The Geometry Of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe Geometry of RelativityTevian DrayDepartment of MathematicsOregon State Universityhttp://www.math.oregonstate.edu/ tevianTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumDifferential GeometryDefinitionA topological manifold is a second countable Housdorff space thatis locally homeomorphic to Euclidean space. A differentiablemanifold is a topological manifold equipped with an equivalenceclass of atlases whose transition maps are differentiable.General Relativity 6 Differential GeometryWhat math is needed for GR?Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumBackgroundDifferential geometry course: Rick SchoenGR reading course: MTWGR course: Sachs–Wudesigned and taught undergrad math course in GR:Schutz, d’Inverno, Wald, Taylor–Wheeler, Hartledesigned and taught undergrad physics course in SRNSF-funded curricular work (math and physics) since 1996national expert in teaching 2nd-year , relativist, curriculum developer, education researcherMathematics, Physics, PER, RUMETevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumMath vs. PhysicsMy math colleagues think I’m a physicsist.My physics colleagues know better.Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumBooksThe Geometry of Special RelativityTevian DrayA K Peters/CRC Press 2012ISBN: ursewikis/GSRDifferential Forms andthe Geometry of General RelativityTevian DrayA K Peters/CRC Press 2014ISBN: wikis/GGRTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplicationsTrigonometrytt’ρρ sinh βAβBβx’β ρ cosh βx Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplicationsLength Contractionttt’x’x’xxℓ′ ℓcosh βℓ′βt’ℓ βℓ′ℓTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplicationsParadoxesA 20 foot pole is moving towards a 10 foot barn fast enough thatthe pole appears to be only 10 feet long. As soon as both ends ofthe pole are in the barn, slam the doors. How can a 20 foot pole fitinto a 10 foot barn?-20-10202010100102030-100-10-10-20-20barn frame10pole frameTevian DrayThe Geometry of Relativity2030

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplicationsRelativistic Mechanicsmc2E0 c cosh Βp0 cΒΒE0Βp 0cp1 cp1 cTevian DrayThe Geometry of RelativityE1hΒsinp0 c sinh ΒΒ0oshΒoshp0 cΒE1E0ccE00ccE0E0Β0Βmc2EE0 c cosh ΒE2p0 c Βp2 cE20E0hΒp2 cΒsinpcΒp0 cp 0cp0 c sinh ΒA pion of (rest) mass m and (relativistic) momentum p 34 mcdecays into 2 (massless) photons. One photon travels in the samedirection as the original pion, and the other travels in the oppositedirection. Find the energy of each photon. [E1 mc 2 , E2 14 mc 2 ]

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplicationsAddition Formulasv c tanh βEinstein Addition Formula:tanh α tanh βtanh(α β) 1 tanh α tanh β(“v w ” v w1 vw /c 2 )Conservation of Energy-Momentum:p mc sinh αE mc 2 cosh αMoving Capacitor:E ′y C cosh(α β) E y cosh β cB z sinh βcB ′z C sinh(α β) cB z cosh β E y sinh βTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumHyperbolic TrigonometryApplications3d spacetime diagrams(v t)2 (c t ′ )2 (c t)2(rising manhole)ct 0ctyx(v t)2 (c t)2 (c t ′ s/3dTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationThe Geometry of General RelativityDoppler effect (SR)Cosmological redshift (GR)Tevian DrayAsymptotic structureThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationLine Elementsaadr 2 r 2 d φ2d θ2 sin2 θ d φ2Tevian Drayd β 2 sinh2 β d φ2The Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationVector Calculusds 2 d r · d rdy d rd rdx {r d d r dx ı̂ dy ̂ dr r̂ r d φ φ̂Tevian DrayThe Geometry of Relativitydr r

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationDifferential Forms in a Nutshell (R3 )Differential forms are integrands: ( 2 1)f f · d rF F · dA F F(0-form)(1-form) f f dV(3-form)(2-form)Exterior derivative: (d 2 0) · d rdf f · dA FdF dV ·Fd F d f 0Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationMaxwell’s Equations 4πρ ·E 0 ·B B 0 E E 4π J B dB Ė 4π J · J ρ̇ 0 d J ρ̇ 0d E 4π ρd B 0dE Ḃ 0 ABB dA Φ AEE d Φ ȦTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationMaxwell’s Equations II 4πρ ·E 0 ·B B 0 E E 4π J B F Ê dt ˆ B̂ F B̂ dt ˆ ÊA Â Φ dtF dA · J ρ̇ 0 d F 4π J AB dF 0 Φ AEd J 0Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationThe Geometry of Differential Formsdx dyr dr x dx y dydxTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationGeodesic EquationOrthonormal basis:d r σ i ê iConnection:( ds 2 d r · d r )ωij ê i · d ê jd σi ωi j σj 0ωij ωji 0Geodesics: v d λ d r v 0Symmetry: · d r 0dX · v const XTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationExample: Polar CoordinatesSymmetry:ds 2 dr 2 r 2 d φ2 d r dr r̂ r d φ φ̂ r φ̂ is Killing · d r r φ̂ · f Idea: df f f φ r φ̂ φCheck: d(r φ̂) dr φ̂ r d φ̂ dr φ̂ r dφ r̂ d rGeodesic Equation: v ṙ r̂ r φ̇ φ̂ r φ̂ · v r 2 φ̇ ℓ 1 ṙ 2 r 2 φ̇2 ṙ 2 Tevian Drayℓ2r2The Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationEinstein’s EquationCurvature:Einstein tensor:Ωi j d ω i j ω i k ω k j1γ i Ωjk (σ i σ j σ k )2iG γ i G i j σ j G i ê i G i j σ j ê iG 0 d GField equation: Λ d r 8π T G(vector valued 1-forms, not tensors)Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumThe MetricDifferential FormsGeodesicsEinstein’s EquationStress-Energy Tensord r σ a ê aVector-valued 1-form: T a b σ b ê aT3-form:τ a T aConservation:d (τ a ê a ) 0 0 d TTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumWhat about Tensors?What tensors are needed to do GR?Metric? Use d r ! (vector-valued 1-form!)Curvature? Riemann tensor is really a 2-form. (Cartan!)Ricci? Einstein? Stress-Energy? Vector-valued 1-forms! only 1 essential symmetric tensor in GR! · d r 0Killing eq: d XStudents understand line elements.ds 2 d r · d rTevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral RelativityCurriculumTopic OrderExamples First!Schwarzschild geometry can be analyzed using vector calculus.Rain coordinates! (Painlevé-Gullstrand; freely falling)Geodesics:EBH: Principle of Extremal AgingHartle: variational principle (Lagrangian mechanics?)TD: “differential forms without differential forms”Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral : invariant objects (no indices)Physicists: components (indices)Relativists: abstract index notation (“indices without indices”)Cartan: curvature without tensors use differential forms?Coordinates:Mathematicians: coordinate basis (usually)Physicists: calculate in coordinates; interpret in orthonormal basisEquivalence problem: 79310 coordinate components reduce to 8690 use orthonormal frames? (d r ?)Tevian DrayThe Geometry of Relativity

IntroductionSpecial RelativityGeneral tivity.geometryof.org/GGRSpecial relativity is hyperbolic trigonometry!General relativity can be described without tensors!BUT: Need vector-valued differential forms.THE ENDTevian DrayThe Geometry of Relativity

Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi