MPP

Mħπ Constants Hierarchy

This document provides a visual representation of the constant hierarchy in the Mħπ system, showing the relationships between primitive and derived constants.

Constant Hierarchy Diagram

graph TD
    %% Primitive Constants
    Pi[π - Pi\nDimensionless]
    P[P - Planck Length\nLength]
    Tau[τ - Planck Time\nTime]
    Hbar[ℏ - Reduced Planck Constant\nEnergy·Time]
    G[G - Gravitational Constant\nL³/MT²]
    K[k - Boltzmann Constant\nEnergy/Temperature]
    E[e - Elementary Charge\nCharge]

    %% Derived Constants
    C[c - Speed of Light\nVelocity]
    Delta[Δ - Planck Time Unit\nTime]

    %% Relationships
    P --> C
    Tau --> C
    C --> Delta
    P --> Delta

    %% Relationship Labels
    Tau -- "c = P/τ" --> C
    P -- "c = P/τ" --> C
    C -- "Δ = P/c" --> Delta
    P -- "Δ = P/c" --> Delta

    %% Styling
    classDef primitive fill:#d4f1f9,stroke:#05386B,stroke-width:2px
    classDef derived fill:#ffedcc,stroke:#05386B,stroke-width:2px

    class Pi,P,Tau,Hbar,G,K,E primitive
    class C,Delta derived

Constant Relationships

Key Identities

  1. Speed of Light:

    c = P / τ

    The speed of light is defined as the Planck length divided by the Planck time.

  2. Planck Time Unit:

    Δ = P / c

    The Planck Time Unit is defined as the Planck length divided by the speed of light.

  3. Derived Identity:

    Δ = τ

    From the above two equations, we can derive that Δ = τ. In the Mħπ system, these are maintained as distinct symbols for notational flexibility, but they represent the same physical quantity.

Dimensional Analysis

The following table shows the dimensions of each constant in the Mħπ system:

Constant Symbol Dimensions Type
Pi π Dimensionless Primitive
Planck Length P Length (L) Primitive
Planck Time τ Time (T) Primitive
Reduced Planck Constant ℏ Energy·Time (ML²/T) Primitive
Gravitational Constant G L³/(M·T²) Primitive
Boltzmann Constant k Energy/Temperature (ML²/T²Θ) Primitive
Elementary Charge e Electric Charge (Q) Primitive
Speed of Light c Velocity (L/T) Derived
Planck Time Unit Δ Time (T) Derived

Natural Units Systems

In natural units systems, certain constants are set to 1:

  1. Natural Units (ℏ=c=1):

    • Reduced Planck constant (ℏ) = 1
    • Speed of light (c) = 1
    • This implies P = τ

  2. Planck Units (ℏ=c=G=k=1):

    • Reduced Planck constant (ℏ) = 1
    • Speed of light (c) = 1
    • Gravitational constant (G) = 1
    • Boltzmann constant (k) = 1
    • This implies P = τ and additional relationships between mass, energy, and temperature

Implementation Notes

In the Mħπ implementation:

  1. The simplifier enforces the relationships c · τ = P and c · Δ = P by automatically simplifying these expressions to P.

  2. The dimensional analysis system ensures that all constants have the correct physical dimensions and that operations maintain dimensional consistency.

  3. The symbols τ and Δ are maintained as distinct for notational flexibility, but they represent the same physical quantity in the current implementation.

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