Stable Big Bang Formation in Solutions to the Einstein-Scalar Field System

Transcription

1 Stable Big Bang Formation in Solutions to the Einstein-Scalar Field System Jared Speck Igor Rodnianski Massachusetts Institute of Technology

2 The Einstein-scalar field eqns. on (T, 1] T 3 Ric µν 1 Rg µν = T µν, g φ = 0 T µν := D µ φd ν φ 1 g µνdφ Dφ The unknowns are the Lorentzian metric g (, +, +, +), the scalar field φ, and the spacetime manifold M Data are: (Σ 1 T 3, g, k, φ 0, φ 1 ) Choquet-Bruhat and Geroch: data verifying constraints launch a maximal globally hyperbolic development (M, g, φ)

3 The Einstein-scalar field eqns. on (T, 1] T 3 Ric µν 1 Rg µν = T µν, g φ = 0 T µν := D µ φd ν φ 1 g µνdφ Dφ The unknowns are the Lorentzian metric g (, +, +, +), the scalar field φ, and the spacetime manifold M Data are: (Σ 1 T 3, g, k, φ 0, φ 1 ) Choquet-Bruhat and Geroch: data verifying constraints launch a maximal globally hyperbolic development (M, g, φ)

4 The Einstein-scalar field eqns. on (T, 1] T 3 Ric µν 1 Rg µν = T µν, g φ = 0 T µν := D µ φd ν φ 1 g µνdφ Dφ The unknowns are the Lorentzian metric g (, +, +, +), the scalar field φ, and the spacetime manifold M Data are: (Σ 1 T 3, g, k, φ 0, φ 1 ) Choquet-Bruhat and Geroch: data verifying constraints launch a maximal globally hyperbolic development (M, g, φ)

5 Hawking-Penrose Theorem (Hawking-Penrose theorem in dimensions) Assume (M, g, φ) is the maximal globally hyperbolic development of data (Σ 1, g, k, φ 0, φ 1 ) Ric αβ X α X β 0 for timelike X (strong energy) tr k < C < 0 Then no past-directed timelike geodesic emanating from Σ 1 is longer than 3 C. Fundamental question: does geodesic incompleteness signify a real singularity (e.g. curvature blow-up), or something more sinister (e.g. Cauchy horizon without curvature blow-up)?

6 Hawking-Penrose Theorem (Hawking-Penrose theorem in dimensions) Assume (M, g, φ) is the maximal globally hyperbolic development of data (Σ 1, g, k, φ 0, φ 1 ) Ric αβ X α X β 0 for timelike X (strong energy) tr k < C < 0 Then no past-directed timelike geodesic emanating from Σ 1 is longer than 3 C. Fundamental question: does geodesic incompleteness signify a real singularity (e.g. curvature blow-up), or something more sinister (e.g. Cauchy horizon without curvature blow-up)?

7 Kasner solutions (Generalized) Kasner solutions on (0, ) T 3 : g KAS = dt + 3 t q j (dx j ), j=1 φ KAS = A ln t Constraints: I) 3 j=1 q j = 1, II) 3 j=1 q j = 1 A Special case - FLRW; all q j = 1/3 g FLRW = dt + t /3 3 (dx j ), φ FLRW = j=1 3 ln t Riem FLRW g FLRW t 4 = Big Bang at {t = 0}

8 Kasner solutions (Generalized) Kasner solutions on (0, ) T 3 : g KAS = dt + 3 t q j (dx j ), j=1 φ KAS = A ln t Constraints: I) 3 j=1 q j = 1, II) 3 j=1 q j = 1 A Special case - FLRW; all q j = 1/3 g FLRW = dt + t /3 3 (dx j ), φ FLRW = j=1 3 ln t Riem FLRW g FLRW t 4 = Big Bang at {t = 0}

9 Kasner solutions (Generalized) Kasner solutions on (0, ) T 3 : g KAS = dt + 3 t q j (dx j ), j=1 φ KAS = A ln t Constraints: I) 3 j=1 q j = 1, II) 3 j=1 q j = 1 A Special case - FLRW; all q j = 1/3 g FLRW = dt + t /3 3 (dx j ), φ FLRW = j=1 3 ln t Riem FLRW g FLRW t 4 = Big Bang at {t = 0}

10 Brief overview Overview of results We slightly perturb the FLRW data (in a Sobolev space) at Σ 1 := {t = 1}. No symmetry or analyticity. We fully describe the entire past of Σ 1 in M. We understand the origin of geodesic incompleteness: Big Bang with curvature blow-up. This yields a proof of Strong Cosmic Censorship for the past-half of the perturbed spacetimes.

11 Brief overview Overview of results We slightly perturb the FLRW data (in a Sobolev space) at Σ 1 := {t = 1}. No symmetry or analyticity. We fully describe the entire past of Σ 1 in M. We understand the origin of geodesic incompleteness: Big Bang with curvature blow-up. This yields a proof of Strong Cosmic Censorship for the past-half of the perturbed spacetimes.

12 Other contributors Many people have investigated solutions to the Einstein equations near spacelike singularities: Partial list of contributors Aizawa, Akhoury, Andersson, Anguige, Aninos, Antoniou, Barrow, Béguin, Berger, Beyer, Chitré, Claudel, Coley, Cornish, Chrusciel, Damour, Eardley, Ellis, Elskens, van Elst, Garfinkle, Goode, Grubišić, Heinzle, Henneaux, Hsu, Isenberg, Kichenassamy, Koguro, LeBlanc, LeFloch, Levin, Liang, Lim, Misner, Moncrief, Newman, Nicolai, Reiterer, Rendall, Ringström, Röhr, Sachs, Saotome, Ståhl, Tod, Trubowitz, Uggla, Wainwright, Weaver,

13 Behavior of solutions near singularities There are no prior rigorous results covering a truly open class of regular Cauchy data. Our solutions are approximately monotonic (non-oscillatory). The Belinskiĭ-Khalatnikov-Lifshitz (oscillatory ODE-type) picture has been confirmed for some matter models in some symmetry classes (e.g. Bianchi IX [Ringström]). Generally speaking, there are many other scenarios that could in principle occur near singularities: (e.g. Taub, spikes [Weaver-Rendall, Lim].

14 Behavior of solutions near singularities There are no prior rigorous results covering a truly open class of regular Cauchy data. Our solutions are approximately monotonic (non-oscillatory). The Belinskiĭ-Khalatnikov-Lifshitz (oscillatory ODE-type) picture has been confirmed for some matter models in some symmetry classes (e.g. Bianchi IX [Ringström]). Generally speaking, there are many other scenarios that could in principle occur near singularities: (e.g. Taub, spikes [Weaver-Rendall, Lim].

15 Behavior of solutions near singularities There are no prior rigorous results covering a truly open class of regular Cauchy data. Our solutions are approximately monotonic (non-oscillatory). The Belinskiĭ-Khalatnikov-Lifshitz (oscillatory ODE-type) picture has been confirmed for some matter models in some symmetry classes (e.g. Bianchi IX [Ringström]). Generally speaking, there are many other scenarios that could in principle occur near singularities: (e.g. Taub, spikes [Weaver-Rendall, Lim].

16 Closely connected results Part I Ringström - (ODE) solutions to Euler-Einstein with p = c s ρ and 0 c s 1 : For Bianchi A spacetimes: curvature blow-up at the initial singularity (except for vacuum Taub) For Bianchi IX spacetimes with c s < 1: there are generically at least 3 distinct limit points Vacuum Type II in the approach towards the singularity ( matter doesn t matter ) For Bianchi A spacetimes with a stiff fluid p = ρ : the solution converges to a singular point ( matter matters )

17 Closely connected results Part I Ringström - (ODE) solutions to Euler-Einstein with p = c s ρ and 0 c s 1 : For Bianchi A spacetimes: curvature blow-up at the initial singularity (except for vacuum Taub) For Bianchi IX spacetimes with c s < 1: there are generically at least 3 distinct limit points Vacuum Type II in the approach towards the singularity ( matter doesn t matter ) For Bianchi A spacetimes with a stiff fluid p = ρ : the solution converges to a singular point ( matter matters )

18 Closely connected results Part I Ringström - (ODE) solutions to Euler-Einstein with p = c s ρ and 0 c s 1 : For Bianchi A spacetimes: curvature blow-up at the initial singularity (except for vacuum Taub) For Bianchi IX spacetimes with c s < 1: there are generically at least 3 distinct limit points Vacuum Type II in the approach towards the singularity ( matter doesn t matter ) For Bianchi A spacetimes with a stiff fluid p = ρ : the solution converges to a singular point ( matter matters )

19 Closely connected results Part II Andersson-Rendall - Quiescent cosmological singularities for Einstein-stiff fluid/scalar field: A-R constructed a large family of spatially analytic solutions to the Einstein equations Same number of degrees of freedom as the general solution ODE-type behavior near the singularity Asymptotic to VTD solutions (i.e., Einstein without spatial derivatives) There is no oscillatory behavior near the singularity

20 We now present the main ideas behind the proof of stable singularity formation. The hard part: existence down to the Big-Bang + estimates.

21 Transported spatial coordinates We consider g in the form g = n dt + g ab dx a dx b n is the lapse g is a Riemannian metric on Σ t The {x i } i=1,,3 are (local) spatial coordinates on the Σ t For FLRW: n = 1, g ab = t /3 δ ab, t φ = t 1, φ = 0 3 := connection of g and is its Laplacian

22 Transported spatial coordinates We consider g in the form g = n dt + g ab dx a dx b n is the lapse g is a Riemannian metric on Σ t The {x i } i=1,,3 are (local) spatial coordinates on the Σ t For FLRW: n = 1, g ab = t /3 δ ab, t φ = t 1, φ = 0 3 := connection of g and is its Laplacian

23 Transported spatial coordinates We consider g in the form g = n dt + g ab dx a dx b n is the lapse g is a Riemannian metric on Σ t The {x i } i=1,,3 are (local) spatial coordinates on the Σ t For FLRW: n = 1, g ab = t /3 δ ab, t φ = t 1, φ = 0 3 := connection of g and is its Laplacian

24 Transported spatial coordinates We consider g in the form g = n dt + g ab dx a dx b n is the lapse g is a Riemannian metric on Σ t The {x i } i=1,,3 are (local) spatial coordinates on the Σ t For FLRW: n = 1, g ab = t /3 δ ab, t φ = t 1, φ = 0 3 := connection of g and is its Laplacian

25 Constant mean curvature gauge We impose trk = t 1 along Σ t, which, upon using the constraints, leads to: [ ] t n t 1 (n 1) = 3 t 1 n 1 t t φ 3 } {{ } crucially important linear term + quadratic error k t g = second fundamental form of Σ t propagation speed synchronizes the singularity

26 Constant mean curvature gauge We impose trk = t 1 along Σ t, which, upon using the constraints, leads to: [ ] t n t 1 (n 1) = 3 t 1 n 1 t t φ 3 } {{ } crucially important linear term + quadratic error k t g = second fundamental form of Σ t propagation speed synchronizes the singularity

27 Parabolic lapse gauge Alternate parabolic lapse gauge: impose λ 1 (n 1) = ttrk + 1 along Σ t, which leads to: λ 1 t n + t n = ( ) λ 1 t 1 (n 1) [ ] + 3 t 1 n 1 t t φ +error 3 } {{ } crucially important linear term First use of a parabolic gauge in GR: Balakrishna, Daues, Seidel, Suen, Tobias, and Wang (CQG 1996) No need to construct CMC hypersurface Note: λ = is CMC gauge

28 Parabolic lapse gauge Alternate parabolic lapse gauge: impose λ 1 (n 1) = ttrk + 1 along Σ t, which leads to: λ 1 t n + t n = ( ) λ 1 t 1 (n 1) [ ] + 3 t 1 n 1 t t φ +error 3 } {{ } crucially important linear term First use of a parabolic gauge in GR: Balakrishna, Daues, Seidel, Suen, Tobias, and Wang (CQG 1996) No need to construct CMC hypersurface Note: λ = is CMC gauge

29 The eqns. in CMC gauge For FLRW: n = 1, g ab = t /3 δ ab, t φ = Constraint equations 3 t 1, φ = 0 R k + t = (n 1 t φ) + φ φ, a k a i = n 1 t φ i φ Evolution equations t g ij = ng ia k a j, t (k i j) = i j n + n { R i j t 1 k i j i φ j φ }, t (n 1 t φ) + n φ = t 1 t φ φ n

30 Energies for the perturbed solution E (φ) (t) := E (g) (t) := Σ t Σ t [ ] n 1 t t φ + t φ 3 dx, {t ˆk + 14 } t g dx The energies vanish for the FLRW solution ˆk is the trace-free part of k The t factors lead to cancellation of some linear terms

31 Energy identity for φ For t < 1, integrating by parts using g φ = 0 yields: E (φ) (t) = E (φ) (1) 4 1 s φ dx ds 3 s=t Σ s [ 1 ] + s 1 (n 1) n 3 s=t Σ 1 s t φ dx ds s 3 } {{ } dangerous quad. integral 1 s φ n dx ds +cubic error 3 s=t Σ } {{ s } dangerous quad. integral

32 Elliptic estimate for n Integrating by parts using ] the lapse PDE [n t 1 1 t 3 t φ = t n t 1 (n 1) +, we 3 have [ ] 3 s 1 (n 1) n 1 s t φ dx Σ s 3 } {{ } dangerous quad. integral = s n + s 1 (n 1) dx Σ s + cubic error

33 Combined identity for t (0, 1] E (φ) (t) = E (φ) (1) 4 1 s φ dx ds 3 s=t Σ s 1 s n + s 1 (n 1) dx ds s=t Σ s 1 s φ n dx ds + cubic error 3 s=t Σ s Note that 3 φ n φ + (/3) n Hence, E (φ) is past-decreasing up to cubic errors!

34 Combined identity for t (0, 1] E (φ) (t) = E (φ) (1) 4 1 s φ dx ds 3 s=t Σ s 1 s n + s 1 (n 1) dx ds s=t Σ s 1 s φ n dx ds + cubic error 3 s=t Σ s Note that 3 φ n φ + (/3) n Hence, E (φ) is past-decreasing up to cubic errors!

35 Energy identity for the metric For t < 1, integrating by parts on the metric evolution equations and using the momentum constraint yields: E (g) (t) = E (g) (1) 1 1 s g dx ds 3 s=t Σ s 1 + s Q( g, n) + Q( g, φ) + Q( φ, n) dx ds s=t Σ } s {{ } dangerous quad. integrals + cubic error

36 Energy inequality for the system Consider the summed energy identity for E (Total) (t) := E (φ) (t) + θe (g) (t). For θ > 0 small, we can absorb all unsigned quadratic integrals into the negative ones! Thus, we conclude that E (Total) (t) is past-decreasing up to cubic error terms! The same estimates hold for the higher derivatives of the solution. The primary remaining difficulty is to bound borderline cubic error integrals that cannot be absorbed.

37 Energy inequality for the system Consider the summed energy identity for E (Total) (t) := E (φ) (t) + θe (g) (t). For θ > 0 small, we can absorb all unsigned quadratic integrals into the negative ones! Thus, we conclude that E (Total) (t) is past-decreasing up to cubic error terms! The same estimates hold for the higher derivatives of the solution. The primary remaining difficulty is to bound borderline cubic error integrals that cannot be absorbed.

38 Energy inequality for the system Consider the summed energy identity for E (Total) (t) := E (φ) (t) + θe (g) (t). For θ > 0 small, we can absorb all unsigned quadratic integrals into the negative ones! Thus, we conclude that E (Total) (t) is past-decreasing up to cubic error terms! The same estimates hold for the higher derivatives of the solution. The primary remaining difficulty is to bound borderline cubic error integrals that cannot be absorbed.

39 Energy inequality for the system Consider the summed energy identity for E (Total) (t) := E (φ) (t) + θe (g) (t). For θ > 0 small, we can absorb all unsigned quadratic integrals into the negative ones! Thus, we conclude that E (Total) (t) is past-decreasing up to cubic error terms! The same estimates hold for the higher derivatives of the solution. The primary remaining difficulty is to bound borderline cubic error integrals that cannot be absorbed.

40 An aside on metric energy estimates In transported coordinates, we have R ij = 1 gab a b g ij + 1 { } i Γ j (g, g) + j Γ i (g, g) + l.o.t. 1 E (g) (t) = E (g) (1) + s=t s nk ab a Γ b dx ds + l.o.t. Σ s Momentum constraint equation + gauge = a k ab = l.o.t Hence, integrating by parts on Σ s, we have s nk ab a Γ b dx = Σ s s n( a k ab )Γ b dx + l.o.t. Σ s } {{ } l.o.t.

41 An aside on metric energy estimates In transported coordinates, we have R ij = 1 gab a b g ij + 1 { } i Γ j (g, g) + j Γ i (g, g) + l.o.t. 1 E (g) (t) = E (g) (1) + s=t s nk ab a Γ b dx ds + l.o.t. Σ s Momentum constraint equation + gauge = a k ab = l.o.t Hence, integrating by parts on Σ s, we have s nk ab a Γ b dx = Σ s s n( a k ab )Γ b dx + l.o.t. Σ s } {{ } l.o.t.

42 An aside on metric energy estimates In transported coordinates, we have R ij = 1 gab a b g ij + 1 { } i Γ j (g, g) + j Γ i (g, g) + l.o.t. 1 E (g) (t) = E (g) (1) + s=t s nk ab a Γ b dx ds + l.o.t. Σ s Momentum constraint equation + gauge = a k ab = l.o.t Hence, integrating by parts on Σ s, we have s nk ab a Γ b dx = Σ s s n( a k ab )Γ b dx + l.o.t. Σ s } {{ } l.o.t.

43 An aside on metric energy estimates In transported coordinates, we have R ij = 1 gab a b g ij + 1 { } i Γ j (g, g) + j Γ i (g, g) + l.o.t. 1 E (g) (t) = E (g) (1) + s=t s nk ab a Γ b dx ds + l.o.t. Σ s Momentum constraint equation + gauge = a k ab = l.o.t Hence, integrating by parts on Σ s, we have s nk ab a Γ b dx = Σ s s n( a k ab )Γ b dx + l.o.t. Σ s } {{ } l.o.t.

44 Borderline error terms We must bound borderline cubic error integrals: E (Total) (t) CE (Total) (1) + 1 s=t ˆk L (Σ s)e (Total) (s) ds + We adopt the Bootstrap Assumption (ɛ > 0 is small): ˆk L (Σ s) ɛs 1 (saturated by Kasner) Then by Gronwall, we find that for t < 1: E (Total) (t) CE (Total) (1)t ɛ Unfortunately, Sobolev embedding + the bound on E (Total) = ˆk L (Σ s) E (Total) (1)s 1 ɛ/ Inconsistent!

45 Borderline error terms We must bound borderline cubic error integrals: E (Total) (t) CE (Total) (1) + 1 s=t ˆk L (Σ s)e (Total) (s) ds + We adopt the Bootstrap Assumption (ɛ > 0 is small): ˆk L (Σ s) ɛs 1 (saturated by Kasner) Then by Gronwall, we find that for t < 1: E (Total) (t) CE (Total) (1)t ɛ Unfortunately, Sobolev embedding + the bound on E (Total) = ˆk L (Σ s) E (Total) (1)s 1 ɛ/ Inconsistent!

46 Borderline error terms We must bound borderline cubic error integrals: E (Total) (t) CE (Total) (1) + 1 s=t ˆk L (Σ s)e (Total) (s) ds + We adopt the Bootstrap Assumption (ɛ > 0 is small): ˆk L (Σ s) ɛs 1 (saturated by Kasner) Then by Gronwall, we find that for t < 1: E (Total) (t) CE (Total) (1)t ɛ Unfortunately, Sobolev embedding + the bound on E (Total) = ˆk L (Σ s) E (Total) (1)s 1 ɛ/ Inconsistent!

47 Borderline error terms We must bound borderline cubic error integrals: E (Total) (t) CE (Total) (1) + 1 s=t ˆk L (Σ s)e (Total) (s) ds + We adopt the Bootstrap Assumption (ɛ > 0 is small): ˆk L (Σ s) ɛs 1 (saturated by Kasner) Then by Gronwall, we find that for t < 1: E (Total) (t) CE (Total) (1)t ɛ Unfortunately, Sobolev embedding + the bound on E (Total) = ˆk L (Σ s) E (Total) (1)s 1 ɛ/ Inconsistent!

48 How to recover the BA for ˆk L (Σ t ) E (8;Total) := the energy of 8 derivatives. Make the BA: E (8;Total) (t) ɛt σ, (ɛ, σ > 0 are small) We use the evolution eqn. t (t ˆk i j ) = & Sobolev embedding to derive a VTD estimate: t (t ˆk i j) t(g 1 ) g + ɛt } {{ } } 1/3 Z {{ } σ tr i loses derivatives j Z Z + = Max. # of factors in tensor products We integrate in time, use 1 s=0 s 1/3 Z σ ds < (for small σ), and the small-data assumption ˆk L (Σ 1 ) ɛ: t ˆk i j ɛ, t (0, 1] = ˆk L (Σ t ) ɛt 1

49 How to recover the BA for ˆk L (Σ t ) E (8;Total) := the energy of 8 derivatives. Make the BA: E (8;Total) (t) ɛt σ, (ɛ, σ > 0 are small) We use the evolution eqn. t (t ˆk i j ) = & Sobolev embedding to derive a VTD estimate: t (t ˆk i j) t(g 1 ) g + ɛt } {{ } } 1/3 Z {{ } σ tr i loses derivatives j Z Z + = Max. # of factors in tensor products We integrate in time, use 1 s=0 s 1/3 Z σ ds < (for small σ), and the small-data assumption ˆk L (Σ 1 ) ɛ: t ˆk i j ɛ, t (0, 1] = ˆk L (Σ t ) ɛt 1

50 How to recover the BA for ˆk L (Σ t ) E (8;Total) := the energy of 8 derivatives. Make the BA: E (8;Total) (t) ɛt σ, (ɛ, σ > 0 are small) We use the evolution eqn. t (t ˆk i j ) = & Sobolev embedding to derive a VTD estimate: t (t ˆk i j) t(g 1 ) g + ɛt } {{ } } 1/3 Z {{ } σ tr i loses derivatives j Z Z + = Max. # of factors in tensor products We integrate in time, use 1 s=0 s 1/3 Z σ ds < (for small σ), and the small-data assumption ˆk L (Σ 1 ) ɛ: t ˆk i j ɛ, t (0, 1] = ˆk L (Σ t ) ɛt 1

51 Summary of main results Theorem (RS; Nonlinear stability of the FLRW Big Bang) Consider near-flrw (H 8 close) data of small size ɛ for the Einstein-scalar field system on Σ 1 = T3. (Gerhardt, Bartnik) a CMC slice Σ 1 near Σ 1 Global energy bound: E (8;Total) (t) ɛt c ɛ, t (0, 1] The past of Σ 1 is foliated by a family of CMC hypersurfaces Σ t of mean curvature 1 3 t 1, t (0, 1] Big Bang: The volume of Σ t collapses to 0 as t 0 Convergence and Stability: t t φ, n, tk i j, t 1 g have finite, near (rescaled) FLRW limits as t 0 SCC: Riem g blows up like t 4 as t 0 H-P: All past-directed timelike geodesics emanating from Σ t are shorter than Ct /3 cɛ VTD: Many spatial derivative terms negligible near t = 0

52 Why are scalar fields special? CMC-lapse equation for a scalar field: { } n (n 1)t = (n 1) R + t + φ φ) + R φ φ = only spatial derivatives The absence of time derivatives is connected to the fact that there is only one characteristic cone in the Einstein-scalar field system. For some other matter models, time derivatives appear, and it is not clear whether or not the low-order spatial derivatives become negligible near {t = 0}; our approach does not apply.

53 Why are scalar fields special? CMC-lapse equation for a scalar field: { } n (n 1)t = (n 1) R + t + φ φ) + R φ φ = only spatial derivatives The absence of time derivatives is connected to the fact that there is only one characteristic cone in the Einstein-scalar field system. For some other matter models, time derivatives appear, and it is not clear whether or not the low-order spatial derivatives become negligible near {t = 0}; our approach does not apply.

54 Future directions How large can q j 1/3 be? (Our proof can be extended to q j 1/3 = δ >> ɛ, but what is the sharp δ?) Other topologies. Other matter models. Stable vs. unstable directions in other regimes.

55 Preprints of the linear and nonlinear article are available at arxiv.org