Quantum gravity (Planck)

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Quantizing gravity from n-body particle-particle gravitational orbital pairs at the Planck scale

An n-body network of particle-to-particle orbital pairs is used to emulate gravitational orbits. Specifically, the orbit is the averaged (over time) sum of individual rotating orbitals embedded on a fine structure constant 'pixel' lattice. Although dimensionless (the model does not use any dimensioned physical constants), orbit period and radius can be measured in Planck units and so this approach is applicable to modeling gravitational orbitals at the Planck scale [1].

All particles under consideration are connected to each other as opposite poles of a rotating orbital, thereby forming an n-body network of particle-to-particle pairs. In dimensioned terms, per unit of Planck time each orbital pair rotates by 1 unit of Planck length at (velocity v = c), all orbitals are then summed and averaged to give the new co-ordinates. Gravitational orbits emerge at the macro-level as the averaged sum of these Planck scale orbital pair rotations.

These orbitals resemble atomic orbitals albeit they act (in dimensioned terms) as physical units of momentum rather than regions of probability where the particle can be found, and as such may be considered as graviton analogs.

As discrete Planck units can be used, with Planck time as the clock-rate, this approach is suitable for programming Simulation Hypothesis models, and in particular Planck scale deep universe simulations.

3 body orbit (3 mass points, 3 orbitals), each point rotating anti-clockwise







Orbital pairing model[edit | edit source]

Particles are assigned a `point' that has co-ordinates within a 4-axis hyper-sphere universe [2]. The hyper-sphere expands in incremental steps (FOR age = 1 TO ...), these steps as the universe simulation clock-rate. The hyper-sphere pulls particles with it as it expands (this expansion as the source of particle motion).

Every particle is connected to every other particle (in the `universe') by a circular orbital. For each value of age, all orbitals rotate by 1 (Planck) step (driven by the hypersphere expansion), they are then summed and averaged giving new particle co-ordinates. As this occurs for every particle in the simulation before incrementing the variable age (the clock-rate), all gravitational interactions can be updated in real time on a serial processor. In Planck terms, the orbital rotates 1 unit of Planck length per 1 unit of Planck time at velocity c.

Although all motion, including orbitals and so gravitational orbits occurs at velocity c, the hyper-sphere expansion is equidistant and so `invisible'. Cylindrical orbits (in hyper-sphere co-ordinates) will therefore register as circular motion on a 2-D plane to local observers. An apparent time dilation effect emerges as a consequence.

Gravitational potential and kinetic energies are measures of the respective alignment of the underlying particle-particle orbitals. The orbital angular momentum of the planetary orbits for example would then derive from the sum of the planet-sun particle-particle orbital angular momentum independent of any angular momentum of the sun itself.


N-body rotations[edit | edit source]

A simulation[3] comprising n-body rotating orbitals is described. Each point in the simulation is assigned initial (x, y) co-ordinates forming orbital pairs that rotate around each other in an anti-clockwise direction on a 2-D plane according to an angle β as defined by the orbital radius. In the following examples, 1 point is assigned as the orbiting point, the remaining points forming the 'central' mass. The results are averaged and summed, the process repeating. After 1 complete orbit, the period tsim (as the number of increments to the simulation clock age) is noted [4].


Simulation variables[edit | edit source]

1. i; number of center points in the orbit

2. j = i*x + 1; number of virtual points in the orbit

3. jmax; mass j to radius r coefficient

4. x, y; start co-ordinates for each point

5. tsim; orbital period

6. ro = radius of orbiting point (from center)

7. ri = radius of center points (from barycenter)

8. orbital constant, alpha = inverse fine structure constant = 137.03599...

9. angle of each rotation β a function of the radius of the individual orbital being calculated.


Orbital formulas[edit | edit source]

, barycenter radius
, point orbit around center of mass
, orbit of center mass around barycenter



4 body orbit; (4 mass points, 6 orbital pairs)

4-point, 6-orbitals (regular)[edit | edit source]

A 4-body (4 points, 6 orbital-pairs) orbit, 1 mass point orbiting i = 3 center points. The number of center points j = 4, 7, 10 ... are increased to simulate a greater (central) mass, up to = 97.

ro = 2090.88...
j tsim tsim - tcalc
4 1445505344 454
25 231280921 139
37 156270894 95
49 118000472 72
61 94787264 59
73 79205796 49
85 68023802 42
97 59608486 37





irregular 4-body orbit, 3 center points around barycenter

4-point, 6-orbitals (irregular)[edit | edit source]

A 4-body (4 points, 6 orbital-pairs) orbit, The 3 center points ( = 4.6) follow an irregular orbit around the barycenter.

ro = 2090.88...
j tsim to
4 1447096726 2090.87
25 231535423 2091.31
37 156442856 2091.28
49 118130320 2091.27
61 94891569 2091.26
73 79292955 2091.26
85 68098655 2091.27
97 59674080 2091.26





8 body orbit; (8 mass points, 28 orbital pairs), 1 external point zoomed to the 7 central points orbiting the barycenter

8-point, 28-orbitals[edit | edit source]

A regular 8-body (8-points, 28 orbital-pairs) orbit. 1 mass point orbiting i = 7 center points. The number of center points j = 8, 15, 22 ... are increased to simulate a greater (central) mass, up to = 225.

ro = 2066.3265
j tsim tsim - tcalc
8 1656794897 1517
29 457046948 498
57 232532567 163
85 155933579 85
113 117295114 7
141 94002481 20
169 78428106 17
196 67280964 15
225 58908222 13



81-point, 3240-orbitals[edit | edit source]

In this example the mass j is fixed and the radius parameter is variable. Setting

i = 81

j = 82

Converting to meters via Planck length

m


Earth - moon orbit

= 384748km
= 0.00887m (Schwarzschild radius)
= 0.000109m

Earth-moon mass ratio via standard gravitational parameters;

From barycenter

Scaling to earth mass


Earth orbit examples[edit | edit source]

For earth orbits, radius rg and Schwarzschild radius λearth are dimensioned and so we need only add a dimensioned quantity to the velocity formula.

Orbital parameters (Earth)
rg (m) ng vg (m/s) Tg (s)
384748000(1) 17791.3 1017.84 0.236 x107
42164170(2) 5889.674 3074.66 86164.09165
6371000(3) 2289.41 7909.79 5060.837
1, moon orbit
2, geosynchronous orbit
3, earth surface


Rotation angle beta[edit | edit source]

We can further modify our orbital radius and velocity without changing period by adding an extra term denoted here as . Thus it is possible for an object to be at a greater (or lesser) radius from the center of mass yet maintain the same orbital period;


8-body (8 mass points, 28 orbitals), the orbit is a function of the start positions of each point

Orbital plane rotation[edit | edit source]

We can also include an orbital plane rotation effect. This can be demonstrated with an elliptical orbit

semi-minor axis:

semi-major axis:

radius of curvature :


arc secs per 100 years:

Mercury = 42.98  
Venus = 8.62  
Earth = 3.84  
Mars = 1.35  
Jupiter = 0.06

The orbital plane becomes

Mercury

years



Illustration of B's cylindrical orbit relative to A's time-line axis

Hyper-sphere orbit[edit | edit source]

An expanding hyper-sphere forms the scaffolding of the `universe'. The hyper-sphere expands in uniform incremental steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates [5]. Time becomes time-line.

While B (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around A (planet), it also follows a cylindrical orbit (from B1 to B11) around the A time-line (vertical expansion) axis (td) in hyper-sphere co-ordinates. A is moving with the universe expansion (along the time-line axis) at (v = c) but is stationary in 3-D space (v = 0). B is orbiting A at (v = c) but the time-line axis motion is equivalent (and so `invisible') to both A and B, as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on A and B. In dimensionless terms;

For object B

(see time dilation)

For object A


Gravitational coupling constant[edit | edit source]

In the above, particles were assigned a mass as a unit of Planck mass. Conventionally, the Gravitational coupling constant αG characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

When converting to SI, `particle' points here are replaced by units of Planck mass, to resolve this the coupling constant is inverted to become a measure of the frequency of occurrence of a unit of Planck mass. If we can allow for wave-particle duality at the Planck level to be represented by an electric-wave to mass-point oscillation [6], where the mass-point is a single discrete unit of Planck mass that occurs for a discrete unit of Planck time (and so is applicable to simulation modeling), then the inverse coupling constant measures this mass-point frequency. As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete mass at the Planck level.

If particles oscillate between an electric wave-state to Planck-mass (for 1 unit of Planck-time) point-state then at any discrete unit of Planck time a number of particles in the universe will simultaneously be in the mass point-state. For example a 1kg satellite orbits the earth, for any given unit of time, satellite (B) will have particles in the point-state. The earth (A) will have particles in the point-state and the number of orbital links between the earth and the satellite will sum to;

This also assigns to gravity a similar magnitude as the strong force (at this, the Planck level) albeit this interaction occurs seldom, and so when averaged over time (the macro level) gravity appears weak.

Examples:

1. The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth).


2. The orbital angular momentum of the planets derives from the angular momentum of the orbital pairs (and so is independent of the orbital angular momentum of the sun).

The orbital angular momentum of the planets;

mercury = .9153 x1039  
venus    = .1844 x1041  
earth    = .2662 x1041 
mars     = .3530 x1040 
jupiter   = .1929 x1044   
pluto   = .365 x1039   


Orbital angular momentum combined with orbit velocity cancels n giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.


Planck force[edit | edit source]

a)

b)


Atomic orbitals[edit | edit source]

The atomic orbital is treated as a distinct unit of rotational momentum (a physical construct rather than a region of probability), and is analogous to a photon albeit of inverse phase whereby photon + orbital can cancel. The orbital is mapped on a grid defined by units of alpha.

The orbiting electron consists of an oscillation between a point-state (with defined co-ordinates on the alpha grid) and a time dimension (the frequency of the particles participating in the orbital). After each frequency cycle, the point-state moves a discrete step on the alpha grid according to the angle of rotation , the frequency component thereby dictating the time interval between these steps, but does not of itself occupy spatial co-ordinates on the grid [7].

Basic orbital:[edit | edit source]


Orbital rotation[edit | edit source]

At each step, the radius of the orbital is used to derive , the angle of rotation (see also gravitational orbitals), thus an orbital of radius will require = 471964.354 steps for 1 complete rotation.


Orbital transition[edit | edit source]

Transitions between orbital levels are mapped using .

such that a photon of wavelength equivalent to an orbital will cancel that orbital after steps, i.e: for the duration of the photon, each step involves

An incoming photon is separated into 2 photons as per the Rydberg formula.

As is minus, the '(+)' photon will decrease the orbital radius in increments as above, however the '- (+)', because of the Rydberg minus term, will conversely increase the orbital radius. And so for the duration of the (+) photon, the orbital radius does not change as the 2 photons cancel;

However the (+) has the longer wavelength, and so for the remaining duration of this photon, the orbital radius will be extended.

As the velocity of rotation derives from the radius of the orbital (from ), the velocity automatically adjusts during the transition phase.


H atom[edit | edit source]

Including the hypersphere relativistic term, the simulation gives a geometrically optimal n=1 to n=2 solution as;

H () = 2466061413187035Hz is the most precisely measured of the spectra and can be used as a reference. Using CODATA and and setting alpha = 137.035999585;

returns an orbital period 471959.956 and H ionization gives 471959.011, a 0.945 wavelength difference [8].



External links[edit | edit source]

References[edit | edit source]

  1. Macleod, Malcolm J.; "2. Quantum gravity n-body orbitals for Planck scale simulation hypothesis". RG. Feb 2011. doi:10.13140/RG.2.2.11496.93445/12. 
  2. Macleod, Malcolm; "3. Programming cosmic microwave background for Planck unit Simulation Hypothesis modeling". RG. 26 March 2020. doi:10.13140/RG.2.2.31308.16004/7. 
  3. https://codingthecosmos.com/gravity-orbitals/
  4. https://www.youtube.com/watch?v=A8Sf9faVbxI
  5. Macleod, Malcolm; "1. Programming relativity for Planck unit Simulation Hypothesis modeling". RG. 26 March 2020. doi:10.13140/RG.2.2.18574.00326/3. 
  6. 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. 
  7. "4. Atomic orbitals in Planck scale simulation hypothesis". RG. Feb 2011. doi:10.13140/RG.2.2.23106.71367/2. 
  8. https://www.youtube.com/watch?v=v-0ykElUAcI