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THE NONLINEAR STABILITY OF MINKOWSKI SPACE FOR SELF-GRAVITATING MASSIVE MATTER Philippe LeFloch Laboratoire Jacques-Louis Lions & CNRS Universit´ e Pierre et Marie Curie (Paris 6, Jussieu) [email protected] GENERAL RELATIVITY § Global geometry of spacetimes pM, g αβ q with signature , `, `, `q § Einstein equations for self-gravitating matter G αβ 8πT αβ § Einstein curvature G αβ R αβ ´pR{2qg αβ § Riccci curvature R αβ “B 2 g `B‹Bg § scalar curvature R :R α α g αβ R αβ 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 T αβ 0 or massless matter fields § Christodoulou - Klainerman (1993), Lindblad - Rodnianski (2010) § Massive matter fields massive matter T αβ , open since 1993 § LeFloch - Yue Ma (2016)

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  • THE NONLINEAR STABILITY OF MINKOWSKI SPACE FORSELF-GRAVITATING MASSIVE MATTER

    Philippe LeFlochLaboratoire Jacques-Louis Lions & CNRS

    Université Pierre et Marie Curie (Paris 6, Jussieu)[email protected]

    GENERAL RELATIVITY

    § Global geometry of spacetimes pM, gαβq with signature p´,`,`,`q§ Einstein equations for self-gravitating matter Gαβ “ 8πTαβ

    § Einstein curvature Gαβ “ Rαβ ´ pR{2qgαβ§ Riccci curvature Rαβ “ B2g ` 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 “ c22φ2, described

    by the energy-momentum tensor

    Tαβ :“ ∇αφ∇βφ´´1

    2gα1β1∇α1φ∇β

    1φ` 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 gravityGeneralized Hilbert-Einstein functional

    § Gravitation mediated by additional fields

    § Functionalż

    M

    ´

    f pRg q ` 16πLrφ, g s¯

    dVg

    § f pRq “ R ` κ2R2 ` κ2OpR3q and κ ą 0

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

  • Critical point equation Nαβ “ 8πTαβ

    Nαβ “f 1pRg qGαβ ´1

    2

    ´

    f pRg q ´ Rg f 1pRg 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 :“ 1

    κln 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:αβ Bβg:αγ “ g:

    αβ Bγg:αβ (following Lindblad-Rodnianski, 2005)

    2R:αβ “ ´rlg:g:αβ ` Qαβ ` Pαβ

    “ ´g:α1β1Bα1Bβ1g:αβ ` Qαβ ` Pαβ

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

    Qαβ : “ g:λλ1

    g:δδ1Bδg:αλ1Bδ1g

    :βλ

    ´ g:λλ1g:δδ1`Bδg:αλ1Bλg

    :βδ1 ´ Bδg

    :βδ1Bλg

    :αλ1

    ˘

    ` g:λλ1g:δδ1`Bαg:λ1δ1Bδg

    :λβ ´ Bαg

    :λβBδg

    :λ1δ1

    ˘

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

    Pαβ :“ ´ 12g:λλ

    1g:δδ1Bαg:δλ1Bβg:λδ1 ` 14g

    :δδ1g:λλ1Bβg:δδ1Bαg:λλ1

    Notation

    § Functions Vκ “ Vκpρq and Wκ “Wκpρq§ defined from f pRq » R ` κ

    2R2

    § quadratic in ρ

    § Quadratic potential Upφq “ c22φ2

  • 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 metricgM “ ´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 :“Lat“ x

    a

    tBt ` Ba

    Hyperboloidal frame B0 :“ Bs Ba “ BaChange of frame Bα “ Φα

    1α Bα1 Bα “ Ψα

    1α Bα1

    Tensor components Tαβ “ Tα1β1Φα1α Φ

    β1

    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

    t2pBtuq2 `

    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

    }u}Hnrss :“ÿ

    |J|ďn

    ÿ

    a“1,2,3

    ´

    ż

    Hs»R3|LJau|2 dx

    ¯1{2

    § completion of smooth and spatially compacted functions

    § In spacetime}u}HN 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

    t3{2 |upt, xq| À }u}H2rss »ÿ

    |I |ď2}LIu}Hs

    LeFloch-Ma (2014)

    For all functions u defined on a hyperboloid Hs :›

    u

    r

    L2fpHs q

    Àÿ

    a

    }Bau}L2fpHs q with Ba “ t

    ´1La

    For all functions defined on the hyperboloidal foliation›

    u

    s

    L2fpHs q

    À›

    u

    s0

    L2fpHs0 q

    `ÿ

    a

    }Bau}L2fpHs q

    `ÿ

    a

    ż s

    s0

    ´

    }Bau}L2fpHs1 q

    ` }ps 1{tqBau}L2fpHs1 q

    ¯ ds 1

    s 1

    Remark.

    § Compute the divergence of the vector field´

    0, t xau2

    p1`r2qs2 χpr{tq¯

    for some smooth

    cut-off function concentrated near the light cone χpyq “#

    0 0 ď y ď 1{31 2{3 ď y

    § Similar to integrating Br`

    u2{r˘

    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 ` x

    a

    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´1{2s3{2ˇ

    ˇpρ´ σqpt, xqˇ

    ˇ` ps{tq´1s1{2ˇ

    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 ` BaBa ´

    xa

    t

    `

    BtBa ` BaBt˘

    § Given the curved metric g:αβ “ gαβM ` H

    αβ , consider the modified waveoperator

    rlg: “ l` HαβBαBβ

    § Decomposition for the curved wave operator´

    pBt ` Br q ´ t2pt ` rq´2H00pBt ´ 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 � 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, xqpt ` rq2 ´ t2H00pt, xq

    Br .

    § 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´1ż t

    t0

    `

    |F | ` |M|˘

    pτ, ϕpτ ; t, xqq dτ

    § t0 is the initial time reached on the hyperboloid H1 from pt, xq§ M :“ r

    ř

    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´νpt ´ rq´1`µ

    for some constants Cf ą 0, 0 ă µ ď 1{2, and 0 ă |ν| ď 1{2. Then, one has

    |upt, xq| À#

    Cfνµpt ´ rqµ´νt´1, 0 ă ν ď 1{2

    Cf|ν|µ pt ´ rq

    µt´1´ν , ´1{2 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 �0 “ �0pNq ą 0 suchthat if the compactly supported initial data satisfy

    }pu0, v0q}HN`1pR3q ` }pu1, v1q}HN pR3q ď �0,

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

    § hierarchy of energy bounds

    Emps, BILJuq1{2 ď C1�skδ, |J| “ k, |I | ` |J| ď N (wave / high-order)

    Emps, BILJuq1{2 ď C1�, |I | ` |J| ď N ´ 4 (wave / low-order)

    Em,c2 ps, BILJvq1{2 ď C1�s1{2`kδ, |J| “ k, |I | ` |J| ď N (KG / high-order)

    Em,c2 ps, BILJvq1{2 ď C1�skδ, |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 �t´1sC�1{2

    ps{tq´2`3δ|BILJφ| ` ps{tq´3`3δ|BILJBKφ| À �s´3{2`C�1{2

  • 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 Bth00

    § Further details given below.

    Recall the notation

    Semi-hyperboloidal frame B0 :“ Bt Ba :“Lat“ x

    a

    tBt ` Ba

    Hyperboloidal frame B0 :“ Bs Ba “ BaChange of frame Bα “ Φα

    1α Bα1 Bα “ Ψα

    1α Bα1

    Tensor components Tαβ “ Tα1β1Φα1α Φ

    β1

    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 �0, δ,C0 ą 0 such that the following propertyholds.

    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(� ď �0)

    }Bc`

    g 0,ab ´ gM,ab˘

    }HN r1s ` }k0,ab ´ kM,ab}HN r1s ď �}Baφ0, φ0, φ1}HN r1s ď �.

    Then the solution exists globally for all times s ě 1

    }Bγ`

    gαβ ´ gM,αβ˘

    }HN r1,ss ď C0�sδ

    }Bαφ, φ}HN r1,ss ď C0�sδ`1{2 (high-order energy)

    }Bαφ, φ}HN´4r1,ss ď C0�sδ (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 sup�Ñ0 δp�q “ 0§ tδ observed by Alinhac (2006) for some semilinear hyperbolic systems

    § sδ`1{2 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 �0, 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,ab}HN r1s ď �

    }Baρ0, ρ0, ρ1}HN r1s ` }Baφ0, φ0, φ1}HN r1s ď �.

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

    }Bγ`

    g:αβ ´ gM,αβ˘

    }HN r1,ss ď C0�sδ

    }Bαρ, ρ}HN r1,ss ` }Bαφ, φ}HN r1,ss ď C0�sδ`1{2 (high-order energy)

    }Bαρ, ρ}HN´4r1,ss ` }Bαφ, φ}HN´4r1,ss ď C0�sδ (low-order energy)

    with constants possibly depending upon κ.

  • Notation: σ :“ 8π`

    g:αβBαφBβφ` c

    2

    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,ab}HN r1s ď �

    }κ1{2 ρ1, κ1{2 Baρ0, ρ0}HN r1s ` }Baφ0, φ0, φ1}HN r1s ď �

    }ρ1 ´ σ1, Bapρ0 ´ σ0q, κ´1{2 pρ0 ´ σ0q}HN´2r1s ď �.

    Then, the solutions exist for all times s ě 1 and all κ Ñ 0, with a constant C0independent of κ

    }Bγ`

    g:αβ ´ gM,αβ˘

    }HN r1,ss ď C0�sδ

    }κ1{2Bαρ, ρ}HN r1,ss ` }Bαφ, φ}HN r1,ss ď C0�sδ`1{2

    }κ1{2Bαρ, ρ}HN´4r1,ss ` }Bαφ, φ}HN´4r1,ss ď C0�sδ

    }Bαpρ´ σq, κ´1{2pρ´ σq}HN´2r1,ss ď C0�sδ`1{2

    }Bαpρ´ σq, κ´1{2pρ´ σq}HN´6r1,ss ď C0�sδ

  • 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 modifiedgravity 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 p0qαβBαφp0qBβφp0q `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´1{2qin 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 relatingthe 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 s1{2 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:1412.8151Comptes 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 |ď3ř

    ZPZ EMps0,ZIuq1{2 ď ε ď ε0

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

    ÿ

    |I |ď3

    ÿ

    ZPZEMps,Z Iuq1{2 ď C1ε

    and the uniform decay estimateˇ

    ˇBαupt, xqˇ

    ˇ ď C1εt pt´|x|q1{2 .

  • 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:δδ1BαhγδBβhγ1δ1 ´

    1

    2g:γγ1

    g:δδ1Bαhγγ1Bβhδδ1 .

    § 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:γγ1g:δδ

    1Bthγγ1Bthδδ1 ` g

    γγ1

    M gδδ1

    M BthγδBthγ1δ1`GShhp0, 0q ` Fp0, 0q ` Cp0, 0q

  • Proof. Observe that

    P00 “1

    4g:γγ1

    g:δδ1BthγδBthγ1δ1 ´

    1

    2g:γγ1

    g:δδ1Bthγγ1Bthδδ1

    » 14g:γγ

    1g:δδ

    1BthγδBthγ1δ1 ´

    1

    2g:γγ

    1g:δδ

    1Bthγγ1Bthδδ1 ` Fp0, 0q,

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

    g:γγ1

    gδδ1BthγδBthγ1δ1 “g:γγ

    1gδδ

    1BthγδBthγ1δ1

    ` g:γγ1g:δδ1

    hγ2δ2Bt`

    Ψγ2

    γ Ψδ2

    δ

    ˘

    Bt`

    Ψγ3

    γ1 Ψδ3

    δ1˘

    hγ3δ3

    ` g:γγ1g:δδ1

    Ψγ2

    γ Ψδ2

    δ Bthγ2δ2Bt`

    Ψγ3

    γ1 Ψδ3

    δ1˘

    hγ3δ3

    ` g:γγ1g:δδ1Bt

    `

    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

    1g:δδ

    1Bthγγ1Bthδδ1 ` 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 “ hα1β1Ψαα1Ψ

    ββ1 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 » ps{tq2Bh ` B h ` t´1h ` h Bh ` t´1h h

  • Corollary. High-order derivative of the key metric component

    ˇ

    ˇBth00ˇ

    ˇ À ps{tq2ˇ

    ˇBhˇ

    ˇ`ˇ

    ˇBhˇ

    ˇ` t´1ˇ

    ˇhˇ

    ˇ`ˇ

    ˇBhˇ

    ˇ

    ˇ

    ˇhˇ

    ˇ

    ˇ

    ˇBILJBth00ˇ

    ˇ`ˇ

    ˇBtBILJh00ˇ

    ˇ

    Àÿ

    |I 1|`|J1|ď|I |`|J||J1|ď|J|

    ´

    ps{tq2ˇ

    ˇBBI1LJ1hˇ

    ˇ`ˇ

    ˇBI1LJ1Bh

    ˇ

    ˇ` t´1ˇ

    ˇBI1LJ1hˇ

    ˇ

    ¯

    `ÿ

    |I1|`|I2|ď|I ||J1|`|J2|ď|J|

    ˇ

    ˇBI1LJ1hˇ

    ˇ

    ˇ

    ˇBBI2LJ2hˇ

    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 ps{tq2which 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, 0qPaβ » 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 ` gγγ1

    M gδδ1M BthγδBthγ1δ1 ` . . .

    § Bth00 » ps{tq2Bh ` B h ` t´1h ` h Bh ` t´1h h§ and a technical calculation leading to

    g:αα1g:ββ

    1Btg:αα1Btg

    :ββ1

    » g:0aB0g:0ag:0bB0g

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