Critical collapse of rotating perfect fluidscjg/talks/3+1rotfluid_AEI2017.pdf · Introduction...

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



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





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





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)





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)




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





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





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





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




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)





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)





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)




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





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∗





Dynamical system: one and two unstable modes

Ζ1

Ζ0

Ζ2

Ζ1

Ζ0

Ζ2

ζ0 spherical mode, ζ1 ballerina mode, ζ2 any other mode




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


Critical collapse of rotating perfect fluidscjg/talks/3+1rotfluid_AEI2017.pdf · Introduction...

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