Puncture Initial Data and Evolution of Black Hole Binaries with High Speed and High Spin

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This Dissertation is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Puncture Initial Data and Evolution of Black Hole Binaries with High Speed and High Spin

Ph.D. Doctor of Philosophy

in Astrophysical Sciences and Technology

Ian Ruchlin

School of Physics and Astronomy

Rochester Institute of Technology

Rochester, New York

August 2015

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Committee:

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Dr. Manuela Campanelli, Committee Member

Dr. Yosef Zlochower, Committee Member

Dr. Lawrence E. Kidder, Committee Member

Dr. Carlos O. Lousto, Thesis Advisor

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Ph.D. DEGREE DISSERTATION

The Ph.D. Degree Dissertation of Ian Ruchlin has been examined and approved by the dissertation committee as satisfactory for the dissertation requirement for the Ph.D. degree in Astrophysical Sciences and Technology.

Dr. Mihail Barbosu, Committee Chair

Dr. Manuela Campanelli, Committee Member

Dr. Yosef Zlochower, Committee Member

Dr. Lawrence E. Kidder, Committee Member

Dr. Carlos O. Lousto, Thesis Advisor

Date

PUNCTURE INITIAL DATA AND EVOLUTION OF BLACK HOLE BINARIES WITH

HIGH SPEED AND HIGH SPIN

Ian Ruchlin

A dissertation submitted in partial fulﬁllment of the

requirements for the degree of Ph.D. in Astrophysical

Sciences and Technology, in the College of Science,

Rochester Institute of Technology.

August, 2015

Approved by

Dr. Andrew Robinson Date

Director, Astrophysical Sciences and Technology

Abstract

This dissertation explores numerical models of the orbit, inspiral, and merger phases of black hole binaries. We focus on the astrophysically realistic case of black holes with nearly extremal spins, and on high energy black hole collisions. To study the evolution of such systems, we form puncture initial data by solving the four general relativity constraint equations using pseudospectral methods on a compactified collocation point domain. The solutions to these coupled, nonlinear, elliptic differential equations represent the desired configuration at an initial moment. They are then propagated forward through time using a set of hyperbolic evolution equations with the moving punctures approach in the BSSNOK and CCZ4 formalisms. To generate realistic initial data with reduced spurious gravitational wave content, the background ansatz is taken to be a conformal superposition of Schwarzschild or Kerr spatial metrics. We track the punctures during evolution, measure their horizon properties, extract the gravitational waveforms, and examine the merger remnant. These new initial data are compared with the well known Bowen-York solutions, producing up to an order of magnitude reduction in the initial unphysical gravitational radiation signature. We perform a collision from rest of two black holes with spins near to the extremal value, in a region of parameter space inaccessible to Bowen-York initial data. We simulate nonspinning black holes in quasi-circular orbits, and perform high energy head-on collisions of nonspinning black holes to estimate the magnitude of the radiated gravitational energy in the limit of infinite momentum. We also evolve spinning black holes in quasi-circular orbits with unequal masses and different spin orientations. These models provide insight into the dynamics and signals generated by compact binary systems. This is crucial to our understanding of many astrophysical phenomena, especially to the interpretation of gravitational waves, which are expected to be detected directly for the first time within the next few years.

i ii Contents

Abstract i

Contents iii

List of Figures vii

List of Tables xv

1 Introduction 1

1.1 Gravitation ...... 1

1.1.1 The Newtonian Theory of Gravitation ...... 2

1.1.2 The Theory of General Relativity ...... 5

1.2 Motivation ...... 7

1.2.1 Black Holes ...... 7

1.2.2 Observations of Black Hole Systems ...... 8

1.2.3 Gravitational Waves ...... 10

1.3 Projects Herein ...... 11

1.3.1 Collisions from Rest of Highly Spinning Black Hole Binaries ...... 11

1.3.2 Quasi-circular Orbits and High Speed Collisions Without Spin ...... 12

1.3.3 Highly Spinning Black Hole Binaries in Quasi-circular Orbits ...... 13

1.3.4 Organization ...... 14

iii 2 Numerical Techniques in Relativity 15

2.1 General Relativity Formalism ...... 15

2.1.1 Geometric Units ...... 16

2.1.2 Manifolds ...... 16

2.1.3 Tensors ...... 17

2.1.4 Diﬀerentiation ...... 23

2.1.5 Curvature ...... 29

2.1.6 The Einstein Field Equation ...... 31

2.1.7 Geodesics ...... 31

2.1.8 Causal Structure and Asymptotic Flatness ...... 32

2.1.9 Black Holes ...... 34

2.2 3 + 1 Formalism ...... 42

2.2.1 The 3 + 1 Decomposition ...... 43

2.2.2 Conformal Decomposition ...... 48

2.2.3 Transverse-traceless Decomposition ...... 49

2.2.4 Thin-sandwich Decomposition ...... 52

2.2.5 Mass, Momentum, and Angular Momentum ...... 53

2.2.6 Black Hole Initial Data ...... 54

2.2.7 Gauge Choices and Evolution Formalisms ...... 58

2.3 Pseudospectral Methods ...... 61

2.4 Cactus and the Einstein Toolkit ...... 64

2.4.1 The TwoPunctures Thorn ...... 64

2.4.2 The HiSpID Thorn ...... 66

3 Collisions from Rest of Highly Spinning Black Hole Binaries 71

3.1 Stationary Kerr Initial Data ...... 74

3.1.1 Conformal Kerr in Quasi-isotropic Coordinates ...... 74

3.1.2 Punctures ...... 78

3.1.3 Attenuation and Superposition ...... 79

iv 3.2 Numerical Techniques ...... 80

3.2.1 Initial Data Solver with Spectral Methods ...... 80

3.2.2 Evolution ...... 81

3.3 Collisions from Rest with Nearly Extremal Spin ...... 82

3.4 Gauge Conditions ...... 86

3.5 Fisheye Transformations ...... 90

3.6 Summary ...... 93

4 Quasi-circular Orbits and High Speed Collisions Without Spin 95

4.1 Lorentz-boosted Schwarzschild Initial Data ...... 97

4.1.1 Conformal Boosted Schwarzschild in Isotropic Coordinates ...... 97

4.1.2 Punctures ...... 102

4.1.3 Attenuation and Superposition ...... 103

4.2 Numerical Techniques ...... 105

4.2.1 Initial Data Solver with Spectral Methods ...... 105

4.2.2 Evolution ...... 105

4.3 Quasi-circular Orbits ...... 107

4.4 Gauge Conditions ...... 112

4.5 Energy Radiated in High Energy Head-on Collisions ...... 115

4.6 Summary ...... 122

5 Highly Spinning Black Hole Binaries in Quasi-circular Orbits 125

5.1 Lorentz-boosted Kerr Initial Data ...... 128

5.1.1 Conformal Boosted Kerr in Isotropic Coordinates ...... 129

5.1.2 Punctures ...... 133

5.1.3 Attenuation and Superposition ...... 135

5.2 Numerical Techniques ...... 137

5.2.1 Initial Data Solver with Spectral Methods ...... 137

5.2.2 Unequal Masses ...... 138

v 5.2.3 Evolution and Gauge Choices ...... 140

5.3 Quasi-circular Orbits ...... 141

5.4 Summary ...... 145

6 Discussion 147

6.1 Summary of Results ...... 147

6.2 Future Research Directions ...... 149

6.2.1 Comparison with Excision Initial Data ...... 149

6.2.2 Conformal Thin-sandwich Initial Data ...... 150

6.2.3 Ultimate Kicks ...... 150

6.2.4 Trumpet Initial Data ...... 151

6.2.5 Multiple Black Hole Systems ...... 151

6.3 Acknowledgements ...... 151

Bibliography 153

vi List of Figures

2.1 A Penrose diagram representation of Minkowski spacetime in compactiﬁed spher-

ical coordinates. The locations of conformal inﬁnity are labeled. Time runs

vertically, from the inﬁnite past at i− and I −, to the inﬁnite future at i+ and

I +. Space extends horizontally from the origin (the vertical line labeled r = 0)

out to spatial inﬁnity at i0. A point on the diagram represents a two-sphere

at that particular time and radius. Light rays move along 45◦ lines, parallel to

I − and I +. Generic surfaces of constant time (dt = 0) and constant radius

(dr = 0) are shown...... 34

2.2 A Penrose diagram of Schwarzschild spacetime. The locations of conformal

inﬁnity are labeled. Time runs vertically, from the inﬁnite past at i− and

I −, to the inﬁnite future at i+ and I +. Light rays move along 45◦ lines,

parallel to I − and I +. A point on the diagram represents a two-sphere at

that particular time and radius. Region I is the ordinary space outside of the

black hole, extending from the event horizon at r = 2M to spatial inﬁnity i0.

Region II is the interior of the black hole, in which all timelike and lightlike

curves end at r = 0, the singularity. Region III is another asymptotically ﬂat

region of space. Since no physical signal can move from I to III, or vice versa,

they are causally disconnected, and represent “separate universes.” Region IV

is the “white hole,” where all timelike and lightlike curves move away from the

r = 0 singularity and are forced into I, II, or III. Generic surfaces of constant

time (dt = 0) and constant radius (dr = 0, r > 2M) are shown...... 38

vii 2.3 An illustration of the 3 + 1 decomposition for two adjacent spatial hypersur-

a faces Σt and Σt+dt. A point in Σt is moved to a point in Σt+dt along t by

a combination of transportation in the direction normal to the surface na and

parallel to the surface βa. The proper time between adjacent surfaces is α dt.. 44

2.4 The TwoPunctures coordinate transformation. The left panel shows the

compactiﬁed (A, B) coordinates with the angular direction suppressed. All

collocation points are distributed uniformly, interior to the outer boundary. The

right panel shows lines of constant A and B in inﬁnite cylindrical coordinates p (x, ρ = y2 + z2). To guide the eye, in both panels the central dashed line of

constant B is made thicker and the locations of the punctures at x = ±b are

marked with dots...... 65

2.5 A comparison of the Hamiltonian constraint residual for a spinning black hole

binary in the xy-plane without (left) and with (right) the x-axis averaging

technique. One of the punctures is located at (x/M, y/M) = (0, 0) and the

other is at (x/M, y/M) = (−12, 0)...... 70

3.1 Masses of the individual black holes from the start of the simulation until merger

(top left). Spins of the individual black holes until merger (bottom left). Masses

of the remnant black hole (top right). Spin of the remnant black hole (bottom

right). BY in blue, HiSpID in red...... 83

3.2 (` = 2, m = 2) mode of Ψ4 at r = 75M (above). (2,0) mode of Ψ4 at r = 75M

(below) for spinning binaries with χ = 0.9. BY data in blue (showing much

larger initial burst), HiSpID in red...... 84

3.3 Convergence of the residuals of the Hamiltonian and momentum constraints

versus number of collocation points N for black hole binaries with χ = 0.99 in

the UU conﬁguration with exponential attenuation parameters ω(±) = 0.2 and p = 4...... 85

viii 3.4 Waveforms and waveform magnitudes of the UU (left) and UD (right) conﬁgu-

rations with highly spinning black holes χ = 0.99. Note the small amplitude of

the initial data radiation content at around t = 75M, compared to the merger

signal after t = 130M...... 87

3.5 The eﬀects of the choice of the initial lapse on the individual black hole masses

(top), spins (middle) and coordinate radii (bottom). The beneﬁts of the new

initial lapse are evident since they follow the higher resolution behavior with

many fewer grid points by (80/125)3...... 88

3.6 The eﬀects of the choice of the initial lapse on the ﬁnal black hole mass (top),

spin (middle) and coordinate radius (bottom). The beneﬁts of the new initial

lapse are evident since they follow the higher resolution behavior with many

fewer grid points...... 89

3.7 The eﬀects of the choice of the initial lapse on the waveforms. The beneﬁts of

the new initial lapse are evident since they show much less initial noise before

the merger...... 90

3.8 Evolution of the mass (left panel) and dimensionless spin parameter (right

panel) for a collision from rest through to merger of two black holes, each with

χ = 0.99. The two curves in each panel represent diﬀerent grid resolutions. The

sudden spike near the end of the curves represents the failure of the isolated

horizon ﬁnder when the black holes are very near to each other...... 91

3.9 Convergence of the residuals of the Hamiltonian and momentum constraints

versus number of collocation points N for black hole binaries in ﬁsheye coor-

dinates with χ = 0.999 in the UU conﬁguration with exponential attenuation

parameters ω(±) = 0.2 and p = 4...... 92

3.10 Evolution of a single black hole with χ = 0.999. The left panel shows the

mass evolution of the puncture, and the right panel shows the dimensionless

spin evolution. The evolution grid has eight reﬁnement levels, the ﬁnest having

resolution M/240 with an extent of 0.2M...... 93

ix 4.1 Convergence of the residuals of the Hamiltonian and momentum constraints

versus number of collocation points N for black hole binaries in a quasi-circular

orbit with exponential attenuation parameters ω(±) = 1.0 and p = 4. Orbital parameters P y = 0.0848M, d = 12M...... 107

4.2 The orbital trajectories of the binary and a comparative radial decay of the

Lorentz-boosted and BY initial data...... 108

4.3 Comparison of the waveforms generated from the Lorentz boost and BY initial

data for the modes (`, m) = (2, 2) and (`, m) = (4, 4). Note the diﬀerence in

initial radiation content at around t = 75M...... 109

4.4 Individual black hole horizon masses (left) and coordinate radii of the horizons

(right) versus time for diﬀerent evolution functions f(α) for the lapse. Initial

4 2 lapse α0 = 2/(1+ψfull) (top row), α0 = 1/ψBL (middle row), α0 = 1/(2ψBL −1) (bottom row) ...... 110

4.5 Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper

left) and the amplitude of those waveforms (upper right). On the lower panel a

zoom-in of the amplitude oscillations for diﬀerent evolution functions f(α) for

4 the lapse. Initial lapse here is α0 = 2/(1 + ψBL)...... 111

4.6 Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper

left) and the amplitude of those waveforms (upper right). The bottom panel

shows a zoom-in of the amplitude oscillations for diﬀerent evolution functions

2 f(α) for the lapse. The initial lapse implemented here is α0 = 1/ψBL...... 112

4.7 Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper

left) and the amplitude of those waveforms (upper right). On the lower panel a

zoom-in of the amplitude oscillations for diﬀerent evolution functions f(α) for

the lapse. Initial lapse here is α0 = 1/(2ψBL − 1)...... 113

4.8 RMS norms of the violations of the Hamiltonian and three components of the

momentum constraints versus time for diﬀerent evolution functions f(α) for the

4 lapse. Initial lapse here is α0 = 2/(1 + ψfull)...... 114

x 4.9 RMS norms of the violations of the Hamiltonian and three components of the

momentum constraints versus time for diﬀerent evolution functions f(α) for the

2 lapse. Initial lapse here is α0 = 1/ψBL...... 115

4.10 RMS norms of the violations of the Hamiltonian and three components of the

momentum constraints versus time for diﬀerent evolution functions f(α) for the

lapse. Initial lapse here is α0 = 1/(2ψBL − 1)...... 116

x 4.11 Horizon mass of the boosted black hole with P /mH = ±2 and waveform after

collision for diﬀerent choices of the initial lapse and evolution f(α) = 2/α... 116

4.12 Hamiltonian and momentum constraints during the free evolution for diﬀerent

choices of the initial lapse and evolution f(α) = 2/α...... 117

x 4.13 Horizon mass (left) of the boosted black hole with P /mH = ±2 and waveform

(right) after collision for initial lapse α0 = 1/(2ψBL − 1) and diﬀerent choices

of the evolution of the lapse...... 117

4.14 Hamiltonian (H) and momentum (M x, M y, and M z) constraints during the

free evolution for initial lapse α0 = 1/(2ψBL − 1) and diﬀerent choices of the

evolution of the lapse...... 118

4.15 The center of mass speeds of the black holes for the set of Lorentz-boosted data

in this paper and Bowen-York data. The latter displays a limitation, reaching

speeds of only v < 0.9c, while the former can reach near the ultrarelativistic

regime. The thin line represent the special relativistic speed expression v = p 1 − 1/γ2...... 118

4.16 Estimated errors for each component of the computation of the radiated Energy.

“Inf Radius” is the extrapolation from the ﬁnite extraction radius robs to inﬁnity.

“Spurious” is the eﬀect of the initial radiation content of the data. “Truncation”

is an estimate of the ﬁnite diﬀerence resolution used in the simulation, and

“1−Mf/MADM vs. Erad/MADM” is a consistency measure of the radiated energy

as computed by the gravitational waveforms or the remnant mass of the ﬁnal

black hole...... 120

xi 4.17 Fits to the energy radiated at inﬁnity for the cases studied in Table 4.3. Upper

plots: Fit using the 1-parameter zero-frequency-limit like ﬁt. Lower plots: An

alternative two-parameter (A and B) ﬁt of the form (y = A exp[−B x]). Both

ﬁts use data assuming a weighting error of the points of 1% and include ﬁts to

both the energy radiated as measured by extraction of radiation (WF) or by

the remnant mass (AH)...... 122

5.1 (Left panel) The residual ADM momentum at each step in the iteration process.

(Right panel) The y-position of one of the punctures as a function of time before

the correction (red) and after (green). After only a few steps, the momentum

drift is eliminated to the levels required by our evolutions. This particular plot

shows the case of Lorentz-boosted Kerr initial data with initial parameters b = y y 6, m(+) = 1, m(−) = 0.5, χ = 0.9, and P(+) = −P(−) = 0.0848. The momentum y y parameters in the ﬁnal iteration are P(+) = 0.0869868 and P(−) = −0.0826132.. 139

5.2 The evolution of the irreducible mass and dimensionless spin of the individual

black holes with q = 2 and χ = 0.8, comparing Lorentz-boosted Kerr and

Bowen-York initial data. The top panels show the larger black hole, and the

bottom panels show the smaller black hole...... 142

5.3 The evolution of the coordinate separation between the black holes in an orbiting

binary with q = 2 and χ = 0.8, comparing Lorentz-boosted Kerr and Bowen-

York initial data...... 142

5.4 Comparison of the waveforms generated from the Lorentz-boosted Kerr and

Bowen-York initial data for the modes (`, m) = (2, 2) and (`, m) = (4, 4). Note

the diﬀerence in initial radiation content at around t = 75M...... 143

5.5 The evolution of the Hamiltonian (H) and momentum (M x, M y, and M z)

constraints for a black hole binary with q = 2 and χ = 0.8, comparing Lorentz-

boosted Kerr and Bowen-York initial data...... 143

xii 5.6 Convergence of the Hamiltonian and momentum constraint residuals with in-

creasing number of collocation points. This example shows an equal mass Kerr

black hole binary in a quasi-circular orbit with spin χ = 0.99, momentum y P(±) = ±0.0958, and separation d = 9M. The initial data superposition used

exponential attenuation with parameters ω(±) = 1.0 and p = 4...... 144 5.7 The evolution of the horizon mass and dimensionless spin of one of the individual

black holes with q = 1 and χ = 0.95...... 145

xiii xiv List of Tables

3.1 Initial data parameters for the equal-mass, collision from rest conﬁgurations.

The punctures are initially at rest and are located at (±b, 0, 0) with spins

S aligned or anti-aligned with the z direction, mass parameters mp, horizon

(Christodoulou) masses m1 = m2 = mH, total ADM mass MADM, and dimen-

2 sionless spins a/mH = S/mH, where a1 = a2 = a. The Bowen-York conﬁgura- tions are denoted by BY, and the HiSpID by HS. Finally, UU or UD denote

the direction of the two spins, either both aligned (UU), or anti-aligned (UD). . 83

3.2 The final mass, final remnant spin, and recoil velocity for each configuration. . 86

4.1 Initial data parameters for the equal-mass, boosted conﬁgurations. The punc-

tures are initially at rest and are located at (±b, 0, 0) with momentum P~ =

x y P xˆ + P yˆ, mass parameters mp, horizon (Christodoulou) masses mH, total

ADM mass MADM, and boost factor γ. The conﬁgurations are Bowen-York

(BY) or Lorentz-boosted (HS) initial data...... 108

4.2 The ﬁnal mass and spin for each conﬁguration...... 108

4.3 Initial parameters and energy radiated. For each system, the initial ADM mass

is normalized to 1...... 119

xv 5.1 Initial data parameters for spinning, orbital conﬁgurations. The punctures are

initially located at (±b, 0, 0), having mass ratio q = m(+)/m(−), with spins S

aligned or anti-aligned with the z direction, mass parameters m(±), total ADM

mass MADM, and dimensionless spins χ. The Bowen-York conﬁgurations are

denoted by BY, and the HiSpID by HS. Finally, UU or UD denote the direction

of the two spins, either both aligned (UU), or anti-aligned (UD)...... 141

5.2 The ﬁnal remnant mass, spin, and recoil velocity. The ﬁnal entry, FF95UU,

lists the remnant parameters estimated from the analytical ﬁtting function. . . 144

xvi Chapter 1

Introduction

We have a theory called general relativity; it is tremendously successful and broad in its application. It unites space and time, links geometry to gravity, and describes large scale phenomena from the big bang to black holes. The purpose of this dissertation is to explore that theory: what it looks like, how we use it to make predictions, and what it has to say about regions of extreme gravitation.

Here, we outline the historical development of the science, including some of the fundamental observations which constitute its foundation. In this chapter, scalar quantities are written as italic letters (e.g. m), three-dimensional vectors with an arrow (e.g. ~v), and four- dimensional tensors in boldface (e.g. g). We end by highlighting the goals and results of the three numerical relativity projects detailed in this report.

1.1 Gravitation

Gravitation is ubiquitous. Out of the four fundamental physical interactions, it is the only one to which all known objects are subject.

In the fourth century BCE, Aristotle [1] espoused that an object’s natural state of motion is to be at rest, and that it will only move under the inﬂuence of a cause. Some substances, the elements ﬁre and air, move towards the heavens because they have rising qualities. Other elements, earth and water, move towards the center of the universe (i.e. the center of Earth)

Chapter 1. Introduction 1 Chapter 1. Introduction because they have falling qualities. Heavy objects fall faster than light ones, at a rate proportional to their weight.

Surprisingly, twenty centuries passed before someone had the idea to actually look carefully at the motions of falling objects to decide whether or not Aristotle’s description was accurate.

Falling objects move rapidly near the surface of Earth, making it difficult to see what is happening. Galileo’s [2] ingenious solution to this problem was to roll balls down inclined planes so as to slow the rate of descent, making measurement easier. He found that objects of all different masses in fact accelerate at the same rate, and reach the bottom of the ramp at the same time. This is also true for objects falling straight down, under the influence of gravity alone. This observational fact is called the universality of free fall. In addition, Galileo was the first to scientifically verify that objects retain their velocities unless acted upon by an outside force, contrary to Aristotle. This is a property of matter called inertia.

1.1.1 The Newtonian Theory of Gravitation

Newton’s monumental contributions laid the framework for theoretical descriptions of gravitational phenomena, and mechanical motion in general [3]. Consider two objects possessing masses m and M, with separation r pointing in the direction of the unit vector ~rˆ from the former to the latter. The eﬀect of gravity on m is modeled as a force, the inverse square law

GMm ˆ F~g = ~r. (1.1.1) r2

The empirically determined universal constant G ≈ 6.67 × 10−11 m3 kg−1 s−2 regulates the overall strength of the interaction [4]. Newton’s third law states that mass M experiences precisely the same force, but pointed in the opposite direction −~rˆ. The result is that the bodies feel a force of attraction.

The total force F~ acting on an object is the vector sum of all individual forces. New- ton’s second law asserts that an object responds to a force by changing its velocity ~v—i.e. accelerating—in the direction of that force at a rate which is proportional to the magnitude

2 1.1. Gravitation 1.1. Gravitation of the force and inversely proportional to its mass

d~v F~ = . (1.1.2) dt m

A body experiencing no forces moves along straight line trajectories at a constant speed.

Restricting our considerations to gravitational forces F~ = F~g, it follows that the center of mass of all objects in the presence of M accelerate in the same manner, irrespective of their internal composition d~v GM = ~r.ˆ (1.1.3) dt r2

This is one form of the weak equivalence principle, and is a mathematical statement of the universality of free fall.

Newtonian gravitation has an alternative formulation as a field theory. Instead of calcu- lating the gravitational force as sourced by point masses, it can be given in terms of a mass density ρ. This determines the gravitational potential field φ, a quantity that fills space and has a scalar value at every location. The mass density is related to the potential by Poisson’s equation

∇2φ = 4πGρ , (1.1.4) where ∇~ φ is the gradient of φ, and taking the divergence gives the Laplace operator ∇2φ ≡

∇·~ (∇~ φ). One consequence of Eq. (1.1.4) is that a change to ρ produces an immediate reaction in φ across any distance. An object with mass m responds to the local potential by accelerating

d~v ∇~ φ ≡ ~v · ∇~ ~v = − . (1.1.5) dt m

The inverse square law model (1.1.1) and the field model (1.1.4) are mathematically equivalent. This means that they produce identical predictions for the motions of objects under the influence of gravity, and so no experiment can hope to help us decide which description is more accurate. However, at the psychological level, they present very different physical pictures.

In the former, objects are responding to a force across extended distances, which compels

Chapter 1. Introduction 3 Chapter 1. Introduction them to accelerate. In the latter, objects sense “hills and valleys” in the potential in their immediate vicinity, and accelerate in the direction of steepest descent. The ﬁeld approach has this attractive feature called locality in space; only information about what is happening in a small volume around the object is needed in order to determine its future motion.

In 1905, Einstein realized that the Galilean coordinate transformations between inertial reference frames—coordinate systems in which objects experiencing no acceleration move along straight line trajectories—are just a special case of a more general class of transformations, now named for Lorentz [5, 6, 7, 8, 9, 10]. Assuming only that inertial frames exist and that the rule for translating coordinate labels from one frame to another should not depend on location or orientation in an absolute sense, then several striking consequences immediately follow: the Lorentz transformations are linear [11], there exists a (possibly inﬁnite) universal speed c upon which all inertial observers agree, and the law governing the composition of velocities conspires to forbid relative motion with speeds exceeding c [12, 13, 14].

It is an empirical fact that c is ﬁnite and that electromagnetic waves propagate through vacuum at precisely this limit, the speed of light. This was initially recognized by Rømer circa 1676 when observations indicated that the motions of the Galilean moons proceed ahead of schedule when Jupiter is near to Earth in its orbit, and behind schedule when Jupiter is distant. This discrepancy is accounted for by the travel time of light across large intervals of space. Soon after, Huygens made the ﬁrst order of magnitude estimation of the speed of light c ≈ 220 000 000 m s−1 [15]. In modern SI units, c = 299 792 458 m s−1 exactly.1

The speed required to escape on a ballistic trajectory from the surface of a spherically symmetric mass M with radius R is

r 2GM vesc = . (1.1.6) R

In 1784, Michell [17] combined the result of setting vesc = c with Newton’s hypothesis on the corpuscular structure of light to speculate that “if there should really exist in nature any

1The SI second is deﬁned to be the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperﬁne levels of the ground state of the stationary Cs-133 atom at absolute zero. The SI meter is then measured to be the distance traversed by light in vacuum in 1/299 792 458 s [16].

4 1.1. Gravitation 1.1. Gravitation bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun . . . their light could not arrive at us.” He went on to suggest that the existence of such “dark stars” could be inferred by observing the orbital motion of a star around an otherwise invisible companion (see Sec. 1.2.2). Laplace considered the same idea a few years later [18].

1.1.2 The Theory of General Relativity

Immediately, we should be concerned by the implications of Eq. (1.1.3). It states that an object moving under the influence of gravity alone experiences acceleration. How can you tell if you are being accelerated? You feel it! It is the sensation of being pressed into your car seat as you speed away from a traffic stop, or go around a turn. It is the weighty pull in your belly during aircraft takeoff, and the jostling of touchdown. It is acutely apparent at the beginning and end of an elevator ride, and indeed we experience it every day living on the surface of

Earth. (Just stand on the nearest scale to be sure!)

In attempting to hone our intuition on the nature of gravity, we have a strong hint to which Einstein was not privy: the manned space program. Consider astronauts aboard the

International Space Station (ISS), orbiting Earth at an average altitude of 412 km, as well as hypothetical explorers in deep space, far from any appreciable concentration of matter.

Do any of them feel accelerated? No. They ﬂoat freely, and perceive no direction in space as special compared to any other—there is no preferred “down.” A ball placed at rest before them remains motionless; when pitched, it travels in a straight line at a constant speed (nearby, at least). These are exactly the qualities of an inertial frame of reference. The ISS astronauts experience no acceleration, so that according to Eq. (1.1.2) there cannot be any “force of gravity” acting upon them, nor, indeed, any of us.2 How, then, do we reconcile this with the observation that the astronauts’ extended trajectories are deﬁnitely not straight lines, but nearly elliptical orbits? The answer is curvature.

2“But I feel accelerated downward by the pull of gravity at this very moment,” I hear you cry. On the contrary, what you feel is the ground accelerating you upwards. That explains why the heaviness sensation is compounded when riding an elevator that is accelerating up, and diminished when the elevator accelerates down. If the ﬂoor were to suddenly vanish, your natural, unaccelerated state of motion—free fall—would move you towards the center of Earth. The ﬂoor has merely gotten in the way of unimpeded coasting.

Chapter 1. Introduction 5 Chapter 1. Introduction

The year 2015 marks the 100th anniversary of the discovery by Hilbert and Einstein of the relativistic field equation describing gravitational phenomena [19, 20]. Among the shortcomings of Newtonian gravity is the prediction that the gravitational influence between masses is transmitted instantaneously across any distance, which conflicts with the speed limit c imposed by special relativity. Einstein’s insight was to see that the universality of free fall permits one to completely eliminate the local influence of gravity by adopting a freely falling frame of reference. He called this his “happiest thought,” and it lead to the construction of a field theory of gravity which is completely consistent with the restrictions of special relativity, and takes on the character of Newtonian gravity in the appropriate weak field, low velocity limits.

Inspired by the principle of general covariance—the precept that the laws of physics should be expressible in a manner independent of the arbitrary coordinate labels designated by humans—the theory is written in the language of tensor equations, which have the property that they preserve their form in all coordinate systems. The scalar ﬁeld φ is generalized to the rank-two tensor ﬁeld g, called the metric. The metric is a function that takes two vectors as arguments and returns a real number; it is a generalization of the vector dot product.

It defines the angles between vectors, and is used to calculate lengths on curved surfaces. In short, the metric contains information about the shape of spacetime. Similarly, the scalar mass density ρ is generalized to the rank-two tensor field T , called the stress-energy. The stress-energy contains information on the local energy density3 and flux, the momentum density and flux, the shear, and the pressure. Mach’s principle states that inertial frames are determined by the large scale distribution of energy and momentum [21, 22], which suggested to Einstein that stress-energy is the source of spacetime curvature. The most general4 function compatible with the properties of T , comprised only of g and its first and second derivatives

3One consequence of the equivalence principle is that we can always choose a reference frame which is locally freely falling. This implies that the stress-energy does not contain a contribution in the form of “local gravitational potential energy.” 4The ﬁeld equation can be made fully general with the addition of a scalar constant Λ 1 8πG R − Rg + Λg = T . 2 c4 This term is called the cosmological constant, and is related to the expansion of the universe on large (mega- parsec) scales [23]. For compact binaries, its eﬀect can be entirely ignored. Throughout this work we adopt Λ = 0.

6 1.1. Gravitation 1.2. Motivation

(contained in R and R), is embodied by the Einstein ﬁeld equation

1 8πG R − Rg = T . (1.1.7) 2 c4

The stress-energy acts as a source for the second order diﬀerential equations describing the curvature, encoded in the metric. Bodies react to curved spacetime by moving along the straightest possible paths in curved geometry, determined by the geodesic equation

(u · ∇) u = 0 . (1.1.8)

Here u is the four-velocity tangent to the trajectory path, and the curvature information is hidden in the derivative operator ∇. This is precisely how orbits can exist even though there is no gravitational force. If there are any forces present (e.g. the electromagnetic Lorentz force), they appear as terms on the right hand side of the geodesic equation. Note the similarities between Eqs. (1.1.4) and (1.1.7), and between Eqs. (1.1.5) and (1.1.8).

1.2 Motivation

General relativity is central to our understanding of many astrophysical processes. In this work, we are concerned with the dynamics of extremely dense, compact objects. Observations give us many examples of such objects, and they are often found to be in a position of great inﬂuence over important structure formation in galactic and stellar environments.

1.2.1 Black Holes

A black hole is a region of space in which gravity is so strong that nothing, not even light, can escape. They are one of the most remarkable predictions of general relativity, although their existence was ﬁrst postulated more than a century prior. They are the consequence of following the rules of gravity to their ultimate. An isolated black hole is extraordinarily simple, and yet possesses an outstandingly rich structure. They are described entirely by only three parameters: mass M, electric charge Q, and dimensionless angular momentum χ = |S~|/M 2

Chapter 1. Introduction 7 Chapter 1. Introduction

(where S~ is the intrinsic angular momentum).

Gravitational collapse occurs when an object’s internal pressures are insuﬃcient to oppose its own gravity. Stars are supported against gravity by nuclear fusion in their cores. When the nuclear fuel is exhausted, thermal pressure decreases, and the core collapses. If the core is not too massive, then electron degeneracy pressure can halt the infall, forming a white dwarf [24, 25]. If the star is more massive, the electrons and protons fuse through inverse beta decay, resulting in the emission of neutrinos, leaving behind a neutron star [26, 27, 28, 29, 30].

If the mass of the stellar remnant exceeds ∼ 3 M , then even the neutron degeneracy pressure

30 cannot save the core from contracting into a black hole [31, 32, 33, 34]. Here, M ≈ 2×10 kg is the solar mass. The spherical perfect ﬂuid solution to the Einstein ﬁeld equation with mass 4c2 M and radius R sets an upper bound M < 9G R before the formation of a black hole becomes inevitable [35].

1.2.2 Observations of Black Hole Systems

Perhaps the strongest evidence for the existence of black holes comes from the radio source

Sagittarius A∗ in the core of the Milky Way galaxy. Infrared observations using adaptive optics have managed to peer through the thick dust near the galactic center and follow the motions of several stars [36]. A few stars orbit a common center, some so rapidly that, in the few decades during which this system has been resolved, multiple complete orbits have been directly observed. Given the size of an orbit and its period, it is possible to deduce the mass at the center of attraction. This method estimates the common center mass to be

6 6 (4.31 ± 0.38) × 10 M [37] and (4.1 ± 0.6) × 10 M [38]. Yet when we look to the location of the center of mass, there is no visible source. The star S2 (also known as S0-2) is one of the best studied. Its period about the galactic center is only 15.24 ± 0.36 yr, and periapsis brings it within 120 AU of the central object [39, 40]. The existence of the supermassive black hole is inferred by the presence of substantial mass concentrated into a small region (on the order of the size of the solar system) generating no visible signal.

The ﬁrst frame-dragging eﬀect to be discovered was Lense-Thirring precession [41, 42, 43].

8 1.2. Motivation 1.2. Motivation

Lunar laser ranging measurements show agreement with general relativity, and report detecting the de Sitter solar geodedic eﬀect to 0.7% [44]. Laser ranging to spacecraft report a 10%–

30% measurement of the frame-dragging eﬀect [45, 46]. Gravity Probe B is a highly precise gyroscope on a polar orbit around Earth, measuring the orientation of the gyroscope relative to a guide star. Over many orbits, the gyroscope’s orientation was measured to drift in the orbital plane (geodedic precession) at a rate of −6601.8±18.3 mas yr−1 and perpendicular to the plane

(frame-dragging) at a rate of −37.2 ± 7.2 mas yr−1, compared with the general relativity predictions −6606.1 mas yr−1 and −39.2 mas yr−1, respectively [47]. These all constitute direct measurements of spacetime curvature around Earth. In the strong field near a black hole horizon, these effects are greatly magnified.

In the real universe, black holes do not exist in a vacuum; they are found deep in globular clusters, orbiting a stellar companion, or surrounded by swirling plasma. Thick disk accretion models suggest that 80% of massive black holes could have spins χ > 0.8 [48]. Fully relativistic magnetohydrodynamics simulations are able to spin a black hole up to χ ∼ 0.93 [49]. Other simulations achieve χ ∼ 0.95 [50]. The x-ray binary source GRB 1915+105 has a black hole with spin χ > 0.98 [51]. Studies of quasars in the redshift range 0.4 < z < 2.1 exhibit near extremal spins χ ∼ 1 [52]. Some models of black hole accretion do not lead to large spins [53, 54, 55]. Therefore, to simulate astrophysically realistic black hole systems, it is imperative to accurately model high spins.

One of the most amazing recent discoveries is that of gravitational recoil [56, 57, 58, 59,

60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. A spinning binary radiates gravitational waves asymmetrically, resulting in a net impulse in some direction.

The result is that, after merger, the remnant black hole may experience a “kick” of up to

5000 km s−1 [78]. This far exceeds the velocity required for most supermassive black holes to escape from the cores of their host galaxies. For lower velocity recoils, it is expected that the merger remnant becomes displaced from the center of the core and oscillates in the galactic potential, until it eventually sinks back to the middle via dynamical friction in the stellar environment [79, 80, 81, 82, 83, 84, 85].

Chapter 1. Introduction 9 Chapter 1. Introduction

Over the last 9 Gyr, large galaxies experienced approximately one merger with another galaxy [86]. The universe is huge and old, and rare events occur all the time, including galactic mergers [87, 88, 89, 90]. This gives a probable scenario for the formation of compact binaries [91]. These mergers have a large impact on galactic evolution, and therefore the study of highly spinning compact binaries is of great theoretical and astrophysical interest.

1.2.3 Gravitational Waves

Accelerating objects can interact with spacetime in such a manner that waves of curvature are able to dissipate energy. These significant predictions of general relativity are called gravitational waves. They are the result of variations in time of second order and higher mass and “mass current” multipole moments. Analogously, electromagnetic waves are the result of changes in the first order and higher electric and magnetic multipole moments. The medium of gravitational waves is spacetime itself, so that the effect of a gravitational wave passing by a set of freely falling test particles is to induce tidal interactions transverse to the direction of propagation. These are waves in the sense that they carry energy and momentum, and satisfy a wave equation in the weak field regime (linearized gravity). They travel at speed c, and have two fundamental polarizations in standard general relativity.

To date, the best evidence for the existence of gravitational waves is only indirect [92,

93, 94, 95]. Several large experiments aim to detect them directly using kilometer-scale laser interferometers, which must be sensitive to changes in arm length to the order of one part in 1022 [96, 97, 98, 99]. This awesome level of precision is comparable to measuring the distance between the Sun and Saturn to within an error smaller than the diameter of an atom. These scientific and engineering collaborations have done a marvelous job meeting design specifications, and the first detections of compact binary merger signals are expected within the next few years. To find and decode the gravitational wave signal amongst the noisy background requires a theoretical understanding of radiation processes. Currently, strong efforts are being made to explore the compact object merger parameter space to produce template waveforms for the experimental searches [100, 101, 102, 103, 104, 105].

10 1.2. Motivation 1.3. Projects Herein

1.3 Projects Herein

This dissertation serves as a report on three numerical relativity projects. Chapters 3, 4, and 5 describe them in detail. In this section, we give a brief overview of each and highlight their main results. They have several goals in common, among them:

• Develop vacuum black hole binary initial data capable of representing astrophysically

relevant conﬁgurations.

• Explore the edges of black hole parameter space with large linear and angular momenta.

In particular, model the orbital inspiral and head-on collision of black hole binaries from

large initial separations through to the plunge, merger, and ringdown phases with full

numerical relativity.

• Compare the spurious initial burst of gravitational radiation produced by the new initial

data with that made by the Bowen-York solutions [106] for boosted and spinning black

holes.

• Measure nonlinear relativistic eﬀects such as frame dragging, orbital and spin precession,

energy loss via gravitational radiation, and black hole kicks.

• Generate accurate gravitational waveforms.

• Implement the initial data solver as an extension to the widely used TwoPunctures

thorn [107] in the Einstein Toolkit [108, 109] of the Cactus framework [110, 111,

112].

1.3.1 Collisions from Rest of Highly Spinning Black Hole Binaries

We solve the general relativity constraint equations to generate initial data representing a pair of Kerr black holes in a collision from rest, each with intrinsic spins up to near the extremal limit. The TwoPunctures thorn is extended to seek spectral solutions to the Hamiltonian and momentum constraints with a conformal superposition of Kerr metrics. We show that the constraint residuals converge with spectral resolution, and measure the solutions’ ADM masses,

Chapter 1. Introduction 11 Chapter 1. Introduction linear momenta, and spins. These data are evolved using the moving punctures approach with the BSSNOK [113, 114, 115] and CCZ4 [116, 117] formalisms in the LaZeV thorn [118]. We track the punctures and horizons through the plunge, merger, and ringdown phases.

The gravitational waveforms extracted from these evolutions are compared with those of the Bowen-York solutions possessing comparable spins χ = 0.9 and no initial momentum. We

ﬁnd that the spurious burst or gravitational radiation resulting from the new initial data is reduced by a factor of ∼ 10 when compared to Bowen-York. The physical gravitational signal originating from the black hole dynamics matches closely between the two methods.

The purpose of conformally Kerr initial data is to break the Bowen-York spin limit, and simulate black hole binaries with χ & 0.93. Using these new solutions, we evolve stably a pair of black holes each with χ = 0.99. We observe strong ﬁeld spin eﬀects, such as frame dragging and black hole kicks.

1.3.2 Quasi-circular Orbits and High Speed Collisions Without Spin

We solve the general relativity constraint equations to generate initial data representing a pair of Schwarzschild black holes, each with arbitrary momentum. To this end, we modify the

TwoPunctures thorn to ﬁnd spectral solutions to the Hamiltonian and momentum constraints with a conformally Lorentz-boosted Schwarzschild superposition ansatz. We demonstrate the convergence of constraint residuals with increasing collocation point resolution, and measure the solutions’ ADM masses, linear momenta, and spins. They are evolved using the

BSSNOK and CCZ4 formalisms in the LaZeV thorn. We study the properties of these data in two main conﬁguration classes: quasi-circular orbits and head-on collisions.

The orbiting binary is of considerable interest in astrophysics. We compare the resulting gravitational waveforms with those produced using Bowen-York black holes with similar parameters. The magnitude of the initial burst of spurious radiation appearing from the new initial data is reduced by a factor of ∼ 3 when compared to Bowen-York. We track the motions of the punctures and their horizons through the quasi-circular inspiral, merger, and ringdown phases.

12 1.3. Projects Herein 1.3. Projects Herein

We study numerically the high energy head-on collision of nonspinning equal mass black holes to estimate the maximum gravitational radiation emitted by these systems. Our simulations include improvements in the full numerical evolutions and computation of waveforms at inﬁnity. We make use of new initial data with notably reduced spurious radiation content, allowing for initial speeds nearing the speed of light, i.e. v ∼ 0.99c. We thus estimate the maximum radiated energy from head-on collisions to be Emax/MADM = 0.13 ± 0.01. This value diﬀers from the second order perturbative (0.164) and zero-frequency-limit (0.17) analytic computations, but is close to those obtained by thermodynamic arguments (0.134) and to previous numerical estimates (0.14 ± 0.03).

1.3.3 Highly Spinning Black Hole Binaries in Quasi-circular Orbits

Astrophysics motivates the ultimate goal of simulating spinning black holes on quasi-circular orbits. We solve the general relativity constraint equations to generate data representing a pair of Kerr black holes with arbitrary initial momenta using a Lorentz transformation. We decompose the physical ﬁelds with the conformal transverse-traceless formalism. The freely chosen background ansatz is written as an attenuated superposition of conformal boosted

Kerr metric and extrinsic curvature terms. Our extended TwoPunctures thorn solves all four constraint equations using pseudospectral methods on a compactiﬁed collocation point domain. We demonstrate that the constraint residuals converge with increasing collocation point resolution, and that the resulting initial data has the desired physical attributes. They are evolved with the moving punctures approach using the BSSNOK and CCZ4 formalisms in the LaZeV thorn.

We compare the new initial data with the well known Bowen-York solutions using quasi- circular configurations with χ = 0.8. We find that the conformally curved ansatz results in a reduction of the initial spurious gravitational wave burst by an order of magnitude compared to the conformally flat solution. We show that the masses and spins of the black holes hold well to their desired values throughout the inspiral. We surpass the Bowen-York limit with an evolution of a black hole binary possessing χ = 0.95 in quasi-circular orbit.

Chapter 1. Introduction 13 Chapter 1. Introduction

1.3.4 Organization

The remainder of this dissertation is organized as follows: in Chapter 2 we review the formalism of general relativity, the methods used to ﬁnd black hole binary solutions to the initial data problem in numerical relativity, and the computer code implementation of these algorithms;

Chapter 3 describes initial data representing the collision from rest of Kerr black holes with intrinsic spins comparable to, and surpassing, those obtainable by the Bowen-York solutions;

Chapter 4 details the construction of Lorentz-boosted Schwarzschild initial data, used for studying quasi-circular orbital conﬁgurations and high energy head-on collisions; Chapter 5 combines all of this with the goal of simulating highly spinning black hole binaries in generic quasi-circular orbits; ﬁnally, Chapter 6 summarizes the results of this work and discusses future research directions.

14 1.3. Projects Herein Chapter 2

Numerical Techniques in Relativity

For all but the simplest, symmetric systems, finding analytical solutions to the Einstein field equation is hopeless. In order to investigate more generic configurations, we are forced to resort to numerical approximations.

In this chapter, we describe methods for extracting physical predictions from the theory of general relativity. We begin by outlining the mathematical formalism of the fully covariant description of curved spacetime. This is followed by a rewriting of the theory into an evolution formulation with constrained initial conditions. Finally, numerical methods and their computer code implementations are discussed.

2.1 General Relativity Formalism

This section is distilled from the invaluable textbook accounts [119, 120, 121, 122, 123, 124].

General relativity is the application of diﬀerential geometry to the study of gravitational phenomena. Linear combinations appear everywhere in the theory, and so it is convenient to drop the sigma sum notation in favor of the Einstein summation convention. Whenever the same index appears as both a superscript and a subscript in an equation, then the following shorthand is taken to be understood

n a X a 0 1 n U Va ≡ U Va = U V0 + U V1 + ... + U Vn . (2.1.1) a=0

Chapter 2. Numerical Techniques in Relativity 15 Chapter 2. Numerical Techniques in Relativity

This operation is called a contraction; in matrix notation, it is pictured as row-column multi- plication. Greek indices are used for spacetime objects (µ = 0,..., 3) and Latin indices from the middle of the alphabet for spatial objects (i = 1,..., 3). Latin indices from the beginning of the alphabet are generic. When the notation is obvious, we sometimes use speciﬁc coordi-

23 yz nate symbols as index labels. For example, in (t, x, y, z) coordinates A0 ≡ At, B ≡ B , and so on.

2.1.1 Geometric Units

Special relativity and the Minkowski model of spacetime put space and time on an equal theoretical footing [125]. It is an unnecessary burden to measure diﬀerent dimensions of the same geometric object—spacetime—using disparate units, e.g. SI seconds and meters. In geometric units, time intervals are interpreted as the distance that is traveled by light in that amount of time. For instance, working in units of seconds and light-seconds (the distance crossed by light in one second) allows us to set the conversion factor c = 1 wherever it appears in equations. This freedom also allows us to set G = 1. These are the standard choices of relativists, and will be used henceforth. With this convention, both distances and durations

30 have units of mass. For a concrete example, take the solar mass [126] 1 M ≈ 2 × 10 kg ≈

1.5 km ≈ 5 µs.

To recover SI expressions from those which have been geometrized, one simply inserts factors of c and G until the desired units are obtained.

2.1.2 Manifolds

We approach the geometry of spacetime with an open mind. The structure of spacetime could be flat like the surface of a table, positively curved like a sphere, negatively curved like a horse’s saddle, or something else entirely. The equivalence principle states that experiments performed in sufficiently small spacetime volumes obey the laws of special relativity, flat spacetime.

A manifold M is a collection of n-dimensional open balls that are connected by smooth

n maps and can be locally mapped to R . At each point p ∈ M , there exists a ﬂat n-dimensional

16 2.1. General Relativity Formalism 2.1. General Relativity Formalism

vector space Tp that represents the local linear approximation to the manifold. The usual results of linear algebra apply in Tp, and we are free to manipulate and compare geometric objects at p. Spacetime appears to have four degrees of coordinate freedom, so a four-dimensional manifold is the natural structure upon which to build general relativity.

The metric g is a linear, symmetric, nondegenerate function that maps two vectors to a scalar

g : Tp × Tp → R . (2.1.2)

For all vectors u, v, w ∈ Tp and scalars a, b ∈ R, linearity means g(au + bv, w) = ag(u, w) + bg(v, w). Symmetry means g(u, v) = g(v, u). Nondegeneracy means that if g(u, v) = 0 for all u ∈ Tp, then v = 0.A positive definite metric satisfies g(u, u) ≥ 0, and attains equality if and only if u = 0.A Riemannian manifold is a one that is equipped with a positive definite metric in each tangent space. If g is not positive definite, then the manifold is referred to as pesudo-Riemannian. A metric that is not positive definite divides vectors into three categories based on the signs of their norms

g(u, u) < 0 =⇒ u is “timelike,”

g(u, u) = 0 =⇒ u is “lightlike/null,” (2.1.3)

g(u, u) > 0 =⇒ u is “spacelike.”

In special relativity, massive objects move along trajectories with timelike tangent vectors, and massless objects (e.g., electromagnetic waves and gravitational waves) move along trajectories with lightlike tangent vectors. No known objects move along trajectories with spacelike tangent vectors, but spacelike surfaces are used in the 3 + 1 decomposition.

2.1.3 Tensors

In order to perform concrete calculations, equations are expressed in terms of coordinates.

We are free to arbitrarily designate an event in spacetime with the labels xµ = (t, x, y, z), µ representing the four degrees of freedom. We could just as well call that same event x0 =

Chapter 2. Numerical Techniques in Relativity 17 Chapter 2. Numerical Techniques in Relativity

µ µ (t0, x0, y0, z0). If we can write one set of coordinates as functions of the others, x0 = x0 (xν), then there is a rule for translating from one set of labels to the other called a coordinate µ transformation with an associated Jacobian transformation matrix ∂x0 /∂xν. The inverse of the Jacobian matrix transforms coordinates in the opposite direction.

We are interested in describing geometric objects, quantities that exist in their own right without regard to coordinate labels. For example, we can imagine a three-vector as an arrow in space with a magnitude and orientation, without explicit reference to components. To write vectors in terms of components, we choose a set of linearly independent vectors ~ea which span the vector space, called a basis [127]. Here the index a labels the individual basis vectors, not vector components. Now the vector can be written in terms of components va as the linear

a ~0 0a~0 combination ~v = v ~ea. In some other coordinate system, with basis e a, we write ~v = v e a.

The component values will transform from one basis to another according to

0a a ∂x va → v0 = vb , (2.1.4) ∂xb but the quantity ~v remains unchanged because the basis vectors transform in exactly the opposite manner ∂xb ~0 ~ea → e a = ~eb . (2.1.5) ∂x0a

These Jacobian matrices are inverses of each other, satisfying

∂xa ∂x0c ∂xc ∂x0a = = δa , (2.1.6) ∂x0c ∂xb ∂x0b ∂xc b where the Kroneker delta (identity matrix)

 1 if a = b a  δb = (2.1.7)  0 if a 6= b has the same components in every coordinate system.

a a a The dual basis e is deﬁned by its operation on the vector basis e (~eb) = δb . The duality e e 18 2.1. General Relativity Formalism 2.1. General Relativity Formalism

b b ∗ relation implies ~ea(e ) = δa. Generic dual vectors w in the dual vector space Tp are expanded in e e a components w = wae . Dual two-vectors are visualized as contour plots, where the magnitude e e is related to how densely the contours are packed. The components of a dual vector transform by b 0 ∂x wa → w a = wb , (2.1.8) ∂x0a and the dual basis by ∂x0a ea → e0a = eb . (2.1.9) ∂xb e e e a 0 0a Thus w = wae = w ae . These transformation properties of the vector and dual vector are e e e the primary strengths of a more general class of geometric objects called tensors.

m A tensor S of rank n is expanded in a basis as

S = Sa1···am~e ⊗ · · · ⊗ ~e ⊗ eb1 ⊗ · · · ⊗ ebn . b1···bn a1 am (2.1.10) e e

m p m+p The tensor product ⊗ combines tensors of rank n and q into a tensor of rank n+q whose m components are the direct product of the input components. The generic rank n components transform in the expected manner

0a1 0am d1 dn a1···am 0a1···am ∂x ∂x ∂x ∂x c1···cm S → S b ···b = ··· ··· S (2.1.11) b1···bn 1 n ∂xc1 ∂xcm ∂x0b1 ∂x0bn d1···dn

0 1 0 A scalar function is a rank 0 tensor. Vectors and dual vectors are rank 0 and rank 1 tensors, respectively.

Tensor index symmetrization, denoted by parentheses, is the average over all permutations of the indices. For example

1 S = (Sabcd + Sacdb + Sadbc + Sacbd + Sabdc + Sadcb) . (2.1.12) a(bcd) 3!

Tensor index antisymmetrization, denoted by square brackets, is the average over all permu-

Chapter 2. Numerical Techniques in Relativity 19 Chapter 2. Numerical Techniques in Relativity tations of the indices, with odd permutations carrying a minus sign. For example

1 S = (Sabcd + Sacdb + Sadbc − Sacbd − Sabdc − Sadcb) . (2.1.13) a[bcd] 3!

0 One of the most important objects in differential geometry is the rank 2 metric tensor a b g = gabe ⊗ e . Given a finite displacement ∆x, the metric is used to define the invariant line e e element

(∆s)2 = g(∆x, ∆x) . (2.1.14)

In the limit of inﬁnitesimal displacements, the line element has components

2 a b ds = gab dx dx . (2.1.15)

This is a generalization to the Pythagorean theorem, and takes on the familiar form when gab = δab. The line element is an invariant quantity upon which all observers agree, regardless of their chosen coordinates and states of motion. The metric also acts as a bijective map from

∗ Tp to Tp . This transforms components according to

a b v → va = gabv , (2.1.16)

a ab va → v = g vb , (2.1.17) where the metric inverse gab is deﬁned by

cb b gacg = δa . (2.1.18)

The metric deﬁnes the inner (dot) product between vectors

a b a a u · v = g(u, v) = gabu v = u va = uav . (2.1.19)

There is a special class of objects called diﬀerential forms. A diﬀerential p-form α is a

20 2.1. General Relativity Formalism 2.1. General Relativity Formalism

0 rank p tensor that is entirely antisymmetric

αa1···ap = α[a1···ap] . (2.1.20)

Since all differential forms have only covariant indices, we are permitted to drop the index notation when no confusion will arise. Differential forms are useful because they can be differentiated and integrated without reference to a metric. A special antisymmetric operation, the wedge product ∧, maps a p-form and a q-form to a (p + q)-form according to the rule

(p + q)! (α ∧ β)a ···a = α β . (2.1.21) 1 p+q p!q! [a1···ap ap+1···ap+q]

On an n-dimensional manifold, of particular utility is the n-form , called the Levi-Civita tensor p a1···an = |g|˜a1···an , (2.1.22) where g = det(gab) and the Levi-Civita symbol is coordinate independent, totally antisymmetric, and deﬁned by

  +1 if a1 ··· an is an even permutation of 01 ··· (n − 1),   ˜a1···an = −1 if a1 ··· an is an odd permutation of 01 ··· (n − 1), (2.1.23)    0 otherwise.

The Levi-Civita tensor fulfills three important roles. First, M is said to be orientable if there exists a continuous, nowhere vanishing n-form field. Second, it acts as the natural volume element for integration. Over some subset of the manifold U ⊆ M , in a coordinate basis, volume integration of a scalar function f is defined as

Z Z p f = f |g| dnx , (2.1.24) U U where the right hand side is evaluated in the normal way as a standard Riemann integral in

Chapter 2. Numerical Techniques in Relativity 21 Chapter 2. Numerical Techniques in Relativity

n R . Third, it deﬁnes the Hodge duality between p-forms and (n − p)-forms by a map called the Hodge star 1 b1···bp (∗α)a ···a = a ···a αb ···b . (2.1.25) 1 n−p p! 1 n−p 1 p

Note that the Hodge star does depend on the metric, which is used to raise and lower indices on the Levi-Civita tensor. This map is a duality in the sense that repeating its operation returns a diﬀerential form to its original rank (up to a minus sign)

∗ ∗α = (−1)s+p(n−p)α , (2.1.26) where s is the number of minus signs in the metric signature. The signiﬁcance of the Hodge duality can be understood in three-dimensional Euclidean space where scalars are dual to volume elements and one-forms are dual to two-forms. In particular, it relates to the fact that a two-dimensional planar hypersurface can be characterized by either a vector normal to its surface or by an oriented area taken as the wedge product of two linearly independent (dual) vectors tangent to the plane. In three-dimensional Euclidean space only, the Hodge dual of the wedge product of two one-forms ui and vi gives another one-form

jk ∗ (u ∧ v)i = i ujvk . (2.1.27)

One-forms are equivalent to vectors in Euclidean space, and this special operator is identiﬁed as the cross product.

The Minkowski metric in Cartesian coordinates

  −1 0 0 0      0 1 0 0   gµν = ηµν ≡   (2.1.28)    0 0 1 0   0 0 0 1 describes the geometry of special relativity. This metric is said to have signature (−, +, +, +), which means that the inner product that it deﬁnes is not positive deﬁnite. The spacetime is

22 2.1. General Relativity Formalism 2.1. General Relativity Formalism divided into regions according to four-velocity norms (2.1.3), which results in the light cone structure. The Minkowski spacetime invariant line element is

ds2 = −dt2 + dx2 + dy2 + dz2 . (2.1.29)

A convenient result of the tensor transformation properties is that we can write any tensor equation without showing the basis explicitly. For example, the Einstein ﬁeld equation (1.1.7) has components 1 Rµν − Rgµν = 8πTµν (2.1.30) 2 and is understood to be in the eµ ⊗ eν basis. Similarly, the geodesic equation (1.1.8) has e e components µ ν u ∇µu = 0 (2.1.31) in the ~eν basis. As tensor equations, these are valid in any coordinate system.

2.1.4 Diﬀerentiation

Unsurprisingly, differentiation is an important topic in differential geometry. When studying curved spaces and dynamics, we are interested how things change in space and time. There are many different ways to define a derivative operator; here we discuss four of the most important for our purposes.

2.1.4.1 The Ordinary Derivative

The ordinary directional derivative is the one about which we learn in introductory calculus

b courses. The eﬀect of this operation is to evaluate a tensor ﬁeld, say Sa, at two nearby points xa and xa + va, and then take the limit → 0. We use the notation

b d d b d c b c ∂ h bi Sa(x + v ) − Sa(x ) v ∂cSa ≡ v Sa = lim (2.1.32) ∂xc →0

Chapter 2. Numerical Techniques in Relativity 23 Chapter 2. Numerical Techniques in Relativity

a b a to represent the derivative with respect to x on the tensor ﬁeld Sa in the direction of v .

We call ∂a the ordinary derivative, and its action on scalar functions ∂af is the gradient. In curved geometry, comparing a geometric object at two different points on the manifold is not well defined. Since the Jacobian transformation matrix will have different values at xa than at xa + va, it is important to remember that this object does not transform like a tensor, except in the case of the ordinary derivative acting on a scalar function. The partial derivative

d d is commutative ∂a∂bSc = ∂b∂aSc .

2.1.4.2 The Covariant Derivative

We seek a derivative operator that transforms like a tensor. It is a linear and Leibniz map m m from tensors of rank n to rank n+1 . For a scalar function f, the covariant derivative ∇a b reduces to the ordinary derivative ∇af = ∂af. Connection coeﬃcients Γac are introduced into the covariant derivative of a vector vb to compensate for the nontensorial nature of the ordinary derivative b b b c ∇av = ∂av + Γacv . (2.1.33)

1 The connection is not a tensor, but the diﬀerence of two connections is. For a rank 1 tensor

c c c d d c ∇aSb = ∂aSb + ΓadSb − ΓabSd . (2.1.34)

This is generalized to tensors of arbitrary rank by adding a connection term for each index of the input tensor. We require that the covariant derivative obeys ∇a∇bf = ∇b∇af, which

c c implies that the connection is torsion free Γab = Γba. We ﬁx the values of the connection by demanding that the covariant derivative be compatible with the metric, meaning ∇agbc = 0.

These are called Christoﬀel symbols and are given by

c 1 cd Γ = g (∂agbd + ∂bgad − ∂dgab) . (2.1.35) ab 2

When moving a geometric quantity along a path, it is often useful to restrict its motion such that it remains parallel to itself at each step. A tensor is parallel transported along a

24 2.1. General Relativity Formalism 2.1. General Relativity Formalism curve with tangent va if a c v ∇aSb = 0 . (2.1.36)

b In words, this equation expresses the constraint that Sa does not change in the direction of va. Use of the covariant derivative here allows this idea to be generalized to curved manifolds.

Parallel transport is a key ingredient in one of the deﬁnitions of curvature. Metric compatibility ensures that the inner product between vectors is preserved during parallel transport.

2.1.4.3 The Lie Derivative

The covariant derivative requires a manifold with a connection. It is possible to construct a derivative that utilizes less structure. The Lie derivative measures how a tensor changes as it is moved along the flow defined by a vector field. It is a coordinate invariant map from tensors m m of rank n to rank n .

Consider a vector field va on the manifold, with integral curves xa(λ) defined by integrating the ordinary differential equations

dxa = va xb(λ) , (2.1.37) dλ where λ is a parameter along the curves. The curves xa(λ) form a family called a congru-

b a ence. We would like to deﬁne the change in a tensor ﬁeld, say Sa, using v . To do this, b c 0b 0c a Sa(x ) is transformed to S a(x ) by a small displacement along v with the active coordinate transformation a xa → x0 = xa + va . (2.1.38)

The Lie derivative is then deﬁned as

" b 0c 0b 0c # b Sa(x ) − S a(x ) LvSa = lim . (2.1.39) →0

b 0c c By a Taylor expanding Sa(x ) about x and taking the limit, we obtain a component expression

Chapter 2. Numerical Techniques in Relativity 25 Chapter 2. Numerical Techniques in Relativity for the Lie derivative b c b c b b c LvSa = v ∂cSa + Sa∂cv − Sc ∂av . (2.1.40)

To generalize to tensors of arbitrary rank, simply append terms involving the derivative of va for each index of the input tensor according to the pattern above. By the symmetry of the connection coeﬃcients, we are free to promote the ordinary derivatives to covariant derivatives

b c b c b b c LvSa = v ∇cSa + Sa∇cv − Sc ∇av . (2.1.41)

For a scalar function f, the Lie derivative reduces to the directional derivative

a Lvf = v ∂af . (2.1.42)

The action of the Lie derivative on a vector is naturally expressed in terms of the commutator

a a b a b a Lvw = [v, w] ≡ v ∂bw − w ∂bv . (2.1.43)

A useful consequence of the linearity of the Lie derivative is that, for a vector va = ua+fwa,

b b b LvSa = LuSa + fLwSa . (2.1.44)

b a A tensor Sa is said to be Lie dragged along v if

b LvSa = 0 .

This operation is handy to use on vectors, where it has a simple geometric interpretation: if

a a a a x (λ) are integral curves of v , then Lvw = 0 implies that w connects points of equal λ along the congruence.

a a We are free to choose coordinates in which the basis vector e0 is aligned with v and all of a a a the other coordinates are constant along v . Then we can set v = δ0 and the Lie derivative

26 2.1. General Relativity Formalism 2.1. General Relativity Formalism reduces to the ordinary partial derivative

b b LvSa = ∂0Sa . (2.1.45)

An important application of the Lie derivative is its action on the metric tensor

c c c Lvgab = v ∇cgab + gcb∇av + gca∇bv . (2.1.46)

If ∇a is metric compatible, then

Lvgab = ∇avb + ∇bva . (2.1.47)

A vector ξa is called a Killing vector if it satisﬁes

Lξgab = 0 . (2.1.48)

From the above, we see that ξa is a solution to Killing’s equation

∇aξb + ∇bξa = 0 . (2.1.49)

The Killing vector represents an isometry whereby the metric components are unchanged along the ξa direction. If the metric can be written in coordinates such that one of the coordinate labels does not appear in the metric components, then there is a Killing vector associated with that coordinate. The two most common examples follow from spacetimes which are time independent and axisymmetric. The former has a Killing vector that is timelike, and the latter has a Killing vector that is spacelike with closed circular orbits (about the symmetry axis).

If a metric possesses a Killing vector that is timelike near inﬁnity, then it is called stationary.

If, in addition, the timelike Killing vector is orthogonal to a family of spacelike hypersurfaces, then the metric is said to be static. Intuitively, a stationary spacetime is one in which the same thing is happening at every instant; for example, consider a top which is spinning at a

Chapter 2. Numerical Techniques in Relativity 27 Chapter 2. Numerical Techniques in Relativity constant rate, at a ﬁxed location. A static spacetime is one in which nothing is happening at all.

Killing vectors imply the existence of conserved quantities. Along a geodesic with tangent

a a vector u , the quantity ξ ua is unchanged

a b u ∇a(ξ ub) = 0 . (2.1.50)

This is a consequence of a combination of the geodesic equation and the antisymmetry of

Eq. (2.1.49).

2.1.4.4 The Exterior Derivative

There is a special operation called the exterior derivative which maps diﬀerential p-forms to

(p + 1)-forms

(dα)ab1···bp = (p + 1)∂[aαb1···bp] , (2.1.51) where ∂a is the ordinary derivative. Unlike the ordinary derivative, the exterior derivative is tensorial. The exterior derivative makes no reference to a metric, and so it requires less structure on the manifold than the covariant derivative. One immediate consequence of this deﬁnition and the commutivity of partial derivatives is the Poincaré lemma

d2α = 0 . (2.1.52)

A p-form is called closed if dα = 0, and exact if there exists a (p−1)-form β such that α = dβ.

A closed differential form is locally exact. However, in general, there does not exist a smooth differential form that satisfies the exact condition globally [128, 129].

The exterior derivative obeys a Leibniz-like rule

d(α ∧ β) = (dα) ∧ β + (−1)pα ∧ (dβ) (2.1.53)

for p- and q-forms α and β. Acting on scalar functions, it reduces to the gradient (df)a = ∂af.

28 2.1. General Relativity Formalism 2.1. General Relativity Formalism

Suppose that there exists an n-dimensional region U with a boundary ∂U , and an (n−1)- form α on U . Then dα is an n-form which can be integrated on U , and α can be integrated on ∂U . These two integrals are related by one of the most powerful results in diﬀerential geometry, Stokes’ theorem Z Z dα = α . (2.1.54) U ∂U

This is the generalization of the fundamental theorem of calculus, the divergence theorem,

Green’s theorem, and the Kelvin-Stokes theorem.

2.1.5 Curvature

We wish to describe the curvature of the manifold. Consider a vector parallel transported around a small closed loop path. If, at the end, the vector is pointing in the same direction as at the beginning, then we say that there is no curvature. This is called a ﬂat geometry, with the picture of a planar surface in mind. If, after marching around the loop, the vector ends up in a diﬀerent orientation, then the geometry is curved. If the vector ends up rotated in the same sense as its motion around the loop, then there is positive curvature; if the vector ends up rotated in the opposite sense as its motion around the loop, then there is negative curvature. The standard two-dimensional examples of positive and negative curvature are the sphere and the hyperboloid of one sheet, respectively.

d The Riemann tensor Rabc is deﬁned by the action of parallel transporting a dual vector around an inﬁnitesimal square loop using the metric compatible covariant derivative

d (∇a∇b − ∇b∇a)wc = Rabc wd . (2.1.55)

Chapter 2. Numerical Techniques in Relativity 29 Chapter 2. Numerical Techniques in Relativity

The Riemann tensor satisﬁes four key symmetry properties

d d Rabc = −Rbac , (2.1.56)

Rabcd = −Rabdc , (2.1.57)

d R[abc] = 0 , (2.1.58)

e ∇[aRbc]d = 0 . (2.1.59)

1 The last property is called the Bianchi identity. As a rank 3 tensor, it is naively expected to have n4 independent components on an n-dimensional manifold. However, the symmetries

1 2 2 reduce this to 12 n (n − 1). In four-dimensional spacetime, it has twenty independent components. A convenient expression for the Riemann tensor in terms of the connection coeﬃcients is given by d d d e d e d Rabc = ∂bΓac − ∂aΓbc + ΓacΓeb − ΓbcΓea . (2.1.60)

Of central importance is a contraction of the Riemann tensor, called the Ricci tensor

c Rab = Racb . (2.1.61)

It turns out that Rab = R(ab). Another crucial quantity is the contraction of the Ricci tensor, called the scalar curvature a R = Ra . (2.1.62)

When R > 0 at a point, the volume of a small ball around that point is less than that of a ball with the same radius in Euclidean space. An example of this can be seen by considering a great circle on the equator of a two-sphere with radius r, for which the line element in 2 2 2 2 2 2 spherical coordinates (θ, ϕ) is dΩ = r dθ + sin (θ) dϕ and R = r2 > 0 everywhere. The radius ρ of the equatorial circle, measured on the surface of the sphere, is the length of a curve 1 connecting one of the poles to the equator along a meridian, ρ = 2 πr. The circumference of 2 the equator is C = 2πr. This implies C = 4ρ, which is smaller than its R counterpart with Euclidean circumference C = 2πρ. The opposite is true when R < 0.

30 2.1. General Relativity Formalism 2.1. General Relativity Formalism

2.1.6 The Einstein Field Equation

The Ricci tensor (2.1.61) and scalar curvature (2.1.62) are combined to form the Einstein tensor 1 Gab = Rab − Rgab (2.1.63) 2

ab with several important properties; it is symmetric Gab = Gba and divergence free ∇aG = 0.

The source term for this diﬀerential equation is the stress-energy tensor Tab, forming the Einstein ﬁeld equation 1 Rab − Rgab = 8πTab . (2.1.64) 2

Since the left hand side is a function only of gab and its derivatives, it satisﬁes the principle of general covariance. The stress-energy tensor is symmetric and divergence free, just like the

Einstein tensor.

A vacuum spacetime is deﬁned by Tab = 0. Taking the trace of Eq. (2.1.64) yields R = 0 so that the vacuum ﬁeld equation reduces to

Rab = 0 . (2.1.65)

2.1.7 Geodesics

With a description of the spacetime curvature (2.1.64) at hand, we wish to know how objects move through the geometry. The nature of manifolds is that they appear ﬂat, locally. This means that we expect objects experiencing no forces to move along the straightest possible lines. We can characterize these trajectories as those which parallel transport their own tangent vectors ua. This type of curve is called a geodesic, and it satisﬁes

a b u ∇au = 0 . (2.1.66)

Points on the geodesic are labeled as xa(λ), where λ is a parameter along the curve. The

a dxa velocity is deﬁned by u = dλ . This turns Eq. (2.1.66) into a diﬀerential equation for the

Chapter 2. Numerical Techniques in Relativity 31 Chapter 2. Numerical Techniques in Relativity curve d2xa dxb dxc + Γa = 0 . (2.1.67) dλ2 bc dλ dλ

Geodesics are stationary curves. This means that they minimize or maximize length between two points on a manifold. The length ` of a curve between points p and q is deﬁned by the invariant line element (2.1.15)

r Z q Z q dxa dxb ` = ds = gab dλ . (2.1.68) p p dλ dλ

Taking variations of the integral and ﬁnding stationary solutions δ` = 0 shows that geodesics represent curves of maximum length in time (for timelike curves), zero length in lightlike directions, and minimum length in space (for spacelike curves).

2.1.8 Causal Structure and Asymptotic Flatness

Due to the finite value of c, spacetime has a causal structure. This is because information can not propagate faster than c, and so there are some regions of the manifold which cannot be influenced by others. The causal future J +(p) of a point p ∈ M is the set of all q ∈ M such that there exists a future-directed causal curve passing through both p and q. A causal curve is one that is nowhere spacelike (i.e. it is timelike or lightlike). The causal past J −(p) is defined in the same way, but with past-directed causal curves. Intuitively, J −(p) is the set of all points from which a signal could possibly emit and reach p. For any subset S ⊂ M , we define [ J ±(S) = J ±(p) . (2.1.69) p∈S

The boundary ∂J ±(S) of J ±(S) is a null hypersurface.

A hypersurface S is called a Cauchy surface if every causal curve in M passes through

S exactly once. This means that S contains all of the information required to predict the future and retrodict the past. It represents complete knowledge of a single “instant of time.”

A spacetime containing a Cauchy surface is called globally hyperbolic.

We are typically interested in studying the properties of isolated systems. Although no

32 2.1. General Relativity Formalism 2.1. General Relativity Formalism real life system is ever truly separated from the rest of the universe, it stands to reason that if the influence of distant perturbers is small, then the isolated model will serve as a good approximation. The goal is to be able to perform tensor calculations “at infinity” in order to extract information from the spacetime, such as gravitational waveforms. This is motivated by the fact that Earth is so far away from astrophysical gravitational wave sources that we are effectively in the asymptotically flat region.

In Cartesian-like coordinates (t, x, y, z), the metric is split into a Minkowski background plus a correction hab

gab = ηab + hab . (2.1.70)

An asymptotically ﬂat spacetime is one in which the curvature vanishes at large distances p r = x2 + y2 + z2, so that its metric components approach the Minkowski geometry

1 lim hab = O , (2.1.71) r→∞ r 1 lim ∂chab = O , (2.1.72) r→∞ r2 1 lim ∂c∂dhab = O . (2.1.73) r→∞ r3

Since this prescription is coordinate dependent, it is often less than desirable. A coordinate independent method for studying asymptotic ﬂatness comes about through a conformal transformation of the metric 2 g¯ab = Ω gab (2.1.74) which maps the physical spacetime manifold M onto an unphysical, conformal manifold M¯.

The conformal factor Ω is an everywhere positive, smooth function. Points corresponding to Ω = 0 are added to the manifold, which allows us to map distant infinity onto a finite location. There, we can perform normal tensor evaluations without applying limits. The points consisting of the boundary of the manifold are called conformal infinity.

By this conformal rescaling, the manifold is extended such that M ⊂ M¯. Conformal infinity is divided into five regions: past timelike infinity, past null infinity, spatial infinity,

Chapter 2. Numerical Techniques in Relativity 33 Chapter 2. Numerical Techniques in Relativity future null inﬁnity, and future timelike inﬁnity. These are designated by the symbols i−, I −, i0, I +, and i+, respectively.1 In Minkowski spacetime, timelike curves begin on i− and end on i+, lightlike curves begin on I − and end on I +, and spacelike curves begin and end on i0.

Using compactiﬁed coordinates, the entire spacetime can be represented by a ﬁnite picture, called a Penrose diagram. An example of such a diagram for Minkowski spacetime is shown in Fig. 2.1.

I +

dt = 0 0 = 0 i r

= 0

r d I −

i−

Figure 2.1: A Penrose diagram representation of Minkowski spacetime in compactified spherical coordinates. The locations of conformal infinity are labeled. Time runs vertically, from the infinite past at i− and I −, to the infinite future at i+ and I +. Space extends horizontally from the origin (the vertical line labeled r = 0) out to spatial infinity at i0. A point on the diagram represents a two-sphere at that particular time and radius. Light rays move along 45◦ lines, parallel to I − and I +. Generic surfaces of constant time (dt = 0) and constant radius (dr = 0) are shown.

2.1.9 Black Holes

A spacetime is said to contain a black hole if M 6⊂ J −(I +). Intuitively, this means that there are regions of space from which light rays cannot escape to inﬁnity. The black hole region B

1The character I is a script Latin “I,” and is read as “scri.”

34 2.1. General Relativity Formalism 2.1. General Relativity Formalism is deﬁned to be B = M − J −(I +). The boundary of B in M , H = ∂J −(I +) ∩ M , is called the event horizon. Black holes, therefore, are a global structure of spacetime since knowledge of the inﬁnite future is required in order to determine their existence.

An important feature of black hole spacetimes is the presence of physical singularities. Sin- gularities are characterized as places where geodesics end in a ﬁnite amount of aﬃne parameter.

Physically, these are locations where tides grow without bound. The cosmic censorship con- jecture states that black hole singularities are always hidden from the outside universe behind a horizon.

Now we describe three of the most important black hole solutions.

2.1.9.1 The Schwarzschild Solution

Deep in the trenches at the front of World War I, a little more than a month after the publication of the Einstein ﬁeld equation, Schwarzschild found the ﬁrst nontrivial solution [130,

131]. In Schwarzschild coordinates, the invariant spacetime line element is given by

2M 2M −1 ds2 = − 1 − dt2 + 1 − dr2 + r2 dθ2 + r2 sin2(θ) dϕ2 . (2.1.75) r r

The coordinate r is called the areal radius, with the property that two-spheres at constant r have proper area 4πr2. Birkhoﬀ’s theorem [132] states that this is the unique spherically symmetric solution to the vacuum ﬁeld equation.

The metric is manifestly independent of t, so that we can immediately deduce the existence

a a of a Killing vector ξ = (∂t) which is timelike near inﬁnity. The metric is also independent

a a of ϕ, so that there is another Killing vector ψ = (∂ϕ) . The lack of cross-terms in the line element indicates that ξa is orthogonal to certain spatial hypersurfaces, speciﬁcally two-spheres surrounding the origin. Therefore, the Schwarzschild spacetime is static.

Suppose that there is a geodesic whose path is xa(λ), where λ is a parameter along the

a dxa curve. The vector tangent to the geodesic is the four-velocity u = dλ . Then (2.1.50) implies

Chapter 2. Numerical Techniques in Relativity 35 Chapter 2. Numerical Techniques in Relativity that there is a conserved energy along the geodesic

a 2M dt E = −ξ ua = 1 − , (2.1.76) r dλ as well as a conserved angular momentum

a 2 dϕ L = ψ ua = r . (2.1.77) dλ

By spherical symmetry, we need only consider equatorial motion, satisfying θ = π/2 and dθ dλ = 0. Substituting the conserved quantities into the line element results in an equation for the radial coordinate 1 dr 2 E2 + V = , (2.1.78) 2 dλ 2 where the eﬀective potential is

1 M L2 ML2 V = κ − κ + − , (2.1.79) 2 r 2r2 r3 and κ = 1 for timelike geodesics and κ = 0 for lightlike geodesics. Upon solving this one- dimensional potential equation for r(λ), the corresponding time and angular coordinates are determined from Eqs. (2.1.76) and (2.1.77). Circular orbits are located at stationary points in ∂V the potential, r satisfying ∂r = 0. If a particle moves towards the center of attraction with √ |L| < 12M, then there are no extrema of V , and the particle will necessarily approach r = 0. √ If |L| > 12M, then there is one stable circular (timelike) orbit and one unstable circular

(timelike or lightlike) orbit. Stable circular orbits are restricted to the range r > 6M, and circular orbits in 3M < r < 6M are unstable. No circular orbits exist for r < 3M.

The Schwarzschild metric has two singularities: one at r = rH = 2M and the other at r = 0. The former is a coordinate singularity and can be eliminated by clever changes of coordinate labels [133, 134, 135, 136, 137, 138, 139]. This is the event horizon. To a freely falling observer, there is nothing special about this location. The singularity at r = 0, however, is a true physical singularity. When r > 2M, the time component of the metric is timelike

36 2.1. General Relativity Formalism 2.1. General Relativity Formalism and the radial component is spacelike, that of the usual Lorentzian type where worldlines are always directed towards increasing t. When r < 2M, these components switch character, and the time component becomes spacelike and the radial component becomes timelike. There, worldlines are directed towards decreasing r. After passing through the event horizon, all timelike trajectories become spacelike and end the at singularity in a finite amount of proper time (affine parameter, for the case of lightlike trajectories).2 A compactification of coordinates allows us to draw the Schwarzschild spacetime as a Penrose diagram, as shown in Fig. 2.2.

2.1.9.2 The Reissner-Nordström Solution

The classical theory of electromagnetism unites electric fields, magnetic fields, charges, cur- rents, and light under a single field framework [140, 141]. In terms of the four-potential differential form A, the electromagnetic field tensor is a two-form defined by the exterior derivative e F = dA with components e e Fab = ∂aAb − ∂bAa . (2.1.80)

For an observer with four-velocity ua, the quantity

a ab E = F ub (2.1.81) is interpreted as the electric ﬁeld measured by that observer, and

a 1 abcd B = − Fbcud (2.1.82) 2

2A curious consequence of this geometry is that the path from the horizon to the singularity possessing the greatest amount of proper time is the geodesic. That is to say, if you are unlucky enough to ﬁnd yourself interior to a black hole horizon, and you wish to live for as long as is possible, then the optimum strategy is to do nothing and free fall to oblivion; any acceleration (via jetpack, say), even if oriented away from the singularity, will increase your rate of fall towards r = 0. It turns out that a timelike geodesic takes precisely τ = πM proper time to make this journey. For a black hole with the mass of the sun, τ = πM ≈ 16 µs. For 9 the supermassive black hole residing in M 87, τ ≈ 6.4 × 10 πM ≈ 28 h. However, prolonging the journey is probably not desirable because the extreme tidal forces work to stretch you longitudinally along the direction to the singularity, and compress you transversely. This process is known as “spaghettiﬁcation.”

Chapter 2. Numerical Techniques in Relativity 37 Chapter 2. Numerical Techniques in Relativity

i+ r = 0 i+

+ r II = 0 + I = 2 r I M d M = 2 r dt = 0

i0 III I i0

r M = 2 = 2 M I − r I − IV

i− r = 0 i−

Figure 2.2: A Penrose diagram of Schwarzschild spacetime. The locations of conformal infinity are labeled. Time runs vertically, from the infinite past at i− and I −, to the infinite future at i+ and I +. Light rays move along 45◦ lines, parallel to I − and I +. A point on the diagram represents a two-sphere at that particular time and radius. Region I is the ordinary space outside of the black hole, extending from the event horizon at r = 2M to spatial infinity i0. Region II is the interior of the black hole, in which all timelike and lightlike curves end at r = 0, the singularity. Region III is another asymptotically flat region of space. Since no physical signal can move from I to III, or vice versa, they are causally disconnected, and represent “separate universes.” Region IV is the “white hole,” where all timelike and lightlike curves move away from the r = 0 singularity and are forced into I, II, or III. Generic surfaces of constant time (dt = 0) and constant radius (dr = 0, r > 2M) are shown. is the magnetic field. In curved spacetime, Maxwell’s equations take on the remarkably simple form

ab b ∇aF = j (2.1.83)

∇[aFbc] = 0 , (2.1.84) where ja is the current density four-vector of electric charge. The ﬁrst set of equations are the constraints relating the divergence of the ﬁelds to the sources, and imply the continuity

38 2.1. General Relativity Formalism 2.1. General Relativity Formalism

a equation (conservation of charge) ∇aj = 0. The second set of equations follow from the fact that F is exact, and describe the ﬁeld evolution, relating time derivatives to vector curls. A e particle with charge q and mass m accelerates due to the Lorentz force

a b q b a u ∇au = F au . (2.1.85) m

The electromagnetic ﬁeld makes a contribution to the stress-energy

c 1 cd Tab = FacFb − gabFcdF . (2.1.86) 4

The Reissner-Nordström metric is a solution to the Einstein field equation in spherical symmetry with an electromagnetic source. It represents a static black hole with mass M and electric charge Q. In spherical coordinates (t, r, θ, ϕ), the only nonvanishing components of the electromagnetic field tensor are Ftr and Fθϕ, representing the radial electric field and radial magnetic field, respectively. Supposing that there are no magnetic monopoles [142], then Fθϕ = 0. What remains is the electric monopole term

Q Frt = . (2.1.87) r2

With this source, the Einstein ﬁeld equation can be solved in closed form for the Reissner-

Nordström metric [143, 144]. The invariant spacetime line element is

2M Q2 2M Q2 −1 ds2 = − 1 − + dt2 + 1 − + dr2 + r2 dθ2 + r2 sin2(θ) dϕ2 . (2.1.88) r r2 r r2

p 2 2 The Reissner-Nordström metric is singular at r = 0 and r = r±H = m ± m − Q . The former is a true physical singularity. The character of the event horizons depends on the black hole charge-to-mass ratio. When Q/M > 1, r±H becomes complex, which means that the horizons vanish, leaving a naked singularity. This is typically considered to be unphysical.

The case Q/M = 1 is called the extreme solution and has a single horizon at r±H = M.

Inside the horizon, all components have have the same timlelike or spacelike character as on

Chapter 2. Numerical Techniques in Relativity 39 Chapter 2. Numerical Techniques in Relativity the outside. The extremal state is unstable because any small increase in mass will result in

Q/M < 1, the most physically realistic case. The resulting black hole has an inner and outer horizon. If r−H < r < r+H then the time component of the metric becomes spacelike and the radial component becomes timelike, but is otherwise unaltered. In call cases, the singularity is timelike and never intersects timelike geodesics. Lightlike and nongeodesic timelike paths are able to reach r = 0.

It is expected that astrophysical black holes will have negligible electric charge. If there were a black hole with signiﬁcant charge, it would quickly neutralize due to opposing charges in the ambient plasma. In the limit Q → 0, the Reissner-Nordström metric reduces to the

Schwarzschild metric.

2.1.9.3 The Kerr-Newman Solution

After the charged black hole solution was found, nearly 50 years passed before the rotating Kerr metric was discovered [145]. It represents a black hole with mass M and angular momentum

J = aM, where a is called the spin parameter and is the angular momentum per unit mass.

Shortly afterward, it was generalized to the charged, rotating Kerr-Newman solution [146,

147]. A black hole with mass M, spin a, and electric charge Q in spherical Boyer-Lindquist coordinates has the invariant spacetime line element

∆ − a2 sin2(θ) 2a sin2(θ)(r2 + a2 − ∆) ds2 = − dt2 − dt dϕ ρ2 ρ2 (r2 + a2)2 − ∆a2 sin2(θ) ρ2 + sin2(θ) dϕ2 + dr2 + ρ2 dθ2 , (2.1.89) ρ2 ∆ with the deﬁnitions

ρ2 = r2 + a2 cos2(θ) , (2.1.90)

∆ = r2 + a2 + Q2 − 2Mr . (2.1.91)

40 2.1. General Relativity Formalism 2.1. General Relativity Formalism

The corresponding electromagnetic four-potential is

Qr 2 Aa = − (dt)a − a sin (θ)(dϕ)a . (2.1.92) ρ2

The angular momentum is aligned with the θ = 0 symmetry axis. When a → 0, this reduces to the Reissner-Nordström metric. When Q → 0, this reduces to the Kerr metric. When a → 0 and Q → 0, this reduces to the Schwarzschild metric. When M → 0 and Q → 0, this reduces to the Minkowski metric in spheroidal coordinates. The Kerr-Newman spacetime is the most general stationary black hole solution, and contains within it all known stationary black hole solutions.

The Kerr spacetime has two immediately obvious Killing vector ﬁelds: one which is timelike

a a a a ξ = (∂t) , and one which is spacelike with closed circular orbits ψ = (∂ϕ) . Geodesics with four-velocity uµ have a conserved energy and angular momentum

2 a 2Mr dt 2Mar sin (θ) dϕ E = −ξ ua = 1 − + , (2.1.93) ρ2 dλ ρ2 dλ 2 2 2 2 2 2 a 2Mar sin (θ) dt (r + a ) − ∆a sin (θ) 2 dϕ L = ψ ua = − + sin (θ) . (2.1.94) ρ2 dλ ρ2 dλ

These alone are suﬃcient only to determine motion in the equatorial plane θ = π/2, dθ = 0.

The Kerr metric also possesses a Killing tensor [148]

2 2 κab = 2ρ l(anb) + r gab , (2.1.95) where the principle null vectors have components

2 2 a r + a a a a a l = (∂t) + (∂ϕ) + (∂r) , (2.1.96) ∆ ∆ 2 2 a r + a a a a ∆ a n = (∂t) + (∂ϕ) − (∂r) . (2.1.97) 2ρ2 2ρ2 2ρ2

Chapter 2. Numerical Techniques in Relativity 41 Chapter 2. Numerical Techniques in Relativity

It follows that there exists an additional integral of motion called the Carter constant

a b C = κabu u (2.1.98) which allows for explicit integration of the geodesic equation in generic, nonequatorial trajectories [149].

−18 It would be very difficult for an astrophysical object to maintain Q/M & 10 [150], so we restrict our attention to the Q = 0 Kerr spacetime. As in the Reissner-Nordström solution, Kerr has a physical singularity at r = 0 and is separated into three classes based on the value of the dimensionless spin parameter χ = a/M. The case χ > 1 contains a naked singularity, and is typically considered to by unphysical. If χ = 1, then Kerr is said to have extreme spin. When χ < 1, the singularity is hidden behind two horizons at r = √ 2 2 r±H = M ± M − a . The singularities at the horizons are coordinate singularities which can be removed by the appropriate gauge choice [151, 149]. In the region r+H < r < M + p 2 2 2 a a M − a cos (θ), the norm of the Killing field ξ = (∂t) becomes positive. This part of the spacetime is called the ergosphere and has the property that the time translation Killing field

ξa becomes spacelike, which implies that an object would require superluminal velocities in order to not orbit prograde with the black hole angular momentum. This eﬀect, referred to as frame dragging, extends beyond the ergosphere, weakening with distance. This provides an interesting contrast with the predictions of Newtonian gravity. In the context of gravitational forces, a particle starting from rest in the equatorial plane of a rotating axisymmetric mass falls radially inward along a straight line towards the center of the mass. In general relativity, frame dragging results in a deviation from radial infall in the direction of rotation of the mass.

2.2 3 + 1 Formalism

General relativity is beautiful in its symmetry. Its covariant nature has great utility in con- structing gauge invariant quantities and in developing new physical models. The most widely used method for performing numerical simulations of spacetime dynamics breaks this sym-

42 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism metry by picking a preferred time direction along which to evolve a set of initial conditions.

This Hamiltonian formulation of general relativity was rapidly laid out by Arnowitt, Deser, and

Misner (ADM) in a series of landmark articles [152, 153, 154, 155, 156, 157, 158, 159, 160, 161].

This section will follow the excellent reviews and textbooks by Cook, Poisson, Alcubierre,

Baumgarte, Shapiro, and Gourgoulhon [162, 163, 164, 165, 166].

2.2.1 The 3 + 1 Decomposition

The basic idea of the 3+1 decomposition is to split four-dimensional spacetime into a series of three-dimensional spatial slices, each representing one moment along the one-dimensional time direction. This is achieved through a foliation, wherein spacetime is divided into “parallel” spatial hypersurfaces Σt, parametrized by a universal time function t. Each event sits on exactly one slice. These are Cauchy surfaces, and one in particular Σ0 ≡ Σt=0 will serve as the initial (boundary) conditions to a set of hyperbolic evolution equations. Our task is to construct a model of the desired physical scenario at some moment in time, called the initial data, to serve as the initial boundary condition.

a a At each point in Σt, there is a vector n = −α∇ t orthogonal to the surface (on which t

a is constant). By construction, the normal is a timelike unit vector n na = −1. The normal vector deﬁnes the spatial metric induced on each slice γab as a projection from the spacetime metric, down along the normal direction

γab = gab + nanb . (2.2.99)

This deﬁnition acts as a projection operator, allowing us to split tensors into spatial and

a a b normal components. For example, given any spacetime vector v , the vector γ bv is purely a b spatial, i.e. γ bv na = 0. The spatial covariant derivative Da associated with γab is simply the projection of its spacetime counterpart

c d e c f DaSb = γa γb γf ∇dSe . (2.2.100)

Chapter 2. Numerical Techniques in Relativity 43 Chapter 2. Numerical Techniques in Relativity

We will always use the metric compatible derivative Daγbc = 0. Between adjacent slices Σt and

Σt+dt, an observer moving in the normal direction experiences proper time dτ = α dt, where

α is the lapse function. From one moment to the next, points can move from one location to another by an amount designated by the shift vector βa. The shift vector is spatial, meaning

a that it is orthogonal to the normal β na = 0 and tangent to Σt, and has only three independent components. The time vector ta connecting events between slices is

ta = αna + βa . (2.2.101)

A schematic of the decomposition is shown in Fig. 2.3. The four components of α and βa are the manifestation of the four-dimensional coordinate freedom in general relativity.

a β dt Σt+dt

αna dt ta dt

Σt

Figure 2.3: An illustration of the 3 + 1 decomposition for two adjacent spatial hypersurfaces a Σt and Σt+dt. A point in Σt is moved to a point in Σt+dt along t by a combination of transportation in the direction normal to the surface na and parallel to the surface βa. The proper time between adjacent surfaces is α dt.

In the Cauchy formulation of the Einstein ﬁeld equation, the spatial metric γab is treated as part of the initial conditions, sometimes referred to as the ﬁrst fundamental form. The

Einstein ﬁeld equation is second-order in time, which implies that, in addition to “initial position” information, “initial velocity” data is also required. This is managed by the extrinsic curvature Kab, also called the second fundamental form. The extrinsic curvature is built by considering how the spatial metric changes as it is dragged along the normal direction

1 Kab = − Lnγab . (2.2.102) 2

44 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism

There is an equivalent alternate interpretation of the extrinsic curvature as the projection onto the spatial hypersurface of the gradient of the normal vectors

c d Kab = −γa γb ∇cnd . (2.2.103)

These are spatial vectors which point in the direction that the hypersurface will deform when dragged along the normal direction. It is easy to see that Kab as deﬁned by Eq. (2.2.102) is symmetric, since γab is symmetric. The case of Eq. (2.2.103) is not as clear. The normal vector is proportional to the gradient of a function, so that it satisﬁes the condition of hypersurface orthogonality n[a∇bnc] = 0. This is tedious to demonstrate in component notation, so we use the language of one-forms n = −α dt. The hypersurface orthogonality condition is then e

n ∧ dn = (−α dt) ∧ d(−α dt) (2.2.104) e e = (α dt) ∧ (dα ∧ dt + α d2t) (2.2.105)

= α dt ∧ dα ∧ dt (2.2.106)

= 0 , (2.2.107) where we went from the second line to the third by the Poincaré lemma, and from the third to the fourth by the complete antisymmetry of the wedge product. If this is contracted with

d e a the normal vector and projected onto the hypersurface, γb γc n n[a∇dne] = 0, a little algebra yields ∇[anb] = 0. Thus, only the symmetric part of Kab in Eq. (2.2.103) remains.

ab The trace of the extrinsic curvature K = γ Kab is called the mean curvature. Taking the trace of Eq. (2.2.102) √ K = −Ln ln( γ) , (2.2.108) where γ = det(γab), shows that the mean curvature measures the fractional rate of change of proper spatial volume elements as the hypersurface is dragged along the normal direction.

Chapter 2. Numerical Techniques in Relativity 45 Chapter 2. Numerical Techniques in Relativity

2.2.1.1 The Constraint Equations

If we wish to simulate systems in the context of general relativity, we cannot pick initial data arbitrarily. The theory imposes constraints on what are the allowable shapes of physical

Cauchy surfaces. The Hamiltonian constraint is a result of projecting two indices of the spacetime Riemann tensor along the normal direction, taking the trace, and plugging it into the Einstein ﬁeld equation. The result is

2 ab R + K − KabK = 16πρ , (2.2.109)

a b where R is the scalar curvature associated with γab, and ρ ≡ Tabn n is the local energy density measured by an observer moving on a normal trajectory. Similarly, the momentum constraint is obtained by projecting one index of the spacetime Riemann tensor along the normal direction, taking the trace, and substituting into the Einstein ﬁeld equation

ab ab a Db(K − γ K) = 8πS , (2.2.110)

b c where Sa ≡ −γa n Tbc is the momentum density measured by the observer with four-velocity na.

2.2.1.2 The Evolution Equations

From Eq. (2.2.102) and the properties of the Lie derivative comes the evolution equation for the spatial metric

Ltγab = −2αKab + Lβγab . (2.2.111)

The remaining components of the projected spacetime Riemann tensor yield the evolution equation for the extrinsic curvature

c 1 LtKab = −DaDbα+α(Rab−2KacK b+KKab)−8πα Sab − (S − ρ)γab +LβKab , (2.2.112) 2

c d a where the spatial stress and its trace are deﬁned as Sab ≡ γa γb Tcd and S ≡ Sa , respectively.

46 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism

2.2.1.3 The 3 + 1 Basis

It is convenient to choose a basis that conforms to the 3 + 1 decomposition. To do this, we

a pick three spatial basis vectors e(i) (the subscript i = 1, 2, 3 distinguishes the vectors, not their a a components) which are tangent to the hypersurface nae(i) = 0. Since e(i) span Σt and are linearly independent, then ni = 0. From Eq. (2.2.99), γij = gij. The spatial basis vectors are moved from one slice to another by Lie dragging along ta

a Lte(i) = 0 . (2.2.113)

a a The remaining basis vector is chosen to be e(0) = t . This implies that Lt ≡ ∂t. In this basis, spatial tensors have vanishing time components.

In terms of these 3 + 1 variables, the spacetime line element is decomposed into

2 2 2 i i j j ds = −α dt + γij dx + β dt dx + β dt , (2.2.114)

i where x are spatial coordinates in Σt. It is easy to see that, in this basis, the spacetime metric has components   −α2 + β βk β  k i  gµν =   . (2.2.115) βj γij

The inverse spatial metric γij satisﬁes

kj j γikγ = δi , (2.2.116) and hence can be used to raise and lower spatial indices of spatial tensors. For example,

j βi = γijβ .

The constraint Eqs. (2.2.109) and (2.2.110) become

2 ij R + K − KijK = 16πρ , (2.2.117)

ij ij i Dj(K + γ K) = 8πS , (2.2.118)

Chapter 2. Numerical Techniques in Relativity 47 Chapter 2. Numerical Techniques in Relativity and the evolution Eqs. (2.2.111) and (2.2.112) become

∂tγij = − 2αKij + Diβj + Djβi , (2.2.119) 1 ∂ K = − D D α + α(R − 2K Kk + KK ) − 8πα S − (S − ρ)γ t ij i j ij ik j ij ij 2 ij

k k k + β DkKij + KikDjβ + KkjDiβ . (2.2.120)

2.2.2 Conformal Decomposition

As they stand, the constraint Eqs. (2.2.117) and (2.2.118) are not well suited to solution by numerical methods. They can be turned into elliptic diﬀerential equations by a conformal transformation of the spatial metric

4 γij = ψ γ˜ij , (2.2.121) where ψ is the conformal factor, and all objects with a tilde are associated with the conformally

ij −4 ij related spatial metric γ˜ij. The inverse is defined by γ = ψ γ˜ , so that the conformal spatial metric can be used to raise and lower conformal spatial indices. Inserting Eq. (2.2.121) into the expression for the spatial covariant derivative Christoffel symbols (see Eq. (2.1.35)) yields the transformation rule for the conformal connection coefficients

i ˜i h i i il i Γjk = Γjk + 2 δj∂k ln(ψ) + δk∂j ln(ψ) − γ˜jkγ˜ ∂l ln(ψ) . (2.2.122)

This gives rise to the conformal metric compatible covariant derivative

˜ Diγ˜jk = 0 . (2.2.123)

In the same way, the Ricci tensor transforms as

h kl i h kl i Rij = R˜ij −2 D˜iD˜j ln(ψ) +γ ˜ijγ˜ D˜kD˜l ln(ψ) +4 ∂i ln(ψ)∂j ln(ψ) − γ˜ijγ˜ ∂k ln(ψ)∂l ln(ψ) ,

(2.2.124)

48 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism and the scalar curvature as

R = ψ−4R˜ − 8ψ−5D˜ 2ψ . (2.2.125)

The extrinsic curvature is split into a trace K and trace-free part Aij

1 Kij = Aij + γijK. (2.2.126) 3

ij That is, γ Aij = 0. The trace-free part is conformally rescaled by

−2 Aij = ψ A˜ij , (2.2.127) and the mean curvature is kept conformally invariant K = K˜ .

Under these conformal transformations, the constraint Eqs. (2.2.117) and (2.2.118) become

2 1 1 5 2 1 −7 ij 5 D˜ ψ − ψR˜ − ψ K + ψ A˜ijA˜ = −16πψ ρ , (2.2.128) 8 12 8 ij 2 6 ij 10 i D˜jA˜ − ψ γ˜ D˜jK = 8πψ S . (2.2.129) 3

The Hamiltonian constraint is now a second-order diﬀerential equation for the conformal factor.

2.2.3 Transverse-traceless Decomposition

Any symmetric, trace-free tensor can be written as the sum of a transverse-traceless part and a longitudinal part. A transverse-traceless tensor is symmetric, trace-free, and divergence- free. A longitudinal tensor is symmetric, trace-free, and curl-free, which implies that it can be expressed as the symmetric, trace-free gradient of a vector. This is a generalization of the

Helmholtz decomposition [167, 168, 169]

ij ij ij A˜ = Q˜ + (L˜X) , (2.2.130)

Chapter 2. Numerical Techniques in Relativity 49 Chapter 2. Numerical Techniques in Relativity where Q˜ij is a symmetric, trace-free tensor satisfying

ij D˜iQ˜ = 0 , (2.2.131) and the longitudinal part is

˜ ij ˜ i j ˜ j i 2 ij ˜ k ( X) ≡ D X + D X − γ˜ DkX . (2.2.132) L 3

i Here, X is a vector potential, and L˜ is the vector gradient, also known as the longitudinal ab a derivative. If (L˜ξ) = 0, then ξ is called a conformal Killing vector. This allows us to write the divergence of the trace-free extrinsic curvature as

˜ ˜ij ˜ i DjA = ∆LX , (2.2.133)

˜ where the vector Laplacian ∆L is deﬁned by

i ˜ ij 2 i 1 i j i j ∆˜ X ≡ D˜j( X) = D˜ X + D˜ D˜jX + R˜ X . (2.2.134) L L 3 j

A transverse tensor is constructed out of a symmetric, trace-free tensor M˜ ij by subtracting oﬀ the longitudinal part ij ij ij Q˜ ≡ M˜ − (L˜Y ) . (2.2.135)

˜ij ˜ i ˜ ˜ ij i The fact that Q is transverse implies ∆LY = DjM , which is solved for Y . Using the linearity of the vector gradient and vector Laplacian, and deﬁning

ij ij ij A˜ = M˜ + (L˜V ) (2.2.136) and V i = Xi − Y i, the momentum constraint Eq. (2.2.129) is turned into an elliptic diﬀeren- tial equation for the vector potential V i, called the conformal transverse-traceless decomposition [170, 171] i ij 2 6 ij 10 i ∆˜ V + D˜jM˜ − ψ γ˜ D˜jK = 8πψ S . (2.2.137) L 3

50 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism

Alternatively, we can decompose the physical trace-free part of the extrinsic curvature using the Helmholtz theorem ij ij ij A = Q + (LW ) , (2.2.138)

ij ij where Q is symmetric, trace-free, and transverse (DjQ = 0). We can express this in terms of an arbitrary symmetric, trace-free tensor by deﬁning

ij −10 ij ij Q ≡ ψ M˜ − (LZ) (2.2.139) and V i = W i − Zi. Then Aij becomes

ij −6 ij ij A = ψ M˜ + (L˜V ) . (2.2.140)

ij −4 ij It is useful to remember the identity (LW ) = ψ (L˜W ) . Equation (2.2.129) thus turns into an elliptic diﬀerential equation for the vector potential V i, called the physical transverse- traceless decomposition [172, 173, 174]

i ˜ ij −6 ij 2 ij 4 i ∆˜ V + 6( V ) D˜j ln(ψ) + ψ D˜jM˜ − γ˜ D˜jK = 8πψ S . (2.2.141) L L 3

The spatial hypersurface began with twelve degrees of freedom, six each in γij and Kij.

These were then decomposed into smaller pieces with special properties. Four of the twelve degrees of freedom in ψ and V i are fixed by the constraint equations. Four others correspond to the freedom to chose coordinates, three spatial in γ˜ij and one temporal in K. Four remain, two in the other components of γ˜ij and two in the transverse part of M˜ ij. These are freely specifiable, and correspond to the dynamical degrees of freedom in the gravitational field. The conformal and physical transverse-traceless decompositions are very similar, and both allow us to produce valid solutions to the constraint equations. However, identically specified free data yields differing physical solutions.

Chapter 2. Numerical Techniques in Relativity 51 Chapter 2. Numerical Techniques in Relativity

2.2.4 Thin-sandwich Decomposition

One shortcoming of the transverse-traceless decomposition is that the constraints impose no restriction on the kinematic variables α and βi. The conformal thin-sandwich decomposition [175] takes the approach of considering the evolution of the spatial metric between two neighboring spatial hypersurfaces. Instead of providing initial data for γij and Kij on a single time slice, we provide γij on two time slices, in the limit of inﬁnitesimal separation. Four additional degrees of freedom α and βi are introduced.

The ﬁrst step is to deﬁne uij as the trace-free part of the time derivative of the spatial metric 1/3 −1/3 uij = γ ∂t γ γij , (2.2.142) which, in terms of Eq. (2.2.119), is

uij = −2αKij + (Lβ)ij . (2.2.143)

4 ij From the conformal rescaling uij = ψ u˜ij, it can be shown that u˜ij = ∂tγ˜ij and γ˜ u˜ij = 0.

With the additional rescaling α = ψ6α˜, the trace-free part of the conformal extrinsic curvature is 1 h ˜ i A˜ij = ( β)ij − u˜ij . (2.2.144) 2˜α L

This turns Eq. (2.2.129) into an elliptic diﬀerential equation for the shift vector

i ˜ ij −1 ij 4 6 ij 10 i ∆˜ β − ( β) D˜j ln(˜α) − α˜D˜j α˜ u˜ − αψ˜ γ˜ D˜jK = 16παψ˜ S . (2.2.145) L L 3

In the thin-sandwich decomposition, there are a total of sixteen degrees of freedom, of which twelve can be specified freely: five each in γ˜ij and u˜ij, and one each in α˜ and K. The remaining four in ψ and βi are fixed by the constraint equations. Alternatively, it is possible to specify ∂tK rather than α˜, and use the evolution equation for the mean curvature to solve

52 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism for α in the extended thin-sandwich decomposition [176, 177, 178, 179]

7 5 1 D˜ 2(αψ) = αψ ψ−8A˜ A˜ij + ψ4K2 + R˜ + 2πψ4(ρ + 2S) − ψ5∂ K + ψ5βiD˜ K. 8 ij 12 8 t i (2.2.146)

2.2.5 Mass, Momentum, and Angular Momentum

The equivalence principle implies that there is no coordinate independent construction of the local energy density of the gravitational field in general relativity. For asymptotically flat systems, it is possible to define globally conserved quantities associated with the total energy

(mass), momentum, and angular momentum. One useful measure of the energy is provided by the ADM mass MADM [161]. The ADM mass is deﬁned as an integral over the two-dimensional surface at inﬁnity ∂Σ∞, bounding the spatial slice Σ [180]

Z 1 ij kl √ MADM = γ γ (∂kγjl − ∂jγkl) γ dSi . (2.2.147) 16π ∂Σ∞

p ∂Σ 2 i Here dSi = σi γ ∞ d z is the outward-oriented surface element, where z are coordinates on

∂Σ∞ i ∂Σ∞, γij is the induced metric on ∂Σ∞, and σ is the unit normal to ∂Σ∞. This requires that the metric approach Minkowski space suﬃciently rapidly

gab = ηab + O(1/r) . (2.2.148)

This deﬁnition is not covariant and is valid only in asymptotically Cartesian coordinates.

Supposing that

ψ = 1 + O(1/r) (2.2.149) and using the deﬁnition of the spatial Christoﬀel symbols and the conformal decomposition rules, the ADM mass is split into two integrals

Z Z 1 jk ˜i ij ˜k ˜ 1 ˜ ˜i MADM = γ˜ Γjk − γ˜ Γjk dSi − Diψ dS . (2.2.150) 16π ∂Σ∞ 2π ∂Σ∞

Chapter 2. Numerical Techniques in Relativity 53 Chapter 2. Numerical Techniques in Relativity

In this expression, we took advantage of the fact that dSi = dS˜i on ∂Σ∞.

Similarly, in asymptotically ﬂat Cartesian coordinates, the ADM linear momentum is de-

ﬁned by Z i 1 ij ij PADM = K − γ K dSj , (2.2.151) 8π ∂Σ∞ and the ADM angular momentum is deﬁned by

ijk Z i l JADM = xj (Kkl − γklK) dS , (2.2.152) 8π ∂Σ∞ where xi is a coordinate vector.

2.2.6 Black Hole Initial Data

We wish to model the dynamics of black holes in vacuum, so we set all of the matter terms

i ρ = 0, S = 0, Sij = 0, and S = 0 (i.e. we are looking for solutions to Rµν = 0). The initial data described here rely on the choices of time symmetry and conformal ﬂatness.

2.2.6.1 Time Symmetric Solutions

The simplest initial data representing a black hole is the static Schwarzschild solution. Time symmetric Cauchy surfaces have vanishing extrinsic curvature Kij = 0, and hence vanishing mean curvature K = 0. Surfaces satisfying K = 0 are called maximal, and they possess extremal volume.3 The momentum constraint (2.2.129) is satisﬁed trivially, and the Hamiltonian constraint (2.2.128) reduces to 1 D˜ 2ψ = ψR.˜ (2.2.153) 8

Any spherically symmetric, three-dimensional space can be transformed into coordinates in which it is conformally ﬂat. Let fij be the ﬂat space metric. Then the assumption γ˜ij = fij

3In a pseudo-Remiannian manifold, a maximal spatial hypersurface has the maximum amount of proper volume. In a Riemannian manifold, a minimal surface has the least amount of proper volume. Soap ﬁlms are a favorite example of minimal surfaces in nature. Both maximal and minimal surfaces are characterized by K = 0.

54 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism implies R˜ = 0, and the Hamiltonian constraint becomes

D˜ 2ψ = 0 , (2.2.154) where D˜ 2 is now the ﬂat space Laplace operator. If we impose the asymptotic ﬂatness boundary condition

lim ψ = 1 , (2.2.155) r→∞ then the Laplace equation has the solution

M ψ = 1 + , (2.2.156) 2r where M/2 is a constant of integration chosen such that we measure the resulting spacetime to have total energy M. This ﬁxes the Cauchy initial data for a single black hole. Since γ˜ij = δij ˜k in Cartesian coordinates, then Γij = 0 and the ADM mass (2.2.150) reduces to

Z 1 ˜ ˜i MADM = − DiψS = M. (2.2.157) 2π ∂Σ∞

In general, M is only a parameter, and does not coincide with the mass of the black hole or the

i i total mass of the system. It is easy to see for this spatial slice that PADM = 0 and JADM = 0.

To obtain a full solution to the Einstein ﬁeld equation, we require a lapse function and shift vector. With the assumption ∂tK = 0, Eq. (2.2.146) simpliﬁes to

D˜ 2(αψ) = 0 . (2.2.158)

This is solved for α supposing that the lapse vanishes on the horizon r = M/2 and goes to unity as r → ∞

lim α = 0 , lim α = 1 . (2.2.159) M r→∞ r→ 2

Chapter 2. Numerical Techniques in Relativity 55 Chapter 2. Numerical Techniques in Relativity

These boundary conditions result in the solution

M 1 − 2r α = M . (2.2.160) 1 + 2r

Finally, if we choose βi = 0, then the evolution equations (2.2.119) and (2.2.120) are satisﬁed.

This completes the solution of the Einstein ﬁeld equation. The invariant line element for this spacetime is

M !2 4 2 1 − 2r 2 M 2 2 2 2 2 2 ds = − dt + 1 + dr + r dθ + r sin (θ) dϕ . (2.2.161) M 2r 1 + 2r

The Schwarzschild line element (2.1.75) (with the areal radial coordinate designated by r¯) is recovered through the radial coordinate transformation

M 2 r¯ = r 1 + . (2.2.162) 2r

Here, r is called the isotropic radial coordinate. The spatial geometry is invariant under the transformation M 2 r → . (2.2.163) 4r

This solution represents two asymptotically ﬂat, causally disconnected spaces connected at the horizon. Therefore, r = 0 is an image of spatial inﬁnity in the other universe, and is only coordinate singularity.

Brill-Lindquist initial data provides a straight forward generalization to time symmetric, multiple black hole conﬁgurations [181, 182]. By the linearity of Eq. (2.2.154), the conformal factor can be extended to represent N black holes at positions Cσ with parameters µσ

N X µσ ψ = 1 + . (2.2.164) |r − C | σ=1 σ

56 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism

2.2.6.2 Bowen-York Solutions

The constraint of time symmetry is very limiting. Initial data representing black holes with linear or angular momentum are nontrivial solutions to the momentum constraint. York and his collaborators found analytical solutions for the case of conformally ﬂat, maximal, asymptotically ﬂat space. With these assumptions and the choice M˜ ij = 0, Eq. (2.2.137) reduces to 2 i 1 i j D˜ V + D˜ D˜jV = 0 . (2.2.165) 3

Remarkably, this is satisﬁed analytically by the Bowen-York solution [106, 183, 184]

i 1 i i j 1 ijk V = − 7P + n njP + njSk , (2.2.166) 4r r2 where P i and Si are vector parameters, r is a coordinate radius, ni = xi/r is an outward pointing unit normal of a two-sphere in conformal space, and ijk is the three-dimensional

Levi-Civita tensor. This gives the trace-free conformal extrinsic curvature

˜ 3 3 h l k l k i Aij = [Pinj + Pjni − (fij − ninj)] + kilS n nj + kjlS n ni . (2.2.167) 2r2 r3

Measured at inﬁnity, P i is the linear momentum and Si is the angular momentum of the black hole [185]. Equation (2.2.165) is linear, implying that solutions representing multiple black holes can be constructed by simple linear combination of copies of Eq. (2.2.167).

The Hamiltonian constraint (2.2.128) becomes

2 1 −7 ij D˜ ψ + ψ A˜ijA˜ = 0 . (2.2.168) 8

One method for assigning boundary conditions to solutions of this elliptic equation is to construct an inversion-symmetric extrinsic curvature, and apply boundary conditions to ψ across the inversion surface (the event horizon) and at spatial inﬁnity [106, 186, 187, 188,

189]. In the multiple black hole case, the expression for inversion-symmetric A˜ij is a series solution [190].

Chapter 2. Numerical Techniques in Relativity 57 Chapter 2. Numerical Techniques in Relativity

Another method for assigning boundary conditions to Eq. (2.2.168), developed by Brandt and Brügmann [191], removes the singularities by writing the conformal factor as

N X µσ ψ = 1 + Ψ + u , Ψ ≡ . (2.2.169) |r − C | σ=1 σ

For asymptotic ﬂatness, this requires u = O(1/r) for large r. This is called the puncture method, and Cσ are the coordinate locations of the punctures. Equation (2.2.168) becomes

u −7 D˜ 2u + η 1 + = 0 , (2.2.170) Ψ

1 −7 ˜ ˜ij where η = 8 Ψ AijA . Near to any of the punctures, the components of the conformal 3 extrinsic curvature (2.2.167) diverge as A˜ij ∼ O(1/r ) and Ψ ∼ O(1/r). Thus, η ∼ O(r), and the source term is ﬁnite at the punctures. This implies that Eq. (2.2.170) has a unique C2 solution.

A major shortcoming of the Bowen-York solutions is that they contain spurious gravitational wave content from the onset. This is due largely to the assumption of conformal ﬂatness, which is not satisﬁed by the Kerr solution [192] or for black holes with linear momentum [185].

This radiation contaminates the spacetime with an undesired signal burst. Some of it ﬂies away to inﬁnity, appearing as a glitch in the extracted gravitational waveforms [193, 194, 195, 196].

The remainder plunges into the black holes. Since the radiation is axisymmetric in the spinning case, it imparts no angular momentum onto the black hole. However, it possesses energy, so the black hole mass increases. This limits Bowen-York solutions to evolutions with spins

χ . 0.93, which eventually settle down to the Kerr solution [106, 197, 198, 199]. Similarly, they are limited in their velocities to v/c . 0.9.

2.2.7 Gauge Choices and Evolution Formalisms

The ADM evolution Eqs. (2.2.119) and (2.2.120) are only weakly hyperbolic, meaning that they are ill-posed. The characteristic of the evolution system can be changed to strongly

i ij hyperbolic by introducing auxiliary variables Γ˜ = −∂jγ˜ , the conformal connection functions.

58 2.2. 3 + 1 Formalism 2.2. 3 + 1 Formalism

This results in a well-posed formalism, where solutions do not grow faster than an exponential function.

1 −4φ The conformal factor φ = 12 ln(γ) ﬁxes the conformal metric γ˜ij = e γij to unit determi- −4φ nant γ˜ = 1. The trace free extrinsic curvature is transformed in the same way A˜ij = e Aij.

It is advantageous to evolve with the variable χ = e−4φ [200] (not to be confused with the dimensionless spin parameter). With the deﬁnition

1 R = − 2D˜ D˜ φ − 2˜γ D˜ kD˜ φ + 4D˜ φD˜ φ − 4˜γ D˜ kφD˜ φ − γ˜lm∂ ∂ γ˜ ij i j ij k i j ij k 2 l m ij ˜k ˜ ˜k lm ˜k ˜ ˜k ˜ +γ ˜k(i∂j)Γ + ΓkΓ(ij) +γ ˜ 2Γl(iΓj)km + ΓimΓklj (2.2.171) we get the Baumgarte-Shapiro-Shibata-Nakamura-Oohara-Kojima (BSSNOK) evolution formalism [113, 114, 115]

∂tγ˜ij = − 2α + Lβγ˜ij , (2.2.172)

2 i i ∂tχ = χ(αK − ∂iβ ) + β ∂iχ , (2.2.173) 3 ˜ TF ˜ ˜ ˜k ˜ ∂tAij = χ(−DiDjα + αRij) + α(KAij − 2AikAj ) + LβAij , (2.2.174) i ij 1 2 i ∂tK = − D Diα + α A˜ijA˜ + K + β ∂iK, (2.2.175) 3 1 2 ∂ Γ˜i =γ ˜jk∂ ∂ βi + γ˜ij∂ ∂ βk + βj∂ Γ˜i − Γ˜j∂ βi + Γ˜i∂ βj t j k 3 j k j j 3 j ij i jk ij 2 ij − 2A˜ ∂jα + 2α Γ˜ A˜ + 6A˜ ∂jφ − γ˜ ∂jK , (2.2.176) jk 3 where TF represents the trace-free part.

An alternative formulation measures the deviation of the solution from satisfying the Ein- stein ﬁeld equation with a vector Zµ. Satisfying the constraints perfectly amounts to Zµ = 0.

i i ij µ With the additional deﬁnitions f = 3/4, Γˆ = Γ˜ + 2˜γ Zj, and Θ = n Zµ, the constraint

Chapter 2. Numerical Techniques in Relativity 59 Chapter 2. Numerical Techniques in Relativity damping CCZ4 evolution formalism is [117]

˜ k 2 k k ∂tγ˜ij = − 2αAij + 2˜γ ∂ β − γ˜ij∂kβ + β ∂kγ˜ij , (2.2.177) k(i j) 3 2 TF ∂tA˜ij = φ [−DiDjα + α(Rij + DiZj + DjZi] + αA˜ij(K − 2Θ)

˜ ˜k ˜ k 2 ˜ k k ˜ − 2αAikA + 2A ∂ β − Aij∂kβ + β ∂kAij , (2.2.178) j k(i j) 3 1 1 k k ∂tφ = αφK − φ∂kβ + β ∂kφ , (2.2.179) 3 3 i i 2 j ∂tK = − D Diα + α(R + 2DiZ + K − 2ΘK) + β ∂jK − 3ακ1(1 + κ2)Θ , (2.2.180) 1 2 ∂ Θ = α R + 2D Zi − A˜ A˜ij + K2 − 2ΘK t 2 i ij 3

i k − Z ∂iα + ∂ ∂kΘ − ακ1(2 + κ2)Θ , (2.2.181) ∂ φ 2 2 ∂ Γˆi = 2α Γ˜i A˜jk − 3A˜ij j − γ˜ij∂ K + 2˜γki α∂ Θ − Θ∂ α − αKZ t jk φ 3 j k k 3 k 1 2 − 2A˜ij∂ α +γ ˜kl∂ ∂ βi + γ˜ik∂ ∂ βl + Γ˜i∂ βk − Γ˜k∂ βi j k l 3 k l 3 k k 2 ij k jk i k ˆi ij + 2κ3 γ˜ Zj∂kβ − γ˜ Zj∂kβ + β ∂kΓ − 2ακ1γ˜ Zj , (2.2.182) 3

k ∂tα = − 2α(K − 2Θ) + β ∂kα , (2.2.183)

i i k i ∂tβ = fB + β ∂kβ , (2.2.184)

i ˆi k ˆi k i i ∂tB = ∂tΓ − β ∂kΓ + β ∂kB − ηB . (2.2.185)

All of the constraint-related modes are damped when κ1 > 0 and κ2 > −1 [201]. For black hole spacetimes κ3 = 1/2 [117].

The punctures can be evolved in two ways. The ﬁxed puncture method factors the singular parts out of the conformal factor and evolves only the regular part. This is the simpler of the two methods, but is disadvantaged because the punctures do not move on the grid. Allowing the full conformal factor to evolve lead to the moving puncture method in the black hole binary simulation breakthrough of 2005 [200, 202]. These utilize the “1 + log” slicing condition and the “gamma-driver” shift condition for the gauge evolution [203].

60 2.2. 3 + 1 Formalism 2.3. Pseudospectral Methods

2.3 Pseudospectral Methods

Consider the differential equation Lu(x) = s(x), where L is some differential operator and s is a source function. The pseudospectral method for evaluating differential equations amounts to expanding u in a truncated series of basis functions {φn(x)}

N X uN (x) = anφn(x) , (2.3.186) n=0 where an are constant coeﬃcients and

u(x) = lim uN (x) (2.3.187) N→∞

(i.e. {φn} is complete). With the truncated solution, we construct the residual function

R(x) = |LuN (x) − s(x)| . (2.3.188)

The goal is to choose an such that R is minimized.

The set of basis functions satisﬁes the orthogonality condition on the interval x ∈ [a, b]

Z b 2 φi(x)φj(x)w(x) dx = kφik δij , (2.3.189) a where w is a weighting function speciﬁc to the particular set of basis functions. From this, we can extract the series coeﬃcients via

Z b an = u(x)φn(x)w(x) dx . (2.3.190) a

An alternative way of determining an is to use the collocation method. If we are representing our function by an Nth order truncated series, we choose a set of N + 1 collocation points

{xi} in the domain of u and require R(xi) = 0. We now have a set of N + 1 equations (one for each collocation point) and N + 1 unknowns (the spectral coeﬃcients). Depending on the nature of the particular problem at hand, we can use any number of linear algebra algorithms

Chapter 2. Numerical Techniques in Relativity 61 Chapter 2. Numerical Techniques in Relativity to solve this system of equations. Although we can choose the collocation points arbitrarily, some choices are better than others, given the set of basis functions. √ inx If u is periodic, we typically choose the basis functions φn(x) = e (where i = −1 is the imaginary unit, not an index) such that the solution is expanded as a Fourier series. Here, we use the collocation points xi = πi/N. For nearly any other form of u, we choose to expand the solution in terms of the Chebyshev polynomials

Tn(x) = cos (n arccos(x)) . (2.3.191)

Since Tn(x) is deﬁned only for x ∈ [−1, 1], it is necessary to map the domain of u to this interval. We have explicit closed-form expressions for the derivatives of Tn, so that it is easy to represent the derivatives of u in terms of the original an. There are two optimal choices for the collocation points, given a Chebyshev polynomial basis. The ﬁrst are the Gauss-Chebyshev points (2i − 1)π xi = cos . (2.3.192) 2n

These are the roots of the Chebyshev polynomials. The second are the Gauss-Lobatto points

iπ xi = cos . (2.3.193) n

These are the extrema of the Chebyshev polynomials. The Chebyshev basis satisﬁes the orthogonality condition

Z 1 Tm(x)Tn(x) π √ dx = (1 + δ0m) δmn . 2 (2.3.194) −1 1 − x 2

k Let k ∈ N and u : D → R. Then u(x) is said to be of differentiability class C on the dku dk+1u dku domain D if and only if dxk is continuous and dxk+1 is not continuous. If dxk is continuous ∞ for all k ∈ N, then u is of differentiability class C and is called smooth. If u is smooth and can be represented as a convergent power series in D, then it is of differentiability class Cω and is called analytic.

62 2.3. Pseudospectral Methods 2.3. Pseudospectral Methods

The Fourier cosine series of u is

∞ X u(x) = an cos(nx) . (2.3.195) n=0

k k+2 ∞ If u is a C function and is periodic in its derivatives, then |an| . 1/n . If u is a C function, −n then the coeﬃcients will behave like |an| ∼ e , converging faster than any polynomial [204,

205]. Alternatively, we can express u as a Chebyshev expansion

∞ X u(y) = anTn(y) . (2.3.196) n=0

If we make the transformation y = cos(x), then this becomes

∞ X u(cos(x)) = an cos(nx) . (2.3.197) n=0

That is, the Fourier cosine series coefficients representing u(cos(x)) are the same as the Cheby- shev series coefficients for u(y). This implies that the derived properties of the coefficients for the Fourier cosine series are also applicable to the coefficients of the Chebyshev series. From here, we can estimate the convergence rate of the residual as a function of the number of terms in the truncated Fourier or Chebyshev expansion. We find

∞ ∞ X X R = |u − uN | = anTn(x) ≤ |an| , (2.3.198) n=N+1 n=N+1 where the ﬁnal step follows from the triangle inequality and the fact that |Tn(x)| ≤ 1 for all n.

Hence, we have obtained an upper bound on R which converges to lowest order like |aN+1|.

Thus, if u is Ck, then R ∼ 1/N k+2. If u is C∞, then we recover exponential convergence in R.

It should be noted that all stated convergence behavior is in the regime N 1.

Chapter 2. Numerical Techniques in Relativity 63 Chapter 2. Numerical Techniques in Relativity

2.4 Cactus and the Einstein Toolkit

The Cactus framework is an open source problem solving environment designed for scientists and engineers [112]. It has a modular infrastructure, with the central core (“ﬂesh”) connecting to application modules (“thorns”). The Einstein Toolkit [109] is a collection of routines widely used by the numerical relativity community [206, 207, 110, 111, 208]. It provides support for a wide range of tools used to investigate black hole and neutron star initial data

(TwoPunctures and Lorene), numerical evolution with BSSNOK and CCZ4 formalisms

(LaZeV), relativistic magnetohydrodynamics (GRHydro and IllinoisGRMHD), method of lines and Runge-Kutta integration, unigrid and adaptive mesh reﬁnement (the PUGH and Carpet drivers), black hole horizons (AHFinderDirect and EHFinder), parallelized computing (MPI and OpenMP), data handling and visualization (HDF5), etc.

Cactus allows us to implement our own thorns, giving us the ability to extend and incorporate modules that are already included. The source code can be written in C, Fortran, or using wrappers.

2.4.1 The TwoPunctures Thorn

The Brandt and Brügmann technique for generating multiple black holes initial data separates the conformal factor into singular analytical parts plus a ﬁnite correction function u [191]. They show that the Hamiltonian constraint becomes a nonlinear elliptic equation for u, and that it reduces to the Laplace equation at the punctures. They assign an outer boundary condition on u to guarantee asymptotic ﬂatness, and then there show that there a unique solution which is C2 at the punctures and C∞ elsewhere.

Spectral series expansion is a powerful method for solving elliptic differential equations to high accuracy [209]. It is best to use on C∞ functions, where it enjoys exponential convergence with the number of terms in the truncated series. For functions only Ck, it converges at polynomial rates. Spectral methods would find terrific application solving the Hamiltonian constraint, if only the solution were smoother.

Ansorg, Brügmann, and Tichy [107] discovered that if one adopts coordinates that become

64 2.4. Cactus and the Einstein Toolkit 2.4. Cactus and the Einstein Toolkit spherical at the puncture, then u becomes C∞ everywhere. They construct a map from an inﬁnite Cartesian domain (x, y, z) to a compactiﬁed domain in prolate spheroidal coordinates

(A, B, ϕ) [210] with the property that coordinates become spherical near to two points on the boundary of the compactified domain. These are the locations of the two punctures. The interior of the compactified domain is filled uniformly with NA × NB × Nϕ collocation points.

Upon remapping to the Cartesian grid, the punctures sit on the x-axis at x = ±b. The transformation (A, B, ϕ) 7→ (x, y, z) is given explicitly by

A2 + 1 2B x = b , (2.4.199) A2 − 1 1 + B2 2A 1 − B2 y = b cos(ϕ) , (2.4.200) 1 − A2 1 + B2 2A 1 − B2 z = b sin(ϕ) . (2.4.201) 1 − A2 1 + B2

Images of these coordinate lines are shown in Fig. 2.4.

1 3

2.5 0.5 2 /b

B 0 1.5 ρ

1 -0.5 0.5

-1 0 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 A x/b

Figure 2.4: The TwoPunctures coordinate transformation. The left panel shows the com- pactiﬁed (A, B) coordinates with the angular direction suppressed. All collocation points are distributed uniformly, interior to the outer boundary. The right panel shows lines of constant p A and B in inﬁnite cylindrical coordinates (x, ρ = y2 + z2). To guide the eye, in both panels the central dashed line of constant B is made thicker and the locations of the punctures at x = ±b are marked with dots.

There are no collocation points on the x-axis, nor at inﬁnity. Inﬁnity and the punctures comprise the three boundaries of this elliptic equation, but they are not on the collocation point grid. Therefore, the solution is written to take boundary conditions into account explicitly.

Chapter 2. Numerical Techniques in Relativity 65 Chapter 2. Numerical Techniques in Relativity

The unknown function is expanded as a truncated Chebyshev series in the A and B directions with NA and NB terms, respectively. A truncated Fourier series with Nϕ terms is used in the ϕ direction. The task, now, is to determine the values of the series coefficients. We demand that the residual of the Hamiltonian constraint vanishes at every collocation point when expressed in terms of the series expansion solution. This forms a system of NANBNϕ nonlinear equations and NANBNϕ unknowns. An application of the chain rule yields a linear approximation of the Hamiltonian constraint. This allows for the series coefficients to be found iteratively using the multidimensional Newton-Raphson method. At each step, the solution to these linear equations is obtained via the biconjugate gradient stabilized method. This process is repeated until the maximum constraint residual over all collocation points is below the user specified tolerance, typically ∼ 10−9.

The resulting Bowen-York initial data is fast to generate, and can achieve high accuracy with relatively few collocation points.

2.4.2 The HiSpID Thorn

We take the TwoPunctures thorn as the starting point to develop our initial data solver.

The main limitations of the Bowen-York solutions stems from the requisite conformal ﬂatness.

We can build more realistic initial data with conformally curved background ansatz, solving all four constraint equations numerically. The HiSpID thorn began as a direct copy of the

TwoPunctures thorn from the May 2011 stable release of the Einstein Toolkit, and incorporated updates from the May 2013 release. We added functionality so as to evaluate all of the background ﬁelds to represent black holes with high spin and high speed, calculate the linear and nonlinear forms of all four constraint equations, measure the physical parameters of the resulting initial data solution, interpolate the physical ﬁelds onto the Carpet grid, and designate the initial gauge conditions for the lapse and shift.

66 2.4. Cactus and the Einstein Toolkit 2.4. Cactus and the Einstein Toolkit

2.4.2.1 Implementation of Conformally Curved Initial Data

In the place of a conformally ﬂat spatial metric, we construct initial data beginning with conformal superpositions of Schwarzschild and Kerr spatial metrics. We evaluate the corresponding extrinsic curvature tensors, connection coeﬃcients, Ricci and scalar curvature tensors, conformal factors, and all of their derivatives analytically.

Conformally curved spatial metrics force us to abandon the analytical Bowen-York solutions of the momentum constraint in favor of numerical solutions. If our ansatz happens to satisfy the maximal slicing condition (K = 0), then the momentum constraint decouples from the Hamiltonian constraint. Otherwise, they are coupled nonlinearly. In general, then, we wish to solve all four constraint equations simultaneously. This follows as a simple extension to the TwoPunctures algorithm. We are now solving 4NANBNϕ nonlinear equations for

4NANBNϕ unknowns. The momentum constraint is linearized using the chain rule in exactly the same manner as the Hamiltonian constraint. The multidimensional Newton-Raphson method is used to solve the resulting linear system of equations for the spectral series co- eﬃcients. The linear system is inverted using the biconjugate gradient stabilized method.

The Newton-Raphson iterations are repeated until the maximum constraint residual over all collocation points is below the user speciﬁed tolerance, typically ∼ 10−9.

2.4.2.2 Measuring Physical Parameters

A black hole binary conﬁguration is determined by ﬁfteen parameters: the separation b, the

i i puncture masses m(±), the local momenta P(±), and the local angular momenta S(±). They q (±) i 2 obey the cosmic censorship restriction Si S(±) < m(±), but can otherwise be given arbitrary values, in principle. However, it is important to remember that spacetime curvature implies that these parameters do not in general have the physical interpretation that their names and labels suggest. Therefore, it is important to extract measurements of the mass, momentum, and angular momentum of the initial data solution to quantify the physical state of the conﬁguration. This is done with the ADM integrals (2.2.150), (2.2.151), and (2.2.152).

After the constraint equations are solved, the spectral solutions are evaluated on a large

Chapter 2. Numerical Techniques in Relativity 67 Chapter 2. Numerical Techniques in Relativity spherical N × N grid. The ADM integrals are deﬁned over a sphere at spatial inﬁnity, but in practice we use the large (compared to the binary scale) coordinate radius r = 106b.

The integrals are approximated using Simpson’s method [211, 212]. The grid resolution N is speciﬁed by the user. For Simpson’s rule, N must be even, so if the user gives and odd number, we simply increment N → N + 1. By default N = 256. We use the spherical variables (θ, ϕ) in the asymptotic Cartesian coordinates. The sphere is divided into area elements by

π 2π θi = i , ϕj = j , (2.4.202) N N where i, j = 0,...,N. The oriented area element on the sphere has magnitude

h+f 2 dS = 2 r sin(θi) dθ dϕ , (2.4.203)

with     0 if i = 0 or i = N 0 if j = 0 or j = N     h = 1 if i ≡ 0 mod 2 , f = 1 if j ≡ 0 mod 2 . (2.4.204)       2 otherwise 2 otherwise

This routine is parallelized with OpenMP.

2.4.2.3 Populating the Evolution Grid

After the spectral solutions are solved on the NA × NB × Nϕ collocation point grid, the series

i solutions for u and b are evaluated on a Nx × Ny × Nz coordinate grid. We then work backwards to reconstruct the physical ﬁelds

4 γij = ψ γ˜ij , (2.4.205)

−2 1 4 Kij = ψ A˜ij + ψ γ˜ijK. (2.4.206) 3

These are saved to a Carpet grid, which is passed on to other thorns for processing and evolution.

68 2.4. Cactus and the Einstein Toolkit 2.4. Cactus and the Einstein Toolkit

The ﬁnal step in the TwoPunctures coordinate transformation goes from cylindrical coordinates (with the x-axis being the symmetry axis) to Cartesian coordinates ρi = (x, ρ, ϕ) 7→ xi = (x, y, z) by the relabeling

y = ρ cos(ϕ) , (2.4.207)

z = ρ sin(ϕ) . (2.4.208)

Conversely,

p ρ = y2 + z2 , (2.4.209) z ϕ = arctan . (2.4.210) y

Degeneracy in the ϕ coordinate when ρ = 0 results in a coordinate singularity along the x-axis in the Jacobian transformation matrix

  1 0 0 i   ∂ρ   = 0 cos(ϕ) sin(ϕ) (2.4.211) ∂xj    sin(ϕ) cos(ϕ)  0 − ρ ρ

In the original TwoPunctures solver, only the value of the scalar correction function u is 1 needed to reconstruct the physical solution with the conformal factor ψ = 1 + χ + u (see Eq. (2.2.169)). However, in the HiSpID solver, we also require the ﬁrst derivatives of the i ˜ ˜ ˜ correction vector b to build the conformal trace-free extrinsic curvature Aij = Mij +(Lb)ij (see Eq. (2.2.136)). Therefore, we must use the Jacobian matrix, and its gradient, which introduces the coordinate singularity into the initial data. Note that this occurs when interpolating the spectral solutions onto the Carpet grid for evolution, not when solving the constraints. This is because the x-axis is on the boundary of the compactiﬁed space (see Fig. 2.4), to which all collocation points are interior.

Our typical grid resolutions are such that only points directly on the x-axis suﬀer from the coordinate singularity, and all others are far enough removed that numerical error is not

Chapter 2. Numerical Techniques in Relativity 69 Chapter 2. Numerical Techniques in Relativity an issue. Nevertheless, we get around this problem by averaging a ﬁnite neighborhood of the x-axis. Suppose we want to evaluate bi(x, y, z), but ρ < ≡ TP_Tiny = 10−4 M, where

TP_Tiny is a free parameter. Then we construct the four-point average

i 1 i i i i b (x, y, z) = b (x, +, 0) + b (x, −, 0) + b (x, 0, +) + b (x, 0, −) , (2.4.212) 4 where all of the evaluations on the right hand side are ﬁnite. The same procedure is used for all of the derivatives of the correction functions, as well. The result is that our Carpet grid data is signiﬁcantly cleaner, as can be seen in an example in Fig. 2.5.

6 10-2 6 10-2 -4 -4 4 10 4 10 10-6 10-6 2 2 10-8 10-8 0 0 y/M 10-10 y/M 10-10 -2 -2 10-12 10-12

-4 10-14 -4 10-14

-6 10-16 -6 10-16 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x/M x/M

Figure 2.5: A comparison of the Hamiltonian constraint residual for a spinning black hole binary in the xy-plane without (left) and with (right) the x-axis averaging technique. One of the punctures is located at (x/M, y/M) = (0, 0) and the other is at (x/M, y/M) = (−12, 0).

70 2.4. Cactus and the Einstein Toolkit Chapter 3

Collisions from Rest of Highly Spinning Black Hole Binaries

General relativity is central to the modern understanding of much of astrophysics, from cosmological evolutions down to the end-state of large stars. Crucial to this is the correctness of the theory itself and the elucidation of its predictions. While much can be done using analytic techniques, one of the most interesting regimes—the merge-phase of compact-object binaries—requires the use of large-scale numerical relativity simulations. In order to evolve these systems, one needs appropriate initial data that allow for the simulation of binaries with astrophysically realistic parameters. Perhaps most important of all is the inclusion of large spins.

Highly spinning black holes are thought to be common. For example, supermassive black holes with high intrinsic spins are fundamental to the contemporary understanding of active galaxies and galactic evolution, in general.

In geometric units, a black hole’s spin magnitude S (i.e., intrinsic angular momentum) is bounded by its mass m, where the maximum dimensionless spin is given by χ ≡ S/m2 = 1.

While it is actually hard to have an accurate measure of astrophysical black hole spins, in a few cases the spins have been measured [213] and were found to be near the extremal value.

Since galactic mergers are expected to lead to mergers of highly spinning black holes, it is

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 71 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries important to be able to simulate black hole binaries with high spins in order to model the dynamics of these ubiquitous objects.

Spin can greatly affect the dynamics of a merging black hole binary. Important spin-based effects include the hangup mechanism [61], which delays or prompts the merger of the binary according to the sign of the spin-orbit coupling; the flip-flop of spins [214] due to a spin-spin coupling effect that is capable of completely reversing the sign of individual spins; and finally, highly spinning binaries may recoil at thousands of km s−1 [65, 215] due to asymmetrical emission of gravitational radiation induced by the black hole spins [216, 217]. These effects are maximized for highly spinning black holes.

No less important is the modeling of gravitational waveforms from compact object mergers.

Unlike low-spin binaries, highly spinning binaries can radiate more than 11% of their rest mass [218, 219], the majority of which emanates during the last moments of merger, down to the formation of a ﬁnal single spinning black hole. Eﬀorts to interpret gravitational wave signals from such systems require accurate model gravitational waveforms [220, 221, 222, 223,

224].

The moving punctures approach [200, 202] has proven to be very eﬀective in evolving black hole binaries with similar masses and relatively small initial separations, as well as small mass ratios [225] and large separations [226]. It is also eﬀective for more general multiple black hole systems [227], hybrid black-hole-neutron-star binaries [228], and gravitational collapse [229].

However, numerical simulations of highly spinning black holes have proven to be very challenging. The most commonly used initial data to evolve those binaries, which are based on the Bowen-York (BY) ansatz [106], use a conformally ﬂat three-metric. This method has a fundamental spin upper limit of χ = 0.928 [230, 231]. Even when relaxing the BY ansatz

(while retaining conformal ﬂatness), the spin is still bounded above by χ = 0.932 [232].

In order to exceed this limit and approach extremely spinning black holes with χ = 1, one must allow for a more general three-metric that captures the conformally curved aspects of the Kerr geometry. Dain showed [233] that it is possible to ﬁnd solutions to the initial value problem representing a pair of Kerr-like black holes. Dain’s method was tested for the case of

72 collisions from rest of spinning black holes using the moving punctures approach, which does not employ excision. These are compared to the BY data with spins up to χ = 0.90 [234].

Here, we revisit the problem of ﬁnding solutions to the puncture initial value problem representing two nearly extremely spinning black holes, and the subsequent evolution using the moving punctures approach—the most widespread method to evolve black hole binaries, implemented in the open source Einstein Toolkit [108, 109, 112, 110].

To solve for these new data, we construct a superposition of two conformally Kerr three- metrics with the corresponding superposition of Kerr extrinsic curvatures. Note that we do not use the Kerr-Schild slice [235] but the Boyer-Lindquist slice, amenable for puncture evolution. To regularize the problem, the superposition is such that very close to each black hole, the metric and extrinsic curvature are exactly Kerr [236]. We then simultaneously solve the Hamiltonian and momentum constraints for an overall conformal factor for the metric and nonsingular correction to the extrinsic curvature using a modiﬁcation of the TwoPunc- tures [107] spectral initial data solver. We refer to this extension as HIghly SPinning Initial

Data (HiSpID, pronounced “high speed”).

We evolve these data sets and ﬁnd that the spurious initial radiation is signiﬁcantly reduced compared to BY initial data for χ ≤ 0.9. This is a desirable feature, if one cannot incorporate the exact radiation content of the initial data (for post-Newtonian inspired radiation content into the initial data ansatz, see for example [237, 238, 239, 240]). We also perform accurate evolutions of highly spinning black holes with χ = 0.99, which has not been possible before for moving puncture codes.

This chapter is organized as follows. In Sec. 3.1 we present the formalism to solve for the initial data. We choose the standard transverse-traceless version since it provides the simplest set of equations and allows one to achieve both a reduction of the spurious initial radiation content and overcome the technical limits of the conformally ﬂat initial data reaching highly spinning black holes. We describe the explicit conformal decomposition and attenuation functions used to regularize the superposition of conformal Kerr black holes in the puncture approach.

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 73 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries

In Sec. 3.2 we describe the numerical techniques to solve for the initial data as an extension of those used to solve the Hamiltonian constraint with the TwoPunctures code. We also provide a summary of the evolution techniques used in the regime of parameters previously unexplored with the moving punctures approach.

In Sec. 3.3 we exhibit the convergence with spectral collocation points of the spinning initial data to levels of accuracy acceptable for evolution. We compare waveforms from the new HiSpID initial data to those of the standard spinning BY solution for χ = 0.90. We then evolve highly spinning black holes with χ = 0.99 from rest and discuss the results for the radiated energy and momenta as well as the horizon measures of mass and spin for the individual and ﬁnal black holes.

In Sec. 3.4 we study how the initial choice of the lapse and its subsequent gauge evolution [241] aﬀects the accuracy of the simulation at the typical marginal resolutions used to evolve highly spinning black holes in collisions from rest.

In Sec. 3.5 we brieﬂy experiment with a radial coordinate transformation which supports the horizon from collapsing in the extremal spin limit. The goal is to reduce the number of collocation points needed to resolve horizon features, and to evolve a black hole with χ = 0.999.

We summarize the results in Sec. 3.6.

3.1 Stationary Kerr Initial Data

To construct initial data representing a pair of spinning black holes, initially at rest with respect to each other, we use the conformal transverse-traceless (CTT) decomposition of the

Einstein ﬁeld equation. See Sec. 2.2.3 for details about CTT methods.

3.1.1 Conformal Kerr in Quasi-isotropic Coordinates

In spherical quasi-isotropic coordinates, the Kerr conformal spatial line element is [242, 233]

˜2 i j 2 2 2 2 4 4 2 d` =γ ˜ij dx dx = dr + r dΩ + a hr sin (θ) dϕ , (3.1.1)

74 3.1. Stationary Kerr Initial Data 3.1. Stationary Kerr Initial Data where m is the puncture mass, a is the angular momentum per unit mass, r is the quasi- isotropic radial coordinate, dΩ is the unit sphere line element, and [234]

m2 − a2 r¯ = r + m + , (3.1.2) 4r ρ2 =r ¯2 + a2 cos2(θ) , (3.1.3) 2mr¯ σ = , (3.1.4) ρ2 1 + σ h = . (3.1.5) ρ2r2

The Boyer-Lindquist coordinate r¯ appears as “r” (with Q = 0) in Eq. (2.1.89).

The nonvanishing components of the conformal extrinsic curvature associated with this metric are given by [233, 243]

2 HE sin (θ) A˜rϕ = , (3.1.6) r2 ˜ HF sin(θ) Aθϕ = , (3.1.7) r with the deﬁnitions

ρ2 e−2q = , (3.1.8) r¯2 + a2 1 + σ sin2(θ)

−q am 2 2 2 2 2 2 HE = e r¯ − a ρ + 2¯r r¯ + a , (3.1.9) ρ4 3 −q a mr¯ 2 2 2 2 HF = e m − a − 4r cos(θ) sin (θ) . (3.1.10) 2rρ4

The Kerr metric in quasi-isotropic coordinates admits spatial hypersurfaces satisfying the maximal slicing condition K = 0 [233].

The quasi-isotropic Kerr conformal factor is

rρ ψ = . (3.1.11) r

Ultimately, only the asymptotic behavior is important, so sometimes just the lowest order

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 75 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries terms of ψ are used [234] √ m2 − a2 ψ ≈ 1 + . (3.1.12) 2r

However, we ﬁnd that HiSpID initial data implementing ψ converges to the desired tolerance in approximately 1/3 as many iterations as compared to the approximation. Therefore, we utilize the exact conformal factor in Eq. (3.1.11).

Near the puncture, the conformal scalar curvature R˜ is ﬁnite

96a2 R˜ = − + O(r2) . (3.1.13) (m2 − a2)2

On the other hand, the Laplacian of the conformal factor diverges at the puncture

6a2 12ma2 D˜ 2ψ = − − + O(r) . (3.1.14) (m2 − a2)3/2r (m2 − a2)5/2

It follows that 1 288m2a2r sin2(θ) D˜ 2ψ − ψR˜ = − + O(r2) (3.1.15) 8 (m2 − a2)7/2 vanishes at the puncture.

All ﬁelds are transformed to a Cartesian basis, with coordinates related by

x = r sin(θ) cos(ϕ) ,

y = r sin(θ) sin(ϕ) , (3.1.16)

z = r cos(θ) .

The conformal spatial metric takes the form

2 γ˜ij = δij + a hvij , (3.1.17)

76 3.1. Stationary Kerr Initial Data 3.1. Stationary Kerr Initial Data

where δij is the Kronecker delta and

  y2 −xy 0     vij = −xy x2 0 . (3.1.18)     0 0 0

The nonvanishing Cartesian components of the trace-free conformal extrinsic curvature tensor are

2H1 sin(ϕ) cos(ϕ) A˜xx = −A˜yy = − , (3.1.19) r3 H1 cos(2θ) A˜xy = , (3.1.20) r3 H2 sin(θ) sin(ϕ) A˜xz = , (3.1.21) r3 H2 sin(θ) cos(ϕ) A˜yz = − , (3.1.22) r3 where

2 H1 = HF cos(θ) + HE sin (θ) , (3.1.23)

H2 = HF − HE cos(θ) . (3.1.24)

At the puncture, these functions have series expansion

2 2 H1 ∼ 3am sin (θ) + O(r ) , (3.1.25)

2 H2 ∼ −3am cos(θ) + O(r ) . (3.1.26)

3 Thus, in Cartesian coordinates A˜ij ∼ O 1/r at the puncture.

At this point, the spin is parallel to the z-axis. The speciﬁed local angular momentum

i p i vector has Cartesian components S , so that a = S Si/m. All of the ﬁelds are then rotated

i i ijk z through an angle φ = arccos(δzSi) about the axis u = δj Sk such that they are oriented in i i i i the desired direction, spin aligned with S . The rotation matrix R j from orientation δz to S

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 77 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries has components i i i k i R j = cos(φ)δj − sin(φ) jku + [1 − cos(φ)]u uj . (3.1.27)

The rotation matrix is orthogonal, meaning that its transpose is its inverse, and it is a proper

i transformation det(R j) = 1.

3.1.2 Punctures

In the puncture approach, the conformal factor is written as singular parts plus a ﬁnite correction [191]

ψ = Ψ + u , (3.1.28) where Ψ contains the conformal factors associated with the individual, isolated black holes

(see Sec. 3.1.3). This turns the Hamiltonian constraint (2.2.128) into an elliptic diﬀerential equation for u ij (Ψ + u)R˜ A˜ijA˜ D˜ 2u + D˜ 2Ψ − + = 0 . (3.1.29) 8 8(Ψ + u)7

Inspired by Eqs. (2.2.136) and (3.1.28), we write our ansatz for the extrinsic curvature as

˜ ˜ ˜ Aij = Mij + (Lb)ij , (3.1.30) where the first term is a freely specifiable symmetric, trace-free tensor [176], and the second term is the vector gradient of the correction functions bi (see Eq. (2.2.132)). It contains the known singular, symmetric, trace-free extrinsic curvature tensors for individual Kerr black holes (see Sec. 3.1.3). This turns the momentum constraint (2.2.137) into differential equations for the vector components bi ˜ i ˜ ˜ ij ∆Lb + DjM = 0 , (3.1.31) where the first term is the vector Laplacian of bi (see Eq. (2.2.134)).

We impose Robin boundary conditions

i lim ∂r(ru) = 0 and lim ∂r(rb ) = 0 . (3.1.32) r→∞ r→∞

78 3.1. Stationary Kerr Initial Data 3.1. Stationary Kerr Initial Data

These are enforced numerically by noting that the TwoPunctures compactiﬁed coordinate

A obeys A → 1 as r → ∞, thus allowing the asymptotic behavior to be factored out (e.g. write u = (A−1)U and solve for the auxiliary function U) [107]. Thus, the Robin boundary condition is enforced as a Dirichlet boundary condition at A = 1 on the compactiﬁed domain. The remaining boundary conditions are “behavioral,” in that the spectral basis functions possess automatic regularity properties [209]. For example, solutions represented by Fourier series are intrinsically periodic.

3.1.3 Attenuation and Superposition

The two punctures “(+)” and “(−)” are initially located at coordinates (x, y, z) = (±b, 0, 0), respectively. The coordinate distance to each puncture is

p 2 2 2 r(±) = (x ∓ b) + y + z . (3.1.33)

We write all of the background ﬁelds as simple superpositions

(+) (−) γ˜ij = δij + f(+) γ˜ij − δij + f(−) γ˜ij − δij , (3.1.34) ˜ ˜(+) ˜(−) Mij = Aij + Aij , (3.1.35)

Ψ = ψ(+) + ψ(−) − 1 (3.1.36)

˜(±) ˜ with scalar functions f(±). Since Aij are symmetric and trace-free, then so too is Mij.

In an effort to tame the singularities in Eqs. (3.1.29) and (3.1.31), we introduce attenuation functions. We seek functions that have the following properties: at the location of (+), the contribution from (−) falls to zero sufficiently fast, and vice versa; towards spatial infinity, the spatial metric becomes flat and the extrinsic curvature vanishes. We achieve this with

p r(∓) f(±) = 1 − exp − , (3.1.37) ω(±)

where ω(±) are parameters controlling the steepness of the attenuation. We take the smallest

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 79 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries possible power index p = 4. This guarantees that the gradient of the attenuation function vanishes at least as fast as O(r3) at the punctures, which cancels the O(1/r3) divergence in the trace-free extrinsic curvature. (±) ˜ ˜ (±) ˜ ˜ As r(±) → 0, then γ˜ij → γ˜ij , and consequently Di → Di and R → R(±). Consider ˜ 2 ˜ Eq. (3.1.29). Near to the punctures, D(±)ψ(±) − ψ(±)R(±)/8 cancels to a finite value by ˜ 2 ˜ −7 ˜ ˜ij virtue of Eq. (3.1.15), while D(±)ψ(∓) − ψ(∓) − 1 + u R(±)/8 is finite. The term ψ AijA vanishes at least linearly (see Sec. 2.2.6.2). Consider, now, Eq. (3.1.31). Near to the punctures, ˜ (±) ˜ij ˜ (±) ˜ij Dj A(±) = 0 is an exact solution to the single black hole momentum constraint, and Dj A(∓) is finite.

Even with attenuation, round-off error makes the divergence in the source term in (3.1.31) difficult to calculate numerically near either of the punctures (outside of the stuffing region).

We alleviate this by rewriting the source with the divergence removed explicitly

˜ ˜ij ˜j ˜(±)j ˜ki ˜i ˜(±)i ˜jk DjA(±) = Γjk − Γjk A(±) + Γjk − Γjk A(±) . (3.1.38)

˜(±)i We define Γjk as the Christoffel symbols associated with the isolated, unattenuated conformal (±) ˜ij metric γ˜ij . Equation (3.1.38) has the additional benefit that no derivatives of A(±) need be calculated.

3.2 Numerical Techniques

Now that the background ﬁelds have been chosen, it is time to solve the constraints and evolve the initial data. The elliptic constraint equations are solved with a spectral series solution, where the nonlinearities handled by the Newton-Raphson method. The gauge is ﬁxed and the initial data are evolved with the moving punctures approach.

3.2.1 Initial Data Solver with Spectral Methods

The TwoPunctures thorn [107] generates conformally ﬂat (γ˜ij = δij) initial data via a spectral expansion of the Hamiltonian constraint on a compactiﬁed collocation point grid.

80 3.2. Numerical Techniques 3.2. Numerical Techniques

The residual values of the constraint are minimized at the collocation points, yielding a series solution that is interpolated onto a Carpet grid [110] to be evolved. The three momentum constraints are satisﬁed analytically for conformally ﬂat initial data by the BY solutions representing black holes possessing linear and angular momentum [183, 244].

Conformally Kerr initial data requires that all four constraint equations be solved numerically. This is achieved by extending TwoPunctures to solve for u and bi simultaneously at each collocation point. The solver handles the nonlinearities in the constraint equations by using a linearized Newton-Raphson method. The constraint Eqs. (3.1.29) and (3.1.31) are linearized using the chain rule

ij ! 1 7A˜ijA˜ A˜ij D˜ 2du − + R˜ du + (˜ db)ij = 0 , (3.2.39) 8 ψ8 4ψ7 L

˜ i ∆L db = 0 , (3.2.40) where du and dbi represent small changes in u and bi, respectively. The terms in the linearized constraints have the behavior described in Sec. 3.1.

3.2.2 Evolution

We use the extended TwoPunctures thorn to generate puncture initial data [191] for the black hole binary simulations. These data are characterized by mass parameters mp (which are not the horizon masses), as well as the momentum and spin of each black hole, and their initial coordinate separation. We evolve these black hole binary data sets using the LazEv [118] implementation of the moving punctures approach with the conformal function W = exp(−2φ) suggested by Ref. [69]. For the runs presented here, we use centered, eighth-order finite differencing in space [227] and a fourth-order Runge-Kutta time integrator. (Note that we do not upwind the advection terms.) Our code uses the Cactus/EinsteinToolkit [112, 109] infrastructure. We use the Carpet mesh refinement driver to provide a “moving boxes” style of mesh refinement.

We locate the apparent horizons using the AHFinderDirect code [245] and measure the

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 81 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries horizon spin using the isolated horizon (IH) algorithm detailed in [246].

For the computation of the radiated angular momentum components, we use formulas based on “ﬂux-linkages” [247] and explicitly written in terms of Ψ4 in [248, 249].

We obtain accurate, convergent waveforms and horizon parameters by evolving this system in conjunction with a modiﬁed 1+log lapse and a modiﬁed Gamma-driver shift condition [203,

200, 250]. The lapse and shift are evolved with

i 2 (∂t − β ∂i)α = −α f(α)K, (3.2.41a)

a 3 a a ∂tβ = Γ˜ − ηβ . (3.2.41b) 4

In the original moving punctures approach we used f(α) = 2/α and an initial lapse α(t =

−2 4 0) = ψBL [200] or α(t = 0) = 2/(1 + ψBL) [61], where ψBL = 1 + m(+)/(2r(+)) + m(−)/(2r(−)).

3.3 Collisions from Rest with Nearly Extremal Spin

To assess the characteristics of this superposed Kerr black hole initial, data we compare with the corresponding BY-type. For given binary separations and spin parameters, the horizon masses and spins are not identical, as shown in Fig. 3.1, since the initial radiation content and distortions are not the same. However, we consider them close enough for comparisons of physical quantities such as the gravitational waveforms.

We study a few test cases of equal-mass black hole binary conﬁgurations starting from rest with spins aligned (UU) or counter aligned (UD) with each other, and perpendicular to the line joining the black holes. We evolve both black hole binaries with the HiSpID data and the standard BY choice (for spins within the BY limit). We also evolve black hole binaries with near-extremal spin, χ = 0.99, a regime unreachable for BY initial data. Table 3.1 gives the initial data parameters of these black hole binary conﬁgurations.

Figure 3.2 shows a comparison of waveforms rΨ4 extracted at an observer location r =

75M. We clearly see that the initial radiation content (located around t ∼ 80M) of the BY data for equal mass spinning black hole binaries with χ = 0.9 has an amplitude comparable to

82 3.3. Collisions from Rest with Nearly Extremal Spin 3.3. Collisions from Rest with Nearly Extremal Spin

Table 3.1: Initial data parameters for the equal-mass, collision from rest configurations. The punctures are initially at rest and are located at (±b, 0, 0) with spins S aligned or anti-aligned with the z direction, mass parameters mp, horizon (Christodoulou) masses m1 = m2 = mH, 2 total ADM mass MADM, and dimensionless spins a/mH = S/mH, where a1 = a2 = a. The Bowen-York configurations are denoted by BY, and the HiSpID by HS. Finally, UU or UD denote the direction of the two spins, either both aligned (UU), or anti-aligned (UD). 2 Configuration b/M mp S/M a/mH mH MADM/M BY90UU 6 0.191475 0.225 0.8977 0.500702 0.982362 HS90UU 6 0.5 0.225 0.8958 0.501287 0.982353 BY90UD 6 0.191475 0.225 0.8977 0.500702 0.982396 HS90UD 6 0.5 0.225 0.8955 0.501208 0.982388 HS99UU 6 0.5 0.2475 0.9896 0.500162 0.980124 HS99UD 6 0.5 0.2475 0.9887 0.499981 0.980163

0.502 0.9818

0.501 0.9816

0.500 0.9814

0.499 0.9812 /M /M 0.498 f (+/-) M

m 0.9810 0.497 0.9808 0.496

0.9806 0.495

0.920 0.4655

0.4654 0.915 0.4653

0.4652

0.910 f χ (+/-)

χ 0.4651

0.905 0.4650

0.4649 0.900 0.4648

0.895 0.4647 0 10 20 30 40 50 60 100 120 140 160 180 200 220 240 t/M t/M

Figure 3.1: Masses of the individual black holes from the start of the simulation until merger (top left). Spins of the individual black holes until merger (bottom left). Masses of the remnant black hole (top right). Spin of the remnant black hole (bottom right). BY in blue, HiSpID in red. that of the physical merger. On the other hand, our Kerr-like initial data has greatly reduced initial radiation content (one order of magnitude smaller). Although not apparent in these

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 83 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries plots, the much lower initial radiation content is not only more physical, but also leads to more accurate computations of waveforms. This initial pulse reﬂects from the reﬁnement boundaries

(since they are not perfectly transmissive) leading to high frequency errors and convergence issues when looking at much ﬁner details of the waveform phase [251, 252].

0.010

0.005 ]

4 2,2 0.000 Ψ

r M Re[ -0.005

-0.010

0.015 ]

4 2,0 0.005 Ψ r M Re[

-0.005

-0.015 0 50 100 150 200 250 t/M

Figure 3.2: (` = 2, m = 2) mode of Ψ4 at r = 75M (above). (2,0) mode of Ψ4 at r = 75M (below) for spinning binaries with χ = 0.9. BY data in blue (showing much larger initial burst), HiSpID in red.

One of our main motivations to study a new set of initial data is to be able to simulate highly

84 3.3. Collisions from Rest with Nearly Extremal Spin 3.3. Collisions from Rest with Nearly Extremal Spin spinning black holes, beyond the BY (or conformally ﬂat) limit, χ ≈ 0.93 [253, 231, 232]. In

Fig. 3.3 we show the level of satisfaction of the constraints for our new initial data for spinning black hole binaries with equal masses and spin parameters χ = 0.99. The physical ﬁelds are reconstructed using interpolations of the correction function series approximations, evaluated on a grid with Nx, Ny, and Nz points in the x-, y-, and z-directions, respectively. We observe an exponential convergence of the root-mean-square (RMS) norm of the constraint violations with spectral collocation points down to a level of ≈ 10−8, which is accurate enough to start black hole binary evolutions. The RMS is deﬁned by

v u NxNyNz u 1 X 2 RMS = t [R(xi)] , (3.3.42) N N N x y z i where the sum is over all NxNyNz grid points, and R(xi) are the constraint residuals at grid point xi. Events inside the black hole horizons do not eﬀect the outside space. We do not consider points interior to the horizons in our RMS calculations. If one requires greater satisfaction of the constraints, one can ﬁne-tune the attenuation functions to that end.

10-1 H -2 10 Mx y -3 M 10 Mz 10-4 10-5 10-6 -7

Norm 10 10-8 10-9 10-10 10-11 10-12 0 16 32 48 64 80 96 112 N

Figure 3.3: Convergence of the residuals of the Hamiltonian and momentum constraints versus number of collocation points N for black hole binaries with χ = 0.99 in the UU conﬁguration with exponential attenuation parameters ω(±) = 0.2 and p = 4.

The evolution of black holes with χ = 0.99 requires high resolution, particularly during the ﬁrst 10M of evolution, but otherwise proceeds with the standard moving punctures set

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 85 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries

Table 3.2: The final mass, final remnant spin, and recoil velocity for each configuration. −1 Configuration Mrem/M χrem V [km s ] BY90UU 0.98053 0.46554 0 HS90UU 0.98162 0.46483 0 BY90UD 0.98073 0 35.90 HS90UD 0.98181 0 36.01 HS99UU 0.97971 0.52501 0 HS99UD 0.97900 0 38.40 up [200]. Even if the initial data had no initial orbital angular momentum, the case of a black hole binary from rest in the UU configuration radiates angular momentum due to mutual frame dragging effects in the opposite direction, as observed in [62]. On the other hand, the

UD conﬁguration leads to a recoil velocity, that can be modeled as [217]

X j Vrecoil = kj∆ , (3.3.43) j=1,3,5...

2 where ∆ = (S(−) − S(+))/(m(+) + m(−)) and the kj are ﬁtting constants (this form applies only to equal mass binaries with vanishing total spin). A summary of the properties of the

ﬁnal merger remnant black hole are listed in Table 3.2.

Figure 3.4 shows the waveforms of the UU and UD cases for highly spinning black holes.

It confirms that the initial radiation content has a much smaller amplitude than the merger waveform, significantly reducing contamination of the physical signals by unresolved high- frequency reflections.

3.4 Gauge Conditions

Notably, the simple choice of the initial lapse α0 = 1/(2ψBL −1) has advantages over the other choices studied for the entire evolution by providing increased accuracy and computational eﬃciency. Here, we display the results of the evolution from rest of black hole binaries with intrinsic spin χ = 0.99. Figure 3.5 shows that we can achieve comparably accurate results with many fewer grid points. The curves follow closely to each other, but with the new lapse we use 80 points per dimension compared to the 125 needed with the original initial lapse.

86 3.4. Gauge Conditions 3.4. Gauge Conditions

0.012 ]

4 2,2 0.004 Ψ

-0.004 r M Re[

-0.012

0.012 |

4 2,2 0.008 Ψ

r M | 0.004

0 0 50 100 150 200 50 100 150 200 250 t/M t/M

Figure 3.4: Waveforms and waveform magnitudes of the UU (left) and UD (right) conﬁgura- tions with highly spinning black holes χ = 0.99. Note the small amplitude of the initial data radiation content at around t = 75M, compared to the merger signal after t = 130M.

This provides a speed up factor of (125/80)4 ∼ 6.

The improvement of the new initial lapse also translates into a more accurate description of the ﬁnal remnant black hole, as shown in Fig. 3.6. Note that even at lower resolutions we observe a similar gain.

We interpret these results as indicating that a better choice of the initial lapse leads to a better coordinate evolution. The intensities of the initial gauge waves are reduced, thus allowing a better distribution of grid points, resulting in a more eﬃcient numerical computation.

See for instance the horizon coordinate radius evolution in Figs. 3.5 and 3.6.

Figure 3.7 displays the eﬀects of the initial lapse on the waveform. We see the notable reduction of the unphysical oscillations pre-merger while reproducing accurately the physical merger waveform for the dominant modes (`, m) = (2, 0) and (`, m) = (2, 2). Note that this reduction of the errors due to improved gauge choices is in addition to and independent from the reduction of the initial burst of radiation (with respect to BY data) that has a physical content, despite being an undesirable eﬀect.

Figs. 3.5, 3.6, and 3.7 show three diﬀerent resolutions for the waveforms and horizon quantities for the highly spinning χ = 0.99 case. The mass and spin parameters for this higly spinning conﬁguration are tracked all the way to merger in Fig. 3.8. The convergence order

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 87 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries

0.501

0.500

0.499 /M (+/-) m 0.498

0.497

0.990

0.985

(+/-) 0.980 χ

0.975

0.970

0.11

0.10

0.09 /M

(+/-) 0.08

0.07

Average r 0.06 2/(1+ψ4) 80 0.05 2/(1+ψ4) 100 2/(1+ψ4) 125 0.04 1/(2ψ-1) 64 1/(2ψ-1) 72 1/(2ψ-1) 80 0.03 0 10 20 30 40 50 60 t/M

Figure 3.5: The effects of the choice of the initial lapse on the individual black hole masses (top), spins (middle) and coordinate radii (bottom). The benefits of the new initial lapse are evident since they follow the higher resolution behavior with many fewer grid points by (80/125)3. was calculated for each of these quantities. For the individual black hole spin, irreducible mass, horizon mass, and dimensionless spin, we find average convergence orders of 7.6, 6.2, 8.2, and

8.2, respectively. The same quantities for the ﬁnal remnant black hole have convergence orders between 3.3 and 4.3. For the amplitude and phase of the 2,2 mode, we ﬁnd a convergence

88 3.4. Gauge Conditions 3.4. Gauge Conditions

0.982

0.981

0.980

0.979

0.978 /M rem

m 0.977

0.976

0.975

0.974

0.530

0.520

0.510

rem 0.500 χ

0.490

0.480

0.470

1.18

1.16

1.14

/M 1.12 rem 1.10

1.08 Average r 1.06 2/(1+ψ4) 80 4 1.04 2/(1+ψ ) 100 2/(1+ψ4) 125 1/(2ψ-1) 64 1.02 1/(2ψ-1) 72 1/(2ψ-1) 80 1.00 80 100 120 140 160 180 200 220 240 t/M

Figure 3.6: The effects of the choice of the initial lapse on the final black hole mass (top), spin (middle) and coordinate radius (bottom). The benefits of the new initial lapse are evident since they follow the higher resolution behavior with many fewer grid points.

order between 3 and 4 after the black holes merge.

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 89 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries

0.012

0.008

0.004 ] 22 4 Ψ 0

r M Re[ -0.004

-0.008

-0.012

2/(1+ψ4) 80 4 0.012 2/(1+ψ ) 100 2/(1+ψ4) 125 1/(2ψ-1) 64 0.008 1/(2ψ-1) 72 1/(2ψ-1) 80

] 0.004 20 4 Ψ 0

r M Re[ -0.004

-0.008

-0.012

80 100 120 140 160 180 200 220 240 t/M

Figure 3.7: The eﬀects of the choice of the initial lapse on the waveforms. The beneﬁts of the new initial lapse are evident since they show much less initial noise before the merger.

3.5 Fisheye Transformations

While it is possible to evolve black hole binary initial data in initially quasi-isotropic coordinates, Liu et al. [254] found that it was more efficient to perform a fisheye transformation [255] to the background Kerr metric to open up the throats of the holes in the numerical coordinates. The radial quasi-isotropic coordinate suffers from the fact that the black hole horizon √ 2 2 at rH = m − a /2 shrinks to zero in the a → m limit. We need collocation points inside the horizon to capture the dynamics accurately, especially for high spins. We could simply ramp up the number of collocation points N in each direction, but this becomes computationally

90 3.5. Fisheye Transformations 3.5. Fisheye Transformations

0.502 0.99

0.5 0.985 0.498

0.496 0.98 (+/-)

0.494 (+/-) χ m 0.492 0.975

0.49 0.97 0.488 80 80 96 96 0.486 0.965 0 50 100 150 200 250 0 50 100 150 200 250 t/M t/M

Figure 3.8: Evolution of the mass (left panel) and dimensionless spin parameter (right panel) for a collision from rest through to merger of two black holes, each with χ = 0.99. The two curves in each panel represent diﬀerent grid resolutions. The sudden spike near the end of the curves represents the failure of the isolated horizon ﬁnder when the black holes are very near to each other.

expensive rather quickly, since the number of simultaneous equations solved at each step scales like O(N 3). An alternative method is to redistribute the coordinates near the puncture so that the horizon settles to a ﬁnite coordinate radius, even at extremal spin. This is accomplished by the use of the coordinate rˆ, deﬁned implicitly by [254]

2 r¯+ r¯ =r ˆ 1 + , (3.5.44) 4ˆr

√ 2 2 where r¯± = m ± m − a are the Boyer-Lindquist radii of the inner (−) and outer (+) horizons of the black hole.

The conformal spatial line element in these ﬁsheye coordinates is

2 rˆ + r¯+ ˜2 4 2 2 2 A 2 2 d` = dˆr +r ˆ dθ + 4 rˆ sin (θ) dϕ , (3.5.45) rˆ(¯r − r¯−) ρ with

A = (¯r2 − a2)2 − (¯r2 − 2mr¯ + a2)a2 sin2(θ) . (3.5.46)

This choice of conformal metric utilizes (3.1.11). The horizon is located at rˆH =r ¯+/4, and has the value rˆH = M/4 > 0 when χ = 1. The nonvanishing components of the trace-free,

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 91 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries conformal extrinsic curvature tensor are

2 ma sin (θ) 4 2 2 4 2 2 2 2 r¯+ A˜rϕˆ = 3¯r + 2a r¯ − a − a (¯r − a ) sin (θ) 1 + , 2p (3.5.47) rρˆ rAˆ (¯r − r¯−) 4ˆr 3 3 ˜ 2a mr¯cos(θ) sin (θ) r¯+ √ Aθϕ = − √ rˆ − r¯ − r¯− . (3.5.48) rρˆ 2 rAˆ 4

The slice remains maximal, K = 0, and we use the stationary Kerr conformal factor (3.1.11) with the ﬁsheye radial coordinate.

We construct fisheye black hole binary initial data using the same superposition and attenuation methods as outlined in Secs. 3.1.2 and 3.1.3. We find solutions to the Hamiltonian and momentum constraints as in Sec. 3.2. The constraint residual as a function of the number of collocation points for a black hole binary, each with χ = 0.999, is shown in Fig. 3.9. The evolution of the mass and dimensionless spin of a single black hole using the fisheye coordinate is shown in Fig. 3.10.

10-2 H x -3 M 10 My Mz 10-4

10-5

10-6 Norm 10-7

10-8

10-9

10-10 0 16 32 48 64 80 96 N

Figure 3.9: Convergence of the residuals of the Hamiltonian and momentum constraints versus number of collocation points N for black hole binaries in ﬁsheye coordinates with χ = 0.999 in the UU conﬁguration with exponential attenuation parameters ω(±) = 0.2 and p = 4.

92 3.5. Fisheye Transformations 3.6. Summary

0.7275 0.9996 0.727 0.9994 0.7265 0.726 0.9992 0.7255 0.725 0.999 /M χ irr 0.7245 M 0.9988 0.724 0.7235 0.9986 0.723 0.9984 0.7225 0.722 0.9982 0 10 20 30 40 50 60 0 10 20 30 40 50 60 t/M t/M

Figure 3.10: Evolution of a single black hole with χ = 0.999. The left panel shows the mass evolution of the puncture, and the right panel shows the dimensionless spin evolution. The evolution grid has eight reﬁnement levels, the ﬁnest having resolution M/240 with an extent of 0.2M.

3.6 Summary

In this chapter we have been able to implement puncture initial data for highly spinning black hole binaries by attenuated superposition of conformal Kerr metrics. We modiﬁed the

TwoPunctures thorn, in the Cactus/Einstein Toolkit framework, to solve the Hamil- tonian and momentum constraint equations simultaneously for highly spinning black holes.

We verified the validity of the data by showing convergence of the Hamiltonian and momentum constraint residuals with the number of collocation points in the spectral solver. We then showed, by evolving this data, that the radiation content of these initial data was much lower than the standard conformally flat choice at spins χ = 0.9. This produced a more accurate and realistic computation of gravitational radiation waveforms. These cleaner initial data allowed us to explore different choices of the moving punctures gauge (initial lapse and its evolution).

We go on to produce the ﬁrst simulation of the collision from rest of a black hole binary with

χ = 0.99 using the moving punctures approach.

Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries 93 Chapter 3. Collisions from Rest of Highly Spinning Black Hole Binaries

94 3.6. Summary Chapter 4

Quasi-circular Orbits and High Speed Collisions Without Spin

We consider Lorentz-boosted Schwarzschild black holes in a quasi-circular orbital configuration. It is important to minimize the spurious radiation content of the initial data in order to achieve a very accurate gravitational waveform computation. We present here the results of using superposed boosted black holes instead of the traditional Bowen-York (BY) solution for the puncture initial data approach. BY solutions are also limited in the maximum boost black holes can achieve (up to P/MADM = 0.897, see Ref. [253]), while our new data do not suffer this restriction since it does not use the conformally flat three-metric ansatz.

The problem of the high energy collision of two black holes is of interest from both the theoretical point of view, to test gravity in its most extreme regime, and experimentally, since increasingly high energy particle collisions could eventually produce a nonnegligible probability for generating black hole pairs (see Ref. [256] for a review).

The production of gravitational waves and the properties of the ﬁnal remnant after the collision of two black holes has been the subject of theoretical study for over half a century, with notable results such as the area theorems by Hawking and Penrose [257, 258] and their application to bounds on the energy radiated via gravitational waves. For instance, they place the maximum radiated energy from a head-on collision of nonspinning black holes within 29%

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 95 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin of the total mass.

More detailed estimates of the radiated energy have been computed by applying perturbation theory [259] to the collision of ultrarelativistic black holes represented by shock waves [260].

Those computations reduce the above bound to 25% for ﬁrst order corrections and to 16.4% for second order corrections of the maximum energy radiated away from a nonspinning head-on collision of two equal mass black holes.

Complete full numerical simulations of such collisions are now possible thanks to the break- throughs in numerical relativity [261, 200, 202]. The ﬁrst full numerical study of the head-on collision of black holes [262] places the maximum eﬃciency of gravitational radiation at 14±3%.

Those studies have been extended to grazing collisions [263], leading to an estimate of 35% for the maximum energy radiated at a critical impact parameter. Further studies including black hole spins [264] show that, at high energies, the structure (i.e. the spin) of the holes tends to be irrelevant for the collision outcomes.

The latest theoretical computations of the energy radiated by the head-on collision of two, equal mass, nonspinning black holes include an estimate of 13.4% based on black hole thermo- dynamics arguments [265] and 17% based on a multipolar analysis of the zero-frequency-limit

(ZFL) approach [266].

We revisit the full numerical head-on computation incorporating new techniques that notably improve the accuracy of the simulations. Those techniques include new initial data with reduced spurious radiation content [267], improved extraction techniques with second order perturbative extrapolation [268], and the use of new gauges [241] and evolution systems [117] in the moving puncture approach [200].

This chapter is organized as follows. In Sec. 4.1 we present the formalism to solve for the initial data. We choose the standard transverse-traceless version since it provides the simplest set of equations and allows one to achieve both a reduction of the spurious initial radiation content and overcome the technical limits of conformally ﬂat, maximally sliced initial data, allowing for cleaner simulations of black holes in quasi-circular orbits and in high energy head- on collisions. We describe the explicit conformal decomposition and attenuation functions used

96 4.1. Lorentz-boosted Schwarzschild Initial Data to regularize the superposition of Lorentz-boosted Schwarzschild black holes in the puncture approach.

In Sec. 4.2 we describe the numerical techniques used to solve for the initial data as an extension of those used to solve the Hamiltonian constraint with the TwoPunctures code.

We also provide a summary of the evolution techniques used in the regime of parameters previously unexplored with the moving punctures approach.

In Sec. 4.3 we show the convergence with spectral collocation points of the initial data for nonspinning Lorentz-boosted black holes. We compare waveforms for our new initial data with the standard boosted BY solution in quasi-circular orbit to highlight the beneﬁts of lower initial spurious radiation of our data.

In Sec. 4.4 we study how the initial choice of the lapse and its subsequent gauge evolution [241] aﬀects the accuracy of the simulation at the typical marginal resolutions used to evolve highly spinning black holes in head-on collisions.

In Sec. 4.5 we evolve black hole binaries in high energy head-on collisions and examine the radiated gravitational wave signals. We extrapolate to the inﬁnite momentum conﬁguration, and estimate the maximum radiated energy from boosted head-on collisions.

We summarize the results in Sec. 4.6.

4.1 Lorentz-boosted Schwarzschild Initial Data

To construct initial data representing a pair of black holes with linear momentum, we use the conformal transverse-traceless (CTT) decomposition of the Einstein ﬁeld equation. See

Sec. 2.2.3 for details about CTT methods.

4.1.1 Conformal Boosted Schwarzschild in Isotropic Coordinates

The Schwarzschild metric can be brought into isotropic coordinates through the radial coordinate transformation M 2 R = 1 + r . (4.1.1) 2r

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 97 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

This is accompanied by a transformation from spherical coordinates (r, θ, ϕ) to Cartesian coordinates (x, y, z) by

x = r sin(θ) cos(ϕ) , (4.1.2)

y = r sin(θ) sin(ϕ) , (4.1.3)

z = r cos(θ) . (4.1.4)

The spacetime metric has components

  −α2 0 0 0      0 ψ4 0 0    gµν =   (4.1.5)  4   0 0 ψ 0    0 0 0 ψ4 and line element becomes

M !2 4 2 1 − 2r 2 M 2 2 2 ds = − dt + 1 + dx + dy + dz . (4.1.6) M 2r 1 + 2r

M 1− 2r In terms of the 3 + 1 line element variables, we read oﬀ the lapse function α = M , shift 1+ 2r i 4 M vector β = 0, spatial metric γij = ψ δij, and conformal factor ψ = 1 + 2r .

In isotropic coordinates, it is easy to identify many important features of the Schwarzschild black hole. The spatial metric is conformally flat, which results in tremendous simplification of the conformal constraint and evolution equations. The vanishing shift and time independent spatial metric indicates that the initial time slice represents a moment of time symmetry, so that the extrinsic curvature vanishes on each spatial hypersurface, Kij = 0. The conformal spatial scalar curvature is, naturally, R˜ = 0. The Hamiltonian constraint reduces to the flat space Laplace equation i ∂ ∂iψ = 0 , (4.1.7) and the momentum constraint is satisfied by the trivial solution. Exploiting the linearity of

98 4.1. Lorentz-boosted Schwarzschild Initial Data 4.1. Lorentz-boosted Schwarzschild Initial Data the ﬂat Laplace operator, we can immediately generalize to initial data representing N black holes beginning from rest by writing

N X Mn ψ = 1 + , (4.1.8) 2r n n where rn is the coordinate distance to the center of the black hole with mass Mn.

To describe a black hole with arbitrary linear momentum P i, we begin with the Schwarzschild line element in isotropic Cartesian coordinates (4.1.6). Next, we perform a Lorentz transformation in the y-direction, which has components

  γ 0 −vγ 0      0 1 0 0 µ   Λ ν =   , (4.1.9)   −vγ 0 γ 0   0 0 0 1 where v is the magnitude of the local velocity vector

P i vi = p 2 j (4.1.10) m + P Pj and 1 γ = √ (4.1.11) 1 − v2 is the Lorentz factor. The associated transformation of coordinate labels xµ = (t, x, y, z) is given by µ µ ν x → Λ νx , (4.1.12)

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 99 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin and the spacetime metric has components

  −γ2 α2 − v2ψ4 0 vγ2 α2 − ψ4 0      0 ψ4 0 0  σ τ   gµν → Λ µΛ νgστ =   . (4.1.13)  2 2 4 2 2 2 4   vγ α − ψ 0 −γ v α − ψ 0    0 0 0 ψ4

The lapse function p α = γψ ψ2 − 1 (4.1.14) is read oﬀ from the spacetime metric. Afterward, all of the ﬁelds are rotated such that they are oriented in the desired direction, momentum aligned with P i.

From the boosted spacetime metric, we extract the lapse function, shift vector, and spatial metric. The only non-vanishing component of the shift is

mv(m2 + 6mr + 16r2)(m3 + 6m2r + 8mr2 + 16r3) βy = − , (4.1.15) B2 with p B = (m + 2r)6 − 16(m − 2r)2r4v2 . (4.1.16)

p 2 2 2 2 On the t0 = 0 hypersurface, r0 → r = x + y γ + z and the conformal factor is

m ψB = 1 + (4.1.17) 2r

The conformal spatial line element on Σ0 is

16(m − 2r)2r4v2 d`˜2 = dx2 + γ2 1 − dy2 + dz2 . (4.1.18) (m + 2r)6

Near to the puncture the scalar curvature vanishes

32v2 7 + cos(2θ) + 2 sin2(θ) cos(2ϕ) r2 R˜ = + O(r3) , (4.1.19) m4

100 4.1. Lorentz-boosted Schwarzschild Initial Data 4.1. Lorentz-boosted Schwarzschild Initial Data as does the Laplacian of the conformal factor

8v2 cos2(θ) + sin2(θ) cos2(ϕ) r ˜ 2 2 D ψB = + O(r ) , (4.1.20) m3 with coordinates θ and ϕ deﬁned by Eq. (3.1.16).

The evolution equation for the spatial metric gives us an expression for the extrinsic curvature 1 Kij = (Diβj + Djβi − ∂tγij) . (4.1.21) 2α

The mean curvature is

32γmv (m + 2r)7 − 32(m − 2r)2(m − r)r4v2 r2y K = . (4.1.22) (m + 2r)3B3

The non-vanishing components of the trace-free, conformal extrinsic curvature tensor are

γmv(m − 4r)(m + 2r)3BCy A˜xx = A˜zz = , (4.1.23) 3Dr4 γmv(m − 4r)(m + 2r)3x A˜xy = − , (4.1.24) 2Br4 2γ3mv(m − 4r)Cy A˜yy = − , (4.1.25) 3(m + 2r)3Br4 γmv(m − 4r)(m + 2r)3z A˜yz = − , (4.1.26) 2Br4 with

C = (m + 2r)6 − 8(m − 2r)2r4v2 , (4.1.27)

D = (m + 2r)12 − 32(m − 2r)2r4(m + 2r)6v2 (4.1.28)

+ 256(m − 2r)4r8v4 . (4.1.29)

3 3 Again, we see in Cartesian coordinates that A˜ij ∼ O 1/r and K ∼ O(r ) at the puncture.

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 101 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

4.1.2 Punctures

In the puncture approach, the conformal factor is written as singular parts plus a ﬁnite correction [191]

ψ = Ψ + u , (4.1.30) where Ψ contains the conformal factors associated with the individual, isolated black holes

(see Sec. 4.1.3). This turns the conformal Hamiltonian constraint (2.2.128) into an elliptic diﬀerential equation for u

5 2 ij ψR˜ ψ K A˜ijA˜ D˜ 2u + D˜ 2Ψ − − + = 0 . (4.1.31) 8 12 8ψ7

Inspired by (2.2.136) and (4.1.30), we write our ansatz for the extrinsic curvature as

˜ ˜ ˜ Aij = Mij + (Lb)ij , (4.1.32) where the ﬁrst term is a freely speciﬁable symmetric, trace-free tensor [176]. It contains the known singular, symmetric, trace-free extrinsic curvature tensors for individual Kerr or

Lorentz-boosted Schwarzschild black holes (see Sec. 4.1.3). This turns (2.2.137) into diﬀerential equations for the vector bi

i ij 2 6 ij ∆˜ b + D˜jM˜ − ψ γ˜ D˜jK = 0 . (4.1.33) L 3

We impose Robin boundary conditions

i lim ∂r(ru) = 0 and lim ∂r(rb ) = 0 . (4.1.34) r→∞ r→∞

These are enforced numerically by noting that the TwoPunctures compactiﬁed coordinate

A obeys A → 1 as r → ∞, thus allowing the asymptotic behavior to be factored out (e.g. write u = (A − 1)U and solve for the auxiliary function U) [107]. The remaining boundary conditions are “behavioral,” in that the spectral basis functions possess automatic regularity

102 4.1. Lorentz-boosted Schwarzschild Initial Data 4.1. Lorentz-boosted Schwarzschild Initial Data properties [209]. For example, solutions represented by Fourier series are intrinsically periodic.

4.1.3 Attenuation and Superposition

The two punctures “(+)” and “(−)” are initially located at coordinates (x, y, z) = (±b, 0, 0), respectively. The coordinate distance to each puncture is

p 2 2 2 r(±) = (x ∓ b) + y + z . (4.1.35)

We write all of the background ﬁelds as simple superpositions

(+) (−) γ˜ij = δij + f(+) γ˜ij − δij + f(−) γ˜ij − δij , (4.1.36) ˜ ˜(+) ˜(−) Mij = g(+)Aij + g(−)Aij , (4.1.37)

K = f(+)g(+)K(+) + f(−)g(−)K(−) , (4.1.38)

Ψ = ψ(+) + ψ(−) − 1 (4.1.39)

QI with scalar functions f(±) and g(±). For the conformally Kerr case, ψ(±) = ψ(±). For the B ˜(±) conformally boosted Schwarzschild case, ψ(±) = ψ(±). Since Aij are symmetric and trace- free, then so too is M˜ ij.

In an effort to tame the singularities in (4.1.31) and (4.1.33), we introduce attenuation functions. We seek functions that have the following properties: at the location of (+), the contribution from (−) falls to zero sufficiently fast, and vice versa; towards spatial infinity, the spatial metric becomes flat and the extrinsic curvature vanishes. We achieve this with

p r(∓) f(±) = 1 − exp − , (4.1.40) ω(±)

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 103 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

(±) ˜ ˜ (±) ˜ ˜ As r(±) → 0, then γ˜ij → γ˜ij , and consequently Di → Di and R → R(±). Consider ˜ 2 ˜ Eq. (4.1.31). Near to the punctures, D(±)ψ(±) − ψ(±)R(±)/8 cancels to a ﬁnite value by ˜ 2 ˜ virtue of Eqs. (4.1.19) and (4.1.20), while D(±)ψ(∓) − ψ(∓) − 1 + u R(±)/8 is ﬁnite. The 5 2 5 2 2 −7 ˜ ˜ij terms ψ K → ψ g(±)K(±) and ψ AijA vanish at least linearly. In Eq. (4.1.33), the term 6 ij ˜ (±) ψ γ˜(±)Dj f(∓)g(∓)K(∓) is eliminated by the attenuation.

Within a small coordinate distance of each puncture (well inside the black-hole horizon),

M˜ ij and K are modiﬁed by the function

  0 if r(±) ≤ λrH g(±) = . (4.1.41)  1 if r(±) > λrH

The horizon radius is denoted by rH and 0 < λ ≤ 1, with typical value λ = 0.2. This technique, known as “stuffing” [269, 270, 271], introduces matter sources which violate the vacuum assumption, but remain trapped behind the horizon [272, 273]. ˜ (±) ˜ij 2 6 ij ˜ (±) Consider Eq. (4.1.33) as r(±) → 0. The terms Dj g(±)A(±) − 3 ψ γ˜(±)Dj f(±)g(±)K(±) ˜ (±) ˜ij are suppressed by the stuffing procedure. What remains is the finite piece Dj g(∓)A(∓) .

In addition to writing g(±) as a discontinuous step function, we experimented with polynomials [274], Gaussians akin to f(±), and a smooth step function. However, we ﬁnd that the discontinuous step function produces solutions with the most accurate independently measured energies, momenta, spins, etc.

Even with attenuation, round-off error makes the divergence in the source term in (4.1.33) difficult to calculate numerically near either of the punctures (outside of the stuffing region).

We alleviate this by rewriting the source with the divergence removed explicitly

ij j (±)j ki i (±)i jk 2 6 ij D˜jA˜ = Γ˜ − Γ˜ A˜ + Γ˜ − Γ˜ A˜ + ψ γ˜ ∂jK . (4.1.42) (±) jk jk (±) jk jk (±) 3 (±) (±) (±)

˜(±)i We define Γjk as the Christoffel symbols associated with the isolated, unattenuated conformal (±) ˜ij metric γ˜ij . Equation (4.1.42) has the additional benefit that no derivatives of A(±) need be calculated.

104 4.1. Lorentz-boosted Schwarzschild Initial Data 4.2. Numerical Techniques

4.2 Numerical Techniques

4.2.1 Initial Data Solver with Spectral Methods

The TwoPunctures thorn [107] generates conformally ﬂat (γ˜ij = δij) initial data via a spectral expansion of the Hamiltonian constraint on a compactiﬁed collocation point grid.

ij 4 2 ! 7A˜ijA˜ R˜ 5ψ K A˜ij D˜ 2du − + + du + (˜ db)ij = 0 , (4.2.43) 8ψ8 8 12 4ψ7 L

˜ i 5 ij ˜ ∆L db − 4ψ γ˜ DjK du = 0 , (4.2.44) where du and dbi represent small changes in u and bi, respectively. The terms in the linearized constraints have the behavior described in Sec. 4.1.3.

4.2.2 Evolution

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 105 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin coordinate separation. We evolve these black hole binary data sets using the LazEv [118] implementation of the moving punctures approach with the conformal function W = exp(−2φ) suggested by Ref. [69]. For the runs presented here, we use centered, eighth-order finite differencing in space [227] and a fourth-order Runge-Kutta time integrator. (Note that we do not upwind the advection terms.) Our code uses the Cactus/EinsteinToolkit [112, 109] infrastructure. We use the Carpet mesh refinement driver to provide a “moving boxes” style of mesh refinement.

We locate the apparent horizons using the AHFinderDirect code [245] and measure the horizon spin using the isolated horizon (IH) algorithm detailed in [246].

For the computation of the radiated angular momentum components, we use formulas based on “ﬂux-linkages” [247] and explicitly written in terms of Ψ4 in [248, 249].

We obtain accurate, convergent waveforms and horizon parameters by evolving this system in conjunction with a modiﬁed 1+log lapse and a modiﬁed Gamma-driver shift condition [203,

200, 250]. The lapse and shift are evolved with

i 2 (∂t − β ∂i)α = −α f(α)K, (4.2.45a)

a 3 a a ∂tβ = Γ˜ − ηβ . (4.2.45b) 4

In the original moving punctures approach we used f(α) = 2/α and an initial lapse α(t =

−2 4 0) = ψBL [200] or α(t = 0) = 2/(1 + ψBL) [61], where ψBL = 1 + m(+)/(2r(+)) + m(−)/(2r(−)).

In Secs. 4.3 and 4.4 we also use α(t = 0) = 1/(2ψBL − 1) which seems better suited for highly boosted black hole evolutions. There, we explore other gauge conditions for the lapse in the form of f(α) = 1/α (gauge speed = 1) and f(α) = 8/(3α(3 − α)) (shock avoiding) [241] which prove to be more convenient when dealing with highly boosted moving punctures.

We have found that the choice f(α) = 8/(3α(3 − α)) (approximate shock avoiding [241]) proves to be more stable and convenient when dealing with highly boosted moving punctures at relatively short separations, ≈ 100M. This is due to its better ability to handle the large amplitude gauge waves generated by those initial conﬁgurations. For the initial form of the lapse we use α(t = 0) = 1/(2ψBL − 1), where ψBL = 1 + m(+)/(2r(+)) + m(−)/(2r(−)).

106 4.2. Numerical Techniques 4.3. Quasi-circular Orbits

4.3 Quasi-circular Orbits

One of the most astrophysically important applications of numerical relativity is the evolution of black hole binaries in quasi-circular orbits. Figure 4.1 shows the convergence rate of the initial data solution as nearly exponential with the number of collocation points for a typical set of orbital parameters. Hamiltonian and momentum constraint residuals reach levels below

10−7 above 80 collocation points. This is accurate enough for most applications.

10-3 H Mx y 10-4 M Mz

10-5

10-6 Norm

10-7

10-8

10-9 16 32 48 64 80 96 112 128 144 N

Figure 4.1: Convergence of the residuals of the Hamiltonian and momentum constraints versus number of collocation points N for black hole binaries in a quasi-circular orbit with exponential y attenuation parameters ω(±) = 1.0 and p = 4. Orbital parameters P = 0.0848M, d = 12M.

In order to evaluate these initial data we perform a numerical evolution of a binary in the merger regime and compare our Lorentz-boosted data with the traditional BY solution. We chose initial parameters that lower the eccentricity for each set of data, as given by Table 4.1.

The black holes orbit nearly ﬁve times before merging (see Fig. 4.2), and at t ∼ 700M, merge to a spinning remnant black hole with the properties given in Table 4.2.

Figure 4.3 shows the waveforms of a binary with the Lorentz boost and BY initial data for the modes (`, m) = (2, 2) and (`, m) = (4, 4). While most of the waveforms superpose, the most notable diﬀerence lies in the initial burst of radiation (located at around t = 75M). The BY data has a nearly factor 2 larger amplitude for the initial burst relative to the Lorentz-boosted data for the leading (2, 2) mode, and this ratio grows for the (4, 4) mode to a factor ∼ 5.

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 107 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

Table 4.1: Initial data parameters for the equal-mass, boosted configurations. The punctures are initially at rest and are located at (±b, 0, 0) with momentum P~ = P xxˆ + P yyˆ, mass parameters mp, horizon (Christodoulou) masses mH, total ADM mass MADM, and boost factor γ. The configurations are Bowen-York (BY) or Lorentz-boosted (HS) initial data. x y Configuration b/M mp P P mH MADM/M BYQC 4.7666 0.48523 -0.001153 0.09932 0.5 0.98931 HSQC 4.7666 0.48745 -0.001138 0.09794 0.5 0.98914

5 10 BY 4 9 HS 3 8 2 7 1 6 0 5 r/M y/M -1 4 -2 3 -3 2 -4 1 -5 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 0 100 200 300 400 500 600 700 800 x/M t/M

Figure 4.2: The orbital trajectories of the binary and a comparative radial decay of the Lorentz-boosted and BY initial data.

This burst of initial radiation may have consequences in the cases when high accuracy of the waveforms is needed (in particular on the phase at late times) as it generates errors reflecting on the refinement boundaries of the grid [251]. The Lorentz-boosted method has the property that the resulting initial data matches closely with the input parameters. This makes it easy for the user to construct the desired configuration. We also note that there is a phase and amplitude mismatch among Lorentz-boosted and BY waveforms at merger. This may be due to a combination of slightly different initial orbital parameters and the above mentioned disparity in the initial radiation content.

Here, we study the eﬀects of lapse evolution choices on the case study of equal mass, non-

Table 4.2: The final mass and spin for each configuration. Configuration Mrem/M χrem BYQC 0.95162 0.68643 HSQC 0.95155 0.68646

108 4.3. Quasi-circular Orbits 4.3. Quasi-circular Orbits

0.08

0.06

0.04

] 0.02 775 800 825 850 875 22 4 Ψ 0

r M Re[ -0.02

-0.04

-0.06 50 75 100 125 150 BY HS 0.015

0.01

] 0.005

44 4 775 800 825 850 875 Ψ 0

r M Re[ -0.005

-0.01

-0.015 50 75 100 125 150

0 200 400 600 800 1000 t/M

Figure 4.3: Comparison of the waveforms generated from the Lorentz boost and BY initial data for the modes (`, m) = (2, 2) and (`, m) = (4, 4). Note the diﬀerence in initial radiation content at around t = 75M.

spinning, orbiting black hole binaries. The Lorentz-boosted initial data has a lower radiation content than the boost BY data and allows us to see more clearly the eﬀects of the initial choice and evolution of the lapse.

Figure 4.4 displays the eﬀects of gauge versus resolution on physical quantities like the horizon mass (left column) and horizon radius (right column). We expect the horizon mass to be essentially conserved during the orbital period up to merger. We can see that this physical observable varies very little with diﬀerent gauge choices. On the other hand, we observe that the coordinate radius varies with the evolution of the lapse choice, but not as much with the

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 109 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin initial lapse. After a sudden growth, typical of a gauge settling, the horizon radius reaches a constant value. The original moving punctures choice, f(α) = 2/α, keeps the value of the horizon coordinate closer to its original value which could be beneﬁcial for setting up the initial mesh reﬁnement levels.

0.57 0 0.54 -1

0.51 -2 0.48 -3 0.45

0.57 0 5

10 0.54 × -1 0.51

/M - 0.5) -2

(+/-) 0.48

(m -3 Average Horizon Radius 0.45

f=2/α f=8/3α(3-α) 0.57 α 0 f=1/ 0.54 -1

0.51 -2 0.48

-3 f=2/α 0.45 f=8/3α(3-α) f=1/α 20 55 90 125 160 195 230 265 300 20 55 90 125 160 195 230 265 300 t/M t/M

Figure 4.4: Individual black hole horizon masses (left) and coordinate radii of the horizons (right) versus time for diﬀerent evolution functions f(α) for the lapse. Initial lapse α0 = 4 2 2/(1 + ψfull) (top row), α0 = 1/ψBL (middle row), α0 = 1/(2ψBL − 1) (bottom row) .

Figure 4.5 displays the waveform as seen by an observer at r = 90M from the sources for

4 diﬀerent evolution functions f(α) for the lapse. The initial lapse here is α0 = 2/(1 + ψfull). While physical quantities like the waveform and its amplitude are essentially independent of the gauge choices, numerical errors, which produce the high frequency noise, are not. The

110 4.3. Quasi-circular Orbits 4.3. Quasi-circular Orbits bottom panel of the ﬁgure shows a close-up view of the amplitude during the post initial pulse period. We observe that overall the choice f(α) = 2/α produces a lower amplitude of this high frequency noise.

1e-4 2e-5

1e-5 1e-5 ] | 2,2 4 2,2 4 Ψ 0 Ψ | Re[ -1e-5 1e-6

-2e-5 1e-7 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 6.4e-6 f=2/α f=8/3α(3-α) 6.0e-6 f=1/α

5.6e-6 |

2,2 4 5.2e-6 Ψ | 4.8e-6

4.4e-6

4.0e-6 140 160 180 200 220 240 260 280 300 t/M

Figure 4.5: Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper left) and the amplitude of those waveforms (upper right). On the lower panel a zoom-in of the amplitude oscillations for diﬀerent evolution functions f(α) for the lapse. Initial lapse here is 4 α0 = 2/(1 + ψBL).

Figures 4.6 and 4.7 display a similar behavior for the waveforms, but their close-up view of the noise shows a smaller amplitude, which suggests that the choice of the initial lapses

2 α0 = 1/ψBL or α0 = 1/(2ψBL − 1) lead to smaller amplitude gauge waves.

Since the moving punctures approach is a free evolution of the general relativistic ﬁeld equations, a very important method to monitor its accuracy is to verify the satisfaction of the

Hamiltonian and momentum constraints. We also monitor the BSSNOK constraints, which are on the order of 10−7 throughout the duration of the evolution.

Figures 4.8–4.10 display the RMS norm (3.3.42) of the nonvanishing values of the Hamil- tonian and momentum components of the constraints. We observe that the propagation of √ p errors travel at diﬀerent speeds, associated with the gauge velocities 2, 4/3, and 1 for

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 111 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

1e-4 2e-5

1e-5 1e-5 ] | 2,2 4 2,2 4 Ψ 0 Ψ | Re[ -1e-5 1e-6

-2e-5 1e-7 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 6.4e-6 f=2/α f=8/3α(3-α) 6.0e-6 f=1/α

5.6e-6 |

2,2 4 5.2e-6 Ψ | 4.8e-6

4.4e-6

4.0e-6 140 160 180 200 220 240 260 280 300 t/M

Figure 4.6: Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper left) and the amplitude of those waveforms (upper right). The bottom panel shows a zoom-in of the amplitude oscillations for diﬀerent evolution functions f(α) for the lapse. The initial 2 lapse implemented here is α0 = 1/ψBL. f(α) = 2/α, 8/(3α(3 − α)), and 1/α, respectively. We also observe slightly larger violations

4 for the choice f(α) = 1/α, and α0 = 2/(1 + ψBL). We thus conclude that while all three evolution choices for the lapse are viable to evolve typical black hole binary simulations, the original moving punctures choice f(α) = 2/α and

2 initial lapse α0 = 1/ψBL or α0 = 1/(2ψBL − 1) are somewhat preferred. This study suggests there might be even more optimal choices of α0 and f(α), as well as shift evolution gauge conditions. We also note that in the independent study of Ref. [252], a higher gauge velocity is preferred for the early stage of evolution.

4.4 Gauge Conditions

Since we observe a notable benefit on using the initial lapse α0 = 1/(2ψBL − 1) in evolutions of highly spinning black holes (see Sec. 3.4, we would like to explore their effect on another extreme configuration: high energy relativistic collisions of black holes. The collisions were

112 4.4. Gauge Conditions 4.4. Gauge Conditions

1e-4 2e-5

1e-5 1e-5 ] | 2,2 4 2,2 4 Ψ 0 Ψ | Re[ -1e-5 1e-6

-2e-5 1e-7 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 6.4e-6 f=2/α f=8/3α(3-α) 6.0e-6 f=1/α

5.6e-6 |

2,2 4 5.2e-6 Ψ | 4.8e-6

4.4e-6

4.0e-6 140 160 180 200 220 240 260 280 300 t/M

Figure 4.7: Waveforms extracted at an observer location r = 90M. Real part of Ψ4 (upper left) and the amplitude of those waveforms (upper right). On the lower panel a zoom-in of the amplitude oscillations for diﬀerent evolution functions f(α) for the lapse. Initial lapse here is α0 = 1/(2ψBL − 1). studied in Refs. [262, 275, 263, 264] with regard to potential applications to collider-generated mini black holes. Here we will consider them as test case for comparing diﬀerent gauge conditions.

In Fig. 4.11 we use physical observables such as the individual horizon masses and the gravitational radiation waveforms as indicators of the numerical accuracy of the evolutions.

We observe that the initial lapse α0 = 1/(2ψBL − 1) gives the best behavior for the horizons mass (i.e., most constant) and a waveform with reduced noise.

The preferred behavior of the initial lapse α0 = 1/(2ψBL − 1) is also conﬁrmed with regard

2 to the constraint preservation as shown in Fig. 4.12, closely followed by the choice α0 = 1/ψfull.

In these evolutions we have taken the standard choice for the moving puncture evolution of the lapse, f(α) = 2/α in Eq. (4.2.45a). It is also worthwhile to explore alternative evolutions of f(α) = 1/α, with gauge speed equal to 1, and f(α) = 8/(3α(3 − α)), with approximate shock avoiding properties [241]. The results of such evolutions are displayed in Figs. 4.13 and

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 113 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 x H M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 y z M M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M

Figure 4.8: RMS norms of the violations of the Hamiltonian and three components of the momentum constraints versus time for diﬀerent evolution functions f(α) for the lapse. Initial 4 lapse here is α0 = 2/(1 + ψfull).

x 4.14 where we have taken an initial separation of the binary d = 66M, P /mH = ±2, and used the initial lapse α0 = 1/(2ψBL − 1).

We ﬁrst observe that the results of Figs. 4.13 and 4.14 indicate that with our numerical setup the evolution f(α) = 1/α fails to complete (i.e., crashes) generating large errors, while the form f(α) = 8/(3α(3 − α)) is stable, but less accurate than the standard f(α) = 2/α.

However, we ﬁnd that for larger initial P/mH values, the lapse evolution equation characterized by f(α) = 2/α fails to complete the evolution while the (approximate) shock avoiding form f(α) = 8/(3α(3 − α)) always succeeds. In these cases, a large amplitude gauge wave is generated by the high energy collision initial data which leads to an inability for the numerics to resolve the waves and stabilize the system. While one can try to ﬁne tune parameters of the evolution or change the evolution equations (for instance to a Z4-type [117]) the form f(α) = 8/(3α(3 − α)) represents a valid alternative to the standard f(α) = 2/α evolution

(which can still be used by starting collisions further apart or slightly grazing).

114 4.4. Gauge Conditions 4.5. Energy Radiated in High Energy Head-on Collisions

1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 x H M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 y z M M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M

Figure 4.9: RMS norms of the violations of the Hamiltonian and three components of the momentum constraints versus time for diﬀerent evolution functions f(α) for the lapse. Initial 2 lapse here is α0 = 1/ψBL.

4.5 Energy Radiated in High Energy Head-on Collisions

The ﬁrst observation about the high-energy collinear collision of black holes has to do with the properties of the initial data used in this paper with respect to previous work [262] using

Bowen-York initial data. The latter is limited [253] to black holes moving at speeds v < 0.9c, as shown in Fig. 4.15. The limitation is due to the condition of conformal ﬂatness imposed by the Bowen-York data. This produces a distortion on a moving black hole metric that, upon evolution, will lead not only to spurious initial data radiation, but mostly to absorption of energy by the moving black hole, which limits their initial speeds to the value given above.

The situation is similar to that observed in highly spinning black holes, where the conformally

ﬂat ansatz for the three-metric leads to a limitation [253, 231, 232] in the maximum intrinsic spin of the black hole of around χ ≈ 0.93.

On the other hand, Fig. 4.15 shows that the new data we use here is not limited by this condition and can reach velocities closer to the speed of light, i.e. v ∼ 0.99c. This is due

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 115 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 x H M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M 1e-5 1e-5 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) f=1/α f=1/α

1e-6 1e-6 y z M M

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 300 0 50 100 150 200 250 300 t/M t/M

Figure 4.10: RMS norms of the violations of the Hamiltonian and three components of the momentum constraints versus time for diﬀerent evolution functions f(α) for the lapse. Initial lapse here is α0 = 1/(2ψBL − 1).

0.238 0.002 4 4 2/(1+ψfull) 2/(1+ψfull) 4 4 0.237 2/(1+ψBL) 2/(1+ψBL) 1/(2ψ-1) 0.001 1/(2ψ-1) 0.236 2 2

1/ψ ] 1/ψ /M 2,2 4

0.235 Ψ 0 (+/-) m Re[ 0.234 -0.001 0.233

0.232 -0.002 0 10 20 30 40 50 0 50 100 150 200 250 t/M t/M

x Figure 4.11: Horizon mass of the boosted black hole with P /mH = ±2 and waveform after collision for diﬀerent choices of the initial lapse and evolution f(α) = 2/α. to the much lower initial radiation and distortion content of the data. We will exploit this characteristic of the initial data to obtain a more accurate estimate of the output gravitational radiation of the head-on collision of two equal mass, nonspinning, black holes.

In order to explore the dependence of the radiated energy with the initial momentum magnitude, and then extrapolate the results to the ultrarelativistic limit, we set up a series of

116 4.5. Energy Radiated in High Energy Head-on Collisions 4.5. Energy Radiated in High Energy Head-on Collisions

1e-2 1e-2 4 4 2/(1+ψfull) 2/(1+ψfull) 2/(1+ 4 ) 2/(1+ 4 ) 1e-3 ψBL 1e-3 ψBL 1/(2ψ-1) 1/(2ψ-1) 2 2 1e-4 1/ψ 1e-4 1/ψ x H M 1e-5 1e-5

1e-6 1e-6

1e-7 1e-7 0 50 100 150 200 250 0 50 100 150 200 250 t/M t/M 1e-2 1e-2 4 4 2/(1+ψfull) 2/(1+ψfull) 2/(1+ 4 ) 2/(1+ 4 ) 1e-3 ψBL 1e-3 ψBL 1/(2ψ-1) 1/(2ψ-1) 2 2 1e-4 1/ψ 1e-4 1/ψ y z M M 1e-5 1e-5

1e-6 1e-6

1e-7 1e-7 0 50 100 150 200 250 0 50 100 150 200 250 t/M t/M

Figure 4.12: Hamiltonian and momentum constraints during the free evolution for diﬀerent choices of the initial lapse and evolution f(α) = 2/α.

0.238 0.002 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) 0.237 f=1/α f=1/α 0.001 0.236 ] /M 2,2 4

0.235 Ψ 0 (+/-) m Re[ 0.234 -0.001 0.233

0.232 -0.002 0 10 20 30 40 50 0 50 100 150 200 250 t/M t/M

x Figure 4.13: Horizon mass (left) of the boosted black hole with P /mH = ±2 and waveform (right) after collision for initial lapse α0 = 1/(2ψBL − 1) and diﬀerent choices of the evolution of the lapse. simulations as summarized in Table 4.3. We have chosen a relatively large initial separation of the black holes to establish initial data that allow us to identify the individual properties of the holes (consider them approximately isolated, with small interaction energy). The initial separations have also been chosen not too large to allow for short and accurate simulations.

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 117 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

1e-2 1e-2 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) 1e-3 f=1/α 1e-3 f=1/α

1e-4 1e-4 x

H 1e-5 1e-5 M

1e-6 1e-6

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 0 50 100 150 200 250 t/M t/M 1e-2 1e-2 f=2/α f=2/α f=8/3α(3-α) f=8/3α(3-α) 1e-3 f=1/α 1e-3 f=1/α

1e-4 1e-4

y 1e-5 z 1e-5 M M

1e-6 1e-6

1e-7 1e-7

1e-8 1e-8 0 50 100 150 200 250 0 50 100 150 200 250 t/M t/M

Figure 4.14: Hamiltonian (H) and momentum (M x, M y, and M z) constraints during the free evolution for initial lapse α0 = 1/(2ψBL −1) and diﬀerent choices of the evolution of the lapse.

0.9

0.8

0.7 ADM 0.6 2P/M 0.5

0.4 Bowen-York 0.3 Lorentz-Boosted 1 2 3 4 5 γ

Figure 4.15: The center of mass speeds of the black holes for the set of Lorentz-boosted data in this paper and Bowen-York data. The latter displays a limitation, reaching speeds of only v < 0.9c, while the former can reach near the ultrarelativistic regime. The thin line represent p the special relativistic speed expression v = 1 − 1/γ2.

118 4.5. Energy Radiated in High Energy Head-on Collisions 4.5. Energy Radiated in High Energy Head-on Collisions

Table 4.3: Initial parameters and energy radiated. For each system, the initial ADM mass is normalized to 1. P/MADM mirr/MADM P/mirr γ d/MADM Erad/MADM δErad/MADM 0.1439 0.4807 0.30 1.0440 100 0.0011 6.8×10−7 0.2245 0.4498 0.50 1.1180 100 0.0030 6.1×10−6 0.3558 0.3559 1.00 1.4142 200 0.0183 1.2×10−4 0.4530 0.2263 2.00 2.2361 200 0.0592 4.7×10−4 0.4800 0.1594 3.01 3.1717 300 0.0859 1.1×10−3 0.4908 0.1217 4.03 4.1231 400 0.0988 9.7×10−4

Finally, the separations should be great enough to produce radiated energies that are almost insensitive to the speciﬁc initial separations.

Given the the relatively large initial separations, we extracted the radiation at accordingly large distances. For instance, for the cases in Table 4.3, starting at separations d = 100M, we have extraction radii as far out as robs = 250 − 275M. In this case, the extrapolation formula

2 of Ref. [268] to O(1/robs) gives a very robust set of values. To provide a generous bound, we used those two radii as estimates of the inﬁnite radius energy radiated, and call the diﬀerence

“Inf Radius” error in Fig. 4.16.

An additional source of error that we seek to keep under control is the initial radiation content. We monitored this spurious bursts for all of the Lorentz-boosted CCZ4 runs by comparing the full waveforms to waveforms truncated up to just before the beginning of the bursts and to waveforms truncated up to just after the end of the bursts. We ﬁnd that, while removing all of the precursor and spurious parts of the waveforms leads to changes of

0.016 Erad, the contributions of the spurious bursts per se are only 0.001 Erad. Based on this, we see that most of the diﬀerence in Erad between waveforms with and without the spurious bursts comes from data on the detector before any physical signals could arrive there. A bound on the eﬀect of this spurious radiation on the accuracy of the total radiated energy is displayed in Fig. 4.16 under the label “Spurious.”

It is worth noting here that the waveforms are extracted by a multipole decomposition at the observer location. In practice a few of the lower modes are necessary for an accurate account of the total radiation. For instance, the `-mode contributions to the CCZ4 simulations

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 119 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

4 Inf Radius 3.5 Spurious 1-Mf/MADM vs Erad/MADM Truncation 3

2.5

1.5 % Uncertainty 1

0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

P/mirr

Figure 4.16: Estimated errors for each component of the computation of the radiated Energy. “Inf Radius” is the extrapolation from the finite extraction radius robs to infinity. “Spurious” is the effect of the initial radiation content of the data. “Truncation” is an estimate of the finite difference resolution used in the simulation, and “1 − Mf/MADM vs. Erad/MADM” is a consistency measure of the radiated energy as computed by the gravitational waveforms or the remnant mass of the final black hole.

for a P/mirr = 3 run (with initial separation d = 100M) at robs = 275M gives that ` = 2 contains 90%, ` = 4 contains 8.3%, and ` = 6 contains 1.68% of the total energy radiated.

Thus our results will include modes up to ` = 6.

We have performed a few convergence studies for runs with initial P/mirr = 2 and 3 for resolutions increasing by a factor 1.2 with respect to our standard simulations (with up to 12 levels of reﬁnement and higher grid resolutions around the horizon of the holes of M/368.)

The eﬀects of ﬁnite resolution on the computation of the total radiated energy are displayed in Fig. 4.16 showing that this error is under control.

It is important to consider the effect of highly distorted black holes on the effectiveness of the moving puncture gauges, Eq. (4.2.45a). We find that, at relatively short initial separations, the standard BSSNOK formalism possesses under-resolved gauge errors for the resolutions that we use, which leads to numerical instabilities in the simulation. This problem is resolved by starting the black holes at greater initial separations, allowing the large gauges waves to sufficiently dissipate before the collision. An alternative solution is to use the CCZ4 formalism, which has stronger damping of the constraint violations than BSSNOK. We also found it

120 4.5. Energy Radiated in High Energy Head-on Collisions 4.5. Energy Radiated in High Energy Head-on Collisions

beneﬁcial to use an initial lapse of the form α0 = 1/(2ψBL − 1) and the approximate shock avoiding gauge proﬁle f(α) = 8/(3α(3 − α)).

To weight the statistical signiﬁcance of each computed energy as a function of the initial momentum we use an average error of 1%. We choose the abscissa variable mirr/P , where mirr stands for the irreducible mass of each initially boosted black hole with momentum ±P .

The upper panels in Fig. 4.17 displays the result of the ﬁtting, assuming the dependence of the energy radiated is given by the ZFL behavior [276, 262]

! E 1 + 2γ2 (1 − 4γ2) log (γ + pγ2 − 1) = E∞ + , (4.5.46) M 2γ2 2γ3pγ2 − 1

p 2 where the only ﬁtting parameter is E∞, and γ = 1 + (P/mirr) .

The residuals displayed in Fig. 4.17 (labeled as % diﬀ Erad/MADM) show that the relative deviations are mostly below 10%, and in particular are around 2% for the most energetic simulated collision.

To assess the dependence with the chosen fitting function, we have assumed a fit of the form (y = A exp[−B x]) with two fitting parameters (A and B), y and x being the independent and dependent variables, i.e. Erad/MADM and mirr/P , respectively. The results of this fit are displayed in the lower panels of Fig. 4.17. In spite of introducing two fitting parameters, we observe that the residuals are larger than the fit using the ZFL form (4.5.46), thus rendering further support to this behavior. We have also experimented with fittings of the form (y =

A exp[−B xC ]), introducing a third parameter C in the ﬁtting function, and also assuming

C = 2, but none of these options displayed better behavior than the ZFL choice.

In either case of the ﬁts shown in Fig. 4.17, the estimated maximum radiated energy is around 13%, which provides a robust estimate, all errors considered, of the form Emax/MADM =

0.13 ± 0.01.

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 121 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

0.14 100 AH Emax/MADM = 0.130 +\- 0.014 0.12 WF Emax/MADM = 0.130 +\- 0.014

0.1 10 ADM 0.08 /M ADM rad /M 0.06 rad E 1

0.04 % diff E

0.02

0 0.1 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

mirr/P mirr/P

0.14 100 AH Emax/MADM = 0.134 +\- 0.040 0.12 WF Emax/MADM = 0.132 +\- 0.039

0.1 10 ADM 0.08 /M ADM rad /M 0.06 rad E 1

0.04 % diff E

0.02

0 0.1 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5

mirr/P mirr/P

Figure 4.17: Fits to the energy radiated at infinity for the cases studied in Table 4.3. Upper plots: Fit using the 1-parameter zero-frequency-limit like fit. Lower plots: An alternative two-parameter (A and B) fit of the form (y = A exp[−B x]). Both fits use data assuming a weighting error of the points of 1% and include fits to both the energy radiated as measured by extraction of radiation (WF) or by the remnant mass (AH).

4.6 Summary

In this chapter we have been able to implement puncture initial data for highly boosted black hole binaries by attenuated superposition of conformal Lorentz-boosted Schwarzschild metrics. We veriﬁed the validity of the data by showing convergence of the Hamiltonian and momentum constraint residuals with the number of collocation points in the spectral solver.

We then showed, by evolving this data, that the radiation content of these initial data was much lower than the standard conformally ﬂat choice. This produced a more accurate and realistic computation of gravitational radiation waveforms. These cleaner initial data allowed us to explore diﬀerent choices of the moving puncture gauge (initial lapse and its evolution).

Using improved full numerical techniques, we have been able to provide a more accu-

122 4.6. Summary 4.6. Summary rate determination of the maximum gravitational radiation produced in the head-on collision of nonspinning black holes. These techniques utilize initial data for highly boosted black holes [267] with much less radiation content than the Bowen-York counterparts, and reach near the ultrarelativistic regime with speeds much closer to c. We have successfully extrapo- lated the extracted waveforms to inﬁnity observer locations with the techniques of Ref. [268], and added up to ` = 6 modes in the computation of the radiated energy. The evolutions of the initial data have been carried out using the moving puncture approach with a choice of the initial lapse and a shock avoiding gauge adapted to these high energy collisions and the

CCZ4 system to further damp the constraints in our free evolution of black hole binaries.

The result of this computation leads to a maximum radiated energy of 13 ± 1% of the total mass of the system, with most of the errors coming from the functional fitting and extrapolation to infinite boost. This result is in close agreement with the analytic estimates of 13.4% of Ref. [265] using thermodynamic arguments and the previous numerical estimate of 14 ± 3% in Ref. [263]. However, they seem to be in conflict with the analytic estimates of 16.4% from second order perturbations [259] and 17% from the multipolar analysis of the zero-frequency-limit [266].

Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin 123 Chapter 4. Quasi-circular Orbits and High Speed Collisions Without Spin

124 4.6. Summary Chapter 5

Highly Spinning Black Hole Binaries in Quasi-circular Orbits

The ultimate goal of this project is to accurately model the orbit, inspiral, and merger phases of black hole binaries with nearly extremal spins. Such systems are important in the end stages of the evolution of massive stars, and in galactic mergers. In both cases, black holes can form gravitationally bound pairs. Emission of gravitational radiation tends to circularize their orbits, which motivates us to begin our simulations with black holes on quasi-circular trajectories. Full numerical relativity is necessary to capture all of the strong field effects, especially in the merger phase of the evolution. These simulations are rather computationally expensive, so only the last few orbits before the final black hole coalescence are considered.

The topic of this chapter is the speciﬁcation of appropriate initial conditions to represent black hole binaries in with astrophysically relevant parameters—quasi-circular orbits, unequal masses, and nearly extremal spins.

In geometric units, a black hole’s spin magnitude S (i.e., intrinsic angular momentum) is bounded by its mass m, where the maximum dimensionless spin is given by χ ≡ S/m2 = 1.

Conservation of angular momentum during stellar core collapse and matter accretion tend to produce spinning black holes, with some existing at nearly the extremal limit. While it is actually hard to have an accurate measure of astrophysical black hole spins, in a few cases the

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 125 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits spins have been measured [213] and were found to be near the extremal value. Since galactic mergers are expected to lead to mergers of highly spinning black holes, it is important to be able to simulate black hole binaries with high spins in order to model the dynamics of these ubiquitous objects.

No less important is the modeling of gravitational waveforms from compact object mergers.

224].

(while retaining conformal ﬂatness), the spin is still bounded above by χ = 0.932 [232].

In order to exceed this limit and approach extremely spinning black holes with χ = 1, one must allow for a more general three-metric that captures the conformally curved aspects

126 of the Kerr geometry. Dain showed [233] that it is possible to ﬁnd solutions to the initial value problem representing a pair of Kerr-like black holes. This proposal was implemented for the case of thin-sandwich initial data with excision of the black hole interiors, and produces stable evolutions of orbiting black hole binaries with χ ∼ 0.97 [277, 218], and more recently

χ ∼ 0.998 [278, 279]. Dain’s method was also tested for the case of collisions from rest of spinning black holes using the moving punctures approach, which does not employ excision.

These are compared to the BY data with spins up to χ = 0.90 [234].

We find solutions to the puncture initial value problem representing two nearly extremely spinning black holes on quasi-circular trajectories, and the subsequent evolution using the moving punctures approach—the most widespread method to evolve black hole binaries, implemented in the open source Einstein Toolkit [108, 109, 112, 110]. To solve for these new data, we construct a superposition of two Lorentz-boosted conformally Kerr three-metrics, with the corresponding superposition of Kerr extrinsic curvature fields. Note that we do not use the Kerr-Schild slice [235] but the Boyer-Lindquist slice, amenable for puncture evolution. To regularize the problem, the superposition is such that very close to each black hole, the metric and extrinsic curvature are exactly Kerr [236]. We then simultaneously solve the Hamiltonian and momentum constraints for an overall conformal factor for the metric and nonsingular correction to the extrinsic curvature using a modification of the TwoPunc- tures [107] spectral initial data solver. We refer to this extension as HIghly SPinning Initial

Data (HiSpID, pronounced “high speed”).

We evolve these data sets and ﬁnd that the spurious initial radiation is signiﬁcantly reduced compared to BY initial data for χ = 0.8. We also perform accurate evolutions of highly spinning black holes with χ = 0.95, which has not been possible before for moving puncture codes.

This chapter is organized as follows. In Sec. 5.1 we present the formalism to solve for the initial data. We perform a conformal transverse-traceless decomposition on the Kerr spacetime metric, boosted into a moving frame of reference with a Lorentz transformation. We discuss the transformation laws for vectors and spin. A black hole binary ansatz is constructed using

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 127 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits a punctured, attenuated superposition of conformal boosted Kerr spatial metric and extrinsic curvature ﬁelds.

In Sec 5.2 we outline the numerical techniques used to solve the Hamiltonian and momentum constraint equations with pseudospectral methods. We introduce and test a procedure to eliminate residual momentum in unequal mass binaries. The evolution formalism and gauge choices are brieﬂy discussed.

In Sec 5.3 we exhibit the convergence of the constraint residuals as a function of the number of pseudospectral collocation points. We evolve an unequal mass q = 2 black hole binary with

χ = 0.8 on quasi-circular orbits through to merger and compare with the widely used Bowen-

York initial data solutions. Both the BSSNOK and CCZ4 formalisms were utilized. We evolve a black hole binary on quasi-circular orbits with χ = 0.95, beyond the Bowen-York spin limit.

The results are summarized in Sec. 5.4.

5.1 Lorentz-boosted Kerr Initial Data

The ﬁrst step in simulating black hole binaries is to construct initial data which serves as the initial boundary condition for the numerical evolution. The covariant spacetime metric is split into spatial hypersurfaces, parametrized by the gradient of a global time function. Each time slice is endowed with an intrinsic spatial metric tensor and an extrinsic curvature tensor, the

first and second fundamental forms. This particular foliation avoids the physical singularity at the black hole center. It represents two asymptotically flat, causally disconnected spaces, connected by an inversion symmetry at the black hole horizon. There exists a coordinate singularity at the origin of coordinates, representing spatial infinity in the “other universe.”

This singularity can be handled using the puncture method, conformally transforming the physical spatial metric into components which are regular at the origin. A black hole binary initial data ansatz is constructed from a simple attenuated superposition of isolated black hole

ﬁelds.

128 5.1. Lorentz-boosted Kerr Initial Data 5.1. Lorentz-boosted Kerr Initial Data

5.1.1 Conformal Boosted Kerr in Isotropic Coordinates

0 We start with a conformal rescaling of the stationary Kerr spacetime metric gµν

0 −4 0 g˜µν = ψ gµν (5.1.1) where the conformal factor is rρ ψ = (5.1.2) r

µ and the conformal Kerr spacetime metric in a Cartesian basis with coordinate labels x0 =

(t0, x0, y0, z0) has components

  2 2 2 2 (σ − 1)r /ρ aσy0/ρ −aσx0/ρ 0    2 2 2 2   aσy0/ρ 1 + a hy −a hx0y0 0 0  0  g˜µν =   (5.1.3)  2 2 2 2   −aσx0/ρ −a hx0y0 1 + a hx0 0   0 0 0 1 with the deﬁnitions [199, 234]

q 2 2 2 r = x0 + y0 + z0 , (5.1.4) m + a m − a r¯ = r 1 + 1 + , (5.1.5) 2r 2r az0 2 ρ2 =r ¯2 + , (5.1.6) r 2mr¯ σ = , (5.1.7) ρ2 1 + σ h = . (5.1.8) ρ2r2

The quasi-isotropic radial coordinate is denoted by r, and the Boyer-Lindquist radial coordi-

i nate by r¯. The mass parameter m and the spin parameters S0 set the angular momentum per q 0 i unit mass a = Si S0/m.

We want to describe a black hole with arbitrary linear momentum P i. The stationary

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 129 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits spacetime metric is transformed into a reference frame with relative velocity

P i vi = . p 2 j (5.1.9) m + P Pj

The Lorentz transformation matrix has components

    Λ0 Λ0 γ −γv µ 0 j j Λ ν =   =   (5.1.10)  i i   i γ2 i i  Λ 0 Λ j −γv 1+γ v vj + δj

γ = √ 1 g˜ and 1−v2 is the Lorentz factor. The boosted conformal Kerr spacetime metric µν is formed from the linear combination

ρ τ 0 g˜µν = Λ µΛ νg˜ρτ . (5.1.11)

µ The coordinate labels of the stationary frame x0 are transformed to those of the boosted frame xµ in the usual way µ µ ν x = Λ νx0 . (5.1.12)

We evaluate all tensors on the t = 0 spatial hypersurface.

The functions comprising the 3 + 1 spacetime element (2.2.115) are simply read oﬀ from the components of Eq. (5.1.11)

γ˜ij =g ˜ij , (5.1.13) ˜ βi =g ˜ti , (5.1.14) q −4 ˜ ˜i α˜ = ψ βiβ − g˜tt . (5.1.15)

The spin vector is not a tensor, but a dual tensor (not to be confused with a dual vector, also known as a one-form). Here, we are referring to the Hodge duality. This is motivated by considering the angular momentum vector, which in Newtonian mechanics is deﬁned as

L~ = ~r × ~p, (5.1.16)

130 5.1. Lorentz-boosted Kerr Initial Data 5.1. Lorentz-boosted Kerr Initial Data where ~r and ~p are some position and momentum vectors, respectively. Suppose we have a black hole spinning with its rotation axis aligned with the z-axis, and then we perform a Lorentz transformation in the y-direction. Naively, one might expect the angular momentum to be unchanged since it is perpendicular to the boost, which normally leaves vectors free of Lorentz contraction. However, Eq. (5.1.16) implies that the angular momentum vector is composed of two other vectors perpendicular to itself. As long as L~ 6= 0, then ~r and ~p are not parallel, and at least one of them must have nonzero components in the y-direction. Under a boost, then, we expect the z-aligned angular momentum to be modiﬁed, and conclude that it is not a true vector, but a pseudovector.

To see how the angular momentum behaves under a Lorentz transformation, we start by promoting the three-dimensional pseudovector to a four-dimensional two-form. The cross product is a duality of the wedge product in three-dimensions (see Eq. (2.1.27)). The simplest generalization that we can write down combines the four-position xµ and the four-momentum pµ into the four-angular-momentum tensor

Lµν = x[µpν] . (5.1.17)

In analogy with the magnetic ﬁeld Eq. (2.1.82), we deﬁne the spin angular momentum four- pseudovector Sµ, as measured by an observer with four-velocity uµ, to be

µ 1 µνρσ S = − Lνρuσ . (5.1.18) 2

µ µ If we adopt the coordinates of the observer u = δ0 , the spin reduces to

i 1 ijk S = − Lij . (5.1.19) 2

µνρσ µ µ By the symmetry of uµuσ and the antisymmetry of , uµS = 0 for any u . Thus, an observer always sees the spin as a purely spatial quantity (i.e. perpendicular to their timelike direction). Since Lµν is formed from the wedge product of two one-forms, it is tensorial. Then

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 131 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits the observer with four-velocity uµ sees the boosted spinning system with angular momentum

µ 1 µνρσ τ κ S = − Λ νΛ ρLτκuσ . (5.1.20) 2

µ With the components of Λ ν listed in Eq. (5.1.10), this reduces to the linear transformation

2 i i γ i j S = γδ − v vj S . (5.1.21) j 1 + γ 0

Compare this to the analogous transformation rule for an ordinary three-vector

2 i i γ i j w = δ + v vj w . (5.1.22) j 1 + γ 0

However, the user speciﬁes the ﬁnal angular momentum parameters Si, so we need a way to

i recover the stationary values S0. We seek an inverse to Eq. (5.1.21), that is a matrix which sends it to the identity. Let us suppose

2 i i k γ k i (Aδ + Bv vk) γδ − v vj = δ (5.1.23) k j 1 + γ j and try to ﬁnd the values of A and B. Grouping like terms gives

i i Aγδj = δj , (5.1.24) 2 2 γ 2 γ i −A + Bγ − Bv v vj = 0 . (5.1.25) 1 + γ 1 + γ

1 γ The solution to this system of equations is A = γ and B = 1+γ , so the inverse transformation has components i 1 i γ i j S = δ + v vj S . (5.1.26) 0 γ j 1 + γ

i i To align S0 with the z-axis, we rotate all of the ﬁelds using the spatial rotation matrix R j, as in Eq. (3.1.27).

i i We allow the user to specify the black hole binary parameters b, m(±), P(±) and S(±), q (±) i 2 subject to the cosmic censorship constraint Si S(±) < m(±).

132 5.1. Lorentz-boosted Kerr Initial Data 5.1. Lorentz-boosted Kerr Initial Data

For the single black hole case, the input parameters are measured in the expected way by the ADM integrals (2.2.151), (2.2.151), and (2.2.152)

MADM = γm , (5.1.27)

i i PADM = P (5.1.28)

i i x ijk SADM = S + bδj Pk . (5.1.29)

We recognize the ADM mass as the relativistic energy of a moving mass in special relativity, the ADM momentum is the three-momentum, and the ADM angular momentum is the sum of the black hole intrinsic spin and the “lever arm” angular momentum of the black hole about the origin of coordinates.

Given the conformal lapse function, shift vector, and spatial metric, we construct the conformal trace-free extrinsic curvature using

˜ 1 ˜ ˜ 1 kl Aij = ( β)ij + γ˜ijγ˜ ∂tγ˜kl − ∂tγ˜ij . (5.1.30) 2˜α L 3

The corresponding mean curvature is given by

−6 ψ ˜i −1 ˜i 1 ij K = D˜iβ + 6ψ β D˜iψ − ∂tψ − γ˜ ∂tγ˜ij . (5.1.31) α˜ 2

The conformal lapse passes through zero at the location of the horizon, although both A˜ij and

K are analytically ﬁnite there in the conformal Kerr geometry. In the numerical solver, we are evaluating these functions only at the collocation points, which typically fall far enough from the horizon for A˜ij and K to evaluate to a ﬁnite value.

5.1.2 Punctures

In the puncture approach, the conformal factor is written as singular parts plus a ﬁnite correction [191]

ψ = Ψ + u , (5.1.32)

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 133 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits where Ψ contains the conformal factors associated with the individual, isolated black holes

(see Sec. 5.1.3). This turns the conformal Hamiltonian constraint (2.2.128) into an elliptic diﬀerential equation for u

5 2 ij ψR˜ ψ K A˜ijA˜ D˜ 2u + D˜ 2Ψ − − + = 0 . (5.1.33) 8 12 8ψ7

Inspired by (2.2.136) and (5.1.32), we write our ansatz for the extrinsic curvature as

˜ ˜ ˜ Aij = Mij + (Lb)ij , (5.1.34) where the ﬁrst term is a freely speciﬁable symmetric, trace-free tensor [176]. It contains the known singular, symmetric, trace-free extrinsic curvature tensors for individual Kerr or

Lorentz-boosted Schwarzschild black holes (see Sec. 5.1.3). This turns (2.2.137) into diﬀerential equations for the vector bi

i ij 2 6 ij ∆˜ b + D˜jM˜ − ψ γ˜ D˜jK = 0 . (5.1.35) L 3

We impose Robin boundary conditions

i lim ∂r(ru) = 0 and lim ∂r(rb ) = 0 . (5.1.36) r→∞ r→∞

These are enforced numerically by noting that the TwoPunctures compactiﬁed coordinate

134 5.1. Lorentz-boosted Kerr Initial Data 5.1. Lorentz-boosted Kerr Initial Data

5.1.3 Attenuation and Superposition

The two punctures “(+)” and “(−)” are initially located at coordinates (x, y, z) = (±b, 0, 0), respectively. The coordinate distance to each puncture is

p 2 2 2 r(±) = (x ∓ b) + y + z . (5.1.37)

We write all of the background ﬁelds as simple superpositions

(+) (−) γ˜ij = δij + f(+) γ˜ij − δij + f(−) γ˜ij − δij , (5.1.38) ˜ ˜(+) ˜(−) Mij = g(+)Aij + g(−)Aij , (5.1.39)

K = f(+)g(+)K(+) + f(−)g(−)K(−) , (5.1.40)

Ψ = ψ(+) + ψ(−) − 1 (5.1.41)

˜(±) with scalar functions f(±) and g(±). Since Aij are symmetric and trace-free, then so too is

M˜ ij.

In an effort to tame the singularities in (5.1.33) and (5.1.35), we introduce attenuation functions. We seek functions that have the following properties: at the location of (+), the contribution from (−) falls to zero sufficiently fast, and vice versa; towards spatial infinity, the spatial metric becomes flat and the extrinsic curvature vanishes. We achieve this with

p r(∓) f(±) = 1 − exp − , (5.1.42) ω(±)

where ω(±) are parameters controlling the steepness of the attenuation. We take the smallest possible power index p = 4. This guarantees that the gradient of the attenuation function vanishes at least as fast as O(r3) at the punctures, which cancels the O(1/r3) divergence in the trace-free extrinsic curvature. (±) ˜ ˜ (±) ˜ ˜ As r(±) → 0, then γ˜ij → γ˜ij , and consequently Di → Di and R → R(±). Consider ˜ 2 ˜ Eq. (5.1.33). Near to the punctures, D(±)ψ(±) − ψ(±)R(±)/8 cancels to a ﬁnite value by virtue ˜ 2 ˜ of Eqs. (3.1.15), (4.1.19), and (4.1.20), while D(±)ψ(∓) − ψ(∓) − 1 + u R(±)/8 is ﬁnite. The

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 135 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

5 2 5 2 2 −7 ˜ ˜ij terms ψ K → ψ g(±)K(±) and ψ AijA vanish at least linearly. In Eq. (5.1.35), the term 6 ij ˜ (±) ψ γ˜(±)Dj f(∓)g(∓)K(∓) is eliminated by the attenuation.

Within a small coordinate distance of each puncture (well inside the black-hole horizon),

M˜ ij and K are modiﬁed by the function

  0 if r(±) ≤ λrH g(±) = . (5.1.43)  1 if r(±) > λrH

˜ (±) ˜ij 2 6 ij ˜ (±) Consider Eq. (5.1.35) as r(±) → 0. The terms Dj g(±)A(±) − 3 ψ γ˜(±)Dj f(±)g(±)K(±) ˜ (±) ˜ij are suppressed by the stuﬃng procedure. What remains is the ﬁnite piece Dj g(∓)A(∓) .

Even with attenuation, round-off error makes the divergence in the source term in (5.1.35) difficult to calculate numerically near either of the punctures (outside of the stuffing region).

We alleviate this by rewriting the source with the divergence removed explicitly

ij j (±)j ki i (±)i jk 2 6 ij D˜jA˜ = Γ˜ − Γ˜ A˜ + Γ˜ − Γ˜ A˜ + ψ γ˜ ∂jK . (5.1.44) (±) jk jk (±) jk jk (±) 3 (±) (±) (±)

˜(±)i We define Γjk as the Christoffel symbols associated with the isolated, unattenuated conformal (±) ˜ij metric γ˜ij . Equation (5.1.44) has the additional benefit that no derivatives of A(±) need be calculated.

136 5.1. Lorentz-boosted Kerr Initial Data 5.2. Numerical Techniques

5.2 Numerical Techniques

The nonlinearity of the Einstein ﬁeld equation dictates that the solution representing a black hole binary is not simply the sum of individual black hole solutions. Rather, there are interaction terms which must be accounted for numerically. With the black hole binary superposition ansatz, we modify slightly the corrections to the conformal factor the trace-free extrinsic curvature until the residuals of the Hamiltonian and momentum constraints are satisfactorily minimized. This completes the initial data solution, which is then propagated forward in time.

5.2.1 Initial Data Solver with Spectral Methods

The TwoPunctures thorn [107] generates conformally ﬂat (γ˜ij = δij) initial data via a spectral expansion of the Hamiltonian constraint on a compactiﬁed collocation point grid.

ij 4 2 ! 7A˜ijA˜ R˜ 5ψ K A˜ij D˜ 2du − + + du + (˜ db)ij = 0 , (5.2.45) 8ψ8 8 12 4ψ7 L

˜ i 5 ij ˜ ∆L db − 4ψ γ˜ DjK du = 0 , (5.2.46) where du and dbi represent small changes in u and bi, respectively. The terms in the linearized constraints have the behavior described in Sec. 5.1.3.

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 137 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

5.2.2 Unequal Masses

There is unambiguous dynamical evidence for the existence of supermassive black holes in many galaxies [280]. In astrophysical systems, the probability of ﬁnding a compact binary with exactly equally massive companions is vanishingly small. The population of supermassive black holes at galactic centers is diverse in its mass distribution. Some galaxies have central black

6 holes with masses as small as ∼ 10 M [281, 282, 38, 283], while others contain black holes

10 with masses up to & 10 M [284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294]. This range of four orders of magnitude suggests that galactic mergers will frequently lead to the formation of compact binaries with unequal masses. At cosmological scales, supermassive black hole mergers are most likely to occur in the mass ratio range q ≡ m(+)/m(−) = 1/10–1/100 [295]. Supermassive systems constitute a strong signal source for space-based, low frequency gravitational wave observatories [296], while intermediate and stellar mass binaries are expected to be detected in the near future by ground-based observatories [297, 298]. This provides strong motivation for simulating accurate gravitational waveforms from the inspiral and merger of unequal mass black hole binaries.

Numerical relativity simulations of spinning binaries with q = 1/8 [299] and nonspinning binaries with q = 1/10 [300] have been published. So, too, have long term evolutions with q = 1/10 and q = 1/15 [301, 302]. black hole binaries with up to q = 1/100 have been studied [225].

5.2.2.1 Center of Mass Drift

In general, comparing vectors at diﬀerent points on a curved manifold it is not well deﬁned.

We do not expect our input parameters to necessarily correspond to the desired physical quantities. In the case of unequal masses, we ﬁnd that when we specify equal and opposite

i i momenta P(+) = −P(−) the ADM integrals measure a net momentum drift in the initial data solution. y Consider the case with parameters b = 6, m(+) = 1, m(−) = 0.5, χ(±) = 0.9, and P(+) = y −P(−) = 0.0848. This results in approximately quasi-circular orbits. The solution is measured

138 5.2. Numerical Techniques 5.2. Numerical Techniques

i −16 −3 −16 to have net ADM momentum PADM ≈ (−4.4 × 10 , −4.7 × 10 , −3.3 × 10 ). The x- and z-components are zero to within round-oﬀ error, but the y-component has appreciable momentum. This results in a drift of the center of mass during evolution, which introduces

Doppler errors into the gravitational waveform signal.

This is remedied by an iterative procedure, where we change slightly the input momentum parameters, solve again the initial data, measure the net momentum, and repeat until the residual is satisfactorily small. At each iteration step, we take the net ADM momentum, split it in two, and subtract one part from each of the momentum parameters

P i P i → P i − ADM . (5.2.47) (±) (±) 2

We found that this method works quite well at reducing the spurious momentum in our initial data, as shown in Fig. 5.1. The iteration process becomes expensive for a large number of collocation points. Luckily, we are able to employ the iterations on a relatively few number of collocation points, and then simply feed the resulting parameters into a high resolution run.

10-2 0.5 0 10-3 -0.5

10-4 -1

| -1.5 -5

ADM 10 y/M

|P -2

10-6 -2.5 -3 10-7 -3.5

10-8 -4 0 1 2 3 4 5 0 200 400 600 800 1000 1200 1400 Iteration t/M

Figure 5.1: (Left panel) The residual ADM momentum at each step in the iteration process. (Right panel) The y-position of one of the punctures as a function of time before the correction (red) and after (green). After only a few steps, the momentum drift is eliminated to the levels required by our evolutions. This particular plot shows the case of Lorentz-boosted Kerr initial y y data with initial parameters b = 6, m(+) = 1, m(−) = 0.5, χ = 0.9, and P(+) = −P(−) = 0.0848. y y The momentum parameters in the ﬁnal iteration are P(+) = 0.0869868 and P(−) = −0.0826132.

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 139 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

5.2.3 Evolution and Gauge Choices

We locate the apparent horizons using the AHFinderDirect code [245] and measure the horizon spin using the isolated horizon (IH) algorithm [246].

For the computation of the radiated angular momentum components, we use formulas based on “ﬂux-linkages” [247] and explicitly written in terms of Ψ4 [248, 249].

We obtain accurate, convergent waveforms and horizon parameters by evolving this system in conjunction with a modiﬁed 1+log lapse and a modiﬁed Gamma-driver shift condition [203,

200, 250]. The lapse and shift are evolved with

i 2 (∂t − β ∂i)α = −α f(α)K, (5.2.48a)

a 3 a a ∂tβ = Γ˜ − ηβ . (5.2.48b) 4

In the original moving punctures approach we used f(α) = 2/α and an initial lapse α(t =

−2 4 0) = ψBL [200] or α(t = 0) = 2/(1 + ψBL) [61], where ψBL = 1 + m(+)/(2r(+)) + m(−)/(2r(−)).

We also use the “trumpet” initial lapse α(t = 0) = 1/(2ψBL − 1) which seems better suited for highly spinning black hole evolutions.

140 5.2. Numerical Techniques 5.3. Quasi-circular Orbits

Table 5.1: Initial data parameters for spinning, orbital configurations. The punctures are initially located at (±b, 0, 0), having mass ratio q = m(+)/m(−), with spins S aligned or anti- aligned with the z direction, mass parameters m(±), total ADM mass MADM, and dimensionless spins χ. The Bowen-York configurations are denoted by BY, and the HiSpID by HS. Finally, UU or UD denote the direction of the two spins, either both aligned (UU), or anti-aligned (UD). y y Configuration b/M q m(+) m(−) χ P(+) P(−) MADM/M BY80UD 5.4489 2 0.4078 0.1998 0.8 0.081882 -0.081882 0.991789 HS80UD 5.4489 2 0.6667 0.3333 0.8 0.086168 -0.083831 0.996224 HS95UU 4 1 0.5 0.5 0.95 0.103289 -0.103289 0.986854

5.3 Quasi-circular Orbits

To assess the characteristics of this superposed Lorentz-boosted Kerr black hole initial data, we compare with the corresponding BY-type. We study a few test cases of unequal mass black hole binary conﬁgurations starting on quasi-circular orbits with antiparallel (UD) spins, perpendicular to the line joining the black holes. We evolve both black hole binaries with the

HiSpID data and the standard BY choice (for spins within the BY limit). We also evolve black hole binaries with nearly extremal parallel (UU) spins, χ = 0.95, a regime unreachable for BY initial data. Table 5.1 gives the initial data parameters of these black hole binary conﬁgurations. Antiparallel spins result in antisymmetric emission of gravitational radiation, leading to a recoil in the merger remnant.

For given binary separations, spin parameters, and momentum parameters, the horizon masses and spins are not identical, as shown in Fig. 5.2, since the initial radiation content and distortions are not the same. However, we consider them close enough for comparisons of physical quantities such as the puncture separations shown in Fig. 5.3 and the gravitational waveforms shown in Fig. 5.4. Figure 5.5 shows that the conformally curved initial data yields better evolved constraint satisfaction than the conformally ﬂat case. The ﬁnal measured parameters are shown in Table 5.2.

One of our main motivations to study a new set of initial data is to be able to simulate highly spinning black holes, beyond the BY (or conformally ﬂat) limit, χ ≈ 0.93 [253, 231, 232].

In Fig. 5.6 we show the level of satisfaction of the constraints for our new initial data for

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 141 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

0.67 0.81 BY HS 0.669 0.808 0.806 0.668 0.804 0.667 0.802 (+) 0.666 (+) χ m 0.8 0.665 0.798 0.664 0.796 0.663 BY 0.794 HS 0.662 0.792 0 200 400 600 800 1000 0 200 400 600 800 1000 t/M t/M

0.336 0.81 BY 0.3355 HS 0.335 0.805 0.3345 0.334 0.8 (-) 0.3335 (-) χ m 0.333 0.795 0.3325 0.332 0.79 0.3315 BY HS 0.331 0.785 0 200 400 600 800 1000 0 200 400 600 800 1000 t/M t/M

Figure 5.2: The evolution of the irreducible mass and dimensionless spin of the individual black holes with q = 2 and χ = 0.8, comparing Lorentz-boosted Kerr and Bowen-York initial data. The top panels show the larger black hole, and the bottom panels show the smaller black hole.

12 BY HS 10

6 r/M 4

0 200 400 600 800 1000 1200 t/M

Figure 5.3: The evolution of the coordinate separation between the black holes in an orbiting binary with q = 2 and χ = 0.8, comparing Lorentz-boosted Kerr and Bowen-York initial data.

142 5.3. Quasi-circular Orbits 5.3. Quasi-circular Orbits

0.0006 0.00015

0.0004 0.0001

0.0002 5e-05 ] ] 22 4 44 4 Ψ Ψ 0 0 r M Re[ r M Re[ -0.0002 -5e-05

-0.0004 -0.0001 BY BY HS HS -0.0006 -0.00015 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 t/M t/M

Figure 5.4: Comparison of the waveforms generated from the Lorentz-boosted Kerr and Bowen- York initial data for the modes (`, m) = (2, 2) and (`, m) = (4, 4). Note the diﬀerence in initial radiation content at around t = 75M.

10-5 10-6 BY BY HS HS

10-6 10-7

-7 x -8

H 10 10 M

10-8 10-9

10-9 10-10 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 t/M t/M

-6 10-6 10 BY BY HS HS

-7 10-7 10

y -8 z 10-8

M 10 M

-9 10-9 10

-10 10-10 10 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 t/M t/M

Figure 5.5: The evolution of the Hamiltonian (H) and momentum (M x, M y, and M z) constraints for a black hole binary with q = 2 and χ = 0.8, comparing Lorentz-boosted Kerr and Bowen-York initial data. spinning black hole binaries with equal masses and spin parameters χ = 0.99. We observe an exponential convergence of the RMS norm (see Eq. (3.3.42)) of the constraint violations with

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 143 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

Table 5.2: The final remnant mass, spin, and recoil velocity. The final entry, FF95UU, lists the remnant parameters estimated from the analytical fitting function. −1 Configuration Mrem/M χrem V [km s ] BY80UD 0.96831 0.41037 420 ± 2 HS80UD 0.96815 0.41008 414 ± 7 HS95UU 0.8933 0.9401 0 FF95UU 0.8940 0.9404 0 spectral collocation points down to a level of ≈ 10−7. We do not consider points interior to the horizons in our RMS calculations. If one requires greater satisfaction of the constraints, one can fine-tune the attenuation functions to that end.

Figure 5.6: Convergence of the Hamiltonian and momentum constraint residuals with increasing number of collocation points. This example shows an equal mass Kerr black hole binary in y a quasi-circular orbit with spin χ = 0.99, momentum P(±) = ±0.0958, and separation d = 9M. The initial data superposition used exponential attenuation with parameters ω(±) = 1.0 and p = 4.

An evolution of the mass and spin parameters for an equal mass binary with χ = 0.95 is shown in Fig. 5.7. The initial and ﬁnal parameters for this run are listed in the ﬁnal rows of Tables 5.1 and 5.2, respectively. To check the validity of these results, we compare the

final mass and spin, calculated during the numerical simulation using the apparent horizon and isolated horizon formalism, to an analytic fitting formula (“FF95UU” in Table 5.2) [219]. These analytic fitting formulas were developed using a set of 37 aligned and anti-aligned spinning unequal mass systems, as well as an additional 38 simulations from the SXS catalog [303], which

144 5.3. Quasi-circular Orbits 5.4. Summary

included aligned systems with spins up to χ = 0.98. The ﬁtting formulas give Mrem/M =

0.8940 and χrem = 0.9404, diﬀering from our measured results by about 0.08% in the mass and 0.03% in the spin.

0.5018 0.948

0.5016

0.5014 0.9475

0.5012 0.947 0.501 (+/-) (+/-) χ m 0.5008 0.9465 0.5006

0.5004 0.946 0.5002

0.5 0.9455 0 100 200 300 400 500 0 100 200 300 400 500 t/M t/M

Figure 5.7: The evolution of the horizon mass and dimensionless spin of one of the individual black holes with q = 1 and χ = 0.95.

5.4 Summary

In this chapter we have been able to implement puncture initial data for highly spinning, orbiting black hole binaries by attenuated superposition of conformal Lorentz-boosted Kerr metrics.

We modified the TwoPunctures thorn, in the Cactus/Einstein Toolkit framework, to solve the Hamiltonian and momentum constraint equations simultaneously for highly spinning black holes. We verified the validity of the data by showing convergence of the Hamiltonian and momentum constraint residuals with the number of collocation points in the spectral solver. We then showed, by evolving this data, that the radiation content of these initial data was much lower than the standard conformally flat choice at spins χ = 0.8. This produced a more accurate and realistic computation of gravitational radiation waveforms. We go on to simulate a black hole binary on quasi-circular orbit with χ = 0.95, beyond the Bowen-York limit. The mass and spin of the resulting remnant black hole agrees with the analytic fitting function estimates to within 0.08% and 0.03%, respectively.

Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits 145 Chapter 5. Highly Spinning Black Hole Binaries in Quasi-circular Orbits

146 5.4. Summary Chapter 6

Discussion

In this ﬁnal chapter, we summarize the results of the three projects discussed in this dissertation and outline future research directions.

6.1 Summary of Results

General relativity has revolutionized our picture of the universe. black holes, one of its most fantastic predictions, are expected to often possess incredible intrinsic angular momentum.

They occasionally ﬁnd themselves orbiting in bound pairs, with their separation continuously shrinking by loss of energy through emission of gravitational radiation. To extract predictions of their dynamics from the theory, the highly nonlinear, strongly coupled diﬀerential equations describing the spacetime curvature for black hole binary systems must be solved using numerical methods.

Over the course of three numerical relativity projects, we extended the widely used TwoP- unctures thorn in the Cactus/Einstein Toolkit framework to solve the four elliptic constraint equations with pseudospectral methods in the conformal transverse-traceless de- compositon of the Einstein ﬁeld equation. Series solutions are generated using the collocation point method on a compactiﬁed domain. These initial data constitute the initial boundary conditions for a set of hyperbolic evolution equations in the LaZeV thorn.

In Chapter 3 we began with the goal of simulating the collision from rest of black hole

Chapter 6. Discussion 147 Chapter 6. Discussion binaries with nearly extremal spin. We constructed the initial data using an attenuated superposition of the maximally sliced Kerr spacetime. This decouples the Hamiltonian and momentum constraint equations. In quasi-isotropic coordinates, we are able to factor the singular behavior out of the Kerr spatial metric by a conformal transformation. This turns the Hamiltonian constraint into an elliptic differential equation for the regular part of the conformal factor. The extrinsic curvature is split into longitudinal and transverse parts, and the momentum constraint becomes an elliptic differential equation for a three-vector potential. We evolve the initial data solutions using the moving punctures approach. We compare the new initial data with the widely used Bowen-York solutions at spins χ = 0.9 and find a decrease in the initial spurious burst of gravitational radiation by a factor of ∼ 10. We surpass the Bowen-York spin limit of χ ≈ 0.93 and simulate the collision from rest of black holes with

χ = 0.99. We experiment with a ﬁsheye coordinate transformation which prevents the horizon from collapsing to zero size in the extremal spin limit, and evolve a black hole with χ = 0.999.

In Chapter 4 we simulate black holes with linear momentum and no spin. We constructed the initial data using an attenuated superposition of the Lorentz-boosted Schwarzschild spacetime. This slice has nonvanishing mean curvature. Thus, the Hamiltonian and momentum constraints are nonlinearlly coupled, and must be solved simultaneously. In isotropic coordinates, we are able to factor the singular behavior out of the boosted Schwarzschild spatial metric by a conformal transformation. This turns the Hamiltonian constraint into an elliptic differential equation for the regular part of the conformal factor. The extrinsic curvature is split into longitudinal and transverse parts, and the momentum constraint becomes an elliptic differential equation for a three-vector potential. We evolve the initial data solutions using the moving punctures approach. We compare the new initial data with the Bowen-York solutions y in quasi-circular orbit configurations, with initial linear momenta of the order P(±) ≈ ±0.08, and find that the initial spurious burst of gravitational radiation is reduced by a factor of ∼ 3.

We use the new initial data to surpass the Bowen-York velocity limit v ≈ 0.9c, and perform high energy head-on collisions of black hole binaries with v ∼ 0.99c. We measure the resulting gravitational radiation, and extrapolate to the inﬁnite momentum limit. We estimate that the

148 6.1. Summary of Results 6.2. Future Research Directions

maximum radiated energy is Emax/MADM = 0.13 ± 0.01.

In Chapter 5 we simulate black holes possessing both linear and angular momentum, with the goal of exploring astrophysically relevant quasi-circular orbital configurations of black hole binaries with nearly extremal spin. We constructed the initial data using an attenuated superposition of the Lorentz-boosted Kerr spacetime. Again, the mean curvature is nonvanishing, so that the constraint equations are coupled. In quasi-isotropic coordinates, we are able to factor the singular behavior out of the boosted Kerr spatial metric by a conformal transformation.This turns the Hamiltonian constraint into an elliptic differential equation for the regular part of the conformal factor. The extrinsic curvature is split into longitudinal and transverse parts, and the momentum constraint becomes an elliptic differential equation for a three-vector potential. We evolve the initial data solutions using the moving punctures approach. We compare the new initial data with the Bowen-York solutions in unequal mass y quasi-circular orbit configurations, with initial linear momenta of the order P(±) ≈ ±0.08 and spin χ = 0.8, and find that the initial spurious burst of gravitational radiation is reduced by a factor of ∼ 10. We go beyond the Bowen-York spin limit, and simulate a black hole binary on quasi-circular orbit with χ = 0.95. The mass and spin of the resulting remnant black hole agrees with the analytic fitting function estimates to within 0.08% and 0.03%, respectively.

6.2 Future Research Directions

We have shown that puncture initial data are well suited for simulating black hole binaries with high speed and high spin. When evolved, the conformally curved ansatz produces cleaner gravitational waveforms when compared to the standard conformally ﬂat case, and allows for the investigation of the edges of black hole parameter space. With this foundation, there are several avenues to explore.

6.2.1 Comparison with Excision Initial Data

An alternative method for generating black hole initial data eliminates the singularities through the method of excision. On a closed surface just interior to the horizon, the ﬁelds are required

Chapter 6. Discussion 149 Chapter 6. Discussion to satisfy the boundary conditions of an apparent horizon. The numerical grid inside the boundary is not evolved. The Simulating Extreme Spacetimes (SXS) collaboration has seen fantastic success with this method, using the Spectral Einstein Code (SpEC). They are able to produce stable evolutions of black hole binaries with spins χ ∼ 0.97 [277, 218], and more recently χ ∼ 0.998 [278, 279]. We will generate moving puncture evolutions with comparable parameters in an eﬀort to conﬁrm the SXS results.

6.2.2 Conformal Thin-sandwich Initial Data

We have constructed initial data using the conformal transverse-traceless decomposition of the

Einstein ﬁeld equation. It is advantageous in its simplicity, but the choices of free data are not always clear in their physical motivation. It also says nothing about the gauge choices

(the lapse function and shift vector) which should be used during evolution. The conformal thin-sandwich formalism is an alternative method of finding initial data which fixes both of these shortcomings by considering the spatial metric on two neighboring spatial hypersurfaces, as opposed to the spatial metric and extrinsic curvature on a single hypersurface. The gauge functions become solutions to elliptic differential equations, coupled to the constraints. This allows for the initial data choices to be more easily motivated by the physical configuration one wishes to model. This can be implemented as an extension to the HiSpID thorn.

6.2.3 Ultimate Kicks

Highly spinning black holes can experience gravitational wave recoils up to about 5000 km s−1 [78].

The Lorentz-boosted Kerr initial data allows for us to simulate a wide range of black hole binary systems. This enables us to search through the high spin parameter space for the ultimate kick conﬁguration. We can place upper limits on astrophysical black hole recoil velocities, which is important to the understanding of globular cluster populations and the end result of galactic mergers.

150 6.2. Future Research Directions 6.3. Acknowledgements

6.2.4 Trumpet Initial Data

Our choice of (quasi-)isotropic coordinates represents a “wormhole” slice of spacetime; the initial spatial hypersurface begins and ends on conformal spatial inﬁnity i0. During evolution with the moving puncture approach, the asymptotic end represented by the puncture becomes detached from i0 and moves towards i+ [304]. After a short time—on the order of t ∼ 50M for spinning black holes—the spatial slice settles down to the “trumpet” geometry, named for its appearance in an embedding diagram [305, 306]. Dennison, Baumgarte, and Montero recently found analytical solutions for the trumpet slice of a Kerr black hole [307]. Thus, it might be advantageous to construct initial data already in the trumpet slice. It is important to see that choosing trumpet initial data does not destabilize the evolution or reduce its accuracy.

6.2.5 Multiple Black Hole Systems

Three- and four-body interactions are expected to be common in globular clusters and galactic cores hosting supermassive black holes [308, 309]. There also exists a possible triple quasar system [310]. This motivates the study of multiple black hole configurations. Approximate analytical initial data [227] and accurate multigrid solvers [311] have been successful in simulating triple black hole configurations. This requires a departure from the TwoPunctures coordinate transformation, which is specifically designed to handle black hole binaries. Evolving multiple black hole systems with large spins is of significant astrophysical interest.

6.3 Acknowledgements

I would like to thank my dissertation advisor Carlos Lousto for his great enthusiasm, guidance, and patience. I thank Manuela Campanelli, James Healy, and Yosef Zlochower for many helpful discussions. I also gratefully acknowledge the National Science Foundation for ﬁnancial support from Grants PHY-1305730, PHY-1212426, PHY-1229173, AST-1028087, PHY-0969855,

OCI-0832606, and DRL-1136221. Computational resources were provided by XSEDE alloca- tion TG-PHY060027N, and by NewHorizons and BlueSky Clusters at Rochester Institute of

Technology, which were supported by National Science Foundation grant No. PHY-0722703,

Chapter 6. Discussion 151 Chapter 6. Discussion

DMS-0820923, AST-1028087, and PHY-1229173. This work was supported in part by the

2013–2014 Astrophysical Sciences and Technology Graduate Student Fellowship, funded in part by the New York Space Grant Consortium, administered by Cornell University.

I am eternally grateful towards the faculty, postdoctoral, and graduate student members of the Center for Computational Relativity and Gravitation research group and the Astrophysical

Sciences and Technology program for their instruction and support in what has been one of the most tremendous opportunities of my life. I will strive to continue their example of excellence wherever I may go.

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