IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Critical collapse of rotating perfect fluids
Carsten Gundlach (work with Thomas Baumgarte)
Mathematical SciencesUniversity of Southampton
AEI, 1 March 2017
C. Gundlach Rotating critical collapse 1 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
What is critical collapse?Numerical experimentsDynamical systems pictureSelf-similarity
What is critical collapse?
Initial data near the threshold of black hole formation, butotherwise generic
Pick a 1-parameter family of initial data and find the criticalvalue p∗ by bisection
For (approximately) scale-invariant physics: “type-II criticalphenomena”
arbitrarily small black hole mass M ∼ (p − p∗)γ
arbitrarily large curvature, eg. maxR ∼ (p∗ − p)−2γ
Naked singularities are codimension one in the space ofinitial data
C. Gundlach Rotating critical collapse 2 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
What is critical collapse?Numerical experimentsDynamical systems pictureSelf-similarity
History of numerical experiments
Choptuik 1993: massless spherically symmetric scalar field
Discrete self-similarity (DSS)
Since then, much more in spherical symmetry
perfect fluid P = kρ (CSS)massive scalar, wave maps, YM, vectors, spinors, Vlasov...Higher and lower dimensions, Λ > 0 and Λ < 0
Axisymmetric vacuum
Abrahams and Evans 1994attempts to repeat this have failed
With angular momentum
with Baumgarte P = kρ (this talk)with Joanna Ja lmuzna scalar field e imθΦ(t, r) in 2+1
C. Gundlach Rotating critical collapse 3 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
What is critical collapse?Numerical experimentsDynamical systems pictureSelf-similarity
Dynamical systems picture
GR as a dynamical system on space of initial data (= phasespace)
Asymptotically flat regular data can
form a starcollapse to a black holedisperse
Threshold between collapse and dispersion
empirically a hypersurfaceitself a dynamical system
Attractors in collapse threshold
static/stationary L ∂∂tg = 0 (type I)
continuously/discretely self-similar L ∂∂τg = −2g (type II)
C. Gundlach Rotating critical collapse 4 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
What is critical collapse?Numerical experimentsDynamical systems pictureSelf-similarity
C. Gundlach Rotating critical collapse 5 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
What is critical collapse?Numerical experimentsDynamical systems pictureSelf-similarity
Self-similarity
Adapted coordinates (τ, x i ) and variables Z :example perfect fluid Z = (gµν , ρ, v
i )
gµν(τ, x) = e−2τ gµν(x, τ) ρ(τ, x) = e2τ ρ(x, τ), v i (τ, x)
Z scale-invariant, e−τ measures scale
any length ∼ e−τ , R ∼ e2τ , M ∼ e−(D−3)τ
But we can choose τ to also be a time coordinate
CSS if and only if Z (x, τ) = Z (x)
. . . and DSS if and only if Z (x, τ + ∆) = Z (x, τ)
C. Gundlach Rotating critical collapse 6 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Initial dataCSS solutionsEvolution near the critical solutionScaling laws
Initial data for rotating perfect fluids
Perfect fluid with “ultrarelativistic” equation of state P = kρ
Time evolutions of asymptotically flat initial data
Consider 2-parameter families of initial data with “strength” pand “rotation” q
M(p,−q) = M(p,q)
J(p,−q) = −J(p,q)
For example, we can define q→ −q to be a reflection
From now on restrict to axisymmetry
C. Gundlach Rotating critical collapse 7 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Initial dataCSS solutionsEvolution near the critical solutionScaling laws
CSS solutions with P = kρ
CSS critical solution Z∗(x) exists for 0 < k < 1
Linear stability depends on k
l = 0 l = 1 l ≥ 2
0 < κ . 0.0105 stable 1 unstable stable(?)
0.0105 . κ < 19 1 unstable 1 unstable stable
19 < κ . 0.49 1 unstable stable stable
0.49 . κ < 1 1 unstable stable many unstable
C. Gundlach Rotating critical collapse 8 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Initial dataCSS solutionsEvolution near the critical solutionScaling laws
One unstable mode: evolution near the critical solution
Intermediate phase near Z∗
Z (x, τ) ' Z∗(x) + ζ0(τ)Z0(x) + ζ1(τ)Z1(x) + other decaying
whereζ0 = P(p, q) eλ0τ , ζ1 = Q(p, q) eλ1τ
From q → −q symmetry, P is even in q, Q is odd. Hence
P ' (p − p∗)− Kq2
Q ' q
to leading order in p, q2
Black hole threshold at P = 0 ⇒ p ' p∗ + Kq2
C. Gundlach Rotating critical collapse 9 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Initial dataCSS solutionsEvolution near the critical solutionScaling laws
Scaling laws
Onset of nonlinearity at τ∗(p, q) defined by
|P|eλ0τ∗ = 1 ⇒ δ := Q|P|−λ1λ0
AH forms or solution disperses depending on sign of Z0
Z (x, τ∗) ' Z∗(x) + P(p, q) eλ0τZ0(x) + Q(p, q) eλ1τZ1(x)
' Z∗(x)±Z0(x) + δ Z1(x)
Intermediate Cauchy data at τ = τ∗ characterised by overallscale e−τ∗ , sign ± and dimensionless parameter δ
Black hole forms for P > 0, with
M ' e−τ∗FM(δ) ' P1λ0 1 ' (p − p∗ − Kq2)
1λ0
J ' e−2τ∗FJ(δ) ' P2λ0 δ ' (p − p∗ − Kq2)
2−λ1λ0 q
C. Gundlach Rotating critical collapse 10 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
OverviewScaling at constant ΩScaling at constant η
-0.2
0.0
0.2
W
1.02
1.03
1.04
1.05
Η
0.00
0.05
0.10
0.15
M
-0.2
0.0
0.2
W
1.02
1.03
1.04
1.05
Η
-0.005
0.000
0.005J
Black hole mass M (left) and angular momentum J (right),against η (strength) and Ω (rotation) of initial data
initial fluid density ρ ∼ ηe−r2
initial angular velocity ∼ Ω/(1 + r2)
C. Gundlach Rotating critical collapse 11 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
OverviewScaling at constant ΩScaling at constant η
Scaling at constant initial rotation Ω
10- 6 10- 5 10- 40.001 0.01 0.1
H Η Η * L - 1
0.01
0.02
0.05
0.10
0.20
0.50
M
10- 5 10- 40.001 0.01 0.1
H Η Η * L - 1
10- 4
0.001
0.01
0.1
J
M (left) and J (right) against ηη∗− 1 (log-log plots)
C. Gundlach Rotating critical collapse 12 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
OverviewScaling at constant ΩScaling at constant η
Scaling at constant initial density η
0.001 0.01 0.11-H W W * L 2
0.10
0.50
0.20
0.30
0.15
0.70
M
0.001 0.01 0.11-H W W * L 2
0.005
0.010
0.050
0.100
0.500
JΗ
0.500
M (left) and J (right) against 1− ΩΩ∗
(log-log plots)
C. Gundlach Rotating critical collapse 13 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Evolution near the critical solutionScaling lawsDynamical systems picture
Two unstable modes: evolution near the critical solution
As before, intermediate phase near Z∗
Z (x, τ) ' Z∗(x) + P(p, q) eλ0τZ0(x) + Q(p, q) eλ1τZ1(x)
As before,
δ := ζ1|ζ0|−λ1λ0 = Q|P|−
λ1λ0
is constant during linear perturbation phase
Onset of nonlinearity at τ = τ∗, defined for example by
ζ20 + ζ2
1 ' 1
Intermediate Cauchy data at τ = τ∗ characterised by overallscale e−τ∗ , sign of Z0 and dimensionless parameter δ
C. Gundlach Rotating critical collapse 14 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Evolution near the critical solutionScaling lawsDynamical systems picture
Scaling laws
Putting this together, we get as before
M = (hom.func.deg.one)(|P|
1λ0 , |Q|
1λ1
)= |P|
1λ0 FM(δ)
J = (hom.func.deg.two)(. . . ) = |P|2λ0 FJ(δ)
J
M2= (hom.func.deg.zero)(. . . ) = FJ/M2(δ)
But we now explore large values of δ
The attracting manifold of the critical solution now hascodimension two
But 0 < λ1 λ0, so q does not have to be very small
The black hole threshold has always codimension one. It mustbe at some δ = δ∗
C. Gundlach Rotating critical collapse 15 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Evolution near the critical solutionScaling lawsDynamical systems picture
Dynamical system: one and two unstable modes
Ζ1
Ζ0
Ζ2
Ζ1
Ζ0
Ζ2
ζ0 spherical mode, ζ1 ballerina mode, ζ2 any other mode
C. Gundlach Rotating critical collapse 16 / 17
IntroductionRotating fluids: one unstable mode
Numerics for P = ρ/3 (radiation fluid)Two unstable modes
Other things I am working on
Collapse in 2+1 generally (role of Λ < 0)
Rotating critical collapse in 2+1 (with Ja lmuzna)
Rotating black holes from point particle mergers in 2+1 (withSkenderis, Hartnett, Iannetta)
Critical collapse in Einstein-Vlasov
C. Gundlach Rotating critical collapse 17 / 17