Mathematical general relativity Gantumur Tsogtgerel McGill - PowerPoint PPT Presentation

Mechanics Electrodynamics Gravitation Winter school in pure and applied math Mathematical general relativity Gantumur Tsogtgerel McGill University 8-10 January 2010

Mechanics Electrodynamics Gravitation Harmonic oscillator 3 v = − x ˙ x = − x ¨ 2 or x = v ˙ 1 d dt ( x 2 + v 2 ) = 2 x ˙ x + 2 v ˙ v 0 = 2 xv − 2 vx = 0 � 1 x ( 0 ) 2 + v ( 0 ) 2 = C � 2 ⇓ x ( t ) 2 + v ( t ) 2 = C � 3 � 3 � 2 � 1 0 1 2 3

Mechanics Electrodynamics Gravitation Physical pendulum 3 v = − sin x ˙ x = − sin x ¨ or x = v ˙ 2 1 d dt (− 2 cos x + v 2 ) = 2 ( sin x ) ˙ x + 2 v ˙ v = 2 ( sin x ) v − 2 v sin x 0 = 0 � 1 − 2 cos x ( 0 ) + v ( 0 ) 2 = C � 2 ⇓ − 2 cos x ( t ) + v ( t ) 2 = C � 3 � 3 � 2 � 1 0 1 2 3

Mechanics Electrodynamics Gravitation Physical pendulum 6 v = − sin x ˙ x = − sin x ¨ or 4 x = v ˙ 2 d dt (− 2 cos x + v 2 ) = 2 ( sin x ) ˙ x + 2 v ˙ v = 2 ( sin x ) v − 2 v sin x 0 = 0 � 2 − 2 cos x ( 0 ) + v ( 0 ) 2 = C � 4 ⇓ − 2 cos x ( t ) + v ( t ) 2 = C � 6 � 6 � 4 � 2 0 2 4 6

Mechanics Electrodynamics Gravitation Constrained pendulum x + x = 0 ¨ d · x = 0 where d ∈ R 2 . For any y ∈ R 2 , x = ( I − dd T ) y satisfies d · x = 0 . We have ( I − dd T )( ¨ y + y ) = 0 Let d = e 2 . Then y 1 + y 1 = 0 ¨ but no equation for y 2 ! x does not depend on y 2 , so y 2 = y 2 ( t ) can be anything, e.g., take y 2 = y 1

Mechanics Electrodynamics Gravitation Maxwell’s equations ∂ t B = − ∇ × E , ∇ · B = 0, ∂ t E = ∇ × B , ∇ · E = 0. ∇ · B = 0 ⇒ B = ∇ × A ∂ t B = − ∇ × E ⇒ ∇ × ( ∂ t A + E ) = 0 ⇒ ∂ t A + E = − ∇ ϕ C := ∂ t ( ∇ · A ) + ∆ϕ = 0, − ∂ t ( ∂ t A + ∇ ϕ ) = ∇ × ∇ × A E := ∂ 2 ∇ × ∇ × A = ∇ ( ∇ · A ) − ∆A ⇒ t A − ∆A + ∇ ( ∂ t ϕ + ∇ · A ) = 0 ∂ t C = ∇ · ∂ 2 t A + ∆∂ t ϕ ∇ · E = ∇ · ∂ 2 t A − ∇ · ∆A + ∇ · ∇ ∂ t ϕ + ∇ · ∇ ( ∇ · A ) = ∂ t C

Mechanics Electrodynamics Gravitation Gauge freedom � ∂ 2 [ ∂ t ( ∇ · A ) + ∆ϕ ] t = 0 = 0, t A − ∆A + ∇ ( ∂ t ϕ + ∇ · A ) = 0 � ∂ t ( ∇ · A ) + ∆ϕ = 0 ⇒ B = ∇ × A , − E = ∇ ϕ + ∂ t A A ′ = A + ∇ λ ∇ × A ′ = ∇ × A + ∇ × ∇ λ = B ⇒ ϕ ′ = ϕ − ∂ t λ ∂ t A ′ + ∇ ϕ ′ = ∂ t A + ∂ t ∇ λ + ∇ ϕ − ∇ ∂ t λ = − E ⇒ ∂ t ϕ ′ + ∇ · A ′ = ∂ t ϕ − ∂ 2 t λ + ∇ · A + ∆λ ∂ t ϕ ′ + ∇ · A ′ = 0 ∂ 2 t λ − ∆λ = ∂ t ϕ + ∇ · A ⇒

Mechanics Electrodynamics Gravitation Einstein’s equations The Lorentzian manifold ( M , g ) satisfies Ric ( g ) = 0. ( E ) Suppose M = R × Σ , each Σ t = { t } × Σ is spacelike. On each Σ t , one has g + ( tr g K ) 2 = 0, R ( g ) − | K | 2 ( C ) div g K − d ( tr g K ) = 0. Conversely, if ( C ) holds on Σ 0 , and ( E ) holds in M , then ( C ) holds for all Σ t . Ric ( g ) = � g + N ( ∂ g , ∂ g ) + ∂ � x α .

Mechanics Electrodynamics Gravitation Einstein’s equations • Special solutions: Minkowski, Schwarzschild, de Sitter, Friedmann, Kerr, ... • Local existence for smooth initial data: Choquet-Bruhat ’52 • Incompleteness theorems: Penrose, Hawking ∼ ’60 • Unique maximal development: Choquet-Bruhat, Geroch ’69 • Local existence for initial metric in H 5 / 2 + ǫ : Hugh, Kato, Marsden ’74 • Nonlinear stability of Minkowski space: Christodoulou, Klainerman ∼ ’90 • Local existence for initial metric in H 2 + ǫ : Klainerman, Rodnianski ∼ ’00 • Black hole formation in vacuum: Christodoulou ’08

Mechanics Electrodynamics Gravitation Black hole stability problem Prove that any nearby solution to a Kerr solution will stay close and asymptotically converge to a Kerr solution. Progress: • Linear wave equations on Kerr background: Rodnianski, Dafermos, Blue, Sterbenz, Tataru, ... • Local uniqueness of the Kerr family: Klainerman, Alexakis, Ionescu

Mechanics Electrodynamics Gravitation Einstein’s constraint equations g + ( tr g K ) 2 = 0, R ( g ) − | K | 2 div g K − d ( tr g K ) = 0. • Positive mass theorem: Schoen, Yau, Witten ∼ ’80 • Conformal method: Lichnerowisz, York, Isenberg, Maxwell, ... • Riemannian Penrose inequality: Huisken, Ilmanen, Bray ’97-99 • Gluing: Corvino, Schoen, ...

Mechanics Electrodynamics Gravitation Books • S EAN C ARROLL . Spacetime and Geometry: An Introduction to General Relativity • R OBERT W ALD . General Relativity • N ORBERT S TRAUMANN . General Relativity: With Applications to Astrophysics • A LAN R ENDALL . Partial Differential Equations in General Relativity • D EMETRIOS C HRISTODOULOU . Mathematical Problems of General Relativity • Y VONNE C HOQUET -B RUHAT . General Relativity and the Einstein Equations