creating effectively an optical horizon.

C - AT = 0 (Equation 6)

The Anderson acceleration boosts orbital velocities according to the following equation:

(Equation 7)

This makes negligible differences at planetary distances, but at galactic distances it makes significant differences, and it obviates the need for arbitrarily modified gravity theories or dark matter.

Part 4. Closed Time Curves.

Gödel's rotating universe idea became discarded as unphysical for two reasons. Firstly no axis of rotation seemed observable. However in a hypersphere the axis lies at right angles to 3d space and remains unobservable except as curvature.

Secondly the Gödel universe contains closed time curves and anything travelling around the universe at lightspeed would in theory eventually catch up with its own past, in the sense that it would arrive back just as it began to set off.

In the vorticitating hyperspherical universe exactly this happens, but it does not create a causality problem, rather it solves the problem of causality by making everything the cause of everything. However no form of radiation or matter could in practise survive the 22 billion year trip and expect to arrive in the same form it departed in.

Part 5. Hyperspherical Particles.

Equation 3, for the angular velocity of a hypersphere,

contains a further surprise.

It reduces to , and substituting , to find the frequency f, and then substituting yields:

(Equation 8)

Now if we identify L with wavelength then this equation also represents the basic unit of fermion particle spin, where one half of frequency times wavelength equals lightspeed. This also explains why fermions have to rotate through 720 rather than 360 degrees to restore their original orientation.

Thus it seems that fundamental particles consist of vorticitating hyperspheres as well. This seems inevitable if they have the rotational freedom described by HD8.

Thus Equation 3 unites the Microcosm and the Macrocosm.

I suspect that Hermes Trismegistus would have appreciated that.

I suspect that the Sufis would also appreciate confirmation that everything spins, including the universe itself.