definition 38.12 Proper time

open in the book · parts/04-special-relativity/02-lorentz-transformations.tex:503 · p. 1213

ground object -- no derivation owed

Rests on

depends_on definition 38.6 Minkowski metric and interval ¶
- depends_on proposition 38.5 The invariant interval ¶
  - 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 ¶
depends_on proposition 38.5 The invariant interval ¶ ↺

Supports

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 proposition 44.37 Energy conditions for the perfect fluid ¶
    - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      - depends_on corollary 44.21 Dust flows on geodesics ¶
    - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
      - depends_on example 49.11 The uniform-density star ¶
      - depends_on phenomenon 49.19 Neutron stars reach two solar masses ¶
  - depends_on theorem 44.27 Newtonian correspondence of the field equations ¶
    - depends_on corollary 44.35 The field equations are essentially unique ¶
    - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶
    - depends_on phenomenon 49.16 Pulsars: clocks at nuclear density ¶
    - depends_on phenomenon 49.14 A neutron star is observed being born ¶
    - depends_on proposition 44.29 Linearized 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 example 40.13 Threshold energy for pair production ¶
  - depends_on phenomenon 99.3 Electron–positron pair creation ¶
  - depends_on postulate 40.11 Conservation of four-momentum ¶
    - depends_on example 40.14 Compton scattering ¶
    - depends_on example 40.13 Threshold energy for pair production ¶ ↺
  - depends_on theorem 40.5 Energy–momentum relation ¶
    - depends_on example 40.14 Compton scattering ¶ ↺
    - depends_on phenomenon 99.3 Electron–positron pair creation ¶ ↺
    - depends_on proposition 26.29 Reparametrization invariance and the mass-shell constraint ¶
    - depends_on proposition 49.2 Ground-state pressure of the ideal Fermi gas ¶
      - depends_on phenomenon 49.6 A maximum mass for a cold degenerate star ¶ ↺
      - depends_on phenomenon 49.8 The white-dwarf luminosity function has an edge ¶
      - depends_on phenomenon 49.1 Matter at a billion kilograms per cubic metre ¶
      - depends_on theorem 49.7 The Chandrasekhar mass ¶
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 ¶
    - cites (evidence) derived bib Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole ¶
    - cites (evidence) derived bib Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole ¶ ↺
    - cites (evidence) derived bib 7897 Resolving the gravitational redshift across a millimetre-scale atomic sample ¶
    - cites (evidence) derived bib 7897 Resolving the gravitational redshift across a millimetre-scale atomic sample ¶ ↺
    - cites (evidence) derived bib 5999 Optical Clocks and Relativity ¶
    - cites (evidence) derived bib 5999 Optical Clocks and Relativity ¶ ↺
    - cites (method) derived bib Kernresonanzfluoreszenz von Gammastrahlung in Ir$^{191}$ ¶
    - cites (method) derived bib Kernresonanzfluoreszenz von Gammastrahlung in Ir$^{191}$ ¶ ↺
    - cites (method) derived bib 7 Apparent Weight of Photons ¶
    - cites (evidence) derived bib 7 Apparent Weight of Photons ¶ ↺
    - cites (method) derived bib 3B Effect of Gravity on Gamma Radiation ¶
    - cites (evidence) derived bib 3B Effect of Gravity on Gamma Radiation ¶ ↺
    - … 2 more
- 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 ¶

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`	→	Minkowski metric and interval	declared	parts/04-special-relativity/02-lorentz-transformations.tex:514
`depends_on`	→	The invariant interval	declared	parts/04-special-relativity/02-lorentz-transformations.tex:514
`assumes`	←	Frisch and Smith: Mount Washington to sea level	declared	parts/04-special-relativity/05-exp-time-dilation.tex:182
`assumes`	←	Ives and Stilwell: the moving atomic clock	declared	parts/04-special-relativity/05-exp-time-dilation.tex:22
`assumes`	←	Rossi and Hall: muon survival with altitude	declared	parts/04-special-relativity/05-exp-time-dilation.tex:119
`depends_on`	←	Four-force	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:180
`depends_on`	←	Four-velocity	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:43
`depends_on`	←	Time dilation	declared	parts/04-special-relativity/02-lorentz-transformations.tex:524
`depends_on`	←	Differential ageing	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:334