Lecture 05: General Relativity - Physics & Astronomy - Northeastern ...

NortheasternIllinoisUniversityRelativity VGeneral RelativityGreg AndersonDepartment of Physics & AstronomyNortheastern Illinois UniversityJanuary, 2008c○2004, 2007 G. Anderson Physics III – slide 1 / 37

NortheasternIllinoisUniversityOverviewSpecialRelativityGeneralRelativityCosmologySpecial RelativityGeneral RelativityCosmologyc○2004, 2007 G. Anderson Physics III – slide 2 / 37

NortheasternIllinoisUniversitySpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologyRecap: SpecialRelativityc○2004, 2007 G. Anderson Physics III – slide 3 / 37

NortheasternIllinoisUniversityPostulates of Special RelativitySpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologySpecial Relativity (1905)The Relativity Postulate:The basic laws of physics arethe same in all inertial referenceframes.Albert Einstein (1879-1955) at age 26.Speed of Light Postulate:The speed of light in vacuumhas the same value c in allinertial reference frames.From the Emilio Segré Visual Archivesc○2004, 2007 G. Anderson Physics III – slide 4 / 37

NortheasternIllinoisUniversityThe Lorentz FactorSpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmology9.08.07.06.05.04.03.02.01.00γ0.0 0.5 1.0Lorentz Factor γγ =v/c1√1 − v 2 /c 2√v = c 1 − γ −2Limiting Caseslim γ = 1v→0lim γ = ∞ v→cc○2004, 2007 G. Anderson Physics III – slide 5 / 37

NortheasternIllinoisUniversityLorentz TransformationsSpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologyyFor two inertial reference frames wherethe second frame moves at v = v 0 î with respect to the first.Sy ′ S ′vE(x, y, t) or (x’,y’,t’)xx ′t ′ = γ(t − vx/c 2 ) t = γ(t ′ + vx ′ /c 2 )x ′ = γ(x − vt) x = γ(x ′ + vt ′ )y ′ = y z = z ′c○2004, 2007 G. Anderson Physics III – slide 6 / 37

NortheasternIllinoisUniversityLorentz TransformationsSpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologyyFor two inertial reference frames wherethe second frame moves at v = v 0 î with respect to the first.Sx 2 + y 2 = c 2 t 2x ′2 + y ′2 = c 2 t ′2y ′ S ′vE(x, y, t) or (x’,y’,t’)xx ′t ′ = γ(t − vx/c 2 ) t = γ(t ′ + vx ′ /c 2 )x ′ = γ(x − vt) x = γ(x ′ + vt ′ )y ′ = y z = z ′c○2004, 2007 G. Anderson Physics III – slide 6 / 37

NortheasternIllinoisUniversityLorentz Transformations for E & BSpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologyFor two inertial reference frames where the second framemoves at v = vê ‖ with respect to the first.t ′ = γ(t − vx ‖ /c 2 ) t = γ(t ′ + vx ′ ‖ /c2 )x ′ ‖= γ(x ‖ − vt) x ‖ = γ(x ′ ‖ + vt′ )x ′ ⊥ = x ⊥ x ⊥ = x ′ ⊥E‖ ′ = E ‖ E ‖ = E‖′E ′ ⊥ = γ(E ⊥ + v × B ⊥ ) E ⊥ = γ(E ′ ⊥ − v × B′ ⊥ )B‖ ′ = B ‖ B ‖ = B‖′B ′ ⊥ = γ(B ⊥ − v × E ⊥ /c 2 ) B ⊥ = γ(B ′ ⊥ + v × E′ ⊥ /c2 )Maxwell’s equations are preserved by the LorentzTransformations.c○2004, 2007 G. Anderson Physics III – slide 7 / 37

NortheasternIllinoisUniversityLorentz Transformations for E & pSpecialRelativityPostulatesLorentz FactorLorentzTransformationsLT for E & BLT for E & pGeneralRelativityCosmologyyFor two inertial reference frames wherethe second frame moves at v = v 0 î with respect to the first.Sy ′ S ′v(p, E) or (p’,E’)xx ′E ′ = γ(E − vp x ) E = γ(E ′ + vp ′ x)p ′ x = γ(p x − vE/c 2 ) p x = γ(p ′ x + vE ′ /c 2 )p ′ y = p y p z = p ′ zc○2004, 2007 G. Anderson Physics III – slide 8 / 37

NortheasternIllinoisUniversitySpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarGeneral RelativityCosmology c○2004, 2007 G. Anderson Physics III – slide 9 / 37

NortheasternIllinoisUniversityHistorical Milestones in RelativitySpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary Pulsar1860 Maxwell’s Equations1881 Michelson’s Interferometer1887 Michelson-Morley Experiment1903 Lorentz Transformations1905 Albert Einstein’s Special Relativity1916 Albert Einstein’s General Relativity1919 Eddington observes “bending” of light.2008 You explain relativity to your friends and family.Cosmology c○2004, 2007 G. Anderson Physics III – slide 10 / 37

NortheasternIllinoisUniversityLight in an Accelerated FrameSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarA homogeneous gravitational field is equivalent to auniformly accelerated reference frame.⋆v ≠ 0a = 0⋆⋆Cosmology c○2004, 2007 G. Anderson Physics III – slide 12 / 37⋆⋆⋆⋆

NortheasternIllinoisUniversityLight in an Accelerated FrameSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarA homogeneous gravitational field is equivalent to auniformly accelerated reference frame.⋆a ≠ 0⋆⋆Cosmology c○2004, 2007 G. Anderson Physics III – slide 12 / 37⋆⋆⋆⋆

NortheasternIllinoisUniversityGeneral RelativitySpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarPredictions and consequences of General Relativity:◆◆◆◆◆◆◆Curvature of space-timeGravitational lensingPrecession of perihelion of MercuryGravitational Red ShiftBlack HolesGravity WavesExpansion of the UniverseCosmology c○2004, 2007 G. Anderson Physics III – slide 13 / 37

NortheasternIllinoisUniversityPositive CurvatureSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarPositive curvatureNon Euclidean Geometry:Cosmology c○2004, 2007 G. Anderson Physics III – slide 14 / 37◆◆Parallel lines can meet∑ Interior Angles ≠ παγAβα + β + γ = π + A R 2C < 2πrA < 4πr 2(2D)(3D)

NortheasternIllinoisUniversityNegative CurvatureSpecialRelativityzGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarαγAβxyCosmology c○2004, 2007 G. Anderson Physics III – slide 15 / 37

NortheasternIllinoisUniversityCurvature & Energy DensitySpecialRelativityEinstein’s EquationGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarAnalogy⎛⎜⎝Small Masscurvatureofspace-time⎞⎛⎟ ⎜⎠ = G ⎝energydensity ofspace-time⎞⎟⎠Cosmology c○2004, 2007 G. Anderson Physics III – slide 16 / 37

NortheasternIllinoisUniversityCurvature & Energy DensitySpecialRelativityEinstein’s EquationGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarAnalogy⎛⎜⎝Bigger Masscurvatureofspace-time⎞⎛⎟ ⎜⎠ = G ⎝energydensity ofspace-time⎞⎟⎠Cosmology c○2004, 2007 G. Anderson Physics III – slide 16 / 37

NortheasternIllinoisUniversityCurvature & Energy DensitySpecialRelativityEinstein’s EquationGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarAnalogy⎛⎜⎝Even Bigger Masscurvatureofspace-time⎞⎛⎟ ⎜⎠ = G ⎝energydensity ofspace-time⎞⎟⎠Cosmology c○2004, 2007 G. Anderson Physics III – slide 16 / 37

NortheasternIllinoisUniversityGravity from CurvatureSpecialRelativityGravity is a manifestation of curvatureGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 17 / 37

NortheasternIllinoisUniversityGravity from CurvatureSpecialRelativityGravity is a manifestation of curvatureGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 17 / 37

NortheasternIllinoisUniversity“Bending” of LightSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarPredicted by Einstein in 1915. Confirmed by ArthurEddington in 1919 by measuring deflection of starlightduring a solar eclipse.⋆⋆αα = 4 GM ⊙c 2 R ⊙≈ 1.75 arcsecondsObservation:α = 1.61 ± 0.30 arcsecondsR ⊙ = distance of closest approachCosmology c○2004, 2007 G. Anderson Physics III – slide 18 / 37

NortheasternIllinoisUniversity“Bending” of Light IISpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarSunCosmology c○2004, 2007 G. Anderson Physics III – slide 19 / 37

NortheasternIllinoisUniversityGravitational LensingSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarNASA PhotoHubble space telescope image showing Gravitational Lensingin the Galactic Cluster Abell 2218.Cosmology c○2004, 2007 G. Anderson Physics III – slide 20 / 37

NortheasternIllinoisUniversityGravitational Lensing IISpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarSTScI PhotoThe Einsten Cross Gravitational Lens. Galaxy lenses abackground quasar.Cosmology c○2004, 2007 G. Anderson Physics III – slide 21 / 37

NortheasternIllinoisUniversityPrecession of perihelionSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarPrecession of perihelion of Mercury 43 arc seconds percentury.Cosmology c○2004, 2007 G. Anderson Physics III – slide 22 / 37

NortheasternIllinoisUniversityGravitational Red ShiftSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary Pulsarhff′hf − hf ′hfE ′ = mc 2 − G Mmr+hgE = mc 2 − G Mmr= E′ − EE≈ GMm rNear the Earth’s surfaceg = G M r 2Gravitational Red Shift(f − f ′ )/f ≈ gh/c 2Confirmed by Pound et. al.in 1960.− G Mmr+hmc 2≈ GMc 2hr ≈ gh2 c 2Cosmology c○2004, 2007 G. Anderson Physics III – slide 23 / 37

NortheasternIllinoisUniversityBlack HoleszSpecialRelativityyGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarRsingularitySchwarzschild Black Holeevent horizonR = 2 GMc 2 , T = ¯hc 38πGMk BRegion of spacetime with so much mass,even light cannot escape.xCosmology c○2004, 2007 G. Anderson Physics III – slide 24 / 37

NortheasternIllinoisUniversityBlack Holes IISpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary Pulsar◆◆◆Super-massive black holes containing billions of solarmasses are thought to be responsible for activegalactic nuclei.Bekenstein-Hawking EntropyS = kAc34G¯hTolman-Oppenheimer-Volkoff (TOV) limitCosmology c○2004, 2007 G. Anderson Physics III – slide 25 / 37

SpecialRelativityNortheasternIllinoisUniversityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarPulsarPulsar = Pulsating star. Highly magnetized, rotatingneutron-stars which emit pulses of radio waves.Optical & X-ray image of crab nebulaCosmology c○2004, 2007 G. Anderson Physics III – slide 26 / 37

NortheasternIllinoisUniversityAriceboSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 27 / 37

NortheasternIllinoisUniversityGravitational WavesSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarGravitational Waves◆◆Ripples in the fabric of space-timeCarry away energy, angular momentum1993 Nobel Prize: 1974 Hulse-Taylor pulsar (PSRB1913+16), a pair of neutron stars losing energy throughgravitational waves.Cosmology c○2004, 2007 G. Anderson Physics III – slide 28 / 37

NortheasternIllinoisUniversityBinary PulsarSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 29 / 37

NortheasternIllinoisUniversityBinary PulsarSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 29 / 37

NortheasternIllinoisUniversityBinary PulsarSpecialRelativityGeneralRelativityMilestonesPrinciple ofEquivalenceGeneralRelativityPositiveCurvatureNegativeCurvatureCurvature &Energy DensityGravity fromCurvature“Bending” ofLightGravitationalLensingPrecession ofperihelionGravitationalRed ShiftBlack HolesBlack Holes IIPulsarGravitationalWavesBinary PulsarCosmology c○2004, 2007 G. Anderson Physics III – slide 29 / 37

NortheasternIllinoisUniversitySpecialRelativityGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextCosmologyc○2004, 2007 G. Anderson Physics III – slide 30 / 37

NortheasternIllinoisUniversityThe Expanding UniverseSpecialRelativityThe Doppler effect for lightGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextvλ = λ 0√ √√√1 + v/c1 − v/c , ∆λλ 0≈ v c(v ≪ c)“The universe is expanding”’ Edwin Hubble (1929)v = HddThe Hubble parameter: H ≈ 70 km/s/Mpcc○2004, 2007 G. Anderson Physics III – slide 31 / 37

•••••••••••••••••••••••••••••••••••••••••••NortheasternIllinoisUniversityExpansion of the UniverseSpecialRelativityzGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextxz• •• •• •• •yx• ••y• •c○2004, 2007 G. Anderson Physics III – slide 32 / 37

SpecialRelativityGeneralRelativityNortheasternIllinoisUniversityFriedmann-Robertson-Walker Cosmology[ds 2 = dt 2 − a(t) 2 dr 2 ]1 − kr + 2 r2 dΩ 2CosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextclosedflatopenk = +1k = 0k = −1c○2004, 2007 G. Anderson Physics III – slide 33 / 37

NortheasternIllinoisUniversityCosmological ParametersSpecialRelativityGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextH 0 = ȧaHubble parameterq 0 = −äa H−2 0deceleration parameterλ 1 λ 0⋆⋆t 1 t 0Redshift vs. Expansion1 + z ≡ λ 0= a(t 0)λ 1 a(t 1 )a(t 0 )a(t 1 ) = 1 + H 0(t 0 − t 1 ) + 1 2 (q 0 + 2)H 2 0(t 0 − t 1 ) 2 + . . .c○2004, 2007 G. Anderson Physics III – slide 34 / 37

NortheasternIllinoisUniversitySingle Component UniversesSpecialRelativityGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNexta(t)9876543210Λ-2 -1 0 1 2 3 4 5 6 7 8H 0 (t − t 0 )curvature (empty)matterradiationFriedmann Equation:H 2 =(ȧa) 2= 8πG3 ρ − ka−2c○2004, 2007 G. Anderson Physics III – slide 35 / 37

NortheasternIllinoisUniversitySingle Component Universes IISpecialRelativityGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextρ ∝ a(t) ∝ t 0 äa −4 t 1/2 1 2 H−1 0 < 0 radiationa −3 t 2/3 2 3 H−1 0 < 0 mattera −2 , 0 t H −10 = 0 curvaturea 0 e Ht > 0 vacuumc○2004, 2007 G. Anderson Physics III – slide 36 / 37

NortheasternIllinoisUniversityNextSpecialRelativityGeneralRelativityCosmologyThe ExpandingUniverseExpansion of theUniverseFriedmann-Robertson-WalkerCosmologyCosmologicalParametersSingleComponentUniversesNextLecture 06: Quantization◆◆◆◆Quantization of Charge■ J.J. Thomson’s Experiment■ Millikan’s Oil DropBlackbody Radiation■ Blackbody Radiation■ Planck’s quantum hypothesisThe photoelectric effectX-rays & Compton scatteringc○2004, 2007 G. Anderson Physics III – slide 37 / 37