(relativity and gravity in a Planck universe) Programming relativity and gravity via a discrete pixel space in Planck level Simulation Hypothesis models Malcolm Macleod E-mail: maclem@platoscode.com Outlined here is a simulation hypothesis approach that uses an expanding (the simulation clockrate measured in units of Planck time) 4-axis hyper-sphere and mathematical particles that oscillate between an electric wave-state and a mass (unit of Planck mass per unit of Planck time) pointstate. Particles are assigned a spin axis which determines the direction in which they are pulled by this (hyper-sphere pilot wave) expansion, thus all particles travel at, and only at, the velocity of expansion (the origin of c), however only the particle point-state has definable co-ordinates within the hyper-sphere. Photons are the mechanism of information exchange, as they lack a mass state they can only travel laterally (in hypersphere co-ordinate terms) between particles and so this hypersphere expansion cannot be directly observed, relativity then becomes the mathematics of perspective translating between the absolute (hypersphere) and the relative motion (3D space) coordinate systems. A discrete 'pixel' lattice geometry is assigned as the gravitational space. Units of ~c 'physically' link particles into orbital pairs. As these are direct particle to particle links, a gravitational force between macro objects is not required, the gravitational orbit as the sum of these individual orbiting pairs. A 14.6 billion year old hyper-sphere (the sum of Planck black-hole units) has similar parameters to the cosmic microwave background. The Casimir force is a measure of the background radiation density. 1 Introduction It was proposed in an article on the mathematical electron [1] that the Planck units for mass, length, time and charge could be constructed as geometrical objects (notes, fig.23) as per mathematical universe hypothesis, yet these objects would arguably be indistinguishable from the mass, length, time and charge of the physical universe, thus laying a theoretical basis for the universe as a simulation where the Planck units are mathematical structures at the Planck level. In this article, part II, the mathematical particle is embedded within an expanding hyper-sphere black-hole 'universe' whereby relativity and gravity are introduced as naturally emerging properties of underlying particle geometries. The sum universe is a 4-axis hyper-sphere expanding in incremental discrete Planck units, this expansion as the origin of Planck-time (the simulation clock-rate), the arrow of time, velocity c (the velocity of expansion) and particle motion. The mathematical particle oscillates between an electric wave-state and a (unit of Planck mass per unit of Planck time) mass point-state. In section 2, the particles are pulled along by this (pilot-wave) hyper-sphere expansion according to their spin axis. In hypersphere co-ordinates all particles travel at, and only at, the speed of expansion c, however information between them is exchanged via electro-magnetic waves, these, being mass-less are restricted to lateral motion within this hyper-sphere resulting in an observable 3-D space of relativistic motion, relativity as the mathematics of perspective translating between these 2 coordinate systems. In section 3, all particles that are simultaneously in the mass point-state for any given unit of (Planck) time are linked to each other to form orbital pairs as units of mP v 2r = ~c within a discrete pixel lattice gravitational space. The gravitational orbit is the sum of these underlying orbital pairs which correspond to atomic orbitals. In section 4, the simulation clock-rate is measured in discrete 'pixels' (comprising the Planck units). Although measured in constant increments, a 14.6 billion year old Planck black-hole (the sum of these pixels) has similar parameters to the cosmic micro-wave background.The Casimir force equates to the background radiation energy density. 2 Relativity 2.1. Wave-mass duality In an article on a mathematical electron [1], localized Planck units may emerge from a unit-less mathematical electron (see notes) oscillating between an electric wave-state (duration = electron frequency in units of Planck time) and a unit of Planck-mass (per 1 unit of Planck-time) point-state. This oscillation is driven by the expansion of the hyper-sphere pilot-wave. 2.2. Space-time Particle A is mapped onto a space-time graph (fig.1). A does not move in space (v = 0), but it does move in time. 1 2 Relativity (relativity and gravity in a Planck universe) Fig. 1: particle A, v = 0 Particle B, v = 0.866c is added (fig.2). After 1s B will have traveled 0.866 x 299792458 = 259620km from A along the horizontal space axis. Fig. 2: particle B, v = 0.886c Particles A and B both have a frequency = 6; 5tp in the electric wave-state then 1tp in the Planck mass pointstate. As the A point-state occurs once every 6tp, mass of A mA = mP /6, however the point-state of B occurs after 3tp along the A time-line and so mB = mP /3 (fig.3). Fig. 3: particle B, relative mass As each step on the time axis involves a 1tp step, there are 6 possible velocity solutions, this also means that mB can attain mP , but B (v = vmax, mB = mP , fig.4) can never attain the (horizontal axis) velocity c. Fig. 4: particle B, maximum velocity The vertical axis would be measured as 1/γ. For a particle that has only 6 divisions (6 steps from point to point), the maximum γ = 6. To determine the maximum velocity that a particle can attain (y-axis = v/c) we simply calculate when that particle will have reached Planck mass, because from there it can go no faster. A small particle such as an electron has more divisions and so a higher γ and so can go faster in 3-D space than a larger particle such as a proton with a smaller γ (a smaller number of divisions). 1 γ = √ 1− v 2 c2 (1) γelectron = mP /me, γproton = mP /mp 2.3. Hyper-sphere 2.3.1. Replacing the above with a 4-axis co-ordinate system, to illustrate are shown (h, x) axis with particles represented as semi-circles (cross-section). Depicted is particle B at some arbitrary universe time t. B begins at origin O and is pulled along by the hyper-sphere pilot wave expansion (fig.5, 6, 7). Fig. 5: t = 1 2 2 Relativity (relativity and gravity in a Planck universe) Fig. 6: t = 2 Fig. 7: t = 5 At t= 6, B collapses into the mass point state and has defined co-ordinates within the hypersphere (fig.8) which then becomes the new origin O', the above repeating ad infinitum t = 7, 8, . . . (fig.9, 10). Fig. 8: t = 6, point-state Fig. 9: t = 6+1 Fig. 10: t = 6+2 The process also repeats for A (fig.11). The universe hypersphere itself is then analogous to a particle presently in the wave-state whose origin O was the big bang. Fig. 11: Origin points; A, B 2.3.2. In the space-time diagram (fig.3) was depicted for A; v = 0, mA = mP /6 and for B; v = 0.866c,mB = mP /3. However in the (h, x) graphs we find that as A and B have the same frequency, f = 6, the lengths OA = OB = 6, this is because the hyper-sphere expands radially. As a consequence B can rightly claim that it is A whose velocity is at v = 0.866c and for B velocity v = 0 (fig.12). Fig. 12: relative mass B to A Both A and B are traveling at the speed of expansion (which translates to c) from the origin O. In the virtual coordinate system everything travels at, and only at, the speed of expansion as this is the origin of all motion, particles and planets do not have any inherent motion of their own, they are simply pulled along by this expansion. After 1 second both A and B will therefore have traveled the equivalent of 299792458m in virtual co-ordinates 3 2 Relativity (relativity and gravity in a Planck universe) from origin O (fig.13). Each of the 11 depicted solutions are equally valid as the radii are the same. Fig. 13: radial expansion 2.3.3. Particles are assigned an N-S spin axis (fig.14). As the universe expands, it stretches particle A (position and motion of the wave-state are undefined). When t = 6, the wave-state collapses to the defined point-state, as determined by the N. This means that of all the possible solutions, it is the particle N-S axis which determines where the point-state will actually occur, with the hypersphere acting as a pilot-wave. Fig. 14: N-S axis; A v = 0, B v = 0.886c Thus if we can change the N-S axis angle of B compared to A, then as the universe expands the B wavestate will be stretched as with A. But the point of collapse will now reflect the new N-S axis angle. B does not need to have an independent motion; B is simply being dragged by the universe in a different direction as the universe expands. We can thus simulate a transfer of physical momentum to B by simply changing the N-S axis. The radial universe expansion does the rest. 2.3.4. Information between particles is exchanged by photons. Photons do not have a mass point-state, only a wave-state and so have no means to travel the time-line axis (they are 'time-stamped', i.e.: a photon reaching us from the sun is 8 mins old). Instead they travel horizontally (and thus at the speed of light in 3-D space). The period required for particles to emit and to absorb photons is proportional to wavelength. In the following diagram (fig.15) A emits a photon. B travels towards A, as such it will take B less time to absorb that photon than if B was parallel to or moving away from A. If the xaxis length x = v/c, then the h-axis length h = √ 12–x2 and the common relativistic Doppler equation can be written; Fig. 15: Doppler shift vobserved = vsource. √ 1− v2c2 1− vc = vsource. h 1− x (2) Ewave = hv applies to both particle and photon wave states but Emass applies only to the particle mass pointstate. For each particle oscillation there is 1 Planckenergy wave-state followed by 1 Planck-mass point-state; and thus Ewave = Emass, however as particle mass is the average frequency of occurrence of units of Planck mass then the formula E = mc2 in the context of this model cannot be used as E = mc2 requires particles to have a constant property defined as mass [2]. 2.3.5. Returning to our ABC particles, if photons (information) can only be exchanged along the horizontal axis which are the (x, y, z) axis, ABC will only 'see' this horizontal information if ABC relies on the electromagnetic spectrum. Instead of virtual co-ordinates OA, OB and OC and a constant time and velocity, the (x, y, z) axis will be able to measure only the horizontal AB, BC and AC (fig.16) as a 3-D space. Fig. 16: 3-axis hyper-sphere surface As for ABC there is no 'depth' (time-line axis) perception, particle space will appear as a 3D surface phe4 2 Relativity (relativity and gravity in a Planck universe) nomena of a hyper-sphere that has a dimension-less interior leading to the singularity paradox. Furthermore time for ABC translates as motion, if there is no motion in the (x, y, z) axis there will be no means to measure time, thus although the dimension of time for the 3-D space ABC world derives from the expansion of the universe (the universe clock-rate, as measured in units of Planck time) and may equate to universe time, it is actually a measure of particle motion (a change of information states). 3 Gravitational Orbitals As particles are treated as oscillations between an electric wave-state (the particle frequency) to a discrete unit of Planck-mass (at unit Planck-time) mass point-state, mass is not treated as a constant property of the particle, consequently for objects whose mass is less than Planck mass there will be units of Planck time when the object has no particles in the point-state and so no mass. Gravity is treated as a (unit of) Planck-mass to (unit of) Planck-mass (particle to particle) interaction and so is also not a constant property but rather a discrete event, the magnitude of the gravitational interaction per unit time approximating the magnitude of the strong force, the gravitational coupling constant representing a measure of the frequency of these interactions and not the magnitude of the gravitational force itself. Each particle that is in the mass point-state per unit of Planck time is linked to every other particle simultaneously in the mass point-state by a unit of Planck mass mP , velocity v 2 and distance r whereby mP v 2r = ~c (defined here as a gravitational orbital). The velocity of a gravitational orbit is summed from these individual particle-particle v2. Orbital angular momentum of the planetary orbits derives from the sum of the planet-sun particle-particle orbital angular momentum irrespective of the angular momentum of the sun itself and the rotational angular momentum of a planet includes particle-particle rotational angular momentum. As orbits are the result of summed particle-particle orbitals, information regarding macro orbiting objects is not required. 3.1 Gravitational coupling constant The gravitational coupling constant αG characterizes the gravitational attraction between a given pair of elementary particles in terms of the electron mass to Planck mass ratio; αG = Gm2e ~c = m2e m2P = 1.75...x10−45 (3) 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 t, satellite (A) will have 1kg/mP = 45.9 x10 6 particles in the point-state. The earth (B) will have 5.97 x1024kg/mP = 0.274 x10 33 particles in the point-state. For any given unit of Planck time the number of links between the earth and the satellite will sum to; Nlinks = mAmB m2P = 0.126 x1041 (4) If A and B are respectively Planck mass particles then N = 1. If A and B are respectively electrons then the probability that any 2 electrons are simultaneously in the mass point-state for any chosen unit of Planck time becomes N = αG and so a gravitational interaction between these 2 electrons will occur only once every 1045 units of Planck time. 3.2 Planck unit formulas (inverse) fine structure constant α = 137.03599... np = pixel number (fig. 17, 18) λobject = Schwarzschild radius / particle wavelength Fig. 17: alpha pixel, np = 1 Fig. 18: lattice geometry, np = 2 d = √ 2αnp (5) distance between 2 rotating point mass r = d2lp = 2αnp 2lp (6) orbital velocity v = c d = c√ 2αnp (7) orbital period Between objects A and B (mass A >> B) 5 3 Gravitational Orbitals (relativity and gravity in a Planck universe) np ∗ = average np (average of all particle to particle links between A and B) Npoints = number of particles in the Planck mass pointstate per unit of Planck time Npoints = MA mP (8) ng = np ∗√ Npoints (9) converting to wavelength rg = 2α(n ∗ p) 2lp = Npoints2αn 2 glp = αn 2 gλM (10) gravitational orbit velocity from summed point velocities vg = √ Npointsv = c√ 2αng (11) gravitational acceleration ag = v2g rg (12) gravitational orbital period Tg = 2πrg vg (13) orbital angular momentum Loam = Nlinksng h 2π √ 2α (14) rotational angular momentum Lram = ( 2 5 )Nlinks nrot h 2π (15) 3.2.1 Example Earth orbits Npoints = Mearth mP (16) Earth surface orbits rg = 6371.0 km ag = 9.820 m/s 2 Tg = 5060.837 s vg = 7909.792 m/s Geosynchronous orbit rg = 42164.0 km ag = 0.2242 m/s 2 Tg = 86163.6 s vg = 3074.666 m/s Moon orbit (d = 84600s) rg = 384400 km ag = .0026976 m/s 2 Tg = 27.4519 d vg = 1.0183 km/s 3.2.2. Example Planetary orbits Npoints = Msun mP (17) mercury rg = 57 (10 9)m,Tg = 87.969d, vg = 47.87km/s venus rg = 108 (10 9m,Tg = 224.698d, vg = 35.02km/s earth rg = 149 (10 9)m,Tg = 365.26d, vg = 29.78km/s mars rg = 227 (10 9)m,Tg = 686.97d, vg = 24.13km/s jupiter rg = 778 (10 9)m,Tg = 4336.7d, vg = 13.06km/s pluto rg = 5.9 (10 12)m,Tg = 90613.4d, vg = 4.74km/s The energy required to lift a 1kg satellite into a geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). Eorbital = hc 2πr6371 − hc 2πr42164 (18) Nlinks = (Mearthmsatellite)/m 2 P = 0.126x10 41 Etotal = EorbitalNlinks = 53MJ/kg 3.3. Angular momentum Nsun = Msun mP (19) Nplanet = Mplanet mP (20) Nlinks = NsunNplanet (21) 3.3.1 Orbital angular momentum Loam Loam = 2π Mr2 T = NsunNplanet h 2π √ 2α Nsun np = Nlinks ng h 2π √ 2α, kgm2 s (22) Orbital angular momentum of the planets are independent of the orbital angular momentum of the sun. 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 6 3 Gravitational Orbitals (relativity and gravity in a Planck universe) separation and a corresponding reduction in orbit velocity accordingly. Loamvg = Nlinks hc 2π , kgm3 s2 (23) 3.3.2 Rotational angular momentum Lram Nlinks = (Nplanet) 2 (24) Rotational angular momentum contribution to planet rotation. vrot = √ Npoints c 2αnp = c 2αnrot (25) Trot = 2πr vrot (26) Lram = ( 2 5 ) 2πMr2 T = ( 2 5 )Nlinks nrot h 2π , kgm2 s (27) Earth (r = 6371km, n = 2289.4) Trot = 83847.7s (86400 observed) vrot = 477.8m/s (463.3) Lram = .727 x10 34 kgm 2 s (.705) Mars (r = 3390km, n = 5094.7) Trot = 99208s (88643) vrot = 214.7m/s (240.29) Lram = .187 x10 33 kgm 2 s (.209) Rotational angular momentum combined with vrot Lramvrot = ( 2 5 )Nlinks hc 2π2α , kgm3 s2 (28) Correlation with Bohr atomic orbitals (n as principal quantum number) wavelength of atomic orbital in units of lp; r = 2αn2plp (29) orbital velocity associated with atomic orbital v = c 2αnp (30) fo, a dimensionless orbital function n = np√ fo (31) ra = αn 2fo2lp = αn 2λo (32) va = √ fov = c 2αn (33) mP va 2ra = mP lpc 2 2α = hc 2π2α (34) 3.4. Orbital plane rotation In section 2., objects are pulled along by the expansion of the hyper-sphere irrespective of any motion in 3-D space. As such, while B (satellite) has a circular orbit in 3-D space co-ordinates, it follows a cylindrical orbit (from B′ to B′′) around the A (planet) time-line axis in hyper-sphere co-ordinates. If A is moving with the universe expansion (albeit stationary in 3-D space) then td naturally emerges along the A time-line axis (fig. 19). B is traveling at the speed of light in this cylindrical orbit in incremental pixel steps, but this is obscured when measured along a circular plane. to = 2πrg c = 2π2αn2plp c (35) td = Tg √ 1− v2g c2 = √ T 2g − t2o = to √ 2αn2g − 1 (36) Fig. 19: B's orbit relative to A's time-line axis We can also derive td by combining the velocities of B (vg) and the A-B orbital plane (cylinder). vplane = c 2( √ 2α)3n3 (37) td = 2πrg (vg + vplane) (38) The ellipticity of the B orbit around A semi-minor axis: b = αl2λsun semi-major axis: a = αn2λsun radius of curvature L: L = b2 a = al4λA n2 (39) 3λA 2L = 3n2 2αl4 (40) Combining L with vplane gives the precession : Tprecession = 4π √ 2αα2nl4λsun 3c (41) 1296000 arc secs: Mercury Tprecession = 3 015 373 yrs Earth Tprecession = 33 763 000 yrs arc secs per 100 years: Mercury = 42.9814 Venus = 8.6248 Earth = 3.8388 7 3 Gravitational Orbitals (relativity and gravity in a Planck universe) Mars = 1.3510 Jupiter = 0.0623 3.5. Fp = Planck force; Fp = mP c 2 lp Ma = mPλa 2lp , mb = mPλb 2lp (42) Fg = MambG R2 = λaλbFp 4R2g = λaλbFp 4α2n4(λa + λb)2 (43) a) Ma = mb Fg = Fp (4αn2) 2 (44) b) Ma >> mb Fg = λbFp (2αn2) 2 λa = mbc 2 2α2n4λa = mbag (45) 3.6 Orbital transition Atomic electron transition is the change of an electron from one energy level to another. The following redefines the Rydberg formula in terms of 'physical' orbitals, where transition is an orbital replacement, the electron plays no role. Consider the Hydrogen Rydberg formula for transition between and initial i and a final f orbit. The incoming photon λR causes the electron to 'jump' from the n = i to n = f orbit. λR = R.( 1 n2i − 1 n2f ) = R n2i − R n2f (46) The above can be interpreted as referring to 2 photons; λR = (+λi)− (+λf ) Let us suppose a region of space between a free proton p+ and a free electron e− which we may define as zero. This region then divides into 2 waves of inverse phase which we may designate as photon (+λ) and anti-photon (−λ) whereby (+λ) + (−λ) = zero The photon (+λ) leaves (at the speed of light), the antiphoton (−λ) however is trapped between the electron and proton and forms a standing wave orbital. Due to the loss of the photon, the energy of (p+ + e− + −λ) < (p+ + e− + 0) and so is stable. Let us define an (n = i) orbital as (−λi). The incoming Rydberg photon λR = (+λi) − (+λf ) arrives in a 2-step process. First the (+λi) adds to the existing (−λi) orbital. Table 1: Age (billions of years) 14.624 Age (units of Planck time) 0.4281 x 1061tp Cold dark matter density 0.21 x 10−26kg.m−3 Radiation density 0.417 x 10−13kg.m−3 Hubble constant 66.86 km/s/Mpc CMB temperature 2.7269K CMB peak frequency 160.2GHz Cosmological constant 1.0137 x 10123 (−λi) + (+λi) = zero The (−λi) orbital is canceled and we revert to the free electron and free proton; p+ + e− + 0 (ionization). However we still have the remaining −(+λf ) from the Rydberg formula. 0− (+λf ) = (−λf ) From this wave addition followed by subtraction we have replaced the n = i orbital with an n = f orbital. The electron has not moved (there was no transition from an ni to nf orbital), however the electron region (boundary) is now determined by the new n = f orbital (−λf ). 4 Planck unit black-hole A micro black-hole 'Planck unit pixel' (fig. 22) is defined here as a discrete entity that embodies the Planck units. For a discussion of these units as geometrical objects refer 6.2, 6.3. The simulation begins with a single pixel, time tage = 1. A second pixel is added, tage = 2 and so on ... tage as the clock rate of the simulation and measured in units of Planck time tp, the sum black-hole (the sum of these pixels) growing in these Planck steps accordingly. The only variable required is tage. Table 1., gives the parameters for a black-hole universe; age = 14.6 billion years (the peak frequency 160.200 GHz was used as reference to obtain the value for tage). The velocity of the universe expansion is constant and is the origin of the speed of light. It is also this expansion that gives the omni-directional (forward) arrow of time. When the black-hole has reached the limit of its expansion (when it is 1 Planck step above absolute zero), the simulation clock will stop. If the clock reverses, the above will reverse, the black-hole universe shrinking pixel by pixel accordingly. 4.1. Mass density Assume that for each expansion step, to the sum black-hole is added a Planck unit pixel = unit of Planck time tp, Planck mass mP and Planck (spherical) volume (Planck length = lp), such that we can calculate the 8 4 Planck unit black-hole (relativity and gravity in a Planck universe) mass, volume and so density of this black-hole for any chosen time by setting tage; the age of the black-hole as measured in units of Planck time and tsec the age of the black-hole as measured in seconds. tp = 2lp c (s) mass : mbh = 2tagemP (kg) volume : vbh = 4πr 3/3, r = 4lptage = 2ctsec (m) mbh vbh = 2tagemP . 3 4π(4lptage) 3 = 3mP 27πt2agel 3 p ( kg m3 ) (47) Via the Friedman equation, replacing p with the above mass density formula, √ λ = r = 2ctsec reduces to the black-hole radius (G = c2lp/mP ); λ = 3c2 8πGp = 4c2t2sec (48) 4.2. Temperature Measured in terms of Planck temperature = TP ; Tbh = TP 8π √ tage (49) The mass/volume formula uses t2age, the temperature formula uses √ tage. We may therefore eliminate the age variable tage and combine both formulas into a single constant of proportionality that resembles the radiation density constant. Tp = mP c 2 kB = √ hc5 2πGkB 2 (50) mbh vbhT 4bh = 253π3mP l3pT 4 P = 283π6k4B h3c5 (51) 4.3. Radiation density From Stefan Boltzmann constant σSB σSB = 2π5k4B 15h3c2 (52) 4σSB c .T 4bh = c2 1440π . mbh vbh (53) 4.4. Casimir The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them, where dc2lp = distance between plates in units of Planck length; −Fc A = πhc 480(dc2lp) 4 (54) if dc = 2π √ tage then eq.53 = eq.55, equating the Casimir force with the background radiation energy density. −Fc A = c2 1440π . mbh vbh (55) fig.20 plots Casimir length dc2lp against radiation energy density pressure measured in mPa for different tage with a vertex around 1Pa, fig.21 plots temperature Tbh. A radiation energy density pressure of 1Pa gives tage ∼ 0.8743 1054tp (2987 years), length = 189.89nm and temperature Tbh = 6034 K . Fig. 20: y-axis = mPa, x-axis = dc2lp (nm) Fig. 21: y-axis = mPa, x-axis = Tbh (K) 4.5. Hubble constant 1 Mpc = 3.08567758 x 1022m. H = 1Mpc tagetp (56) 4.6. Black body peak frequency xex ex − 1 − 3 = 0, x = 2.821439... (57) fpeak = kBTbhx h = x 8π2 √ tagetp (58) 4.6. Entropy SBH = 4πtage 2kB (59) 4.7. Cosmological constant Riess and Perlmutter (notes) using Type 1a supernovae calculated the end of the universe tend ∼ 1.7 x 10−121 ∼ 0.588 x 10121 units of Planck time; tend ∼ 0.588x10121 (60) 9 4 Planck unit black-hole (relativity and gravity in a Planck universe) The maximum temperature Tmax would be when tage = 1. What is of equal importance is the minimum possible temperature Tmin that temperature 1 Planck unit above absolute zero, for in the context of this model, this temperature would signify the limit of expansion (the black-hole could expand no further). For example, if we simply set the minimum temperature as the inverse of the maximum temperature; Tmin ∼ 1 Tmax ∼ 8π TP ∼ 0.177 10−30 K (61) This would then give us a value 'the end' in units of Planck time (∼ 0.35 1073 yrs) which is close to Riess and Perlmutter; tend = T 4 max ∼ 1.014 10123 (62) The mid way point (Tmid = 1K) becomes T 2max ∼ 3.18 1061 ∼ 108.77 billion years. 4.8. Spiral By expanding according to a Theodorus spiral pattern (fig. 22) the universe can rotate with respect to itself differentiating between an L and R universe without recourse to an external reference. The integer dimensions (mass, volume) follow a linear progression (spiral circumference), the radiation components a sqrt progression (spiral arm). tage = number of pixels (see also fig.18). Fig. 22: spiral lattice geometry 5 Comments The Mathematical Universe Hypothesis states that our physical world is a mathematical structure [3]. In this paper and the paper on the mathematical electron [1] I have described a simulation hypothesis method that may reconcile the mathematical universe with the physical universe. The principal assumption being that our universe operates at the Planck level and at this level mass, space and time are mathematical structures. All events occur at the Planck level, the quantum level is an averaging of these events as the macro world is a statistical averaging of the quantum world. The physical world has definable co-ordinates. It is the geometries of the particles that naturally result in orbits, half-life (see notes) etc. ..., the universe guided by geometrical imperatives rather than abstract laws, and as these motions follow repeating patterns they can be described using mathematical formulas (the laws of Physics). Notes: 6.1 Half-life: We drop coffee cups, they break only when landing on the handle (the fracture point), an event which occurs on average every 16 drops. If we start with 16 cups and drop them simultaneously, pick up the remaining unbroken cups, drop and repeat until all cups are broken, then we will derive the half-life formula. If particles have a geometrical structure and this structure has 1 or more fracture points then a half-life will emerge naturally from that geometry. Conversely the electron formula (eq. 62) suggests the electron is perfectly symmetrical and so has no fracture points and so it could have a quark substructure [1] but this would not be detected as the quarks would be identical and the electron structure immutable. 6.2 The Planck units as geometrical objects and the relationships between them [1], fig.23. Fig. 23: Planck units as geometrical objects 6.3 The formula for the mathematical electron fe (α = inverse fine structure constant); fe = 4π 2r3 (r = 263π2αΩ5), units = (AL)3 T = 1 (63) 6.4 Hypersphere: The wavelength (oscillation cycle) derives from felp (in Planck units along the universe ex10 5 Comments (relativity and gravity in a Planck universe) pansion time-line). For example, the wavelength of a sin wave from the circumference of a circle 2πr (fig.24). Fig. 24: sine wave A (2D) sin wave from the (dimensionless) surface area of a sphere 4πr2 (fig.25). Fig. 25: 2D sine wave A (3D) sin wave from the (dimensionless) surface area of a 4-axis hyper-sphere 2π2r3 (fig.26). Fig. 26: 3D sine wave However fe = 4π 2r3 (units = 1) is equivalent to 2 wavelengths, this can be resolved by rotating the 4D electron sphere such that the sine wave returns to the original position after 720 degrees. References 1. Macleod, Malcolm J., Programming Planck units from a virtual electron; a Simulation Hypothesis Eur. Phys. J. Plus (2018) 133: 278 2. Macleod, Malcolm J., Source Code of the Gods, online edition http://platoscode.com/ 3. M. Tegmark 2014, Our Mathematical Universe, Knopf 11 5 Comments