Critical Phenomena in Gravitational Collapse

Transcription

1 Critical Phenomena in Gravitational Collapse José M. Martín-García Laboratoire Univers et Theories & Institut d Astrophysique de Paris GRECO Seminar, IAP, Paris 6 October 2008 JMMG (LUTH & IAP) CritPhen 6 October / 30

2 Summary 1 Historical introduction Christodoulou Choptuik Other results 2 The current model Self-similarity GR as a dynamical system Mass scaling 3 Interesting results Non-spherical systems Global structure of the critical solution Chaos 4 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

3 Summary 1 Historical introduction Christodoulou Choptuik Other results 2 The current model Self-similarity GR as a dynamical system Mass scaling 3 Interesting results Non-spherical systems Global structure of the critical solution Chaos 4 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

4 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: JMMG (LUTH & IAP) CritPhen 6 October / 30

5 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: Example: ḟ (t) = f 2 (t) Stability of f (t) = 0? JMMG (LUTH & IAP) CritPhen 6 October / 30

6 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: Example: ḟ (t) = f 2 (t) Stability of f (t) = 0? Linearize: ḟ (t) 0 stability (not asymptotic). JMMG (LUTH & IAP) CritPhen 6 October / 30

7 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: Example: Stability of f (t) = 0? ḟ (t) = f 2 (t) Linearize: ḟ (t) 0 stability (not asymptotic). Nonlinear: f (t) = f 0 1 f 0 t singular at t = f 1 0. JMMG (LUTH & IAP) CritPhen 6 October / 30

8 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: Example: Stability of f (t) = 0? ḟ (t) = f 2 (t) Linearize: ḟ (t) 0 stability (not asymptotic). Nonlinear: f (t) = f 0 1 f 0 t singular at t = f 1 0. Cosmic censorship: is it possible to form a naked (visible to far observers) singularity starting from smooth initial conditions in a self-gravitating system which is regular without gravity? JMMG (LUTH & IAP) CritPhen 6 October / 30

9 1.1. Motivation from mathematics Nonlinear stability of Minkowski in vacuum GR: Example: Stability of f (t) = 0? ḟ (t) = f 2 (t) Linearize: ḟ (t) 0 stability (not asymptotic). Nonlinear: f (t) = f 0 1 f 0 t singular at t = f 1 0. Cosmic censorship: is it possible to form a naked (visible to far observers) singularity starting from smooth initial conditions in a self-gravitating system which is regular without gravity? JMMG (LUTH & IAP) CritPhen 6 October / 30

10 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. JMMG (LUTH & IAP) CritPhen 6 October / 30

11 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. Birkhoff theorem in 3+1 no grav. freedom JMMG (LUTH & IAP) CritPhen 6 October / 30

12 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. Birkhoff theorem in 3+1 no grav. freedom Add massless real scalar field φ(t, r), obeying Klein-Gordon eq. JMMG (LUTH & IAP) CritPhen 6 October / 30

13 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. Birkhoff theorem in 3+1 no grav. freedom Add massless real scalar field φ(t, r), obeying Klein-Gordon eq. Results (CMP 86): Small finite data Minkowski is stable. JMMG (LUTH & IAP) CritPhen 6 October / 30

14 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. Birkhoff theorem in 3+1 no grav. freedom Add massless real scalar field φ(t, r), obeying Klein-Gordon eq. Results (CMP 86): Small finite data Minkowski is stable. Large data Schwarzschild end state. JMMG (LUTH & IAP) CritPhen 6 October / 30

15 1.1. Christodoulou Address both problems in a simpler setting: spherical symmetry. Birkhoff theorem in 3+1 no grav. freedom Add massless real scalar field φ(t, r), obeying Klein-Gordon eq. Results (CMP 86): Small finite data Minkowski is stable. Large data Schwarzschild end state. JMMG (LUTH & IAP) CritPhen 6 October / 30

16 1.1. Christodoulou (II) General question: What happens in between? Naked singularity? JMMG (LUTH & IAP) CritPhen 6 October / 30

17 1.1. Christodoulou (II) General question: What happens in between? Naked singularity? Particular question (1987): is it possible to form arbitrarily small black holes? M R BH, curvature at surface 1 R 2 BH JMMG (LUTH & IAP) CritPhen 6 October / 30

18 1.1. Christodoulou (II) General question: What happens in between? Naked singularity? Particular question (1987): is it possible to form arbitrarily small black holes? M R BH, curvature at surface 1 R 2 BH Intuition: need for self-similarity near the centre (cf. Ori & Piran PRL 87): φ(t, r) = f ( t/r) + κ log( t) JMMG (LUTH & IAP) CritPhen 6 October / 30

19 1.1. Christodoulou (II) General question: What happens in between? Naked singularity? Particular question (1987): is it possible to form arbitrarily small black holes? M R BH, curvature at surface 1 R 2 BH Intuition: need for self-similarity near the centre (cf. Ori & Piran PRL 87): φ(t, r) = f ( t/r) + κ log( t) Goldwirth and Piran, PRD 87: We present a numerical study of the gravitational collapse of a massless scalar field. We calculate the future evolution of new initial data, suggested by Christodoulou, and we show that in spite of the original expectations these data lead only to singularities engulfed by an event horizon. JMMG (LUTH & IAP) CritPhen 6 October / 30

20 1.2. Choptuik (PhD): scalar field spherical collapse code. Cauchy, fully constrained : Improve accuracy and convergence: adaptive mesh refinement and Richardson extrapolation. JMMG (LUTH & IAP) CritPhen 6 October / 30

21 1.2. Choptuik (PhD): scalar field spherical collapse code. Cauchy, fully constrained : Improve accuracy and convergence: adaptive mesh refinement and Richardson extrapolation. Choptuik, Goldwirth and Piran CQG 92: compare codes [CA Cauchy (Choptuik s code). CH Characteristic (GP s code).]... although the levels of error in the CA and CH results at a given resolution were quite comparable at early retarded times (...), the CA values were significantly more accurate than the CH data once the pulse of scalar field had reached r = 0. JMMG (LUTH & IAP) CritPhen 6 October / 30

22 1.2. Choptuik (PhD): scalar field spherical collapse code. Cauchy, fully constrained : Improve accuracy and convergence: adaptive mesh refinement and Richardson extrapolation. Choptuik, Goldwirth and Piran CQG 92: compare codes [CA Cauchy (Choptuik s code). CH Characteristic (GP s code).]... although the levels of error in the CA and CH results at a given resolution were quite comparable at early retarded times (...), the CA values were significantly more accurate than the CH data once the pulse of scalar field had reached r = 0. And then in JMMG (LUTH & IAP) CritPhen 6 October / 30

23 1.2. Choptuik s setup The system: Φ = ds 2 = α 2 (t, r)dt 2 + a 2 (t, r)dr 2 + r 2 dω 2, Φ φ, Π a φ/α ( α ) a Π 1 (, Π = r 2 r 2 α ) a Φ α, α = a 1 a +a2 = 2πr(Π 2 +Φ 2 ). r JMMG (LUTH & IAP) CritPhen 6 October / 30

24 1.2. Choptuik s setup The system: Φ = ds 2 = α 2 (t, r)dt 2 + a 2 (t, r)dr 2 + r 2 dω 2, Φ φ, Π a φ/α ( α ) a Π 1 (, Π = r 2 r 2 α ) a Φ α, α = a 1 a +a2 = 2πr(Π 2 +Φ 2 ). r One-parameter (p) families of initial conditions with the property: Small p leads to no BH formation (small finite data). Large p produces a BH (large data). JMMG (LUTH & IAP) CritPhen 6 October / 30

25 1.2. Choptuik s setup The system: Φ = ds 2 = α 2 (t, r)dt 2 + a 2 (t, r)dr 2 + r 2 dω 2, Φ φ, Π a φ/α ( α ) a Π 1 (, Π = r 2 r 2 α ) a Φ α, α = a 1 a +a2 = 2πr(Π 2 +Φ 2 ). r One-parameter (p) families of initial conditions with the property: Small p leads to no BH formation (small finite data). Large p produces a BH (large data). Example (pure ingoing): φ(0, r) = φ 0 r 3 exp ( [(r r 0 )/δ] q ) p = φ 0, r 0, δ, q Φ q r JMMG (LUTH & IAP) CritPhen 6 October / 30

26 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): JMMG (LUTH & IAP) CritPhen 6 October / 30

27 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). JMMG (LUTH & IAP) CritPhen 6 October / 30

28 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. JMMG (LUTH & IAP) CritPhen 6 October / 30

29 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. JMMG (LUTH & IAP) CritPhen 6 October / 30

30 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. Oscillations in the central region, accumulating at (r = 0, t = 0). JMMG (LUTH & IAP) CritPhen 6 October / 30

31 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. Oscillations in the central region, accumulating at (r = 0, t = 0). Discrete self-similarity: φ(t, r) φ(t/e, r/e ) JMMG (LUTH & IAP) CritPhen 6 October / 30

32 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. Oscillations in the central region, accumulating at (r = 0, t = 0). Discrete self-similarity: φ(t, r) φ(t/e, r/e ) Universality: γ 0,37, 3,44, same profile φ (t, r) for all families. JMMG (LUTH & IAP) CritPhen 6 October / 30

33 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. Oscillations in the central region, accumulating at (r = 0, t = 0). Discrete self-similarity: φ(t, r) φ(t/e, r/e ) Universality: γ 0,37, 3,44, same profile φ (t, r) for all families. Conjecture: φ exact solution with high symmetry and attractive properties. JMMG (LUTH & IAP) CritPhen 6 October / 30

34 1.2. Choptuik s results Bisection in p (prec ) to BH formation threshold. He found (PRL 93): There is always a well defined value p separating BH formation from dispersal (the threshold is not fractal). It is possible to form arbitrarily small black holes. Scaling: M BH (p) (p p ) γ for p p. Oscillations in the central region, accumulating at (r = 0, t = 0). Discrete self-similarity: φ(t, r) φ(t/e, r/e ) Universality: γ 0,37, 3,44, same profile φ (t, r) for all families. Conjecture: φ exact solution with high symmetry and attractive properties. Comment: Self-similarity is dynamically found, but in a more general form! JMMG (LUTH & IAP) CritPhen 6 October / 30

35 1.3. Further results JMMG (LUTH & IAP) CritPhen 6 October / 30

36 1.3. Further results Independent confirmations: Gundlach (PRL 95): φ as solution of eigenvalue problem. Hamadé & Stewart (CQG 96): higher precision collapse. Naked! JMMG (LUTH & IAP) CritPhen 6 October / 30

37 1.3. Further results Independent confirmations: Gundlach (PRL 95): φ as solution of eigenvalue problem. Hamadé & Stewart (CQG 96): higher precision collapse. Naked! Phenomenology confirmed in more than 20 other systems: Abrahams & Evans (PRL 93): axisymmetric vacuum (DSS). Evans & Coleman (PRL 94): perfect fluid, p = ρ/3 (CSS). Choptuik, Chmaj & Bizoń (PRL 96): SU(2) Yang-Mills (DSS). Liebling & Choptuik (PRL 96): Brans-Dicke (CSS/DSS). JMMG (LUTH & IAP) CritPhen 6 October / 30

38 1.3. Further results Independent confirmations: Gundlach (PRL 95): φ as solution of eigenvalue problem. Hamadé & Stewart (CQG 96): higher precision collapse. Naked! Phenomenology confirmed in more than 20 other systems: Abrahams & Evans (PRL 93): axisymmetric vacuum (DSS). Evans & Coleman (PRL 94): perfect fluid, p = ρ/3 (CSS). Choptuik, Chmaj & Bizoń (PRL 96): SU(2) Yang-Mills (DSS). Liebling & Choptuik (PRL 96): Brans-Dicke (CSS/DSS). Proca, Dirac, sigma fields,..., Vlasov(?) With/without mass, charge, conformal couplings,... Different equations of state for fluids. Other dimensions. JMMG (LUTH & IAP) CritPhen 6 October / 30

39 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. JMMG (LUTH & IAP) CritPhen 6 October / 30

40 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. Christodoulou (AM 94): Naked singularities in scalar field collapse. JMMG (LUTH & IAP) CritPhen 6 October / 30

41 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. Christodoulou (AM 94): Naked singularities in scalar field collapse. Christodoulou (AM 99): They are unstable! JMMG (LUTH & IAP) CritPhen 6 October / 30

42 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. Christodoulou (AM 94): Naked singularities in scalar field collapse. Christodoulou (AM 99): They are unstable! Cosmic censorship is modified: no stable naked singularities. JMMG (LUTH & IAP) CritPhen 6 October / 30

43 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. Christodoulou (AM 94): Naked singularities in scalar field collapse. Christodoulou (AM 99): They are unstable! Cosmic censorship is modified: no stable naked singularities. 1997: Hawking concedes defeat in his famous bet. JMMG (LUTH & IAP) CritPhen 6 October / 30

44 1.3. Back to mathematics Christodoulou & Klainerman 93: Minkowski is nonlinearly stable. Christodoulou (AM 94): Naked singularities in scalar field collapse. Christodoulou (AM 99): They are unstable! Cosmic censorship is modified: no stable naked singularities. 1997: Hawking concedes defeat in his famous bet. New bet! JMMG (LUTH & IAP) CritPhen 6 October / 30

45 Summary 1 Historical introduction Christodoulou Choptuik Other results 2 The current model Self-similarity GR as a dynamical system Mass scaling 3 Interesting results Non-spherical systems Global structure of the critical solution Chaos 4 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

46 2.1. Self-similarity JMMG (LUTH & IAP) CritPhen 6 October / 30

47 2.1. Self-similarity Invariance under change of scale, or absence of a preferred scale. JMMG (LUTH & IAP) CritPhen 6 October / 30

48 2.1. Self-similarity Invariance under change of scale, or absence of a preferred scale. Any continuous symmetry has a discrete version. JMMG (LUTH & IAP) CritPhen 6 October / 30

49 2.1. Geometry of self-similarity JMMG (LUTH & IAP) CritPhen 6 October / 30

50 2.1. Geometry of self-similarity CSS: Homothetic Killing vector ξ a : L ξ g ab = 2 g ab JMMG (LUTH & IAP) CritPhen 6 October / 30

51 2.1. Geometry of self-similarity CSS: Homothetic Killing vector ξ a : L ξ g ab = 2 g ab ¼ Ü Ü Cauchy horizon of the singularity In spherical symmetry, define adapted coordinates: ¾ x r t, t τ log t 0 ÈË Ö Ö ÔÐ Ñ ÒØ point singularity identify regular center Ü Ü past light cone of the singulari Ü Ü Ô JMMG (LUTH & IAP) CritPhen 6 October / 30

52 2.1. Geometry of self-similarity CSS: Homothetic Killing vector ξ a : L ξ g ab = 2 g ab ¼ Ü Ü Cauchy horizon of the singularity In spherical symmetry, define adapted coordinates: ¾ x r t, Then any metric is: t τ log t 0 e 2τ ( Adτ 2 + 2Bdτdx + Cdx 2 + FdΩ 2) with ξ = τ. ÈË Ö Ö ÔÐ Ñ ÒØ point singularity regular center Ü Ü identify past light cone of the singulari Ü Ü Ô JMMG (LUTH & IAP) CritPhen 6 October / 30

53 2.1. Geometry of self-similarity CSS: Homothetic Killing vector ξ a : L ξ g ab = 2 g ab ¼ Ü Ü Cauchy horizon of the singularity In spherical symmetry, define adapted coordinates: ¾ x r t, Then any metric is: t τ log t 0 e 2τ ( Adτ 2 + 2Bdτdx + Cdx 2 + FdΩ 2) with ξ = τ. ÈË Ö Ö ÔÐ Ñ ÒØ CSS: A, B, C, F functions of x only. DSS: also periodic in τ, period. point singularity regular center Ü Ü identify past light cone of the singulari Ü Ü Ô JMMG (LUTH & IAP) CritPhen 6 October / 30

54 2.2. GR as a dynamical system (Evans & Coleman PRL 94; Koike, Hara& Adachi PRL 95; Gundlach PRD 97) JMMG (LUTH & IAP) CritPhen 6 October / 30

55 2.2. GR as a dynamical system (Evans & Coleman PRL 94; Koike, Hara& Adachi PRL 95; Gundlach PRD 97) State S {γ, K, Ψ}. Ṡ = F[S] with some initial S(0) = S 0. JMMG (LUTH & IAP) CritPhen 6 October / 30

56 2.2. GR as a dynamical system (Evans & Coleman PRL 94; Koike, Hara& Adachi PRL 95; Gundlach PRD 97) State S {γ, K, Ψ}. Ṡ = F[S] with some initial S(0) = S 0. Evolution in ( -dim) phase space: CRITICAL SURFACE p>p* supercritical solution subcritical solution Choptuik Schwarzschild(M) p<p* curve of initial data Minkowski JMMG (LUTH & IAP) CritPhen 6 October / 30

57 2.2. GR as a dynamical system (Evans & Coleman PRL 94; Koike, Hara& Adachi PRL 95; Gundlach PRD 97) State S {γ, K, Ψ}. Ṡ = F[S] with some initial S(0) = S 0. Evolution in ( -dim) phase space: CRITICAL SURFACE p>p* supercritical solution subcritical solution Choptuik Schwarzschild(M) p<p* curve of initial data Minkowski Warnings: Which functional space? Asymptotic properties of the spacetimes. Which foliations? Which coordinates? Meaning of attraction? JMMG (LUTH & IAP) CritPhen 6 October / 30

58 2.2. The emerging idea JMMG (LUTH & IAP) CritPhen 6 October / 30

59 2.2. The emerging idea For any system in GR: Find global attractors of evolution: Minkowski, stars, black holes. Restrict to the boundaries among basins of attractors. Find attractors on the boundaries ( critical solutions ). JMMG (LUTH & IAP) CritPhen 6 October / 30

60 2.2. The emerging idea For any system in GR: Find global attractors of evolution: Minkowski, stars, black holes. Restrict to the boundaries among basins of attractors. Find attractors on the boundaries ( critical solutions ). Critical Phenomena Study of basins boundaries in GR phase space. JMMG (LUTH & IAP) CritPhen 6 October / 30

61 2.2. The emerging idea For any system in GR: Find global attractors of evolution: Minkowski, stars, black holes. Restrict to the boundaries among basins of attractors. Find attractors on the boundaries ( critical solutions ). Critical Phenomena Study of basins boundaries in GR phase space. Same mathematical ideas and techniques used in Statistical Mechanics. We believe there is no physical connection. JMMG (LUTH & IAP) CritPhen 6 October / 30

62 2.2. The emerging idea For any system in GR: Find global attractors of evolution: Minkowski, stars, black holes. Restrict to the boundaries among basins of attractors. Find attractors on the boundaries ( critical solutions ). Critical Phenomena Study of basins boundaries in GR phase space. Same mathematical ideas and techniques used in Statistical Mechanics. We believe there is no physical connection. Attraction Forget initial details Highly symmetric solutions: Spherical or axisymmetric Static ( type I ) or self-similar ( type II ). Both continuous or discrete. JMMG (LUTH & IAP) CritPhen 6 October / 30

63 2.2. The emerging idea For any system in GR: Find global attractors of evolution: Minkowski, stars, black holes. Restrict to the boundaries among basins of attractors. Find attractors on the boundaries ( critical solutions ). Critical Phenomena Study of basins boundaries in GR phase space. Same mathematical ideas and techniques used in Statistical Mechanics. We believe there is no physical connection. Attraction Forget initial details Highly symmetric solutions: Spherical or axisymmetric Static ( type I ) or self-similar ( type II ). Both continuous or discrete. Two approaches to criticality: The nonlinear way: Evolution code and fine tune IC families to critical surface. Search for critical phenomena (mainly universality). The linear way: Construct candidate critical solution. Check there is a unique unstable linear mode. JMMG (LUTH & IAP) CritPhen 6 October / 30

64 2.3. Mass scaling JMMG (LUTH & IAP) CritPhen 6 October / 30

65 2.3. Mass scaling Linearize evolution around critical solution (CSS): S p (τ, x) Z (x) + C i (p)e λ i τ s i (x) i=0 JMMG (LUTH & IAP) CritPhen 6 October / 30

66 2.3. Mass scaling Linearize evolution around critical solution (CSS): S p (τ, x) Z (x) + C i (p)e λ i τ s i (x) For large τ only the unstable mode (i = 0) survives. By def C 0 (p ) = 0. S p (τ, x) Z (x) + K (p p )e λ0τ s 0 (x), K dc 0 dp i=0 p JMMG (LUTH & IAP) CritPhen 6 October / 30

67 2.3. Mass scaling Linearize evolution around critical solution (CSS): S p (τ, x) Z (x) + C i (p)e λ i τ s i (x) For large τ only the unstable mode (i = 0) survives. By def C 0 (p ) = 0. S p (τ, x) Z (x) + K (p p )e λ0τ s 0 (x), K dc 0 dp The linear approximation breaks when i=0 K (p p )e λ 0τ p s 0 (x) ɛ p JMMG (LUTH & IAP) CritPhen 6 October / 30

68 2.3. Mass scaling Linearize evolution around critical solution (CSS): S p (τ, x) Z (x) + C i (p)e λ i τ s i (x) For large τ only the unstable mode (i = 0) survives. By def C 0 (p ) = 0. S p (τ, x) Z (x) + K (p p )e λ0τ s 0 (x), K dc 0 dp The linear approximation breaks when i=0 K (p p )e λ 0τ p s 0 (x) ɛ At that time τ p the system has forgotten everything except for the scale ( t p ) = t 0 e τp (p p ) 1/λ 0, M BH (p p ) 1/λ 0, m«ax R (p p ) 2/λ 0 p JMMG (LUTH & IAP) CritPhen 6 October / 30

69 Summary 1 Historical introduction Christodoulou Choptuik Other results 2 The current model Self-similarity GR as a dynamical system Mass scaling 3 Interesting results Non-spherical systems Global structure of the critical solution Chaos 4 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

70 3.1. Massless scalar field JMMG (LUTH & IAP) CritPhen 6 October / 30

71 3.1. Massless scalar field Perturbative results: JMMG (LUTH & IAP) CritPhen 6 October / 30

72 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). JMMG (LUTH & IAP) CritPhen 6 October / 30

73 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). Garfinkle, Gundlach & JMMG PRD 99 : Conjectured scaling law for angular momentum, exponent 0.762(2). JMMG (LUTH & IAP) CritPhen 6 October / 30

74 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). Garfinkle, Gundlach & JMMG PRD 99 : Conjectured scaling law for angular momentum, exponent 0.762(2). (Gundlach & JMMG PRD 96 : Conjectured charge scaling, exponent 0.884(1), confirmed by Hod & Piran PRD 96.) JMMG (LUTH & IAP) CritPhen 6 October / 30

75 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). Garfinkle, Gundlach & JMMG PRD 99 : Conjectured scaling law for angular momentum, exponent 0.762(2). (Gundlach & JMMG PRD 96 : Conjectured charge scaling, exponent 0.884(1), confirmed by Hod & Piran PRD 96.) Non-linear results: JMMG (LUTH & IAP) CritPhen 6 October / 30

76 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). Garfinkle, Gundlach & JMMG PRD 99 : Conjectured scaling law for angular momentum, exponent 0.762(2). (Gundlach & JMMG PRD 96 : Conjectured charge scaling, exponent 0.884(1), confirmed by Hod & Piran PRD 96.) Non-linear results: Choptuik et al PRD 03, axisymmetry: unstable l = 2 polar mode, exponent Critical solution cascade. JMMG (LUTH & IAP) CritPhen 6 October / 30

77 3.1. Massless scalar field Perturbative results: JMMG & Gundlach PRD 99 : All nonspherical perturbations of the Choptuik spacetime decay. Slowest decaying mode is l = 2 polar, with λ = (2)+ i 0.55(9). Garfinkle, Gundlach & JMMG PRD 99 : Conjectured scaling law for angular momentum, exponent 0.762(2). (Gundlach & JMMG PRD 96 : Conjectured charge scaling, exponent 0.884(1), confirmed by Hod & Piran PRD 96.) Non-linear results: Choptuik et al PRD 03, axisymmetry: unstable l = 2 polar mode, exponent Critical solution cascade. Choptuik et al PRL 04: ansatz φ(t, ρ, z, φ) = e imφ ψ(t, ρ, z). DSS criticality. Isolated m sectors. Which unstable? JMMG (LUTH & IAP) CritPhen 6 October / 30

78 3.1. Fluids JMMG (LUTH & IAP) CritPhen 6 October / 30

79 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : JMMG (LUTH & IAP) CritPhen 6 October / 30

80 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). JMMG (LUTH & IAP) CritPhen 6 October / 30

81 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. JMMG (LUTH & IAP) CritPhen 6 October / 30

82 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. k >0.49: many unstable polar modes. JMMG (LUTH & IAP) CritPhen 6 October / 30

83 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. k >0.49: many unstable polar modes. Note: spherically-stable naked singularity for k <0.01 (Harada & Maeda PRD 03, Snajdr CQG 06). JMMG (LUTH & IAP) CritPhen 6 October / 30

84 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : Non-linear results: k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. k >0.49: many unstable polar modes. Note: spherically-stable naked singularity for k <0.01 (Harada & Maeda PRD 03, Snajdr CQG 06). Jin & Suen PRL 07: BH threshold in neutron stars head-on collision. JMMG (LUTH & IAP) CritPhen 6 October / 30

85 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : Non-linear results: k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. k >0.49: many unstable polar modes. Note: spherically-stable naked singularity for k <0.01 (Harada & Maeda PRD 03, Snajdr CQG 06). Jin & Suen PRL 07: BH threshold in neutron stars head-on collision. Signs of type I criticality. Critical solution: oscillating spherical neutron star, probably a finite perturbation of an unstable TOV star (Noble & Choptuik 08). JMMG (LUTH & IAP) CritPhen 6 October / 30

86 3.1. Fluids Perturbative results: Gundlach PRD 01, CSS p = kρ : Non-linear results: k < 1/9 (analytical): l = 1 axial unstable (ballerina effect). 1/9 < k <0.49: stable nonspherical modes. k >0.49: many unstable polar modes. Note: spherically-stable naked singularity for k <0.01 (Harada & Maeda PRD 03, Snajdr CQG 06). Jin & Suen PRL 07: BH threshold in neutron stars head-on collision. Signs of type I criticality. Critical solution: oscillating spherical neutron star, probably a finite perturbation of an unstable TOV star (Noble & Choptuik 08). Universality is unlikely. JMMG (LUTH & IAP) CritPhen 6 October / 30

87 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: JMMG (LUTH & IAP) CritPhen 6 October / 30

88 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. JMMG (LUTH & IAP) CritPhen 6 October / 30

89 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. N circular orbits before merging or dispersing. JMMG (LUTH & IAP) CritPhen 6 October / 30

90 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. N circular orbits before merging or dispersing. e N (p p ) γ, γ JMMG (LUTH & IAP) CritPhen 6 October / 30

91 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. N circular orbits before merging or dispersing. e N (p p ) γ, γ % total energy radiated per orbit. non-stationary criticality. JMMG (LUTH & IAP) CritPhen 6 October / 30

92 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. N circular orbits before merging or dispersing. e N (p p ) γ, γ % total energy radiated per orbit. non-stationary criticality. m«ax N limited by kinetic energy available. JMMG (LUTH & IAP) CritPhen 6 October / 30

93 3.1. Glancing BH collisions Pretorius & Khurana CQG 07: Equal mass BHs. Fine tune boost. N circular orbits before merging or dispersing. e N (p p ) γ, γ % total energy radiated per orbit. non-stationary criticality. m«ax N limited by kinetic energy available. Possible self-similar criticality for zero mass BHs. JMMG (LUTH & IAP) CritPhen 6 October / 30

94 3.2. Global structures for a self-similar spacetime JMMG (LUTH & IAP) CritPhen 6 October / 30

95 3.2. Global structures for a self-similar spacetime Recall structure e 2τ g µν (x). Central singularity at τ =. JMMG (LUTH & IAP) CritPhen 6 October / 30

96 3.2. Global structures for a self-similar spacetime Recall structure e 2τ g µν (x). Central singularity at τ =. Self-similarity horizons: null homothetic lines. JMMG (LUTH & IAP) CritPhen 6 October / 30

97 3.2. Global structures for a self-similar spacetime Recall structure e 2τ g µν (x). Central singularity at τ =. Self-similarity horizons: null homothetic lines. Building blocks: fan and splash (Gundlach & JMMG PRD 03) JMMG (LUTH & IAP) CritPhen 6 October / 30

98 3.2. Global structures for a self-similar spacetime Recall structure e 2τ g µν (x). Central singularity at τ =. Self-similarity horizons: null homothetic lines. Building blocks: fan and splash (Gundlach & JMMG PRD 03) Example: JMMG (LUTH & IAP) CritPhen 6 October / 30

99 3.2. High precision numerical Choptuik spacetime JMMG & Gundlach PRD 03 JMMG (LUTH & IAP) CritPhen 6 October / 30

100 3.2. High precision numerical Choptuik spacetime JMMG & Gundlach PRD 03 Three regions ¼ Ü Ü Cauchy horizon of the singularity ¾ ÈË Ö Ö ÔÐ Ñ ÒØ point singularity identify regular center past light cone of the singularity Ü Ü Ü Ü Ô JMMG (LUTH & IAP) CritPhen 6 October / 30

101 3.2. High precision numerical Choptuik spacetime JMMG & Gundlach PRD 03 Three regions Psedospectral code. Fourier in τ; 4 th order FD in x. ¾ ¼ Ü Ü Cauchy horizon of the singularity ÈË Ö Ö ÔÐ Ñ ÒØ point singularity identify regular center Ü Ü past light cone of the singularity Ü Ü Ô JMMG (LUTH & IAP) CritPhen 6 October / 30

102 3.2. The inner patch Impose DSS and regularity at centre and past light cone. JMMG (LUTH & IAP) CritPhen 6 October / 30

103 3.2. The inner patch Impose DSS and regularity at centre and past light cone. = 3, (3) Ü Ü ÈË Ö Ö ÔÐ Ñ ÒØ Í Ü Î Ü JMMG (LUTH & IAP) CritPhen 6 October / 30

104 3.2. All together JMMG (LUTH & IAP) CritPhen 6 October / 30

105 3.2. All together Oscillations pile up at the Cauchy Horizon, but decay. JMMG (LUTH & IAP) CritPhen 6 October / 30

106 3.2. All together Oscillations pile up at the Cauchy Horizon, but decay. Curvature is continuous but non-differentiable. Continuation not unique: one free function (radiation from the singularity). JMMG (LUTH & IAP) CritPhen 6 October / 30

107 3.2. All together Oscillations pile up at the Cauchy Horizon, but decay. Curvature is continuous but non-differentiable. Continuation not unique: one free function (radiation from the singularity). Unique DSS continuation with regular center (nearly flat): Í ÈË Ö Ö ÔÐ Ñ ÒØ Î Ú Ù ¾ Ú Ù ¾ 0 JMMG (LUTH & IAP) CritPhen 6 October / 30

108 3.2. Global structure of the Choptuik spacetime All other continuations produce a negative mass singularity at the centre, with no new self-similarity horizon: JMMG (LUTH & IAP) CritPhen 6 October / 30

109 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 JMMG (LUTH & IAP) CritPhen 6 October / 30

110 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. JMMG (LUTH & IAP) CritPhen 6 October / 30

111 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. Take ds 2 = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 [ ] e 2B σ1 2 + e 2C σ2 2 + e 2(B+C) σ3 2 σ 1 + i σ 2 = e iψ (cos θ dφ + i dθ), σ 3 = dψ sin θ dφ JMMG (LUTH & IAP) CritPhen 6 October / 30

112 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. Take ds 2 = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 [ ] e 2B σ1 2 + e 2C σ2 2 + e 2(B+C) σ3 2 σ 1 + i σ 2 = e iψ (cos θ dφ + i dθ), σ 3 = dψ sin θ dφ Triaxial symmetry: exchange of the σ i. 6-copy solutions. JMMG (LUTH & IAP) CritPhen 6 October / 30

113 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. Take ds 2 = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 [ ] e 2B σ1 2 + e 2C σ2 2 + e 2(B+C) σ3 2 σ 1 + i σ 2 = e iψ (cos θ dφ + i dθ), σ 3 = dψ sin θ dφ Triaxial symmetry: exchange of the σ i. 6-copy solutions. DSS criticality with B = C (biaxial 3-copy solutions). JMMG (LUTH & IAP) CritPhen 6 October / 30

114 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. Take ds 2 = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 [ ] e 2B σ1 2 + e 2C σ2 2 + e 2(B+C) σ3 2 σ 1 + i σ 2 = e iψ (cos θ dφ + i dθ), σ 3 = dψ sin θ dφ Triaxial symmetry: exchange of the σ i. 6-copy solutions. DSS criticality with B = C (biaxial 3-copy solutions). 3 critical solutions and basins of attraction. JMMG (LUTH & IAP) CritPhen 6 October / 30

115 4+1 vacuum collapse Bizoń et al PRD 05, PRL 05, PRL 06 No Birkhoff theorem in 4+1: gravitational waves in spherical symmetry. Take ds 2 = Ae 2δ dt 2 + A 1 dr 2 + r 2 4 [ ] e 2B σ1 2 + e 2C σ2 2 + e 2(B+C) σ3 2 σ 1 + i σ 2 = e iψ (cos θ dφ + i dθ), σ 3 = dψ sin θ dφ Triaxial symmetry: exchange of the σ i. 6-copy solutions. DSS criticality with B = C (biaxial 3-copy solutions). 3 critical solutions and basins of attraction. Boundaries among those are controled by triaxial DSS codim-2 sols. JMMG (LUTH & IAP) CritPhen 6 October / 30

116 Chaos on the critical surface Szybka & Chmaj PRL 08 JMMG (LUTH & IAP) CritPhen 6 October / 30

117 Chaos on the critical surface Szybka & Chmaj PRL 08 Quadruple precision (32 digits) to fine tune two modes. JMMG (LUTH & IAP) CritPhen 6 October / 30

118 Chaos on the critical surface Szybka & Chmaj PRL 08 Quadruple precision (32 digits) to fine tune two modes. Chaotic evolution within the critical surface: which of three DSS end-state? Reported fractal dim JMMG (LUTH & IAP) CritPhen 6 October / 30

119 Chaos on the critical surface Szybka & Chmaj PRL 08 Quadruple precision (32 digits) to fine tune two modes. Chaotic evolution within the critical surface: which of three DSS end-state? Reported fractal dim κ-family of ICs. Possible end-states h = 1, 1/2, 2 or 0 (unknown). JMMG (LUTH & IAP) CritPhen 6 October / 30

120 Summary 1 Historical introduction Christodoulou Choptuik Other results 2 The current model Self-similarity GR as a dynamical system Mass scaling 3 Interesting results Non-spherical systems Global structure of the critical solution Chaos 4 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

121 Conclusions and open questions JMMG (LUTH & IAP) CritPhen 6 October / 30

122 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. JMMG (LUTH & IAP) CritPhen 6 October / 30

123 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. JMMG (LUTH & IAP) CritPhen 6 October / 30

124 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. JMMG (LUTH & IAP) CritPhen 6 October / 30

125 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. JMMG (LUTH & IAP) CritPhen 6 October / 30

126 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. JMMG (LUTH & IAP) CritPhen 6 October / 30

127 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. JMMG (LUTH & IAP) CritPhen 6 October / 30

128 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. What happens outside spherical symmetries? No 3D simulations so far. JMMG (LUTH & IAP) CritPhen 6 October / 30

129 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. What happens outside spherical symmetries? No 3D simulations so far. Show existence of the Choptuik spacetime. JMMG (LUTH & IAP) CritPhen 6 October / 30

130 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. What happens outside spherical symmetries? No 3D simulations so far. Show existence of the Choptuik spacetime. Can we approximate critical exponents analytically? Holography? JMMG (LUTH & IAP) CritPhen 6 October / 30

131 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. What happens outside spherical symmetries? No 3D simulations so far. Show existence of the Choptuik spacetime. Can we approximate critical exponents analytically? Holography? Quantum and astrophysical relevance? JMMG (LUTH & IAP) CritPhen 6 October / 30

132 Conclusions and open questions It is easy to form a naked singularity: fine tune to the BH threshold. Process controlled by an exact solution ( choptuon ), as gravitational collapse is controlled by a black hole. Route to visible regions with arbitrarily high curvature. First qualitative pictures of GR phase space. Chaos. Numerical Relativity can add new physics to mainstream GR. Note the importance of very high precision numerics. Dynamical understanding of the process missing. What happens outside spherical symmetries? No 3D simulations so far. Show existence of the Choptuik spacetime. Can we approximate critical exponents analytically? Holography? Quantum and astrophysical relevance? Gundlach & JMMG, Living Reviews Relativity JMMG (LUTH & IAP) CritPhen 6 October / 30