Recoil of Binary Black Hole systems and the multipolar structure of gravitational wave signals.

MASTER’S THESIS

RECOIL OF BINARY BLACK HOLE SYSTEMS AND THE MULTIPOLAR STRUCTURE OF

GRAVITATIONAL WAVE SIGNALS

Rafel Jaume Amengual

Master’s Degree in Advanced Physics and Applied Mathematics Relativity and Astrophysics

Centre for Postgraduate Studies

Academic Year 2019-20

ii

Recoil of Binary Black Hole systems and the

multipolar structure of gravitational wave signals Rafel Jaume Amengual

Master’s Thesis

Centre for Postgraduate Studies University of the Balearic Islands

Academic Year 2019-20

Key words:

Recoil, Black Hole, Binary Black Hole, Gravitational Wave, Waveform Modelling, General Relativity, Numerical Relativity

Thesis Supervisor’s Name: Sascha Husa

iii

Abstract

This thesis concerns the gravitational wave signal emitted by merging black holes, as is described by the theory of general relativity, which is our modern theory of gravity. The detection of tens of such gravitational wave signals since 2015 has started the new field of gravitational wave astronomy. The success in the detection of these tiny oscillations on the fabric of the spacetime is a motivation for constant upgrading of the actual ground-based detectors. In the future, space- based and a third generation of ground-based observatories will join efforts. So improved sensitivity and accuracy to the detected gravitational waves will provide an increase in the the quality and quantity of the observed signals, allowing us to extract more physical information from complex sources.

A particularly interesting quantity, which is the subject of this thesis is the gravitational recoil. After an introduction to the theory of general relativity, and the identification of solutions of the linearized equations with gravitational waves, the multipolar structure of the gravitational wave signal is discussed in terms of spin weighted spherical harmonics. A brief introduction is also given to gravitational wave detectors and the process of detecting such waves, and to the theoretical modelling of the signals, which is crucial for the identification of the sources, e.g. for their masses, spins, and location in the sky.

Complex gravitational signals require complex waveform models for the efficient detection and accurate measurement of the source parameters. The gravitational wave recoil has proven to be a very useful tool to test and improve the models. The general calculation of the recoil of a binary black hole system from the multipolar structure of the waves is discussed in detail, and an example numerical simulation is presented, where the recoil is computed and discussed. Finally, an outlook is given to the future relevance of the recoil for waveform models.

Contents

Abstract ii

Abbreviations iii

1 Introduction 1

1.1 Introduction to General Relativity 3

1.1.1 Covariance 4

1.1.2 Curvature 5

1.1.3 Einstein tensor 6

1.2 Gravitational waves 8

1.2.1 Weak field approximation 8 1.2.2 The effect of gravitational waves on free particles 11

1.2.3 Generation of gravitational waves 13

1.2.4 Physics encoded in gravitational waves 15

1.3 Sources of gravitational waves 17

1.4 Gravitational waves detectors 19

1.4.1 Laser Interferometer Gravitational wave Observatory 19

1.4.2 Interferometric detectors 22

1.4.3 Some goals of the gravitational waves astronomy 24

2 The two body problem 25

2.1 Black holes 25

2.2 Parameters of a binary black hole 27

2.3 Stages of a binary black hole, waveforms 30

2.4 Numerical relativity 34

2.4.1 Comments on Hamiltonian formulation of GR 34

2.4.2 ADM 3+1 formalism 35

2.4.3 Spacetime 3+1 38

2.4.4 Comments on BSSNOK formulation 40

2.4.5 Comments on the BAM code 40

2.5 Extracted and derived quantities 42

2.5.1 The Weyl scalar 42

2.5.2 Gravitational waves 44

2.5.3 Apparent horizons 47

2.5.4 Final spin 47

2.5.5 Final mass 48

2.5.6 Radiated energy and momentum 48

2.6 Phenomenological waveform models 49

ψ

3.1 Radiated linear momentum, multipolar expansion 53

3.2 Computing the recoil of a non-precessing BBH 58

4 Conclusions 62

5 Appendix A Mixed terms for 63

References 67

2 ≤ l

≤ 6

Chapter 1

Introduction

In 1915 Einstein’s theory of General Relativity (GR) [1, 2, 3] has replaced Newton’s law of gravitational as our modern theory of gravity. While Newton’s theory explains gravitation in terms of forces, the new theory makes us understand and predict the motion of objects due to gravity from a completely different point of view, still classical but revolutionary. Now it is the geometry of the spacetime which rules the motion of its content and its content shapes that geometry, avoiding the concepts of forces and instantaneity of interaction strongly tied to the Newton’s non-relativistic mechanics and his theory of gravity, the marvellous Newton's Law of Universal Gravitation.

More than a century later GR is still the best theory that we have to describe gravity. Not only does it provide a more accurate picture of the dynamics previously explained by Newton, it expands the physical landscape to new behaviors and exotic objects like black holes (BH), gravitational waves (GWs), light bending, wormholes, or the expansion of the Universe.

Over more than hundred years GR has passed all the tests it was subjected to, starting with explaining the anomalous  precession  of the perihelion of Mercury [4] previously attributed to a possible ghost planet; the bending of light observed during the 1919 solar eclipse.

General relativity is built upon two fundaments, Einstein’s theory of special relativity, which abandons the idea of instant exchange of information, and the principle of equivalence, which traces back to Galilei. It is the latter, the equivalence of gravitational mass and inertial mass, which is the root of general relativity’s geometric interpretation of gravitation: if all objects experience the same acceleration regardless of their composition, then their trajectories can be described in terms of some universal geometry, as expressed by the words of John Archibald Wheeler : 1

“Spacetime tells matter how to move, matter tells spacetime how to curve.”

Gravitation (1973) [11], Misner, Thorne and Wheeler, page 5, right-hand margin, first sketch of this famous quote attributable

1

individually to Wheeler.

FIG 1.1: Massive bodies warp spacetime.

T. Pyle/Caltech/MIT/LIGO

The finite speed of the propagation of information is responsible for the appearance of wave phenomena in general relativity, rather similar in mathematical nature to electromagnetic waves in electrodynamics. In the geometric framework of GR, where the geometry of the spacetime is understood as the physical entity which provides the gravitational dynamics, its is the fabric of spacetime itself which can oscillate and give wave to wave phenomena. This was realized by Einstein early on, who checked and calculated that GR [5, 6] predicts that any non-axisymmetric accelerated density of mass and/or energy creates oscillations of the spacetime. These ripples travel at the speed of light and we refer to them as GWs. While electromagnetic waves can interact strongly with matter - depending on their frequency, GWs interact with matter only extremely weakly. The energy that they emit face thus travels through the universe essentially freely, and the information they carry about their source is not modified - at least in the theory of general relativity. This greatly simplifies the analysis of gravitational wave data.

A hundred years after their theoretical prediction, these GWs were detected for first time on the 14th of September 2015 by the LIGO-Virgo collaboration [7 ,8] and the source was identified as the last orbits and merger of a binary black hole (BBH). Since then, tens of other GW signals have been detected by the advanced gravitational wave detector network [9, 10], with all being consistent with the merger of two compact objects - black holes or neutron stars. This has allowed to start to understand the population of compact objects in the Universe, and has given rise to new tests of the theory of general relativity in the strong field and highly dynamical regime, which have been impossible before [cite some LVC testing GR papers]. Again, our theory of gravitation has passed all these tests to date.

The source of GWs of interest for this work are Compact Binary Coalescences (CBC), specifically the coalescence of BBHs. This is the type of source which by far dominates the event rate, and which shows the strongest effects of general relativity. For such events one can roughly distinguish three phases: the inspiral, merger, and ringdown. In the inspiral the two BHs are orbiting each other as the system is slowly losing energy due to the emission of GWs, resulting in the two objects getting closer and losing energy faster. The peak of radiated power is reached at the merger, when a single black hole forms. The remnant black hole then relaxes to a stationary state through damped oscillations in the “ringdown phase”, which can be described by black hole perturbation theory.

When both black holes are equal, the gravitational waves emitted in the coalescence process only carry away energy and angular momentum. For a generic BBH the emission of GWs carries however also carries away linear momentum, due to the anisotropic and asymmetric emission of gravitational waves. The remnant BH thus receives a recoil. The corresponding “kick velocity” can reach up to several thousand km/

s, as we will see below. The information of this recoil is encoded inside the waveform generated during all that process, and can in principle be measured from the gravitational wave signal if the signal received in the detectors is strong enough.

The recoil is of significant astrophysical interest: First, large recoils can eject a merger remnant form its host galaxy, and even moderate recoils can eject the remnant from globular clusters, thus reducing the chance of subsequent dynamical formation of another BBH and coalescence. Second, if the BBH is surrounded by an accretion disk, then the recoil can cause a collision of the remnant with the accretion disk and result in an electromagnetic counterpart to the GW signal. Furthermore, the recoil is also an interesting parameter for testing and comparing waveform models of BBH waveforms. Both for the astrophysical interpretation and for testing waveform models it is useful to study the dependence of the recoil on the parameter space, and to develop analytical models of the recoil. This is the aim of this work.

1.1 Introduction to General Relativity ²

General relativity can be formulated in a surprisingly simple and compact form in terms of a metric tensor and the tools of differential geometry on manifolds. The metric tensor is related to the line element or infinitesimal distance between points by

(1.1)³

where the are the coordinates used to represent the metric tensor. GR can then be formulated by a single tensor equation, the Einstein field equation:

(1.2)

The left hand side of (1.2) is the geometrical description of the spacetime where is the Einstein tensor which is computed from the metric tensor. The right hand side of (1.2) describes the matter content of the spacetime, where is the stress-energy tensor of the energy-matter field. This tensor describes the flux and the density of energy and momentum inside the spacetime. If we take (1.2) to the Newtonian limit we find that is required to recover Newtonian gravity, where is Newton's gravitational constant, and is the speed of light, which converts (1.1) to the form

(1.3)

Just by inspection, the order of magnitude of is 10^-45 which represents a huge suppression factor, in other words, deforming the spacetime geometry requires a very large amount of energy-matter. For simplicity we choose to work in geometrical units were , unless we state explicitly that we use SI units.4 The tensors , and are symmetric rank 2 tensors, the tensorial Einstein equation can thus be interpreted as set of 10 second order nonlinear partial differential equations (PDE) for the metric components. The value of the metric contains the effects of the curved spacetime, understood as the gravitational effects, but at the cost of solving this really complicated equations system. Einstein was pessimistic about finding exact solutions, however a first and indeed very important class of solutions was obtained by Karl Schwarzschild in 1916 [12], the general solution for a spherically symmetric and static vacuum solution. The Schwarzschild metric constrains a lot the physical system to study, but indeed has wide applications in astrophysics: it describes the geometry surrounding fluid bodies with small angular momentum, in particular astrophysical objects like planets and many stars. For more general complex systems like the BBHs we study here, exact solutions can typically not be found, and a large part of the history of GR has been devoted to developing the mathematical tools to find and analyze more general spacetimes, often using numerical approximations.

g

d s

d s

= d x

d x

g

, x

G

= 8π κ T

.

G

T

κ = G /c

G

c

G

= 8π G c

T

.

κ

G = c = 1 g

G

T

References consulted [11, 13-24, 26-28]

2

Einstein notation: repeated, or paired, indices are implicitly summed over, this notation will be used over all this work.

3

International System of Units.

4

1.1.1 Covariance

We will be interested in the standard framework of GR, where the metric is defined on a smooth four- dimensional Lorentzian manifold (other choices are of interest in quantum or string theory).

As in any other physical law, general relativity is formulated in the language of tensor fields in a coordinate invariant way. Furthermore, the principle of general covariance is one of the axioms of GR. This principle demands that in addition to the theory being formulated in terms of tensors, the geometry of spacetime enters the theory only through the metric. The theory thus takes the same mathematical form independently of the coordinate system chosen, and no tensor fields appear in the theory, which cannot be derived from the metric or be interpreted as dynamical matter fields. Understanding the meaning of general covariance has been a significant challenge in formulating general relativity. Allow me to cite another quote from [11]:

“Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.”

Then (1.1) has to be invariant under the election of smooth and differentiable coordinates, keeping constant the value of the length element. In (1.1) we have the infinitesimal displacements of which have to transform contravariantly, and which has to operate and transform as a covariant tensor. Ruling that a change of coordinates transform as following is needed for fulfilling the requirements of invariance:

(1.4)

(1.5)

Now it is needed to keep this requirements to the core of GR, the relationship which generates the gravity dynamics, variations of the metric related to variations on the distribution of mass/energy in the spacetime, and vice versa. For that we need to know how to differentiate correctly vectors and tensors. In flat spaces equipped with the flat Euclidean or Minkowski metric, considering Cartesian coordinates it is straight ⁵ forward, we only have to differentiate the components. In flat spaces two vectors are compared just by dragging one on top to the other without changes on its direction and magnitude, this situation satisfies the parallel transport. For more general curved spaces, and in particular the curved spacetime considered in general relativity, the change of geometry also has to be considered when taking the derivative of tensor fields. This is solved by the covariant derivative,

(1.6)

(1.7)

x

g

x

= ∂x

∂x

x

, g

=

( ∂x

∂x

∂x

∂x

) g

.

∇

x

= ∂

x

+ Γ

x

, ∇

x

= ∂

x

− Γ

x

, Γ

= 1

2 g

( ∂

g

+ ∂

g

− ∂

g

) ,

Euclidean metric: Minkowski metric:

5 g_μν=d i a g(1,1,1) g_μν=d i a g(−1,1,1,1)

where the are the Christoffel Symbols, which are the connection induced by the metric in the coordinate system chosen (1.7). corrects the partial derivative for terms that are due to the change of the coordinate basis vectors, the lower left index tells us which basis vector is considered, the lower right which coordinate is varied, caused by a change of the basis vector, and the upper one which basis vector is pointing to the direction of that derivative component. This connection prescribes how vectors are correctly

“parallely-transported” in a curved space. A simple situation to that can be considered to illustrate a curved geometry is a 2D sphere embedded in a 3D flat space. In this geometry we can visualize a set of tangent planes to the sphere fully covering it, and a vector centered in one of this “touching” points laying in one of these tangent planes. The embedding of the curved sphere into the 3D flat space provides a natural notion of comparing the direction of vectors in two points of the sphere, and for transporting a vector along a curve without changing direction on the sphere. Without the embedding however, no natural prescriptions for taking derivatives or parallel transport exist. Such a prescription is provided by the connection, or equivalently, Christoffel symbols, which give rise to the covariant derivative. All Christoffel symbols are zero for flat spaces in Cartesian coordinates, but not necessary zero if we choose another coordinate system.

Summarizing, the Christoffel symbols contain the information about the curvature of the space and about the curvature of the coordinates system chosen. For higher rank tensors, and tied to its nature the covariant derivative takes different shapes, it is easy to define just adding terms for each contravariant index and subtracting one for covariant ones. Let us write it for the metric, and the local condition to fully recover the physics described by Newton's laws and special relativity:

(1.8)

(1.9)

1.1.2 Curvature

We have seen the deep entanglement of GR and geometry, where we must distinguish physical phenomena due to the effects of spacetime curvature from the effects of the choice of the coordinates to represent it.

The way to unambiguously quantify curvature it is via the Riemann tensor. A natural way to define the Riemann tensor is via the commutator of the covariant derivative applied to a vector

(1.10) where the value of the commutator is zero if the order of two operations has no effects on the result. For a flat space the order makes no difference, and consequently the value of the commutator is zero, remembering us the parallel transport. Knowing that, any other result comes from the curvature of the space, showing a measure of the non-Euclideanity of it. Therefore for any metric there is a tensor field called Riemann or curvature tensor defined from (1.10)

(1.11)

Now we can rewrite (1.11) using (1.6) and renaming some index to read the Riemann tensor as

(1.12)

Γ

Γ

∇

g

= ∂

g

− Γ

g

− Γ

g

,

∇

g

= 0.

[ ∇

, ∇

]x

= ( ∇

∇

− ∇

∇

)x

,

g

R

x

= ( ∇

∇

− ∇

∇

)x

.

R

= ∂

Γ

− ∂

Γ

+ Γ

Γ

− Γ

Γ

.

At one point of the space it is always possible to choose a coordinate system such that all the Christoffel symbols are zero, but if this space is not flat all the Christoffel symbols for other locations will not be zero.

Then

(1.13) is a necessary and sufficient condition to completely define a flat space. So the tensorial behavior of (1.12) ensures us that if the Riemann tensor is not zero for a coordinate system, it is not zero for any other choice.

This fact give us a way to distinguish the curvature (physics) of the space to the effects of the coordinates choice.

From the Riemann tensor we can derive other curvature tensors just contracting pairs of indexes. Such as the Ricci curvature tensor

(1.14) and the Ricci scalar curvature

(1.15) Since the Riemann tensor is a four rank tensor, it has 4⁴=256 components, which are reduced to only 20 independent components as the result of a variety of algebraic symmetries. In addition the Riemann tensor also satisfies differential identities, in particular the the second Bianchi identity

(1.16)

which plays an important role in defining the Einstein tensor.

1.1.3 Einstein tensor

We start with remembering the form of the Einstein’s field equations (1.1), collecting all constants in a single constant :

(1.17) The left hand side is the Einstein tensor encoding gravity into the metric tensor of the spacetime, and the right hand side is the stress-energy tensor describing the spacetime contents of energy-matter which generates the gravitational field. A simple example would be “dust”, i.e. a pressure-less perfect fluid, which has a stress-energy tensor that can be expressed as

(1.18)

where is the four-velocity of the fluid elements and the density in its local rest frame. Then the local conservation of energy-momentum in a flat spacetime is expressed as

(1.19)

R

= 0

R

= R

= ∂

Γ

− ∂

Γ

+ Γ

Γ

− Γ

Γ

,

R = g

R

= R

.

∇

R

+ ∇

R

+ ∇

R

= 0.

κ

G

= κ T

.

T

= ρu

u

,

u

ρ

∂

T

= 0.

In a curved spacetime the local conservation of energy-momentum can be expressed in any coordinate system via the covariant divergence through the continuity equation

(1.20)

which is an important constraint for (1.17).

A tempting first thought for (1.17) would be identify the Ricci tensor (1.14) to the stress-energy tensor multiplied by a constant. Unfortunately in general

(1.21) which would create tension with the right hand side, since the energy-momentum tensor is always divergence free according to (1.20). However this gives us the path to get the right tensor. So we want that the Einstein tensor to satisfy

(1.22) Now it is the moment to call the second Bianchi identity (1.16), where contracting two pairs of indexes using the metric twice we got an expression appearing two curvature tensors, the Ricci tensor (1.14) and the Ricci scalar (1.15).

(1.23)

This expression can be arranged just using the metric again for rising an index and a bit of algebra

(1.24)

to get the Tensor that Einstein proposed for the geometrical part of GR

(1.25)

Now having an explicit expression for the geometry of the spacetime we can rewrite the Einstein field equations:

(1.26)

From a mathematical point of view, from all possible rank 2 tensors that we can derive by contracting the Riemann tensor, the Einstein tensor is the only one which satisfies the Bianchi identities and (1.22). The Einstein equations can then be interpreted as a set of 10 second order nonlinear PDEs for the metric components.

∇

T

= 0,

∇

R

≠ 0,

∇

G

= 0.

2 ∇

R

− ∇

R = 0.

∇

( R

− 1 2 g

R

) = 0,

G

= R

− 1 2 g

R .

R

− 1

2 g

R = 8π T

.

g

1.2 Gravitational waves

As we have seen along the previous section, the physics of gravity is contained inside the metric of the spacetime. Then GWs are a perturbation of the metric, propagating at the speed of light across the spacetime and thus modifying its geometry. Since the gravitational interaction is in general weaker than other interactions an event involving an extreme quantity of mass-energy is generally required to detect or perceive the effects of GWs. As a curiosity the non axisymmetric motion of the system Sun-Earth is emitting energy via GWs just to approach them barely a proton diameter per day, and the corresponding GW emission of about 200W is still very far from a possibility of detection. For greater and detectable GWs we have to consider much more powerful sources like compact objects, in particular black holes or neutron stars. These sources, luckily for life on earth and unluckily for science, are really far away, so as an approximation we can consider this GWs as a small perturbation of the flat spacetime metric. Of course nearby the source due to strong gravitational field regime, this is by far more complicated.

1.2.1 Weak field approximation

The simplest way to derive GWs from GR is via the weak field approximation. This situation happens when the metric of the spacetime is close to be flat, the gravitational field is “weak”. This assumption will in practice always be satisfied far away from the source, e.g. at the detector on earth, and for sources such as non-compact binary stars. The assumption will also be sufficient to understand some qualitative general aspects of gravitational wave emission. For the motion of the sources we will make the additional assumption that the relative velocities within the source are much smaller than the speed of light. We will now rewrite the Einstein field equations (1.26) in the weak field limit. Since this is an introductory chapter only a brief of the derivations will be shown .⁶

This begins with the assumption of a flat perturbed metric

(1.27)

were is the Minkowski spacetime metric (flat) and is the small and slow perturbation. Here rises the advantage of the weak field regime, it is easier to deal with the differential equations “hidden” inside the Einstein field equations (1.26). Now we can expand it in powers of and keep only the linear terms, which for a transient GW from a far source or for the gravitational dynamics of a solar system holds good results without a big loss of accuracy. This way of facing (1.26) is called “the linearized theory of gravity”.

g

= η

+ h

| h

| ≪ 1,

η

= d i ag (−1,1,1,1) h

h

References consulted [11, 16, 17, 19-25 , 27-29]

6

FIG 1.2: Visualization of a BBH numerical simulation. S. Husa & R. Jaume.

After apply this perturbation, some algebra and keeping only the linear terms on the Einstein tensor reads7

(1.28)

where (1.29)

is the trace-reversed metric perturbation which reduces the increased number of terms.

Then we have the linearized field equations

(1.30) By inspection we find that the third term of (1.30) is the flat space d’Alembertian operator

(1.31)

which points us toward the wave equation character of the Einstein equations. Now applying the Lorentz gauge conditions8

(1.32)

(1.30) takes the form of a wave equation describing metric waves propagating at the speed of light with as source

(1.33) In order to study the propagation of the waves in empty spacetime, we simplify it to the vacuum situation, meaning , just the emitted wave traveling far from the source,

(1.34) This allows us to find the easiest set of solutions, a superposition of plane waves

(1.35)

where is the amplitude tensor and is the wave vector. But plain waves have two degrees of freedom for the amplitude, two polarizations, and has in principle 16 independent components. So we have to reduce it to only 2.

h

G

= 1

2 ( ∂

h ¯

+ ∂

h ¯

− ∂

h ¯

− η

∂

h ¯

) , h ¯

= h

− 1

2 η

h

∂

h ¯

+ ∂

h ¯

− ∂

h ¯

− η

∂

h ¯

= 16π T

.

□ = ∂

∂

= η

∂

∂

= ∂

∂t

− ∇

,

∂

h ¯

= 0,

T

− □ h ¯

= 16π T

.

T

= 0

□ h ¯

= 0.

h ¯

= Re[A

e

],

A

k

h ¯

Along the derivation instead of for rising or lowering indexes is used .

7 g_μν η_μν

Analogous to the applied at the electromagnetic vector potential.

8

The first reduction comes from the fact that is symmetric, then is symmetric. This reduces to 10 independent components. The second comes from the Lorentz gauge, which provides this two conditions for the solutions:

(1.36a) (1.36b)

From (1.36a) we check that the wave solution travels at the speed of light since is a null vector, also named light-like vector. The expression (1.36b) can be understood as the wave amplitude must be orthogonal to the wave vector, this reduces to 6 independent components. So we have a plane wave traveling along null vectors.

The last comes from the choice of another gauge, the traverse-traceless gauge (TT). Accomplishing the Lorentz conditions too this choice makes the trace of become zero. Meaning that while the perturbation is traveling the volume elements of the spacetime suffer stretching and squeezing but its volume is preserved. This reduces to desired 2 the independent components.

Setting an arbitrary propagation direction, parallel to the z-axis , we can write a more manageable and ⁹ understandable solution

(1.37)

On (1.37) only two components of the amplitude are independent, we can expand it to see it more clearly,

(1.38)

Where two independent polarization tensors clearly appear. Which are denoted as + (plus) and x (cross)

(1.39)

Then, these waves of the metric or GWs traveling parallel to the z-axis can be expressed as the linear combination of these two polarization states and , which are functions of .

(1.40)

h ¯

A

h ¯

k

k

= 0, A

k

= 0.

k

A

h ¯

= A

cos(ω (t − z)) .

h¯^{(T T}_μν ⁾=

0 0 0 0

0 h_{x x} h_{x y} 0 0 h_{x y} −h_{x x} 0

0 0 0 0

=

0 0 0 0

0 A_{x x} A_{x y} 0 0 A_{x y} −A_{x x} 0

0 0 0 0

cos(ω(t−z)) .

e₊=

0 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 0

, e_×=

0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

.

h

h

t − z

h¯^{(T T}_μν ⁾=

0 0 0 0

0 h₊ h_× 0 0 h_× −h₊ 0

0 0 0 0

=h₊e₊+h_×e_×

So we have chosen a Cartesian coordinates system for the inertial frame of the GW.

9

1.2.2 The e ff ect of gravitational waves on free particles

After describing a wave solution to the linearized Einstein’s equations it is interesting to understand the effects on matter when a GW is passing through, effects ruled by the 2 polarizations presented in the last section. So for simplicity we will treat the case of a GW facing free particles. If we think about a free particle at rest it will remain at rest for ever if nothing disturbs it, this is equally valid for two or more non interacting free particles at rest. But into the GR context this only means that the coordinates chosen to describe ¹⁰ their position do not change. Well this looks the same from a classical dynamics description, but if a GW passes trough them, they will remain at rest only from a non-GR point of view. The effect of a GW on this set of free particles is to modify the proper distance ¹¹

(1.41)

between them. This change in proper distances caused by the oscillating amplitude of the GW is the base concept of the GWs detectors .¹²

Before check the effects of the two polarizations on a free particle let us review some dynamical concepts.

For classical dynamics the equations of motion for time-independent Lagrangians are Lagrange equations

(1.42a, b)

It is well known that the Lagrange equations are invariant to coordinate transformations. Let us cite a ¹³ paragraph of the classical reference from Professor Cornelius Lanczos [30]:

“The remarkable invariance of the Lagrangian equations of motion with respect to arbitrary point- transformations give these equations a unique position in the development of mathematical thought. These equations stand out as the first example of that “principle of invariance” which was one of the leading ideas of the 19th century mathematics, and which has become of dominant importance in contemporary physics”

Moving us from non-relativistic mechanics to particle motion in GR one to rewrite (1.42) for a free falling particle in a general coordinate system. Where is the proper time facing the role of the classical time. 14

(1.43a, b)

This equations are called geodesic equations, and their solutions geodesics. The geodesics generalize the concept of a straight line in classical mechanics and flat spaces to curved geometries and the GR context.

L =

∫

g

d x

d x

L (q

, · q

; t) d d t

∂L

∂ q ·

− ∂L

∂q

= 0.

τ

L = 1

2 g

x ·

x ·

d dτ

∂L

∂ x ·

− ∂L

∂x

= 0.

inertial frame of reference.

10

In a curved spacetime can be more than one “straight” path so proper distance is along an arbitrary one P.

11

The detectors as interferometers work in the frequency domain, some details will be provided in next sections.

12

[30] “We use the word “covariant” rather than “invariant” if a whole system of equations is involved”

13

In a GR context proper time is the time measured by a clock following the geodesic of the particle.

14

As an example, the projection of a geodesic of the curved spacetime caused by the Sun is the path of the Earth orbiting it.

Back to the situation of the set of free particles, the proper distance between two of them is measured along a geodesic between them. Now we can rewrite (1.42b) using the formalism of GR

(1.44)

Here we rename for simplicity from here we rename and .

Applying the weak field equations we got the expression for the acceleration that cause the oscillatory metric perturbation on the free particles we obtain the solution [11].

(1.45)

(1.46)

For simplicity we consider only two particles, A and B. We choose to analyze the oscillations of the separation between them from the viewpoint of A. Then the proper reference frame of particle A is the coordinate system chosen. To see easily the behavior of this solution we choose an orientation that makes zero one of the polarizations,

(1.47)

d

x

dτ

+ Γ

d x

dτ

d x

d τ = 0.

h

:= ¯ h

t := τ

h

d

x

d t

= 1 2

d

h

d t

x

x

(t ) =

( δ

+ 1 2 h

(t)

) x

h

= 0 x

(t) =

( 1 + 1 2 h

(t )

)

x

y

(t) =

( 1 − 1 2 h

(t )

)

y

.

FIG 1.3: Effect of the two independent polarizations of a GW traveling parallel to the z-axis (perpendicular to the image) on a ring of free or test particles lying on the x-y plane. Top panel shows the polarization ( ), and the bottom one the polarization ( ). Over time both polarization images are showing only some states. For a better statical image we have the next figure. For an animation is recommended this youtube video:

HTTPS://WWW.YOUTUBE.COM/WATCH?V=R4YFGKM25VQ Image: R. Jaume // Video: ESA/C.Carreau

+ h_×= 0

× h₊= 0

FIG 1.4: Same effect explained on previous figure now on a tube of test particles. HTTPS://WWW.EINSTEIN-ONLINE.INFO/

Max Planck Institute for Gravitational Physics.

So while the wave is stretching in one coordinate direction, we are shrinking in the other. Then if we do the same calculations for we get a couple of equations behaving equally but rotated 45º. Next figures are showing this effect on a ring Fig. 1.3 , and on a tube Fig. 4, of test particles.

The TT choice can be a bit a mind constriction for a general coordinate system. So we can define a projector operator [19]

(1.48) Where is a unit vector pointing towards the propagation direction of the GW. Then, using it on the metric perturbation onto the spatial components of ensures the transverse and traceless requirements, all in a generic coordinates system

(1.49)15

So we can rewrite (1.46) assuming z-axis propagation to get the motion equations for the metric and see how the test particles have to move due to GW passing through

(1.50)

Here we rename as before, for simplicity from here we rename

1.2.3 Generation of gravitational waves

Along the last section we have obtained from a small perturbation on a flat the metric, and a , a plane wave traveling over the metric at the speed of light, and it effects on test masses at rest on a flat spacetime. Let us go one step further setting a non zero . So based on the electromagnetism again we know that a general solution for (1.33) can be obtained by a retarded Green function [11, 19]. Since we have chosen Cartesian coordinates for deriving the previous solution, we will continue tied to it for simplicity. So setting t as , as the position of the source and describing the geometry of the source

(1.33) (1.51)

Equation (1.51) can be simplified significantly in the far zone, see Fig. 1.5. Noticing that the detectable astrophysical or cosmological objects are really far, and since the a GW is only defined within the wave zone. So the distance from the detector to the source’s mass-energy distribution is by far larger than the geometry of the source and the wavelength that is being emitted. And the difference between and

, which is the distance of the source’s center of mass to the defector, is absolutely negligible.

h

= 0

P

= δ

− ̂ n

n

̂ .

̂

n

h

h

=

[ P

P

− 1

2 P

P

] h

.

·· x = 1 2 ( ··

h

x + ··

h

y ) ·· y = 1 2 ( ··

h

x − ··

h

y )

h

:= h

T

= 0 T

x

x

y

− □ h ¯

= 16π T

⟶ ψ

(t, x

) = 4

∫

T

( t − |x

− y

| , y

)

| x

− y

| d

y .

x

y

λ x

r

Note that latin indexes are used for spatial components and & are functions of

15 h₊ h_× t−z

(1.52)

(1.53)

Now can be Taylor expanded around the Minkowski retarded time . Keeping only the first term, using the stress-energy tensor for a perfect fluid (1.18) again, and the conservation equation (1.19) we obtain

(1.54)

where the is the mass-quadrupole moment of the source. As we made in the last section let us extract the GW part of (1.54) using the projection tensor (1.48)

(1.55)

We see that the gravitational signal corresponds to the trace-free part of the quadrupole moment, or the reduced quadrupole moment. Equation (1.55) is known as the quadrupole formula of gravitational waves.

Consequently, sources with a vanishing second derivative of the quadrupole will not emit GWs, in particular perfectly spherical sources. Axisymmetric can emit gravitational waves, but only one of two polarization states, essentially perfectly axisymmetric sources will only radiate GWs if their mass-distribution changes along the axis, but not if an axisymmetric body rotates around its symmetry axis. As an example, an apparently axisymmetric neutron star but with a tiny mis-alignment with the rotation axis, can emit monochromatic GWs with enough luminosity to be detected in the near future [31]. Unluckily this quadrupole formula is incomplete for describing composed and/or self-gravitating objects like Neutron stars (NS) and CBCs due to all considerations and approximations. Other methods will be necessary for that, and some will be treated further.

r ∼ | x

| ∼ | x

− y

| > > | λ | > | y

|

ψ

(t, x

) = 4

r ∫ T

( t − |x

− y

| , y

) d

y

T

t′ = t − r

ψ

(t, x

) = 2

r ∂

∂

∫ T

x

x

d

x = 2 r

d

d t

∫ ρ x

x

d

x = 2 r

d

d t

I

(t′), I

I ˜

= I

− 1 3 δ

∑

k

I

, h

=

[ P

P

− 1 2 P

P

] 2 r

d

d t

I ˜

(t − r) .

FIG 1.5: Detector frame

1.2.4 Physics encoded in gravitational waves

After introducing GWs as a traveling metric perturbation and the easiest source, now it is the moment to introduce the flux of energy and the carried momentum from the source by GWs. But at this moment rises a profound problem in GR, energy and the momentum of a gravitational field cannot be defined locally because there is no local expression . But globally is possible to define this quantities for a system in an ¹⁶ asymptotically flat spacetime.

A first consideration from the previous sections and facing only outgoing GWs from the source is that the metric perturbation has the following functional form

(1.56)

So asymptotically (1.57)

A second one comes from the the fact that is very large in comparison to other magnitudes as we state at (1.52). Then GWs can be locally treated as plane waves, meaning that we can neglect angular derivatives compared to radial ones in the next calculations .17

(1.58)

For computing the flux of energy and momentum, avoiding the local indefiniteness, we can consider to use an effective stress-energy tensor of GWs. This was solved in the late sixties by Isaacson [33, 34] proposing an stress-energy tensor which allows to describe GWs energy and momentum over the average of several wavelengths, this is valid for the weak filed approximation. An easy to follow modern derivation can be found in [19]. So let us start with the Isaacson stress-energy tensor

(1.59)

where means an average over time for at least several GW periods or wavelengths. Now using the form of GWs in the TT gauge (1.40) we can rewrite (1.59) in the polarization terms presented in section 1.2.1., complete calculations can be found in [17, 32]

(1.60)

Defining this useful complex quantity called strain¹⁸

(1.61)

h ∼ 1

r f (r − t ) .

∂

h ∼ − ∂

h = − · h r

∂

H ∼ x

r ∂

H

T

= 1

32π ⟨∂

h

∂

h

⟩

⟨ ⟩

T

= 1

16π ⟨∂

h

∂

h

+ ∂

h

∂

h

⟩ .

H = h

− i h

,

This is discussed in [11] section 20.4, summarizing: Local curvature weights zero at the right hand side of the Einstein Field

16

equations.

The angular momentum carried away by GWs will not be treated in this work, its derivation can be found [17, 32,19 ]

17

Will be used in the next chapter. The Weyl scalar , a quantity provided by numerical relativity, can be written in terms of .

18 Ψ₄ ··

H

(1.60) can be expressed as19

(1.62)

Using this equation we can extract the flux of energy and linear momentum carried by GWs depending on the parts of considered. With the contravariant form in mind, we can review its components [11]:

= density of mass-energy (1.63a)

= density of i-component of momentum (1.63b)

= j-component of energy flux (1.63c)

= j-component of flux of i-component of momentum (1.63d)

Then the energy flux of the GW along the radial direction knowing (1.63c) and (1.57)

(1.61)

where is the element of area of a spherical area enclosing the system. Now only remains to integrate over the sphere and take the limit to provide validity on previous assumptions. So we can write the expression for the total energy flux carried by GWs leaving the system

(1.62)

Since we are integrating over the sphere, and the integrand is going to be integrated over time before compute the energy, the averaging operation is not necessary any more.

Now considering (1.63d), (1.57) and (1.58) we can write the flux of linear momentum over the radial direction defined in a flat spacetime

(1.63a)

(1.63b)

So we can compute the flux of linear momentum carried by GWs integrating over the sphere with the same comments of the energy flux

(1.64)

This is the quantity which rules the recoil of a CBC, when its anisotropic emission radiates away linear momentum asymmetrically, appears this “rocket effect”. Now we have an expression for the radiated linear momentum. But is expressed in terms of which is not easy to extract form a numerical

T

= 1

16π Re⟨∂

H ∂

H ⟩,

T

T

T

T

T

T

d E

d t d A = T

= 1

16π Re⟨∂

H ∂

H ¯ ⟩ = 1 16π ⟨ ·

H ·

H ⟩ = 1 16π ⟨ | ·

H |

⟩

d A = r

d Ω

d E

d t = lim

r→∞

r

16π ∮ | ·

H |

d Ω .

l

l

= x

r = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) d P

d t d A = T

= 1

16π Re⟨∂

H ∂

H ¯ ⟩ = 1 16π

x

r ⟨ ·

H ·

H ⟩ = 1 16π l

⟨ | ·

H |

⟩ .

d P

d t = lim

r→∞

r

16π ∮ l

| ·

H |

d Ω .

H = h

− i h

Notation: means real part and the complex conjugate of .

19 Re(a) a a