# The Planck Units And The Sub-Planck Units

The four base Planck units are the smallest units of measurement as calculated by Max Planck. These are physical units measuring length, time, mass and electric charge. In the excerpt below from the recently published book on Amazon “Universe Binary Constructors- Strings & Singularities” we explain how Max Planck derived their values and why there are many conundrums in physics that indicate the existence of Sub-Planck units. For example, the Planck mass is too large and no subatomic particles could be found with that level of mass. To appreciate the excerpt below you need to keep in mind that at the fundamental level all subatomic particles are constructed of the proposed two hidden particles with binary properties. For more detail, you are referred to chapters 2 and 3 of the book. The book also assumes that the vacuum is the positive,y curved platform in which the physics universe manifests itself. It is constructed of Spacetime Particles (SPs), which are in turn composed of the two hidden particles. The maths below shows how Max Planck calculated the Planck units and why they are larger than the Sub-Planck units by a value equal approximately to 10¹⁷.

# 4.3) Sub-Planck units of measurements

To appreciate the roles of the SPs in the working of the Universe and deriving the laws of physics from the first principle, we need to introduce the relevant sub-Planck base units associated with them. We will show in this section that the currently calculated Planck mass is approximately more than the Higgs boson mass by a factor of 10¹⁷. This divergence is due to ignoring the required escape velocity of standard packets of coupled strings as they get emitted by electromagnetic fields.

# ▪️The hierarchy problem

A hierarchy problem occurs when a fundamental value of a physical parameter, such as a coupling constant or a mass, is vastly different from the value measured in an experiment. The Renormalization process is usually employed to correct the difference. Typically the renormalized values of parameters are close to their fundamental values, but in some cases, there is a vast difference, hence the term “Hierarchy problem”.

Our suggestions regarding the SPs base units provide a possible answer to the puzzling hierarchy problem associated with the links between the Higgs field and the mass-energy scale used in calculating the gravitational constant. The magnitude of the puzzling difference is estimated to be 10¹⁷ approximately.

# ▪️sub-Planck base units of measurements

*The four sub-Planck base units associated with a vacuum spacetime particle are:

- 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

A large number of spacetime particles’ strings construct standard packets of coupled strings representing a Planck constant. They are emitted by the interacting virtual electrons and positrons (that construct the electromagnetic fields) once their aggregate mass reaches escape velocity level (m₀c₀²). On reaching m₀c₀², the radius of the spacetime particle expands by a factor of c₀² in line with the centripetal force law:

- 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:

a) The uncertainty between energy and mass (known in physics as the width problem) due to the Higgs boson being very unstable, hence its large mass shift. A shift in mass means that it loses some of its energy before measurement by the currently available technology. A Higgs boson means lifetime is estimated to be 1.6 10-²², while the fastest measuring device is 10-¹⁸ second.

b) The value of a Higgs boson mass is calculated within Earth’s gravitational field. The measured Higgs boson would have contributed some of their energy in the process of creating Earth mass.

c) Part of the SPs strings (used as the vacuum gluing force, the proposed fifth force) are not included in the Higgs boson mass.

# ▪️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.