the zero-mode stability of kaluza klein spacetimes...the cauchy problem in grkaluza klein...

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

The zero-mode stability of Kaluza Kleinspacetimes

Zoe Wyatt

MIGSAA & University of Edinburgh

27 April 2018

Based on arXiv:1706.00026


The Cauchy Problem in GR

Einstein equations (1915):(3 + 1)-dim spacetime manifold (M, g), Lorentzian metric g of signature(−,+,+,+). The spacetime is dynamical with EoM

Rµν [g ]− 1

2gµνR[g ] = Tµν . (EE)

Restrict to vacuum (VEE) and fix coordinate redundancy via wave gauge

∇ν∇νxµ = 0

Then VEE become system of quasilinear wave equations

�ggµν := gρσ∂ρ∂σgµν = Nµν(g , ∂g) . (1)

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.

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


Nonlinear stability of Minkowski

Idea: For initial data (R3, γ,K ) ‘sufficiently close’ to trivial data, themaximal 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) , (2)

where the O(∂h)2 terms are P and Qµν , and cubic terms are Gµν .

η = diag(−1, 1, 1, 1), µ, ν ∈ {0, 1, 2, 3}


Kaluza Klein spacetimes

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

Rµν [G ] = 0 . (3)

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

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

Gµν =

(ηab 00 δab

). (4)

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}


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 .(5)

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 for d ≥ 1. Contrast this to stability of minimally coupledEE-Maxwell fields [Choquet-Bruhat & Chrusciel & Loizelet, ‘07]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

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

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 .

(6)

Assume initial data Σ0 ' R3 n Td and ‘asymptotically KK’.Zero-mode perturbations equivalent to TIP (take Fourier expansion incompact directions, very small radii)

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


Wave equation on a Minkowski Background

Simple example: fix background Minkowski geometry and only one fieldψ satisfying the free wave equation (c = 1) with C∞c (R3) sph. sym.initial data

�ηψ := ηµν∂µ∂νψ = 0 ,

η = diag(−1, 1, 1, 1) .(7)

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

LL(rψ) = 0 (8)

where L := ∂t + ∂r , L := ∂t − ∂r .



Integrate along v in the wave-zone regionR = {(u, v) : u0 ≤ u ≤ u1, v ≥ v0}.

L(rψ)(u, v) = L(rψ)(u, v0)⇒ |L(rψ)| ≤ C (9)

Then integrate along u from u0 to obtain

rψ(u, v) = 0 +

∫ u

u0

du′L(rψ)⇒ |rψ| ≤ C (10)

From rLψ = L(rψ) + ψ we obtain |rLψ| ≤ C . Similarly |r2Lψ| ≤ C .

In terms of decay, the Lψ derivative decays the slowest.



Similar story without spherical symmetry.

L = ∂t + ∂r , L = ∂t − ∂r , θA , θB

Then denote full derivatives and derivatives tangent to light cones by

∂ := {L , L , θ} , ∂ := {L , θ} (11)

Get different decay rates along different directions.

⇒ |Lψ| . 1

r, |∂ψ| . 1

r2. (12)

Problems (eg, for GWP) coming from direction with ‘bad decay’.


Semilinear equations on a Minkowski Background

Two semilinear wave equations with C∞c (R3) sph. sym. I.D

�ηψ = (∂tψ)2 − |∇ψ|2 = −ηµν∂µψ∂νψ , (13)

�ηψ = −(∂tψ)2 . (14)

Look at PDE wrt null frame

(1)→ LL(rψ) = r(Lψ)(Lψ) , (15)

(2)→ LL(rψ) =r

2(Lψ − Lψ)2 ∼ r(Lψ)(Lψ) . (16)

For (15) use the following bootstrap assumptions

|rψ| ≤ Cε , |rLψ| ≤ Cε , |r2Lψ| ≤ Cε

Repeat linear argument with bootstrap assumptions

|L(rψ)| ≤ ε+

∫ v

v0

|rLψ||r2Lψ|dvr2≤ ε+ C 2ε2

∫ v

v0

dv1

r2≤ 2ε

µ, ν ∈ {0, 1, 2, 3}


Semilinear equations on a Minkowski Background

(13) has global solutions for small initial data.

(14) has finite time blow up for arbitrarily small initial data. [John, ‘81]

More generally, equation (13) has a null structure.

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

�ηψi = Q(∂ψ, ∂ψ) +O(|ψ|3 + |∂ψ|3) (17)

where Q is a linear combination of null forms

Q0(∂φ, ∂φ) = ηµν∂µφ∂νφ

Qαβ(∂φ, ∂χ) = ∂αφ∂βχ− ∂αχ∂βφ .


A generalised PDE system

Return to our system: Let gµν = ηµν + hµν satisfy the wave-gauge,and gµν and ψk , for k ∈ K , satisfy

�ghµν = Pµν + Qµν + Gµν ,

�gψk = Qk + Gk ,(18)

where

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

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

Main result: (hµν , ∂thµν , ψk , ∂tψk )|t=0 sufficiently small, the perturbedsolution exists for all time and decays.

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

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


Brief comments on the proof

Notation: 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 .

(1) Bootstrap argument for weighted energy norms of Z Ih1µν ,Z

Iψk ,

where h1µν(t) = hµν(t)− χ(r , t) M

r δµν and Z Minkowski vector fields.

Prove estimates for inhomogeneous terms, then cose by Gronwall andcontinuation criterion based on finiteness of energy norms.


(2) Worried about Pµν , hLL, |∂h|UU .Non-null form P decomposes into null-frame

P(∂µh, ∂νh) = ηρσηλτ(

1

4∂µhρσ∂νhλτ −

1

2∂µhρλ∂νhστ

)= −1

8

(∂µhLL∂νhLL + ∂µhLL∂νhLL

)+ c ijkl∂µhTi Uj∂νhTk Ul

.

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}


Concern for bad component

|P(∂h, ∂h)|UU . |∂h|2T U + |∂h|LL|∂h|UU . (19)

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 . (20)

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 .

(21)

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

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


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 |UU ∼ |P|UU , |F |K ∼ |F |T U ∼ |P|T U .

The simplest L∞ estimates are then

|∂h|T U + |∂ψ|K .ε

t, |∂h|UU .

ε ln t

t.

Quasilinear �g = �η + Hµν∂µ∂ν estimates entirely unchanged from [L &R].

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


Conclusion

Summary: The Kaluza-Klein model with toroidal independentperturbations is stable to small ID perturbations. Hence theEinstein-Maxwell-Scalar field system arising from the n = 0 modetruncation of (3 + d + 1)−dim VEE over a Td is non-linearly 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, mass terms appear in the metric :-(


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the zero-mode stability of kaluza klein spacetimes...the cauchy problem in grkaluza klein...

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