Cosmic censorship and the collapse of a scalar field in cylindrical - PowerPoint PPT Presentation

Cosmic censorship and the collapse of a scalar field in cylindrical symmetry Eoin Condron Dublin City University, Ireland Supported by the Irish Research Council for Science, Engineering and Technology April 4, 2012 Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 1 / 1

Outline The model Summary of solutions to the past of the past null cone of the origin Σ Field equations Solutions to the future of the past null cone of the origin Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 2 / 1

b Conformal diagram of the spacetime N ( u = 0) O II Σ( v = 0) η = 1 I Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 3 / 1

The Model The line element for cylindrical symmetry with self-similarity in double null coords with η = v / u ds 2 = −| u | − 1 e 2 γ ( η )+2 φ ( η ) dudv + | u | e 2 φ ( η ) S 2 ( η ) d θ + | u | e − 2 φ ( η ) dz 2 . (1) The energy-momentum tensor for self-interacting scalar field: T ab = ∇ a ψ ∇ b ψ − 1 2 g ab ∇ c ψ ∇ c ψ − g ab V ( ψ ) (2) ψ = F ( η ) + k V ( ψ ) = V 0 e − 2 k ψ 2 ln | u | Different values of V 0 and k represent different matter models Regular axis provides initial data surface Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 4 / 1

b b b Summary of solutions in region I Σ( v = 0) Σ( v = 0) Σ( v = 0) I + u = u 0 u = u 0 u = u 0 u = − 1 u = − 1 u = − 1 V 0 > 0 , k 2 ≥ 2 V 0 < 0 , k 2 ≥ 2 k 2 < 2 Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 5 / 1

Field equations in region II for V 0 < 0 Field equations for V 0 < 0 1 ( t ) = x 1 − x 2 − x 2 x ′ (3a) 1 � 1 � x ′ 2 ( t ) = Lx 2 2 − x 3 (3b) 3 ( t ) = 1 2 x 3 + 1 2 x 1 − 2 x ′ k 2 x 2 − x 1 x 3 (3c) t →−∞ e − Lt / 2 ( x 1 , x 2 , x 3 ) = ( a 1 , b 1 , c 1 ) lim (3d) L , a 1 , b 1 , c 1 depend on k 2 Σ is a singular point, and fixed point, of the equations Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 6 / 1

Three possible outcomes Case I lim t →∞ � x = (1 , 0 , 1) Corresponds to naked singularity Case II � � � | λ | / 2 , k 2 / 8 , 1 / 2 lim t →∞ � x = 1 / 2 − Corresponds to censored singularity Case III | � x | → ∞ in finite time Corresponds to censored singularity Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 7 / 1

b b b The three pictures O O O v = 0 v = 0 v = 0 η = 1 η = 1 η = 1 Case I Case II Case III Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 8 / 1

Some results Lemma 1 Suppose there exists t 1 > −∞ such that x 1 ( t 1 ) = 0 . Then there exists t 2 > t 1 such that lim t → t 2 x 1 = −∞ . Lemma 2 Suppose there exists t 1 such that x 1 ( t 1 ) , x 3 ( t 1 ) < 1 / 2 and x ′ ( t 1 ) , x ′ 3 ( t 1 ) < 0 . Then there exists t 2 > t 1 such that x 1 ( t 2 ) = 0 . ... Lemma 7 Suppose that for 1 < k 2 < 4 / 3 there exists some t 1 such that x ′ 3 ( t 1 ) , x ′′ 3 ( t 1 ) , x ′′′ 3 ( t 1 ) , u ′′′ 1 ( t 1 ) < 0 , and Lx 3 ( t 1 ) / 2 < x 1 ( t 1 ) < 2 x 2 ( t 1 ) , x ′ 1 ( t 1 ) < 2 x ′ 2 ( t 1 ) , x ′′ 1 ( t 1 ) < 2 x ′′ 2 ( t 1 ) . Then there exists some time t 2 > t 1 such that x 1 ( t 2 ) = 0 . Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 9 / 1

Formal series solutions ∞ ∞ ∞ � � � a n e nLt / 2 b n e nLt / 2 c n e nLt / 2 x 1 = x 2 = x 3 = (4) n =1 n =1 n =1 Recurrence relations for coefficients   � − 1 n − 1 � nL �  , a n = 2 − 1  − b n − a j a n − j (5a) j =1 n − 1 2 � b n = − b j c n − j , (5b) n − 1 j =1   � − 1 n − 1 � nL 2 − 1  a n 2 − 2 b n �  . c n = k 2 − a j c n − j (5c) 2 j =1 Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 10 / 1

Field equations in region II for V 0 < 0 Field equations for V 0 < 0 1 ( t ) = x 1 − x 2 − x 2 x ′ (6a) 1 � 1 � x ′ 2 ( t ) = Lx 2 2 − x 3 (6b) 3 ( t ) = 1 2 x 3 + 1 2 x 1 − 2 x ′ k 2 x 2 − x 1 x 3 (6c) t →−∞ e − Lt / 2 ( x 1 , x 2 , x 3 ) = ( a 1 , b 1 , c 1 ) lim (6d) L , a 1 , b 1 , c 1 depend on k 2 Σ is a singular point, and fixed point, of the equations Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 11 / 1

Theorem 8 Suppose that | a j | , | b j | , | c j | ≤ κδ j − 1 / j 2 for all j < n ∗ , for some n ∗ , κ, δ . Suppose further that � � n ∗ > 2 1 + 2 k 2 + κ δ s n ∗ , (7) L where s n ∗ = 2 H 2 , n ∗ − 1 + 4 H 1 , n ∗ − 1 . (8) n ∗ Then | a n ∗ | , | b n ∗ | , | c n ∗ | ≤ κδ n ∗ − 1 / n 2 ∗ . Corollary 9 If theorem 8 holds for some κ, δ, n ∗ , then it holds for κ, δ and all n > n ∗ . Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 12 / 1

Uniform convergence & truncation error Theorem 10 Let τ = e Lt / 2 . Suppose that there exist κ, δ such that | a n | , | b n | , | c n | ≤ κδ n − 1 / n 2 for all n ≥ 1 . Then the series (4) converge uniformly on τ ∈ [0 , 1 /δ ) . Lemma 11 Suppose that there exist κ, δ such that | a n | , | b n | , | c n | ≤ κδ n − 1 / n 2 for all n ≥ 1, and that the series (4) are approximated by the first N terms. Then the truncation errors e i for these approximations satisfy the bound � ( δτ ) N +1 κ � e i ≤ . (9) N 2 δ 1 − δτ Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 13 / 1

Numerical data Let j ∈ [1 , N ] { j 2 ( | a j | , b j | , | c j | ) /κδ j − 1 } m = max (10) k 2 δ κ N m τ 1 ¯ x 1 ( τ 1 ) e ( τ 1 ) < 10 6 10 5 9 × 10 − 7 − 2 . 4 × 10 − 5 10 − 24 0 . 01 412 0 . 796 10 5 9 × 10 − 6 − 3 . 04 × 10 − 4 10 − 14 0 . 02 25 , 000 224 0 . 792 . . . . . . . . . . . . . . . . . . . . . . . . 10 − 13 0 . 32 90 80 80 0 . 846 0 . 008 − 0 . 0527 10 − 9 0 . 33 80 80 80 0 . 976 0 . 01 − 0 . 0832 . . . . . . . . . . . . . . . . Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 14 / 1

Thanks for listening. Any questions? Eoin Condron (DCU) Cosmic censorship and the collapse of a scalar field in cylindrical symmetry April 4, 2012 15 / 1