Stable Big Bang Formation in General Relativity: The Complete - PowerPoint PPT Presentation

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Stable Big Bang Formation in General Relativity: The Complete Sub-Critical Regime Jared Speck Vanderbilt University October 20, 2020

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Cauchy Problem for Einstein’s equations Ric µν − 1 2 Rg µν = T µν := D µ φ D ν φ − 1 2 g µν D φ · D φ, � g φ = 0 Some results I will describe hold when φ ≡ 0 Data on Σ 1 = T D are tensors (˚ g , ˚ k , ˚ φ 0 , ˚ φ 1 ) verifying the Gauss and Codazzi constraints Our data will be Sobolev-close to Kasner data Choquet-Bruhat and Geroch: data verifying constraints launch a unique maximal globally hyperbolic development ( M , g , φ )

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Cauchy Problem for Einstein’s equations Ric µν − 1 2 Rg µν = T µν := D µ φ D ν φ − 1 2 g µν D φ · D φ, � g φ = 0 Some results I will describe hold when φ ≡ 0 Data on Σ 1 = T D are tensors (˚ g , ˚ k , ˚ φ 0 , ˚ φ 1 ) verifying the Gauss and Codazzi constraints Our data will be Sobolev-close to Kasner data Choquet-Bruhat and Geroch: data verifying constraints launch a unique maximal globally hyperbolic development ( M , g , φ )

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Cauchy Problem for Einstein’s equations Ric µν − 1 2 Rg µν = T µν := D µ φ D ν φ − 1 2 g µν D φ · D φ, � g φ = 0 Some results I will describe hold when φ ≡ 0 Data on Σ 1 = T D are tensors (˚ g , ˚ k , ˚ φ 0 , ˚ φ 1 ) verifying the Gauss and Codazzi constraints Our data will be Sobolev-close to Kasner data Choquet-Bruhat and Geroch: data verifying constraints launch a unique maximal globally hyperbolic development ( M , g , φ )

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Cauchy Problem for Einstein’s equations Ric µν − 1 2 Rg µν = T µν := D µ φ D ν φ − 1 2 g µν D φ · D φ, � g φ = 0 Some results I will describe hold when φ ≡ 0 Data on Σ 1 = T D are tensors (˚ g , ˚ k , ˚ φ 0 , ˚ φ 1 ) verifying the Gauss and Codazzi constraints Our data will be Sobolev-close to Kasner data Choquet-Bruhat and Geroch: data verifying constraints launch a unique maximal globally hyperbolic development ( M , g , φ )

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Cauchy Problem for Einstein’s equations Ric µν − 1 2 Rg µν = T µν := D µ φ D ν φ − 1 2 g µν D φ · D φ, � g φ = 0 Some results I will describe hold when φ ≡ 0 Data on Σ 1 = T D are tensors (˚ g , ˚ k , ˚ φ 0 , ˚ φ 1 ) verifying the Gauss and Codazzi constraints Our data will be Sobolev-close to Kasner data Choquet-Bruhat and Geroch: data verifying constraints launch a unique maximal globally hyperbolic development ( M , g , φ )

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Goal Goal : Understand the formation of stable spacelike singularities in ( M , g , φ ) . Math problem: For which open sets of data does Riem αβγ δ Riem αβγ δ blow up on a spacelike hypersurface? “Dynamic stability of the Big Bang”

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Goal Goal : Understand the formation of stable spacelike singularities in ( M , g , φ ) . Math problem: For which open sets of data does Riem αβγ δ Riem αβγ δ blow up on a spacelike hypersurface? “Dynamic stability of the Big Bang”

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Goal Goal : Understand the formation of stable spacelike singularities in ( M , g , φ ) . Math problem: For which open sets of data does Riem αβγ δ Riem αβγ δ blow up on a spacelike hypersurface? “Dynamic stability of the Big Bang”

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Some sources of inspiration Hawking–Penrose “singularity” theorems. Explicit solutions, especially FLRW and Kasner. Heuristics from the physics literature. Numerical work on singularities. Rigorous results in symmetry and analytic class. Dafermos–Luk.

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future “Generalized” Kasner solutions D � t 2 q I dx I ⊗ dx I , g KAS = − dt ⊗ dt + φ KAS = B ln t I = 1 The q I ∈ ( − 1 , 1 ] and B ≥ 0 verify the Kasner constraints: D D � � ( q I ) 2 = 1 − B 2 q I = 1 , I = 1 I = 1 Riem αβγ δ Riem αβγ δ = Ct − 4 where C > 0 (unless one q I equals 1 and the rest vanish) “Big Bang” singularity at t = 0

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future “Generalized” Kasner solutions D � t 2 q I dx I ⊗ dx I , g KAS = − dt ⊗ dt + φ KAS = B ln t I = 1 The q I ∈ ( − 1 , 1 ] and B ≥ 0 verify the Kasner constraints: D D � � ( q I ) 2 = 1 − B 2 q I = 1 , I = 1 I = 1 Riem αβγ δ Riem αβγ δ = Ct − 4 where C > 0 (unless one q I equals 1 and the rest vanish) “Big Bang” singularity at t = 0

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future “Generalized” Kasner solutions D � t 2 q I dx I ⊗ dx I , g KAS = − dt ⊗ dt + φ KAS = B ln t I = 1 The q I ∈ ( − 1 , 1 ] and B ≥ 0 verify the Kasner constraints: D D � � ( q I ) 2 = 1 − B 2 q I = 1 , I = 1 I = 1 Riem αβγ δ Riem αβγ δ = Ct − 4 where C > 0 (unless one q I equals 1 and the rest vanish) “Big Bang” singularity at t = 0

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future “Generalized” Kasner solutions D � t 2 q I dx I ⊗ dx I , g KAS = − dt ⊗ dt + φ KAS = B ln t I = 1 The q I ∈ ( − 1 , 1 ] and B ≥ 0 verify the Kasner constraints: D D � � ( q I ) 2 = 1 − B 2 q I = 1 , I = 1 I = 1 Riem αβγ δ Riem αβγ δ = Ct − 4 where C > 0 (unless one q I equals 1 and the rest vanish) “Big Bang” singularity at t = 0

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Hawking’s incompleteness theorem Theorem (Hawking) Assume ( M , g , φ ) is the maximal globally hyperbolic g , ˚ k , ˚ φ 0 , ˚ development of data (˚ φ 1 ) on Σ 1 ≃ T D tr ˚ k < − C < 0 Then no past-directed timelike geodesic emanating from Σ 1 is longer than C ′ < ∞ . • Hawking’s theorem applies to perturbations of Kasner: tr ˚ k KAS = − 1.

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Hawking’s incompleteness theorem Theorem (Hawking) Assume ( M , g , φ ) is the maximal globally hyperbolic g , ˚ k , ˚ φ 0 , ˚ development of data (˚ φ 1 ) on Σ 1 ≃ T D tr ˚ k < − C < 0 Then no past-directed timelike geodesic emanating from Σ 1 is longer than C ′ < ∞ . • Hawking’s theorem applies to perturbations of Kasner: tr ˚ k KAS = − 1.

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Hawking’s incompleteness theorem Theorem (Hawking) Assume ( M , g , φ ) is the maximal globally hyperbolic g , ˚ k , ˚ φ 0 , ˚ development of data (˚ φ 1 ) on Σ 1 ≃ T D tr ˚ k < − C < 0 Then no past-directed timelike geodesic emanating from Σ 1 is longer than C ′ < ∞ . • Hawking’s theorem applies to perturbations of Kasner: tr ˚ k KAS = − 1.

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Why? Glaring question: Why are the timelike geodesics incomplete? For Kasner, incompleteness ↔ Big Bang, but what about perturbations?

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Why? Glaring question: Why are the timelike geodesics incomplete? For Kasner, incompleteness ↔ Big Bang, but what about perturbations?

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Potential sources of incompleteness Curvature blowup/crushing singularities à la Kasner Cauchy horizon formation à la Kerr black hole interiors

Cauchy problem Goals Kasner Incompleteness Oscillatory vs Monotonic Results The Gauge Proof Hints Future Potential sources of incompleteness Curvature blowup/crushing singularities à la Kasner Cauchy horizon formation à la Kerr black hole interiors