Ulrich Sperhake

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Numerical simulations of high-energy collisions
Of black hole
Ulrich Sperhake Friedrich-Schiller Universität
Jena
Work with E.Berti, V.Cardoso, J.A.Gonzalez,
F.Pretorius
Relativity Seminar Austin, 18th March 2008
Work supported by the SFB TR7
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Overview

• Black holes in physics

• General Relativity

• Ingredients of numerical relativity

• Results

• High energy collisions

• Summary

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1. Black holes in physics
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Black Holes predicted by GR

• Black holes predicted by Einsteins theory of
  relativity

• Term Black hole by John A. Wheeler 1960s

• Vacuum solutions with a singularity

• For a long time mathematical curiosity

• valuable insight into theory

• but real objects in the universe?

• That picture has changed dramatically!

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How to characterize a black hole?

• Consider light cones

• Outgoing, ingoing light

• Calculate surface area

of outgoing light

• ExpansionRate of

change of that area

• Apparent horizon

Outermost surface with zero expansion

• Light cones tip over due to curvature

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Black Holes in astrophysics

• Black holes are important in astrophysics

• Black holes found at
• centers of galaxies

• Structure of galaxies

• Important sources of
• electromagnetic radiation

• Structure formation in
• the early universe

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Fundamental physics of black holes

• Allow for unprecedented tests of fundamental
  physics

• Strongest sources of Gravitational Waves (GWs)
• Test alternative theories of gravity
• No-hair theorem of GR
• Production in Accelerators

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Gravitational wave physics

• Accelerating bodies produce GWs

• Weber 1960s

• Bar detector
• Claimed detection probably not real

• GWs displace particles

• GW observatories GEO600, LIGO, TAMA, VIRGO
• Bar detectors

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The big picture
Detectors
Physical system
observe
test
Provide info
describes
Help detection

• Model
• GR (NR)
• PN
• Perturbation theory
• Alternative Theories?

• External Physics
• Astrophysics
• Fundamental Physics
• Cosmology

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2. General relativity
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The framework General Relativity

• Curvature generates
• acceleration

geodesic deviation
No force !!

• Description of geometry

Metric
Connection
Riemann Tensor
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The metric defines everything

• Christoffel connection

• Covariant derivative

• Riemann Tensor

• Geodesic deviation

• Parallel transport

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How to get the metric?

• The metric must obey the Einstein Equations

• Ricci-Tensor, Einstein-Tensor, Matter tensor

Trace-reversed Ricci
Matter

• Einstein Equations

• Take metric
• Calculate
• Use that as matter tensor

• Solutions Easy!

• Physically meaningful solutions Difficult!

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The Einstein equations in vacuum

• Spacetime tells matter how to move,
• matter tells spacetime how to curve

• Field equations

Second order PDEs for the metric components
Invariant under coordinate (gauge) transformations

• System of equations extremely complex Pile
  of paper!

Analytic solutions Minkowski,
Schwarzschild, Kerr,
Robertson-Walker,
Numerical methods necessary for general
scenarios!
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3. The basics of numerical relativity
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A list of tasks

• Target Predict time evolution of BBH in GR

• Einstein equations

• Cast as evolution system
• Choose specific formulation
• Discretize for Computer

• Choose coordinate conditions Gauge

• Fix technical aspects

• Mesh-refinement / spectral domains
• Excision
• Parallelization
• Find large computer

• Construct realistic initial data

• Start evolution and wait

• Extract physics from the data

Gourgoulhon gr-qc/0703035
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3.1. The Einstein equations
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31 decomposition

• GR Space and time exist together as
  Spacetime

• Numerical relativity reverse this process!

• ADM 31 decomposition

Arnowitt, Deser, Misner (1962)
York (1979)
Choquet-Bruhat, York (1980)
3-metric
lapse
shift

• lapse, shift Gauge

• Einstein equations

6 Evolution equations
4 Constraints

• Constraints preserved under evolution!

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The ADM equations

• Time projection

Hamiltonian constraint

• Mixed projection

Momentum constraints

• Spatial projection

Evolution equations
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The structure of the ADM equations

• Constraints

• They do not contain time derivatives

• They must be satisfied on each slice

• The Bianchi identities propagate the constraints

If they are satisfied initially, they are always
satisfied

• Evolution equations

• Commonly written as first order system

• Gauge

• Equations say nothing about lapse and shift
  !

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The ADM equations as an initial value problem

• Entwicklungsgleichungen

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Alternatives to the ADM equations

• Unfortunately the ADM eqs. do not seem to work
  in NR !!

Weak hyperbolicity Nearby initial data can
diverge
From each other super-exponentially

• Many alternative formulations have been
  suggested

• Two successful families so far

• ADM based formulations BSSN

Shibata Nakamura 95, Baumgarte Shapiro 99

• Generalized harmonic formulations

Choque-Bruhat 62, Garfinkle 04, Pretorius 05
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The BSSN equations
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3.2. Gauge choices
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The gauge freedom

• Remember Einstein equations say nothing about

• Any choice of lapse and shift gives a solution

• This represents the coordinate freedom of GR

• Physics do not depend on

So why bother?

• Avoid coordinate singularities!

• Stop the code from running into the physical
  singularity

• No full-proof recipe, but

• Singularity avoiding slicing

• Use shift to avoid coordinate stretching

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3.3. Initial data
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Initial data problem
Two problems
Constraints, realistic data

• York-Lichnerowicz split
• Rearrange degrees of freedom

• Conformal transverse traceless
• Physical transverse traceless
• Thin sandwich

York, Lichnerowicz
OMurchadha, York
Wilson, Mathews York

• Conformal flatness Kerr is NOT conformally
  flat!

Non-physical GWs problematic for high energy
collisions!
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2 families of initial data

• Generalized analytic solutions

Isotropic Schwarzschild
Time-symmetric, -holes
Brill-Lindquist, Misner (1960s)
Spin, Momenta
Bowen, York (1980)
Punctures
Brandt, Brügmann (1997)

• Excision Data IH boundary conditions on
  excision surface

Meudon group Cook, Pfeiffer Ansorg

• Quasi-circular

• Effective potential
• PN parameters
• helical Killing Vektor

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4. Extracting physics
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Basic assumptions

• Extracting physics in NR is non-trivial !!

• Newtonian quantities are not always
  well-defined !!

• We assume that the ADM variables

Lapse
Shift
3-metric
Extrinsic curvature
are given on each hypersurface

• Even when using other formulations, the ADM
  variables

are straightforward to calculate
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Global quantities

• ADM mass Global energy of the spacetime

• Total angular momentum of the spacetime

By construction all of these are time-independent
!!
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Local quantities

• Often impossible to define !!

• Isolated horizon framework

Ashtekar and coworkers

• Calculate apparent horizon

• irreducible mass, momenta associated with horizon

• Total BH mass

Christodoulou

• Binding energy of a binary

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Gravitational waves

• Most important result Emitted gravitational
  waves (GWs)

• Newman-Penrose scalar

Complex 2 free functions

• GWs allow us to measure

• Radiated energy

• Radiated momenta

• Angular dependence of radiation

• Predicted strain

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Angular dependence

• Waves are normally extracted at fixed radius

are viewed from the source frame !!

• Decompose angular dependence

• Modes

• Spin-weighted spherical harmonics

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5. Status of black hole simulations
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A very brief history

• First attemts Hahn and Lindquist 1960s,
  Eppley and Smarr 1970s

• Breakthrough Pretorius 2005, Brownsville,
  NASA Goddard 2005

• Problems

• Computer resources

• AMR

• Instabilities Gauge, Equations

• Now about 10 codes

• GHG

• Moving puncture

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The BBH breakthrough

• Simplest configuration

• GWs circularize orbit
• quasi-circular initial data

Pretorius 05

• BBH breakthrough

• Initial data scalar field

• Radiated energy

25 50 75 100 4.7 3.2 2.7 2.3

• Eccentricity

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Astrophysical binaries basics

• 4 stages of BBH merger

• Newtonian inspiral (not GW driven)

• Post-Newtonian (PN) inspiral phase (GW driven)

• merger (Numerical Relativity)

• Ring-down (Perturbation Theory)

• Free parameters

• 1 Total mass (merely a scaling factor)

• 1 Mass ratio

• 6 Spin

• 1 Eccentricity GWs circularize orbit

• NR can only simulate the last orbits

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Astrophysical binaries Results

• Equal-mass, non-spinning binaries

• of the total mass of the system

• GWs quadrupole dominated

• Good agreement with PN predictions for GW
  emission

• Recoil or kick

• Unequal mass and/or spin kick

• Certain spin configurations

Enough to eject BHs from galaxies
Observations such large kicks not generic

• Spin-flip

• Merger spin realignment

• X-shaped radio sources

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Zoom whirl orbits
Pretorius Khurana 07

• 1-parameter family of initial data linear
  momentum

• Fine-tune parameter

Threshold of immediate merger

• Analogue in gedodesics !

• Reminiscent of

Critical phenomena
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6. High energy collision
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Motivation

• No known mechanism to accelerate BHs to

• Such collisions unlikely to occur in
  astrophysical scenarios

• But

• Probe GR in the most dynamic range

• Near the speed of light, structure is lost

Collisions might describe generic collisions

• Test cosmic censorship

• Threshold of immediate merger Pretorius 2006

Indication of unexpected phenomena
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Setup of the problem

• Take two black holes

Total rest mass
Initial position
Linear momentum

• For now Equal-mass, head-on

• Target GW energy

Mode distribution of GW energy
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Analytic studies of a single boosted BH

• Analytically studied case

• Schwarzschild boosted with

Aichelburg-Sexl metric

• Planar shock wave travelling along

• Minkowskian everywhere else

• No event horizon!

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Analytic studies of a BH binary

• Superpose two Aichelburg-Sexl metrics

Shock waves collide Penrose, Eardley Giddings

• Future trapped surface!

• Max. gravitational radiation
  Penrose

• Perturb superposed Aichelburg-Sexl metrics

DEath Payne 90s
First correction term reduces GWs to
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Analytic studies of a BH binary

• Instant Collision approximation

Applied to superposed Aichelburg-Sexl metric
Smarr Adler Zeks
Energy spectrum
i) Flat at sufficiently low
ii) Radiation isotropic in the limit
iii) Functional relation
iv) Multipole structure

• Numerical simulations Grand Challenge 90s

Limited accuracy, mild boosts
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Numerical simulations

• LEAN code Sperhake 07

• Based on the Cactus computational toolkit

• BSSN formulation

• Puncture initial data Brandt Brügmann 1996

• Elliptic solver TwoPunctures Ansorg 2005

• Mesh refinement Carpet Schnetter 04

• Numerically very challenging!

• Length scales

Horizon Lorentz-contracted Pancake

• Mergers extremely violent

• Substantial amounts of unphysical junk
  radiation

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Case example

• Boost

• initial separation

• Total radiated energy

• Mode energy

• Junk radiation

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Modes
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Modes
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Modes
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Modes
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Modes
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Modes
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Energy spectrum

• Junk radiation affects spectrum for small
  separations

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Conclusions

• Introduction to Numerical Relativity

• NR has solved the binary problem

• Key results on kicks, spin-flips, PN, data
  analysis

• High energy collisions

Penrose limit
Numerical simulations very difficult (junk
radiation!)
Merger radiation for

• Todo Improve accuracy, larger

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Gravitational recoil

• Anisotropic emission of GWs radiates momentum
• recoil of remaining system

• Leading order Overlap of Mass-quadrupole
• with
  octopole/flux-quadrupole
  
• Bonnor Rotenburg 61, Peres 62,
  Bekenstein 73

• Merger of galaxies

Merger of BHs
Recoil
BH kicked out?
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Gravitational recoil

• Escape velocities

Globular clusters
dSph
dE
Giant galaxies
Merrit et al 04

• Ejection or displacement of BHs has
  repercussions on

• Structure formation in the universe

• BH populations

IMBHs via ejection?

• Growth history of Massive Black Holes

• Structure of galaxies

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Kicks of non-spinning black holes

• Simulations PSU 07, Goddard 07

• Parameter study Jena 07

• Target Maximal Kick

• Mass ratio

• 150,000 CPU hours

• Maximal kick
• for

• Convergence 2nd order

• Spin

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Super Kicks

• Test Rochester-Prediction Jena 07

• Two models

Lean Bam

• Model I

• Lean code
• Moving puncture method version
• Not quasi-circular
• Resolutions

• Model II Safety check with independent code
  BAM

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Convergence

• Discretization error