The Planck Units And The Sub-Planck Units

4.3) Sub-Planck units of measurements

▪️The hierarchy problem

▪️sub-Planck base units of measurements

  • m₀, the mass of spacetime particle strings, expressed in kilograms, Kg.
  • r₀, the radius of its orbiting strings, expressed in meters, m.
  • t₀, the elapsed time of spacetime particle strings moving at their intrinsic speed of light velocity to complete a distance equal to r₀. Every 2πt₀ represents a proper microsecond.
  • e₀ is a sub-Planck electric charge of six singularities spinning in one direction.

▪️Assumptions made and the relevant laws used

  • To find the values of the sub-Planck base units, we assume that a spacetime particle mass is equal to that of a Higgs boson. Our calculations based on this assumption show that the sub-Planck base units are equal to the Planck base units divided by c₀², which is the square dimensionless speed of light (2.99792458 x 10⁸)² ≈10¹⁷. This value represents the escape velocity associated with the emission of electromagnetic radiation, as will be shown below. In calculating the sub-Planck units, we need to keep in mind the following:

i- A spacetime particle centripetal force (F₀)

  • The centripetal force and centripetal acceleration generated by the 12 singularities of a spacetime particle: *
  • F₀ = m₀c² / r₀
  • α₀ = r₀ / t₀²
  • These two relationships are the most fundamental laws in physics. They explain the quantum mechanics between SPs and between SPs and fermion particles. *

ii- Understanding Max Planck approach in calculating the Planck units

  • The Planck base units are length (lp), time (tp), mass (mp), and e.
  • To calculate the Planck units Max started with some known fundamental constants, namely:
  • Gravitational constant: G = (distance x distance x distance) / (time x time x mass). G = lp³ / **http://tp².mp* (http://tp².mp)*. It is a proportionality constant.
  • Speed of light: c = distance / time. c = lp / tp. It is a proportionality constant.
  • Reduced Planck constant: ħ = mass x distance x distance / time. ħ = mp.lp² / tp. It is not a proportionality constant. The numerator consists of three base units of measurement, and the denominator consists only of one base unit.

Note 1:

  • Proportionality constant values are the same whether we use the Planck units or the proposed sub-Planck units of measurements.

Note 2:

  • Max Planck observed that there is only one way to combine these constants to obtain the value of the Planck distance:
  • lp = √(ħG / c³). Once the Planck distance was calculated, from the known values of ħ, G and c, Planck time and Planck mass were calculated using the relationships:
  • tp = lp / c and
  • mp = lp³ / Gtp²
  • Note 3: The chosen relationship by Max,
  • lp = √(ħG / c³), which is derived from the law that governs the escape velocity of mass in orbital momentum.

▪️Packets of strings. are emitted on reaching escape velocity level

  • a) Escape velocity law: vₑ = √(2Gm / r).
  • Since the velocity of the strings is constant, then the escape velocity of the coupled strings composing a Planck constant can be written as c² = Gmp. / lp. This makes lp = Gmp / c²….. equation (a)
  • b) since ħ =mpc.lp then mp = ħ / clp ……. equation (b)
  • c) From equations (a) and (b) we get:
  • lp = √(ħG/c³). It is the equation used by Maxwell to derive the Planck length.

▪️Emission of a Planck constant

  • F₀ = (m₀c²) / r₀ = (m₀c²)c₀² / r₀c₀²
  • The emitted strings change their momentum from orbital to linear and construct a Planck constant:
  • h = (m₀c)c₀².(2πr₀)c₀² = (m₀c²)c₀².(2πt₀)c₀²
  • h = mpc.2πlp = mpc².2πtp

▪️Frequency of emitted packets of Planck constants

  • The speed at which the standard packets of coupled strings are accumulated and emitted is subject to the level of the moved charged particles and the kinetic energy used to move them. These two factors determine the frequencies of the photons and their energy levels.*
  • h = mpc².2πtp = hf

▪️Calculations of the sub-Planck units

  • Our focus is on calculating the sub-Planck base units associated with a spacetime particle. The value of the base unit of an electric charge is invariant since it represents the constant electrostatic force of singularities. On the other hand, the other three base units, (mass, length, and time) do vary with the changing geometry of a spacetime particle.
  • The four base units of measurements are the only quantities from which all other units are derived. For example:
  • The strings intrinsic speed of light, c = r₀ / t₀
  • The spacetime particle strings momentum, p₀ = m₀c
  • The spacetime particle energy, E₀ = m₀c²
  • The spacetime particle centripetal force, F₀ = E₀ / r₀
  • The spacetime particle acceleration, a₀ = c / t₀
  • The gravitational constant, G = F₀r₀² / m₀², and so on.
  • Below is our approximate calculation of the four fundamental sub-Planck units associated with a spacetime particle, using the Standard. International Units. *

▪️Higgs bosons mass, mH⁰

  • *Physicists are still trying to measure with a high degree of confidence the mass of the Higgs boson. According to a quote from a recent paper:
  • “We compute the Higgs mass in a model for the electroweak interactions based on a confining theory. This model is related to the standard model by the complementarity principle. A dynamical effect due to the large typical scale of the Higgs boson shifts its mass above that of the W- bosons. We obtain mH⁰ = 129.6 GeV/c².”
  • The reported Higgs boson. mass ranges between 125 GeV/c² and 129.6 GeV/c². Our proposed value is 135.31 GeV/c² based on our postulation that the Planck units are equal to the sub-Planck units time. c₀². The discrepancy *between 129.6 and 135.31 GeV/c² is approximately equal to 4.4%. It is accounted for as follows:

▪️Calculating the centripetal force

  • Since a space particle centripetal force, time the gravitational constant is equal to c⁴, then we can calculate F₀ as shown below:
  • F₀ = c⁴ / G = (2.99792458)⁴/ 6.674×10−¹¹ ≈ 1.2103×10⁴⁴ Kg.m.s-². This formulation is in line with the Planck centripetal force, Fp.
  • By applying a Higgs boson mass of 135.31 GeV/c² (or m₀ = 2.42 x 10-²⁵ kg) we calculate the spacetime particle radius:
  • r₀ = m₀c² / F₀ = (2.42 x 10-²⁵ ) x (2.9979 x 10⁸)² / (1.21 × 10⁴⁴) ≈ 1.8 x 10-⁵² m.
  • Therefore, the approximate values of the sub-Planck base units are:
  • r₀ ≈ 1.8 x 10-⁵² m. Distance in Planck units: lp =1.616. x 10-³⁵ m)
  • m₀ ≈ 2.42 x 10-²⁵ Kg. Mass in Planck units: mp = 2.176 x 10-⁸ Kg.
  • t₀ = r₀/c ≈ 0.6 x 10-⁶⁰ s. Time in Planck units: tp = 5.39 x 10-⁴⁴ s.
  • e₀ = e/c₀². e = (8.98755 x 10¹⁶) e₀. *Since e = 1.602*10-¹⁹C then e₀ =1.7825*10-³⁶C. *Measured in (Coulomb) or A.s (Ampere second). The number 6 associated with a unit of an electron charge is taken care of by the vacuum fine structure constant (1/137.036), which keeps popping up in many calculations. More on that later in chapter 21.

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