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 p´, `, `, `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
Universite 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 ´ 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
§ 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
§ 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.
§ 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
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