Gravitational Waves and Coalescing Black Holes

Theoretical Physics

Gravitational Waves and Coalescing Black Holes

Anna Karpinska, Joanna Ekehult annakarp@kth.se, ekehult@kth.se

SA114X Degree Project in Engineering Physics, First Level Department of Physics

KTH Royal Institute of Technology

May 20, 2017

Abstract

Gravitation is a manifestation of the space-time curvature and gravitational waves are distortions traveling through the fabric of space-time. Scientists have tried to detect gravitational waves for decades, but the first direct detection of gravitational waves and the first observation of a binary black hole merger was made in 2015. The purpose of this report is to introduce General Relativity and binary black holes and to calculate the time to coalescence of two back holes with circular or elliptical orbits. We find that a binary black hole in an elliptical orbit with large eccentricity coalesces faster than in a circular orbit.

Sammanfattning

Gravitation är en manifestation av rumstidens krökning och gravitationsvågor är krus-

ningar i rumtidens metrik. Forskare har länge försökt detektera gravitationsvågor, men

den första direkta observationen av gravitationsvågor och hur ett binärt svart hål gener-

era gravitationsvågor gjordes 2015. Syftet med denna rapport är att introducera allmän

relativitet och binära svarta hål och att beräkna tiden det tar för två svarta hål att kol-

lidera. Vi finner att två svarta hål i en elliptisk bana kolliderar snabbare än två svarta

hål i en cirkulär bana.

Contents

1 Introduction 2

2 Background Material 4

2.1 Introduction to General Relativity . . . . 4

2.2 Einstein’s Field Equations . . . . 5

2.3 Black Holes . . . . 7

2.3.1 Binary Black Holes . . . . 8

2.4 Linearized Solution to Einstein’s Field Equations . . . . 9

2.5 Gravitational Waves . . . . 12

3 Investigation 14 3.1 Problem . . . . 14

3.2 Model . . . . 14

3.3 Analytical Calculations . . . . 18

3.3.1 Circular Orbit . . . . 19

3.3.2 Non-Circular Orbit . . . . 20

3.4 Results . . . . 22

3.5 Discussion . . . . 23

4 Summary and Conclusions 24 5 Appendix 25 5.1 Notation . . . . 25

5.2 Covariance and Contravariance . . . . 25

5.3 Raising and Lowering Indices . . . . 26

5.4 Christoffel Symbol . . . . 26

Chapter 1 Introduction

In theoretical physics scientists try to answer some of the most fundamental questions of nature and describe natural phenomena using mathematics. To comprehend general relativity and its predictions for the dynamical interactions of two black holes has been an long-standing unresolved problem in theoretical physics. Gravity is the weakest inter- action known in physics, still it is significant in astrophysics and cosmology. The study of gravitationally bound binary systems is necessary since binaries are important gravi- tational wave sources. There are hopes that more knowledge about gravitational waves will improve the understanding of relativity, astrophysics, particle physics, and nuclear physics and scientists hope to find a theory that combines quantum physics and general relativity [1, 2].

The classical concepts of inertia, the resistance a physical object has to any change in its state of motion, are found in the famous three laws of motion of Isaac Newton.

They describe the relationship between a body and the forces acting upon it, and the motions of the body in response to those forces, and lay the foundation for classical mechanics [3, 4]. The classic field theory, how one or more physical fields interact with matter through field equations, made it possible for scientists to describe phenomena such as electromagnetism. The electromagnetic field theory is the study of electricity and magnetism phenomena caused by electric charges. Some of the most important equations concerning electromagnetism are Maxwell’s equations. They describe electromagnetic fields, the magnetic scalar potential, magnetic mass and magnetic conductivity [4, 5, 6].

The Special Theory of Relativity, where Einstein postulated The Principle of (Special) Relativity: that all physical laws are the same in all inertial frames of reference [7]. It was based on the principle of relativity of Galileo Galilei, which said that all observers in inertial motion are equally privileged, and no preferred state of motion can be attributed to any particular inertial observer. Einstein independently derived and radically rein- terpreted the Lorentz transformations by changing the fundamental definitions of space and time intervals, while abandoning the absolute simultaneity of Galilean kinematics.

The special theory of relativity describes the dynamics of particles moving at speeds comparable with c, the speed of light in vacuum, and the equivalence of mass and energy illustrated by the formula E = mc

, among many other phenomena [7].

The merging of special relativity (SR) and quantum mechanics produced the relativistic

Quantum Field Theory, on which the standard model of particle physics is based. It is a theoretical framework for constructing quantum mechanical models of subatomic particles, particle physics, quasiparticles and condensed matter physics [8].

SR as a theory of flat Lorentzian geometry was a step towards general relativity (GR), where Einstein attempted to formulate a theory of gravity that is compatible with the ideas of SR [9]. He argued that physical laws should not depend on the frame of reference, since these these do not exist a priori in nature. If they did, they would not satisfy the requirement of causality of a physical system [7]. His requirement of general covariance not only changed our concept of gravity but also changed our concept of a straight line.

In Einstein’s theory all particles, including photons will follow the straightest possible line - a geodesic - along the curvature of space-time [10]. This concept allows even elec- tromagnetic radiation to be influenced by gravity. Einstein’s theory led to the conclusion that gravitation is a manifestation of the space-time curvature, with gravitational waves being distortions traveling through the "fabric" of space time [9]. The existence of grav- itational waves was debated for over a century, until the first successful direct detection of gravitational waves was carried out in 2015 by a group of scientists at The Laser Interferometer Gravitational-Wave Observatory (LIGO) in Washington, USA [2]. The first signal, named GW150914, was detected on September 14th 2015 [11]. The detection was made by measuring the tiny disturbances the gravitational waves make to space and time as they pass through the earth. GW150914 matches the waveform predicted by GR for the gravitational wave from two coalescing black holes of 36

M

and 29

M

, with their orbits shrinking, exactly as predicted, due to the emission of gravitational wave energy [11]. Two black holes form a binary black hole when they cannot escape each other’s gravity, so they orbit and merge, creating one black hole [2, 11].

The fact that gravitational waves interact weakly, due to weakness of the coupling of gravity to matter cause them to remain almost unscathed when traveling through all matter, and allows them to be detected from great distances. The black hole binary that was the source to the first direct detection of gravitational waves (GW150914) was more than one billion light years away [12]. The gravitational waves carry with them information about their origins and the history of the Universe, and will hopefully lead to the capability of detecting phenomena unobservable by other means [11]. There are hopes that further measurements could increase the knowledge about the expansion of the Universe and about the properties of black holes [1, 2].

In Chapter 2 we will introduce the reader to some of the principles of GR, including the

Einstein’s Field Equations, black holes, especially in binaries, and gravitational waves. A

solution to the linearized field equations will be derived. In Chapter 3 we will calculate

the time to coalescence for a binary system of black holes, following a discussion of the

results. In Chapter 4 we will summarize the results and discussions from Chapter 3. In

the Appendix the reader will find some of the formulas used throughout this report.

Chapter 2

Background Material

In this Chapter we will present some of the basics of GR. This includes the Einstein field equations and its different components, black holes, and gravitational waves. We will also find solutions to the field equations by linearization. For a more comprehensive overview of the theory we refer to the books [9, 13, 14, 15, 16]. as well as the lecture notes [10]. The linearization is also performed in [9, 14, 15].

2.1 Introduction to General Relativity

Einstein’s theory of general relativity is a metric theory of gravity in modern physics, which not only generalizes SR and Newton’s law of gravitation, but also quantum electro- dynamics and Maxwell’s equations [10]. In GR gravity manifests itself by a tensor field.

A tensor is a mathematical object that is a generalization of the terms scalar, vector and linear operator, and a tensor field is an allocation of a tensor to each point of a space.

The theory of general relativity describes the interaction between space-time and matter, and is based on the equivalence principle; that the local effects of a gravitational field and of acceleration of an inertial system are indistinguishable [9], as well as the postulate of general covariance; that physical laws remain the same under an arbitrary differentiable transformation of space-time coordinates. In order to express the theory of GR as a general covariant theory the use of infinitesimal calculus from SR was replaced with a new formalism called ’absolute differential calculus’ [17], also known as tensor calculus.

GR describes gravity as a consequence of the curvature of space-time, arising from the presence of matter, energy and momentum [15, 16, 10]. Newton’s law of gravitation is most familiar as a mutual interaction between two bodies, but can also be presented as a classical field theory, with a gravitational scalar field Φ,

∇

Φ = 4πGρ. (2.1)

Here G is Newton’s gravitational constant and ρ is the mass density, which acts as a

source term for the scalar field [15]. In GR the gravitational field is instead characterized

by a tensor field, the metric field, described by the metric tensor g

. The source term is

represented by a rank-2 tensor, called the energy-momentum tensor, T

, both of which

will be discussed in further detail in Section 2.2.

The equations that govern the gravitational field in GR are called Einstein’s field equa- tions. These equations compose a system of partial differential equations that describe the geometry, or curvature of space. Solving Einstein’s field equations involves specify- ing the energy-momentum tensor, usually by imposing certain energy conditions for a particular event in space-time, and then solving for the metric tensor [1]. Einstein found that the linearized weak-field equations have wave-like solutions, known as gravitational waves. These are transverse waves of space-time warpage traveling through Universe at the speed of light, generated by time variations of the mass quadrupole moment of the source [11].

The fact that GR is a non-linear makes it difficult to find exact solutions to its field equations, especially since some of the exact solutions lack any physical meaning [18].

The non-linearity arises from the fact that the gravitational field itself possesses energy- momentum, and thus will act as an additional source term for the energy-momentum tensor [15]. There are however some fundamental exact solutions that have contributed greatly to the evolvement of GR [18], some of them describing the gravitational field from black holes. The first exact solution was derived just a year after Einstein published his work on GR [11], and will be discussed further in Section 2.3. For weak, quasi-static gravitational fields and non-relativistic velocities, the predictions of GR converge to those of Newton’s law of gravitation [16], an important property that advocates for the validity of the theory. In this report we will follow the convention used in most of the literature, using the signature (−, +, +, +) for the Minkowski space and reserve the use of Greek indicies to four-dimensional space-times, by convention, the indices µ and ν go from 0 to 3 where 0 is the time coordinate and 1, 2 and 3 represent the space coordinates.

2.2 Einstein’s Field Equations

The Einstein field equations are given by

R

+ 1

2 Rg

+ Λg

= 8πG

c

T

, (2.2)

where R

is the Ricci curvature tensor, R, the curvature scalar, g

, the metric tensor and T

is the energy momentum tensor. Further on, Λ is the cosmological constant, c is the speed of light and G is Newton’s gravitational constant. Using Einstein’s summation convention, summing over the repeated indices µ and ν, the above equation gives us ten highly non-linear (with respect to the metric and its first derivative), coupled partial differential equations of the second order [19].

An alternate way of representing the field equations can be achieved by introducing the Einstein tensor, G

= R

−

g

, then one can formulate the above equation as

G

= 8πT

, (2.3)

where we have introduced geometrized units, c = G = 1. Notice that the term Λg

has

vanished from the equation. The cosmological constant, which can be seen as the energy

density of vacuum i.e. the energy density without any matter present has been estimated

through observations of supernovae to Λ ∼ 10

[20]. Predictions from quantum field

theory suggests that Λ is about 10

times larger, and is still a major unresolved problem in physics. [15].

The Metric Tensor

The metric tensor, g(u, v), is a second-rank symmetric tensor which takes in a pair of tangent vectors u, v and returns their scalar product

g(u, v) ≡ u · v. (2.4)

If we consider an infinitesimal displacement ds on a N-dimensional Riemannian manifold we can see that [15]

ds

= g(dx, dx). (2.5)

In some sense, the metric thus tells us how distance varies on the manifold. Equation (2.4) can be written in component form,

ds

= g

dx

dx

. (2.6)

Note that ds

is invariant under transformation, and that for the special case of flat space-time, we have the Minkowski metric, η

such that

η = diag(−1, 1, 1, 1), (2.7)

where we again, have used the convention of setting c = 1.

The Ricci Tensor and the Ricci Scalar

The Ricci tensor can be derived through contraction of the Riemann tensor. A simple contraction of a tensor, in terms of components, means setting an upper index equal to a lower index and summing over them. This operation will reduce the tensor rank [15].

The Ricci tensor gives us information about the curvature of space-time in the sense that it tells us how much matter converges or diverges in time [15].

The Ricci tensor can be expressed in terms of the Christoffel symbols (see Appendix (A10),

R

≡ Γ

− Γ

+ Γ

Γ

− Γ

Γ

. (2.8) A second contraction of the Riemann tensor gives us the Ricci scalar,

R ≡ g

R

= R

= R. (2.9)

The Energy-momentum tensor

The energy-momentum tensor, T

, sometimes called the energy-stress tensor [10] is a

symmetric rank-2 tensor that contains all forms of momentum and energy present, and

tells us how they are distributed. The components of the tensor are the sources of gravity,

and are not restricted to matter-energy but also, for example, electromagnetic radiation,

neutrino fields and meson fields [14, 15].

We can now see that the Einstein’s field equations on the form (2.3) balance curvature (on the left-hand side) with matter and energy (on the right-hand side). Some of its most important solutions predict the existence of black holes, which will be discussed in the next Section.

2.3 Black Holes

A black hole is a region in space-time that induces such a strong gravitational attraction, such strongly curved space-time, that no particles or radiation, not even light, can escape its gravity. Such a strong gravitational field is often the product of a massive dying star collapsing due to depletion of nuclear fuel [21]. Even in Newtonian physics it’s possible for an object to be so dense that the escape velocity would exceed the speed of light, c.

Newton showed that an object, with mass M and gravitational radius r

, could have an escape speed, v

, equal to the speed of light, c [16]:

v

=  2GM r



= c ⇒ r

= 2GM

c

, (2.10)

where G is Newton’s gravitational constant. A black hole is formed when a gravitating object of mass M becomes smaller than its gravitational radius [21]. To describe black hole’s consistently one however needs the tools provided by GR. Black hole’s are relatively simple to describe mathematically, in general relativity. The simplest one’s consisting of only a singularity surrounded by an event horizon. The event horizon is often described as the "black hole’s surface" but is in reality only a mathematical surface inside of which a velocity greater than the speed of light is needed to escape the gravitational attraction, causing any information that passes the event horizon to be irreversibly lost for any external observer [22].

An important solution to Einstein’s field equations is the Schwarzschild solution, which played a major role in the early development of GR and is even today regarded as a solution of fundamental importance [9]. This solution describes

a point mass M placed at r = 0 surrounded by an event horizon at r = r

, where r

is the Schwarzschild radius. This radius coincides with the gravitational radius of (2.10) and provides an effective size for the black hole [16]. A black hole with the mass of the sun would for example have a Scwarzschild’s radius r

∼ 1.5 km. Schwarzschild’s spherically symmetric solution describes describes the gravitational field outside of a non-rotating static black hole [9, 23]. The Schwarzschild metric [9],

ds

= −(1 − r

r )c

dt

+  1 − r

r



dr

+ r

(dθ

+ sin

θdφ

). (2.11) is only characterized by its mass, M . We can see that (2.11) has two singularities; one at r = 0, which is an actual physical singularity, and the second at r = r

being a coordinate singularity, meaning it depends on the choice of coordinate system. Above we have described the simplest type of black holes, where we only need to know one parameter, the mass of the black hole to to determine the metric g

. If we want to

1assuming the use of Schwarzschild coordinates

consider a general black hole we need two additional parameters; charge, Q and angular momentum J [9]. Charged black holes are usually only of theoretical interest since a charged black hole will be neutralized rapidly by the plasma surrounding it [21, 23].

A black hole with angular momentum J > 0, and with zero charge is known as the Kerr solution, which generalizes the Schwarzschild solution to rotating sources. Rotating black holes cause space-time to drag with them as they rotate, a phenomena known as frame dragging [21].

2.3.1 Binary Black Holes

A binary system of black holes consists of two black holes so close to each other that their gravitational attraction causes them to become gravitationally bound. The loss of energy and momentum, due to the emission of gravitational waves will cause the separation between them to decrease, eventually leading them to coalesce into a single black hole. Binary system are not restricted to black holes, also stars, asteroids and planets form binary systems [23].

There are three stages of a binary black hole coalescence: the inspiral, the merger and the ringdown stage. The inspiral stage is when the black holes are two well-separated bodies that lose energy and angular momentum when spiraling together until they have reached a stage of dynamical instability. The binaries lose energy at a much higher rate compared to the loss of angular momentum [22]. This leads to circularization of an initially eccentric orbit, assuming that the binaries masses are of comparable size [22]. The inspiral is the slowest stage, and in the early stages of the inspiral one can consider the progression as quasi-adiabatic, meaning that the change in orbit much smaller compared to the orbital period. As the orbit shrinks the gravitational wave emission increases and causes the orbit to shrink more rapidly. The produced gravitational waves have a characteristic form of a chirp, a sine wave with frequency and amplitude increasing with time [13]. The phase of the inspiral is affected by the magnitude of the mass ratio and spin of the black holes and the orbital eccentricity and frequency. In particular, the spin projections along the direction of orbital angular momentum affect the inspiral rate of the binary. The binary black holes will eventually reach their innermost stable circular orbit (ISCO), after of which we can no longer speak of stable or circular orbits. In Schwarzschild coordinates the radial distance between the black holes at the ISCO can be approximated by

r

= 6Gm

c

, (2.12)

with m being the total mass of the system [13]. The second stage is the merging of two black holes, during which the black holes "meet" and form a single event horizon.

In this stage the dynamics of the binary are described by the strong-field, dynamical part of general relativity [23]. In the merger stage the binaries will generate the most prominent gravitational waves, and the dynamics of this stage are highly non-linear.

The merger stage is affected by the mass and the spin of the final black hole, and is only

stage not amenable with perturbation techniques [23]. In the ringdown stage, following

immediately after the single black hole is created, the black hole oscillates and is damped

by the emission of gravitational waves until in a stable state [23].

2.4 Linearized Solution to Einstein’s Field Equations

We start by considering a weak gravitational field. This gravitational field could for instance be caused by gravitational waves emitted from a distant source, passing through a detector on earth. In this region we can, for some coordinate system, with coordinates x

, decompose the metric into two terms: the first one being the Minkowski metric, η

and the second a small perturbation, h

, i.e.

g

= η

+ h

, (2.13)

where |h|  1. These coordinates are so-called quasi-Minkowskian [15]. It is clear that since η

is symmetric (with respect to swapping indicies) h

must be symmetric as well.

We are looking for solutions in vacuum, outside of the source, so T

= 0.

The Ricci scalar is defined as the trace of the Ricci tensor. Tracing in Minkowski space is done with the Minkowski metric:

R = η

R

= η

R

= R

. (2.14) This reduces equation (2.3) to simply

R

= 0. (2.15)

The outline of the linearization is as follows: To solve for (2.15) we start by linearizing the Christoffel symbol, Γ, by expressing it in terms of the metric and only considering terms linear in h. Then, we express R

in terms of Γ and again throw away terms non-linear in h. This will give us ten equations to solve. However, using some of the symmetries [13]("gauge transformations"), we can eliminate some of the redundant degrees of free- dom. We will then impose gauge conditions to deal with the residual degrees of freedom.

It can be shown that the Christoffel symbol can be expressed as [12]

Γ

= 1

2 g

 ∂g

∂x

+ ∂g

∂x

− ∂g

∂x



= 1

2 g

g

+ g

− g

, (2.16) where commas represent partial derivatives. Since the Minkowski metric is constant, the Christoffel symbol only depend on g

and different derivatives of h, i.e.

Γ

= 1

2 g

(h

+ h

− h

) . (2.17) The contravariant metric is given by g

= η

+ h

, where η

= η

. Note that raising the indices (see Appendix (A8)) of h

can be performed with the Minkowski metric instead of g

, since the components of h are small. So the contravariant metric takes the form

g

= η

− h

η

η

. (2.18) Now, inserting (2.17) into (2.18), we find

Γ

= 1

2 h

+ h

− h

+ O(h

) ⇒ δΓ

= 1

2 h

+ h

− h

. (2.19)

We use δ to emphasize that this is the linearized expression of the Christoffel symbols. We

are now ready to linearize the Ricci tensor. The Ricci tensor expressed in the Christoffel

symbol, which we stated in the previous section (see equation 2.8), contains two terms linear in Γ and two quadratic in Γ, and can thus be written as

R

= Γ

− Γ

+ O(Γ

), (2.20) i.e.

δR

= δΓ

− δΓ

. (2.21) We can now express the Ricci tensor in terms of the perturbation by inserting (2.14) into (2.16) and calculating the derivatives. The linearized form of the Ricci tensor now becomes

δR

= 1

2 (h

− h

− h

+ h

). (2.22) We can further simplify this expression by defining h ≡ h

= η

h

and  = ∂

∂

, which is the d’Alembertian operator,

 = ∂

∂

= 1 c

∂

∂t

−  ∂

∂x

+ ∂

∂y

+ ∂

∂z



= 1 c

∂

∂t

+ ∇

. (2.23)

 represents the Laplace operator in Minkowski space. Furthermore, we define h ¯

= h

− 1

2 h

. (2.24)

¯ h

is a so-called trace-reversed perturbation variable (since the trace is simply ¯ h

= −h).

The linearized Einstein field equations in vacuum then take the form

δR

= −h

+ ¯ h

+ ¯ h

= 0. (2.25) It can be shown that introducing "ripples" in the coordinates ,

x

→ x

: x

= x

+ E

, (2.26) where E

is some arbitrary vector field such that |E

|  1, generates small changes in the forms of all tensor and vector fields, as well as scalars [24]. These changes however are small enough to be ignored for all relevant quantities except the metric [12] Furthermore, it can be shown by using the transformation properties of the metric that the perturbation functions in the two coordinate systems are related by

h

(x) = h

− E

− E

. (2.27) This gives the metric on the form

g

= η

+ h

. (2.28)

This is done in detail in [12]. What this means is that there are solutions that satisfy (2.25), where h

is not caused by physical changes, but "wiggles" in the coordinates.

Hence the linearized field equations are invariant under the transformation[15, 9].

h

→ h

− E

− E

. (2.29)

Note that we do not view this as a coordinate transformation, but as a gauge transforma-

tion, i.e. we are working in the same coordinates as before [15]. This can be compared to

the familiar gauge ambiguity in electromagnetism. In Maxwell’s equations, gauge trans- formations induce different potentials but the same electric field. In linearized gravity, gauge transformations lead to different perturbations h, but the same linearized Ricci tensor, δR

We use the same approach as in electromagnetism and solve this by fixing the gauge [15]. A standard choice is the gravitational analogue to the Lorenz gauge [15].

It can be shown that the coordinates can be specified (for any physical situation) in such a way so that [14]

¯ h

= 0. (2.30)

Using the gauge freedom (2.30) reduces (2.25) to

h

= 0, (2.31)

under the condition

h

− 1

2 h

= 0. (2.32)

Here h

≡ η

h

. It is now easy to verify that the solutions to the linearized field equations in vacuum are plane waves,

h

= α

exp(ik

x

) + α

exp(ik

x

). (2.33) Here α is the symmetric polarization tensor that contains information about the different states of polarization and k

is the 4-wave vector, with the general form

, k
[9].

For h

to satisfy (2.31) we require

k

= k

k

= −ω

c

+ k

= 0. (2.34)

The fact that k is a null vector means that the waves travel at the speed of light.

The gauge condition (2.32) requires that k

α

= 1

2 k

α

, (2.35)

where k

= η

k

. Note that since (2.35) gives us four conditions, this will reduce the number of degrees of freedom to six, from the initial ten. The Lorentz gauge does not fix the gauge completely, we still have some residual degrees of freedom left. This can be solved by using yet another gauge condition. The transverse traceless gauge (TT-gauge) is defined by choosing

h

= 0, h

= h

, Tr(h) ≡ h

= 0, and h

= 0, (2.36) which for solutions on the form (2.33) corresponds to

α

= 0 and Tr(α) = 0 (2.37)

Using (2.36) together with (2.32), and choosing the direction of propagation along the z-axis we get

h

(t, z) =









0 0 0 0

0 α

α

0 0 α

−α

0

0 0 0 0









µν

e

. (2.38)

We see that there now are only two degrees of freedom left. The terms α

= −α

are called the cross-polarization of the wave, denoted h

and the terms α

= α

are called the plus-polarization, denoted h

. Since a physical wave has to be real we take the real part of (2.38) and simplify;

h

= h

h

h

−h



ab

cos [ω(t − z/c)] , (2.39)

where a, b are indicies in the (x, y)-plane.

2.5 Gravitational Waves

Another common type of approximation is the post-Newtonian (PN) approximation. PN approximations are often used in general relativity, especially when studying the dynam- ics of gravitationally bound systems [13]. These approximations are based on weak fields and non-relativistic velocities and can be derived by formulating the classical Newtonian equations and expanding in terms of (v/c)

, where v is the typical speed of the system [25]. The first order expansion is known as the quadrupole formula, which describes the gravitational radiation from a system with a time-varying quadrupole moment. It can be shown that the amplitude of the gravitational waves can be expressed in terms of the second-mass moment of the mass distribution [13, 25]

h

∼ 1 r

Q ¨

. (2.40)

Q

is also known as the quadrupole tensor, and is given by Q =

Z

dx

ρ(t, x)x

x

. (2.41)

. For a more accurate description, expansions of higher order terms are needed.

In the linearized regime gravitational waves have certain characteristics that are very similar to the ones of electromagnetic waves. Gravitational waves - like electromagnetic waves - are characterized by their wavelengths and frequencies. Electromagnetic radia- tion is caused by accelerating charges and changing currents, while gravitational radiation is caused by accelerating masses [13, 14]. Electromagnetic radiation is to leading order produced by oscillating electric and magnetic dipoles

, gravitational radiation is to the lowest order caused by an alternating mass- and current quadrupole moment [13]. Con- servation of total mass-energy guarantees that there can be no monopole gravitational radiation and conservation of linear and angular momenta removes any possibility of dipo- lar gravitational waves

, making the leading-order contribution to gravitational radiation quadrupolar [13]. This shows that gravitational waves are weaker than electromagnetic waves [10]. Also, because of the small gravitational cross-section, scattering and absorp- tion is minimal [13]. The weak coupling of gravitational radiation to matter enables the

2In the far field region

3Keep in mind that we are talking about the linearized region. Generally mass and momentum are not conserved, still monopole- and dipole moments do not oscillate outside of the linearized region [13]

gravitational waves to travel relatively unaffected through space-time and making them detectable from great distances. This is however not the case in the near vicinity of a black hole, where the absorption will be extensive [13, 26]. The only significant change as they propagate is the decrease in amplitude while they travel away from their source [13], as seen in (2.40). Gravitational waves have only two independent states of polarization in Einstein’s theory: the "plus" polarization and the "cross" polarization, h

and h

[9].

The angle between the two polarization states is π/4. In GR, a gravitational wave far from the source can be approximated by a superposition of these states, which oscillates in time and propagates with the speed of light, written as a complex waveform strain h [9]

h = h

+ ih × . (2.42)

Each state produces tidal forces (stretching and squeezing forces), perpendicular to the wave propagation direction, on any object which it passes [16]. The way LIGO, and other gravitational wave detectors, detect gravitational waves is to measure the very small compressions and elongations that are caused by passing gravitational waves [16].

The linearized solution to Einstein’s field equation is applicable for gravitational waves

since we assume gravitational waves to be weak. We expect g

= η

+ h

to be a valid

approximation to the full gravitational theory when the metric departs only slightly from

that of flat space time. The equation of motion for linear perturbations to flat space-time

in the Lorenz gauge is a wave equation, with the energy momentum tensor T

acting as

a source term. In vacuum, the metric perturbation solutions will be waves, and where

matter present it can generate gravitational waves.

Chapter 3 Investigation

3.1 Problem

Black-hole binaries are one of the strongest sources of gravitational waves it is legitimate to wonder in what timespan the binaries merge [23]. In this Chapter we will analytically calculate the time it takes for two black holes to coalesce using the physical laws for energy, angular momentum and gravitation. Since gravitational radiation is caused by alternating mass quadrupole moments, this is done by finding the quadrupole momentum tensor Q

and expressing the energy and angular momentum radiation and angular distributions in terms of time derivatives of Q

. These equations are then applied to a system of two point masses, m

and m

, expecting that the binary is moving in an elliptical orbit as if it simply experienced Newtonian gravity, and then integrating these equations along that path. There is no net-contribution of oscillatory changes after each cycle, hence only secular decays are considered. The secular decays of the orbit’s semi- major axis and eccentricity are found as functions of time, and are then integrated to express the time to coalescence as functions of the orbital initial conditions.

We refer to the book [13] for an overview as well as a different derivation of the time to coalescence. In this report the results from [27, 28] are used, and we refer to these two articles for more detailed calculations and derivations.

3.2 Model

Einstein’s theory of general relativity predicts black-hole binary dynamics and the gen-

erating of gravitational-waves at all stages of the merger process. But since finding

exact solutions is difficult, modeling the merging of two black holes requires different

approximations and numerical simulations. The early inspiral stage can be calculated

analytically by using the PN approximation, which is a low-velocity and weak-field per-

turbation expansion of the Einstein field equations in terms of (v/c)

, where v is the

typical dynamical speed inside the system and c is the speed of light [25]. To assure

convergence we have the requirement that v  c, which means that this expansion is

only valid in the early- to mid-stages of the inspiral, since increasing velocities cause the errors to grow and thus making the PN-approximation invalid [23].

As the black holes get closer, we reach the late inspiral stage and subsequently the merger phase. This is when the gravitational wave frequencies are in detection range and the PN-approximation breaks down, instead the ringdown is often approximated using black hole perturbation theory. To model the entire evolution requires large-scale numerical relativity simulations, that can simulate space-time [23].

We will model the motion of two black holes resulting from gravitational radiation by as- suming the binary is well-separated, and therefore treating the objects like point masses.

We will solve the two-body problem for the two point masses in a Keplerian orbit, which is the problem of finding motion and gravitational radiation of self-gravitating relativistic systems. A binary system in a Keplerian orbit radiates both energy and angular mo- mentum [13]. In our approximation of point-like bodies the system loses its energy and angular momentum and therefore the orbital motion changes, both in its semi-major axis a and in its ellipticity e, until the system enters the merging phase and collapses.

Newton’s law of universal gravitation states that the force of gravitational attraction between two point-like bodies, with masses m

and m

and distance r = |r

− r

|, is equal to

F

= G m

m

r

, (3.1)

where G is the universal constant of gravitation [3]. We combine this with Newton’s second law of motion: F = ma = m¨ r [3], and get the differential equations

m

¨ r

= Gm

m

r

− r

r

and m

¨ r

= Gm

m

r

− r

r

. (3.2)

We derive the differential equation for the relative motion by starting with combining the above equations:

¨

r = −G(m

+ m

) r

− r

r

, (3.3)

here r

− r

is the relative coordinate. By then taking the cross product of equation (3.3) with r we get that

r × ¨ r = 0, (3.4)

and since

d

dt (r × ˙r) = ˙r × ˙r + r × ¨ r = 0, (3.5) it follows that r× ˙r = constant. Therefore, the angular momentum, L = r×(m

+m

) ˙r, is also constant, which implies that the orbit lies in a plane. Since r and ˙r are perpendicular to the angular momentum, the orbit is in a plane orthogonal to the angular momentum.

If A is the area over which the line joining the two bodies has swept up to a specific time, the rate of change of this area is then

dA dt = 1

2 |r × ˙r| = 1

2(m

+ m

) |L| = constant. (3.6)

The quantity solving equation (3.3) may be obtained by taking the cross-product of the angular momentum and equation (3.3):

L × ¨ r = −G(m

+ m

) 1

r

hL × ri = −G(m

+ m

)

d dt

 r r

, (3.7)

and then integrating this equation with respect to time, we find that L × ˙r = −G(m

+ m

)

 r

r + e

, (3.8)

where e is the constant of integration called the eccentricity vector. The solution is obtained by taking the scalar product of r with (3.10). One can perform the following computations.

r · hL × ˙ri = r · −G(m

+ m

)

 r r + e

= − 1

(m

+ m

) L

. So the squared angular momentum can be expressed as:

L

= G(m

+m

)

r·  r r +e

⇒ r · r

r +r·e = L

G(m

+ m

)

⇒ r+re cos θ = L

G(m

+ m

)

where θ, the true anomaly, is the angle between e and r. We see that

r = L

G(m

+ m

)

1

1 + e cos θ = p

1 + e cos θ , (3.9)

which is the polar formula of a conic section. Here p is the semi-latus rectum p =

, which is the distance from the focal point to the curve, measured perpendicular to the eccentricity vector e. The shape of an ellipse is characterized by the semi-major axis a and the eccentricity e. Any point on the ellipse can be characterized by the true anomaly θ or by the eccentricity anomaly E. The radius of a specific point on the ellipse is given by the shape equation

r = a(1 − e cos E). (3.10)

Since

p = L

G(m

+ m

)

= a(1 − e

), (3.11) the equation of relative orbit of the motion of two point masses resulting from gravita- tional radiation is

r(θ) = a 1 − e

1 + e cos θ , (3.12)

where a is the semi-major axis and e is the eccentricity. The eccentricity describes the

shape of an orbit, with e = 0 being a circular orbit, 0 ≤ e < 1 an elliptic orbit and

e > 1 a parabolic orbit. In figure 3.1 the definitions for an elliptical orbit are shown.

Figure 3.1: Elliptical orbit and its definitions.

The energy and angular momentum radiation and angular distributions in a binary sys- tem are expressed in terms of time derivatives of the quadrupole moment tensor in the non-relativistic limit (velocities much smaller than c) [27]. The quadrupole formula is the lowest-order PN approximation for the emitted radiation. The quadrupole moment tensor for a discrete system of point masses is can in analogy with (2.41) be expressed as [25]

Q

= X

a

m

x

x

. (3.13)

We evaluate the components of the quadrupole momentum tensor. The non-vanishing Q

are [28]

Q

= µr

(θ) cos

θ, Q

= µr

(θ) sin

θ, Q

= Q

= µr

(θ) cos θ sin θ,

where µ = m

m

/(m

+ m

) is the reduced mass. The derivations for the time derivatives

of the quadrupole tensor who give the energy and angular momentum radiation and

angular distributions are done in [27, 28], in which they use the PN approximation,

also discussed in Section 2.5. The resulting equations from these papers are used in the

calculations in the section below.

3.3 Analytical Calculations

In GR a and e are functions of time, which will be slowly varying in the non-relativistic limit. They are related to the total energy E

and the relative angular momentum L through [13]

a = − Gm

m

2E

and L

= Gm

m

(m

+ m

) a(1 − e

), (3.14) note that that a is independent of the angular momentum L, so orbits with the same energy have the same value of the semi-major axis a. The angular velocity ˙ θ vary along the orbit axis according to:

dθ dt = 1

r

[G(m

+ m

)a(1 − e

)]. (3.15) The power radiated in gravitational waves can be obtained with the quadrupole formula, see Eq. (2.40) [27]

P = − dE

dt = G

45c

... Q

= G 5c

 d

Q

dt

d

Q

dt

− 1

3 d

Q

dt

d

Q

dt



, (3.16)

The calculations for the time derivatives are done by P. C. Peters and J.Mathews [27, 28]

as mentioned above. With their results we obtain

− dE

dt = 8G

m

m

(m

+ m

)

15a

c

(1 − e

)

(1 + e cos θ)

[12(1 + e cos θ)

+ e

sin

θ]. (3.17) We average this over one period of the elliptical motion, since a particle in a Keplerian elliptic orbit emits gravitational waves at frequencies which are integer multiple of the frequency ω

[13], where

ω

= G(m

+ m

)

a

. (3.18)

And therefore, the period of the gravitational waves is a fraction of the orbital period T, T = 2π

ω

. (3.19)

So we take the average of dE/dt over one period T [13]. The angle brackets indicate an average over an orbit. We get [27]

 dE

dt

time

= − 32 5

G

c

m

m

(m

+ m

) a

(1 − e

)



1 + 73

24 e

+ 37 96 e



, (3.20)

It is convenient to define

E

(e) = 1 +

e

+

e

(1 − e

)

. (3.21)

E

is known as the Peters and Mathews enhancement factor [27]. Figure 3.2 shows the

rapid increase of the enhancement factor with increasing eccentricity of the orbit, note

that this implies a rapid increase in the energy loss rate.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 eccentricity e

0 10⁰

10¹ 10² 10³ 10⁴ 10⁵

0

Figure 3.2: Enhancement factor

3.3.1 Circular Orbit

For a circular binary, without eccentricity with initial semi-major axis a

we observe that (3.20) becomes [27]

−  dE

dt

= 32G

5c

a

m

m

(m

+ m

). (3.22) Assuming that the masses are constant we can write

 dE

dt

= G

c

∂

∂t



m

+ m

− m

m

2a



= G

c

m

m

2a

˙a, (3.23) we then obtain

 da dt

= − 64 5

G

c

m

m

(m

+ m

)

a

. (3.24)

Since the rate of inspiral due to gravitational wave emission is proportional to a

, it follows that as the two bodies approach each other, they inspiral faster and faster. We find the time to coalescence by taking

da

dt = 4a

da

dt = − 256 5

G

c

m

m

(m

+ m

), (3.25) and we see from this that the inspiral reaches a = 0 in a finite time, the time to coales- cence:

t

= 5 256G

a

c

m

m

(m

+ m

) . (3.26)

3.3.2 Non-Circular Orbit

For a non-circular orbit, e

6= 0, the derivation is not as simple, but still straight forward.

Even if the orbit initially was elliptical, the emission of gravitational radiation will in any case quickly circularize the orbit [28]. From equation (3.14) we obtain

da = −Gm

m

2E

dE

= −2a

Gm

m

dE

⇒ da

dE

= − 2a

Gm

m

. (3.27) Since

da dE

dE

dt = da

dt , (3.28)

we get that

 da dt

= − 2a

Gm

m

 dE

dt

= − 64 5

G

c

m

m

(m

+ m

) a

(1 − e

)



1 + 73

24 e

+ 37 96 e



. (3.29) To find the differential equation relating a to e, during a decay of the orbit for which gravitational radiation is the only energy loss mechanism, we need an expression for the average angular momentum carried by the gravitational waves over one period. This is [28]

 dL dt

= − 2G 45c

eh ...

Q

Q ¨

i = − 32 5

G

c

m

m

(m

+ m

)

a

(1 − e

)

 1 + 7

8 e



. (3.30) From this equation we can derive

 de dt

= − 304 15

G

c

em

m

(m

+ m

) a

(1 − e

)



1 + 121 304 e



. (3.31)

Note that for e = 0 then de/dt = 0, therefore a circular orbit remains circular and for e > 0 then de/dt < 0 and therefore, an elliptic orbit becomes more and more circular because of the emission of gravitational waves. Since

da dt

dt de = da

de , (3.32)

we obtain

 da de

= 64 5

G

c

m

m

(m

+ m

) a

(1 − e

)

15c

a

(1 − e

)

304G

em

m

(m

+ m

)

1 +

e

+

e

1 +

e

=

= 12 19 a e

[1 + (73/24)e

+ (37/69)e

]

(1 − e

)[1 + (121/304)e

] . (3.33) These equations are enough to determine the orbital decay uniquely. Starting from a given orbit with parameters a

and e

, we find a(e) by analytically integrating (3.33):

a(e) = c

e

(1 − e

)



1 + 121 304 e



. (3.34)