Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Stable Big Bang Formation in Solutions
to the Einstein-Scalar Field System
Jared Speck
Igor Rodnianski
Massachusetts Institute of Technology
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
The Einstein-scalar field eqns. on (T , 1] × T3
1
Ricµν − Rgµν = Tµν ,
2
g φ = 0
1
Tµν := Dµ φDν φ − gµν Dφ · Dφ
2
The unknowns are the Lorentzian metric
g ∼ (−, +, +, +), the scalar field φ, and the spacetime
manifold M
Data are: (Σ1 ' T3 , g̊, k̊ , φ̊0 , φ̊1 )
Choquet-Bruhat and Geroch: data verifying
constraints launch a maximal globally hyperbolic
development (M, g, φ)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
The Einstein-scalar field eqns. on (T , 1] × T3
1
Ricµν − Rgµν = Tµν ,
2
g φ = 0
1
Tµν := Dµ φDν φ − gµν Dφ · Dφ
2
The unknowns are the Lorentzian metric
g ∼ (−, +, +, +), the scalar field φ, and the spacetime
manifold M
Data are: (Σ1 ' T3 , g̊, k̊ , φ̊0 , φ̊1 )
Choquet-Bruhat and Geroch: data verifying
constraints launch a maximal globally hyperbolic
development (M, g, φ)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
The Einstein-scalar field eqns. on (T , 1] × T3
1
Ricµν − Rgµν = Tµν ,
2
g φ = 0
1
Tµν := Dµ φDν φ − gµν Dφ · Dφ
2
The unknowns are the Lorentzian metric
g ∼ (−, +, +, +), the scalar field φ, and the spacetime
manifold M
Data are: (Σ1 ' T3 , g̊, k̊ , φ̊0 , φ̊1 )
Choquet-Bruhat and Geroch: data verifying
constraints launch a maximal globally hyperbolic
development (M, g, φ)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Hawking-Penrose
Theorem (Hawking-Penrose theorem in 1 + 3 dimensions)
Assume
(M, g, φ) is the maximal globally hyperbolic
development of data (Σ1 , g̊, k̊ , φ̊0 , φ̊1 )
Ricαβ Xα Xβ ≥ 0 for timelike X (strong energy)
trk̊ < −C < 0
Then no past-directed timelike geodesic emanating from
Σ1 is longer than C3 .
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)?
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Hawking-Penrose
Theorem (Hawking-Penrose theorem in 1 + 3 dimensions)
Assume
(M, g, φ) is the maximal globally hyperbolic
development of data (Σ1 , g̊, k̊ , φ̊0 , φ̊1 )
Ricαβ Xα Xβ ≥ 0 for timelike X (strong energy)
trk̊ < −C < 0
Then no past-directed timelike geodesic emanating from
Σ1 is longer than C3 .
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)?
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Kasner solutions
• (Generalized) Kasner solutions on (0, ∞) × T3 :
2
gKAS = −dt +
3
X
t 2qj (dx j )2 ,
φKAS = A ln t
j=1
• Constraints: I)
P3
j=1
qj = 1,
II)
P3
j=1
qj2 = 1 − A2
• Special case - FLRW; all qj = 1/3
gFLRW
3
X
2
2/3
= −dt + t
(dx j )2 ,
j=1
r
φFLRW =
2
ln t
3
|RiemFLRW |2gFLRW ∼ t −4 =⇒ Big Bang at {t = 0}
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Kasner solutions
• (Generalized) Kasner solutions on (0, ∞) × T3 :
2
gKAS = −dt +
3
X
t 2qj (dx j )2 ,
φKAS = A ln t
j=1
• Constraints: I)
P3
j=1
qj = 1,
II)
P3
j=1
qj2 = 1 − A2
• Special case - FLRW; all qj = 1/3
gFLRW
3
X
2
2/3
= −dt + t
(dx j )2 ,
j=1
r
φFLRW =
2
ln t
3
|RiemFLRW |2gFLRW ∼ t −4 =⇒ Big Bang at {t = 0}
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Kasner solutions
• (Generalized) Kasner solutions on (0, ∞) × T3 :
2
gKAS = −dt +
3
X
t 2qj (dx j )2 ,
φKAS = A ln t
j=1
• Constraints: I)
P3
j=1
qj = 1,
II)
P3
j=1
qj2 = 1 − A2
• Special case - FLRW; all qj = 1/3
gFLRW
3
X
2
2/3
= −dt + t
(dx j )2 ,
j=1
r
φFLRW =
2
ln t
3
|RiemFLRW |2gFLRW ∼ t −4 =⇒ Big Bang at {t = 0}
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Brief overview
Overview of results
We slightly perturb the FLRW data (in a Sobolev
space) at Σ1 := {t = 1}. No symmetry or analyticity.
We fully describe the “entire” past of Σ1 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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Brief overview
Overview of results
We slightly perturb the FLRW data (in a Sobolev
space) at Σ1 := {t = 1}. No symmetry or analyticity.
We fully describe the “entire” past of Σ1 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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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, · · ·
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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 Belinskiı̆-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].
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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 Belinskiı̆-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].
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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 Belinskiı̆-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].
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Closely connected results Part I
Ringström - (ODE) solutions to Euler-Einstein with
p = cs2 ρ and 0 ≤ cs2 ≤ 1 :
For Bianchi A spacetimes: curvature blow-up at the
initial singularity (except for vacuum Taub)
For Bianchi IX spacetimes with cs2 < 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”)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Closely connected results Part I
Ringström - (ODE) solutions to Euler-Einstein with
p = cs2 ρ and 0 ≤ cs2 ≤ 1 :
For Bianchi A spacetimes: curvature blow-up at the
initial singularity (except for vacuum Taub)
For Bianchi IX spacetimes with cs2 < 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”)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Closely connected results Part I
Ringström - (ODE) solutions to Euler-Einstein with
p = cs2 ρ and 0 ≤ cs2 ≤ 1 :
For Bianchi A spacetimes: curvature blow-up at the
initial singularity (except for vacuum Taub)
For Bianchi IX spacetimes with cs2 < 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”)
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
We now present the main ideas behind the proof of stable
singularity formation.
The hard part: existence “down” to the Big-Bang
+ estimates.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Transported spatial coordinates
We consider g in the form
g = −n2 dt 2 + gab dx a dx b
n is the “lapse”
g is a Riemannian metric on Σt
The {x i }i=1,2,3 are (local) spatial coordinates on the Σt
q
For FLRW: n = 1, gab = t 2/3 δab , ∂t φ = 23 t −1 , ∇φ = 0
∇ := connection of g and ∆ is its Laplacian
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Transported spatial coordinates
We consider g in the form
g = −n2 dt 2 + gab dx a dx b
n is the “lapse”
g is a Riemannian metric on Σt
The {x i }i=1,2,3 are (local) spatial coordinates on the Σt
q
For FLRW: n = 1, gab = t 2/3 δab , ∂t φ = 23 t −1 , ∇φ = 0
∇ := connection of g and ∆ is its Laplacian
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Transported spatial coordinates
We consider g in the form
g = −n2 dt 2 + gab dx a dx b
n is the “lapse”
g is a Riemannian metric on Σt
The {x i }i=1,2,3 are (local) spatial coordinates on the Σt
q
For FLRW: n = 1, gab = t 2/3 δab , ∂t φ = 23 t −1 , ∇φ = 0
∇ := connection of g and ∆ is its Laplacian
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Transported spatial coordinates
We consider g in the form
g = −n2 dt 2 + gab dx a dx b
n is the “lapse”
g is a Riemannian metric on Σt
The {x i }i=1,2,3 are (local) spatial coordinates on the Σt
q
For FLRW: n = 1, gab = t 2/3 δab , ∂t φ = 23 t −1 , ∇φ = 0
∇ := connection of g and ∆ is its Laplacian
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Constant mean curvature gauge
We impose trk = −t −1 along Σt , which, upon using the
constraints, leads to:
"
r #
2
2
t∆n − t −1 (n − 1) = 2
t −1 n−1 t∂t φ −
3
3
{z
}
|
crucially important linear term
+ quadratic error
r
k ∼ ∂t g = second fundamental form of Σt
∞ propagation speed → synchronizes the singularity
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Constant mean curvature gauge
We impose trk = −t −1 along Σt , which, upon using the
constraints, leads to:
"
r #
2
2
t∆n − t −1 (n − 1) = 2
t −1 n−1 t∂t φ −
3
3
{z
}
|
crucially important linear term
+ quadratic error
r
k ∼ ∂t g = second fundamental form of Σt
∞ propagation speed → synchronizes the singularity
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Parabolic lapse gauge
Alternate parabolic lapse gauge: impose
λ−1 (n − 1) = ttrk + 1 along Σt , which leads to:
1 −1 −1
−1
λ ∂t n + t∆n = 1 + λ
t (n − 1)
3
"
r #
r
2 −1 −1
2
t
n t∂t φ −
+error
+ 2
3
3
|
{z
}
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
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Parabolic lapse gauge
Alternate parabolic lapse gauge: impose
λ−1 (n − 1) = ttrk + 1 along Σt , which leads to:
1 −1 −1
−1
λ ∂t n + t∆n = 1 + λ
t (n − 1)
3
"
r #
r
2 −1 −1
2
t
n t∂t φ −
+error
+ 2
3
3
|
{z
}
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
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
The 1 + 3 eqns. in CMC gauge
For FLRW: n = 1, gab = t
2/3
δab , ∂t φ =
q
2 −1
t ,
3
∇φ = 0
Constraint equations
R − |k |2 + t −2 = (n−1 ∂t φ)2 + ∇φ · ∇φ,
∇a k ai = −n−1 ∂t φ∇i φ
Evolution equations
∂t gij = −2ngia k aj ,
∂t (k ij ) = −∇i ∇j n
+ n R i j − t −1 k ij − ∇i φ∇j φ ,
−∂t (n−1 ∂t φ) + n∆φ = t −1 ∂t φ − ∇φ · ∇n
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Energies for the perturbed solution

r #2

2
−1
2
2
E(φ) (t) :=
n t∂t φ −
+ t |∇φ|
dx,

3
Σt 
Z 1 2
2
2
2
E(g) (t) :=
t |k̂ | + t |∂g| dx
4
Σt

Z "
The energies vanish for the FLRW solution
k̂ is the trace-free part of k
The t factors lead to cancellation of some linear terms
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Energy identity for φ
For t < 1, integrating by parts using g φ = 0 yields:
E(φ) (t)
Z
Z
4 1
= E(φ) (1) −
s
|∇φ|2 dx ds
3 s=t
Σs
"
r #
r Z 1
Z
2
2
(n − 1) n−1 s∂t φ −
s−1
dx ds
+2
3 s=t
3
Σs
|
{z
}
dangerous quad. integral
r Z 1 Z
2
−2
s
∇φ · ∇n dx ds +cubic error
3 s=t
Σs
|
{z
}
dangerous quad. integral
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Elliptic estimate for n
Integrating
using
q
h by parts q
i the lapse PDE
2 −1
2
−1
2 3t
n t∂t φ − 3 = t∆n − t −1 (n − 1) + · · · , we
have
"
r #
r
Z
2 −1
2
(n − 1) n−1 s∂t φ −
2
s
dx
3
3
Σs
|
{z
}
dangerous quad. integral
Z
=−
s|∇n|2 + s−1 (n − 1)2 dx
Σs
+ cubic error
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Combined identity for t ∈ (0, 1]
E(φ) (t)
Z Z
4 1
= E(φ) (1) −
s|∇φ|2 dx ds
3 s=t Σs
Z 1 Z
−
s|∇n|2 + s−1 (n − 1)2 dx ds
s=t Σs
r Z 1 Z
2
s
∇φ · ∇n dx ds + cubic error
−2
3 s=t
Σs
q2
• Note that 2 3 ∇φ · ∇n ≤ |∇φ|2 + (2/3)|∇n|2
Hence, E(φ) is past-decreasing up to cubic
errors!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Combined identity for t ∈ (0, 1]
E(φ) (t)
Z Z
4 1
= E(φ) (1) −
s|∇φ|2 dx ds
3 s=t Σs
Z 1 Z
−
s|∇n|2 + s−1 (n − 1)2 dx ds
s=t Σs
r Z 1 Z
2
s
∇φ · ∇n dx ds + cubic error
−2
3 s=t
Σs
q2
• Note that 2 3 ∇φ · ∇n ≤ |∇φ|2 + (2/3)|∇n|2
Hence, E(φ) is past-decreasing up to cubic
errors!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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)
Z
Z
1 1
s
|∂g|2 dx ds
−
3 s=t
Σs
Z 1 Z
+
s
Q(∂g, ∇n) + Q(∂g, ∇φ) + Q(∇φ, ∇n) dx ds
s=t
Σs
|
{z
}
dangerous quad. integrals
+ cubic error
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
An aside on metric energy estimates
In transported coordinates, we have
o
1n
1
∂i Γj (g, ∂g) + ∂j Γi (g, ∂g) + l.o.t.
Rij = − g ab ∂a ∂b gij +
2
2
E(g) (t) = E(g) (1) + 2
Z
1
s=t
Z
s2 nk ab ∂a Γb dx ds + l.o.t.
Σs
Momentum constraint equation + gauge =⇒
∂a k ab = l.o.t
Hence, integrating by parts on Σs , we have
Z
Z
2
ab
2
s nk ∂a Γb dx = −2
s2 n(∂a k ab )Γb dx + l.o.t.
{z
}
Σs
Σs |
l.o.t.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
An aside on metric energy estimates
In transported coordinates, we have
o
1n
1
∂i Γj (g, ∂g) + ∂j Γi (g, ∂g) + l.o.t.
Rij = − g ab ∂a ∂b gij +
2
2
E(g) (t) = E(g) (1) + 2
Z
1
s=t
Z
s2 nk ab ∂a Γb dx ds + l.o.t.
Σs
Momentum constraint equation + gauge =⇒
∂a k ab = l.o.t
Hence, integrating by parts on Σs , we have
Z
Z
2
ab
2
s nk ∂a Γb dx = −2
s2 n(∂a k ab )Γb dx + l.o.t.
{z
}
Σs
Σs |
l.o.t.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
An aside on metric energy estimates
In transported coordinates, we have
o
1n
1
∂i Γj (g, ∂g) + ∂j Γi (g, ∂g) + l.o.t.
Rij = − g ab ∂a ∂b gij +
2
2
E(g) (t) = E(g) (1) + 2
Z
1
s=t
Z
s2 nk ab ∂a Γb dx ds + l.o.t.
Σs
Momentum constraint equation + gauge =⇒
∂a k ab = l.o.t
Hence, integrating by parts on Σs , we have
Z
Z
2
ab
2
s nk ∂a Γb dx = −2
s2 n(∂a k ab )Γb dx + l.o.t.
{z
}
Σs
Σs |
l.o.t.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
An aside on metric energy estimates
In transported coordinates, we have
o
1n
1
∂i Γj (g, ∂g) + ∂j Γi (g, ∂g) + l.o.t.
Rij = − g ab ∂a ∂b gij +
2
2
E(g) (t) = E(g) (1) + 2
Z
1
s=t
Z
s2 nk ab ∂a Γb dx ds + l.o.t.
Σs
Momentum constraint equation + gauge =⇒
∂a k ab = l.o.t
Hence, integrating by parts on Σs , we have
Z
Z
2
ab
2
s nk ∂a Γb dx = −2
s2 n(∂a k ab )Γb dx + l.o.t.
{z
}
Σs
Σs |
l.o.t.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Borderline error terms
We must bound borderline cubic error integrals:
Z 1
E(Total) (t) ≤ CE(Total) (1) +
kk̂ kL∞ (Σs ) E(Total) (s) ds + · · ·
s=t
We adopt the Bootstrap Assumption ( > 0 is small):
√
kk̂ kL∞ (Σs ) ≤ s−1
(saturated by Kasner)
Then by Gronwall, we find that for t < 1:
E(Total) (t) ≤ CE(Total) (1)t −
√
Unfortunately, Sobolev embedding + the bound on E(Total)
√
p
=⇒ kk̂ kL∞ (Σs ) . E(Total) (1)s−1− /2
Inconsistent!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Borderline error terms
We must bound borderline cubic error integrals:
Z 1
E(Total) (t) ≤ CE(Total) (1) +
kk̂ kL∞ (Σs ) E(Total) (s) ds + · · ·
s=t
We adopt the Bootstrap Assumption ( > 0 is small):
√
kk̂ kL∞ (Σs ) ≤ s−1
(saturated by Kasner)
Then by Gronwall, we find that for t < 1:
E(Total) (t) ≤ CE(Total) (1)t −
√
Unfortunately, Sobolev embedding + the bound on E(Total)
√
p
=⇒ kk̂ kL∞ (Σs ) . E(Total) (1)s−1− /2
Inconsistent!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Borderline error terms
We must bound borderline cubic error integrals:
Z 1
E(Total) (t) ≤ CE(Total) (1) +
kk̂ kL∞ (Σs ) E(Total) (s) ds + · · ·
s=t
We adopt the Bootstrap Assumption ( > 0 is small):
√
kk̂ kL∞ (Σs ) ≤ s−1
(saturated by Kasner)
Then by Gronwall, we find that for t < 1:
E(Total) (t) ≤ CE(Total) (1)t −
√
Unfortunately, Sobolev embedding + the bound on E(Total)
√
p
=⇒ kk̂ kL∞ (Σs ) . E(Total) (1)s−1− /2
Inconsistent!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Borderline error terms
We must bound borderline cubic error integrals:
Z 1
E(Total) (t) ≤ CE(Total) (1) +
kk̂ kL∞ (Σs ) E(Total) (s) ds + · · ·
s=t
We adopt the Bootstrap Assumption ( > 0 is small):
√
kk̂ kL∞ (Σs ) ≤ s−1
(saturated by Kasner)
Then by Gronwall, we find that for t < 1:
E(Total) (t) ≤ CE(Total) (1)t −
√
Unfortunately, Sobolev embedding + the bound on E(Total)
√
p
=⇒ kk̂ kL∞ (Σs ) . E(Total) (1)s−1− /2
Inconsistent!
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
How to recover the BA for kk̂ kL∞(Σt )
E(8;Total) := the energy of ≤ 8 derivatives. Make the BA:
q
E(8;Total) (t) ≤ t −σ ,
(, σ > 0 are small)
• We use the evolution eqn. ∂t (t k̂ ij ) = · · · & Sobolev
embedding to derive a VTD estimate:
|∂t (t k̂ ij )| . |t(g −1 )2 ∂ 2 g| + · · · .
{z
}
|
∼tR i j
−1/3−Z σ
t
| {z }
loses derivatives
Z ∈ Z+ = Max. # of factors in tensor products
R1
We integrate in time, use s=0 s−1/3−Z σ ds < ∞ (for small
σ), and the small-data assumption kk̂ kL∞ (Σ1 ) . :
|t k̂ ij | . , t ∈ (0, 1]
=⇒ kk̂ kL∞ (Σt ) . t −1
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
How to recover the BA for kk̂ kL∞(Σt )
E(8;Total) := the energy of ≤ 8 derivatives. Make the BA:
q
E(8;Total) (t) ≤ t −σ ,
(, σ > 0 are small)
• We use the evolution eqn. ∂t (t k̂ ij ) = · · · & Sobolev
embedding to derive a VTD estimate:
|∂t (t k̂ ij )| . |t(g −1 )2 ∂ 2 g| + · · · .
{z
}
|
∼tR i j
−1/3−Z σ
t
| {z }
loses derivatives
Z ∈ Z+ = Max. # of factors in tensor products
R1
We integrate in time, use s=0 s−1/3−Z σ ds < ∞ (for small
σ), and the small-data assumption kk̂ kL∞ (Σ1 ) . :
|t k̂ ij | . , t ∈ (0, 1]
=⇒ kk̂ kL∞ (Σt ) . t −1
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
How to recover the BA for kk̂ kL∞(Σt )
E(8;Total) := the energy of ≤ 8 derivatives. Make the BA:
q
E(8;Total) (t) ≤ t −σ ,
(, σ > 0 are small)
• We use the evolution eqn. ∂t (t k̂ ij ) = · · · & Sobolev
embedding to derive a VTD estimate:
|∂t (t k̂ ij )| . |t(g −1 )2 ∂ 2 g| + · · · .
{z
}
|
∼tR i j
−1/3−Z σ
t
| {z }
loses derivatives
Z ∈ Z+ = Max. # of factors in tensor products
R1
We integrate in time, use s=0 s−1/3−Z σ ds < ∞ (for small
σ), and the small-data assumption kk̂ kL∞ (Σ1 ) . :
|t k̂ ij | . , t ∈ (0, 1]
=⇒ kk̂ kL∞ (Σt ) . t −1
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Summary of main results
Theorem (RS; Nonlinear stability of the FLRW Big Bang)
Consider near-FLRW (H 8 −close) data of small size for the
Einstein-scalar field system on Σ01 = T3 .
(Gerhardt, Bartnik) ∃ a CMC slice Σ1 near
Σ01
√
−c Global energy bound: E(8;Total) (t) . t
, t ∈ (0, 1]
The past of Σ1 is foliated by a family of CMC
hypersurfaces Σt of mean curvature − 13 t −1 , t ∈ (0, 1]
Big Bang: The volume of Σt collapses to 0 as t ↓ 0
√
Convergence and Stability: t∂t φ, n, tk ij , t −1 g have
finite, near (rescaled) FLRW limits as t ↓ 0
SCC: |Riem|2g blows up like t −4 as t ↓ 0
H-P: All past-directed timelike geodesics emanating
from Σt are shorter than Ct 2/3−c
VTD: Many spatial derivative terms negligible near t = 0
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Why are scalar fields special?
• CMC-lapse equation for a scalar field:
n
o
∆n − (n − 1)t −2 = (n − 1) R + t 2 + ∇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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Why are scalar fields special?
• CMC-lapse equation for a scalar field:
n
o
∆n − (n − 1)t −2 = (n − 1) R + t 2 + ∇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.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Future directions
How large can |qj − 1/3| be? (Our proof can be
extended to |qj − 1/3| = δ >> , but what is the sharp
δ?)
Other topologies.
Other matter models.
Stable vs. unstable directions in other regimes.
Intro The Equations Context Proof Setup Approximate Monotonicity Transported Coordinates Nonlinear Analysis Summary
Preprints of the linear and nonlinear article are
available at arxiv.org