Applications of Dynamical Horizons in Numerical Relativity E. - - PowerPoint PPT Presentation

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applications of dynamical horizons in numerical relativity
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Applications of Dynamical Horizons in Numerical Relativity E. - - PowerPoint PPT Presentation

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Applications of Dynamical Horizons in Numerical Relativity E. Schnetter 2 B. Krishnan 1 F. Beyer 1 1 Max Planck Institut fr

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Applications of Dynamical Horizons in Numerical Relativity

  • E. Schnetter2
  • B. Krishnan1
  • F. Beyer1

1Max Planck Institut für Gravitationsphysik

Albert Einstein Institut D-14476 Golm, Germany

2Center for Computation and Technology

Louisiana State University Baton Rouge, LA 70803, USA

Paris, November 22, 2006

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Outline

1

Motivation and Background Trapped Surfaces The trapping boundary

2

Dynamical horizons

3

Horizon Multipole Moments

4

Example Numerical Simulations Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Trapped Surfaces The trapping boundary

Definition of a trapped surface

For a sphere in flat space

Outgoing light rays are diverging: Θ(ℓ) = ˜ qab∇aℓb > 0 Ingoing light rays are converging: Θ(n) = ˜ qab∇anb < 0

For a trapped surface, both sets of null rays are converging: Θ(ℓ) < 0 and Θ(n) < 0 Trapped surfaces are signatures of black holes:

Existence of trapped surface = ⇒ singularity in future Trapped surfaces lie inside the event horizon For cross sections of stationary EHs Θ(ℓ) = 0, Θ(n) < 0

Future Marginally Outer Trapped Surface (FMOTS):

Θ(ℓ) = 0, Θ(n) < 0

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Trapped Surfaces The trapping boundary

The trapping boundary

Trapping boundary is boundary of region containing trapped surfaces There are spherically symmetric trapped surfaces right up to the Schwarzschild event horizon

  • I+

I− i+ i0 E S

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Trapped Surfaces The trapping boundary

The trapping boundary

I+ i+ i− i0 E v = 0 r = 0 H Vaidya: sph. symmetric trapped surfaces only up to H Suggestion by Eardley: event horizon is the trapping boundary

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Trapped Surfaces The trapping boundary

The trapping boundary

We can look for marginally trapped surfaces on non-symmetric surfaces using apparent horizon finders In Vaidya we can push marginally trapped surfaces arbitrarily close to the EH Marginal surfaces can also extend into flat region Recent analytic proof by Ben-Dov

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 z x horizon flat region MS

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Trapped Surfaces The trapping boundary

Evolution of MTSs in Time

It is observed numerically that MTSs evolve smoothly in time Apparent horizons may jump due to outermost condition Smooth world tube of MTSs is a Marginally Trapped Tube MTT shown to exist if MTS is strictly stably outermost

Andersson et al, PRL 95 111102 (2005)

Untrapped Trapped

Linear outward deformation makes S untrapped: δfrΘ(ℓ) > 0 for f ≥ 0 In practice we look for surfaces with Θ(ℓ) = ǫ > 0 and check that it lies outside S

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Definitions

Dynamical Horizon: Spacelike MTT with Θ(n) < 0 Outermost MTT usually forms a DH MTT with |σ(ℓ)|2 = 0 or Tabℓaℓb = 0 somewhere are spacelike if they are SSO (Andersson et al.) Timelike Membrane: Timelike MTT Cannot be the black hole surface Inner MTTs might form timelike membranes Isolated Horizon: Null MTT (BH in equilibrium) Other cases: MTTs with mixed signature also possible

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Basic Properties

Cross section is topologically S2 Area increases along ˆ r a for a DH

Consequence of Θ(ℓ) = 0 and Θ(n) < 0 Area increases in time if t.r > 0 Area decreases for a TLM

Foliation of DH is unique (Ashtekar & Galloway, gr-qc/0503109)

Implies that changing Σ leads to different DH Other restrictions on occurence of MTS in presence of DH

Event horizon is probably the boundary of the trapped region (Eardley 1998, Schnetter & Krishnan 2006, Ben-Dov 2006)

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Multipole Moments

We are interested in source multipole moments for black hole In classical electrodynamics we have charge and current multipole moments for sources For a black hole we have mass and angular momentum multipole moments Mn and Jn J0 vanishes by absence of monopole charges (here NUT charge) M0 is mass and J1 is angular momentum In Kerr, M0 and J1 determine all higher moments In Schwarzschild, only M0 = 0

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Multipole Moments

Given rotational Killing vector ϕa construct coordinate system (θ, φ) on S

φ ∈ [0, 2π) is affine parameter along ϕa ζ = cos θ ∈ [−1, 1] is defined by ˜ Daζ = 1 R2

S

˜ ǫbaϕa ,

  • S

ζ = 0

Use spherical harmonics for (θ, φ) to define multipoles Mn = Rn

SMS

8π

  • S
  • ˜

RPn(ζ)

  • d2V

Jn = Rn−1

S

8π

  • S

P′

n(ζ) ¯

KabϕaRb d2V

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Multipole Moments

Properties of the multipole moments Mn and Jn are coordinate independent They characterize geometry of DH at any given time Need only data on MTS to calculate them Coincide with corresponding isolated horizon formulae (Ashtekar et. al, CQG 21 2549 (2004)) Useful for characterizing rate of approach to Kerr

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Brill-Lindquist Initial Data

Describes head on collision for two BH case Σ is R3 with two “punctures” Time symmetric: ¯ Kab = 0 Conformally flat: ¯ qab = ψ4δab ∆ψ = 0, ψ → 1 as r → ∞ ψ = 1 + α1 2r1 + α2 2r2 Single BH case is Schwarzschild in isotropic coordinates mADM = 2α1 + 2α2 and punctures are asymptotic regions Take units such that mADM = 1

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Brill-Lindquist Initial Data

Equal mass: 2α1 = 2α2 = 0.5 punctures initially at z = ±0.5 Explicit octant symmetry and extent upto x, y, z = 96 4th order spatial differencing and 3rd order Runge-Kutta Use mesh refinement: h = 1.6 at boundary and h = 0.0125 at horizon Horizon diameter contains 32 points initially About 10 grid points excised around punctures AEI BSSN formulation (inconsistent boundary conditions!) 1 + log slicing with α = 1 initially; zero shift Common MTS forms at t ≈ 0.5 We use Jonathan Thornburg’s AHFinderDirect.

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Horizon Shapes

  • 0.5

0.5 1

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 x z Horizon shapes at t=1 individual horizons inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Signature of MTT

  • 1

1 2 3 4 5 6 7 π π/2 det q θ Horizon metric determinant at t=0.6 individual horizon inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Signature of MTT

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 π π/2 det q θ Horizon metric determinant at t=1 individual horizon inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Horizon Mass

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M t Irreducible mass inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Behavior of the MTTS

Outer MTT is spacelike and growing Individual MTTs are essentially isolated Inner MTT is initially spacelike but soon becomes timelike Inner MTT has decreasing area

time

Area Mixed Timelike Spacelike

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Behavior of the MTTS

What is the geometry of the full MTT – “inverted pair of pants”? Lose track of inner MTT because of resolution and because it may not be star shaped How, if at all, does the inner MTT merge with the individual horizons? Area is monotonic radially outwards

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Mass Quadrupole Moment

  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M2 t Mass quadrupole moment inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Mass Multipole M4

  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

1 2 3 4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M4 t Mass multipole moment l=4 inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Mass Multipole M4

  • 5

5 10 15 20 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 M6 t Mass multipole moment l=6 inner horizon

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Behavior of Multipole Moments

All Jns vanish Mn = 0 for odd n All higher moments for outer horizon vanish asymptotically But inner MTT does not seem to approach Schwarzschild

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Shear

0.05 0.1 0.15 0.2 0.25 π π/2 |σ|2 θ Shear at t=0.6

  • uter horizon
  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Axisymmetric gravitational collapse

Uniformly rotating perfect fluid Used Whisky K = 100, Γ = 2 polytrope (p = KρΓ) Model D4 in Baiotti et al (PRD, 2005): MNS = 1.86M⊙, ρc = 1.934 × 1015 g cm−3, and JNS = 0.543M2

NS.

Ratio of polar to equatorial coordinate radius is 0.65 Rotational frequency 1295.34 Hz Equatorial radius 14.22 km Configuration is dynamically unstable – reduce pressure to induce collapse We use Jonathan Thornburg’s AHFinderDirect.

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Horizon Shapes

0.5 1 1.5 2 2.5 3 130 140 150 160 170 180 190 200 r t Average coordinate radius

  • uter horizon

inner horizon

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Signature of the MTT

0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 π π/2 det q θ Horizon metric determinant at t=138.24

  • uter horizon

inner horizon

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Signature of the MTT

1e-05 0.0001 0.001 0.01 0.1 1 10 130 140 150 160 170 180 190 200 average of det q t Horizon metric determinant

  • uter horizon

inner horizon

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Horizon Area

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 130 140 150 160 170 180 190 200 R t Areal radius

  • uter horizon

inner horizon

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Horizon Angular Momentum

  • 0.5

0.5 1 1.5 2 130 140 150 160 170 180 190 200 J t Angular momentum

  • uter horizon

inner horizon ADM J

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Mass

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 130 140 150 160 170 180 190 200 M t Total mass

  • uter horizon

inner horizon ADM mass

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Mass Quadrupole Moment M2

  • 2
  • 1.8
  • 1.6
  • 1.4
  • 1.2
  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

130 140 150 160 170 180 190 200 M2 t Mass quadrupole moment

  • uter horizon

M2 for Kerr

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse

Angular Momentum Multipole Moment J3

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

130 140 150 160 170 180 190 200 J3 t Angular momentum octupole moment

  • uter horizon

J3 for Kerr

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR

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Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary

Summary

We have looked at various types of trapped surfaces, multipole moments, etc. in some example simulations Shows how a black hole grows and settles down to a Kerr horizon Can be applied in current stable evolutions

Are non-spinning punctures really non-spinning? Observe spin-orbit coupling? How far is initial data from Kerr? . . .

  • E. Schnetter, B. Krishnan, F

. Beyer DHs and NR