definition 38.6 Minkowski metric and interval

open in the book · parts/04-special-relativity/02-lorentz-transformations.tex:223 · p. 1208

ground object -- no derivation owed

Rests on

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 ¶

Supports

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 proposition 44.37 Energy conditions for the perfect fluid ¶
      - depends_on proposition 44.20 Perfect-fluid equations of motion ¶
      - depends_on theorem 49.9 Tolman–Oppenheimer–Volkoff equations ¶
    - 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 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 ¶
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 theorem 109.44 Jost ¶
      - depends_on theorem 109.49 Spin and statistics ¶
  - 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 remark 38.24 Boosts alone do not form a group ¶
  - depends_on theorem 105.34 $CPT$ ¶
    - assumes experiment CP Violation ¶
      - cites (data) derived bib 22 First Observation ofCPViolation in the Decays ofBs0Mesons ¶
      - cites (data) derived bib 22 First Observation ofCPViolation in the Decays ofBs0Mesons ¶ ↺
      - cites (evidence) derived bib 22 First Observation ofCPViolation in the Decays ofBs0Mesons ¶ ↺
      - cites (evidence) derived bib 22 First Observation ofCPViolation in the Decays ofBs0Mesons ¶ ↺
      - cites (evidence) derived bib 21 Observation of CP Violation in Charm Decays ¶
      - cites (data) derived bib 21 Observation of CP Violation in Charm Decays ¶ ↺
      - cites (evidence) derived bib 21 Observation of CP Violation in Charm Decays ¶ ↺
      - cites (evidence) derived bib 9 Observation of LargeCPViolation in the NeutralBMeson System ¶
      - cites (data) derived bib 9 Observation of LargeCPViolation in the NeutralBMeson System ¶ ↺
      - cites (data) derived bib 9 Observation of LargeCPViolation in the NeutralBMeson System ¶ ↺
      - cites (evidence) derived bib 9 Observation of LargeCPViolation in the NeutralBMeson System ¶ ↺
      - cites (data) derived bib 8 Measurement of the Permanent Electric Dipole Moment of the Neutron ¶
      - … 109 more
    - 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 ¶
      - assumes experiment CP Violation ¶ ↺
      - depends_on proposition 105.39 A direct $CP$ asymmetry needs two amplitudes and two kinds of phase ¶
    - depends_on theorem 105.36 Equal masses ¶
      - assumes experiment CP Violation ¶ ↺
      - depends_on phenomenon 111.17 The universe is made of matter ¶
    - depends_on theorem 105.41 Sakharov's conditions ¶
      - depends_on phenomenon 111.17 The universe is made of matter ¶ ↺
      - depends_on phenomenon 103.44 The universe contains matter and almost no antimatter ¶
  - 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 ¶
      - tests experiment Dyson, Eddington and Davidson: the eclipse of 1919 ¶
    - depends_on phenomenon 53.1 Measured anomalous advance of Mercury's perihelion ¶
      - tests experiment Le Verrier and the perihelion of Mercury ¶
    - 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 phenomenon 45.24 Compact objects exist above every cold-support limit ¶
    - depends_on theorem 45.23 Oppenheimer–Snyder collapse ¶
      - depends_on phenomenon 45.24 Compact objects exist above every cold-support limit ¶ ↺
- depends_on theorem 40.5 Energy–momentum relation ¶ ↺

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`	→	The invariant interval	declared	parts/04-special-relativity/02-lorentz-transformations.tex:235
`depends_on`	←	Causal classification	declared	parts/04-special-relativity/02-lorentz-transformations.tex:288
`depends_on`	←	Proper time	declared	parts/04-special-relativity/02-lorentz-transformations.tex:514
`depends_on`	←	Lorentz group	declared	parts/04-special-relativity/03-minkowski-lorentz-poincare.tex:65
`depends_on`	←	Four-vector	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:27
`depends_on`	←	Differential ageing	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:334
`depends_on`	←	Normalization of the four-velocity	declared	parts/04-special-relativity/04-relativistic-dynamics.tex:54