Gravitational Waves in General Relativity

Aaron Bello

June 20, 2017 Abstract

In this paper, we write a summary about general relativity and, in particular, gravitational waves. We start by discussing the mathematics that general relativity uses, as well as the geometry in general relativity’s spacetime. Af- terwards, we explain linearized general relativity and derive the linearized versions of Einstein’s equations. From here, we construct wave solutions and explain the polarization of gravitational waves. The quadrupole formula is derived, and generation and detection of gravitational waves is brieﬂy discussed. Finally, LIGO and its latest discovery of gravitational waves is reviewed. Contents

1 Introduction to General Relativity 3 1.1 From Newton to Einstein ...... 3

2 The Mathematics behind General Relativity 5 2.1 Einstein’s index notation ...... 5 2.2 Euclidean space ...... 6 2.2.1 Coordinate systems ...... 6 2.2.2 Which basis should we use? ...... 7 2.2.3 Surfaces in Euclidean space ...... 8 2.3 Manifolds, metrics and tensors ...... 9

3 The Geometry of Spacetime 11 3.1 The Metric Tensor ...... 11 3.2 Geodesics ...... 12 3.2.1 The Christoffel Symbols ...... 13 3.3 Curvature ...... 13 3.3.1 Parallel transport and covariant differentiation . . . . . 13 3.3.2 The curvature tensor ...... 14 3.3.3 The Ricci and the Einstein tensors ...... 15 3.4 The stress tensor ...... 16 3.5 Einstein’s field equations ...... 16

4 Linearization of General Relativity 18 4.1 Linearized gravity ...... 18 4.1.1 Linearized Einstein’s ﬁeld equations ...... 19

5 Gravitational Waves 22 5.1 Plane waves in spacetime ...... 22

1 5.1.1 How many polarizations? ...... 23 5.1.2 Eﬀects on test masses ...... 24 5.2 The quadrupole formula ...... 25 5.3 Generation of gravitational waves ...... 27 5.3.1 Black holes ...... 28 5.3.2 Supernovae and pulsars ...... 28 5.3.3 Binary stars ...... 29 5.4 Detection of gravitational waves ...... 29 5.4.1 Weber bars ...... 30 5.4.2 Pulsar timing arrays ...... 30 5.4.3 Laser interferometry ...... 30 5.5 LIGO’s interferometer and ﬁrst observation of gravitational waves...... 32

6 Summary 34

References 36

2 Chapter 1

Introduction to General Relativity

1.1 From Newton to Einstein

Newton’s theory of gravity had always been characterized by describing the movement of particles under Earth’s gravitational field and the motion of planets with great accuracy. As a matter of fact, in the whole Solar System, Newton’s theory only failed at describing one motion: Mercury’s orbit around the Sun1, and even here, there is only a one part in 107 discrepancy (see [5]). Galilean invariance applies to Newton’s theory of gravity. In other words, if Newton’s equations can be applied to a particular inertial frame of reference, they can be applied to any other inertial frame by just using a transformation of coordinates. However, the nature of Newton’s equations has another implication: the invariance under a uniform acceleration. This means that, to switch from one frame of reference to another one that has an acceleration relative to the first one, we can do a change of coordinates subtracting the acceleration from the gravitational field ~g. Hence, by making the acceleration equal the gravitational force, an object will feel weightless. This is due to the fact that, as it can be seen in Newton’s equations, the ratio of the inertial mass to the weight of a body is the same for all objects. This was proven first by Galileo and centuries later by Eötvös,who, performing a very

1Newton’s theory of gravity cannot explain the precession of the orbit of Mercury. This precession is known as perihelion precession, and although all planets have it, it was ﬁrst noted in Mercury, challenging Newton’s theory at that time.

3 ingenious experiment, showed that the inertial mass and the gravitational mass of an object is equal to at least to one part in 109[2]. The consequence of this is that the effects of gravity and acceleration become indistinguishable from one another (except for the fact that the magnitude of the gravitational field becomes zero at very large distances from the source). This is known as the equivalence principle. For example, if you stayed in a box, you would not be able to tell whether you are on Earth under the influence of the gravitational field, or the box is accelerating towards the top with the value of ~g2. And then the 19th century arrived, the century in which James Clark Maxwell published his famous equations. Maxwell’s set of equations seemed to challenge Newton’s physics’ invariance. It appeared as if some reference frames had preference over others. However, Einstein’s work half a century later solved this problem. The mathematical framework of special relativity was able to make coordinates transformation from different non-accelerating reference frames using what is known as Lorentz transformations. Depending on the reference frame, time and distance could actually change, as opposed to Newton and Galileo’s absolute time and space. Einstein based his special theory of relativity on two simple postulates: Postulate. The speed of light in vacuum is the same for all observers. Postulate. The laws of physics are invariant in all non-accelerating reference frames. However, his theory was not complete yet. Einstein took the final step towards his theory of gravity by reasoning that the curvature of space-time was related to the energy and momentum of matter. He published his theory of general relativity in 1915 building on two principles, in the same that he did with special relativity[14]: Postulate. Special relativity can be applied to inertial frames over short distances and times. Postulate. Gravity (in the form of tidal forces) shows up as the relative acceleration of nearby inertial frames. In a way, general relativity signified a transition from special relativity’s Minkowski’s space-time towards a curved space-time, which we will discuss here later. 2Unless you measured and compared nearby geodesics, as this only holds locally. We will study geodesics in later chapters.

4 Chapter 2

The Mathematics behind General Relativity

Even if we have not made use of any mathematical formula or equations throughout the introduction, the truth is that a correct and deep under- standing of general relativity requires using a lot of algebra. For this reason, we will focus in this chapter on the mathematics and notation that are used in Einstein’s theory of gravity.

2.1 Einstein’s index notation

All throughout this document, we will be using what is known as Einstein notation. Introduced by Albert Einstein in 1916 [10], it is a very clever way to achieve certain brevity when it comes to writing the math needed in general relativity and other ﬁelds. There are two steps to follow Einstein’s notation [5]:

1. First, we must assume that whenever we see a suffix that belongs to one of the letters from the middle of the alphabet such as i, j, k . . ., they will be running through the values 1, 2, 3,... and hence we will be able to remove the usual clarification written between parenthesis, i.e., (i =1, 2, 3,...). Pn 2. Second, we will eliminate the summation symbol i=1 when a suffix shows up twice, once as a superscript and another time as a subscript

5 2.2 Euclidean space

In this section, we will do a introduction to the widely known three-dimensional Euclidean space, and we will study the features of vector ﬁelds in this setting.

2.2.1 Coordinate systems To describe a Eucledian space, we can use the famous Cartesian system of coordinates (x, y, z) and the unit vectors {i, j, k}. If we were to have another system of coordinates (u, v, w) other than the Cartesian ones, we could always express the Cartesian coordinates in terms of this non-Cartesian system,

x = x(u, v, w) y = y(u, v, w) z = z(u, v, w) (2.1) so that the position vector r can always be expressed as:

r = xi + yj + zk (2.2)

From this equation, we can diﬀerentiate partially with respect to the coordinates (u, v, w), so that we obtain tangent vectors to the coordinate curves: ∂r ∂r ∂r e = , e = , e = . (2.3) u ∂u v ∂v w ∂w We see that we were able to construct a basis using the coordinates (u, v, w) for any point using the tangents to the coordinates curves. {eu, ev, ew} is called the normal basis. If, instead of the tangents, we use the normals to the coordinate surfaces, we obtain an alternate basis, known as the dual basis: eu = ∇u, ev = ∇v, ew = ∇w (2.4) Using the chain rule, we can verify that

i i e · ej = δj, (2.5)

i where δj is the Kronecker delta, which equals 0 if i and j are diﬀerent from each other, and 1 if they are the same. We can now, using Einstein’s notation explained in section 2.1, express a vector ﬁeld λ, in terms of the natural basis,

i λ = λ ei (2.6)

6 or in terms of the dual basis: i λ = λie (2.7) The components λi that appear in equation 2.6 are known as the contravari- ant components, whereas the λi in equation 2.7 are the covariant components. For example, if we do the dot product of two vectors λ and µ using the con- travariant components, we get that

i j λ· µ = gijλ µ , (2.8) where gij ≡ ei· ej. (2.9) We can also do the dot product using the covariant components, where we would deﬁne, in a similar way as before:

gij ≡ ei· ej. (2.10)

ij The quantities gij can be used to “lower” the suﬃx, whereas g have the power to “raise” it.

2.2.2 Which basis should we use? We have seen that we have two possible bases that we can use to locate points in a Euclidean space. It is natural to now wonder which one of these two bases, the natural or the dual basis, we should use. Are they both equally valid? Can we use them interchangeably? Actually, we ﬁnd that it is more convenient to use the natural basis when dealing with tangents to curves and the dual basis when dealing with gradients [5]. As an example of the use of the natural basis, let us calculate the formula to obtain the length of a curve γ. If the curve is parameterized by t, which will run from t1 to t2, its length L will be given by:

Z t2 dr L = dt, (2.11) t1 dt where r is the position vector of points on the curve γ. If we can express the Cartesian coordinates (x, y, z) in terms of (u(t), v(t), w(t)), then we can apply the chain rule to dr/dt: dr ∂r du ∂r dv ∂r dw = + + =u ˙ i(t)e , (2.12) dt ∂u dt ∂v dt ∂w dt i 7 whereu ˙ i(t) is the derivative of ui(t) with respect to time. Now, using the expression 2.9, we can obtain the formula for the distance between two points,

2 i j ds = gijdu du , (2.13) as well as the square of dr/dt:

dr2 = g u˙ iu˙ j, (2.14) dt ij which will lead us to the following expression for the length of a curve γ:

Z t2 p i j L = giju˙ u˙ dt (2.15) t1 On the other hand, if we were to calculate the gradient of a function, the dual basis would be the most appropriate ones to express it. Using the chain rule and Einstein notation, we can write the gradient of V as:

i ∇V = ∂iV e , (2.16)

i where ∂i ≡ ∂/∂u .

2.2.3 Surfaces in Euclidean space We can express the Cartesian coordinates as function of the parameters u and v,

x = x(u, v) y = y(u, v) z = z(u, v), and hence obtain what is called a surface. At any point on a surface, there are two parametric curves given by

r1 = x(u, v0)i + y(u, v0)j + z(u, v0)k (2.17)

r2 = x(u0, v)i + y(u0, v)j + z(u0, v)k (2.18) where (u0, v0) are the coordinates of the point. Taking this into consideration, we can assign to each point a vector tangential to the surface:

u v λ = λ eu + λ ev, (2.19)

8 where {eu,ev} constitute the natural basis for the tangent plane: ∂r ∂r e = 1 , e = 2 . (2.20) u ∂u v ∂v

Usually, the natural basis can be denoted as {eA} = {eu, ev}. We can also deﬁne a dual basis {eA} = {eu, ev}, in a way that the following relation applies: A A e · eB = δB. (2.21)

We can now deﬁne a metric tensor gAB, following equation 2.9:

gAB ≡ eA· eB. (2.22)

Analogously, we obtain gAB:

gAB ≡ eA· eB. (2.23)

2.3 Manifolds, metrics and tensors

Now that we have gained some basic knowledge about Euclidean space, we can take one step forward and introduce manifolds. What are manifolds? Basically, manifolds are spaces in which points can be labelled by a system of coordinates, so that each point corresponds to one label, and viceversa. If we have, for example, the surface of a sphere, we need two coordinates to describe points. If, instead, we are dealing with relativity, we need four space-time coordinates. Therefore, an N-dimensional manifold will be described by a system of N coordinates (x1, x2, x3, ..., xN ). It might actually not be possible to cover the whole manifold with just one coordinate system. Sometimes, different coordinate systems may overlap. Apart from a manifold, we can also use another structure to characterize a surface: a metric. The metric of a surface determines its geometry by giving the distance ds between nearby points. Doing calculations involving the metric coefficients in many dimensions can become quite tedious. A compact way to simplify this can be done using tensor calculus. A tensor is 0 0 a1...ar a mathematical object that assigns quantities τ 0 0 to each local coordinate b1...bs 0 0 a1...ar system [14]. τ 0 0 is considered a tensor of type (r, s). A manifold with b1...bs a metric tensor field with components gab will satisfy gab = gba (symmetry

9 relation), and, because the tensor is nonsingular, there will be another tensor which satisﬁes:

bc c gabg = δa (2.24) Under a change of coordinates, the components of a tensor transform according to a0 ...a0 0 0 1 r a1 ar d1 ds c1...cr τ 0 0 = Xc ...Xc X 0 ...X 0 τ , (2.25) b1...bs 1 r b1 bs d1...ds where we assume that the two sets of coordinates are related by diﬀerentiable equations, that allow us to express

a0 0 ∂x Xa ≡ , (2.26) b ∂xb as well as a a ∂x X 0 ≡ . (2.27) b ∂xb0

10 Chapter 3

The Geometry of Spacetime

Now that we have gained knowledge on the algebra that we will be using in the following chapters, we can proceed to the study of spacetime, the arena in which general relativity is explained. As we have mentioned in section 2.3, spacetime can be described by a manifold of four dimensions. What makes Einstein’s theory of general relativity so powerful is the fact that it can explain gravity just by studying the curvature of spacetime. In this curved space, the metric tensor is fundamental for the calculations, and it is given the symbol of gµν, where µ and ν can take the values 0, 1, 2 and 3.

3.1 The Metric Tensor

The metric tensor gµν is one of the most important objects in relativity. gµν is a symmetric tensor, and it is usually taken to be nondegenerate (i.e. its determinant does not vanish). This allows us to deﬁne an also symmetric tensor, gµν. As we explained in one of the postulates in section 1.1, one of the require- ments for spacetime is that special relativity can be applied to inertial frames over short distances. Hence, in for any point there is a system in which, 1 approximately, gµν is equal to the metric tensor of special relativity, ηµν. Due to the fact that the spacetime of special relativity is a four-dimensional pseudo-Riemannian manifold, we assume that the spacetime of general relativity is a four-dimensional pseudo-Riemannian manifold as well. However,

1We assume that the reader is familiarized with special relativity.

11 whereas the spacetime of special relativity is flat, the spacetime of general relativity is curved. The metric tensor is used to define the distance between two infinitely close points:

2 µ ν ds = gµνdx dx . (3.1)

3.2 Geodesics

What is the shortest distance between two points? In a three dimensional flat space, the shortest distance will be a straight line. However, we can also generalize this concept to curved spaces by introducing the concept of a geodesic, the analog of a straight line. Geodesics are very important in general relativity, as they describe the motion of free falling particles. Even though we could define a geodesic as the shortest curve that joins two points, this can be tricky as there are curves with zero length. Therefore, we will use the fact that a straight path r has tangent vectors λ with constant direction: dλ = 0, (3.2) ds where s is the arclength, which will allow us to parametrize the curve. Now, k let us define the quantities Γij at each point of space as

k ∂jei = Γijek. (3.3) Using the symbols deﬁned in (3.3) and plugging equation (2.6) in (3.2), we can arrive, after some manipulations, to the geodesic equation: d2ui duj duk + Γi = 0. (3.4) ds2 jk ds ds This is, basically, the famous Euler-Lagrange equations that are used in Classical Mechanics. Before continuing, I will introduce a change of notation that will allow us to go from the classical three-dimensional Euclidean space to spacetime. For this, we will use xµ instead of ui to express the coordinates, and, instead of s, we will use the proper time τ to parametrize our curves, so that

2 2 µ ν c dτ ≡ gµνdx dx . (3.5)

12 The proper time τ, however, will not be appropriate if we are dealing with, for example, a massless particle, as τ stays constant along its path. We will need another parameter for these cases. Thus, with this new notation, equation 3.4 will transform into:

d2xµ dxν dxσ + Γµ = 0. (3.6) dτ 2 νσ dτ dτ 3.2.1 The Christoﬀel Symbols

µ We have introduced the symbols Γνσ, which are called the Christoffel symbols (or the connection coefficients). They are very important in general relativity, as they indicate, through the geodesic equation, the path that a free particle will take. They can be calculated as 1 Γµ = gµρ(∂ g + ∂ g − ∂ g ). (3.7) νσ 2 ν ρσ σ ρν ρ νσ The Christoffel symbols satisfy the symmetry relation

µ µ Γνσ = Γσν. (3.8)

3.3 Curvature

In this section we will be dealing with how to workout curvature in General Relativity. As the material in this section can be applied to any manifold, we will keep using the subscripts a, b, c, ...

3.3.1 Parallel transport and covariant diﬀerentiation How can we know whether space is ﬂat or curved? In order to answer this question, we need two basic concepts of General Relativity: parallel transport of a vector and covariant derivative.

Parallel transport of a vector If we move an arbitrary vector λ along a curve so that it stays constant in the diﬀerent spaces tangent to the space where the curve is, we will be doing what is called a parallel transport. If the curve is a geodesic, it can be shown

13 that the angle between the vector and the tangent of the geodesic remains constant. This technique of parallel transport allows us to distinguish between a curved and a ﬂat space:

• Curved space: the parallel transport of a vector along a closed trajectory that arrives to the starting point changes the vector, in a way that depends of such trajectory.

• Flat space: for any curved trajectory, the parallel transport of a vector that arrives in the starting point does not change the vector.

Covariant derivative The absolute derivative of a vector ﬁeld λa(u) along a curve is given by Dλa dλa dxc ≡ + Γa λb . (3.9) du du bc du However, we know that dλa ∂λa dxc = , (3.10) du ∂xc du so we can write equation 3.9 as:

Dλa ∂λa dxc = + Γa λb . (3.11) du ∂xc bc du The expression between parenthesis is known as the covariant derivative of λa:

a a a b λ;c ≡ λ,c + Γbcλ , (3.12) a a where λ;c is how we denote the covariant derivative of λ , and the subscript “, c” represents the partial derivative with respect to xc.

3.3.2 The curvature tensor Let us take the expression 3.12 and use it to take the repeated covariant diﬀerentiation of a covariant vector ﬁeld λa[5]:

14 d d e d λa;bc = ∂c∂bλa − (∂cΓab)λd − Γab∂cλd − Γab(∂bλe − Γebλd) e d (3.13) −Γbc(∂eλa − Γaeλd). We can exchange b and c and take the diﬀerence:

d λa;bc − λa;cb = Rabcλd. (3.14) d Rabc is known as the curvature tensor (or Riemann tensor), and it equals:

d d d e d e d Rabc ≡ ∂bΓac − ∂cΓab + ΓacΓeb − ΓabΓec. (3.15) The curvature tensor encodes derivatives of the Christoﬀel symbols. In a way, this tensor lets us know the diﬀerence in the acceleration between points that are close from each other [14]. The number of the components of the Riemann tensor can be cut down from 44 = 256 to 20 using the symmetry properties: Rabcd = −Rbacd = −Rabdc = Rcdab, (3.16) as well as the cyclic identity:

a a a Rbcd + Rcdb + Rdbc = 0. (3.17)

Another important property satisﬁed by this tensor is the Bianchi identity:

a a a Rbcd;e + Rbde;c + Rbec;d = 0 (3.18)

The Riemann tensor can be used in order to decide whether a manifold is a flat or curved: if Rbcd = 0 the manifold is flat, otherwise, it is curved. As an example, special relativity’s flat spacetime has coordinate systems in which µ gµν = ηµν, and hence Γνσ = 0, causing the curvature tensor to equal zero.

3.3.3 The Ricci and the Einstein tensors We can deﬁne two new symmetric tensors from the curvature tensor: the Ricci tensor and the Einstein tensor. The Ricci tensor is simply deﬁned as

c R ≡ Rabc, (3.19) whereas the Einstein tensor is given by:

15 Rg G ≡ R − ab . (3.20) ab ab 2 Here, R is the curvature scalar:

ab R ≡ g Rab. (3.21)

3.4 The stress tensor

To finish off this section, we must introduce the tensor which describes the density and flux of momentum and energy. It is called the energy-momentum- stress tensor, and it is given for a perfect fluid by p dxµ dxν T µν ≡ ρ + − pgµν, (3.22) c2 dτ dτ where ρ represents the density and p the pressure. This tensor contains the information on the content of matter of spacetime, and its derivative equals zero because of the classical equation of motion and continuity equation:

µν T;µ = 0. (3.23)

3.5 Einstein’s ﬁeld equations

Einstein’s ﬁeld equations, more commonly known as Einstein’s equations, are a set of 10 equations that explain how the curvature of spacetime (due to mass and energy) causes gravitational interaction. They were derived by Albert Einstein in 1915. As we saw in the previous chapter, the importance of the stress tensor T µν lies on the fact that the distribution of matter in spacetime is summa- rized in it, so this tensor can be taken as the source of the gravitational ﬁeld in Einstein’s equation, in the same way as the density of mass is the source of gravity in Newton’s classical equations. This, and the fact that Einstein’s equations must reduce to Poisson’s equation (∇2V = 4πGρ) in the Newto- nian limit, led Einstein to summarize his set of equations into an elegant one: 8πG Gµν = − T µν. (3.24) c4 16 Actually, we can also obtain equation 3.23 by combining the Bianchi identities with 3.24.

17 Chapter 4

Linearization of General Relativity

In this chapter, we will be studying the linearization of general relativity. For this, we will start by simplifying Einstein’s ﬁeld equations in order to obtain the linearized versions of them.

4.1 Linearized gravity

For weak gravity, we can approximate Einstein’s equations to be linear, and hence ignore all nonlinear contributions. This leads to what is known as linear gravity. In the linearization of general relativity, the metric is approximated as the sum of an exact solution to Einstein’s equations (for most cases, Minkowski spacetime) and a small perturbation:

gµν = ηµν + hµν (4.1)

Due to the fact that hµν (and all its derivatives) are very small, we will from now on ignore all products of quantities whose kernel is letter h. An- other rule that we will follow is that, in order to lower or raise suﬃxes, we µν µν will not be using g or gµν, but instead η and ηµν.

18 4.1.1 Linearized Einstein’s field equations Now that we know the method to linearize gravity, we can proceed and try to obtain the linearized version of Einstein’s field equations, which are valid for weak gravitational fields. Using equation 2.24, we can express gµν to first order as

gµν = ηµν − hµν. (4.2) We can now derive the value of the Christoﬀel symbols and the curvature tensors.

The Christoﬀel symbols in linearized general relativity In the linearized metric, we can easily calculate the Christoﬀel symbols from equation 3.7: 1 Γµ = hµ + hµ − h,µ , (4.3) νσ 2 σ,ν ν,σ νσ where we used:

µβ ,µ η hνσ,β = hνσ. (4.4)

Curvature tensors in linearized general relativity From equation 3.15, we can calculate the Ricci tensor as:

α α β α β α Rµν = Γµα,ν − Γµν,α + ΓµαΓβν − ΓµνΓβα α α = Γµα,ν − Γµν,α (4.5) 1 = h − hα − hα + hα 2 ,µν ν,µα µ,να µν,α µ µν where we threw away the ΓΓ terms and h ≡ hµ = η hµν. The curvature scalar is, from equation 3.21:

α αβ R = h,α − h,αβ. (4.6) Applying these last results to equation 3.20, we can obtain that the Ein- stein tensor in linearized general relativity is: 1 G = h − hα − hα + hα − η hα − hαβ (4.7) µν 2 ,µν ν,µα µ,να µν,α µν ,α ,αβ 19 This allows us to express equation 3.24 as 16πG h¯α + η h¯αβ − h¯α − h¯α = − T , (4.8) µν,α µν ,αβ ν,µα µ,να c4 µν where we used what is known as the trace-reserved perturbation variable[12], deﬁned by: 1 h¯ ≡ h − hη . (4.9) µν µν 2 µν

Gauge transformations ¯ It would seem possible now to try to solve for hµν and solve Einstein’s equations. In order to do this, we must first find suitable coordinates that make 4.8 easy to work with. Imagine two coordinate systems, xµ and xµ0 , different µ from each other by a deviation of ξ , which is about as small as hµν:

xµ0 ≡ xµ + ξµ. (4.10) The metric of these two coordinate systems will be related by, using equation 2.27:

0 α β gµ0ν0 (x ) = Xµ0 Xν0 gαβ(x) α α β β 0 0 σ = δµ0 − ξ,µ0 δν0 − ξ,ν0 [gαβ(x ) − gαβ,σ(x )ξ ] (4.11)

α β σ: ≈ 0 to first order = gµν(x) − ξ,µgαν − ξ,µgµβ −gµν,σ ξ . As a consequence of this, a very small change of coordinates due to ξ changes the metric like this:

¯ α ∆hµν = −ξν,µ − ξµ,ν + ξ,αηµν. (4.12) In order to ﬁx in a unique way the coordinate system, we must introduce what is known as gauge conditions. One of the most appropriate gauge conditions for general relativity is the Lorentz gauge1:

¯µα h,α = 0 (4.13)

1There is always a choice of gauge that satisﬁes equation 4.13. For a proof of this, see, for example, [12]

20 If we now introduce the d’Alembertian operator 22:

2 αβ 2 ≡ −η δαδβ, (4.14) we can apply the Lorentz gauge to 4.8 and obtain the linearized Einstein equation in the Lorentz gauge: 16πG 22h¯µν = − T (4.15) c4 µν

21 Chapter 5

Gravitational Waves

We now reach the main goal of this thesis: gravitational waves. In order to introduce them, we must further work on linearized Einstein’s ﬁeld equations. Once we prove how gravitational waves are generated, we will discuss diﬀerent detection methods and review the latest discovery of these waves, announced by LIGO in February 2016.

5.1 Plane waves in spacetime

Let us now apply equation 4.15 to empty spacetime1:

22h¯µν = 0, (5.1) which is a set of 10 wave equations, so we can look for solutions of the form:

¯µν µν α h = Re[A exp(ikαx )], (5.2) where Re refers to the real part of the bracketed expression, [Aµν] is the µ µα amplitude matrix and k = η kα is the wave vector. The d’Alembertian α operator acting on an exponential like 5.2 brings down a −kαk , so in order to satisfy 5.1, we need the wave vector to be a null vector:

µ kµk = 0. (5.3) 1Empty spacetime implies T µν = 0

22 5.1.1 How many polarizations? As there was a set of 10 wave equations, it can be tempting to think that there are 10 diﬀerent polarizations. However, is this really the case? Let us remember the Lorentz gauge condition (equation 4.13), which implies a restriction on the amplitude components:

µν A kν = 0. (5.4) This generates four conditions that cut down the 10 equations down to six. However, this is not the end yet. We can still reduce the number of equations all the way to only two! In order to do this, I will follow in the next lines the procedure used in [6]. Let us start by introducing the following gauge transformation:

µ µ α ξ = Re[iB exp(ikαx )]. (5.5)

. From equation 4.12, we can see that the gravitational wave amplitude µν µν µ ν ν µ α changes from (A ) to (A + k B + k B − kαB ηµν). The following expression allows us to see how the quantities change:

 1 α   1 α     0  2 Aα 2 Aα −ω −k1 −k2 −k3 B 01 01 1  A   A   k1 −ω 0 0   B   02  →  02  +    2  . (5.6)  A   A   k2 0 −ω 0   B  03 03 3 A A k3 0 0 −ω B

α 0 We can choose a coordinate system in which Aα and one of the A co- eﬃcients are zero because the 4x4 matrix is invertible. This is called the TT gauge2 and it gives us four more conditions on the amplitude matrix Aµν, which leaves with only two possible polarizations. Hence, Aµν can be expressed as a linear combination of two linear polarization matrices:

µν µν µν A = C+e+ + C×e× , (5.7) where C+ and C× are complex constants and they represent two modes of polarization, and:

2TT stands for transverse traceless.

23  0 0 0 0   0 0 0 0  µν  0 1 0 0  µν  0 0 1 0  [e+ ] =   [e× ] =   . (5.8)  0 0 −1 0   0 1 0 0  0 0 0 0 0 0 0 0

5.1.2 Eﬀects on test masses We will now focus on what happens when gravitational waves strike matter. For this, we consider a wave propagating in the 3-direction, hence

α ikαx = i(k0ct − k0z) = ik0(ct − z), (5.9) with kα = (k0, 0, 0, −k0). In this case, the metric will be given by:

2 2 α 1 2 α 1 2 ds =dt + (1 + Re[C+exp(ikαx )])(dx ) + 2Re[C×exp(ikαx )]dx dx α 2 2 3 3 + (1 − Re[C+exp(ikαx )])(dx ) + (dx ) . (5.10) What does this mean? Let us place a bunch of test particles at different coordinates, (x1, x2, x3), and imagine a wave striking on them that is linearly polarized type “+” (which means C× = 0). Particles along the direction of propagation x3 will not feel anything. However, different particles with the same x1 coordinate will drift apart, whereas particles with same x2 will get closer. After a short period of time, the opposite will happen: particles with different x1 coordinate will come closer, whereas those with the same x2 will move farther from each other. If we imagine a circle in the x1x2 plane, we will see how, first, it becomes squashed in the x1 direction and elongated in the x2 direction, and later it will do the opposite. This is why we define this polarization as +. If, instead of a type + linearly polarized wave we have a type × linearly polarized wave, the same thing will happen. However, the only difference is that, this time, the axes along which elongation/compression happens will be rotated 45o with respect to the previous ones, and hence the reason why the symbol × is used for this case. We can see that in both cases, we are dealing with transverse waves.

24 5.2 The quadrupole formula

We can try to apply an equivalent of the formulae of Electromagnetism in order to grasp how gravitational waves work. One of the things that we should do is to replace the charge of the electron by the mass. In electromagnetic theory, the power output of an electric dipole can be expressed as

1 2d¨2 L = 3 , (5.11) 4π0c 3 where d is the dipole moment. In gravity, the equivalent of an electric dipole moment is the mass dipole moment, whose first derivative is the linear momentum. The first derivative of the linear momentum (and therefore, the second derivative of the mass dipole moment) is zero, because of conser- vation of momentum. As a consequence, there is no such thing as dipole radiation in general relativity. Although it is not subject of this paper, the gravitational field has spin 2. This means that gravitational radiation is quadrupolar to lowest non-vanishing order. Using electromagnetic theory as a reference once again, we can express the solution for equation 4.15 as:

−4G Z T µν(x0 − |x − x0|, x0) h¯µν(x0, x) = dV 0 (5.12) c4 |x − x0| which is a retarded integral in which x represents the ﬁeld point and x0 the source. From equations 4.15 and 4.13, it can be seen that

µν T,ν = 0. (5.13) This leads to

00 0k T,0 + T,k = 0 (5.14) and

i0 ik T,0 + T,k = 0. (5.15) In order to continue, we will need the following identity: Z Z Z ik j ik j ij (T x ),kdV = T,k x dV + T dV. (5.16)

25 Knowing that, from Gauss’s theorem, the left side of this equation is zero, we can apply this last equation to 5.15. Z Z ij i0 j T dV = T,0 x dV. (5.17) We repeat this procedure, but replacing i with j, and add up the results:

Z Z T i0xj + T j0xi 1 d Z T ijdV = ,0 ,0 dV = (T i0xj + T j0xi)dV (5.18) 2 2c dt Using again identity 5.16:

Z Z Z 0k i j 0k i j 0i k 0j i (T x x ),kdV = T,k x x dV + (T x + T x )dV, (5.19) where, because of Gauss once again, the left side is null. We now apply 5.14: Z 1 d Z (T 0ixk + T 0jxi)dV = T 00xixjdV. (5.20) c dt We can now take the derivative on both sides of this equation, and use 5.18:

Z 1 d2 Z 1 d2 Z T ij = T 00xixjdV ≈ ρxixjdV, (5.21) 2c2 dt2 2 dt2 where we used that T 00 ≈ ρc2, valid for slow source particles (with ρ rep- resenting the density). But before plugging this into equation 5.22, we can approximate 5.12 by −4G Z h¯µν(ct, x) = T µν(ct − r, x0)dV 0. (5.22) c4r which is valid, once again, for slow source particles. Finally, we obtain:

2G d2 Z h¯ij(ct, x) = − ρxixjdV , (5.23) c4r dt2 where the integral is taken at t − r/c, which is the retarded time. This equation tells us at which rate and amplitude gravitational waves are generated from a system of masses. If we now used the ﬂux from a Landau-Lifshitz

26 pseudo-energy-momentum tensor (in a similar way to what is done in Electro- dynamics, see [3]), we can integrate over all angles to obtain the quadrupole formula: G d3Q 2 L = ij , (5.24) GW 5c5 dt3 where Qij is the quadrupole moment,

Z 1 Q (t) = d3x ρ xixj − δijx2 , (5.25) ij 3 and whose trace is zero: δijQij = 0. (5.26) Equation 5.24 gives us the total power in the gravitational waves emitted by a source.

5.3 Generation of gravitational waves

The quadrupole formula is a way to understand how gravitational waves are produced. However, just by looking at it it is hard to have an idea of what type of events have the power to generate gravitational waves. The truth is that the events in our universe capable of producing measurable gravitational waves are of astronomical magnitude. We need very massive objects, such as neutron stars or black holes orbiting each other. Let us try to express LGW in terms of the mass and radius of the astrophysical sources. First, we can say that

d3Q ij ∼ ω3MR2, (5.27) dt3 where ω is some typical eigenfrequency, and it can be determined from the equations of motion. For example, for two masses M circulating each other:

Ma = Fgrav, (5.28)

GM and hence, if Mf = c2 , we can express 5.24 as !5 Mf L ∼ L , (5.29) GW R 0

27 where L0 is a conversion factor: c5 L ≡ . (5.30) 0 G 5.3.1 Black holes Black holes are one of the most important sources of gravitational waves. For example, it is presumed that matter being swallowed by a black hole can generate this type of waves. Looking at equation 5.29, the fact that the radius is raised to a power of −5 tells us that when matter reaches the black hole it can emit a strong burst. It has been estimated (see [4]) that the energy output E of such burst would be

0.0104m2 E ≈ c2, (5.31) M m being the mass of the matter falling, and M the mass of the black hole. However, as we mentioned earlier, matter falling into black holes is not the only way for black holes to emit gravitational waves. If there are two massive black holes colliding against each other, they will also produce radiation.

5.3.2 Supernovae and pulsars Supernovae, the explosion of very massive stars when they reach the end of their lives, are very catastrophic events in our universe and thus, they also have the power to create gravitational waves. Estimations say that the power output of the explosion of a star of mass M is (see [12])

E ≈ 0.1Mc2. (5.32) However, a large part of this probably happens through neutrino emission (see [8]). Pulsars, which are rotating neutron stars or white dwarfs with huge mag- netic ﬁelds and emitting electromagnetic radiation, are also sources of gravitational waves. In order for a pulsar to emit gravitational radiation it needs to have low axial symmetry. This is because axial symmetry implies a constant quadrupole moment, which forbids gravitational radiation emissions.

28 5.3.3 Binary stars As it has been seen over the last decades, a big fraction of the stars are actually part of binary systems. And because binary star systems can also create gravitational waves, they become the biggest generators of these waves, as the number of them is so great. If we introduce the reduced mass of a binary system of two stars of masses m1 and m2 as m m µ = 1 2 , (5.33) m1 + m2 the power output can be expressed as ([12])

(m + m )3µ2 L ≡ e 1 e 2 e L , (5.34) 4a5 0 where a is the semi-major axis. Actually, binary systems have also played an important role in testing general relativity, as the discovery of pulsar PSR 1913+16 in 1974 by Russel Hulse and Joseph Taylor in a binary star system became the ﬁrst indirect support of gravitational waves. Hulse and Taylor’s Nobel Prize awarded discovery showed how the period of the pulsar decreased over time, implying a loss of energy, probably in the form of gravitational waves (see [7]).

5.4 Detection of gravitational waves

The ﬁrst evidence of gravitational wavesBeing able to detect gravitational waves is clearly not an easy task for several reasons. One of them is that the events that generate suﬃciently large waves can be very uncommon in the universe. As well, in case one of this events happens to reach us, it can have a very short length in time, so we need to be paying attention constantly. However, probably the biggest challenge when it comes to detecting these phenomena is the experimental challenge they give rise to. An experiment that aims to detect gravitational waves needs to have one of the greatest sensitivities in all experiments ever performed, as the perturbation in space time that they create is extremely small. Nevertheless, there are some clever methods to detect these elusive phenomena.

29 5.4.1 Weber bars Weber was one of the pioneers in gravitational waves detection. The experiments he performed, back in the 1960s, were based on a large metallic bar which, when struck by a gravitational wave, would oscillate. These os- cillations would happen in the same way that we saw in subsection 5.1.2. Moreover, if the frequency of the gravitational wave is close to one of the eigenfrequencies of the bar, ampliﬁcation can occur. As reported in [13], sensitivities of 10−18 can be achieved with the We- ber bars. However, if we calculate the amplitude of the gravitational waves emitted by, for example, the previously mentioned pulsar PSR 1913+16, and try to measure it using Weber bars, we would be trying to ﬁnd a change in length of one hundredth the size of an atomic nucleus. It seems obvious that it is essential that the bar is perfectly isolated from outside vibrations.

5.4.2 Pulsar timing arrays Another possible way to detect gravitational waves is based on what is known as pulsar timing arrays. The fundamentals behind this is using a series of pulsars whose rotational period is of the order of milliseconds. If a gravitational wave were to go through our planet, we would measure a very slight change in the signals coming from the pulsars. Pulsar timing arrays are very often used in order to detect stochastic gravitational wave background (especially originated from super-massive black hole binary systems, see [11]).

5.4.3 Laser interferometry The method in which the biggest eﬀorts are being put lately is laser interferometry. Figure 5.4.3 contains a schematic diagram of a basic laser interferometer. The way this works is the following: there are arms placed at an angle of 90o and mirrors are placed along them. Then, a beam of light is split so that it travels both arms’ length until reaching the mirrors at the end of each arm. Then, each beam gets reﬂected and they both recombine, creating an interference pattern. If a gravitational wave reached our interferometer, the length of the arms would very slightly change, and hence the laser beams would arrive out of phase, and we would be able to detect this in the interference pattern.

30 Figure 5.1: Diagram of how a Michelson interferometer works: the (red) laser beam reaches the splitter (BS), so that it gets divided into two beams, with one (1) going towards the mirror 1, and the other (2) towards the mirror 2. After bouncing oﬀ the mirror, they reach BS again, recombine (1+2) and hit the screen. If the arms had a variation of length, an interference pattern should be projected onto the screen. SOURCE OF THE IMAGE: Werner Boeglin, Dept. of Physics, FIU, Miami

When we say ”slightly change”, we are talking about one part in approximately 1021. To get an idea of how ridiculous this amount is, if we were to measure the diameter of our galaxy3 this way, we would be looking for changes of only one meter! This means that, when doing laser interferometry, we need very long distances in the arms of our detector.

3It is estimated that the Milky Way has a diameter of, approximately, 100, 000 light years.

31 5.5 LIGO’s interferometer and ﬁrst observation of gravitational waves

LIGO (Laser Interferometer Gravitational-Wave Observatory) is the largest gravitational wave detector. It is made up of two laser interferometers, one located in Hanford, Washington, and the other one in Livingston, Louisiana. This means that they are located 3000 km away from each other. Why do we need two detectors located so far away from each other? The answer is noise. Even though these detectors are located in very quiet areas, there is always environmental noise and even the possibility of earthquakes, which would result in an interferometer pattern. However, if there are two detectors in completely different locations, the noise will be totally different, so we will be able to identify it and remove it. If a gravitational wave arrived in the two detectors, it would do it almost simultaneously, and we would be able to distinguish it from the background noise. Nonetheless, noise is not the only reason why more than one detector is used. Using two detectors allows us to discover from which direction the gravitational wave came from4. The arms of LIGO have a length of 4 kilometers, which means that, referring to the one part in 1021 previously mentioned, the arms would vary by a distance of one ten thousandth a proton’s width. Such magnitudes of precision require the smoothest mirrors ever constructed. The arms are also in a vacuum, to avoid air interfering with measurements. As well, in order to obtain detectable interference pattern, the laser needs to have a very precise wavelength and a lot of power, to have a good enough resolution. As a matter of fact, LIGO’s laser power is 750 kilowatts5 (see [9]). And for the wavelength, LIGO uses infrared light, which means that the variations of length are only a trillionth of a wavelength, leading to extremely small shifts of the interference pattern. All this clever engineering made it possible to detect, on September 14th, 2015, and for the first time, gravitational waves. The results are published in [1]. In order to identify the signal, named GW150914, in all the data that LIGO had, two different methods of search were used, as explained in [1]: one of them used predictions of general relativity in order to filter waveforms

4For a complete determination of the direction of the gravitational wave, three detectors would be needed. 5However, LIGO’s laser beams has a power of 200 kilowatts before arriving in the interferometer. In order to increase the power, a clever design of mirrors is used.

32 from the fusion of two massive objects, whereas the other did not make any assumptions on the form of the wave, looking for all types of signals. These methods were able to calculate the likelihood of obtaining a signal like GW150914, and it was found that an event like this has a probability of happening randomly once every 203 000 years, which is equivalent to a significance over 5.1σ. In [1], we can see that the signal GW150914 arrived first at the Liv- ingston detector at 09:50:45 UTC, and 6.9 ms later, it reached Hanford. The frequency of the signal was observed to start at 35 Hz, and then it went up until 250 Hz, lasting 0.2 s and completing a total of 8 cycles. Not only the frequency increased during this time, so did the amplitude. But what was the event that caused GW150914? In order to obtain fre- quencies of the order of 100 Hz, we need, as explained in [1], very compact objects orbiting around each other at a very close distance. Out of the astrophysical objects that we mentioned in section 5.4 with the power to generate measurable gravitational waves, the most compact ones are neutron stars and black holes. Calculations showed that the masses of the two objects were, respectively, 36 and 29 solar masses. However, the Tolman-Oppenheimer- Volkoff limit sets an upper boundary to the mass of neutron stars, and it is way below these results, showing that the objects responsible for event GW150914 were two black holes. This discovery was very important as, apart from being the first direct detection of gravitational waves, it was also the first observation of a binary black hole merger ([1]). It also provided evidence of the existence of stellar- mass black holes whose mass is greater than 25 solar masses. But what is probably most important about GW150914 is that it has opened a new field of astronomy: gravitational-wave astronomy.

33 Chapter 6

Summary

As it has been seen, a very big part of general relativity deals with geometry and math. For that reason, we decided to start this thesis introducing the math and geometry needed to understand some basic principles of Einstein’s theory of gravity. Once this was done, we were able to introduce the elegant Einstein’s ﬁeld equations: 8πG Gµν = − T µν. c4 Afterwards, we linearized gravity, starting oﬀ by approximating the metric as the sum of an exact solution to Einstein’s equations and a perturbation:

gµν = ηµν + hµν. This allowed us to reach the linearized version of Einstein’s equations: 16πG 22h¯µν = − T . c4 µν Applying this equation to empty space, which means equaling the right hand side to zero, we were able to ﬁnd solution in the shape of waves, which is what we call gravitational waves:

¯µν µν α h = Re[A exp(ikαx )]. After a set of conditions were applied, we found out that the number of possible polarizations for these gravitational waves is two. The quadrupole formula, the equation that describes the generation of gravitational waves, was derived as:

34 2G d2 Z h¯ij(ct, x) = − ρxixjdV . c4r dt2 Finally, we saw the diﬃculties that detecting gravitational waves implies, as well as seeing how LIGO was able to battle them, achieving the ﬁrst direct detection of these events in our universe.

35 References

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