Global properties of solutions to the Einstein-matter equations

Transcription

1 Global properties of solutions to the Einstein-matter equations Makoto Narita Okinawa National College of Technology Tokyo

2 Singularity theorems and two conjectures Theorem 1 (Penrose) Suppose the following conditions hold: (1) a Cauchy surface Σ is noncompact, (2) the null convergence condition, (3) Σ contains a closed trapped surface. Then the corresponding maxmal future development D + (Σ) is incomplete. Theorem 2 (Hawking) Suppose the following conditions hold: (1) a Cauchy surface Σ is compact, (2) the timelike convergence condition, (3) the generic condition. Then the corresponding maxmal Cauchy development D(Σ) is incomplete. These theorems say physically reasonable spacetimes have singularities in general.

3 However, the theorem does not say us nature of singularity. predictability is breakdown if singularity can be seen. Conjecture 1 (Belinskii-Khalatnikov-Lifshitz (BKL) conjecture) Solutions to the Einstein(-matter) equations should be Kasner-like ones near spacetime singularity. Conjecture 2 (Strong cosmic censorship (SCC) conjecture) Generic Cauchy data sets have maximal Cauchy developments which are locally inextendible as Lorentzian manifolds.

4 To prove the conjectures, we need to show global existence theorems of solutions to the Einstein(-matter) equations in suitable coordinates, existence of Kasner-like solutions near spacetime singularity in generic, and inextendibility of spacetimes.

5 To solve global problems for the Einstein(-matter) equations, some assumptions will be needed, because the equations are very complicated nonlinear PDEs and then we have less mathematical tools to analyze such equations. Existence of some spacelike Killing vectors will be assumed (so called Gowdy spacetimes, which are the most simplest inhomogeneous ones including dynamical degree of freedom of gravity), Maxwell field as matter will be assumed.

6 Recent results concerning SCC conjecture (my personal selection): Local existence (Choquet-Bruhat, Klainerman-Rodnianski) Global nonlinear stability of Minkowski space (Christodoulou-Klainerman, Lindblad-Rodnianski) Spherically symmetric gravitational collapse (Andreasson, Christodoulou, Dafermos, Rein, Rendall, MN) Gowdy spacetime (Ringström, Rendall, Andreasson, MN)

7 Recent results concerning BKL conjecture (my personal selection): Bianchi spacetime (Rendall, Ringström) Bianchi spacetime with magnetic field (Weaver, Liebscher-Rendall- Tchapnda) Gowdy spacetime (Kichenassamy-Rendall, MN, Beyer-LeFloch) U(1)-symmetric spacetime (Isenberg-Moncrief)

8 Einstein-Maxwell system in Gowdy symmetric spacetimes The action is S = d 4 x g [ R + F 2]. (1) The Einstein-Maxwell equations are R µν = 2F µρ Fν ρ 1 2 g µνf 2, (2) µ ( F µν ) = 0 (3) µ F µν = 0, (4) where F µν := 2 [µ A ν], F µν := 2 1 gϵµνδρ F δρ.

9 The metric is given by g = e 2(η U) αdt 2 + e 2(η U) dθ 2 + e 2U (dx + Ady) 2 + e 2U t 2 dy 2, (5) where η, α, U and A are functions of t (0, ) and θ T 1. assume We F = Bdx dy.

10 Constraint equations η t = U 2 + αu 2 + e4u 4t 2 (Ȧ2 + αa 2 ) + αe2(η U) B 2 t 4, (6) η t = 2 UU + e4u 2t 2 ȦA α 2tα, (7) α = 4α2 e 2(η U) B 2 t 3. (8) Dot and prime denote derivative with respect to t and θ, respectively. Note that B should be constant by the Maxwell equations.

11 Evolution equations Ü αu = U t + α U 2α + α U 2 + e4u 2t 2 (Ȧ2 αa 2 ) + αe2(η U) B 2 t 4, (9) Ä αa = Ȧ t + αȧ 2α + α A 2 4(Ȧ U αa U ), (10) We call this system magnetic Gowdy system.

12 Map u : (M 2+1, G) (N 2, h) The system of the evolution equations is equivalent with the following system of nonlinear wave equations (wave map): where and S MG = dtdθ G G αβ h AB α u A β u B + αe2(η U) B 2, (11) S 1 G = dt α dθ2 + t 2 dψ 2, 0 θ, ψ 2π, h = du 2 + e4u 4t 2 da2. Every functions depend on time t and θ. t 4

13 The energy-momentum tensor T αβ form: T αβ = h AB ( α u A β u B 1 2 G αβ λ u A λ u B ) for this system is given of the G αβ αe 2(η U) B 2 2t 4. (12)

14 The energies are defined as follows: E(t) = = 1 2 = T S 1 tt S 1 S 1 dθ α h AB ( t u A t u B + θ u A θ u B) + αe2(η U) B 2 E + αe2(η U) B 2 2t 4 dθ α, t 4 dθ α and Ẽ(t) := E dθ. S 1 α

15 Global existence Theorem 3 Let (M, g) be the maximal Cauchy development of C initial data for the magnetic Gowdy system. Then, M can be covered by compact Cauchy surfaces of constant areal time t with each value in the range (0, ).

16 Lemma 1 E and Ẽ decrease monotonically along time t, that is, de(t) dt < 0 and dẽ(t) dt < 0, (13) and E and Ẽ are bounded on (T, T + ), where 0 < T < t i < T + <. Furthermore, there exists numbers, E and Ẽ, satisfying E = lim t T E(t) and Ẽ = lim t T Ẽ(t). (14) From the above lemma, we have boundedness for αα 1. Lemma 2 If αα 1 is bounded, E is bounded on (T, T + ) T 1. The method of the proof is the light cone estimate.

17 Existence of Kasner-like solutions near spacetime singularity Theorem 4 (Ames-Beyer-Isenberg-LeFloch) Suppose that a firstorder PDE system for u : (0, δ] T 1 R d for some integer d 1 and some real δ > 0 S 1 (t, θ, u)t t u(t, θ) + S 2 (t, θ, u)t θ u(t, θ) + N(t, θ, u)u(t, θ) = f(t, θ, u) (15) is a quasilinear symmetric hyperbolic Fuchsian system. If the following conditions 1. positivity of energy, 2. regularity of f(t, θ, u), 3. Lipschitz continuity for remainder w := u u 0 (regular part),

18 are satisfied. Here, u 0 is the leading-order term (singular part). Then, there exists a unique solution u of equation (15) such that u u 0 belong to a suitable function space.

19 The leading-order term for Kasner-like solution: η = 1 4 (1 k(θ))2 ln t + η (θ) +, (16) α = α (θ) +, (17) U = 1 2 (1 k(θ)) ln t + U (θ) +, (18) A = A (θ) + t 2k(θ) A (θ) +, (19)

20 Theorem 5 Suppose that a set of asymptotic data functions k, U, α (> 0), A C (T 1 ) which satisfy the integrability condition 2π ( (1 k(θ))u (θ) 1 ) 2 (ln α ) (θ) dθ = 0 with either 0 1. k(θ) > 3 for arbitrary A (θ), or 2. k(θ) > 3 or k(θ) < 3 for A (θ) 0, at each point θ T 1 Then, there is a δ > 0, and a magnetic Gowdy solution U, A, η, α of the Einstein-maxwell equations of the form (U, A, η, α) = (U 0, A 0, η 0, α 0 ) + W,

21 where the leading-order terms (U 0, A 0, η 0, α 0 ) is given by equations (16)-(19) and η (θ) := η 0 + θ 0 ( (1 k(θ))u (Θ) 1 ) 2 (ln α ) (Θ) dθ. The remainder W is contained in a suitable function space.this solution is unique among all solutions with the same leading-order term and with the remainder. This theorem means that the Kasner-like solution exists near spacetime singularity.

22 Summary We show that 1. global existence theorem of solutions, 2. existence theorem of Kasner-like solutions, to the Einstein-Maxwell equations with magnetic Gowdy spacetimes. These results are support the validity of the SCC and BKL conjectures.