THE NONLINEAR STABILITY OF MINKOWSKI SPACE FORSELF-GRAVITATING MASSIVE MATTER

Philippe LeFlochLaboratoire Jacques-Louis Lions & CNRS

Universite Pierre et Marie Curie (Paris 6, Jussieu)contact@philippelefloch.org

GENERAL RELATIVITY

§ Global geometry of spacetimes pM, gαβq with signature p´,`,`,`q

§ Einstein equations for self-gravitating matter Gαβ “ 8πTαβ§ Einstein curvature Gαβ “ Rαβ ´ pR2qgαβ§ Riccci curvature Rαβ “ B

2g ` B ‹ Bg§ scalar curvature R :“ Rαα “ gαβRαβ

CAUCHY PROBLEM

§ Global nonlinear stability of Minkowski spacetime§ initial data prescribed on a spacelike hypersurface§ small perturbation of an asymptotically flat slice in Minkowski space

§ Vacuum spacetimes Tαβ “ 0 or massless matter fields§ Christodoulou - Klainerman (1993), Lindblad - Rodnianski (2010)

§ Massive matter fields massive matter Tαβ , open since 1993

§ LeFloch - Yue Ma (2016)

CHALLENGES

§ Gravitational waves§ Weyl curvature (vacuum), Ricci curvature (matter)

§ Nonlinear wave interactions§ exclude dynamical instabilities, self-gravitating massive modes§ avoid gravitational collapse (trapped surfaces, black holes)

§ Global dynamics§ (small) perturbation disperses in timelike directions§ asymptotic convergence to Minkowski spacetime§ future timelike geodesically complete spacetime

MAIN STRATEGY

§ Nonlinear wave systems§ Einstein equations in wave gauge§ PDE system which couples wave and Klein-Gordon equations§ no longer scale-invariant, time-asymptotics drastically different

§ Hyperboloidal Foliation Method PLF-YM, Monograph, 2014

§ foliation of the spacetime by asymptotically hyperbolidal slices§ sharp time-decay estimates (metric, matter fields) for wave and

Klein-Gordon equations§ quasi-null structure of the Einstein equations

OUTLINE of the lecture

§ Einstein gravity and f(R)-gravity

§ Nonlinear global stability: geometric statements

§ Overview of the Hyperboloidal Foliation Method

§ Nonlinear global stability: statements in wave coordinates

§ Quasi-null hyperboloidal structure of the Einstein equations

ELEMENTS of proof

§ Second-order formulation of the f(R)-gravity theory

§ Wave-Klein-Gordon systems

§ null interactions (simplest setup, better energy bounds)§ weak metric interactions (simplest setup)§ strong metric interactions (simpler setup)

§ Sharp pointwise bounds for wave-Klein-Gordon equations on curved space

EINSTEIN GRAVITY AND F(R)-GRAVITY

Self-gravitating massive fields

Massive scalar field with potential Upφq, for instance Upφq “ c2

2φ2, described

by the energy-momentum tensor

Tαβ :“ ∇αφ∇βφ´´1

2gα1β1∇α1φ∇β1φ` Upφq

¯

gαβ

Einstein-Klein-Gordon system for the unknown pM, gαβ , φq

Rαβ ´ 8π´

∇αφ∇βφ` Upφq gαβ¯

“ 0

lgφ´ U 1pφq “ 0

Field equations of the f pRq-modified gravity

Generalized Hilbert-Einstein functional

§ Gravitation mediated by additional fields

§ Functionalż

M

´

f pRg q ` 16πLrφ, g s¯

dVg

§ f pRq “ R ` κ2R2` κ2OpR3

q and κ ą 0

§ long history in physics: Weyl 1918, Pauli 1919, Eddington 1924, . . .

Critical point equation Nαβ “ 8πTαβ

Nαβ “f1pRg qGαβ ´

1

2

´

f pRg q ´ Rg f1pRg q

¯

gαβ `´

gαβ lg ´∇α∇β¯

`

f 1pRg q˘

§ If f linear, Nαβ reduces to Gαβ .

§ Vacuum Einstein solutions are vacuum f(R)-solutions

§ Fourth-order derivatives of g

Gravity/matter coupling

Bianchi identities (geometry) ∇αRαβ “12∇βR

§ imply ∇αGαβ “ 0, but also ∇αNαβ “ 0.

§ Euler equations ∇αTαβ “ 0

Energy-momentum tensor of a massive field

§ Jordan’s coupling

Tαβ :“ ∇αφ∇βφ´´1

2∇γφ∇γφ` Upφq

¯

gαβ lgφ´ U 1pφq “ 0

§ Einstein’s coupling

Tαβ :“ f 1pRg q

´

∇αφ∇βφ´´1

2∇γφ∇γφ` Upφq

¯

gαβ¯

ill-posed PDE for φ

WAVE-KLEIN-GORDON FORMULATION

Field equations in coordinates

Einstein equations Gαβ “ 8πTαβ

§ Second-order system with no specific PDE type

§ Wave coordinates lgxα“ 0

§ Second-order system of 11 nonlinear wave-Klein-Gordon equations

§ Hamiltonian-momentum Einstein’s contraints

Modified gravity equations Nαβ “ 8πTαβ

§ Fourth-order system with no specific PDE type

§ The augmented formulation

§ conformal transformation g:αβ :“ f 1pRg qgαβ

§ evolution equation for the scalar curvature ρ :“ 1κ

ln f 1pRg q

(new degree of freedom)

§ Wave coordinates lg:xα“ 0

§ Second-order system of 12 nonlinear wave-Klein-Gordon equations

§ More involved algebraic structure, and additional constraints

Ricci curvature in wave gauge

2 g:αβBβg

:αγ “ g:

αβBγg

:αβ (following Lindblad-Rodnianski, 2005)

2R:αβ “ ´rlg:g:αβ ` Qαβ ` Pαβ

“ ´g:α1β1

Bα1Bβ1g:αβ ` Qαβ ` Pαβ

(i) terms satisfying Klainerman’s null condition (good decay in time)

Qαβ : “ g:λλ1

g:δδ1

Bδg:αλ1Bδ1g

:βλ

´ g:λλ1

g:δδ1`Bδg

:αλ1Bλg

:βδ1 ´ Bδg

:βδ1Bλg

:αλ1

˘

` g:λλ1

g:δδ1`Bαg

:λ1δ1Bδg

:λβ ´ Bαg

:λβBδg

:λ1δ1

˘

` . . . . . . . . . . . .

(ii) “quasi-null terms” (need again the gauge conditions)

Pαβ :“ ´ 12g:λλ1

g:δδ1

Bαg:δλ1Bβg

:λδ1 `

14g:δδ1

g:λλ1

Bβg:δδ1Bαg

:λλ1

Notation

§ Functions Vκ “ Vκpρq and Wκ “Wκpρq

§ defined from f pRq » R ` κ2R2

§ quadratic in ρ

§ Quadratic potential Upφq “ c2

2φ2

f(R)-gravity for a self-gravitating massive field

rlg:g:αβ “Fαβpg

:, Bg:q ´ 3κ2BαρBβρ` κVκpρqg

:αβ

´ 8π`

2e´κρBαφBβφ` c2φ2e´2κρ g:αβ˘

3κ rlg:ρ´ ρ “κWκpρq ´ 8π´

g:αβBαφBβφ`

c2

2e´κρφ2

¯

rlg:φ´ c2φ “ c2`

e´κρ ´ 1˘

φ` κg:αβBαφBβρ

§ wave gauge conditions g:αβ

Γ:λαβ “ 0

§ curvature compatibility eκρ “ f 1pRe´κρg: q

§ Hamiltonian and momentum constraints(propagating from a Cauchy hypersurface)

In the limit κÑ 0 one has g: Ñ g and ρÑ 8π`

gαβ∇αφ∇βφ` c2

2φ2

˘

Einstein system for a self-gravitating massive field

rlggαβ “Fαβpg , Bgq ´ 8π`

2BαφBβφ` c2φ2 gαβ˘

rlgφ´ c2φ “ 0

Main issues

§ Time-asymptotic decay (energy, sup-norm), global existence theory§ Dependence in f and singular limit f pRq Ñ R

NONLINEAR GLOBAL STABILITY: geometric statements

Earlier works on vacuum spacetimes or massless matter

§ Christodoulou-Klainerman 1993

§ fully geometric proof, Bianchi identities, geometry of null cones§ null foliation, maximal foliation

§ Lindblad-Rodnianski 2010

§ first global existence result in coordinates§ wave coordinates (despite an “instability” result by Choquet-Bruhat)§ asymptotically flat foliation§ their proof relies strongly on the scaling field rBr ` tBt of Minkowski

spacetime

§ Extensions to massless models

§ same time asymptotics, same Killing fields§ Bieri (2009), Zipser (2009), Speck (2014)

Initial value problem

§ geometry of the initial hypersurface pM0 » R3, g0, k0q

§ matter fields φ0, φ1

§ initial data sets close to a spacelike, asympt. flat slice in Minkowskispacetime

Dynamics of self-gravitating massive matter

§ Spatially compact problem

§ compactly supported massive scalar field

§ Positive mass theorem

§ no solution can be exactly Minkowski “at infinity”§ coincides with a slice of Schwarzschild near space infinity, with ADM mass

m ăă 1

§ Compact Schwarzschild perturbation

Theorem 1. Nonlinear stability of Minkowski spacetime with self-gravitatingmassive fields

Consider the Einstein-massive field system when the initial data set pM0 »

R3, g0, k0, φ0, φ1q is a compact Schwarzschild perturbation satisfying the Einsteinconstraint equations. Then, the initial value problem

§ admits a globally hyperbolic Cauchy development,

§ which is foliated by asymptotically hyperbolic hypersurfaces.

§ Moreover, this spacetime is future causally geodesically complete andasymptotically approaches Minkowski spacetime.

Theorem 2. Nonlinear stability of Minkowski spacetime in f(R)-gravity

Consider the field equations of f pRq-modified gravity when the initial data setpM0 » R3, g0, k0,R0,R1, φ0, φ1q is a compact Schwarzschild perurbation satisfy-ing the constraint equations of modified gravity. Then, the initial value problem

§ admits a globally hyperbolic Cauchy development,

§ which is foliated by asymptotically hyperbolic hypersurfaces.

§ Moreover, this spacetime is future causally geodesically complete andasymptotically approaches Minkowski spacetime.

Limit problem κÑ 0

§ relaxation phenomena for the spacetime scalar curvature

§ passing from a second-order wave equation to an algebraic equation

Theorem 3. f(R)-spacetimes converge toward Einstein spacetimes

The Cauchy developments of modified gravity in the limit κÑ 0

when the nonlinear function f “ f pRq (the integrand in theHilbert-Einstein action) approaches the scalar curvature function R

converge (in every bounded time interval, in a sense specified quantitatively inSobolev norms) to Cauchy developments of Einstein’s gravity theory.

OVERVIEW OF THE HYPERBOLOIDAL FOLIATION METHOD

§ LeFloch & Ma, monograph published by World Scientific, 2014

§ Earlier work by Klainerman and Hormander for Klein-Gordon

Foliations by asymptotically hyperboloidal hypersurfaces

§ global coordinate chart pxαq “ pt, xaq with a “ 1, 2, 3

§ boosts La :“ xaBt ` tBa associated with the Minkowski metric

gM “ ´dt2`ř

a“1,2,3 dxa

§ foliation of the interior of the light cone by hyperboids

Notation

§ foliation of the future light cone from pt, xq “ p1, 0q

§ level sets of constant Lorentzian distance from the origin p0, 0q

§ hyperboloids Hs :“

pt, xqL

t ą 0; t2´ |x |2 “ s2

(

§ parametrized by their hyperbolic radius s ě 1

§ data prescribed on Hs0 for some s0 ą 1

Vector frames in addition to the frame pBt , Baq

Semi-hyperboloidal frame B0 :“ Bt Ba :“La

t“

xa

tBt ` Ba

Hyperboloidal frame B0 :“ Bs Ba “ Ba

Change of frame Bα “ Φα1

α Bα1 Bα “ Ψα1

α Bα1

Tensor components Tαβ “ Tα1β1Φα1

α Φβ1

β

Energy functional on hyperboloids of Minkowski spacetime

§ Semi-hyperboloidal decomposition of the wave operator

lu “ ´s2

t2B0B0u ´

3

tBtu ´

xa

t

`

B0Bau ` BaB0u˘

`ÿ

a

BaBau

§ For instance, for the linear Klein-Gordon operator in Minkowski space

lu ´ c2 u

u “ upt, xq “ ups, xq with s2 “ t2 ´ r2 and r2 “ř

apxaq2

Em,c rs, us :“

ż

Hs

˜

s2

t2pBtuq

2 `

3ÿ

a“1

´ xa

tBtu ` Bau

¯2`

c2

2u2

¸

dx

“

ż

Hs

˜

s2

s2 ` r2pB0uq

2 `

3ÿ

a“1

`

Bau˘2`

c2

2u2

¸

dx

Functional analysis for hyperboloidal foliations

§ Decompose the wave operators, the metric, etc. in various frames

§ Good commutator properties

§ Weighted norms based on the translations Bα and the Lorentzian boostsLa only

Weighted norms

§ On each hypersurface

uHnrss :“ÿ

|J|ďn

ÿ

a“1,2,3

´

ż

Hs»R3

|LJau|

2 dx¯12

§ completion of smooth and spatially compacted functions

§ In spacetimeuHN rs0,s1s

:“ supsPrs0,s1s

ř

|I |`nďN

›

›BIu›

›

Hnrss

§ Here, 1 ď s0 ă s1 ă `8 and N ě 0§ for each s P rs0, s1s and for all multi-index |I | “ m ď N, one hasBIups, ¨q P HN´mrss.

Klaineman-Sobolev inequality on hyperboloids, Hardy inequalities for hyperboloids

Klainerman (1985), Hormander (1997)

For functions u defined on Hs Ă R3`1 (with t2 “ s2 ` |x |2):

suppt,xqPHs

t32 |upt, xq| À uH2rss »ÿ

|I |ď2

LIuHs

LeFloch-Ma (2014)

For all functions u defined on a hyperboloid Hs :›

›

›

u

r

›

›

›

L2fpHs q

Àÿ

a

BauL2fpHs q

with Ba “ t´1La

For all functions defined on the hyperboloidal foliation›

›

›

u

s

›

›

›

L2fpHs q

À

›

›

›

u

s0

›

›

›

L2fpHs0

q`

ÿ

a

BauL2fpHs q

`ÿ

a

ż s

s0

´

BauL2fpHs1 q

` ps 1tqBauL2fpHs1 q

¯ ds 1

s 1

Remark.

§ Compute the divergence of the vector field´

0, t xau2

p1`r2qs2 χprtq¯

for some smooth

cut-off function concentrated near the light cone χpyq “

#

0 0 ď y ď 13

1 23 ď y

§ Similar to integrating Br`

u2r˘

for the classical Hardy inequality

Initial value problem

§ Initial data prescribed on an asymptotically hyperbolic hypersurface,identified with Hs0 in our coordinates

§ Energy estimates expressed in domains limited by two hyperboloids

Bootstrap

§ time-integrability of the source terms

§ total contribution of the interaction terms contribute only a finite amountto the growth of the total energy

Model formally extracted from the Einstein-massive field system

§ handle “strong interactions” between the metric and the matter, say

´lu “ PαβBαvBβv ` Rv 2

´lv ` u HαβBαBβv ` c2v “ 0

§ hierarchy of energy bounds of various order of differentiation / growth in s

§ successive improvements of the sup-norm bounds, via successiveapplications of sup-norm estimates

Pointwise decay of solutions§ sup-norm bounds for nonlinear wave equations and nonlinear Klein-Gordon

equations on curved space§ sharp rates required§ several L8-L8 estimates

§ integration along radial rays (curved space, Klein-Gordon)§ integration along characteristics (curved space, wave equation)§ Kirchhoff explicit formula (flat space, wave equation)

1. A sharp L8 estimate for Klein-Gordon equations on curved space§ Introduce the vector field BK :“ Bt `

xa

tBa

§ orthogonal to the hyperboloids and proportional to the scaling vector field§ second-order ODE’s along radial rays from the origin§ Klainerman (1985), but not optimal time rate; Delort etal. (2004) in two

spatial dimensions

Proposition

Solutions of the Klein-Gordon equation 3κ rlg:ρ´ ρ “ σ

κ´12s32ˇ

ˇpρ´ σqpt, xqˇ

ˇ` pstq´1s12ˇ

ˇBKpρ´ σqpt, xqˇ

ˇ À V pt, xq

§ the implied constant independent of κ P r0, 1s

§ V “ V ps, xq determined from the metric g: and the right-hand side σ

2. A sharp L8 estimate for wave equations on curved space

§ Start with a decomposition of the flat wave operator l in thesemi-hyperboloidal frame as

l “ ´s2

t2BtBt ´

3

tBt ` B

aBa ´

xa

t

`

BtBa ` BaBt˘

§ Given the curved metric g:αβ“ gαβM ` Hαβ , consider the modified wave

operatorrlg: “ l` HαβBαBβ

§ Decomposition for the curved wave operator´

pBt ` Br q ´ t2pt ` rq´2H00

pBt ´ Br q

¯´

`

Bt ´ Br˘

pruq¯

“ ´r rlgu ` rÿ

aăb

`

r´1Ωab

˘2u ` H00X rus ` r Y ru,Hs

§ H00 is the p0, 0q-component of the metric perturbation in thesemi-hyperboloidal frame

§ X rus and Y ru,Hs involve§ derivatives tangential to the hyperboloids§ metric component Haα

§ independent of H00

Solutions to the wave equation ´lu ´ HαβBαBβu “ F

§ in which |H00pt, xq| ď ε t´r`1t

§ given any point ps1, x1q ” pt1, xq on an arbitrary hyperboloid Hs1

§ denote by s P r1, s1s ÞÑ`

s, ϕps; s1, x1q˘

the characteristic integral curve leavingfrom ps1, x1q associated with the vector field

Bt `pt ` rq2 ` t2H00pt, xq

pt ` rq2 ´ t2H00pt, xqBr .

§ Integation along this curve

(Lindblad and Rodnianski used a different vector field)

Proposition

|pBt ´ Br qupt, xq| À t´1 supH1

´

|pBt ´ Br qpruq|¯

` t´1|upt, xq|

` t´1

ż t

t0

`

|F | ` |M|˘

pτ, ϕpτ ; t, xqq dτ

§ t0 is the initial time reached on the hyperboloid H1 from pt, xq

§ M :“ rř

aăb

`

r´1Ωab

˘2u ` H00X rus ` rY ru,Hs

3. A sharp L8 estimate for the wave equation in Minkowski space

§ in Minkowski spacetime, we can rely on Kichhoff formula

Proposition

Let u be a spatially compactly supported to the wave equation

´lu “ f ,

u|t“2 “ 0, Btu|t“2 “ 0,

where the source f is spatially compactly supported and satisfies

|f | ď Cf t´2´ν

pt ´ rq´1`µ

for some constants Cf ą 0, 0 ă µ ď 12, and 0 ă |ν| ď 12. Then, one has

|upt, xq| À

#

Cfνµpt ´ rqµ´νt´1, 0 ă ν ď 12

Cf|ν|µpt ´ rqµt´1´ν , ´12 ď ν ă 0

The Wave-Klein-Gordon Model

´lu “ PαβBαvBβv ` Rv 2´lv ` u HαβBαBβv ` c2v “ 0

Theorem. Global existence theory for the wave-Klein-Gordon model

Given any integer N ě 8, there exists a positive constant ε0 “ ε0pNq ą 0 suchthat if the compactly supported initial data satisfy

pu0, v0qHN`1pR3q ` pu1, v1qHN pR3q ď ε0,

then the associated Cauchy problem admits a global-in-time solution.

§ hierarchy of energy bounds

Emps, BILJuq12 ď C1εs

kδ, |J| “ k, |I | ` |J| ď N (wave / high-order)

Emps, BILJuq12 ď C1ε, |I | ` |J| ď N ´ 4 (wave / low-order)

Em,c2 ps, BILJvq12 ď C1εs12`kδ, |J| “ k, |I | ` |J| ď N (KG / high-order)

Em,c2 ps, BILJvq12 ď C1εskδ, |J| “ k, |I | ` |J| ď N ´ 4 (KG / low-order)

§ successive improvements of the sup-norm bounds, via successive applications ofthe sup-norm estimates above

§ refined pointwise estimate |LJhαβ | À εt´1sCε12

pstq´2`3δ|BILJφ| ` pstq´3`3δ|BILJBKφ| À εs´32`Cε12

Structure of the Einstein equations

§ Quadratic nonlinearities in Bg : involved algebraic structure

§ Einstein and f(R)-gravity do not satisfy the null condition(Lindblad-Rodnianski, 2005)

§ detailled analysis of the field equations

§ here, we work with a hyperboloidal foliation

§ The quasi-null structure (P00,Paβ)§ The wave gauge condition used to control the component Bth

00

§ Further details given below.

Recall the notation

Semi-hyperboloidal frame B0 :“ Bt Ba :“La

t“

xa

tBt ` Ba

Hyperboloidal frame B0 :“ Bs Ba “ Ba

Change of frame Bα “ Φα1

α Bα1 Bα “ Ψα1

α Bα1

Tensor components Tαβ “ Tα1β1Φα1

α Φβ1

β

NONLINEAR STABILITY: Statements in wave coordinates

Theorem 1. Nonlinear stability of Minkowski spacetime for self-gravitating mas-sive fields

Consider the Einstein-massive field system in wave coordinates. Given any sufficiently

large integer N, there exist constants ε0, δ,C0 ą 0 such that the following property

holds.

Consider an asymptotically hyperboloidal initial data set pR3, g 0, k0, φ0, φ1q co-inciding with Schwarzschild outside a compact set and satisfying Einstein’sHamiltonian and momentum constraints together with the smallness conditions(ε ď ε0)

Bc`

g 0,ab ´ gM,ab

˘

HN r1s ` k0,ab ´ kM,abHN r1s ď ε

Baφ0, φ0, φ1HN r1s ď ε.

Then the solution exists globally for all times s ě 1

Bγ`

gαβ ´ gM,αβ˘

HN r1,ss ď C0εsδ

Bαφ, φHN r1,ss ď C0εsδ`12 (high-order energy)

Bαφ, φHN´4r1,ss ď C0εsδ (low-order energy)

First global stability theory

§ large class of spacetimes containing massive matter

§ Einstein gravity, as well as f(R)-gravity theory (see below)

§ sufficient decay so that the spacetime is future geodesically complete

§ smallness conditions on both g , φ necessary (gravitational collapse)

Energy may grow in time

§ exponent such that lim supεÑ0 δpεq “ 0

§ tδ observed by Alinhac (2006) for some semilinear hyperbolic systems

§ sδ`12 for the scalar field

Work in preparation

§ asymptotically Schwarzschild data

gab “ δab

´

1` 2mr

¯

` Opr´1´δq, kab “ Opr´2´δq, φ “ Opr´1´δq

§ spacetime weight outside the light cone

w “ 1 for r ď t, while w “ p1` |r ´ t|q1`δ for r ě t

Theorem 2. Nonlinear stability of Minkowski spacetime in modified gravity

Consider the field equations of modified gravity in the augmented conformal formulation

and in conformal wave coordinates. Given any sufficiently large integer N and some fixed

κ P r0, 1s, there exist constants ε0, δ,C0 ą 0 such the following property holds.

Consider an asymptotically hyperboloidal initial data coinciding with

Schwarzschild outside a a compact set, pR3, g:0, k:

0, ρ0, ρ1, φ0, φ1q, satisfyingthe constraints of modified gravity with and

Bc`

g:0,ab ´ gM,ab

˘

HN r1s ` k:

0,ab ´ kM,abHN r1s ď ε

Baρ0, ρ0, ρ1HN r1s ` Baφ0, φ0, φ1HN r1s ď ε.

Then, the solution exists globally for all times s ě 1

Bγ`

g:αβ ´ gM,αβ˘

HN r1,ss ď C0εsδ

Bαρ, ρHN r1,ss ` Bαφ, φHN r1,ss ď C0εsδ`12 (high-order energy)

Bαρ, ρHN´4r1,ss ` Bαφ, φHN´4r1,ss ď C0εsδ (low-order energy)

with constants possibly depending upon κ.

Notation: σ :“ 8π`

g:αβBαφBβφ`

c2

2e´κρφ2

˘

Theorem 3. The singular limit problem for the modified gravity equations

Consider a sequence of initial data sets depending upon κÑ 0, as follows:

Bc`

g:0,ab ´ gM,ab

˘

HN r1s ` k:

0,ab ´ kM,abHN r1s ď ε

κ12 ρ1, κ12Baρ0, ρ0HN r1s ` Baφ0, φ0, φ1HN r1s ď ε

ρ1 ´ σ1, Bapρ0 ´ σ0q, κ´12

pρ0 ´ σ0qHN´2r1s ď ε.

Then, the solutions exist for all times s ě 1 and all κ Ñ 0, with a constant C0

independent of κBγ

`

g:αβ ´ gM,αβ˘

HN r1,ss ď C0εsδ

κ12Bαρ, ρHN r1,ss ` Bαφ, φHN r1,ss ď C0εs

δ`12

κ12Bαρ, ρHN´4r1,ss ` Bαφ, φHN´4r1,ss ď C0εs

δ

Bαpρ´ σq, κ´12

pρ´ σqHN´2r1,ss ď C0εsδ`12

Bαpρ´ σq, κ´12

pρ´ σqHN´6r1,ss ď C0εsδ

Moreover, if

§ the initial data set`

g:pκq, k:pκq

, ρ:0pκq, ρ:1

pκq, φ:0

pκq, φ:1

pκq˘converges to some

limit`

g p0q, kp0q, ρp0q0 , ρ

p0q1 , φ

p0q0 , φ

p0q1

˘

§ in the norms associated with the uniform bounds above

then

§ the corresponding solutions pg:pκq, ρ:pκq, φ:pκqq to the system of modified

gravity converge to a solution pg p0q, φp0qq of the Einstein-massive field system

with, in particular, in the HN´2 norm on each compact set in time

ρpκq Ñ Rp0q :“ 8π´

g p0qαβBαφp0qBβφ

p0q `c2

2pφp0qq2

¯

as κÑ 0.

Remarks.

§ The convergence property above relates a fourth-order system to a second-ordersystem of PDEs.

§ The highest (pN ` 1q-th order) derivatives of the scalar curvature are Opεκ´12q

in L2 and may blow-up when κÑ 0, while the N–th order derivatives are solelybounded and need not converge in a strong sense.

§ Throughout, the initial data set of modified gravity (and thus the solution to thefield equations) satisfies the compatibility condition eκρ “ f 1pRe´κρg: q relating

the augmented variable ρ to the spacetime scalar curvature.

CONCLUDING REMARKS§ Application of the Hyperboloidal Foliation Method

§ Encompass a large class of nonlinear wave-Klein-Gordon systems withquasi-null coupling

Alinhac, Lindblad on asymptotically Euclidian foliations

§ Fully geometric construction§ Improve the growing rate s12 for the scalar field§ Additional arguments, hyperboloidal foliation based on the curved metric

Q. Wang by generalizing Christodoulou-Klainerman’s geometric method

§ Extension to other massive fields§ Kinetic models (density), Vlasov equa. (collisionless), Boltzmann equa.

Fajman, Joudioux, Smulevici

§ Penrose’s peeling estimates§ Asymptotics for the spacetime curvature along timelike directions

Penrose, Christodoulou-Klainerman

§ Very challenging open problem in wave coordinatesLindblad-Rodnianski

§ Our Hyperboloidal Foliation Method provides a possible path to establishingthe peeling estimates directly in wave gauge.

§ For instance, for nonlinear wave systems with null forms and without metriccoupling

§ proof simpler than the standard one based on flat hypersurfaces§ uniform energy bound for the highest-order energy

P.G. LeFloch and Y. Ma

§ The hyperboloidal foliation method, World Scientific, 2014

§ The nonlinear stability of Minkowski space for self-gravitating massivefields

§ A wave-Klein-Gordon model Comm. Math. Phys. (2016)

ArXiv:1507.01143

§ Analysis of the Einstein equations ArXiv:1511.03324

§ Analysis of the f pRq-theory of modified gravity ArXiv:1412.8151

Comptes Rendus Acad. Sc. Paris (2016)

+ article under completion

WAVE EQUATIONS WITH NULL INTERACTIONS

The simplest model

lu “ PαβBαu Bβu u|Hs0“ u0, Btu|Hs0

“ u1 p‹q

§ initial data u0, u1 compactly supported in the intersection of the spacelikehypersurface Hs0 and the cone K “

pt, xq |x | ă t ´ 1(

with s0 ą 1

§ standard null condition: Pαβξαξβ “ 0 for all ξ P R4 satisfying ´ξ20 `

ř

a ξ2a “ 0

§ hyperboloidal energy EM “ EM,0: Minkowski metric and zero K-G mass

§ admissible vector fields Z P Z : spacetime translations Bα, boosts La

Theorem. Global existence theory for wave equations with null interactions

There exist ε0 ą 0 and C1 ą 1 such that for all initial data satisfyingř

|I |ď3

ř

ZPZ EMps0,ZIuq12 ď ε ď ε0

the Cauchy problem p‹q admits a global-in-time solution, satisfying the uniformenergy bound

ÿ

|I |ď3

ÿ

ZPZ

EMps,ZIuq12 ď C1ε

and the uniform decay estimateˇ

ˇBαupt, xqˇ

ˇ ďC1ε

t pt´|x|q12.

THE QUASI-NULL HYPERBOLOIDAL STRUCTUREOF THE EINSTEIN EQUATIONS

The hierarchy property of quasi-null terms in the hyperboloidal foliation

Proposition

rlgh00 » GShhp0, 0q ` P00 ´

´

2κρ2 ` 8πc2φ2¯

gM,00

´ 3κ2BtρBtρ

´ 16πBtφBtφ` Cp0, 0q

rlgh0a »2

tBah00 ´

2xa

t3h00 ` GShhp0, 0q ` Sρρp0, 0q ` Sφφp0, 0q ` Cp0, 0q

rlghaa »4xa

t2Bah00 `

´ 2

t2´

6|xa|2

t4

¯

h00 `4

tBah0a ´

4xa

t3h0a

´

´

2κρ2 ` 8πc2φ2¯

gM,aa

` GShhp0, 0q ` Sρρp0, 0q ` Sφφp0, 0q ` Cp0, 0q

rlghab »2

t2

´

xbBah00 ` xaBbh00

¯

´6xaxb

t4h00 `

2

tBah0b ´

2xa

t3h0b `

2

tBah0a ´

2xb

t3h0a

´

´

2κρ2 ` 8πc2φ2¯

gM,ab

` GShhp0, 0q ` Sρρp0, 0q ` Sφφp0, 0q ` Cp0, 0q

with a ‰ b.

Observations

§ The quasi-null terms Pαβ in the semi-hyperboloidal frame read

Pαβ “1

4g:γγ1

g:δδ1

BαhγδBβhγ1δ1 ´1

2g:γγ1

g:δδ1

Bαhγγ1Bβhδδ1 .

§ Null terms: g:αβBαuBβv and BαuBβv ´ BβuBαv

§ All the components Pαβ except P00 are “good terms” (null terms):

§ they involve at least one derivative tangential to the hyperboloids,§ plus terms having more decay in time (change of frame).

Proposition

Up to irrelevant multiplicative coefficients, one has

Paβ »GShhp0, 0q ` Cp0, 0q

P00 » g:γγ1

g:δδ1

Bthγγ1Bthδδ1 ` gγγ1

M gδδ1

M BthγδBthγ1δ1

`GShhp0, 0q ` Fp0, 0q ` Cp0, 0q

Proof. Observe that

P00 “1

4g:γγ1

g:δδ1

BthγδBthγ1δ1 ´1

2g:γγ1

g:δδ1

Bthγγ1Bthδδ1

»1

4g:γγ

1

g:δδ1

BthγδBthγ1δ1 ´1

2g:γγ

1

g:δδ1

Bthγγ1Bthδδ1 ` Fp0, 0q,

where we have changed the frame (for instance for the first term):

g:γγ1

gδδ1

BthγδBthγ1δ1 “g:γγ1gδδ

1

BthγδBthγ1δ1

` g:γγ1

g:δδ1

hγ2δ2Bt`

Ψγ2

γ Ψδ2

δ

˘

Bt`

Ψγ3

γ1 Ψδ3

δ1

˘

hγ3δ3

` g:γγ1

g:δδ1

Ψγ2

γ Ψδ2

δ Bthγ2δ2Bt`

Ψγ3

γ1 Ψδ3

δ1

˘

hγ3δ3

` g:γγ1

g:δδ1

Bt`

Ψγ2

γ Ψδ2

δ

˘

hγ2δ2Ψγ2

γ1 Ψδ2

δ1 Bthγ3δ3 .

Moreover, modulo cubic terms, we can also replace the curved metric by theflat one

1

4gγγ

1

M gδδ1

M BthγδBthγ1δ1 ´1

2g:γγ

1

g:δδ1

Bthγγ1Bthδδ1 ` Fp0, 0q ` Cp0, 0q.

l

The (0,0)-component of the metric Bth00

§ Notation

hαβ “ g:αβ´ gαβM hαβ “ g:αβ ´ gM,αβ

hαβ “ g:αβ ´ gαβM hαβ “ g:αβ´ gM,αβ

hαβ “ hα1β1Ψα

α1Ψββ1 hαβ “ hα1β1Φ

α1

α Φβ1

β .

§ The wave gauge condition g:αβ

Γ:γαβ “ 0 reads

g:βγBαg:αβ

“1

2g:αβBγg

:αβ .

Proposition. Key component of the metric

Bth00» pstq2Bh ` B h ` t´1h ` h Bh ` t´1h h

Corollary. High-order derivative of the key metric component

ˇ

ˇBth00ˇ

ˇ À pstq2ˇ

ˇBhˇ

ˇ`ˇ

ˇBhˇ

ˇ` t´1ˇ

ˇhˇ

ˇ`ˇ

ˇBhˇ

ˇ

ˇ

ˇhˇ

ˇ

ˇ

ˇBILJBth

00ˇ

ˇ`ˇ

ˇBtBILJh00

ˇ

ˇ

Àÿ

|I 1|`|J1|ď|I |`|J|

|J1|ď|J|

´

pstq2ˇ

ˇBBI 1LJ1h

ˇ

ˇ`ˇ

ˇBI 1LJ1

Bhˇ

ˇ` t´1ˇ

ˇBI 1LJ1h

ˇ

ˇ

¯

`ÿ

|I1|`|I2|ď|I ||J1|`|J2|ď|J|

ˇ

ˇBI1LJ1h

ˇ

ˇ

ˇ

ˇBBI2LJ2h

ˇ

ˇ.

Observations

§ The “bad” derivative of h00 is bounded by the “good” derivatives arisingin the right-hand side.

§ The “bad” termˇ

ˇBhˇ

ˇ still arise, but it is multiplied by the factor pstq2

which provides us with extra decay and turns this term into a “good” term.

The reduced form of the quasi-null terms

Proposition

P00 » BthaαBthbβ `GShhp0, 0q ` Cp0, 0q ` Fp0, 0q

Paβ » GShhp0, 0q ` Cp0, 0q ` Fp0, 0q

Observations

§ Hence, it only remains to analyze the term BthaαBthbβ , which is donewithin the bootstrap argument.

§ This proposition follows from

§ P00 » g:γγ1g:δδ

1Bthγγ1Bthδδ1 ` gγγ

1

M gδδ1

M BthγδBthγ1δ1 ` . . .

§ Bth00 » pstq2Bh ` B h ` t´1h ` h Bh ` t´1h h

§ and a technical calculation leading to

g:αα1

g:ββ1

Btg:

αα1Btg

:

ββ1

» g:0aB0g:

0ag:0bB0g

:

0b` GShhp0, 0q ` Fp0, 0q ` Cp0, 0q