`dim_T`

definition

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module: IndisputableMonolith.Constants.Dimensions
domain: Constants
line: 48 · github
papers citing: none yet

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IndisputableMonolith.Constants.Dimensions on GitHub at line 48.

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depends on

formal source

  45def dim_L : Dimension := ⟨1, 0, 0⟩
  46
  47/-- Time dimension: [L⁰T¹M⁰] -/
  48def dim_T : Dimension := ⟨0, 1, 0⟩
  49
  50/-- Mass dimension: [L⁰T⁰M¹] -/
  51def dim_M : Dimension := ⟨0, 0, 1⟩
  52
  53/-! ## Physical Constant Dimensions -/
  54
  55/-- Speed of light dimension: [L¹T⁻¹M⁰] -/
  56def dim_c : Dimension := ⟨1, -1, 0⟩
  57
  58/-- Reduced Planck constant dimension: [L²T⁻¹M¹] -/
  59def dim_hbar : Dimension := ⟨2, -1, 1⟩
  60
  61/-- Gravitational constant dimension: [L³T⁻²M⁻¹] -/
  62def dim_G : Dimension := ⟨3, -2, -1⟩
  63
  64/-! ## Dimensional Consistency (Documentation)
  65
  66The Planck length formula has correct dimensions:
  67  ℓ_P = √(ℏG/c³)
  68  [ℏG/c³] = [L²T⁻¹M¹][L³T⁻²M⁻¹]/[L³T⁻³M⁰]
  69          = [L⁵T⁻³M⁰]/[L³T⁻³M⁰]
  70          = [L²T⁰M⁰]
  71  [√(ℏG/c³)] = [L¹] ✓
  72
  73The Planck time formula has correct dimensions:
  74  t_P = √(ℏG/c⁵)
  75  [ℏG/c⁵] = [L²T⁻¹M¹][L³T⁻²M⁻¹]/[L⁵T⁻⁵M⁰]
  76          = [L⁵T⁻³M⁰]/[L⁵T⁻⁵M⁰]
  77          = [L⁰T²M⁰]
  78  [√(ℏG/c⁵)] = [T¹] ✓

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