Dynamics of cold atoms in moving optical lattices (Version Dek8)
Project
1. “I want to know God’s thoughts; the rest are details.”
— Albert Einstein
Universidade de Aveiro Departamento de F´ısica,
2014
Jorge Filipe
M´onico Delgado
Gravitational Waves: Basic Theory and
Experimental Evidence
2.
3. Universidade de Aveiro Departamento de F´ısica,
2014
Jorge Filipe
M´onico Delgado
Gravitational Waves: Basic Theory and
Experimental Evidence
Bachelors project presented to Aveiro University for the acquirement of
the Bachelors degree in Physics, supervised by Dr. Carlos Alberto Ruivo
Herdeiro, Assistant Professor at the Physics Department of Aveiro Univer-
sity, and Dr. Jai Grover, Post Doctoral Researcher at the Physics Depart-
ment of Aveiro University.
4.
5. the jury
Dr. Manuel Ant´onio dos Santos Barroso
Professor Auxiliar da Universidade de Aveiro
Dr. Carlos Alberto Ruivo Herdeiro
Professor Auxiliar Convidado da Universidade de Aveiro (advisor)
Dr. Jai Grover
Post Doctoral Researcher (co-advisor)
Dr. Juan Carlos Degollado Daza
Post Doctoral Researcher (arguente)
Dr. Vitor Hugo da Rosa Bonif´acio
Professor Auxiliar da Universidade de Aveiro
6.
7. acknowledgements First and foremost I offer my greatest gratitude to my supervisor, Dr.
Carlos Alberto Ruivo Herdeiro and to my co-supervisor, Dr. Jai Grover,
whose support was undoubtedly the best one can have. This thesis was
only possible due to all the patience of both to answering all of my
(sometimes, stupid) questions and to all of the encouragement and effort
of both. Without them I would still be stuck in the first calculation.
To Professor Manuel Barroso for his support and guidance throughout the
entire degree.
I like to thank to my roommates Miguel Ferreira and David Pereira for all
the awesome things we did together in the past two years and for all that
maybe come next. Your awesomeness was off the charts and I am really
glad that I lived with you the past two years. I also want to thank my first
roommate Bruno Roda; although we only lived one year together, it was a
year to remember and it was a year to learn how to discuss serious matters.
Without him I would still be an ignorant fool.
I want to thank to all my fellow colleagues from Physics, Physics Engi-
neering and MOG, for all the help that everyone gave me. In particular
to Diamantino Silva, Ana Mota and Ivo Maceira for all their help, their
encouragement and, above all, for the patience to put up with me, I know
that was not easy. I also want to thank particularly my “sister” S´ılvia Reis
for one of the greatest friendships I made in these three years. She always
was there when I needed her and always acted as a real sister to me. To
Rafa, Rita, Jennifer and Novo for our friendship that began three years
ago and that will continue for indefinite years. To my “patr˜ao” Bruno
Rodrigues who, although he was not here in the last two years, always gave
me advice about my future and always was present when I needed him.
Until this day I can not think of a better person to be my “patr˜ao”. To “10
da vida airada” for all the memories I will recall for the rest of my life, for
all the happy times that we spent together, for all the dinners that I made
lasagna and especially for the Bacon Festival. I was the outsider of the
group but you embraced me as a friend and always made me smile whenever
I needed it. To all the freshmen I met this year, I also want to thank you
for the patience to put up with me and for listening to my random gibberish.
Last but definitely not least, I want to express my gratitude to my family
for supporting me at all times and for giving me the opportunity to pursue
my dreams; despite going through tough times, they have always tried to
give me everything they could, and for that I am really thankful for having
a family like mine.
8.
9. Abstract Gravitational waves are one of the most fascinating predictions of the gen-
eral theory of relativity. In this thesis, we present the basic theory and
experimental evidence for gravitational waves. After a brief review of the
formalism of general relativity, we introduce gravitational waves as tenso-
rial perturbations around flat space obeying the wave equation. We then
present the Landau-Lifschitz pseudo-tensor as a meaningful quantity to mea-
sure the energy and momentum carried by gravitational waves. Using it, we
compute the Einstein quadrupole formula for the power emitted in gravi-
tational waves in a system with varying (in time) gravitational quadrupole,
computed in the Keplerian approximation. In order to make contact with
observations we apply the quadrupole formula to a binary system and ob-
tain the Peters-Mathews formula in terms of the orbital parameters. The
application of this formula matches with great accuracy the observational
data for the decay of the orbital period in binary pulsar systems. We briefly
illustrate that gravitational waves can be used to test alternative theories of
gravity. Finally, we remark on the promising future of this field of research.
10.
11. Contents
Contents i
List of Figures iii
List of Tables v
1 Introduction 1
2 Topics of General Relativity 2
3 Gravitational Waves: Basic Theory 4
3.1 Weak Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1.2 Plane Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1.3 Energy Carried by Gravitational Waves . . . . . . . . . . . . . . . . . 6
3.1.4 The Pseudotensor for a Plane Wave . . . . . . . . . . . . . . . . . . . 8
3.2 Gravitational Waves in Curved Space-Time . . . . . . . . . . . . . . . . . . . 9
3.3 The Quadrupole Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Gravitational Waves: Observational Evidence 13
4.1 Dissipation Power of an Astrophysical System . . . . . . . . . . . . . . . . . . 13
4.1.1 Derivation of the Formula for the Total Power Radiated . . . . . . . . 13
4.1.2 Total Radiation of a Binary System . . . . . . . . . . . . . . . . . . . 14
4.2 The Hulse-Taylor Binary Pulsar (B1913+16) . . . . . . . . . . . . . . . . . . 16
4.3 The Double Pulsar (J0737-3039) . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Testing Alternative Theories of Gravitational Waves 21
6 The Future: Direct Detection of Gravitational Waves 23
Appendices 25
A Computation of the Landau-Lifschitz Pseudotensor 26
B Derivation of the Angular Velocity of any Kepler Motion 29
Bibliography
i
15. List of Tables
4.1 Measured orbital parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Measured and derived parameters of pulsars J0737-3039A and B. The number
in parentheses is the standard errors of the last digit(s). [10] . . . . . . . . . . 19
v
17. Chapter 1
Introduction
In 1687, Isaac Newton presented a mathematical law for the gravitational interaction. It
was subsequently understood that this, Newtonian gravity, has the fundamental field equation
2
Φ = 4πGρ, (1.1)
where 2 is the Laplacian operator, Φ is the gravitational potential, G is Newton’s constant
and ρ is the mass density of the sources.
According to this equation, massive bodies source gravitational fields; there is, however,
a striking feature in the way they do it. Since the differential operator on the left hand side
has no time derivatives, any change in time of the sources must change the gravitational field
immediately everywhere. In other words, gravitational effects propagate instantaneously.
In 1905, Albert Einstein published one of his most famous papers, On the Electrodynamics
of Moving Bodies [1]. One of the consequences of this revolutionary work, is the notion of
relativistic causality: information cannot travel faster than the speed of light (in vacuum).
Clearly, Newtonian gravity is inconsistent with this relativistic causality. As such, Einstein’s
1905 work made clear that a new theory of gravity was needed. Such a theory should have a
causal effect through which the gravitational field communicates its variations. That is, there
must be some type of signal carrying the news that gravitational sources have changed, and
such a signal should not propagate faster than the speed of light.
Einstein formulated his relativistic theory of gravity in 1915: General Relativity (GR)
[2]. In 1916 he found that the signal described above exists; changes in the gravitational
field are communicated via ripples in space-time traveling precisely at the speed of light.
These are the Gravitational Waves (GWs). After some initial misunderstanding, Einstein
proposed an essentially correct picture of GWs in 1918. In 1937, however, he changed his
mind and submitted a paper, in collaboration with Nathan Rosen, suggesting GWs do not
exist! Fortunately, the referee rejected Einstein’s (incorrect) paper, which made him think
deeper upon the subject and eventually change his mind again [3]. Einstein’s doubts, however,
lingered in the community and only with the discovery of the Hulse-Taylor binary pulsar
system and subsequent observation of a decrease in its orbital period, which is a consequence
of the emission of GWs, did the scientific community come to agree on the reality of GWs.
In this thesis we will describe the basic formalism of GWs and, in particular, address the
energy loss in a system due to GW emission. This theoretical prediction is compared (and
verified!) in some well-known binary pulsar systems, as we shall discuss. Finally we will
comment on how GWs can be used to constrain alternative theories of gravity.
1
18. Chapter 2
Topics of General Relativity
The field equations of GR state that the appropriately defined curvature of spacetime is
proportional to the energy-momentum of the sources. These are tensor equations involving
4 × 4 symmetric tensors [4]
Gik =
8πk
c4
Tik, (2.1)
where Gik is the Einstein tensor and Tik is the stress-energy-momentum tensor1.
The Einstein tensor Gik contains information about the local curvature of spacetime and
can be calculated through the Ricci tensor Rik and the Ricci scalar R ≡ Ra
a
Gik = Rik −
1
2
gikR. (2.2)
The Ricci tensor represents the amount by which the volume of a geodesic ball in a curved
Riemannian manifold deviates from that of the standard ball in Euclidean space, and is
derived from the Riemann tensor Rm
ilk
Rik = Ra
iak. (2.3)
The Riemann tensor depends on the Christoffel symbols Γm
kl
Rm
ilk = ∂lΓm
ki − ∂kΓm
li + Γm
laΓa
ki − Γm
kaΓa
li. (2.4)
And finally we can relate the Christoffel symbols with the metric of the spacetime
Γm
ik =
1
2
gma
(∂kgia + ∂igka − ∂agik) . (2.5)
This relation is obtained from the conditions that the metric is covariantly constant and that
the torsion is zero.
The stress-energy-momentum tensor Tik describes the density and flux of energy and mo-
mentum of an object in spacetime. It is generated by matter, radiation and non-gravitational
force fields. If we analyse each one of the components we can say that the component T00 is
the density of mass-energy, i.e. the energy density divided by the speed of light squared; the
1
In this thesis we will use the letter k for the gravitational constant, instead of G, to avoid confusion with
the Einstein scalar Ga
a ≡ G.
2
19. components Tµ0
2 equal the density of the µ-component of momentum; the components T0ν
correspond to the ν-component of energy flux. Note that the tensor Tik in GR is symmetric,
Tµ0 = T0µ. The components Tik represent “stresses”.
If one writes explicitly the tensor equation (2.1) in terms of differential equations for the
metric gik, one obtains a set of coupled equations involving gik, ∂lgik and ∂l∂mgik. Thus
equation (2.1) yields a set of ten second order differential equations, which are non-linear.
For this type of differential equation, exact solutions are difficult to obtain, but, under some
assumptions, such as sufficient symmetry for the metric ansatz, and simple energy-momentum
tensors for the sources, exact solutions are possible to obtain.
2
The indices µ and ν range from 1 to 3.
3
20. Chapter 3
Gravitational Waves: Basic Theory
As seen in the previous chapter, GR is a theory whose field equations are non-linear, which
makes the task of solving them exactly, quite challenging. But, just as in any non-linear
theory, one can use first a linear approximation to understand how the system behaves when
the curvature is weak, or just a small deviation from a known exact solution. In this chapter we
will start by considering linear perturbations around flat space and compute quantities up to
the first order in the perturbation of the metric. Subsequently we shall consider perturbations
around a non-trivial background metric. Our goal is to obtain information about the energy
and momentum carried by GWs and the energy loss for a system emitting GWs. All of this
chapter was based on [5] and [6].
3.1 Weak Gravitational Waves
3.1.1 The Wave Equation
We start by considering a flat spacetime where we introduce a weak perturbation. The
spacetime metric gik is then
gik = ηik + hik, (3.1)
where ηik is the Galilean metric1 and hik is the weak perturbation. Then, to terms of first
order in hik, the contravariant metric tensor is
gik
= ηik
− hik
. (3.2)
In the following it will be relevant to see how the metric perturbation hik changes when
an infinitesimal coordinate transformation is performed
xi
−→ x i
= xi
+ ξi
. (3.3)
Then, the metric perturbation hik will change as
hik = hik −
∂ξi
∂xk
−
∂ξk
∂xi
. (3.4)
1
The ”Galilean” metric is the Minkowski metric in a global inertial coordinate system, and in this thesis
we consider the signature (+ − −−). The raising and lowering of indices is made with the metric ηik
4
21. As in Maxwell’s electromagnetic theory, where we have a family of magnetic potentials,
related by transformations of the form A −→ A + ψ, that correspond to the same magnetic
field, we also have this type of gauge freedom in GR, where it amounts to the choice of the
coordinates. For the study of linear perturbations around Minkowski space, the gauge, or
coordinate system, often chosen is called De Donder’s Gauge or Harmonic Gauge and it is
defined by the condition
∂ψk
i
∂xk
= 0, ψk
i = hk
i −
1
2
δk
ih, (3.5)
where h ≡ ha
a. In practice this gauge is attained as follows. Imagine that in the initial
coordinate system {xi} we have
∂
∂xk
hk
i −
1
2
δk
ih ≡ Ai = 0. (3.6)
Then, by performing the coordinate transformation, equations (3.3) and (3.4), we will obtain
Ai = Ai + ∂k∂k
ξi. (3.7)
If we choose the gauge transformation function ξi such that ∂k∂kξi = Ai, then Ai = 0, which
means, in the new coordinates, the harmonic gauge condition is obeyed. But notice this gauge
choice does not exhaust the gauge freedom. A further coordinate transformation such that
∂k∂k ˜ξi = 0 implies Ai = Ai , so that if the original coordinate system obeys the harmonic
gauge condition so will the new.
The advantage of choosing the harmonic gauge becomes clear when considering the Ricci
tensor in linearized theory. In order to obtain the Ricci tensor we first have to write down
the curvature tensor and the Christoffel symbols for the linearized connection, i.e., equation
(3.1)
Γi
jk =
1
2
ηil
(∂khlj + ∂jhlk − ∂lhjk) , (3.8)
Ri
klm =
1
2
ηij
(∂k∂lhjm − ∂j∂lhkm − ∂k∂mhjl + ∂j∂mhkl) . (3.9)
Due to the choice of gauge that we made, the Ricci tensor will be just
Rkm = −
1
2
∂j∂j
hkm =
1
2
hkm, (3.10)
where = −ηjl ∂
∂xj
∂
∂xl , denotes the d’Alembertian operator. Then, writing the vacuum
Einstein’s Equations
Rik = 0 we obtain hik = 0 ⇐⇒ hk
i = 0. (3.11)
So the choice of harmonic gauge allowed us to write the first order (in the metric perturbation)
Einstein equations as a set of decoupled wave equations, one for each component of the metric
perturbation hik. An immediate conclusion is that perturbations of the gravitational field can
propagate in vacuum at precisely the speed of light, similar to electromagnetic waves. These
traveling perturbations are what we call GWs, in the linear theory description.
5
22. 3.1.2 Plane Gravitational Waves
In order to gain some more insight into linear GWs, we need to solve the wave equation.
Let us consider a plane gravitational wave that propagates only in one direction, which we
choose to be x1 ≡ x. The equation (3.11) becomes
∂2
∂x2
−
1
c2
∂2
∂t2
hk
i = 0. (3.12)
The generic solution of this equation is a function of t±x/c, so hk
i = fk
i(t+x/c)+gk
i(t−x/c).
In other words, a part of the wave propagates in the negative x direction fk
i(t + x/c) and
the other propagates in the positive direction gk
i(t − x/c). To simplify let us choose only the
wave propagating in the positive x direction.
Due to the harmonic gauge we can obtain the relation ˙ψ0
i − ˙ψ1
i = 0, where the dot
denotes differentiation with respect to t. Integrating and setting the integration constant to
zero, because we are only interested in the varying part of the gravitational field, we obtain
the following relations
ψ0
0 = ψ1
0 ; ψ0
1 = ψ1
1 ; ψ0
2 = ψ1
2 ; ψ0
3 = ψ1
3. (3.13)
As mentioned before, imposing the harmonic gauge does not exhaust the gauge freedom.
There is a residual gauge freedom of the type xi −→ x i = xi + ξi where ξi = 0. The four
functions ξi can be chosen to make ψ0
1, ψ0
2 and ψ0
3 vanish. Then, from the relations (3.13)
and the fact that the “trace reversed metric perturbation”, ψk
i, is symmetric, all the other
components will also vanish, except for ψ2
2, ψ3
3 and ψ3
2 = ψ2
3. These components cannot
be made to vanish in this choice of reference system because these components do not change
under a transformation like equation (3.3). Moreover, in this gauge, the trace of ψk
i can also
be made to vanish; therefore ψk
i = hk
i.
In conclusion the metric perturbations are
hij =
0 0 0 0
0 0 0 0
0 0 h22 h23
0 0 h23 −h22
. (3.14)
This means a plane gravitational wave is transverse and it has two independent polarizations,
since there are two independent quantities defining it. These two quantities are often denoted
h+ ≡ h22 and h× ≡ h23.
3.1.3 Energy Carried by Gravitational Waves
Our goal now is to compute the energy and momentum carried by a plane gravitational
wave. For this purpose we will derive the expression for the energy-momentum Landau-
Lifschitz (LL) pseudotensor tik. The reason for deriving this pseudotensor is that “matter”
fields, such as electromagnetic fields, may exchange energy and momentum with the gravi-
tational field. Therefore, there must be a mathematical quantity encoding the energy and
momentum of the gravitational field. There is, however, no known true local energy momen-
tum tensor of the gravitational field. The best one can do, but which seems to suffice in
practice, is to define a pseudotensor.
6
23. To obtain the energy momentum pseudotensor for the gravitational field, we first consider
the equation of conservation of energy and momentum for some material field with energy-
momentum tensor Tik. In a covariant form this equation is: kTik = 0, where k denotes
the covariant derivative. If one expands the covariant derivative, one obtains
1
√
−g
∂k Tk
i
√
−g −
1
2
∂i (glk) Tkl
= 0. (3.15)
Due to the second term, this equation is not expressing the familiar energy-momentum con-
servation law in Minkowski space, ∂kTik = 0. But if we consider normal coordinates at
some spacetime point P, then, at that point, the first derivatives of the metric will be zero,
∂kgij = 0, and the equation (3.15) reduces to the usual energy-momentum conservation law.
By using the Einstein equations2 we can obtain an explicit formula for Tik, in terms of the
metric, valid at point P. This formula reads:
Tik
|P = ∂l
c4
16πk
1
(−g)
∂m (−g) gik
glm
− gil
gkm
. (3.16)
Observe that this expression contains second derivatives of the metric; thus it needs not vanish
at P. Also, it is simple to check that ∂kTik|P = 0, using the fact that first metric derivatives
vanish and that (3.16) is anti-symmetric in k ↔ l.
It is now useful to define
λikl
≡
c4
16πk
∂m (−g) gik
glm
− gil
gkm
; (3.17)
then, at P it follows that
∂lλikl
= (−g)Tik
|P. (3.18)
For a general point not in normal coordinates, equation (3.18) does not hold. As such we
introduce
(−g)tik
≡ ∂lλikl
− (−g)Tik
. (3.19)
The quantity tik is what we call the gravitational LL energy-momentum pseudotensor. The
central property of this definition is that tik is symmetric and, most importantly, for a general
point
∂k (−g) Tik
+ tik
= 0 . (3.20)
Thus, the definition of tik allows us to obtain a conservation law for energy-momentum which
includes the gravitational field and describes exchanges of energy-momentum between mate-
rial fields and the gravitational field.
In order to compute a usable formula for this pseudotensor in terms of the metric, we must
use the Einstein equations again, but now for generic coordinates. A manipulation yields the
relation:
2
At the point P, the first derivatives of the metric vanish, but the same does not happen with the second
derivatives. Therefore the Christoffel symbols vanish, but not their derivatives, which entail the curvature. In
other words, the connection can be gauged away at some point, but not the curvature.
7
24. tik
=
c4
16πk
1
(−g)
∂l∂m (−g) gik
glm
− gil
gkm
+
+ gik
glm
− 2gim
gkl
[∂pΓp
lm − ∂mΓp
pl + Γp
npΓn
lm − Γp
nmΓn
pl] . (3.21)
In this form, it seems that the LL pseudotensor contains both first and second derivatives of
the metric. This is however, not true, since all the second derivatives cancel out. This can
be shown by a rather lengthy calculation, of which more details are provided in Appendix A.
Then the following expression for tik is found
(−g)tik
=
c4
16πk
˜gik
,l ˜glm
,m −˜gil
,l ˜gkm
,m +
1
2
˜gik
˜glm˜gln
,p ˜gpm
,n −
− ˜gil
˜gmn˜gkn
,p ˜gmp
,l +˜gkl
˜gmn˜gin
,p ˜gmp
,l + ˜gnp
˜glm˜gil
,n ˜gkm
,p +
+
1
8
2˜gil
˜gkm
− ˜gik
˜glm
(2˜gnp˜gqr − ˜gpq˜gnr) ˜gnr
,l ˜gpq
,m , (3.22)
where ˜gik =
√
−ggik, while the index ,i ≡ ∂i denotes a simple differentiation with respect to
xi. This expression is valid in general, not only in the linear theory. Moreover, it is now
obvious that tik contains only first derivatives of the metric. Therefore it vanishes in normal
coordinates. For this reason it is not a tensor, since any tensorial quantity that vanishes in
one coordinate system must vanish in all. Still, it is a meaningful quantity to describe the
energy and momentum carried by the gravitational field, when properly used.
3.1.4 The Pseudotensor for a Plane Wave
The LL pseudotensor can be specialized for linear theory, using (3.2), that is gik ≈ ηik
and gik,l = −hik,l. Substituting this result in (3.22) yields
tik
=
c4
16πk
hik
,l hlm
,m −hil
,l hkm
,m +
1
2
ηik
ηlmhln
,p hpm
,n −
− ηil
ηmnhkn
,p hmp
,l + ηkl
ηmnhin
,p hmp
,l + ηnp
ηlmhil
,n hkm
,p +
+
1
8
2ηil
ηkm
− ηik
ηlm
(2ηnpηqr − ηpqηnr) hnr
,l hpq
,m . (3.23)
For the plane gravitational wave considered before, due to the choice of gauge and the choice of
the axes of the Galilean system of reference as function of x and t in the combination t ± x
c ,
all the terms will vanish, except one: 1
2ηilηkmηnpηqrhnr,l hpq,m, which can be simplified to:
1
2hn
q,i hq
n,k. Thus
tik
=
c4
32πk
hn
q,i
hq
n,k
. (3.24)
The energy flux in the wave is given by the quantity −cgt0α, but under the conditions we
are considering, we can write: −cgt0α ≈ ct01. Therefore
ct01
=
c5
32πk
hn
q,0
hq
n,1
. (3.25)
8
25. We can transform the x0 ≡ ct and x1 ≡ x derivatives into time derivatives ∂t, and since the
perturbation hi
k is a function of (t − x/c), we can write
hn
q,0
=
1
c
˙hn
q and hn
q,1
=
1
c
˙hn
q .
Adding the fact that, h2
2 = −h3
3 and h2
3 = h3
2, we can write the energy flow for this
gravitational wave as
ct01
=
c3
16πk
˙h23
2
+
1
4
˙h22 − ˙h33
2
. (3.26)
3.2 Gravitational Waves in Curved Space-Time
The study of the propagation of GWs as perturbations of flat spacetime, can be generalized
by considering weak perturbations of an arbitrary (non Galilean) background metric g
(0)
ik . For
this purpose we reconsider equations, (3.1) and (3.2), but now with an arbitrary background
metric3
gik = g
(0)
ik + hik ; gik
= g(0)ik
− hik
. (3.27)
We perform the same procedure as in the previous section, but with the difference that
the derivatives of the background metric do not need to vanish. After writing the Christoffel
symbols, the curvature tensor, the Ricci tensor and using the Einstein equations in vacuum,
Rik = 0, we obtain
hl
i;k;l + hl
k;i;l − hik
;l
;l − h;i;k = 0, (3.28)
where the index ;i ≡ i denotes the covariant derivative.
Next, we consider the gauge fixing. The gauge freedom discussed previously still exists.
So we choose a covariant version of the gauge discussed before
ψk
i;k = 0 ; ψk
i = hk
i −
1
2
δk
ih . (3.29)
The existence of a non-trivial background introduces one characteristic distance scale L
and time scales L/c over which the background geometry changes. Naturally, these scales will
show up once derivatives of the background metric are taken. In many situations of interest,
we want to consider GWs of high frequency, that is when the wavelength λ and the oscillation
period λ/c are small compared to characteristic scales of the background. This means that
each differentiation of the perturbation hik increases the order of the quantity by a factor
of L/λ relative to the derivatives of the unperturbed metric g
(0)
ik . Thus the latter will be
suppressed. With these assumptions, equation (3.28) can be simplified to
hik
;l
;l = 0. (3.30)
Now, as before, we can still subject the choice of coordinates to a transformation of the
type found in equation (3.3), where the small quantities ξi satisfy the equation ξi;k
;k = 0.
3
In this analyse we still ignore the second order terms in the perturbation hik
, since it is a linear analysis
around a fixed, but not flat, background.
9
26. These transformations can be used to impose on the perturbation the condition h = 0.
Therefore, ψk
i = hk
i, so
hk
i;k = 0. (3.31)
The important assumptions in this argument are:
1. λ << L
I.e., the wavelength of the GW is much smaller than the “wavelength” of the background.
2. h ∼ cos(x
λ) and g(0) ∼ cos( x
L)
Since the perturbation and the background are waves with wavelength λ and L, respec-
tively, we can say that they are of the type of cos( x
λ ), where λ is the wavelength.
3. ∂xh ∼ 1
λh and ∂xg(0) ∼ 1
L g(0)
From point 1 it’s easy to see that L/λ >> 1, so each time we derive g, we increase the
order of the quantity by a factor of L/λ.
4. h ∼ 0 ; ∂xh ∼ 0 ; g0 ∼ 0 ; ∂xg(0) ∼ 04
Since h and g0 are in the form of point 2, and its derivatives in the form of point 3, we
see that all the linear terms in h, g0, ∂xh and ∂xg(0) vanish.
We can arrive to the same LL pseudotensor as before, equation (3.23) if we consider these 4
assumptions and if we average all the quantities tik over regions of four-space with dimensions
large compared to λ but small compared to L. Such an averaging will annihilate all quantities
that are linear in the rapidly oscillating quantities hik but will not affect the g
(0)
ik . Thus,
tik
=
c4
16πk
hik
,l hlm
,m − hil
,l hkm
,m +
1
2
g(0)ik
g
(0)
lm hln
,p hpm
,n −
− g(0)il
g(0)
mnhkn
,p hmp
,l + g(0)kl
g(0)
mnhin
,p hmp
,l + g(0)np
g
(0)
lm hil
,n hkm
,p +
+
1
8
2g(0)il
g(0)km
− g(0)ik
g(0)lm
2g(0)
np g(0)
qr − g(0)
pq g(0)
nr hnr
,l hpq
,m . (3.32)
Due to the choice of gauge and all of the assumptions that we made, every term of
the LL pseudotensor will vanish, except one: 1
2 g(0)ilg(0)kmg
(0)
np g
(0)
qr hnr,l hpq,m , which can be
simplified to: 1
2 hn
q,i hq
n,k . Thus
tik
=
c4
32πk
hn
q,i
hq
n,k
. (3.33)
4
We introduce x as the mean of the variable x over the period.
10
27. 3.3 The Quadrupole Formula
We finally arrive to the most important section of this chapter, which is the computation
of the leading term for the energy radiated by moving bodies in the form of GWs, also known
as the quadrupole formula.
Let us consider now a weak gravitational field, produced by arbitrary bodies, moving with
velocities small compared with the velocity of light. Because of the presence of matter, the
equations of the gravitational field, equation (3.11), will be a little different, by having, on the
right side, terms coming from the energy-momentum tensor of the matter. We write these
equations in the form
1
2
ψk
i =
8πk
c4
τk
i, (3.34)
where ψk
i is the same as before, equation (3.5), and τk
i denotes the auxiliary quantities which
are obtained upon going over from the exact equations of gravitation to the case of a weak
field in the approximation we are considering.
Due to the fact that the quantities ψk
i satisfy the condition ∂ψk
i/∂xk = 0, the auxiliary
quantities τk
i will obey the same condition
∂τk
i
∂xk
= 0. (3.35)
In order to solve this problem we must determine the gravitational field in the “wave
zone”, i.e., at distances large compared with the wavelength of the radiated waves.
Under these considerations, we can solve equation (3.34), for which the general solution has
the form
ψk
i = −
4k
c4
τk
i
t−R
c
dV
R
. (3.36)
Since, by assumption, the velocities of the bodies are small compared to the speed of light,
we can write, for the field at large distances from the system,
ψk
i = −
4k
c4R0
τk
i
t−
R0
c
dV, (3.37)
where R0 is the distance from the origin, chosen anywhere in the interior of the system. To
simplify the writing we will omit the index t − (R0/c) in the integrand.
By using the condition (3.35), we can prove that
ταβ dV =
1
2
∂2
∂x2
0
τ00 xα
xβ
dV. (3.38)
The advantage of this computation is that the components τ00 and τ0α, where α correspond
to {1, 2, 3}, are obtained directly from the corresponding components Tik by taking out from
them the terms of the order of magnitude in which we are interested. So, by definition,
Tk
i = µc2ukui, where ui is the four-velocity and µ is the mass density (sum of the rest masses
of the system in a unit volume), but since all the bodies of the system have small velocities,
11
28. we can assume that all the space components vanish and the temporal component is equal to
one. Thus
T00 = τ00 = µc2
. (3.39)
Substituting this result in equation (3.38) and introducing the time t = x0/c, we find for
equation (3.37)
ψαβ = −
2k
c4R0
∂2
∂t2
µxα
xβ
dV. (3.40)
Seeing that, at large distances from the bodies, the waves can be consider as plane waves,
we can use the results that we obtain in section 2.1, in particular, equation (3.25) and the
fact that h23 = ψ23 and h22 − h33 = ψ22 − ψ33. Thus
h23 = −
2k
3c4R0
¨D23 ; h22 − h33 = −
2k
3c4R0
¨D22 − ¨D33 , (3.41)
where we introduced the mass quadrupole tensor
Dαβ = µ 3xα
xβ
− δαβr2
dV. (3.42)
Making the time derivative of hαβ, which is simply adding one more dot in Dαβ, and substi-
tuting in equation (3.26), we obtain
ct01
=
k
36πc5R2
0
...
D
2
23 +
...
D22 −
...
D33
2
2
. (3.43)
As seen in the section 3.1, plane GWs have 2 polarizations, which correspond to the
two terms of this previous equation, but it is more convenient to express the polarization
in an invariant form. So we introduce the three-dimensional unit polarization tensor eαβ,
which determines the nonzero components of hαβ. This tensor is symmetric and satisfies the
conditions
eαα = 0 ; eαβnβ = 0 ; eαβeαβ = 1. (3.44)
where n is a unit vector in the direction of propagation of the wave.
Using this tensor and multiplying equation (3.43) by R2
0 dΩ to obtain the intensity of ra-
diation (power) of a given polarization into a solid angle dΩ, we finally obtain the Quadrupole
Formula, first obtained by Einstein in 1918
dP
dΩ
=
k
72πc5
...
Dαβeαβ
2
. (3.45)
Observe that the natural power in relativistic phenomena goes as k/c5. This amounts to
about 1052 Watts, roughly 1026 solar luminosities, which benchmarks the energy flux to be
expected in highly relativistic GW emission.
12
29. Chapter 4
Gravitational Waves: Observational
Evidence
In the previous chapter we have described some basic mathematical formalism and physical
properties of GWs propagating in a background spacetime. We now will compare the results
obtained with observational evidence gathered throughout the last 40 years using binary
systems with pulsars. We shall start, in the first section, by obtaining an equation that
expresses the power lost due to GWs emission in a binary system.
4.1 Dissipation Power of an Astrophysical System
At the end of the previous chapter we derived the expression of the Quadrupole Formula
(3.45). Now we will apply that formula to an astrophysical system, in order to study the
influence of the emission of GWs in this system.
Consider an astrophysical system as a discrete system of masses, so the mass quadrupole
tensor can be simplified to a sum instead of an integral. With this simplification, the mass
quadrupole tensor, (3.42), can be written as
Dij = 3
α
mαxαixαj, (4.1)
where α labels the masses. Consequently, the quadrupole formula can be simplified to
dP
dΩ
=
k
8πc5
...
Qijeij
2
, (4.2)
where Qij is the tensor Qij = α mαxαixαj.
4.1.1 Derivation of the Formula for the Total Power Radiated
The first thing to do is to expand this new quadrupole formula with the allowed polariza-
tions. We know from before that the only polarizations allowed satisfy the conditions (3.44),
and can be written as [7]
e1 =
1
√
2
(ˆxˆx − ˆyˆy), (4.3)
e2 =
1
√
2
(ˆxˆy − ˆyˆx). (4.4)
13
30. So if one expands the quadrupole formula with these allowed polarizations, one obtains
dP
dΩ
=
k
16πc5
X2
11 + X2
22 − 2X11X22 + X2
12 + X2
21 + 2X12X21 ,
where we use Xij ≡
...
Qij and {t, x, y, z} → {0, 1, 2, 3} to simplify the writing. With some
mathematical manipulation we can simplify the previous equation to
dP
dΩ
=
k
8πc5
XijXij − 2niXijnkXkj −
1
2
(Xii)2
+
1
2
(ninjXij)2
+ XiinjnkXjk , (4.5)
where n is the unit vector in the direction of radiation.
Since we have the angular power, we have to integrate over the solid angle to obtain the
total power. In this way we find the total power emitted to be:
P =
k
5c5
...
Qij
...
Qij −
1
3
...
Qii
...
Qjj . (4.6)
This formula yields the total energy per unit time dissipated due to the emission of GWs in
any discrete system of masses.
4.1.2 Total Radiation of a Binary System
Our goal now is to compute the power dissipated for a specific system - a binary system.
We will consider the bodies as point masses, in order to use the results from the previous
subsection.
Let the masses m1 and m2 have coordinates (d1 cos ψ, d1 sin ψ) and (−d2 cos ψ, −d2 sin ψ),
respectively, in the xy plane, as in Fig. (4.1). The origin of the coordinate system will be
taken to be the center of mass, so that
d1 =
m2
m1 + m2
d, d2 =
m1
m1 + m2
d. (4.7)
Here d is the distance between the two bodies; in order words, the sum of d1 with d2. In this
system, the mass quadrupole tensor will be quite simple
Qxx = µd2 cos2 ψ,
Qyy = µd2 sin2
ψ,
Qxy = µd2 sin ψ cos ψ = Qyx,
where µ denotes the reduced mass, µ = m1m2
m1+m2
.
The only thing remaining, in order to obtain the emitted power, is to differentiate three
times each term of the mass quadrupole tensor. In order to simplify the calculations we use
the Keplerian orbit equation
d =
a(1 − e2)
1 + e cos ψ
, (4.8)
14
31. y
x
d1
d2
m1
m2
ψ
Figure 4.1: Coordinates for the binary system
and the angular velocity of an orbital Keplerian motion1
˙ψ =
k(m1 + m2)a(1 − e2)
1/2
d2
. (4.9)
By using these two results, the third derivatives become simple to compute, and the results
can be simplified to
...
Qxx = β(1 + e cos ψ)2(2 sin 2ψ + 3e sin ψ cos2 ψ), (4.10)
...
Qyy = −β(1 + e cos ψ)2[2 sin 2ψ + e sin ψ(1 + 3 cos2 ψ)], (4.11)
...
Qxy =
...
Qyx = −β(1 + e cos ψ)2[2 cos 2ψ − e cos ψ(1 − 3 cos2 ψ)], (4.12)
where we define β as
β2
=
4k3m2
1m2
2(m1 + m2)
a5(1 − e2)5
.
To get the total power radiated we have to expand equation (4.6) and substitute therein
all the terms in equations (4.11), (4.12) and (4.12). Having done that, one obtains
P =
8
15
k4
c5
m2
1m2
2(m1 + m2)
a5(1 − e2)5
(1 + e cos ψ)4
[12(1 + e cos ψ)2
+ e2
sin2
ψ]. (4.13)
This equation gives the instantaneous power, which depends on ψ, describing the angular
position of the masses. Since the Keplerian motion is periodic, it is more useful to average this
power over one period of the elliptical motion, in order to have the average power dissipated
per orbital period. This way we obtain
P =
32
5
k4
c5
m2
1m2
2(m1 + m2)
a5(1 − e2)7/2
1 +
73
24
e2
+
37
96
e4
. (4.14)
Inspection of this formula reveals that this quantity is small, due to the factor k4/c5,
which is of the order of 10−84. In other words, only systems with objects of great mass,
great eccentricity (e ∼ 1) and small semimajor axis (a) will lose significant amounts of energy
1
The derivation of this expression is given in Appendix B.
15
32. Argument of periapsis
Inclination
Reference
direction
Celestial body
Plane of reference
Orbit
ω0
ψ
i
♈
a
Figure 4.2: Orbital parameters of a binary system
by emission of GWs. This is the reason why GW emission can only lead to observable
consequences in highly relativistic systems, where very large masses are in very close orbits
and hence at very large velocities. Fortunately, Nature has provided an abundance of such
systems in the Universe for us to study.
4.2 The Hulse-Taylor Binary Pulsar (B1913+16)
In this section we will apply the formula (4.14) to a real system: the binary pulsar
B1913+16. This system provided the first, albeit indirect, evidence for the existence of GWs.
Pulsars are highly magnetized, rotating neutron star that emit a beam of electromagnetic
radiation. If the beam of emission is pointing toward the Earth we can detect this radiation,
much in the same way an observer sees a lighthouse. Since neutron stars are very dense, and
have short, regular rotational periods, pulsars have very precise intervals between pulses, that
range from roughly milliseconds to seconds for the known pulsars. Certain types of pulsars
even rival atomic clocks in their ability to keep accurate time.
The binary pulsar B1913+16 was the first binary system discovered in which one of the
members is a pulsar (Hulse & Taylor in 1975) [8]. The other member is a neutron star
which is not seen as a pulsar. Since its discovery, it has been continuously monitored by radio
telescopes, which has allowed astronomers to measure several relativistic phenomena. Thanks
to these observations, the orbital parameters of this system are known with great accuracy: the
projected semimajor axis of the pulsar orbit ap sin i, orbital eccentricity e, epoch of periastron
T0, orbital period Tb, and the argument of periastron ω0 - Fig.4.2. Relativistic effects give
rise to orbital variations, which are described by three other quantities: the mean rate of
advance of periastron ˙ω , gravitational redshift and time-dilation parameter γ, and orbital
period derivative ˙Tb. All these eight parameters have been experimentally measured and are
listed in Table 4.1.
With these eight parameters we can indirectly measure other quantities, such as inclination
i, masses of the system m1 and m2, respectively, the mass of the pulsar and the mass of the
companion, and the semimajor axis a. But the most relevant to us are the masses of the
system: m1 = 1.4414 ± 0.0002 and m2 = 1.3867 ± 0.0002 solar masses.
With the knowledge of these astrophysical parameters, we can compute the average energy
16
33. Fitted Parameters Value
ap sin i (s) 2.3417725 (8)
e 0.6171338 (4)
T0 (MJD)2 52144.90097844 (4)
Tb (d) 0.322997448930 (4)
ω0 (deg) 292.54487 (8)
˙ω (deg/yr) 4.226595 (5)
γ (s) 0.0042919 (8)
˙Tb (10−12 s/s) -2.4184 (9)
Table 4.1: Measured orbital parameters
lost by GW emission, given equation (4.14). Even though GWs are very difficult to measure
directly, by any instrument, their emission leaves an imprint in the system due to the variation
of the orbital period ˙Tb. Indeed the energy loss should make the orbital period decrease, and
the companion stars approach each other. This effect can indeed be measured as displayed
in Table 4.1. Thus, our task is to compare the measured value with the value predicted by
GR, as a consequence of the energy loss due to the emission of GWs. We shall now derive
the necessary formula to make such comparison.
We know from Kepler’s laws that for any planetary motion:
ω2
=
k(m1 + m2)
a3
, Etotal = −
km1m2
2a
. (4.15)
Thus
a =
k(m1 + m2)
ω2
1/3
(4.16)
Etotal = −
1
2
m1m2
k2ω2
(m1 + m2)
1/3
. (4.17)
Taking a time derivative and substituting ω = 2π/Tb and ˙ω = −2π ˙Tb/T2
b , we obtain the
following equation
˙Tb = −
3
2
Tb P
Etotal
, (4.18)
where P is the time derivative of the total energy and is given by equation (4.14). Then, by
using equations (4.14), (4.16) and (4.17), one can obtain the variation of the orbital period
˙Tb = −
192
5
k5/3
c5
m1m2
(m1 + m2)1/3
Tb
2π
−7/2
1 +
73
24
e2
+
37
96
e4
. (4.19)
2
Modified Julian Date - is the integer assigned to a whole solar day in the Julian day count starting from
midnight Greenwich Mean Time, with Julian day number 0 assigned to the day starting at midnight on January
1, 4713 BC.
17
34. 0
−5
−10
−15
−20
−25
−30
−35
−40
Cumulativeperiodshift(s)
1975 1980 1985 1990 1995 2000 2005
Year
Figure 4.3: Cumulative period shift of pulsar B1913+16 since its discovery
Finally, by using the measured quantities in Table 4.1 and putting them in the previous
equation, we obtain the following result
˙Tb,GR = −(2.40242 ± 0.00002) × 10−12
s/s. (4.20)
In order to compare the measured ˙Tb with the theoretical value ˙Tb,GR we have to make
a small correction, ˙Tb,Gal, due to the relative acceleration between our Solar System and
the binary pulsar system, projected onto the line of sight. This correction will change the
measured ˙Tb as follows: ˙Tb,corrected = ˙Tb − ˙Tb,Gal. The real problem is that the correction term
depends on several quantities that are poorly known, such as the distance and proper motion
of the pulsar and the radius of the Sun’s galactic orbit. But, so far, the best available value
is ˙Tb,Gal = −(0.0128 ± 0.0050) × 10−12 s/s, thus ˙Tb,corrected = (2.4056 ± 0.0051) × 10−12 s/s.
With this correction, we can now compare the measured value ˙Tb,corrected with the theo-
retical one ˙Tb,GR
˙Tb,corrected
˙Tb,GR
= 1.0013 ± 0.0021. (4.21)
From this equation we can easily see that the measured value is consistent with the theoretical
value from the general prediction for the emission of GWs with an accuracy of (0.13±0.21)%.
In Fig. 4.3 we can the consecutive measurements of the cumulative period shift of the orbital
period Tb since the binary pulsar was discovered (red points), and the theoretical cumulative
period shift for a system emitting GWs (blue line). It is easy to see how accurate the
theoretical prediction coincides with the measurements made.
18
35. Fitted Parameters J0737-3039A J0737-3039B
Orbital period Tb (day) 0.10225156248(5)
Orbital eccentricity e 0.0877775(9)
Epoch of periastron T0 (MJD) 53155.9074280(2)
Advance of periastron ˙ω (deg/yr) 16.89947(68)
Argument of periastron ω (deg) 87.0331(8) 87.0331 + 180.0
Projected semi-major axis a sin i/c (s) 1.415032(1) 1.5161(16)
Gravitational redshift parameter γ (ms) 0.3856(26)
Shapiro delay3 parameter s = sin i 0.99974(39, +16)
Shapiro delay parameter r (µs) 6.21(33)
Orbital decay ˙Tb (10−12) (s/s) −1.252(17)
Total system mass mA + mB (M ) 2.58708(16)
Mass ratio R ≡ mA/mB 1.0714(11)
Stellar mass (M ) 1.3381(7) 1.2489(7)
Table 4.2: Measured and derived parameters of pulsars J0737-3039A and B. The number in
parentheses is the standard errors of the last digit(s). [10]
4.3 The Double Pulsar (J0737-3039)
Double Neutron Star (DNS) binaries systems like B1913+16 are extremely rare; only six
such systems are known so far, and one of the most important is J0737-3039, because it was
the first (and so far, the only) Double Pulsar Binary, i.e., it is the only binary system in which
the two bodies are pulsars. The DNS J0737-3039 was discovered in 2003 at the Australian
Parkes Observatory by an international team led by the radio astronomer Marta Burgay
during a high-latitude pulsar survey [9]. The advantages of this binary system compared to
the others are that the relativistic effects are more significant, the orbital plane of the binary
system is almost perfectly aligned with the line of sight to Earth, and the distance between
the system and the Earth is smaller than any other DNS binary system (we do not have to
do the correction due to the relative acceleration between our Solar System and the binary
system).
Due to all of these advantages, J0737-3039 is a better laboratory to test GR than B1913+16;
in fact more tests have been made with this binary system in 11 years than with B1913+16
in 39 years.
In Table 4.2 we can see that the advance of periastron is extremely big. For comparison,
Mercury, which is the planet of the Solar System that feels the most relativistic effects due to
the smaller distance to the Sun (and highly eccentric orbit), has an advance of periastron or
42.7 arcsec/century, or 1.186×10−4deg/yr. Thus, the advance of periastron of Mercury is one
hundred thousand times smaller than the advance of periastron of the DNS. Even when we
compare with the pulsar B1913+16, which has an advance of periastron of 4.226595 deg/yr,
the value of the advance of periastron of the DNS J0737-3039 is four times larger. Indeed, the
DNS J0737-3039 is the known system that displays, in general, the largest relativistic effects.
Curiously, however, for this system the orbital decay is smaller than that for the pulsar
B1913+16. This happens because of the small eccentricity. By analysing equation (4.19) one
3
The Shapiro delay is the extra time delay light experiences by travelling past a massive object due to
general relativistic time dilation. It can be described by just two variables, the range r and the shape s = sin i.
19
36. can see that the lower the eccentricity, the lower the orbital decay, and the minimum value
will be when the eccentricity is zero, that is, a circular orbit.
With the values from Table 4.2 and equation (4.19) we can arrive to the theoretical value
for the orbital decay: ˙Tb = −1.24787(13) × 10−12 (s/s). From here we can do the same
comparison that we did in the previous section to the pulsar B1913+16, and find
˙Tb,measured
˙Tb,GR
= 1.003 ± 0.0014. (4.22)
As before, we can see that the measured value is consistent with the theoretical value from
the GR with an accuracy of (0.3 ± 0.14)%. It is not as accuracy as the value predict for
the pulsar B1913+16; but we have to keep in mind that this DNS was found only 11 years
ago, while the pulsar B1913+16 was found 39 years ago and has been tracked since. Still,
scientists have already collected almost as much information about the DNS J0737-3039 as
for the pulsar B1913+16.
20
37. Chapter 5
Testing Alternative Theories of
Gravitational Waves
There are many theories of gravity proposed as alternatives to GR. Some of them are
extensions of GR, whereas others are radically different. Since GR has passed all of the
observational tests that have been devised so far, it is almost universally accepted by the
scientific community as the best available description of gravity. But there are hints, of both
theoretical and observational nature, that it is not the final word about the gravitational
field. Moreover, considering enlarged frameworks is always useful to understand how special
a given theory and its predictions are. Thus considering extensions/alternatives to GR and
their predictions will help us understand how special GR is.
One class of theories that has raised interest for almost 50 years goes by the name of
scalar-tensor theories of gravity. The original example is the Brans-Dicke Theory [11]. This
theory adds a scalar degree of freedom to the metric of GR. Consequently, GWs, besides the
two tensorial polarizations will have another (scalar) polarization. Moreover, in this class of
theories, matter is assumed to be universally coupled to a different metric: ˜gµν ≡ A2(ϕ)gµν,
where A(ϕ) is a non-vanishing function defining the matter-scalar coupling [12].
This theory has many predicted deviations from GR. But the one deviation that we
are more interested in is that scalar waves are also emitted by any binary system, thus
contributing to the observed variation of the orbital period. These scalar waves make a
great contribution in asymmetric systems, i.e. with large mass ratio q ≡ Mp/Mc, because
the main contribution comes from dipolar waves. One can prove that the variation of the
orbital period ˙Tb,D is proportional to k/c3 ∼ 10−35kg−1
s [12], which means that in this type
of system, the orbital decay will be caused mainly by the emission of dipolar GWs because
˙Tb,D >> ˙Tb,GR(∝∼ 10−84).
An asymmetric binary system that was recently found is the pulsar J1738+0333 [12],
which is composed of a pulsar of mass Mp = 1.46+0.06
−0.05M and a white dwarf of mass Mc =
0.181+0.008
−0.007M (where M represents, as before, one Solar mass). It has a very low eccentricity
e < 4 × 10−7 and has a orbital period of 8.5 hours. Measurements of this system yielded an
orbital decay of ˙Tb,Measured = (−25.9 ± 3.2) × 10−15 s/s. From these data and from equation
(4.19) we arrive to the theoretical value from GR: ˙Tb,GR = (−27.7+1.5
−1.9 × 10−15) s/s. This
means that the prediction made by GR fits the date within the error bars.
Since ˙Tb,D >> ˙Tb,GR, Brans-Dicke theory fails this test for a large region in its parameter
space, severely constraining the viability of this theory.
21
38. Although this result appears to invalidate the Brans-Dicke theory, there are many other
alternative theories of gravity that, to date, are still viable. For example there is the the-
ory of Modified Newtonian Dynamics (MoND) [13]. MoND is a theory that was proposed
by Moreghai Milgrom, in 1983, as a solution to the missing mass problem in extragalactic
astronomy [14]. This theory is non-relativistic, but, in 2004, Jacob Bekenstein proposed a
relativistic generalization of MoND [15], and this theory has not yet encountered any fatal
objection; this makes it, for now, a viable contender for an alternative gravity theory.
22
39. Chapter 6
The Future: Direct Detection of
Gravitational Waves
In this thesis we have discussed the theory and experimental evidence for GWs. One of
their properties is that they are only emitted in considerable amounts in highly relativistic
systems. That is they are hard to generate (in large amounts). Simultaneously, they are
extremely hard to detect, since they interact very weakly with matter. This difficulty actually
makes them even more interesting: GWs emitted in extraordinarily far away events will keep
essentially their original form even if they must travel the whole Universe to reach us, as their
interactions along the way will be small. Thus they will keep intact the pristine information
about the system that produced them.
Despite all the odds, in March of this year, an international group of scientists claimed
at the Harvard-Smithsonian Center for Astrophysics in Cambridge, Massachusetts, that they
had measured for the first time GWs [16], more precisely Primordial GWs, i.e. the GWs
caused by the exponential expansion in the early universe, in other words, Inflation [17].
This was great news to the science community because it supported the theory of Inflation
and, in a sense, also GR passed one more test. But, unfortunately, what the Harvard team
has detected may not be Primordial GWs, but “radio loops” caused by dust grains enriched
by metallic iron or ferromagnetic molecules [18].
There are currently various ongoing and proposed experiments to directly detect GWs.
The most advanced systems are the ground based interferometers, such as Advanced Laser
Interferometer Gravitational-Wave Observatory (aLIGO) [19] and Virgo [20], which should
be doing science runs starting next year. On a different front, the European Spatial Agency
(ESA) is currently developing a project called Evolved Laser Interferometer Space Antenna
(eLISA) [21]. It was originally called Laser Interferometer Space Antenna (LISA) and it was
being developed by National Aeronautics and Space Administration (NASA) and ESA, but
in 2011, NASA announced that it would be unable to continue this partnership with ESA,
due to funding limitations.
The main objective of both aLIGO, Virgo and eLISA is to detect and measure GWs
produced by astrophysical systems, like binary systems and black holes. The basic working
principle for these interferometers, is that a passing GW will produce a stretching/squeezing
in orthogonal arms of the interferometer, and such deformations can be very accurately mea-
sured. Yet another direction that may yield results is that of the so called pulsar timing
arrays (PTA). The basic idea is then to use the continuous signals received from the network
23
40. of known pulsars as a ‘many arms’ interferometer, wherein passing GWs will leave imprints
that can be statistically recovered.
Let us close by recalling that in 2015 GR will be celebrating 100 years. It is but a small
demonstration of the richness of this theory that one century after its proposal we are still
trying to extract all observational consequences from it. GWs are one of the extraordinary
predictions of GR and their direct observation will open a new field of research: GW astro-
physics. But more importantly it will open a completely new window to the Universe.
24
42. Appendix A
Computation of the
Landau-Lifschitz Pseudotensor
We start by expanding the expression (3.21) for the LL pseudotensor as follows:
16πk
c4
tik
=
1
(−g)
∂l∂m(−g) gik
glm
− gil
gkm
(A.1)
+
1
(−g)
∂l(−g) ∂m gik
glm
− gil
gkm
(A.2)
+
1
(−g)
∂m(−g) ∂l gik
glm
− gil
gkm
(A.3)
+ ∂l∂m gik
glm
− gil
gkm
(A.4)
+ gik
glm
− 2gim
gkl
∂pΓp
lm (A.5)
− gik
glm
− 2gim
gkl
∂mΓp
pl (A.6)
+ gik
glm
− 2gim
gkl
(Γp
npΓn
lm − Γp
nmΓn
pl) (A.7)
We delineate the structure of the pseudotensor in this fashion so that we can highlight those
lines with second order derivatives of the metric; the pseudotensor only contains first deriva-
tives of the metric and so we expect the second order contributions, (A.1), (A.4), (A.5) and
(A.6), to vanish.
By expanding each of these four lines, we will get, for each line, terms of first and sec-
ond order in derivatives of the metric. We then perform a simple transformation replacing
derivatives of the covariant form of the metric with derivatives of its contravariant form, as
follows
gae
gbf
∂c∂dgef + ∂c∂dgab
= gef ∂dgae
∂cgbf
+ gef ∂cgae
∂dgbf
. (A.8)
26
43. We then obtain
1
(−g)
∂l∂m(−g) gik
glm
− gil
gkm
+ ∂l∂m gik
glm
− gil
gkm
+ gik
glm
− 2gim
gkl
∂pΓp
lm − gik
glm
− 2gim
gkl
∂mΓp
pl
= gik
gqm ∂pglm
∂lgpq
+ ∂lglm
∂pgpq
+ gpq
glm ∂qgil
∂pgkm
+ ∂pgil
∂qgkm
− gil
gqm ∂qgkm
∂lgpq
+ ∂lgkm
∂pgpq
− gkm
glq ∂pgil
∂mgpq
+ ∂mgil
∂pgpq
. (A.9)
From here we see that all the second order terms vanished, and what remains are some
residual first order terms.
Our goal now is to take all the remaining first order terms and try to simplify them.
For that we will combine these last four lines with the first order terms that remained from
lines (A.1), (A.4), (A.5) and (A.6), and with the rest of our initial expression for the LL
pseudotensor (lines (A.2), (A.3) and (A.7)). By analysing each term carefully and by keeping
in mind that our ultimate goal is to arrive at equation (3.22), we arrive at the following
expression
16πk
c4
tik
= gik
glm ∂pgnl
∂ngpm
+ 2gpn
glm ∂ngil
∂pgkm
− gil
gmn ∂lgpm
∂pgkn
− gkl
gmn ∂lgpm
∂pgin
+
2
(−g)
∂l gik√
−g ∂m glm√
−g +
2
(−g)
∂l
√
−g glm
∂mgik
+ gik
∂mglm
+
2
(−g)
gik
glm
∂l
√
−g ∂m
√
−g +
1
2
gik
glm
∂mgpq
∂lgpq
−
1
(−g)
∂l gil√
−g ∂m gkm√
−g −
1
(−g)
∂m gil√
−g ∂l gkm√
−g
−
1
√
−g
∂l
√
−g gkm
∂mgil
+ gil
∂mgkm
∂m
√
−g gil
∂lgkm
+ gkm
∂lgil
−
2
(−g)
gil
gkm
∂l
√
−g ∂m
√
−g
+ gik
glm
− 2gim
gkl
Γp
npΓn
lm − Γp
nmΓn
pl −
1
2
∂pgpq
∂qgml . (A.10)
In the following we denote
˜gik
≡
√
−ggik
, ˜gik ≡
1
√
−g
gik. (A.11)
27
44. We shall now rewrite our expressions for the LL pseudotensor in terms of these 2 quantities
since equation (3.22) is written in this way.
16πk
c4
tik
= −
1
(−g)
−∂l˜gik
∂m˜glm
+ ∂l˜gil
∂m˜gkm
−
1
2
˜gik
˜glm ∂p˜gnl
∂m˜gpm
(A.12)
+ ˜gil
˜gmn ∂p˜gkn
∂l˜gmp
+ ˜gkl
˜gmn ∂p˜gin
∂l˜gmp
− ˜gpn
˜glm ∂n˜gil
∂p˜gkm
(A.13)
+
1
2(−g)
˜gik
˜glm ∂p˜gnl
∂m˜gpm
+
1
(−g)
˜gpn
˜glm ∂ngil
∂pgkm
(A.14)
+
1
(−g)
∂lgik
∂m˜glm
(A.15)
−
2
√
−g
glm
∂l
√
−g ∂mgik
−
1
(−g)
gik
glm
∂l
√
−g ∂m
√
−g (A.16)
+
1
2
gik
glm
∂mgpq
∂lgpq (A.17)
−
1
(−g)
∂m˜gil
∂l˜gkm
(A.18)
+ gik
glm
− 2gim
gkl
Γp
npΓn
lm − Γp
nmΓn
pl −
1
2
∂pgpq
∂qgml . (A.19)
Where the two first lines, A.12 and A.13, are already in their final form.
We now have to expand the connection in order to arrive at the final form of the LL
pseudotensor. Recall that the connection can be expressed in terms of the metric as Γa
bc =
1/2 gad (∂cgbd + ∂bgcd − ∂dgbc). Using this expression in line A.19 and simplifying we finally
arrive at the desired form of the LL pseudotensor
(−g)tik
=
c4
16πk
˜gik
,l ˜glm
,m −˜gil
,l ˜gkm
,m +
1
2
˜gik
˜glm˜gln
,p ˜gpm
,n −
− ˜gil
˜gmn˜gkn
,p ˜gmp
,l +˜gkl
˜gmn˜gin
,p ˜gmp
,l + ˜gnp
˜glm˜gil
,n ˜gkm
,p +
+
1
8
2˜gil
˜gkm
− ˜gik
˜glm
(2˜gnp˜gqr − ˜gpq˜gnr) ˜gnr
,l ˜gpq
,m , (A.20)
where the index ,i ≡ ∂i denotes a simple differentiation with respect to xi.
28
45. Appendix B
Derivation of the Angular Velocity
of any Kepler Motion
We begin by writing down the Lagrangian of the binary system
L =
1
2
m1 ˙x2
1 + ˙y2
1 +
1
2
m2 ˙x2
2 + ˙y2
2 +
km1m2
d1 + d2
. (B.1)
But this coordinate system is complicated to work with. So, as illustrated in Fig. B.1, we
can adopt a new coordinate system
x1 = d1 cos ψ
y1 = d1 sin ψ
,
x2 = −d2 cos ψ
y2 = −d2 sin ψ
. (B.2)
Making use of equation (4.7), we arrive at the following equations
˙x2
1 + ˙y2
1 =
m2
m1 + m2
2
˙d2
+ d2 ˙ψ2
, ˙x2
2 + ˙y2
2 =
m1
m1 + m2
2
˙d2
+ d2 ˙ψ2
. (B.3)
Hence the Lagrangian for the binary system, in these coordinates, is
L =
1
2
µ ˙d2
+ d2 ˙ψ2
+
km1m2
d
, (B.4)
where µ denotes, as before, the reduced mass.
We can calculate the generalized momentum of the angular component Pψ, in order to
obtain a simple equation that expresses the dependence of the angular velocity on the angular
momentum
˙ψ =
j
d2
, (B.5)
where j is the specific relative angular momentum, j = Pψ/µ.
The last thing we have to calculate is the angular momentum. We already know that the
orbital motion of each body will be elliptical, so their equations of motion should yield the
equation of an ellipse. Applying the Lagrange Equations for the coordinate d, we obtain
¨d =
j2
d3
−
k(m1 + m2)
d2
. (B.6)
29
46. y
x
d1
d2
m1
m2
ψ
Figure B.1: Coordinate system of a binary system
In order to solve this second order differential equation, we make a change of variable of the
type
d(t) → u(ψ), u =
1
d
. (B.7)
Performing the second derivative in this new variable, and substituting the results in equation
(B.6), we can obtain:
d2u
dψ2
+ u =
k(m1 + m2)
j2
. (B.8)
The solution of this differential equation is the following
u(ψ) = A cos(ψ − ψ0) +
k(m1 + m2)
j2
, (B.9)
where A and ψ0 are constants of integration, which can be interpreted, respectively, as the
amplitude of the oscillatory motion and the initial phase of the same.
Now that we have the solution, we can return to the origin coordinate d, in terms of which
d(ψ) =
1
u(ψ)
=
j2
k(m1+m2)
1 + j2A
k(m1+m2) cos(ψ − ψ0)
. (B.10)
As noted earlier, this solution represents an ellipse written in polar form. Through a com-
parison with equation (4.8), we arrive at the following equalities
a(1 − e2) =
j2
k(m1 + m2)
e =
j2A
k(m1 + m2)
. (B.11)
The first of these equations can be rewritten as an expression for j
j = a(1 − e2
)k(m1 + m2)
1/2
. (B.12)
Thus, through equation (B.5), we finally obtain the angular velocity of any Keplerian motion
˙ψ =
a(1 − e2)k(m1 + m2)
1/2
d2
. (B.13)
30
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