Critical Phenomena in Gravitational Collapse Thomas Baumgarte - - PowerPoint PPT Presentation

Critical Phenomena in Gravitational Collapse

Thomas Baumgarte Bowdoin College ICERM, Brown University, Oct. 27, 2020

Critical Phenomena Critical Phenomena in gravitational collapse

Critical Phenomena in gravitational collapse

Thomas Baumgarte, Bowdoin College

2

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Consider scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

1 2 3 4 5 6 7 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 α η =0.6 η =0.1

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3 < η∗ < 0.4

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

1 2 3 4 5 6 7 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 α η =0.39 η =0.31

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3 < η∗ < 0.31

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

1 2 3 4 5 6 7 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 α η =0.309 η =0.301

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303 < η∗ < 0.304

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

2 4 6 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 α η =0.3039 η =0.3031

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3033 < η∗ < 0.3034

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

2 4 6 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 α η =0.30339 η =0.30331

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.30337 < η∗ < 0.30338

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

2 4 6 8 t −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 α η =0.303379 η =0.303371

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375 < η∗ < 0.303376

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

5.8 6.0 6.2 6.4 6.6 t 0.0 0.2 0.4 0.6 0.8 α η =0.303379 η =0.303371

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375 < η∗ < 0.303376

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

5.8 6.0 6.2 6.4 6.6 t 0.0 0.2 0.4 0.6 0.8 α η =0.3033759 η =0.3033751

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3033759 < η∗ < 0.3033760

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

5.8 6.0 6.2 6.4 6.6 t 0.0 0.2 0.4 0.6 0.8 α η =0.30337599 η =0.30337591

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.30337599 < η∗ < 0.30337600

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

5.8 6.0 6.2 6.4 6.6 t 0.0 0.2 0.4 0.6 0.8 α η =0.303375999 η =0.303375991

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375994 < η∗ < 0.303375995

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.45 6.50 6.55 6.60 t 0.0 0.2 0.4 0.6 α η =0.303375999 η =0.303375991

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375994 < η∗ < 0.303375995

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.45 6.50 6.55 6.60 t 0.0 0.2 0.4 0.6 α η =0.3033759949 η =0.3033759941

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3033759947 < η∗ < 0.3033759948

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.45 6.50 6.55 6.60 t 0.0 0.2 0.4 0.6 α η =0.30337599479 η =0.30337599471

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.30337599472 < η∗ < 0.30337599473

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.45 6.50 6.55 6.60 t 0.0 0.2 0.4 0.6 α η =0.303375994729 η =0.303375994721

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375994729 < η∗ < 0.303375994730

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.5825 6.5850 6.5875 6.5900 6.5925 t −0.1 0.0 0.1 0.2 0.3 0.4 α η =0.303375994729 η =0.303375994721

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.303375994729 < η∗ < 0.303375994730

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

A numerical experiment...

• Let’s say scalar wave

φ ≡ gab∇a∇b φ = 0 coupled to Einstein’s equations

• Initial data

φ = η exp(−R2/R2

0)

• try out different η...

6.5825 6.5850 6.5875 6.5900 6.5925 t −0.1 0.0 0.1 0.2 0.3 0.4 α η =0.3033759947299 η =0.3033759947291

Have critical value η∗ so that η < η∗ α → 1 end up with flat space η > η∗ α → 0 end up with black hole Black-hole threshold 0.3033759947297 < η∗ < 0.3033759947298

Thomas Baumgarte, Bowdoin College

3

Critical Phenomena Critical Phenomena in gravitational collapse

Critical Solution

• Let’s look at φ for η ≈ η∗ at r = 0

2 4 6 t −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 φc

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical Phenomena in gravitational collapse

Critical Solution

• Let’s look at φ for η ≈ η∗ at r = 0
• plot as function of proper time τ

= ⇒ oscillations “accumulate” at “accumulation” time τ∗ ≈ 1.5698

0.0 0.5 1.0 1.5 τ −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 φc

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical Phenomena in gravitational collapse

Critical Solution

• Let’s look at φ for η ≈ η∗ at r = 0
• plot as function of proper time τ

= ⇒ oscillations “accumulate” at “accumulation” time τ∗ ≈ 1.5698

• plot as function of

T ≡ − log(τ∗ − τ)

0.0 2.5 5.0 7.5 10.0 T −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 φc

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical Phenomena in gravitational collapse

Critical Solution

• Let’s look at φ for η ≈ η∗ at r = 0
• plot as function of proper time τ

= ⇒ oscillations “accumulate” at “accumulation” time τ∗ ≈ 1.5698

• plot as function of

T ≡ − log(τ∗ − τ)

0.0 2.5 5.0 7.5 10.0 12.5 T −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 φc

= ⇒ critical solution performs periodic oscillations in T (discrete self-similarity) = ⇒ “Choptuik spacetime”

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical Phenomena in gravitational collapse

Can we form arbitrarily small black holes? Plot mass M of forming black hole as function of parameter η = ⇒ find power-law scaling M ≃ (η − η∗)γ with critical exponent γ universal (for given matter field)

• reminiscent of critical phenomena in other

fields of physics

• can form arbitrarily small black holes

[Choptuik, 1998]

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical Phenomena in gravitational collapse

Critical Phenomena in Gravitational Collapse Consider initial matter distribution parametrized by η (say density) and evolve... Then critical parameter η∗ separates

• supercritical data: form black hole
• subcritical data: don’t

Close to η∗ observe critical phenomena:

• black hole formed from supercritical data

has mass M ≃ |η − η∗|γ where γ is universal [Choptuik, 1998]

• spacetime approaches self-similar critical solution

[Choptuik, 1993]

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .
• . . . towards accumulation event at τ = τ∗

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .
• . . . towards accumulation event at τ = τ∗
• radius R proportional to τ∗ − τ,

R ≃ (τ∗ − τ) = ⇒ dimensionless quantities are functions of ξ ≡ R τ∗ − τ

• nly, i.e.

Z = Z∗(ξ)

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .
• . . . towards accumulation event at τ = τ∗
• radius R proportional to τ∗ − τ,

R ≃ (τ∗ − τ) = ⇒ dimensionless quantities are functions of ξ ≡ R τ∗ − τ

• nly, i.e.

Z = Z∗(ξ)

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Self-similarity

Self-similarity

• Solution

contracts without changing

• shape. . .
• . . . towards accumulation event at τ = τ∗
• radius R proportional to τ∗ − τ,

R ≃ (τ∗ − τ) = ⇒ dimensionless quantities are functions of ξ ≡ R τ∗ − τ

• nly, i.e.

Z = Z∗(ξ) = ⇒ no preferred global length scale What sets scale of forming black holes?

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Self-similarity

Three phases of evolution

• Phase I:

from initial data to something close to critical solution (how close? depends on degree of fine-tuning)

• Phase II:

critical solution plus perturbation (until perturbation becomes nonlinear)

• Phase III:

collapse to black hole or disperse = ⇒ length scale set by size of self-similar solution at transition from Phase II to III

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution
• assume that only one mode is unstable

= ⇒ grows at rate λ in T = − log(τ∗ − τ) ζ ∝ exp(λT) = (τ∗ − τ)−λ

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution
• assume that only one mode is unstable

= ⇒ grows at rate λ in T = − log(τ∗ − τ) ζ ∝ exp(λT) = (τ∗ − τ)−λ

• to leading order also proportional to η − η∗

ζ ∝ (η − η∗)(τ∗ − τ)−λ

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution
• assume that only one mode is unstable

= ⇒ grows at rate λ in T = − log(τ∗ − τ) ζ ∝ exp(λT) = (τ∗ − τ)−λ

• to leading order also proportional to η − η∗

ζ ∝ (η − η∗)(τ∗ − τ)−λ Mode becomes nonlinear when ζ = const = ⇒ determines length scale R ∝ (τ∗ − τ) ∝ (η − η∗)1/λ

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Phase II: Perturbations of Critical Solutions

• Consider perturbations ζ of critical solution
• assume that only one mode is unstable

= ⇒ grows at rate λ in T = − log(τ∗ − τ) ζ ∝ exp(λT) = (τ∗ − τ)−λ

• to leading order also proportional to η − η∗

ζ ∝ (η − η∗)(τ∗ − τ)−λ Mode becomes nonlinear when ζ = const = ⇒ determines length scale R ∝ (τ∗ − τ) ∝ (η − η∗)1/λ = ⇒ scaling laws, e.g. M ∝ (η − η∗)γ with γ = 1/λ [Koike et.al., 1995; Maison 1995]

10−10 10−8 10−6 10−4 10−2 |η − η∗| 10−2 10−1 100 M, ρ−1/2

max

ρ−1/2

max

M

1/λ = 0.3558 γ = 0.356 [Celestino & TWB, 2018]

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Critical Phenomena Phase II: Perturbations of Critical Solutions

Continuous versus discrete self-similarity For fluid, for example, encounter con- tinuous self-similarity (CSS) For scalar waves, expect “super- imposed” oscillation = ⇒ discrete self-similarity (DSS)

0.0 2.5 5.0 7.5 10.0 T −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 φc −30 −25 −20 −15 −10 ln(η∗ − η) 5 10 15 20 ln(ρmax) ǫ2 = 0

= ⇒ leaves periodic “wiggle” in power- law scaling [Gundlach, 1997; Hod & Piran, 1997]

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Critical Phenomena Key ingredients...

Key ingredients...

• Unique critical solution, either CSS or DSS
• Single unstable mode, Lyapunov exponent λ

= ⇒ Power-law scaling with critical exponent γ = 1/λ

• Pretty well established in spherical symmetry...

= ⇒ ... but what about non-spherical cases??

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Critical Phenomena Critical collapse of gravitational waves

Critical collapse of gravitational waves

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Critical Phenomena Critical collapse of gravitational waves

Numerous attempts to reproduce this... Despite many attempts... [e.g. Alcubierre et.al., 2000; Garfinkle & Duncan, 2001; Santamaria, 2006; Rinne, 2008; Sorkin, 2011; Hilditch et.al., 2013; Hilditch et.al., 2017] ... the results of Abrahams & Evans have yet to be reproduced. Issues...

• Few of the current 3D numerical relativity codes are designed for critical-collapse

simulations

• Some evidence that coordinate conditions that work for other simulations do not

work well for critical collapse of gravitational waves

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical collapse of gravitational waves

Collapse of Brill waves

• Fine-tune Brill waves to black-hole threshold
• Some agreement with Abrahams & Evans
• But lack of clear evidence for DSS...

[Hilditch, Weyhausen, & Br¨ ugmann, 2017]

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical collapse of electromagnetic waves

Critical collapse of electromagnetic waves Solve Einstein-Maxwell system in axisymmetry

• Forms system of equations similar to that for scalar waves
• Does not allow spherically symmetric critical solution

Consider dipolar initial data of the form Eφ = −4η ψ6

• e−(r−r0)2 + e−(r+r0)2

Evolve with code that solves BSSN equations in spherical polar coordinates [Baumgarte et.al., 2013]

• “gravitational gauge”: 1+log slicing; zero shift
• “EM gauge”: choose Φ ≡ naAa = 0

= ⇒ fine-tune parameter η to critical value η∗... [Baumgarte, Gundlach, & Hilditch, 2019]

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical collapse of electromagnetic waves

The critical solution As invariant diagnostic, consider Aξ ≡ ξaAa (ξaξa)1/2 Here

• Aa electrodynamic vector potential
• ξa = ∂/∂ϕ axisymmetric Killing vec-

tor

• T = − ln(τ∗ − τ)
• λ affine parameter along null geodesics

= ⇒ approximate DSS, with period ∆ ≃ 0.55 – but not exact

Thomas Baumgarte, Bowdoin College

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Critical Phenomena Critical collapse of electromagnetic waves

Scaling

• Approximate scaling

ρmax

c

≃ (η∗ − η)−2γ with γ = 0.145 – but not exact

• wiggles not exactly periodic

10−11 10−9 10−7 10−5 10−3 η∗ − η 10−1 100 101 102 ρmax

c

r0 = 0, N = 1 r0 = 0, N = 2 r0 = 0, N = 3 r0 = 0, N = 4 r0 = 3, N = 2 r0 = 3, N = 3 0.08 (η∗ − η)−2γ

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Critical Phenomena Critical collapse of electromagnetic waves

Scaling

• Approximate scaling

ρmax

c

≃ (η∗ − η)−2γ with γ = 0.145 – but not exact

• wiggles not exactly periodic

= ⇒ reminiscent of [Hilditch et.al., 2017]

10−11 10−9 10−7 10−5 10−3 η∗ − η 10−1 100 101 102 ρmax

c

r0 = 0, N = 1 r0 = 0, N = 2 r0 = 0, N = 3 r0 = 0, N = 4 r0 = 3, N = 2 r0 = 3, N = 3 0.08 (η∗ − η)−2γ

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Critical Phenomena Critical collapse of electromagnetic waves

Behavior of lapse

10 20 30 t 10−3 10−2 10−1 100 αc subcrit. supercrit. 31 32 10−1

Gravitational waves Electromagnetic waves [Abrahams & Evans, 1994] = ⇒ No conclusive evidence for strict periodicity in either case

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Critical Phenomena Critical collapse of electromagnetic waves

Is the critical solution unique? Centered (r0 = 0) Off-centered (r0 = 3) = ⇒ No evidence for strict uniqueness

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Critical Phenomena Critical collapse of electromagnetic waves

Is the critical solution unique? Centered (r0 = 0) Off-centered (r0 = 3) = ⇒ No evidence for strict uniqueness = ⇒ Considering more general initial data suggests non-uniqueness of critical solution [Perez Mendoza et.al., in prep]

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Critical Phenomena Summary

Summary

• Numerical simulations of critical collapse of electromagnetic waves suggest...
• ... approximate, but not exact DSS of critical solution
• ... approximate, but not exact power-law scaling
• ... similarities with results for gravitational waves
• Absence of exact DSS and scaling might be caused by...
• ... interplay between gravitational and electromagnetic degrees of freedom

[Gundlach et.al., 2019]

• ... interplay between different multipole moments

= ⇒ appear to be related to non-spherical nature of critical solution

• No evidence for uniqueness of critical solution

[Fern´ andez et.al., 2020] Our notion of critical phenomena in gravitational collapse in- vokes the existence of a unique, strictly self-similar critical solu- tion with a single unstable mode. This notion does not appear to apply for electromagnetic (or gravitational) waves.

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