NR BlackHoles

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Numerical Relativity II:From Discretizing PDEs to

Computational Infrastructure for NR

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Sascha HusaUniversitat de les Illes BalearsChris Engelbrecht Summer SchoolGrahamstown, January 2013


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(Theoretical) computational cost scales with grid size, 3D:

x 2 resolution -> x 16 computational cost

Real world software and hardware may behave in a more complicated way!

To estimate the error, we need to understand convergence. n-th accurate:

3 resolutions determine

logic: is n consistent with what I think my algorithm is?Yes -> estimate X0, estimate error e.g. as X0 - X(best resolution).

break-even for 3D - n=4: x 2 resolution, 16 computational cost, error/16

typical n for binary black holes: 6-10.

spectral code: exponential convergence (SpEC, standard initial datasolvers), more accurate, harder to verify.

mixed order schemes are common, and can lead to subtle problems, e.g.RK4 + 6th order finite differencing + ...

Cost & error II

X(x) = X0 + exn + O(xn+1)

X0,e ,n

x3t1


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smooth anddistributional solutions

More generally: characteristics can cross, typically signifies physicalbreakdown of underlying PDE, like in fluid dynamics.

Unless a PDE is linearly degenerate (speeds independent of solution),shocks can form from smooth data in a finite time.

Vacuum EE: can be written in linearly degenerate form, do not expectphysical shocks, but shocks can form due to bad gauge conditions.

Numerical methods for fluid dynamics are dominated by methods thatdeal with shocks - e.g. propagate shocks at correct physical speed.

Solutions of vacuum GR are smooth except due to bad gauges or

physical singularities, high order FD or spectral ideal!3

Burgers equation: ut = u ux.Characteristic speeds depend on u,

peak velocity overtakes rest of the waveafter some time.


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Numerical

Relativity 3:Black HoleS acetimes

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Sascha HusaUniversitat de les Illes BalearsChris Engelbrecht Summer School

Grahamstown, January 2013


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GWs from stellar binariesBlack hole binaries are the most compact systems generating GWs.

Astrophysical black holes are described by mass and spin only!

=> Extraordinarily clean systems that allow precisionastrophysics and fundamental physics - find new physics?.

At sufficient separation, BH-NS and NS-NS binaries behave likeBH-BH (point particles).

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Kepler: - point particles dynamics scales with mass.

Solving the 2-body problem in GR is hard- but sufficiently advanced for source modelers

to work closely with data analysts!

Morb =R

M

3/2

R20 km

5M

3M

1

/3

f100Hz

2

/3


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How will this work?

Einstein equations form singularities, but not shocks (unless

created by coordinate conditions) -> use finite differencing orspectral methods as appropriate for smooth solutions.

BBH problem has many different scales (BHs are the mostcompact objects) -> need mesh refinement!

2 paradigms for dealing with the singularity inside black holes:

excision (cut a hole in the computational domain, CalTech++)

singularity avoiding gauge conditions (everybody else).

Need coordinate conditions that represent black holes in anappropriate time-independent way.

Crucial accuracy problem: tracking an unstable process - smallinaccuracy in orbital frequency adds up to large phase error!

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Penrose diagramGraph global

causal structureof spacetime,uses conformal

rescalinggunphys = 2 gphys

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black holes andevent horizons

Event horizon: null surface traced backwards in timefrom i+, BH is the spacetime region inside the EH.

Applying the rigorous definition in a numericalspacetime wont work.

Reasonable approximation: trace back spherical nullsurface starting well after merger.

level set method: EH @f(xi,t) = 0 ->

re-initialize to deal with steepening gradients

Diener CQG20 (2003), Cohen+ CQG26 (2009) [geodesics]8

gf = 0

tf=gtiif+

(gtiif)2gttgijifjf

gtt


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... Matzner+ Science 270 (1995) Husa, Winicour, Phys.Rev. D60 (1999) 084019

Cohen+, Phys. Rev. D 85, 024031 (2012) ...


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Apparent horizons: local in time

Characteristic speeds c: AH is a outflow boundary! BH excision.

Singularity theorems:very general conditions: trapped surface => spacetime singularity

Elliptic equation for constant expansion surfaces:

Solve directly with elliptic solver or more robust parabolic flow.

Thornburg, Liv. Rev. Rel. 10 (2007), BUT: flow is fast.10

Trapped surface: spacelike 2-

surface with the property thatthe ingoing and outoingwavefronts decrease in area.Light cones tip over!

AH: outermost marginallytrapped surface.

= K rarbKab Dar

a


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4 phases of inspiral

1. Inspiral not driven by GW backreaction (e.g. gravitationalfriction).

2. Tightening of orbit due to the emission of energy and angularmomentum in the form of gravitational waves millions of years!

At sufficiently large separation point particles and post-Newtonian expansions work very well.

3. For the last few orbits radial infall motion becomes significantand full GR has to be taken into account.

4. After merger the resulting single distorted black hole enters thering-downp hase during which the black hole settles down to astationary, axisymmetric Kerr black hole.

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Lets see what this looks like:

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Difficulty levels of solving EEsMass-energy is dominated by rest mass of objects (astrophysicalBHs).

Mass-energy is dominated by rest mass of objects (highlyrelativistic BHs).

Mass-energy is dominated by gravitational waves (collapse of

GWs).

Limiting situations (like EMRIs, close to extreme Kerr BHs,ultrarelativistic BHs, critical collapse, ...).

Special easy case: spherical symmetry, only 1+1 D, no gravitational

waves, local mass def. exists, ...

Nontrivial topology -> Reula, Tiglio, ...

Matter: adds a lot of complexity, but not for solving EE

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Initial dataNeed to choose 12 quantities gij, Kij subject to 4 Einstein constraintequations.

Key: find out what quantities we can choose freely, set up ellipticproblems for the rest.

Example: Solving the Maxwell constraints

Choose arbitrary 3-vector fields AE, AB, scalar fields E, B:

-> Initial data

Exercise: show Maxwell constraints satisfied for E,B!14

E = 4, B = 0.

3R + (Kaa)

2 = 16 + KabKab

E = 4, B = 0.

Da

K

ab h

abK

= 8J

b

E= E+ AE, B = B + AB


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York conformal procedure

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Brill-Lindquist BH initial dataSimplifications:

Conformal flatness: good approximation for moderate boostsand spins, neutron stars, ...

Schwarzschild conf. flat, Kerr data can be viewed as conf.flat + GW wave.

Moment of time symmetry: Kab = 0.Momentum C. solved, Hamiltonian C. becomes:

Solution contains N+1 spatial infinities, at least N minimalsurfaces -> N AHs -> data for N BHs!

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hab = ab, hab = 4ab

= 0

= 1 +

i

mi

2|r ri|

puncture data: BH asymptoticends are compactified at price of

coordinate singularity.


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Bowen-York (puncture) dataAssume conformal flatness, can find analytical solutions to MC.

Choose free data for constraints:

Simplifies MC to

Bowen-York solution for the 3-vector Wi, P & S constant vectors.

Obtain extrinsic curvature as

Still need to solve the Hamiltonian constraint (numerically)

-> conformal factor blows up at r=0,total momentum P and angular momentum S!

Linearity -> superposition of black holes!

Numerical solution: subtract singular 1/r part of analytically!18

hab = ab, K = 0, Aab = 0, jb

= 0.

a

aW

b+

bW

a

2

3ab

cW

c

= 0

W = 14r

7P+ n

n P

+ 1

r2n S, n =

rr

Kab = (LW)ab

= 18K

abKab

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removing eccentricity

> 2007 longer evolutions & higher accuracy require moresystematic approach - how low should eccentricity is needed?Wanted: circular at large separation (tens of M).

basic ideas:compute PN inspiral, use PN momenta or velocities asapproximate initial data (Husa+ 2007)Search for zero eccentricity data with an iterative scheme(Pfeiffer+ CQG 2007)

combine both ideas. 19

Typical comparable mass inspiral eventswill have negligible eccentricity. How to

construct such data?Early work focused on very close BHbinaries 1-2 orbits before merger:Effective potential, helical Killingvector methods to construct quasi-circular data [Cook, Liv.Rev.Rel.]


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Black hole gauge conditions wish list

Symmetry-seeking: gauge

conditions should findsymmetries, e.g. stationarity,spherical/axial/helical symmetry.

Keep horizon at fixed size.

Choose shift vector outwardpointing, s.t. observers athorizon move out @ lightspeed.

Singularity avoiding slicing: slowdown lapse, s.t. slice never hits

the horizon.

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Choose evolution equations for lapse and shift which drive themtoward a desired equilibrium.

Comment: Stationary puncture slice for Schwarzschild does not

exist.


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Singularity avoiding fixed punctureevolutions

Good idea for BH evolutions:

start with Bowen-York puncture data,

analytically factor out the 1/r singularityof the conformal factor.

To do this, we need the singularity to stayin place in our coordinate system. Good,use co-rotation (using the shift!)

Problem: The stationary solution does nothave a 1/r term.

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Wormholes and Trumpets

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