Gravitational Waves in General Relativity

Aaron Bello

June 20, 2017

Abstract

In this paper, we write a summary about general relativity and, in particu-lar, gravitational waves. We start by discussing the mathematics that generalrelativity uses, as well as the geometry in general relativity’s spacetime. Af-terwards, we explain linearized general relativity and derive the linearizedversions of Einstein’s equations. From here, we construct wave solutionsand explain the polarization of gravitational waves. The quadrupole formulais derived, and generation and detection of gravitational waves is brieflydiscussed. Finally, LIGO and its latest discovery of gravitational waves isreviewed.

Contents

1 Introduction to General Relativity 31.1 From Newton to Einstein . . . . . . . . . . . . . . . . . . . . . 3

2 The Mathematics behind General Relativity 52.1 Einstein’s index notation . . . . . . . . . . . . . . . . . . . . . 52.2 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Coordinate systems . . . . . . . . . . . . . . . . . . . . 62.2.2 Which basis should we use? . . . . . . . . . . . . . . . 72.2.3 Surfaces in Euclidean space . . . . . . . . . . . . . . . 8

2.3 Manifolds, metrics and tensors . . . . . . . . . . . . . . . . . . 9

3 The Geometry of Spacetime 113.1 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 The Christoffel Symbols . . . . . . . . . . . . . . . . . 133.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3.1 Parallel transport and covariant differentiation . . . . . 133.3.2 The curvature tensor . . . . . . . . . . . . . . . . . . . 143.3.3 The Ricci and the Einstein tensors . . . . . . . . . . . 15

3.4 The stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . 16

4 Linearization of General Relativity 184.1 Linearized gravity . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Linearized Einstein’s field equations . . . . . . . . . . . 19

5 Gravitational Waves 225.1 Plane waves in spacetime . . . . . . . . . . . . . . . . . . . . . 22

1

5.1.1 How many polarizations? . . . . . . . . . . . . . . . . . 235.1.2 Effects on test masses . . . . . . . . . . . . . . . . . . 24

5.2 The quadrupole formula . . . . . . . . . . . . . . . . . . . . . 255.3 Generation of gravitational waves . . . . . . . . . . . . . . . . 27

5.3.1 Black holes . . . . . . . . . . . . . . . . . . . . . . . . 285.3.2 Supernovae and pulsars . . . . . . . . . . . . . . . . . . 285.3.3 Binary stars . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4 Detection of gravitational waves . . . . . . . . . . . . . . . . . 295.4.1 Weber bars . . . . . . . . . . . . . . . . . . . . . . . . 305.4.2 Pulsar timing arrays . . . . . . . . . . . . . . . . . . . 305.4.3 Laser interferometry . . . . . . . . . . . . . . . . . . . 30

5.5 LIGO’s interferometer and first observation of gravitationalwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Summary 34

References 36

2

Chapter 1

Introduction to GeneralRelativity

1.1 From Newton to Einstein

Newton’s theory of gravity had always been characterized by describing themovement of particles under Earth’s gravitational field and the motion ofplanets 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 aroundthe 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 refer-ence, they can be applied to any other inertial frame by just using a transfor-mation of coordinates. However, the nature of Newton’s equations has an-other implication: the invariance under a uniform acceleration. This meansthat, to switch from one frame of reference to another one that has an accel-eration relative to the first one, we can do a change of coordinates subtractingthe acceleration from the gravitational field ~g. Hence, by making the accel-eration equal the gravitational force, an object will feel weightless. This isdue to the fact that, as it can be seen in Newton’s equations, the ratio of theinertial mass to the weight of a body is the same for all objects. This wasproven first by Galileo and centuries later by Eotvos, who, performing a very

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

3

ingenious experiment, showed that the inertial mass and the gravitationalmass of an object is equal to at least to one part in 109[2]. The consequenceof this is that the effects of gravity and acceleration become indistinguishablefrom one another (except for the fact that the magnitude of the gravitationalfield becomes zero at very large distances from the source). This is knownas the equivalence principle. For example, if you stayed in a box, you wouldnot be able to tell whether you are on Earth under the influence of the grav-itational 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 ClarkMaxwell published his famous equations. Maxwell’s set of equations seemedto challenge Newton’s physics’ invariance. It appeared as if some referenceframes had preference over others. However, Einstein’s work half a centurylater solved this problem. The mathematical framework of special relativitywas able to make coordinates transformation from different non-acceleratingreference frames using what is known as Lorentz transformations. Dependingon the reference frame, time and distance could actually change, as opposedto 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 refer-ence frames.

However, his theory was not complete yet. Einstein took the final steptowards his theory of gravity by reasoning that the curvature of space-timewas related to the energy and momentum of matter. He published his theoryof general relativity in 1915 building on two principles, in the same that hedid with special relativity[14]:

Postulate. Special relativity can be applied to inertial frames over short dis-tances and times.

Postulate. Gravity (in the form of tidal forces) shows up as the relativeacceleration of nearby inertial frames.

In a way, general relativity signified a transition from special relativity’sMinkowski’s space-time towards a curved space-time, which we will discusshere later.

2Unless you measured and compared nearby geodesics, as this only holds locally. Wewill study geodesics in later chapters.

4

Chapter 2

The Mathematics behindGeneral Relativity

Even if we have not made use of any mathematical formula or equationsthroughout 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 usedin Einstein’s theory of gravity.

2.1 Einstein’s index notation

All throughout this document, we will be using what is known as Einsteinnotation. Introduced by Albert Einstein in 1916 [10], it is a very clever way toachieve certain brevity when it comes to writing the math needed in generalrelativity 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 toone 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 beable to remove the usual clarification written between parenthesis, i.e.,(i =1, 2, 3,...).

2. Second, we will eliminate the summation symbol∑n

i=1 when a suffixshows 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-dimensionalEuclidean 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 ofcoordinates (x, y, z) and the unit vectors {i, j, k}. If we were to have anothersystem of coordinates (u, v, w) other than the Cartesian ones, we could alwaysexpress 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 coordi-nates (u, v, w), so that we obtain tangent vectors to the coordinate curves:

eu =∂r

∂u, ev =

∂r

∂v, ew =

∂r

∂w. (2.3)

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 normalsto the coordinate surfaces, we obtain an alternate basis, known as the dualbasis:

eu = ∇u, ev = ∇v, ew = ∇w (2.4)

Using the chain rule, we can verify that

ei· ej = δij, (2.5)

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

λ = λiei (2.6)

6

or in terms of the dual basis:λ = λie

i (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

λ·µ = gijλiµj, (2.8)

wheregij ≡ ei· ej. (2.9)

We can also do the dot product using the covariant components, where wewould define, in a similar way as before:

gij ≡ ei· ej. (2.10)

The quantities gij can be used to “lower” the suffix, whereas gij have thepower 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 pointsin a Euclidean space. It is natural to now wonder which one of these twobases, the natural or the dual basis, we should use. Are they both equallyvalid? Can we use them interchangeably? Actually, we find that it is moreconvenient to use the natural basis when dealing with tangents to curves andthe dual basis when dealing with gradients [5].

As an example of the use of the natural basis, let us calculate the formulato obtain the length of a curve γ. If the curve is parameterized by t, whichwill run from t1 to t2, its length L will be given by:

L =

∫ t2

t1

dr

dtdt, (2.11)

where r is the position vector of points on the curve γ. If we can expressthe Cartesian coordinates (x, y, z) in terms of (u(t), v(t), w(t)), then we canapply the chain rule to dr/dt:

dr

dt=∂r

∂u

du

dt+∂r

∂v

dv

dt+∂r

∂w

dw

dt= ui(t)ei, (2.12)

7

where ui(t) is the derivative of ui(t) with respect to time. Now, using theexpression 2.9, we can obtain the formula for the distance between two points,

ds2 = gijduiduj, (2.13)

as well as the square of dr/dt:(dr

dt

)2

= gijuiuj, (2.14)

which will lead us to the following expression for the length of a curve γ:

L =

∫ t2

t1

√gijuiujdt (2.15)

On the other hand, if we were to calculate the gradient of a function, thedual basis would be the most appropriate ones to express it. Using the chainrule and Einstein notation, we can write the gradient of V as:

∇V = ∂iV ei, (2.16)

where ∂i ≡ ∂/∂ui.

2.2.3 Surfaces in Euclidean space

We can express the Cartesian coordinates as function of the parameters uand 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, thereare 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:

λ = λueu + λvev, (2.19)

8

where {eu,ev} constitute the natural basis for the tangent plane:

eu =∂r1∂u

, ev =∂r2∂v

. (2.20)

Usually, the natural basis can be denoted as {eA} = {eu, ev}. We can alsodefine a dual basis {eA} = {eu, ev}, in a way that the following relationapplies:

eA· eB = δAB. (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, wecan take one step forward and introduce manifolds. What are manifolds?Basically, manifolds are spaces in which points can be labelled by a system ofcoordinates, so that each point corresponds to one label, and viceversa. If wehave, for example, the surface of a sphere, we need two coordinates to describepoints. If, instead, we are dealing with relativity, we need four space-timecoordinates. Therefore, an N-dimensional manifold will be described by asystem of N coordinates (x1, x2, x3, ..., xN). It might actually not be possibleto 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 characterizea surface: a metric. The metric of a surface determines its geometry bygiving the distance ds between nearby points. Doing calculations involvingthe metric coefficients in many dimensions can become quite tedious. Acompact way to simplify this can be done using tensor calculus. A tensor is

a mathematical object that assigns quantities τa′1...a

′r

b′1...b′s

to each local coordinate

system [14]. τa′1...a

′r

b′1...b′s

is considered a tensor of type (r, s). A manifold with

a metric tensor field with components gab will satisfy gab = gba (symmetry

9

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

gabgbc = δca (2.24)

Under a change of coordinates, the components of a tensor transformaccording to

τa′1...a

′r

b′1...b′s

= Xa′1c1. . . Xa′r

cr Xd1b′1. . . Xds

b′sτ c1...crd1...ds

, (2.25)

where we assume that the two sets of coordinates are related by differentiableequations, that allow us to express

Xa′

b ≡∂xa

′

∂xb, (2.26)

as well as

Xab′ ≡

∂xa

∂xb′. (2.27)

10

Chapter 3

The Geometry of Spacetime

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

3.1 The Metric Tensor

The metric tensor gµν is one of the most important objects in relativity. gµνis a symmetric tensor, and it is usually taken to be nondegenerate (i.e. itsdeterminant does not vanish). This allows us to define an also symmetrictensor, 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 framesover short distances. Hence, in for any point there is a system in which,approximately, gµν is equal to the metric tensor of special relativity, ηµν .

1

Due to the fact that the spacetime of special relativity is a four-dimensionalpseudo-Riemannian manifold, we assume that the spacetime of general rela-tivity 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 generalrelativity is curved.

The metric tensor is used to define the distance between two infinitelyclose points:

ds2 = gµνdxµdxν . (3.1)

3.2 Geodesics

What is the shortest distance between two points? In a three dimensionalflat space, the shortest distance will be a straight line. However, we canalso generalize this concept to curved spaces by introducing the concept ofa geodesic, the analog of a straight line. Geodesics are very important ingeneral relativity, as they describe the motion of free falling particles.

Even though we could define a geodesic as the shortest curve that joinstwo 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 constantdirection:

dλ

ds= 0, (3.2)

where s is the arclength, which will allow us to parametrize the curve. Now,let us define the quantities Γkij at each point of space as

∂jei = Γkijek. (3.3)

Using the symbols defined in (3.3) and plugging equation (2.6) in (3.2), wecan arrive, after some manipulations, to the geodesic equation:

d2ui

ds2+ Γijk

duj

ds

duk

ds= 0. (3.4)

This is, basically, the famous Euler-Lagrange equations that are used inClassical Mechanics.

Before continuing, I will introduce a change of notation that will allow usto go from the classical three-dimensional Euclidean space to spacetime. Forthis, we will use xµ instead of ui to express the coordinates, and, instead ofs, we will use the proper time τ to parametrize our curves, so that

c2dτ 2 ≡ gµνdxµdxν . (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 willneed another parameter for these cases.

Thus, with this new notation, equation 3.4 will transform into:

d2xµ

dτ 2+ Γµνσ

dxν

dτ

dxσ

dτ= 0. (3.6)

3.2.1 The Christoffel Symbols

We have introduced the symbols Γµνσ, 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 particlewill take. They can be calculated as

Γµνσ =1

2gµρ(∂νgρσ + ∂σgρν − ∂ρgνσ). (3.7)

The Christoffel symbols satisfy the symmetry relation

Γµνσ = Γµσν . (3.8)

3.3 Curvature

In this section we will be dealing with how to workout curvature in GeneralRelativity. 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 thisquestion, we need two basic concepts of General Relativity: parallel trans-port 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 inthe different spaces tangent to the space where the curve is, we will be doingwhat 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 remainsconstant.

This technique of parallel transport allows us to distinguish between acurved and a flat space:

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

• Flat space: for any curved trajectory, the parallel transport of a vectorthat 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

du≡ dλa

du+ Γabcλ

bdxc

du. (3.9)

However, we know that

dλa

du=∂λa

∂xcdxc

du, (3.10)

so we can write equation 3.9 as:

Dλa

du=

(∂λa

∂xc+ Γabcλ

b

)dxc

du. (3.11)

The expression between parenthesis is known as the covariant derivative ofλa:

λa;c ≡ λa,c + Γabcλb, (3.12)

where λa;c is how we denote the covariant derivative of λa, 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 covariantdifferentiation of a covariant vector field λa[5]:

14

λa;bc = ∂c∂bλa − (∂cΓdab)λd − Γdab∂cλd − Γeab(∂bλe − Γdebλd)

−Γebc(∂eλa − Γdaeλd).(3.13)

We can exchange b and c and take the difference:

λa;bc − λa;cb = Rdabcλd. (3.14)

Rdabc is known as the curvature tensor (or Riemann tensor), and it equals:

Rdabc ≡ ∂bΓ

dac − ∂cΓdab + ΓeacΓ

deb − ΓeabΓ

dec. (3.15)

The curvature tensor encodes derivatives of the Christoffel symbols. In away, this tensor lets us know the difference in the acceleration between pointsthat are close from each other [14]. The number of the components of theRiemann tensor can be cut down from 44 = 256 to 20 using the symmetryproperties:

Rabcd = −Rbacd = −Rabdc = Rcdab, (3.16)

as well as the cyclic identity:

Rabcd +Ra

cdb +Radbc = 0. (3.17)

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

Rabcd;e +Ra

bde;c +Rabec;d = 0 (3.18)

The Riemann tensor can be used in order to decide whether a manifold isflat or curved: if Ra

bcd = 0 the manifold is flat, otherwise, it is curved. As anexample, special relativity’s flat spacetime has coordinate systems in whichgµν = ηµν , and hence Γµνσ = 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: theRicci tensor and the Einstein tensor. The Ricci tensor is simply defined as

R ≡ Rcabc, (3.19)

whereas the Einstein tensor is given by:

15

Gab ≡ Rab −Rgab

2. (3.20)

Here, R is the curvature scalar:

R ≡ gabRab. (3.21)

3.4 The stress tensor

To finish off this section, we must introduce the tensor which describes thedensity and flux of momentum and energy. It is called the energy-momentum-stress tensor, and it is given for a perfect fluid by

T µν ≡(ρ+

p

c2

) dxµdτ

dxν

dτ− pgµν , (3.22)

where ρ represents the density and p the pressure. This tensor contains theinformation on the content of matter of spacetime, and its derivative equalszero because of the classical equation of motion and continuity equation:

T µν;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 (dueto mass and energy) causes gravitational interaction. They were derived byAlbert Einstein in 1915.

As we saw in the previous chapter, the importance of the stress tensorT 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 fieldin Einstein’s equation, in the same way as the density of mass is the sourceof gravity in Newton’s classical equations. This, and the fact that Einstein’sequations must reduce to Poisson’s equation (∇2V = 4πGρ) in the Newto-nian limit, led Einstein to summarize his set of equations into an elegantone:

Gµν = −8πG

c4T µν . (3.24)

16

Actually, we can also obtain equation 3.23 by combining the Bianchiidentities with 3.24.

17

Chapter 4

Linearization of GeneralRelativity

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

4.1 Linearized gravity

For weak gravity, we can approximate Einstein’s equations to be linear, andhence ignore all nonlinear contributions. This leads to what is known aslinear gravity.

In the linearization of general relativity, the metric is approximated as thesum of an exact solution to Einstein’s equations (for most cases, Minkowskispacetime) and a small perturbation:

gµν = ηµν + hµν (4.1)

Due to the fact that hµν (and all its derivatives) are very small, we willfrom 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, wewill not be using gµν or gµν , but instead ηµν and ηµν .

18

4.1.1 Linearized Einstein’s field equations

Now that we know the method to linearize gravity, we can proceed and tryto obtain the linearized version of Einstein’s field equations, which are validfor 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 curvaturetensors.

The Christoffel symbols in linearized general relativity

In the linearized metric, we can easily calculate the Christoffel symbols fromequation 3.7:

Γµνσ =1

2

(hµσ,ν + hµν,σ − h,µνσ

), (4.3)

where we used:

ηµβhνσ,β = h,µνσ. (4.4)

Curvature tensors in linearized general relativity

From equation 3.15, we can calculate the Ricci tensor as:

Rµν = Γαµα,ν − Γαµν,α + ΓβµαΓαβν − ΓβµνΓαβα

= Γαµα,ν − Γαµν,α

=1

2

(h,µν − hαν,µα − hαµ,να + hαµν,α

) (4.5)

where we threw away the ΓΓ terms and h ≡ hµµ = ηµνhµν . The curvaturescalar is, from equation 3.21:

R = hα,α − hαβ,αβ. (4.6)

Applying these last results to equation 3.20, we can obtain that the Ein-stein tensor in linearized general relativity is:

Gµν =1

2

(h,µν − hαν,µα − hαµ,να + hαµν,α − ηµν

(hα,α − h

αβ,αβ

))(4.7)

19

This allows us to express equation 3.24 as

hαµν,α +(ηµν h

αβ,αβ − h

αν,µα − hαµ,να

)= −16πG

c4Tµν , (4.8)

where we used what is known as the trace-reserved perturbation variable[12],defined by:

hµν ≡ hµν −1

2hηµν . (4.9)

Gauge transformations

It would seem possible now to try to solve for hµν and solve Einstein’s equa-tions. In order to do this, we must first find suitable coordinates that make4.8 easy to work with. Imagine two coordinate systems, xµ and xµ

′, different

from each other by a deviation of ξµ, which is about as small as hµν :

xµ′ ≡ xµ + ξµ. (4.10)

The metric of these two coordinate systems will be related by, using equation2.27:

gµ′ν′(x′) = Xα

µ′Xβν′gαβ(x)

=(δαµ′ − ξα,µ′

) (δβν′ − ξ

β,ν′

)[gαβ(x′)− gαβ,σ(x′)ξσ]

= gµν(x)− ξα,µgαν − ξβ,µgµβ −�����:≈ 0 to first order

gµν,σξσ.

(4.11)

As a consequence of this, a very small change of coordinates due to ξ changesthe metric like this:

∆hµν = −ξν,µ − ξµ,ν + ξα,αηµν . (4.12)

In order to fix in a unique way the coordinate system, we must introducewhat is known as gauge conditions. One of the most appropriate gaugeconditions for general relativity is the Lorentz gauge1:

hµα,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:

22 ≡ −ηαβδαδβ, (4.14)

we can apply the Lorentz gauge to 4.8 and obtain the linearized Einsteinequation in the Lorentz gauge:

22hµν = −16πG

c4Tµν (4.15)

21

Chapter 5

Gravitational Waves

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

5.1 Plane waves in spacetime

Let us now apply equation 4.15 to empty spacetime1:

22hµν = 0, (5.1)

which is a set of 10 wave equations, so we can look for solutions of the form:

hµν = Re[Aµνexp(ikαxα)], (5.2)

where Re refers to the real part of the bracketed expression, [Aµν ] is theamplitude matrix and kµ = ηµαkα is the wave vector. The d’Alembertianoperator acting on an exponential like 5.2 brings down a −kαkα, so in orderto satisfy 5.1, we need the wave vector to be a null vector:

kµkµ = 0. (5.3)

1Empty spacetime implies Tµν = 0

22

5.1.1 How many polarizations?

As there was a set of 10 wave equations, it can be tempting to think thatthere are 10 different polarizations. However, is this really the case? Letus remember the Lorentz gauge condition (equation 4.13), which implies arestriction on the amplitude components:

Aµνkν = 0. (5.4)

This generates four conditions that cut down the 10 equations down tosix. However, this is not the end yet. We can still reduce the number ofequations all the way to only two! In order to do this, I will follow in thenext lines the procedure used in [6]. Let us start by introducing the followinggauge transformation:

ξµ = Re[iBµexp(ikαxα)]. (5.5)

.From equation 4.12, we can see that the gravitational wave amplitude

changes from (Aµν) to (Aµν + kµBν + kνBµ − kαBαηµν). The following ex-pression allows us to see how the quantities change:

12AααA01

A02

A03

→

12AααA01

A02

A03

+

−ω −k1 −k2 −k3k1 −ω 0 0k2 0 −ω 0k3 0 0 −ω

B0

B1

B2

B3

. (5.6)

We can choose a coordinate system in which Aαα and one of the A0 co-efficients are zero because the 4x4 matrix is invertible. This is called theTT gauge2 and it gives us four more conditions on the amplitude matrixAµν , which leaves with only two possible polarizations. Hence, Aµν can beexpressed as a linear combination of two linear polarization matrices:

Aµν = C+eµν+ + C×e

µν× , (5.7)

where C+ and C× are complex constants and they represent two modes ofpolarization, and:

2TT stands for transverse traceless.

23

[eµν+ ] =

0 0 0 00 1 0 00 0 −1 00 0 0 0

[eµν× ] =

0 0 0 00 0 1 00 1 0 00 0 0 0

. (5.8)

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

ikαxα = i(k0ct− k0z) = ik0(ct− z), (5.9)

with kα = (k0, 0, 0,−k0). In this case, the metric will be given by:

ds2 =dt2 + (1 +Re[C+exp(ikαxα)])(dx1)2 + 2Re[C×exp(ikαx

α)]dx1dx2

+ (1−Re[C+exp(ikαxα)])(dx2)2 + (dx3)3.

(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 linearlypolarized type “+” (which means C× = 0). Particles along the direction ofpropagation x3 will not feel anything. However, different particles with thesame x1 coordinate will drift apart, whereas particles with same x2 will getcloser. After a short period of time, the opposite will happen: particles withdifferent x1 coordinate will come closer, whereas those with the same x2 willmove farther from each other. If we imagine a circle in the x1x2 plane, we willsee how, first, it becomes squashed in the x1 direction and elongated in thex2 direction, and later it will do the opposite. This is why we define this po-larization as +. If, instead of a type + linearly polarized wave we have a type× linearly polarized wave, the same thing will happen. However, the onlydifference is that, this time, the axes along which elongation/compressionhappens will be rotated 45o with respect to the previous ones, and hence thereason why the symbol × is used for this case. We can see that in both cases,we are dealing with transverse waves.

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5.2 The quadrupole formula

We can try to apply an equivalent of the formulae of Electromagnetism in or-der to grasp how gravitational waves work. One of the things that we shoulddo is to replace the charge of the electron by the mass. In electromagnetictheory, the power output of an electric dipole can be expressed as

L =1

4πε0c32d2

3, (5.11)

where d is the dipole moment. In gravity, the equivalent of an electric dipolemoment is the mass dipole moment, whose first derivative is the linear mo-mentum. The first derivative of the linear momentum (and therefore, thesecond derivative of the mass dipole moment) is zero, because of conser-vation of momentum. As a consequence, there is no such thing as dipoleradiation in general relativity. Although it is not subject of this paper, thegravitational field has spin 2. This means that gravitational radiation isquadrupolar to lowest non-vanishing order.

Using electromagnetic theory as a reference once again, we can expressthe solution for equation 4.15 as:

hµν(x0,x) =−4G

c4

∫T µν(x0 − |x− x′|,x′)

|x− x′|dV ′ (5.12)

which is a retarded integral in which x represents the field point and x′ thesource. From equations 4.15 and 4.13, it can be seen that

T µν,ν = 0. (5.13)

This leads to

T 00,0 + T 0k

,k = 0 (5.14)

and

T i0,0 + T ik,k = 0. (5.15)

In order to continue, we will need the following identity:∫(T ikxj),kdV =

∫T ik,k x

jdV +

∫T ijdV. (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.∫

T ijdV =

∫T i0,0 x

jdV. (5.17)

We repeat this procedure, but replacing i with j, and add up the results:

∫T ijdV =

∫T i0,0 x

j + T j0,0 xi

2dV =

1

2c

d

dt

∫(T i0xj + T j0xi)dV (5.18)

Using again identity 5.16:

∫(T 0kxixj),kdV =

∫T 0k,k x

ixjdV +

∫(T 0ixk + T 0jxi)dV, (5.19)

where, because of Gauss once again, the left side is null. We now apply 5.14:∫(T 0ixk + T 0jxi)dV =

1

c

d

dt

∫T 00xixjdV. (5.20)

We can now take the derivative on both sides of this equation, and use 5.18:∫T ij =

1

2c2d2

dt2

∫T 00xixjdV ≈ 1

2

d2

dt2

∫ρxixjdV, (5.21)

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 canapproximate 5.12 by

hµν(ct,x) =−4G

c4r

∫T µν(ct− r,x′)dV ′. (5.22)

which is valid, once again, for slow source particles. Finally, we obtain:

hij(ct,x) = −2G

c4r

d2

dt2

(∫ρxixjdV

), (5.23)

where the integral is taken at t− r/c, which is the retarded time. This equa-tion tells us at which rate and amplitude gravitational waves are generatedfrom 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 quadrupoleformula:

LGW =G

5c5

(d3Qij

dt3

)2

, (5.24)

where Qij is the quadrupole moment,

Qij(t) =

∫d3x ρ

(xixj − 1

3δijx2

), (5.25)

and whose trace is zero:δijQij = 0. (5.26)

Equation 5.24 gives us the total power in the gravitational waves emitted bya source.

5.3 Generation of gravitational waves

The quadrupole formula is a way to understand how gravitational waves areproduced. However, just by looking at it it is hard to have an idea of whattype of events have the power to generate gravitational waves.

The truth is that the events in our universe capable of producing mea-surable gravitational waves are of astronomical magnitude. We need verymassive 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 astro-physical sources. First, we can say that

d3Qij

dt3∼ ω3MR2, (5.27)

where ω is some typical eigenfrequency, and it can be determined from theequations of motion. For example, for two masses M circulating each other:

Ma = Fgrav, (5.28)

and hence, if M = GMc2

, we can express 5.24 as

LGW ∼

(M

R

)5

L0, (5.29)

27

where L0 is a conversion factor:

L0 ≡c5

G. (5.30)

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 holecan generate this type of waves. Looking at equation 5.29, the fact that theradius is raised to a power of −5 tells us that when matter reaches the blackhole it can emit a strong burst. It has been estimated (see [4]) that theenergy output E of such burst would be

E ≈ 0.0104m2

Mc2, (5.31)

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 theonly way for black holes to emit gravitational waves. If there are two massiveblack 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 oftheir lives, are very catastrophic events in our universe and thus, they alsohave the power to create gravitational waves. Estimations say that the poweroutput 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 gravi-tational waves. In order for a pulsar to emit gravitational radiation it needsto have low axial symmetry. This is because axial symmetry implies a con-stant 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 areactually part of binary systems. And because binary star systems can alsocreate 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 abinary system of two stars of masses m1 and m2 as

µ =m1m2

m1 +m2

, (5.33)

the power output can be expressed as ([12])

L ≡ (m1 + m2)3µ2

4a5L0, (5.34)

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 RusselHulse and Joseph Taylor in a binary star system became the first indirectsupport of gravitational waves. Hulse and Taylor’s Nobel Prize awardeddiscovery showed how the period of the pulsar decreased over time, implyinga 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 gravitationalwaves is clearly not an easy task for several reasons. One of them is that theevents that generate sufficiently large waves can be very uncommon in theuniverse. As well, in case one of this events happens to reach us, it can havea very short length in time, so we need to be paying attention constantly.However, probably the biggest challenge when it comes to detecting thesephenomena is the experimental challenge they give rise to. An experimentthat aims to detect gravitational waves needs to have one of the greatestsensitivities in all experiments ever performed, as the perturbation in spacetime that they create is extremely small. Nevertheless, there are some clevermethods to detect these elusive phenomena.

29

5.4.1 Weber bars

Weber was one of the pioneers in gravitational waves detection. The ex-periments he performed, back in the 1960s, were based on a large metallicbar 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 theeigenfrequencies 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 wavesemitted by, for example, the previously mentioned pulsar PSR 1913+16, andtry to measure it using Weber bars, we would be trying to find a change inlength of one hundredth the size of an atomic nucleus. It seems obvious thatit 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 knownas pulsar timing arrays. The fundamentals behind this is using a series of pul-sars whose rotational period is of the order of milliseconds. If a gravitationalwave were to go through our planet, we would measure a very slight changein the signals coming from the pulsars. Pulsar timing arrays are very oftenused in order to detect stochastic gravitational wave background (especiallyoriginated 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 in-terferometry. Figure 5.4.3 contains a schematic diagram of a basic laserinterferometer. The way this works is the following: there are arms placedat an angle of 90o and mirrors are placed along them. Then, a beam of lightis split so that it travels both arms’ length until reaching the mirrors at theend of each arm. Then, each beam gets reflected and they both recombine,creating an interference pattern. If a gravitational wave reached our inter-ferometer, the length of the arms would very slightly change, and hence thelaser beams would arrive out of phase, and we would be able to detect thisin the interference pattern.

30

Figure 5.1: Diagram of how a Michelson interferometer works: the (red) laserbeam reaches the splitter (BS), so that it gets divided into two beams, withone (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) andhit the screen. If the arms had a variation of length, an interference patternshould be projected onto the screen. SOURCE OF THE IMAGE: WernerBoeglin, Dept. of Physics, FIU, Miami

When we say ”slightly change”, we are talking about one part in approx-imately 1021. To get an idea of how ridiculous this amount is, if we wereto measure the diameter of our galaxy3 this way, we would be looking forchanges 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 lightyears.

31

5.5 LIGO’s interferometer and first observa-

tion of gravitational waves

LIGO (Laser Interferometer Gravitational-Wave Observatory) is the largestgravitational wave detector. It is made up of two laser interferometers, onelocated in Hanford, Washington, and the other one in Livingston, Louisiana.This means that they are located 3000 km away from each other. Why do weneed 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 alwaysenvironmental noise and even the possibility of earthquakes, which wouldresult in an interferometer pattern. However, if there are two detectors incompletely different locations, the noise will be totally different, so we willbe able to identify it and remove it. If a gravitational wave arrived in thetwo detectors, it would do it almost simultaneously, and we would be able todistinguish it from the background noise. Nonetheless, noise is not the onlyreason why more than one detector is used. Using two detectors allows us todiscover 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 varyby a distance of one ten thousandth a proton’s width. Such magnitudes ofprecision require the smoothest mirrors ever constructed. The arms are alsoin a vacuum, to avoid air interfering with measurements. As well, in order toobtain detectable interference pattern, the laser needs to have a very precisewavelength and a lot of power, to have a good enough resolution. As amatter of fact, LIGO’s laser power is 750 kilowatts5 (see [9]). And for thewavelength, LIGO uses infrared light, which means that the variations oflength are only a trillionth of a wavelength, leading to extremely small shiftsof 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 publishedin [1]. In order to identify the signal, named GW150914, in all the data thatLIGO 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 detectorswould be needed.

5However, LIGO’s laser beams has a power of 200 kilowatts before arriving in theinterferometer. 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 makeany assumptions on the form of the wave, looking for all types of signals.These methods were able to calculate the likelihood of obtaining a signallike GW150914, and it was found that an event like this has a probabilityof happening randomly once every 203 000 years, which is equivalent to asignificance 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. Thefrequency of the signal was observed to start at 35 Hz, and then it went upuntil 250 Hz, lasting 0.2 s and completing a total of 8 cycles. Not only thefrequency 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 compactobjects orbiting around each other at a very close distance. Out of the astro-physical objects that we mentioned in section 5.4 with the power to generatemeasurable gravitational waves, the most compact ones are neutron stars andblack 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 itis way below these results, showing that the objects responsible for eventGW150914 were two black holes.

This discovery was very important as, apart from being the first directdetection of gravitational waves, it was also the first observation of a binaryblack 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 isprobably most important about GW150914 is that it has opened a new fieldof astronomy: gravitational-wave astronomy.

33

Chapter 6

Summary

As it has been seen, a very big part of general relativity deals with geometryand math. For that reason, we decided to start this thesis introducing themath and geometry needed to understand some basic principles of Einstein’stheory of gravity. Once this was done, we were able to introduce the elegantEinstein’s field equations:

Gµν = −8πG

c4T µν .

Afterwards, we linearized gravity, starting off by approximating the met-ric 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:

22hµν = −16πG

c4Tµν .

Applying this equation to empty space, which means equaling the righthand side to zero, we were able to find solution in the shape of waves, whichis what we call gravitational waves:

hµν = Re[Aµνexp(ikαxα)].

After a set of conditions were applied, we found out that the number ofpossible polarizations for these gravitational waves is two. The quadrupoleformula, the equation that describes the generation of gravitational waves,was derived as:

34

hij(ct,x) = −2G

c4r

d2

dt2

(∫ρxixjdV

).

Finally, we saw the difficulties that detecting gravitational waves implies,as well as seeing how LIGO was able to battle them, achieving the first directdetection of these events in our universe.

35

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[3] J. Buitrago, E. Mediavilla, and A. Oscoz. Relativistic Astrophysics andCosmology: Proceedings of the Spanish Relativity Meeting. 1997. isbn:9789814545907.

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