The Geometry of Relativity, & Piecewise Conserved Quantities

Transcription

1 Introduction Piecewise Models Vector Calculus, & Piecewise Conserved Quantities Department of Mathematics Oregon State University

2 Introduction Piecewise Models Vector Calculus Personal History 1987: Indo-American IMSc, TIFR (visits to Pune and RRI) 1988: Returned to RRI and TIFR Collaborated with Sam:, Ravi Kulkarni, and Joseph Samuel, Duality and Conformal Structure, J. Math. Phys. 30, (1989). Piecewise Conserved Quantities

3 Introduction Differential Geometry Definition A topological manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space. A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are differentiable. Differential Geometry What math is needed for GR??

4 Background Introduction Differential geometry course: Rick Schoen GR reading course: MTW GR course: Sachs Wu designed and taught undergrad math course in GR: Schutz, d Inverno, Wald, Taylor Wheeler, Hartle designed and taught undergrad physics course in SR NSF-funded curricular work (math and physics) since 1996 national expert in teaching 2nd-year calculus geometer, relativist, curriculum developer, education researcher Mathematics, Physics, PER, RUME

5 Introduction Mathematics vs. Physics My mathematics colleagues think I m a physicist. My physics colleagues know better.

6 Books Introduction The Geometry of A K Peters/CRC Press 2012 ISBN: Differential Forms and the Geometry of A K Peters/CRC Press 2014 ISBN:

7 Trigonometry Introduction Hyperbolic Trigonometry Applications t t A ρ ρ sinh β β β B x x β ρ cosh β

8 Length Contraction Introduction Hyperbolic Trigonometry Applications t t t t x x x x l = l coshβ l l β l β l

9 Paradoxes Introduction Hyperbolic Trigonometry Applications A 20 foot pole is moving towards a 10 foot barn fast enough that the pole appears to be only 10 feet long. As soon as both ends of the pole are in the barn, slam the doors. How can a 20 foot pole fit into a 10 foot barn? barn frame -20 pole frame

10 Relativistic Mechanics Introduction Hyperbolic Trigonometry Applications A pion of (rest) mass m and (relativistic) momentum p = 3 4 mc decays into 2 (massless) photons. One photon travels in the same direction as the original pion, and the other travels in the opposite direction. Find the energy of each photon. [E 1 = mc 2, E 2 = 1 4 mc2 ] pc p0c sinhβ 0 p 2 c E 2 Β p 0 c Β E 0 E0c coshβ p0c sinhβ p0c p2c 0 Β E2 Β E0 E mc 2 0 E 0 Β E0c coshβ mc 2 0 p0c Β Β E0 E0c coshβ E0c coshβ Β Β p 0 c p0c sinhβ p 1 c E 1 p0c sinhβ p1c E1

11 Addition Formulas Introduction Hyperbolic Trigonometry Applications v = c tanhβ Einstein Addition Formula: tanh(α+β) = tanhα+tanhβ 1+tanhαtanhβ ( v + w = v+w 1+vw/c 2 ) Conservation of Energy-Momentum: p = mc sinhα E = mc 2 coshα Moving Capacitor: E y = C cosh(α+β) = E y coshβ cb z sinhβ cb z = C sinh(α+β) = cb z coshβ E y sinhβ

12 3d spacetime diagrams Introduction Hyperbolic Trigonometry Applications (rising manhole) (v t) 2 +(c t ) 2 = (c t) 2 ct 0 ct y x (v t) 2 (c t) 2 = (c t ) 2 Am. J. Phys. 81, (2013)

13 Introduction The Geometry of The Metric Differential Forms Geodesics Einstein s Equation Doppler effect (SR) Cosmological redshift (GR) Asymptotic structure

14 Line Elements Introduction The Metric Differential Forms Geodesics Einstein s Equation a a dr 2 + r 2 dφ 2 dθ 2 + sin 2 θ dφ 2 dβ 2 + sinh 2 β dφ 2

15 Vector Calculus Introduction The Metric Differential Forms Geodesics Einstein s Equation ds 2 = d r d r Ý Ö Ö Ü ß Ö Ö Ö d r = dx î+dy ĵ = dr ˆr + r dφ ˆφ

16 Introduction The Metric Differential Forms Geodesics Einstein s Equation Differential Forms in a Nutshell (R 3 ) Differential forms are integrands: ( 2 = 1) f = f F = F d r F = F da f = f dv (0-form) (1-form) (2-form) (3-form) Exterior derivative: (d 2 = 0) df = f d r df = F da d F = F dv d f = 0

17 Introduction The Geometry of Differential Forms The Metric Differential Forms Geodesics Einstein s Equation dx + dy r dr = x dx + y dy dx

18 Geodesic Equation Introduction The Metric Differential Forms Geodesics Einstein s Equation Orthonormal basis: d r = σ i ê i (= ds 2 = d r d r) Connection: ω ij = ê i dê j dσ i +ω i j σ j = 0 ω ij +ω ji = 0 Geodesics: v dλ = d r v = 0 Symmetry: d X d r = 0 = X v = const

19 Introduction Example: Polar Coordinates The Metric Differential Forms Geodesics Einstein s Equation Symmetry: ds 2 = dr 2 + r 2 dφ 2 = d r = dr ˆr + r dφ ˆφ = r ˆφ is Killing Idea: df = f d r = r ˆφ f = f φ = r ˆφ = φ Check: d(r ˆφ) = dr ˆφ+r d ˆφ = dr ˆφ r dφˆr d r Geodesic Equation: v = ṙ ˆr + r φ ˆφ = r ˆφ v = r 2 φ = l = 1 = ṙ 2 + r 2 φ 2 = ṙ 2 + l2 r 2

20 Einstein s Equation Introduction The Metric Differential Forms Geodesics Einstein s Equation Curvature: Ω i j = dω i j +ω i k ω k j Einstein tensor: γ i = 1 2 Ω jk (σ i σ j σ k ) G i = γ i = G i j σ j G = G i ê i = G i j σ j ê i = d G = 0 Field equation: G +Λ d r = 8π T (vector valued 1-forms, not tensors)

21 Stress-Energy Tensor Introduction The Metric Differential Forms Geodesics Einstein s Equation d r = σ a ê a Vector-valued 1-form: T = T a bσ b ê a 3-form: τ a = T a Conservation: d (τ a ê a ) = 0 d T = 0

22 Introduction What 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! Killing eq: d X d r = 0 Students understand line elements... ds 2 = d r d r

23 Topic Order Introduction Examples First! Schwarzschild geometry can be analyzed using vector calculus. Rain coordinates! (Painlevé-Gullstrand; freely falling) Geodesics: EBH: Principle of Extremal Aging Hartle: variational principle (Lagrangian mechanics?) TD: differential forms without differential forms

24 Choices Introduction Language: Mathematicians: 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 basis Equivalence problem: coordinate components reduce to 8690 use orthonormal frames? (d r?)

25 Does it work? Introduction I am a mathematician... There is no GR course in physics department. (I developed the SR course.) Core audience is undergraduate math and physics majors. (Many double majors.) Hartle s book: Perfect for physics students, but tough for math majors. My course: 10 weeks differential forms, then 10 weeks GR. (Some physics students take only GR, after crash course.) In this context: YES!

26 Introduction SUMMARY # Special relativity is hyperbolic trigonometry! General relativity can be described without tensors! BUT: Need vector-valued differential forms...

27 Introduction Piecewise Models Vector Calculus Shells Dray & t Hooft (1985) Gravitational shock wave of a massless particle; Shells of matter at horizon of Schwarzschild black hole Clarke & Dray (1987) Junction conditions for null hypersurfaces Dray & Padmanabhan (1988) Conserved quantities from piecewise Killing vectors Dray & Joshi (1990) Glueing Reissner-Nordstrøm spacetimes together Reissner Nordström. Hazboun & Dray (2010) Negative-energy shells in Schwarzschild spacetimes Piecewise Conserved Quantities

28 Introduction Piecewise Models Vector Calculus Signature Change Dray, Manogue, & Tucker (1991); Ellis et al. (1992) Scalar field in the presence of signature change Dray & Hellaby (1994); Hellaby & Dray (1994) Patchwork Divergence Theorem Schray, Dray, Manogue, Tucker, & Wang (1996) Spinors and signature change Dray (1996); Dray, Ellis, Hellaby, & Manogue (1997) Gravity and signature change Dray (1997); Hartley, Tucker, Tuckey, & Dray (2000) Tensor distributions in the presence of signature change Piecewise Conserved Quantities

29 Introduction Piecewise Models Vector Calculus Gluing Spacetimes Together Shells: Signature Change: Piecewise Conserved Quantities

30 Introduction Piecewise Models Vector Calculus Spacelike Boundaries M,g M, g, h g ab = (1 Θ)g ab +Θg+ ab [g ab ] = g + ab Σ g ab Σ = 0 = R ab = (1 Θ)R ab +ΘR+ ab +δ c[γ c ab] δ b [Γ c ac] = (1 Θ)R ab +ΘR+ ab +δρ ab (δ c = δn c = c Θ) Piecewise Conserved Quantities

31 Introduction Piecewise Models Vector Calculus Null Boundaries M, g S M, g Dray & t Hooft (1996) Hazboun & Dray (2010) m > 0 m < 0 Piecewise Conserved Quantities

32 Introduction Piecewise Models Vector Calculus Conserved Quantities Piecewise Killing vector: ξ a = (1 Θ)ξ a +Θξ a + = (a ξ b) = [ ξ (a ] δb) Darmois junction conditions: ([h ij ] = 0 = [K ij ]) = [T ab ] = 0 = a (T ab ξ b ) = ( a T ab )ξ b +T ab a ξ b = 0+T ab [ξ a ]δ b conserved quantity if [ξ a ] = Ξn a & T ab n a n b = 0 Piecewise Conserved Quantities

33 Introduction Piecewise Models Vector Calculus Patchwork Divergence Theorem Divergence Theorem, X smooth: (m σ = ω) = W div(x)ω = L X ω = d(i X ω)+i X (dω) div(x)ω = i X ω = m(x)σ W W X piecewise smooth: (m 0 from M to M + ) div(x)ω = m(x)σ m 0 ([X])σ 0 W W Σ Piecewise Conserved Quantities

34 Introduction Piecewise Models Vector Calculus Shells M, g S M, g ξ = (1 Θ) Conserved quantity: Σ Two Schwarzschild regions: { ds 2 32m3 = r e r/2m dudv +r 2 dω 2 (u α) 32m3 r e r/2m dudv +r 2 dω 2 (u α) [g] = 0 = α m = U(α) u u = MU (α) m = U U M v v u u 4m +Θ V V U U 4M = [ξ] V T uu = δ (M m) = Tvv = 0 γπr2 ( (1 Θ)T t t +ΘT T T ) ds = M m Piecewise Conserved Quantities

35 Introduction Piecewise Models Vector Calculus Signature Change M,g M, g, h ds 2 = { +dt 2 +h ij dx i dx j (t < 0) dt 2 +h ij dx i dx j (t > 0) Volume element is continuous!! ρ := G ab n a n b = 1 ( ) (K c c) 2 K ab K ab ǫr 2 Darmois = [R] = 0 = [K ab ] but [ǫ] 0 = [ρ] = R 0 Energy density at change of signature Piecewise Conserved Quantities

36 Introduction Piecewise Models Vector Calculus Geometric Reasoning Iedereen in deze kamer kan twee talen praten (Everyone in this room is bilingual.) Mathematics Physics Mathematicians teach algebra; Physicists do geometry! Piecewise Conserved Quantities

37 Introduction Piecewise Models Vector Calculus Geometric Reasoning Vector Calculus Bridge Project: Differentials (Use what you know!) Multiple representations Symmetry (adapted bases, coordinates) Geometry (vectors, div, grad, curl) Online text ( Paradigms in Physics Project: Redesign of undergraduate physics major (18 new courses!) Active engagement (300+ documented activities!) Piecewise Conserved Quantities

38 Introduction Piecewise Models Vector Calculus Lorentzian Vector Calculus Minkowski 3-space: dz dx n up dy n down ˆx ˆx = 1 = ŷ ŷ, F = F x ˆx +F y ŷ +F tˆt ˆt ˆt = 1 F = Fx x + Fy y + Ft t ; Divergence Theorem: F dx dy dt = W W F ˆn da where ˆn = outward normal in spacelike directions, but ˆn = inward normal in timelike directions! Piecewise Conserved Quantities

39 Introduction Piecewise Models Vector Calculus SUMMARY Junction conditions at null boundary are surprising! Junction conditions at signature change are surprising! The Divergence Theorem in Minkowski space is surprising! All of these results follow from Patchwork Divergence Thm. THANK YOU Piecewise Conserved Quantities