Critical collapse of rotating perfect fluids

Transcription

1 Critical collapse of rotating perfect fluids Carsten Gundlach (work with Thomas Baumgarte) Mathematical Sciences University of Southampton AEI, 1 March 2017 C. Gundlach Rotating critical collapse 1 / 17

2 What is critical collapse? What is critical collapse? Numerical experiments Self-similarity Initial data near the threshold of black hole formation, but otherwise generic Pick a 1-parameter family of initial data and find the critical value p by bisection For (approximately) scale-invariant physics: type-ii critical phenomena arbitrarily small black hole mass M (p p ) γ arbitrarily large curvature, eg. max R (p p) 2γ Naked singularities are codimension one in the space of initial data C. Gundlach Rotating critical collapse 2 / 17

3 History of numerical experiments What is critical collapse? Numerical experiments Self-similarity 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 1994 attempts to repeat this have failed With angular momentum with Baumgarte P = kρ (this talk) with Joanna Ja lmużna scalar field e imθ Φ(t, r) in 2+1 C. Gundlach Rotating critical collapse 3 / 17

4 What is critical collapse? Numerical experiments Self-similarity GR as a dynamical system on space of initial data (= phase space) Asymptotically flat regular data can form a star collapse to a black hole disperse Threshold between collapse and dispersion empirically a hypersurface itself a dynamical system Attractors in collapse threshold static/stationary L g = 0 (type I) t continuously/discretely self-similar L g = 2g (type II) τ C. Gundlach Rotating critical collapse 4 / 17

5 What is critical collapse? Numerical experiments Self-similarity C. Gundlach Rotating critical collapse 5 / 17

6 Self-similarity What is critical collapse? Numerical experiments Self-similarity Adapted coordinates (τ, x i ) and variables Z: example perfect fluid Z = ( g µν, ρ, v i ) g µν (τ, x) = e 2τ g µν (x, τ) ρ(τ, x) = e 2τ ρ(x, τ), v i (τ, x) Z scale-invariant, e τ measures scale any length e τ, R e 2τ, 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

7 Initial data for rotating perfect fluids Initial data CSS solutions Evolution near the critical solution 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 p and 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

8 CSS solutions with P = kρ Initial data CSS solutions Evolution near the critical solution CSS critical solution Z (x) exists for 0 < k < 1 Linear stability depends on k l = 0 l = 1 l 2 0 < κ stable 1 unstable stable(?) κ < unstable 1 unstable stable 1 9 < κ unstable stable stable 0.49 κ < 1 1 unstable stable many unstable C. Gundlach Rotating critical collapse 8 / 17

9 Initial data CSS solutions Evolution near the critical solution One unstable mode: evolution near the critical solution Intermediate phase near Z Z(x, τ) Z (x) + ζ 0 (τ)z 0 (x) + ζ 1 (τ)z 1 (x) + other decaying where ζ 0 = P(p, q) e λ 0τ, ζ 1 = Q(p, q) e λ 1τ From q q symmetry, P is even in q, Q is odd. Hence P (p p ) Kq 2 Q q to leading order in p, q 2 Black hole threshold at P = 0 p p + Kq 2 C. Gundlach Rotating critical collapse 9 / 17

10 Initial data CSS solutions Evolution near the critical solution Onset of nonlinearity at τ (p, q) defined by P e λ 0τ = 1 δ := Q P λ 1 λ 0 AH forms or solution disperses depending on sign of Z 0 Z(x, τ ) Z (x) + P(p, q) e λ 0τ Z 0 (x) + Q(p, q) e λ 1τ Z 1 (x) Z (x) ±Z 0 (x) + δ Z 1 (x) Intermediate Cauchy data at τ = τ characterised by overall scale e τ, sign ± and dimensionless parameter δ Black hole forms for P > 0, with M e τ F M (δ) P 1 λ 0 1 (p p Kq 2 ) 1 λ 0 J e 2τ F J (δ) P 2 λ 0 δ (p p Kq 2 ) 2 λ 1 λ 0 q C. Gundlach Rotating critical collapse 10 / 17

11 Overview Scaling at constant Ω Scaling at constant η J M Η Η W W Black hole mass M (left) and angular momentum J (right), against η (strength) and Ω (rotation) of initial data initial fluid density ρ ηe r 2 initial angular velocity Ω/(1 + r 2 ) C. Gundlach Rotating critical collapse 11 / 17

12 Overview Scaling at constant Ω Scaling at constant η Scaling at constant initial rotation Ω J Η M (left) and J (right) against η η 1 (log-log plots) C. Gundlach Rotating critical collapse 12 / 17

13 Overview Scaling at constant Ω Scaling at constant η Scaling at constant initial density η JΗ M (left) and J (right) against 1 Ω Ω (log-log plots) C. Gundlach Rotating critical collapse 13 / 17

14 Evolution near the critical solution : evolution near the critical solution As before, intermediate phase near Z Z(x, τ) Z (x) + P(p, q) e λ 0τ Z 0 (x) + Q(p, q) e λ 1τ Z 1 (x) As before, δ := ζ 1 ζ 0 λ 1 λ 0 = Q P λ 1 λ 0 is constant during linear perturbation phase Onset of nonlinearity at τ = τ, defined for example by ζ ζ Intermediate Cauchy data at τ = τ characterised by overall scale e τ, sign of Z 0 and dimensionless parameter δ C. Gundlach Rotating critical collapse 14 / 17

15 Evolution near the critical solution Putting this together, we get as before ) M = (hom.func.deg.one) ( P 1 λ 0, Q 1 λ 1 = P 1 λ 0 F M (δ) J = (hom.func.deg.two)(... ) = P 2 λ 0 F J (δ) J M 2 = (hom.func.deg.zero)(... ) = F J/M 2(δ) But we now explore large values of δ The attracting manifold of the critical solution now has codimension two But 0 < λ 1 λ 0, so q does not have to be very small The black hole threshold has always codimension one. It must be at some δ = δ C. Gundlach Rotating critical collapse 15 / 17

16 Evolution near the critical solution Dynamical system: one and two unstable modes Ζ 2 Ζ 2 Ζ 0 Ζ 0 Ζ 1 Ζ 1 ζ 0 spherical mode, ζ 1 ballerina mode, ζ 2 any other mode C. Gundlach Rotating critical collapse 16 / 17

17 Other things I am working on Collapse in 2+1 generally (role of Λ < 0) Rotating critical collapse in 2+1 (with Ja lmużna) Rotating black holes from point particle mergers in 2+1 (with Skenderis, Hartnett, Iannetta) Critical collapse in Einstein-Vlasov C. Gundlach Rotating critical collapse 17 / 17