proposition 38.5 The invariant interval

open in the book · parts/04-special-relativity/02-lorentz-transformations.tex:198 · p. 1207

Rests on

depends_on equation 38.1 eq:lor-beta-gamma ¶
depends_on theorem 38.4 Lorentz boost ¶
- depends_on notation 38.3 Standard configuration ¶
- depends_on postulate 38.2 Invariance of the speed of light ¶
- depends_on postulate 38.1 Principle of relativity ¶
- proves proof ch:02-lorentz-transformations@proof-1 ¶
proves proof ch:02-lorentz-transformations@proof-2 ¶

Supports

depends_on definition 38.6 Minkowski metric and interval ¶
- depends_on definition 38.8 Causal classification ¶
  - depends_on proposition 38.9 Causal order is absolute ¶
- depends_on definition 38.12 Proper time ¶
  - assumes experiment Frisch and Smith: Mount Washington to sea level ¶
    - cites (evidence) derived bib 5 Measurement of the Relativistic Time Dilation Using $\mu$-Mesons ¶
  - assumes experiment Ives and Stilwell: the moving atomic clock ¶
    - cites (evidence) derived bib 7 An Experimental Study of the Rate of a Moving Atomic Clock ¶
  - assumes experiment Rossi and Hall: muon survival with altitude ¶
    - cites (evidence) derived bib 3 Variation of the Rate of Decay of Mesotrons with Momentum ¶
  - depends_on definition 40.9 Four-force ¶
  - depends_on definition 40.2 Four-velocity ¶
    - depends_on definition 44.15 Dust ¶
      - depends_on definition 44.16 Perfect fluid ¶
      - depends_on theorem 44.27 Newtonian correspondence of the field equations ¶
    - depends_on definition 40.4 Four-momentum ¶
      - depends_on definition 40.9 Four-force ¶ ↺
      - depends_on definition 40.12 Invariant mass ¶
      - depends_on phenomenon 99.3 Electron–positron pair creation ¶
      - depends_on postulate 40.11 Conservation of four-momentum ¶
      - depends_on theorem 40.5 Energy–momentum relation ¶
  - depends_on phenomenon 38.13 Time dilation ¶
    - depends_on phenomenon 53.3 Gravitational shift of gamma rays over a tower ¶
      - tests experiment Pound, Rebka and Snider: redshift in a tower ¶
    - depends_on phenomenon 42.9 The relativistic clock budget of satellite navigation ¶
    - depends_on phenomenon 50.3 Relativistic redshift and precession of a stellar orbit ¶
    - depends_on remark 38.14 The effect is reciprocal, and that is consistent ¶
    - tests experiment Frisch and Smith: Mount Washington to sea level ¶ ↺
    - tests experiment Hafele and Keating: caesium clocks flown around the world ¶
      - cites (history) derived bib 4044 Around-the-World Atomic Clocks: Predicted Relativistic Time Gains ¶
      - cites (history) derived bib 4044 Around-the-World Atomic Clocks: Predicted Relativistic Time Gains ¶ ↺
      - cites (evidence) derived bib 4044 Around-the-World Atomic Clocks: Observed Relativistic Time Gains ¶
      - cites (evidence) derived bib 4044 Around-the-World Atomic Clocks: Observed Relativistic Time Gains ¶ ↺
    - tests experiment Ives and Stilwell: the moving atomic clock ¶ ↺
    - tests experiment Rossi and Hall: muon survival with altitude ¶ ↺
  - depends_on phenomenon 40.15 Differential ageing ¶
- depends_on definition 39.1 Lorentz group ¶
  - depends_on definition 39.4 Improper Lorentz transformations ¶
    - depends_on proposition 39.8 Component structure ¶
  - depends_on definition 39.5 Orthochronous Lorentz group ¶
    - depends_on definition 39.7 Antichronous transformations ¶
    - depends_on lemma 109.42 Total inversion is a complex Lorentz transformation ¶
      - depends_on corollary 109.43 Total inversion is automatic ¶
    - depends_on proposition 39.8 Component structure ¶ ↺
    - depends_on proposition 39.6 $L^{\uparrow}$ and $L^{\uparrow}_{+}$ are subgroups ¶
      - depends_on theorem 38.23 Polar decomposition of a Lorentz transformation ¶
    - depends_on theorem 105.34 $CPT$ ¶
      - assumes experiment CP Violation ¶
      - depends_on phenomenon 111.9 Charge asymmetry in semileptonic kaon decay ¶
      - depends_on proposition 105.40 Greenberg ¶
      - depends_on theorem 105.38 Equal total lifetimes ¶
      - depends_on theorem 105.36 Equal masses ¶
      - depends_on theorem 105.41 Sakharov's conditions ¶
    - depends_on theorem 38.23 Polar decomposition of a Lorentz transformation ¶ ↺
  - depends_on definition 39.10 Poincaré group ¶
    - depends_on proposition 39.11 Invariance of the interval ¶
  - depends_on definition 39.3 Proper Lorentz group ¶
  - depends_on proposition 39.9 The generators ¶
  - depends_on proposition 39.2 $L$ is a Lie group ¶
- depends_on definition 40.1 Four-vector ¶
  - depends_on definition 40.2 Four-velocity ¶ ↺
  - depends_on definition 40.12 Invariant mass ¶ ↺
  - depends_on theorem 115.31 Second-order Fermi acceleration ¶
    - depends_on theorem 115.33 The spectrum from a strong shock ¶
- depends_on phenomenon 40.15 Differential ageing ¶ ↺
- depends_on proposition 40.3 Normalization of the four-velocity ¶
  - depends_on proposition 109.74 The vacuum is thermal in a wedge ¶
  - depends_on proposition 45.6 Radial equation and effective potential ¶
    - depends_on proposition 45.10 Circular orbits and their energy ¶
      - depends_on phenomenon 45.12 Efficiency of accretion onto compact objects ¶
    - depends_on proposition 45.8 Orbit equation ¶
      - depends_on phenomenon 53.2 Measured deflection of starlight at the solar limb ¶
      - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - depends_on phenomenon 45.9 Relativistic advance of the periastron ¶
    - depends_on proposition 45.21 Radial plunge: the cycloid ¶
      - depends_on proposition 45.22 The view from outside: exponential freezing ¶
      - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
  - depends_on theorem 40.5 Energy–momentum relation ¶ ↺
depends_on definition 38.12 Proper time ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:lor-beta-gamma	declared	parts/04-special-relativity/02-lorentz-transformations.tex:204
`depends_on`	→	Lorentz boost	declared	parts/04-special-relativity/02-lorentz-transformations.tex:204
`depends_on`	←	Minkowski metric and interval	declared	parts/04-special-relativity/02-lorentz-transformations.tex:235
`depends_on`	←	Proper time	declared	parts/04-special-relativity/02-lorentz-transformations.tex:514
`proves`	←	ch:02-lorentz-transformations@proof-2	declared	parts/04-special-relativity/02-lorentz-transformations.tex:207