Analytic and geometric aspectsof spacetimes of low regularity
Annegret Burtscher
Faculty of MathematicsUniversity of Vienna
Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie
PhD thesis defense, 13 Jan 2014
Introduction Energy estimates Trapped surface formation Low-regularity metrics
Theory of relativity
Classical mechanics (Newton 1687)
static geometric backgroundgravitational force
General relativity (Einstein 1915)
dynamic spacetimemore accurate on large scale
Key ideaGravitation is a geometric property of space and time.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Einstein equations
The Einstein equations relate the curvature of spacetime tothe matter distribution of the universe.
Einstein equations
Rαβ − 12R gαβ = 8πTαβ
(M,gαβ) 4-dimensional Lorentzian manifoldRαβ Ricci curvature tensorR scalar curvatureTαβ energy-momentum tensor
General relativity is the study of the solutions of this systemof coupled nonlinear partial differentialequations.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Outline
1 Energy estimates with one-sided geometric bounds
2 Trapped surface formation for compressible matter
3 Low-regularity metrics
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Outline
1 Energy estimates with one-sided geometric boundsGlobal existence vs. breakdownBel–Robinson energyEstimate of Bel–Robinson energyRelated results
2 Trapped surface formation for compressible matter
3 Low-regularity metrics
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Initial value formulation
Vacuum Einstein equationsRαβ = 0
Choquet-Bruhat (1952):
initial value formulation and local existenceinitial data (Σ,h,k) must satisfy “constraint equations”
Choquet-Bruhat, Geroch (1969):to each initial vacuum data set exists a unique maximalglobally hyperbolic spacetime (M,g)
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Global existence vs. breakdown
Klainerman, Rodnianski (2010),Wang (2012):
a vacuum globally hyperbolicspacetimeM =
⋃t∈[t0,t1) Σt
foliated by CMC hypersurfacescan be extended beyond t1 ifthe deformation tensor πsatisfies∫ t1
t0‖π‖L∞(Σt )dt =
∫ t1
t0
(‖k‖L∞(Σt ) + ‖∇ log n‖L∞(Σt )
)dt <∞.
Σt+∆t
Σt
nN∂t
Foliation ofM by spacelikehypersurfaces Σt
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Bel–Robinson energy
Definition (Bel–Robinson tensor)
Q[R]αβγδ := RαλγµRβλδµ + ?Rαλγµ
?Rβλδµ
divergence-free in vacuum: ∇αQαβγδ = 0
Definition (Bel–Robinson energy)
Q[R]Σt =∫
ΣtQ[N,N,N,N]dVgt
∫ t1t0‖π‖L∞(Σt )dt <∞ =⇒ Q[R]Σt1
≤ CQ[R]Σt0
important in derivation of breakdown criteria
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Estimate of Bel–Robinson energy
Theorem (B., Grant, LeFloch 2012)
Given a vacuum spacetime endowed with a foliation (Σt )t∈Iwithlapse n and second fundamental form k, one has for t0 ≤ t1,t0, t1 ∈ I,
Q[R]Σt1≤ e3Kn,k(t0,t1)Q[R]Σt0
,
where
Kn,k(t0, t1) :=
∫ t1
t0sup
Σt
ρ(n,k)dt
and ρ(n,k) is the largest eigenvalue of a certainsymmetric tensor field Π(n,k).
requires only one-sided bound on deformation tensor π
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Sketch of proof
Stokes’ theorem and divergence-free property ofBR tensor imply
Q[R]Σt1−Q[R]Σt0
= −32
∫ t1
t0
∫Σt
nQαβ00παβdVgt dt
use standard decomposition of BR tensor inelectric and magnetic parts of curvature tensor to obtain
−12
nQαβ00παβ = tr
((E H
)Π(n,k)
(EH
))≤ ρ(n,k)Q0000
Gronwall type argument implies
Q[R]Σt1≤ Q[R]Σt0
exp
(3∫ t1
t0sup
Σt
ρ(n,k)dt
)
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Related results
Sufficient conditionGershgorin’s circle theorem=⇒ upper bound for ρ(n,k) in terms of n andeigenvalues of k
GeneralizationsWeyl fields (e.g. conformal curvature tensor)vacuum Maxwell fields and Yang Mills fieldsdoes not work for minimally coupled scalar fields
Possible applicationsgeneralization of breakdown criteria to spacetimeswith such one-sided bounds?
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Outline
1 Energy estimates with one-sided geometric bounds
2 Trapped surface formation for compressible matterSpacetimes with matterGravitational collapseCompressible matterExistence of solutions and formation of trapped surfaces
3 Low-regularity metrics
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Spacetimes with matter
Einstein equations with matter
Gαβ := Rαβ − 12R gαβ = 8πTαβ
Typical matter fields:scalar fieldselectromagnetic fields (Maxwell equations)collisionless gas (Vlasov equation)perfect fluids, dust (Euler equation)
Conservation of energy
∇αTαβ = 0
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Theory of gravitational collapse
Goalstudy of the formation of black holes and singularities
for general asymptotically flat initial conditions,where no symmetry conditions are imposed.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Black holes and the point of no return
trapped surface is a spacelike surface with decreasingarea in the direction of the null normalsevent horizon is the “point of no return”, surrounding ablack hole
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Already collapsed matter
time
understandingdust
scalar field
vacuum
Vlasov
Euler
2-phase model
trapped surface weak cosmic
Christodoulou (1990s)
Christodoulou (1995)
Christodoulou (2008)
B., LeFloch (2013)
formation censorship
Andréasson, Rein (2010)Rendall (1991)
Oppenheimer, Snyder (1939)Christodoulou (1980s)
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Compressible matter
Energy-momentum tensor
Tαβ = (µ+ p)uαuβ + pgαβ
µ mass-energy densityp(µ) = k2µ pressure with k ∈ (0,1) speed of sounduα velocity vector, normalized to uαuα = −1∇αTαβ = 0 Euler equations
consider spacetimes that satisfy the Einstein–Euler equationsin the distributional sense
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Main result (summarized)
Theorem (B., LeFloch 2013)There exists a class of untrapped initial dataprescribed on a hypersurface, that evolves to sphericallysymmetric Einstein–Euler spacetimes with bounded variationthat contain trapped surfaces.
Tasks:deal with low regularityfind admissible initial datasolve initial value problem and estimate time of existenceshow that trapping occurs during time of existence
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Full Einstein–Euler system and regularity
Einstein equations for metric coefficients a,bEddington–Finkelstein coordinates with prescribed decaylimr→0 a(v , r) = limr→0 b(v , r) = 1three first-order ODEs (equalities between BVfunctions)one second-order PDE (in sense of distributions)
Euler equations for normalized fluid variables M,Vsystem of two coupled first-order PDEs
M,V ∈ L∞([v0, v∗],BV [0, r∗+∆])∩Lip([v0, v∗],L1[0, r∗+∆])
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Reduced Einstein–Euler system
Proposition (B., LeFloch 2013)The full Einstein–Euler system reduces to a system of twoequations for the fluid,
∂v U + ∂r F (U,a,b) = S(U,a,b),
and two integral formulas for the geometry,
b(v , r) = exp(
4π(1 + k2)
∫ r
0M(v , s)s ds
),
a(v , r) = 1− 4π(1+k2)r
∫ r
0
b(v ,s)b(v ,r) M(v , s)(1− 21−k2
1+k2 V (v , s))s2 ds.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
1. Initial data sets
∃! smooth static solutions (M,V ,a,b)static for any initialvalue µ0 = µ(0) > 0 (Rendall, Schmidt 1991)
compact perturbation around r∗ > 0
µ0 > 0
r∗r∗ − δ r∗ + δr = 0
M0 = Mstatic M0 = Mstatic M0 = Mstatic
V0 = Vstatic V0 = Vstatic
V0 = Vstatic(1 + 1
h
)
r∗ + ∆r∗ −∆
Proposition (B., LeFloch 2013)
There exist positive constants C1,C2 such that for δh ≤
1C1
:
0 < a0(r) ≤ a(0)(r), r ∈ [0, r∗ +∆],
av (v0, r) ≤ −C2δ
h3 , r ∈ [r∗ − δ, r∗ + δ].
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
2. Existence result for initial value problem
based on a generalized random choice scheme on adiscretized grid (LeFloch, Rendall 2011; LeFloch, Stewart2005; Groah, Temple 2004)
consider homogeneous Euler system
∂v U + ∂r F (U) = 0
on uniform geometric background (a,b constant) strictly hyperbolic, genuinely nonlinear system of
conservation lawssolve Riemann problem explicitly for arbitrary dataevolve solution of Riemann problem using ODE system
∂v W = S(W ,a,b) = S + geometry
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
3. Control perturbation during evolution
need control of growth of solution M,V ,a,bdomain of dependence Ω∗
minimal time of existence v∗
Perturbation property
holds if a solution (M,V ,a,b) satisfies in the domain influencedby the perturbation for some constants C,C0,Cb,Λ > 0, κ > 1:
1C0
e−C v−v0hκ ≤ M(v , r) ≤ C0eC v−v0
hκ
1C0
e−C v−v0hκ
(1 +
1h
)≤ −V (v , r) ≤ C0eC v−v0
hκ
(1 +
1h
)−1
h≤ a(v , r) ≤ 1, 1 ≤ b(v , r) ≤ Cb, −Λ
h≤ λi(v , r) ≤ Cb
2.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Existence result
Theorem (B., LeFloch 2013)
Suppose admissible initial data (M0,V0,a0,b0) aregiven. Then there exist constants τ > 0, κ > 1 so thatapproximate solutions are well-defined on the interval[v0, v∗] with v∗ = v0 + τhκ, and satisfy
the perturbation property,a BV property in r , uniformly in v,an L1-Lipschitz property in v.
Consequently, a subsequence converges pointwise toward alimit (M,V ,a,b) which is a bounded variation solution to theEinstein–Euler system in spherical symmetry and satisfiesthe initial conditions and the perturbation property.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Formation of trapped surfaces
Corollary (B., LeFloch 2013)
Fix k ∈ (0,1) such that κ < 2. Suppose the initial data withδh = 1
C1additionally satisfy
8πr∗ > e3ΛC30C1.
Then, if h is chosen sufficiently small, a trapped surface formsin the solution before time v∗, i.e. there exists (v•, r•) ∈ Ω∗ suchthat a(v•, r•) < 0.
κ < 2 possible for k sufficiently smalldomain of dependence does not “close up” too sooninitial data that satisfy additional assumption do exist
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Outline
1 Energy estimates with one-sided geometric bounds
2 Trapped surface formation for compressible matter
3 Low-regularity metricsProblems and geometric approachesLength structure of continuous Riemannian manifolds
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Problems with low regularity
Riemannian metrics with regularity below C1,1 generallydo not have
a classical notion of curvaturelocally unique geodesics (Hartman 1950)length-minimizing geodesics(Hartman, Wintner 1951)a diffeomorphic exponential map(Kunzinger et al., Minguzzi 2013)
Continuous Lorentzian metrics moreover have problems with
causality theory (Chrusciel, Grant 2012)
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Geometric approaches to low regularity structures
Some generalizations:length spaces, or more preciselyAlexandrov spaces with one-sided curvature boundsgeometric measure theory
Some relations to low-regularity Riemannian geometry:Berestovskii, Nikolaev (1980s): Alexandrov spaces withtwo-sided curvature bounds are manifolds with C1,α
(0 < α < 1) Riemannian metricsOtsu, Shioya (1994): Alexandrov spaces with curvaturebounded below inherit a C0-Riemannian structure(everywhere apart from a set of singular points)
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Length spaces
Need a topological space M witha class of curves Aa notion of length L
From this can definea metric d(p,q) = infL(γ) | γ ∈ A connecting p,qa metric length Ld (γ) = sup
∑ni=1 d(γ(ti−1), γ(ti))
In general neither newly defined topology nor length coincidewith original one.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Low regularity Riemannian geometry
Theorem (B. 2012)Let M be a connected manifold with continuousRiemannian metric g. Consider the class of absolutelycontinuous curves Aac together with the usual arc-length Lof curves. Then
(M,Aac,L) is admissible length structure on M,the distance function d induces the manifold topology,g (and dg) can be uniformly approximated by smoothRiemannian metrics gn (and dgn )L = Ld on Aac.
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Introduction Energy estimates Trapped surface formation Low-regularity metrics
Summary
extendability of solutions to the Einstein equations can bestudied via energy estimates, where one-sidedgeometric bounds play an important role
the formation of trapped surfaces occurs duringthe evolution of certain initial data of (compressible) matter
low-regularity metrics also have an impact ongeometric properties of semi-Riemannian manifolds
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Thank you for your attention!
