Self-gravitating massive fields: stability, decay, and regularity - - PowerPoint PPT Presentation

Self-gravitating massive fields: stability, decay, and regularity

Philippe G. LeFloch

Sorbonne Universit´ e

www.philippelefloch.org Email: contact@philippelefloch.org

From analytical methods to computational methods Recent analytical advances on self-gravitating matter dynamics Numerical relativity benefits from ideas and techniques developed in mathematical relativity Toward structure-preserving algorithms for numerical relativity stability, decay, regularity/singularity Topics for this lecture

• 1. Multi-physics/multi-scale nonlinear waves

general background

• 2. Structure-preserving schemes

simple models

• 3. Recent mathematical advances

new structure, decay, scattering Collaborators: F. Beyer, B. Le Floch, Y. Ma, T.-C. Nguyen, G. Veneziano

• 1. Multi-physics and multi-scale nonlinear waves

From first principles of continuum physics

Massive fields and interfaces Klein-Gordon, complex fluids, modified gravity beyond Einstein gravity global dynamics of shocks, moving material interfaces, phase boundaries impulsive gravitational waves, cosmological singularities Fluids, gases, plasmas, solid materials liquid-vapor flows, thin liquid films, combustion waves, bores in shallow water, astrophysical flows, neutron stars, phase transformations Multi-scale wave phenomena many parameters viscosity, surface tension, heat, Hall effect, friction competitive effects several scales (fluid, geometry) fine-scale structure

• scillations, turbulence

Scale-sensitive nonlinear waves regime where one can extract variables with well-defined limits (despite possible oscillations) under-compressive shocks, subsonic liquid-gas boundaries, combustion waves, etc. “driving for” for the global dynamics

Diffusive-dispersive nonlinear wave propagation

∂ ∂t ρ + ∂ ∂x ρ3 = ε ∂2ρ ∂x2 + κ ∂3ρ ∂x3

plane-symmetry, conservation law fluid density ρ = ρ(t, x) in a tube time t 0 space x ∈ R viscosity ε << 1 surface tension/capillarity κ << 1

Intermolecular forces between a liquid and its surroundings

Riemann problem

single initial discontinuity, dam breaking problem

Three possible asymptotic regimes ε → 0 κ << ε2 (viscosity is dominant) no oscillations, single limit κ = α ε2 (balanced regime) mild oscillations, α-dependent limit κ >> ε2 (surface tension is dominant, α fixed) oscillations, no limit Isothermal compressible fluids Van der Waals fluid

two coupled conservation laws

τt − ux = 0, ut + p(v)x = εuxx − κ τxxx specific volume τ velocity u pressure p(τ) =

8T 3τ−1 − 3/τ 2

Typical Riemann wave structure

x u

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

x !

0.2 0.3 0.4 0.5 0.6 0.7 1 1.5 2 2.5 3 3.5

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.5

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.5

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.2

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.95

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.7

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.4

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=.6

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.1

x !

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.3

!

Small-scale dependent nonlinear waves rules for connecting left- and right-hand state values from both sides beyond the standard Rankine-Hugoniot relations! notion of a kinetic function/scattering map for interfaces varying the ratio surface tension/viscosity κ = α ε2 Structure-preserving algorithms front tracking

• shock capturing with well-controled dissipation

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.5

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.5

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.2

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.95

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=0.7

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.4

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=.6

x v

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.1

x !

0.25 0.5 0.75 1 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

uL=1.3

!

• 2. Structure-preserving schemes: simple models

Preserving the asymptotic structure on an inhomogeneous FLRW background

joint with Y. Cao (Paris) and M. Ghazizadeh (Ottawa)

ArXiv:1912.13439 Formulation of the problem 2 + 1 dim., isothermal, relativistic compressible flow p(ρ) = k2ρ FLRW-type cosmological background, with small inhomogeneities future-contracting geometry (t < 0 and t → 0) ρ → +∞ Asymptotic behavior toward the cosmological singularity nonlinear hyperbolic systems on a curved geometry ∂tU + ∂xF(t, x, U) = H(t, x, U)

• two competitive effects

contracting geometry shock propagation, nonlinear interactions

• small-scale structure, driven by the (fixed) background geometry

similar to phase transition dynamics (multiple scales)

Structure-preserving divergence form: finite volume scheme, shock-capturing (speed) high accuracy: 4th-order in time, 2nd-order in space, oscillation-free well-balanced property

• introduce suitably rescaled unknowns

(Fuchsian PDE method)

• enforce the asymptotic state equations at the discrete level
• enforce commutation property

limt→0 lim∆x→0 U = lim∆x→0 limt→0 U Typical behavior: sharp transitions with spikes plots of the rescaled velocity component u and rescaled density ρ

0.0 0.2 0.4 0.6 0.8 1.0

x

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

u

t = -1e-05 t = -1e-06 t = -1e-07 t = -1e-08 t = -1e-09 t = -1e-10 t = -1e-11 t = -1e-12 0.0 0.2 0.4 0.6 0.8 1.0

x

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 t = -1e-05 t = -1e-06 t = -1e-07 t = -1e-08 t = -1e-09 t = -1e-10 t = -1e-11 t = -1e-12

Standard algorithm

Times t = −10−1, −10−3, −10−5 Velocity magnitude V Rescaled density ρ

Well-balanced algorithm

Times t = −10−1, −10−3, −10−5 Velocity magnitude V Rescaled density ρ

Preserving the asymptotic structure on a Kasner background: evolution from the singularity

joint with F. Beyer (Dunedin) ArXiv:2005.13504 Kasner geometry spatially homogeneous, anisotropic vacuum solution g = t(K2−)/2 − dt2 + dx 2 + t1−Kdy 2 + t1+Kdz2M = (0, +∞) × T3 with asymptotic velocity K ∈ R and Kasner exponents p1 = K 2 − 1 K 2 + 3, p2 = 2(1 − K) K 2 + 3 , p3 = 2(1 + K) K 2 + 3 Compressible fluid flow with pressure law p = (γ − 1)ρ with γ ∈ (1, 2) Characteristic exponent Γ = 1 4

• 3γ − 2 − K 2(2 − γ)

∈ (0, 1) which compare the geometry and fluid behaviors Γ > 0 : sub-critical regime dynamically stable Γ 0 : super-critical / critical regimes dynamically unstable

Formally, plug an expansion in power of t and attempt to validate it (Fuchsian asymptotics)

Evolution from the cosmological singularity t = 0 formulate a singular initial value problem B0(U, t, x)∂tU + B1(U, t, x)∂xU = f (U, t, x) suitable “singular initial data” prescribed on t = 0 Fuchsian-type expansions near the cosmological singularity sufficiently regular, shock-free regime Algorithm preserving the Fuchsian structure discretize U(t, x) ≃ V (t) = (Vj(t)) by the pseudo-spectral method of lines ∂tV − AV = h(V , t) high-order Runge-Kutta discretization in time introduce suitably rescaled variables rigorous analysis of the numerical error: take into account the Fuchsian expansion two sources of approximation error: continuum / discrete

• ur proposal : keep the two error sources asymptotically in balance

With this numerical strategy, we numerically demonstrated the nonlinear stability of the flow near the cosmological singularity the sub-critical regime.

Numerical results for a typical evolution from the singularity – fluid density (contour plot) and velocity field (flow lines) – time (vertically) the density ρ is unbounded in the limit t → 0 we carefully checked the numerical error reliable algorithm, despite the solutions being very singular

0.0 0.5 1.0 1.5 x/π 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t 1

− 2

1

−2

10 −2 10

−1

1 0−1 10 0 1 1

1

101 1

2

103 1 04 0.0 0.5 1.0 1.5 2.0 x/π 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t

Numerical results: evolution V ε toward the singularity by adding a perturbation ε on the data at time t = T > 0 regime far from linear perturbation theory compute the perturbed singular data V ε

0 reached on t = 0

compare to the given unperturbed data V0 stability/continuous dependence: V ε

0 → V ε as ε → 0

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 ǫ 50 100 150 200 250 |V ǫ,∞ − V ∗|

Extension to self-gravitating fluids: in progress

• 3. Recent mathematical advances: new structure

– Einstein-matter equations, stability and regularity/singularities – asymptotics for isolated massive gravitational & cosmological solutions

Initial data set with prescribed asymptotics

joint work with T.C. Nguyen (Paris) ArXiv: 1903.00243 Einstein’s constraint equations (M, g, k) from the extrinsic curvature k we define h := k − Tr(k)g matter content scalar field H⋆, vector field J⋆ Hamiltonian and momentum constraints G = (H, M) H(g, h) = Rg + 1 2

• Tr(h)2 − |h|2

= H⋆ M(g, h) = Divgh = J⋆ Many mathematical works nonlinear elliptic system of partial differential equations Lichnerowicz, Choquet-Bruhat, . . . , Corvino, Chrusciel, Delay, Dilts, Galloway,

Holst, Isenberg, Maxwell, Mazzeo, Miao, Pollack, . . . Carlotto and Schoen.

New analytical approach: the seed-to-solution method

• a seed data/approximate solution (M, g1, h1):
• prescribe the asymptotic behavior at infinity

Theorem. The seed-to-solution method (case of vacuum data) —— (LeFloch & Nguyen, 2019) Given a seed data set (M, g1, h1) on a manifold (with one asymptotic end) con- sisting of a Riemannian metric g1 and a symmetric two-tensor h1: 1/2 < pG min(1, pM) and 1/2 < pM < +∞ g1 = gEucl + O(r −pG ) h1 = O(r −pG −1) H(g1, h1) = O(r −pM−2) M(g1, h1) = O(r −pM−2) there exists a solution to Einstein’s constraint equations G(g, h) = 0. sub-critical decay: pM < 1 g = g1 + O(r −pM) h = h1 + O(r −pM−1) critical decay: pM = 1 with H(g1, h1) and M(g1, h1) in L1(M) g = g1 + m/r + o(r −1) h = h1 + O(r −2) super-critical decay: pM > 1 p = min(pG + 1, pM, 2) g = g1 + m/r + O(r −p) h = h1 + O(r −2). in which the “mass corrector” is

• m =

m(g1, h1) = − 1 8π

• M

H(g1, h1) dVg1 + O(G(g1, h1)2)

Iterative construction scheme approximation based on the seed data a fixed-point strategy for nonlinear elliptic equations converging sequence of approximate solutions stability property: continuous dependence w.r.t. the Einstein operator g − g1L2C2,α

p

(M) H(g1, h1) − H⋆L2Cα

p+2(M) + εG M(g1, h1) − M⋆L2C1,α q+1 (M)

h − h1L2C2,α

q

(M) εG H(g1, h1) − H⋆L2Cα

p+2(M) + M(g1, h1) − M⋆L2C1,α q+1 (M)

Structure relevant for numerical relativity construct solutions (M, g, h) with prescribed behavior at infinity control the mass corrector

• m =

m(g1, h1) = − 1 8π

• M

H(g1, h1) dVg1 + O(G(g1, h1)2) “spurious wave” propagating to infinity allow for free parameters to be fitted, produce realistic initial data sets for instance asymptotically localized in angular directions quantitative error bounds in specific functional norms

Global dynamics of self-gravitating matter near Minkowski spacetime

joint with Yue Ma (Xi’an) 2015–2020

Einstein gravity theory

minimally coupled, Klein-Gordon field φ with U(φ) = (c2/2)φ2 Einstein-Klein-Gordon system gφ = U′(φ) Rαβ = 8π ∇αφ∇βφ + U(φ) gαβ

• nonlinear geometric system of partial differential equations (PDEs)

f(R)-modified gravity theory

generalized action

M f (R) dVg

f (R) = R + κ

2 R2 with a large “mass parameter” 1/κ

field equations of modified gravity: Mαβ = 8πTαβ Mαβ = f ′(R) Gαβ − 1 2

• f (R) − Rf ′(R)

gαβ + gαβ g − ∇α∇β

• f ′(R)

up to fourth-order terms... second-order after suitable transformations

Global dynamics near the Minkowski regime ?

Long history beginning in the 90’s: vacuum spacetimes by Christodoulou & Klainerman, Lindblad & Rodnianski.... Recently for Vlasov matter spacetimes: Lindblad, Taylor, Smulevici, Fajman. Independent method of proof for Einstein-KG: Ionescu & Pausader

Spherically symmetric collapse of a massive field asymptotically-flat massive matter spacetimes, stability/instability dispersion of the matter toward the infinite future

• r gravitational collapse and formation of a black hole, or rather

“oscillating soliton stars” ? Goncalves et al. (1997) Okawa, Cardoso, and Pani (2014) highly accurate, multi-scale simulation over a long time ‘total mass’ vs. ‘amplitude’

log(wµ) log(A/µ)

Prompt collapse Delayed collapse Soliton star A*

delayed~µ

• 0.2

A*

II~µ 0.2

~t

• 3/2 decay

A*

I~µ

• 0.2

C A*

soliton~µ

• 1

New mathematical advances (no symmetry restriction!) rigorous proof for the regime of dispersion wave gauge gx α = 0, coupled wave-Klein-Gordon, second-order PDEs f(R)-gravity for a self-gravitating massive field g†g†

αβ = Fαβ(g†, ∂g†) + 8π

− 2e−κρ∂αφ∂βφ + c2φ2e−2κρ g†

αβ

• − 3κ2∂αρ∂βρ + κ O(ρ2)g†

αβ

g†φ − c2φ = c2 e−κρ − 1 φ + κg†αβ∂αφ∂βρ 3κ g†ρ − ρ = κ O(ρ2) − 8πe−κρ g†αβ∂αφ∂βφ + 2c2 e−κρφ2 wave gauge conditions g†αβΓ†λ

αβ = 0

curvature compatibility eκρ = f ′(Re−κρg†) Hamiltonian and momentum constraints of modified gravity propagate from any given Cauchy hypersurface The Euclidian-hyperboloidal foliation method Global stability theorem for massive matter fields gravitational radiation, time and space decay apply to actions

M

• Rg + κ

2 (Rg)2

dVg with κ 0

Main challenge: lack of scale invariance

– Euclidian-hyperboloidal spacetime foliation hyperboloidal slices in the light cone interior

capture the decay in time

asymptotically Euclidian slices in the exterior

capture the decay in space

merged together near the light cone – Energy norms based on symmetries of Minkowski spacetime “asymptotic” Killing fields (translations, spatial rotations, boost) exclude the scaling field S = t∂t + r∂r

does not commute with KG

– Hierarchy/coupling of energy bounds frames of vector fields

semi-hyperboloidal, semi-null

adapted Sobolev, Poincar´ e, Hardy estimates

low-order vs. high-order

differentiation order/growth in s

hyperboloidal-Euclidian time

geometry-matter coupling

nonlinear interaction terms

Structure relevant to numerical relativity work in Euclidian-hyperboloidal foliations

(Rinne, Zenginoglu, Hilditch)

provide energy norms and quantitative error estimates

Universal scattering laws for cosmological singularities

joint work with B. Le Floch (ENS, Paris) and G. Veneziano (CERN, Geneva)

ArXiv:2005.11324 & ArXiv:2005.11324

– Einstein equations coupled to a scalar field – bouncing (contracting/expanding) cosmologies – vicinity of the singularity hypersurface Large literature Penrose, Tod, L¨ ubbe, Turok, Barrow, etc.

symmetric spacetimes and special junctions

General formulation of the problem ? classes of physically meaningful junction conditions degrees of freedom, constraints at the singularity Main results systematic study of (past, future) singularity data (g±, K ±, φ±

0 , φ± 1 )

we fully classify the possible bouncing conditions notion of a singularity scattering map (g−, K −, φ−

0 , φ− 1 ) → (g+, K +, φ+ 0 , φ+ 1 )

we distinguish between:

• (Kepler-like) universal scattering laws
• model-dependent scattering maps

examples of modified gravity theories uncover the relevant structure

Local ADM formulation near a singularity hypersurface M(4) =

τ∈[τ−1,τ1] Hτ: (local) Gaussian foliation of a (small

neighborhood in a) spacetime by spacelike hypersurfaces (diffeom. to H0) g(4) = g(4)

αβ

• = −dτ 2 + g(τ)

g(τ) = gij(τ)dx idx j Einstein’s evolution equations: unknowns g and K ∂τgij = −2 Kij ∂τK i

j = Tr(K)K i j + Ri j − 8π Mi j

Mi

j = 1 2ρgi j + T i j − 1 2Tr(T)gi j

Einstein’s constraint equations R + |K|2 − Tr(K 2) = 16πρ ∇iK i

j − ∇j(TrK) = 8πJj

coupled to the wave equation g(4)φ = 0 for a scalar field Fuchsian approach

(Baouendi, Goulaouic, Kichenassamy, Rendall,...)

– solve from τ = 0 toward the past (τ < 0) or the future (τ > 0) – derive an ODE system based on the so-called “velocity dominated” Ansatz

Singularity data and asymptotic profiles Throughout, H denotes a 3-manifold. Definition

• 1. A set (g−, K −, φ−

0 , φ− 1 ) consisting of two tensor fields and two scalar fields

defined on H is called a singularity initial data set provided: Riemannian metric g− CMC symmetric (1, 1)-tensor K − with Tr(K −) = 1 Hamiltonian constraint 1 − |K −|2 = 8π (φ−

0 )2

momentum constraints Divg−(K −) = 8π φ−

0 dφ− 1

Notation I(H): space of all singularity data 2 The asymptotic profile associated with (g−, K −, φ−

0 , φ− 1 ) ∈ I(H) is the

flow on H τ ∈ (−∞, 0) → g∗, K ∗, φ∗ (τ) g∗(τ) = |τ|2K−g− K ∗(τ) = −1 τ K − φ∗(τ) = φ−

0 log |τ| + φ− 1

Bounce based on a singularity scattering map Singularity hypersurface as an interface between two “phases”, across which the geometry and the matter field encounter a “jump” Fluid dynamics and material science with phase transitions when some (micro-scale) parameters (like viscosity, surface tension, heat conduction, etc.,) are neglected in the modeling macro-scale effects are captured by jump conditions

Rankine-Hugoniot, kinetic relations

Definition A (past-to-future) singularity scattering map on a 3-manifold H: a diffeomorphism-covariant map on the space of singularity data I(H) S : I(H) ∋ g−, K −, φ−

0 , φ− 1

• →

g+, K +, φ+

0 , φ+ 1

• ∈ I(H)

satisfying the locality property: for any open set U ⊂ H the restriction of the image S g−, K −, φ−

0 , φ− 1

• U

depends only on the restriction of the data g−, K −, φ−

0 , φ− 1

• U

Locality property S g−, K −, φ−

0 , φ− 1

• (x) depends upon

g−, K −, φ−

0 , φ− 1

• (x)

and (possibly) derivatives at x only Definition S is an ultra-local map if for all x ∈ H S(g−, K −, φ−

0 , φ− 1 )(x) depends only on (g−, K −, φ− 0 , φ− 1 )(x)

S is a tame-preserving map if it preserves positivity: if K − > 0 then K + > 0, where K + is defined from the image of S. S is a conformal map if g+ and g− only differ by a conformal factor. Observations Quiescent singularities K > 0: motivated by the absence of BKL

• scillations in this case (named after Belinsky, Khalatnikov, and Lifshitz)

Asymptotic profiles with K −, K + > 0 describe a “bounce”: – volume element decreases to zero as τ → 0− – then increases back to finite values for τ > 0

• Theorem. Classification of singularity scattering maps

Ultra-local spacelike singularity scattering of conformal type for self-gravitating scalar fields: only two classes of such maps ! Isotropic conformal bounce Siso

λ,ϕ

g+ = λ2g− K + = δ/3 φ+

0 = 1/

√ 12π φ+

1 = ϕ

parametrized by a conformal factor λ = λ(φ−

0 , φ− 1 , det K −) > 0 and a constant ϕ

Non-isotropic conformal bounce Sani

f ,c

g+ = c2µ2g− K + = µ−3(K − − δ/3) + δ/3 φ+

0 = µ−3φ− 0 /F ′(φ− 1 )

φ+

1 = F(φ− 1 )

parametrized by a constant c > 0 and a function f : R → [0, +∞) µ(φ0, φ1) = 1 + 12π(φ0)2f (φ1)1/6 F(φ1) = φ1

0 (1 + f (ϕ))−1/2dϕ

By working with general spacetimes we characterize physically meaningful classes of junctions

Earlier works: symmetric spacetimes & special junctions

New and physically-motivated classes of junctions

conformal/non-conformal spacelike/timelike; scalar field/stiff fluid

References

Structure-preserving methods 2014 Structure-preserving shock-capturing methods: late-time asymptotics,

curved geometry, small-scale dissipation, and nonconservative products Lectures for the J.-L. Lions’s Spanish-French School

2005.13504 A numerical algorithm for Fuchsian equations and fluid flows on cosmological spacetimes

with F. Beyer

1912.13439 Asymptotic structure of cosmological fluid flows in one and two space dimensions

with Y.-Y. Cao and M.A. Ghazizadeh

Foliations and spacetime decay 2018 The global nonlinear stability of Minkowski space for self-gravitating massive fields, World Scientific Press, Singapore

with Y. Ma

1903.00243 The seed-to-solution method for the Einstein equations and the

asymptotic localization problem with T.-C. Nguyen

Scattering maps

with B. Le Floch and G. Veneziano

2006.08620 Universal scattering laws for bouncing cosmology 2005.11324 Cyclic spacetimes through singularity scattering maps 2020 On the global evolution of self-gravitating matter. Phase boundaries,

scattering maps, and causality (soon on ArXiv) Email: contact@philippelefloch.org Blog: philippelefloch.org