The Cauchy Problem in General Relativity and Kaluza Klein … · 2018-05-27 · The Cauchy Problem in GRKaluza Klein spacetimesWave EquationsComments on the proof The Cauchy Problem

The Cauchy Problem in GR Kaluza Klein spacetimes Wave Equations Comments on the proof

The Cauchy Problem in General Relativity andKaluza Klein spacetimes

Zoe Wyatt

MIGSAA & University of Edinburgh

24 May 2018

Based on arXiv:1706.00026


1 The Cauchy Problem in GR

2 Kaluza Klein spacetimes

3 Wave Equations

4 Comments on the proof


The Einstein Equations of General Relativity

(3 + 1)-dim spacetime manifold (M, g), Lorentzian metric g of signature(−,+,+,+). The spacetime is now dynamical with EoM

Rµν [g ]− 1

2gµνR[g ] = Tµν . (EE)

Restrict to vacuum (VEE)

Rµν [g ] = 0 . (VEE)

Example solution 1: the Minkowski metricds2 = −dt2 + dx2 + dy2 + dz2 .

Example solution 2: the Schwarzschild metric (1917)

gS (M) = −(

1− 2M

r

)dt2 +

(1− 2M

r

)−1

dr2 + r2(dθ2 + sin2 θdφ2) ,

describing the geometry around (r > R) aspherical object of mass M and radius R.

µ, ν ∈ 0, 1, 2, 3


The Einstein Equations of General Relativity

(3 + 1)-dim spacetime manifold (M, g), Lorentzian metric g of signature(−,+,+,+). The spacetime is now dynamical with EoM

Rµν [g ]− 1

2gµνR[g ] = Tµν . (EE)

Restrict to vacuum (VEE)

Rµν [g ] = 0 . (VEE)

Example solution 1: the Minkowski metricds2 = −dt2 + dx2 + dy2 + dz2 .

Example solution 2: the Schwarzschild metric (1917)

gS (M) = −(

1− 2M

r

)dt2 +

(1− 2M

r

)−1

dr2 + r2(dθ2 + sin2 θdφ2) ,

describing the geometry around (r > R) aspherical object of mass M and radius R.

µ, ν ∈ 0, 1, 2, 3


The Cauchy Problem in General Relativity

Firstly recall the wave equation and the summation convention

ηµν∂µ∂νψ = −∂2t ψ + ∂2

i ψ = 0 .

Rµν =1

2gρσ (∂µ∂ρgσν + ∂ν∂ρgσµ − ∂µ∂νgρσ − ∂ρ∂σgµν) + l.o. (1)

VEE in terms of metric have indeterminate PDE type. Fix coordinateredundancy via wave gauge

∇ν∇νxµ = 0⇔ gρσ∂ρgσµ =1

2gρσ∂µgρσ .

Then VEE become system of quasilinear wave equations

ggµν := gρσ∂ρ∂σgµν = Nµν(g , ∂g) . (2)

Aim: view VEE as a well-posed initial value problem (existence,uniqueness up to diffeomorphism, continuous dependence on initial data).

µ, ν ∈ 0, 1, 2, 3


The Cauchy Problem in General Relativity

Firstly recall the wave equation and the summation convention

ηµν∂µ∂νψ = −∂2t ψ + ∂2

i ψ = 0 .

Rµν =1

2gρσ (∂µ∂ρgσν + ∂ν∂ρgσµ − ∂µ∂νgρσ − ∂ρ∂σgµν) + l.o. (1)

VEE in terms of metric have indeterminate PDE type. Fix coordinateredundancy via wave gauge

∇ν∇νxµ = 0⇔ gρσ∂ρgσµ =1

2gρσ∂µgρσ .

Then VEE become system of quasilinear wave equations

ggµν := gρσ∂ρ∂σgµν = Nµν(g , ∂g) . (2)

Aim: view VEE as a well-posed initial value problem (existence,uniqueness up to diffeomorphism, continuous dependence on initial data).

µ, ν ∈ 0, 1, 2, 3


VEE are locally well-posed

Theorem: [Choquet-Bruhat ‘52, C-B & Geroch ‘69]Let (Σ, γ,K ) be a smooth vacuum initial data set satisfying theconstraint equations.

There exists a unique (up to diffeomorphism) smooth (M, g), called themaximal Cauchy development, such that

1 Rµν [g ] = 0 ,

2 (Σ, γ) embeds as a Cauchy hypersurface into (M, g) with secondfundamental form K ,

3 any other smooth spacetime satisfying (1&2) embeds isometricallyinto M.

Note: result only guarantees a local solution, geodesic completeness notguaranteed.

A Cauchy hypersurface is a spacelike hypersurface such that every inextendible causal geodesicintersects the hypersurface exactly once.


Nonlinear stability of Minkowski

Idea: For initial data (R3, γ,K ) ‘sufficiently close’ to trivial data, showthe maximal Cauchy development (M, g) is future-causally geodesicallycomplete and ‘tends’ to Minkowski space along all past and futuredirected geodesics.

[Christodoulou & Klainerman ‘90s]

Wave gauge:Gauge thought to break down [Choquet-Bruhat, ‘73]

Nonetheless, nonlinear stability by [Lindblad & Rodnianski ‘03, ‘04]

Think of perturbation from Minkowski h := g − η, then VEE becomes

ghµν = P(∂µh, ∂νh) + Qµν(∂h, ∂h) + Gµν(h)(∂h, ∂h) , (3)

where the O(∂h)2 terms are P (non-null) and Qµν (null), while Gµν arecubic.

η = diag(−1, 1, 1, 1), µ, ν ∈ 0, 1, 2, 3, g := gρσ∂ρ∂σ .


Nonlinear stability of Minkowski

Idea: For initial data (R3, γ,K ) ‘sufficiently close’ to trivial data, showthe maximal Cauchy development (M, g) is future-causally geodesicallycomplete and ‘tends’ to Minkowski space along all past and futuredirected geodesics.

[Christodoulou & Klainerman ‘90s]

Wave gauge:Gauge thought to break down [Choquet-Bruhat, ‘73]

Nonetheless, nonlinear stability by [Lindblad & Rodnianski ‘03, ‘04]

Think of perturbation from Minkowski h := g − η, then VEE becomes

ghµν = P(∂µh, ∂νh) + Qµν(∂h, ∂h) + Gµν(h)(∂h, ∂h) , (3)

where the O(∂h)2 terms are P (non-null) and Qµν (null), while Gµν arecubic.

η = diag(−1, 1, 1, 1), µ, ν ∈ 0, 1, 2, 3, g := gρσ∂ρ∂σ .


Kaluza Klein spacetimes

String theory: (3 + 1 + d)-dim spacetime (M,Gµν) satisfying

Rµν [G ] = 0 . (4)

Compactify extra dimensions: R1+3 × S1 [Kaluza, Klein, 20s]

More general case: R1+3 × Td with flat background metric

Gµν =

(ηab 00 δab

). (5)

Notation: xa non-compact coords, xa compact coordinate.

Question 1) why did Kaluza and Klein look at this?

Question 2) is this stable to small perturbations of initial data?

µ, ν ∈ 0, . . . , 3 + d


Kaluza Klein spacetimes

String theory: (3 + 1 + d)-dim spacetime (M,Gµν) satisfying

Rµν [G ] = 0 . (4)

Compactify extra dimensions: R1+3 × S1 [Kaluza, Klein, 20s]

More general case: R1+3 × Td with flat background metric

Gµν =

(ηab 00 δab

). (5)

Notation: xa non-compact coords, xa compact coordinate.

Question 1) why did Kaluza and Klein look at this?


µ, ν ∈ 0, . . . , 3 + d


EE-Maxwell-φ system from Kaluza-Klein, d = 1

Aim of compactification: obtain non-minimally coupled EE-Maxwell-φsystem in (3 + 1)−dim.

Assume Gµν(xa, x4) = Gµν(xa), then can take Ansatz:

Gab = e2αφgab + e2βφAaAb ,

Ga4 = e2βφAa , G44 = e2βφ .(6)

Here gab is (3 + 1)−dimensional metric, Aa a vector potential and φ adilaton. Higher-dimensional VEE reduce to

Rab[g ] =1

2∂aφ∂bφ+

1

2e−6αφ

(FacFb

c − 1

4FcdF cdgab

),

∇a(e−6αφFab

)= 0 ,

gφ = −3

2αe−6αφFcdF cd .

Similar idea for d ≥ 1.

a , b ∈ 0, 1, 2, 3, α = −2β = 1/√

12, non-minimal w.r.t scalar and Maxwell fields


EE-Maxwell-φ system from Kaluza-Klein, d = 1

Aim of compactification: obtain non-minimally coupled EE-Maxwell-φsystem in (3 + 1)−dim.

Assume Gµν(xa, x4) = Gµν(xa), then can take Ansatz:

Gab = e2αφgab + e2βφAaAb ,

Ga4 = e2βφAa , G44 = e2βφ .(6)

Here gab is (3 + 1)−dimensional metric, Aa a vector potential and φ adilaton. Higher-dimensional VEE reduce to

Rab[g ] =1

2∂aφ∂bφ+

1

2e−6αφ

(FacFb

c − 1

4FcdF cdgab

),

∇a(e−6αφFab

)= 0 ,

gφ = −3

2αe−6αφFcdF cd .

Similar idea for d ≥ 1.

a , b ∈ 0, 1, 2, 3, α = −2β = 1/√

12, non-minimal w.r.t scalar and Maxwell fields


Nonlinear stability of Kaluza Klein spacetimes to TIP


Theorem: [ZW, ‘17] Kaluza Klein spacetimes are non-linearly stable totorus independent perturbations

hµν := Gµν − Gµν such that ∂ahµν = 0 . (TIP)

Note: TIP still allows for non-orthogonal fibres.Assuming TIP and wave gauge for Gµν ⇒ VEE takes the form

ghab = Pab + Qab + Gab ,

ghaa = Qaa + Gaa ,

ghab = Qab + Gab .

(7)

Assume initial data Σ0 ' R3 × Td and ‘asymptotically KK’.

a, b ∈ 0, . . . , 3, a , b ∈ 4, . . . , 3 + d, µ, ν ∈ 0, . . . , 3 + d


Nonlinear stability of Kaluza Klein spacetimes to TIP


Theorem: [ZW, ‘17] Kaluza Klein spacetimes are non-linearly stable totorus independent perturbations

hµν := Gµν − Gµν such that ∂ahµν = 0 . (TIP)

Note: TIP still allows for non-orthogonal fibres.Assuming TIP and wave gauge for Gµν ⇒ VEE takes the form

ghab = Pab + Qab + Gab ,

ghaa = Qaa + Gaa ,

ghab = Qab + Gab .

(7)

Assume initial data Σ0 ' R3 × Td and ‘asymptotically KK’.

a, b ∈ 0, . . . , 3, a , b ∈ 4, . . . , 3 + d, µ, ν ∈ 0, . . . , 3 + d


Reasoning behind TIP assumption, d = 1

Take d = 1 and Fourier expand on compact coordinate spanning S1 ofradius R:

Gµν(xa, x4) =∑n∈Z

exp

(inx4

R

)G (n)µν (xa) . (8)

Using this expansion, (part of) LHS term GGµν becomes

Gρσ∂ρ∂σGµν =

∑n∈Z

(ηab∂a∂b −

( nR

)2)G (n)µν . (9)

Take R → 0 ⇒ only perceive n = 0 modes at low energies. Reducing to

n = 0 mode implies G(n)µν = 0 for all n 6= 0 and thus

Gµν(xa, x4) = Gµν(xa) . (10)

i.e. the perturbations hµν := Gµν − Gµν satisfy

∂4hµν = 0 , ∀µ, ν ∈ 0, . . . , 3, 4 .

a, b ∈ 0, . . . , 3, a = 4, µ, ν ∈ 0, . . . , 3, 4


Reasoning behind TIP assumption, d = 1

Take d = 1 and Fourier expand on compact coordinate spanning S1 ofradius R:

Gµν(xa, x4) =∑n∈Z

exp

(inx4

R

)G (n)µν (xa) . (8)

Using this expansion, (part of) LHS term GGµν becomes

Gρσ∂ρ∂σGµν =

∑n∈Z

(ηab∂a∂b −

( nR

)2)G (n)µν . (9)

Take R → 0 ⇒ only perceive n = 0 modes at low energies. Reducing to

n = 0 mode implies G(n)µν = 0 for all n 6= 0 and thus

Gµν(xa, x4) = Gµν(xa) . (10)

i.e. the perturbations hµν := Gµν − Gµν satisfy

∂4hµν = 0 , ∀µ, ν ∈ 0, . . . , 3, 4 .

a, b ∈ 0, . . . , 3, a = 4, µ, ν ∈ 0, . . . , 3, 4


Consistent Truncations, d = 1

Consistent truncation: n = 0 mode truncation is ‘consistent’, i.e.setting ∂4Gµν = 0 in action then varying gives solutions that solveoriginal EoM.

Inconsistent truncation: setting ∂4Gµν = 0 and G44 = 0 is notconsistent. See this at the level of the EoM:

gφ = −3

2αe−6αφFcdF cd ⇒ F2 = 0 .

Contrast this to stability of minimally coupled EE-Maxwell fields[Choquet-Bruhat & Chrusciel & Loizelet, ‘07]

Full understanding of consistent truncations not known for more complexmanifolds, e.g. R1+3 × K for K Calabi-Yau.

µ, ν ∈ 0, . . . , 3, 4


A generalised PDE system

Main result: [ZW, ‘17] Let gµν = ηµν + hµν satisfy the wave-gauge, andgµν and ψk , for k ∈ K , satisfy

ghµν = Pµν + Qµν + Gµν ,

gψk = Qk + Gk ,(11)

where

Pµν = P(∂µh, ∂νh) + P(∂µh, ∂νψ) + P(∂µψ, ∂νψ) ,

Qµν = Qµν(∂h, ∂h) + Qµν(∂h, ∂ψ) + Qµν(∂ψ, ∂ψ) .

For (hµν , ∂thµν , ψk , ∂tψk )|t=0 sufficiently small, the perturbed solutionexists for all time and decays.

Case 0: ψk ≡ 0Case 1: Qk = Gk = 0, Blue terms = 0 and |K | = 1 [L & R, 00s]Case 2: Kaluza Klein spacetimes with ψk = haa, hab

Asymptotics: [Lindblad, ‘16], alternative method in [Hintz & Vasy, ‘17]

µ, ν ∈ 0, 1, 2, 3 ,K = 1, . . . ,m for some positive integer m.


Wave equation on a Minkowski Background

Simple example: fix background Minkowski geometry and only one fieldψ satisfying the free wave equation

ηψ := ηµν∂µ∂νψ = 0 ,

η = diag(−1, 1, 1, 1) .(12)

Take C∞c (R3) spherically symmetric initial data

−∂2t ψ + (∂2

x + ∂2y + ∂2

z )ψ = −∂2t ψ + ∂2

r ψ +2

r∂rψ = −∂2

t (rψ) + ∂2r (rψ) .

Use coordinates adapted to the light cones u = t − r , v = t + r .

LL(rψ) = 0 , (13)

where L := ∂t + ∂r , L := ∂t − ∂r .

Other common notation ∂v = L = l, ∂u = L = n



Interested in wave-zone region

R := (u, v) : u0 ≤ u ≤ u1, t ≥ 0 .

Aim: understand solution decay in R. Integrate along v

L(rψ)(u, v) = L(rψ)(u, v0)⇒ |L(rψ)| ≤ C in R (14)

Then integrate along u: rψ(u, v) = 0 +∫ u

u0L(rψ)(u′, v)du′

⇒ |rψ| ≤ C (15)

From rLψ = L(rψ) + ψ we obtain

|rLψ| ≤ C (16)

and from L(rLψ) = L(L(rψ)− ψ) = −Lψ we obtain

|rLψ| ≤ C

r(u, v)(17)

Main point: in terms of decay, the Lψ derivative decays the slowest.



Interested in wave-zone region

R := (u, v) : u0 ≤ u ≤ u1, t ≥ 0 .

Aim: understand solution decay in R. Integrate along v

L(rψ)(u, v) = L(rψ)(u, v0)⇒ |L(rψ)| ≤ C in R (14)

Then integrate along u: rψ(u, v) = 0 +∫ u

u0L(rψ)(u′, v)du′

⇒ |rψ| ≤ C (15)

From rLψ = L(rψ) + ψ we obtain

|rLψ| ≤ C (16)

and from L(rLψ) = L(L(rψ)− ψ) = −Lψ we obtain

|rLψ| ≤ C

r(u, v)(17)

Main point: in terms of decay, the Lψ derivative decays the slowest.



Similar story without spherical symmetry.

L = ∂t + ∂r , L = ∂t − ∂r , θA , θB

Denote full derivatives and derivatives tangent to light cones by

∂ := L , L , θ , ∂ := L , θ (18)

Get different decay rates along different directions.

⇒ |Lψ| . 1

r, |∂ψ| . 1

r2. (19)

Once again: Lψ derivative decays the slowest.Problems (eg, for GWP) coming from direction with ‘bad decay’.


Semilinear equations on a Minkowski Background

Two semilinear wave equations with C∞c (R3) spherical data

ηψ = (∂tψ)2 − |∇ψ|2 = −ηµν∂µψ∂νψ , (20)

ηψ = −(∂tψ)2 . (21)

Look at PDE wrt null frame

(1)→ LL(rψ) = r(Lψ)(Lψ) , (22)

(2)→ LL(rψ) =r

2(Lψ − Lψ)2 ∼ r(Lψ)(Lψ) . (23)

(20) has global solutions for small initial data.

(21) all compactly supported initial data leads to solution break down infinite time [John, ‘81]

µ, ν ∈ 0, 1, 2, 3



Continuous induction: define B ⊂ [0,∞) to be the subset of t ′ suchthat in R∩ t < t ′ the following bootstrap assumptions hold:

|rψ| ≤ Cε , |rLψ| ≤ Cε , |r2Lψ| ≤ Cε , (24)

for some large C = C (C0).Aim: show B is nonempty, connected, closed and open.Method: replace C → C/2 by repeating linear argument with bootstrapassumptions. Integrate LL(rψ) = r(Lψ)(Lψ) to obtain

L(rψ)(u, v) = L(rψ)(u, v0) +

∫ v

v0

(rLψ)(r2Lψ)dv

r2

⇒ |L(rψ)| ≤ C0ε+ C 2ε2 < 2C0ε ≤1

2Cε .

(25)

Similarly |rψ(u, v)| ≤ 2C0ε ≤ 12Cε, etc.



Continuous induction: define B ⊂ [0,∞) to be the subset of t ′ suchthat in R∩ t < t ′ the following bootstrap assumptions hold:

|rψ| ≤ Cε , |rLψ| ≤ Cε , |r2Lψ| ≤ Cε , (24)

for some large C = C (C0).Aim: show B is nonempty, connected, closed and open.Method: replace C → C/2 by repeating linear argument with bootstrapassumptions. Integrate LL(rψ) = r(Lψ)(Lψ) to obtain

L(rψ)(u, v) = L(rψ)(u, v0) +

∫ v

v0

(rLψ)(r2Lψ)dv

r2

⇒ |L(rψ)| ≤ C0ε+ C 2ε2 < 2C0ε ≤1

2Cε .

(25)

Similarly |rψ(u, v)| ≤ 2C0ε ≤ 12Cε, etc.


Null condition

(20) has global solutions for small initial data. The nonlinearity has a nullstructure.

Theorem: [Christodoulou, Klainerman ‘86] for sufficiently small C∞c (R3)initial data the following PDE has a global solution,

ηψi = Q(∂ψ, ∂ψ) , (26)

where Q is a linear combination of null forms

Q0(∂φ, ∂φ) = ηµν∂µφ∂νφ , (27a)

Qαβ(∂φ, ∂χ) = ∂αφ∂βχ− ∂αχ∂βφ . (27b)


Brief comments on the proof

(0) Semilinear example: quadratic nonlinearity

ηφ = N (∂φ, ∂φ) .

Consider ‘energy norms’

EN (t) :=∑|I |≤N

∑Z∈Z

∫Σt

|∂Z Iφ(t, x)|2d3x ,

where Z are the Minkowski vector fields

Z := ∂µ,Ωµν := xµ∂ν − xν∂µ,S := xµ∂µ . (28)

General strategy for small data GWP: have continuation criterionbased on finiteness of energy norms.

For sufficiently small initial data EN (0) = ε obtain local-in-time solutionwith bootstrap assumptions valid on a maximal interval [0,T ] ⊂ [0,∞).


Brief comments on the proof

(0) Semilinear example: quadratic nonlinearity

ηφ = N (∂φ, ∂φ) .

Consider ‘energy norms’

EN (t) :=∑|I |≤N

∑Z∈Z

∫Σt

|∂Z Iφ(t, x)|2d3x ,

where Z are the Minkowski vector fields

Z := ∂µ,Ωµν := xµ∂ν − xν∂µ,S := xµ∂µ . (28)

General strategy for small data GWP: have continuation criterionbased on finiteness of energy norms.

For sufficiently small initial data EN (0) = ε obtain local-in-time solutionwith bootstrap assumptions valid on a maximal interval [0,T ] ⊂ [0,∞).


(0) Semilinear example

Aim: to improve bootstrap by using generalised energy estimates

EN (t) ≤ EN (0) +∑|I |≤N

∑Z∈Z

∫ t

0

‖Z IN (∂φ, ∂φ)‖L2E1/2N (τ)dτ .

From Gronwall’s inequality arrive at

EN (t) ≤ exp

C

∫ t

0

∑|I |≤ N

2 +2

∑Z∈Z

‖∂Z Iφ(t)‖L∞

EN (0) . (29)

Decay: comes from Klainerman-Sobolev inequality [Klainerman ‘85]. Forsufficiently smooth φ(t, x) and at each (t, x) ∈ R1+3

(1 + t + r)3−1

2 (1 + |r − t|)1/2|∂Z Iφ(t, x)| . E|I |+3(t) . (30)

Argument gives EN (t) ≤ Cε in n ≥ 4 and in n = 3 with null condition(for large enough N).


(0) Semilinear example

Aim: to improve bootstrap by using generalised energy estimates

EN (t) ≤ EN (0) +∑|I |≤N

∑Z∈Z

∫ t

0

‖Z IN (∂φ, ∂φ)‖L2E1/2N (τ)dτ .

From Gronwall’s inequality arrive at

EN (t) ≤ exp

C

∫ t

0

∑|I |≤ N

2 +2

∑Z∈Z

‖∂Z Iφ(t)‖L∞

EN (0) . (29)

Decay: comes from Klainerman-Sobolev inequality [Klainerman ‘85]. Forsufficiently smooth φ(t, x) and at each (t, x) ∈ R1+3

(1 + t + r)3−1

2 (1 + |r − t|)1/2|∂Z Iφ(t, x)| . E|I |+3(t) . (30)

Argument gives EN (t) ≤ Cε in n ≥ 4 and in n = 3 with null condition(for large enough N).


(1) Notation for our Einstein PDE

Need to estimate/understand unknown variables

W := UµV νh1µν , ψk : U,V ∈ U , k ∈ K .

Introduction collections of the null-frame (L, L, θA, θB )

U := L , L , θ , T := L , θ , L := L .

Keep track of components and derivatives

|π|VW :=∑

V∈V,W∈W

|V µW νπµν | , |∂π|VW :=∑

V∈V,W∈WU∈U

|V µW νUρ(∂ρπµν)| .

Similarly for |∂π|VW . For example

|∂h|LT =∑

L∈L,T∈TU∈U

|LµT νUρ(∂ρhµν)| ,

|∂h|UU =∑

U,V∈UT∈T

|UµV νT ρ(∂ρhµν)| .

Also define K := 1, . . . ,m, |ψ|K :=∑

k∈K |ψk |.


(2) PDE setup

Subtract off mass contribution from metric perturbation

h1µν(t) := hµν(t)− χ(r , t)

M

rδµν (31)

PDE now becomes

gZIh1µν = Z IF − Z I g

(χ(r , t)

M

rδµν)

+ [Z I , g ]h1µν ,

gZIψk = Z IFk + [Z I , g ]ψk ,

(32)

with unknowns

W := UµV νh1µν , ψk : U,V ∈ U ,K . (33)

Take small initial data ε = EN (0) + M, where L2 energy norms are

EN (t) :=∑|I |≤N

∑Z∈Z

∫Σt

w(|∂Z Ih1|2UU + |∂Z Iψ|2K

)d3x ,

Key change: allow for polynomially growing energy EN ∼ tδ.


(2) PDE setup

Subtract off mass contribution from metric perturbation

h1µν(t) := hµν(t)− χ(r , t)

M

rδµν (31)

PDE now becomes

gZIh1µν = Z IF − Z I g

(χ(r , t)

M

rδµν)

+ [Z I , g ]h1µν ,

gZIψk = Z IFk + [Z I , g ]ψk ,

(32)

with unknowns

W := UµV νh1µν , ψk : U,V ∈ U ,K . (33)

Take small initial data ε = EN (0) + M, where L2 energy norms are

EN (t) :=∑|I |≤N

∑Z∈Z

∫Σt

w(|∂Z Ih1|2UU + |∂Z Iψ|2K

)d3x ,

Key change: allow for polynomially growing energy EN ∼ tδ.


(3) Bootstrap assumption

For fixed 0 < δ < 1/4 the local smooth solution satisfies on somemaximal time interval [0,T ]

EN (t) ≤ 2CNε(1 + t)δ .

Aim: for sufficiently small ε to show EN (t) ≤ CNε2(1 + t)cε ⇒ T =∞.

Problem: δ-energy growth now appears in Klainerman-Sobolev

|∂Z IW (t, x)| ≤ Cε(1 + t)δ

(1 + t + r)(1 + |r − t|)1/2, |I | ≤ N − 2 . (34)

This ‘weak decay’ is not strong enough for the non-null part of theinhomogeneity.



Recall the inhomogeneity

Fµν = P(∂µW , ∂νW ) + Qµν(∂W , ∂W ) + Gµν(W )(∂W , ∂W ) ,

Fk = Qk (∂W , ∂W ) + Gk (W )(∂W , ∂W ) .

Klainerman-Sobolev-type estimates imply good decay for two of theterms

|Q|UU + |G |UU . |∂W ||∂W | ≤ ε2(1 + r + t)−2−2γ(1 + |t − r |)−2γ (35)

Similarly for |Q|K + |G |K .However the non-null term is problematic in the wave zone t ∼ r region

|P|UU . ε2(1 + t)−1−2γ (36)

Goal: better understand Pµν decomposed with respect to the null frame!

U = L, L,A,B



Recall the inhomogeneity

Fµν = P(∂µW , ∂νW ) + Qµν(∂W , ∂W ) + Gµν(W )(∂W , ∂W ) ,

Fk = Qk (∂W , ∂W ) + Gk (W )(∂W , ∂W ) .

Klainerman-Sobolev-type estimates imply good decay for two of theterms

|Q|UU + |G |UU . |∂W ||∂W | ≤ ε2(1 + r + t)−2−2γ(1 + |t − r |)−2γ (35)

Similarly for |Q|K + |G |K .However the non-null term is problematic in the wave zone t ∼ r region

|P|UU . ε2(1 + t)−1−2γ (36)

Goal: better understand Pµν decomposed with respect to the null frame!

U = L, L,A,B


(4) Non-null quadratic terms

Decompose Pµν in null frame

Pµν := ηρσηλτ(1

4∂µhρσ∂νhλτ −

1

2∂µhρλ∂νhστ

)+ P(∂µh, ∂νψ) + P(∂µψ, ∂νψ)

= −1

8

(∂µhLL∂νhLL + ∂µhLL∂νhLL

)+ ∂µWG∂νWG .

Note: new terms contain at most one copy of |∂h|T U .Control for good components

|P(∂h, ∂h)|T U =∑

T∈T ,U∈U

|TµUνP(∂µh, ∂νh)| . |∂h|UU |∂h|UU ,

ends up being the same as for null forms

|Q(∂h, ∂h)|UU . |∂h|UU |∂h|UU .

Similarly for Qµν(∂h, ∂ψ) ,Qµν(∂ψ, ∂ψ) .

U = L, L,A,B , T := L,A,B,L = L, ∂ = L, L, θ, ∂ = L, θ


(4) Non-null quadratic terms

Decompose Pµν in null frame

Pµν := ηρσηλτ(1

4∂µhρσ∂νhλτ −

1

2∂µhρλ∂νhστ

)+ P(∂µh, ∂νψ) + P(∂µψ, ∂νψ)

= −1

8

(∂µhLL∂νhLL + ∂µhLL∂νhLL

)+ ∂µWG∂νWG .

Note: new terms contain at most one copy of |∂h|T U .Control for good components

|P(∂h, ∂h)|T U =∑

T∈T ,U∈U

|TµUνP(∂µh, ∂νh)| . |∂h|UU |∂h|UU ,

ends up being the same as for null forms

|Q(∂h, ∂h)|UU . |∂h|UU |∂h|UU .

Similarly for Qµν(∂h, ∂ψ) ,Qµν(∂ψ, ∂ψ) .



(4) Return to wave gauge

Concern for bad component

|P(∂h, ∂h)|UU . |∂h|2T U + |∂h|LL|∂h|UU . (37)

Improve this by using wave gauge

∂µ

(gµν

√detg

)= ∂µ

(Hµν − 1

2ηµνtrH +Oµν(H2)

)= 0 .

Wave gauge gives control on good components of bad derivatives1

|∂h|LT ≤ |∂h|UU + |h|UU |∂h|UU . (38)

So we obtain better control

|P(∂h, ∂h)|UU . |∂h|2T U + |∂h|LL|∂h|UU. |∂h|2T U + |∂h|UU |∂h|UU + |h|UU |∂h|2UU .

(39)

1c.f. |∂A|L control from Lorenz gauge in [Choquet-Bruhat, Chrusciel & Loizelet].



(4) Return to wave gauge

Concern for bad component

|P(∂h, ∂h)|UU . |∂h|2T U + |∂h|LL|∂h|UU . (37)

Improve this by using wave gauge

∂µ

(gµν

√detg

)= ∂µ

(Hµν − 1

2ηµνtrH +Oµν(H2)

)= 0 .

Wave gauge gives control on good components of bad derivatives1

|∂h|LT ≤ |∂h|UU + |h|UU |∂h|UU . (38)

So we obtain better control

|P(∂h, ∂h)|UU . |∂h|2T U + |∂h|LL|∂h|UU. |∂h|2T U + |∂h|UU |∂h|UU + |h|UU |∂h|2UU .

(39)

1c.f. |∂A|L control from Lorenz gauge in [Choquet-Bruhat, Chrusciel & Loizelet].



(4) Improved decay estimates

New terms P(∂h, ∂ψ),P(∂ψ, ∂ψ) contain at most one copy of |∂h|T U .

|P(∂h, ∂ψ)|UU . |∂h|T U |∂ψ|K|P(∂ψ, ∂ψ)|UU . |∂ψ|2K .

In particular ψk obey estimates like the good components hT U , and so

|F |K + |F |T U ∼ |P|T U . ε2(1 + r + t)−2−2γ(1 + |t − r |)−2γ

|F |UU ∼ |P|UU . |∂W |2G + ε2(1 + r + t)−2−2γ(1 + |t − r |)−2γ

Coupling here means we need independent decay estimates [Lindblad‘90]. The simplest ‘strong decay’ estimates are

|∂h|T U + |∂ψ|K .ε

1 + t, |∂h|UU .

ε ln(1 + t)

1 + t.

Note: quasilinear |[Z I , g ]W | estimates entirely unchanged from [L &R].


(5) Energy inequalities

Apply improved decay estimates to close bootstrap. For example at 0thorder

|F |UU + |F |K . ε|∂h|UU + |∂ψ|K

1 + t.

and so the energy inequality looks like

E0(t) ≤ 8E0(0) + Cε

∫ t

0

E0(τ)

1 + τdτ

+ Cε

∫ t

0

∫Στ

(|F |UU + |F |K )(|∂h|UU + |∂ψ|K )

≤ 8E0(0) + Cε

∫ t

0

E0(τ)

1 + τdτ .

Conclude by Gronwall that E0(t) ≤ Cε2(1 + t)cε < bootstrap.In summary: for sufficiently small ε we can show EN (t) < Cε(1 + t)δ on[0,T ] and thus by continuity of EN (t) we must have T =∞.

U = L, L,A,B , T := L,A,B, ∂ = L, L, θ


Conclusion

Summary: the Kaluza-Klein model is nonlinearly stable to toroidalindependent perturbations. Hence the Einstein-Maxwell-Scalar fieldsystem arising from the n = 0 mode truncation of (3 + d + 1)−dim VEEover a Td is nonlinearly stable.

Other work:

Semiclassical instability of Kaluza Klein (for d = 1) where initialdata has topology R2 × S2. [Witten, ‘82]

Recent stability against TIP result for cosmological Kaluza Kleinspacetimes by [Branding, Fajman, Kroncke ‘18]

M4 × Td with g = −dt2 +t2

9γ + gflat,Td

J. Kier, weak null & heirarchy of unknowns, using rp-method of[Dafermos & Rodnianski ‘09]

Open problems:

more general flat background metric Gµν

Full mode case, Klein-Gordon terms appear in the metric :-(


Conclusion


Other work:




9γ + gflat,Td


Open problems:




Conclusion


Other work:




9γ + gflat,Td


Open problems:




Conclusion


Other work:




9γ + gflat,Td


Open problems:



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The Cauchy Problem in General Relativity and Kaluza Klein … · 2018-05-27 · The Cauchy Problem in GRKaluza Klein spacetimesWave EquationsComments on the proof The Cauchy Problem