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Relativity
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2. Relativity as a translation between 2 co-ordinate systems

In this model series the universe is an incrementally expanding (in discrete Planck unit steps) 4-axis hyper-sphere, with our 3-D space residing on the surface of the hypersphere. Relativity then becomes a translation between the 2 co-ordinate systems, our 3-D space and the 4-axis hyper-sphere. The hyper-sphere expands in steps (the universe is spatially finite (a closed 4-sphere), but it is not a static system for it expands in steps with every unit of Planck time. At each step Planck units of mass

\[ m_{P}\]

, length

\[ l_{p}\]

and time

\[ t_{p}\]

are added, thus forming a scaffolding for the particle universe. As for each unit of Planck time there is a unit of Planck length, this Planck framework is expanding at a constant rate (the speed of light

\[ c=l_{p}/t_{p}\]

). As the hypersphere expands, it also pulls particles with it (at the speed of light), and so all particles and objects are traveling at, and only at, the speed of light (in the hyper-sphere frame of reference there is only 1 velocity, c). However, if we consider 3-D space as the surface of the hyper-sphere then motion between particles is relative. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally across this surface (in 3-D space) and so this incremental hyper-sphere expansion at velocity c cannot be observed directly via the electromagnetic spectrum, relativity then becomes the mathematics of perspective translating between the absolute albeit expanding hyper-sphere background and the relative motion of 3D space [1].

Wave-particle oscillation

Further information: User:Platos_Cave_(physics)/Simulation_Hypothesis/Electron (mathematical)

It is proposed as a central theme of this model that particles at the Planck scale are oscillations between an electric wave-state to (Planck) mass point-state, the wave-state as the duration of particle frequency in Planck time units and is the origin of the particle electric properties, the mass point-state as 1 unit of Planck mass (1mP) for a duration of 1 unit of Planck time. Wave-particle duality at the Planck scale then becomes a wave-point oscillation, the particle does not exist at any 1 unit of (Planck) time, and so can more precisely be defined as an event (1 complete wave-point oscillation). 1 of the dimensions of the particle is therefore time (frequency), thus particle quantum states also therefore do not occur at the Planck scale, but rather are emergent properties (and so quantum state physics cannot be applied to the Planck scale) [2].

Notably at the mass state the particle has defined (point) co-ordinates within the universe hypersphere and thus can be mapped, conversely the wave-state is undefined.

Radial motion

A and B in hypersphere co-ordinates from origin O, (frequency 0A = 0B = 6)

We take 2 particles A (v = 0 in 3D space) and B (v = 0.866c in 3D space). Both have a frequency = 6; 5tp (5 increments to tage) in the wave-state followed by 1tp (1 increment to tage) in the point-state (the point-state is represented by a black dot, diagram right).

The hyper-sphere expands radially at the speed of light. Both particles start at the same location; defined as origin O. After 1 second, B will have traveled 299792458 * 0.866 = 259620km from A in 3-D space (the horizontal axis). However for both A and B the radial axis length is the same OA = OB.

As the surface of the hypersphere constitutes 3-D space, movement along this surface corresponds to distance travelled (in conventional terms). However as the hypersphere is expanding outward (time goes forward), the surface expands with it, and so motion would register as a space-time continuum. We would note that A has travelled 1 second in time but no distance in meters. B has travelled both in time and in space (259620km from stationary A).

In hypersphere co-ordinates however, both A and B have travelled the same radial axis length; OA = OB. There is only 1 velocity and that is the velocity of expansion; radial axis OA will always equal OB. For 3D-surface-dwellers, there is a distinction between time and space because the 3D-surface-dweller has no means to register the hypersphere expansion as motion.

From the perspective of surface-dweller A (v = 0), B will have reached the point-state after 3 units of tage (3 units of Planck time tp) and so will have twice the (relativistic) mass of A. However the hypersphere expands radially from origin O, and so A will also have traveled the equivalent of 299792458m from O (radial axis OA = OB, v = c), and so from the perspective of the hypersphere, B can equally claim that A has traveled 259620km from B in 3-D space terms.

The time-line axis (axis of expansion) maps Planck time (1tp) steps, note that only the particle point-state has defined co-ordinates, and so on this graph A can only have 6 possible time-line divisions (if including v = 0). As the minimum step is 1 unit of Planck time, this means that B can attain Planck mass (mB = mP/1) when at maximum velocity vmax (relative to the A time-line axis), but B can never attain the horizontal axis = velocity c because in the process at least 1 time step (1 unit of Planck time) is required, and so for particles vmax can never attain c. However a small particle such as an electron has more time-line divisions, and so can travel faster in 3-D space than can a larger particle (with a shorter wavelength).

Particle motion

particle wave to point oscillations in hyper-sphere expansion co-ordinates

Depicted is particle B at some arbitrary universe time t = 1. B begins at origin O (top left) and is pulled (stretched) by the hyper-sphere (pilot wave) expansion in the wave-state (top right). At t = 6, B collapses back into the mass point state (bottom left) and now has new co-ordinates within the hypersphere, these co-ordinates becoming the new origin O’.

In hypersphere coordinates everything travels at, and only at, the speed of expansion = c, this is the origin of all motion, particles (and planets) do not have any inherent motion of their own, they are pulled along by this expansion as particles oscillate from (electric) wave-state to (mass) point-state ... ad-infinitum.

Particle N-S axis

particle N-S spin axis orientation mapped onto hyper-sphere

Particles are assigned an N-S spin axis around which particles can rotate (spin left and spin right). The co-ordinates of the point-state are determined by the orientation of the N-S axis. Of all the possible solutions, it is the particle N-S axis which determines where the point-state will next occur.

A, B and C begin together at O, if we can then change the N-S axis angle of A and C compared to B, then as the universe expands the A wave-state and the C wave-state will be stretched as with the B wave-state, but the point state co-ordinates of A (and C) will now reflect their new N-S axis angles of orientation.

A, B, C do not need to have an independent motion; they are being pulled by the universe expansion in different directions (relative to each other). We can thus simulate a transfer of physical momentum to a particle by simply changing the N-S axis. The radial hyper-sphere expansion does the rest.

Photons

Information between particles is exchanged by photons. In this model, photons are unique: they do not have a mass point-state. and because they lack this mass state, they do not travel along the timeline h-axis in the same way as matter. Instead, they are time-stamped and travel laterally across the 3D surface of the expanding hypersphere. This behavior is the key to understanding what we observe. It explains how light moves and why we perceive cosmic redshift.

Matter (like particles A and B) is carried outward with the expansion along the expansion axis. Light (a photon) however travels sideways across this surface.

The photon's total speed through this 4D space—its sideways motion combined with the outward expansion—is always equal to the speed of light, c.

When a photon travels for a long time across this expanding surface, its wavelength is stretched. This effect is what we observe as cosmological redshift.

Unlike a simple Doppler shift seen from an object moving through space, this cosmological redshift is a direct consequence of space itself expanding while the photon is in transit.

Modelling a Planck scale simulation universe using geometrical forms. For AI analysis, the following links have been compiled into a single file [3].

References

  1. ↑ Macleod, Malcolm J.; "2. Relativity as the mathematics of perspective in a hyper-sphere universe". SSRN. Feb 2011. doi:10.2139/ssrn.3334282.
  2. ↑ Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x.
  3. ↑ https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf Compilation of journal articles 1-7