An Introduction to
Gravitational Waves
Adriana Marina Cardini
November 24, 2017
Bishop’s University

 A Little General Relativity
Gravitational Waves
LIGO and Virgo
Sources of Gravitational Waves

 A Little General Relativity
Manifold is one of the most fundamental concepts in mathematics and
physics. We are all used to the properties of n-dimensional Euclidean
space, 𝑹 𝒏
= 𝑥1
, 𝑥2
, … … . , 𝑥 𝑛
, often equipped with a flat positive-definite
metric. Mathematicians have developed calculus in 𝑹 𝒏
(differentiation,
integration, etc.). However, there are other spaces which we think of as
“curved” or topologically complicated on which we would like to perform
analogous operations.
To address this problem we invent the notion of a manifold, which
corresponds to a space that may be curved and has a complicated topology,
but in local regions looks like 𝑹 𝒏.
– What is a Manifold?

The Metric
The metric tensor is an important object in curved space that it is given by the
symbol 𝒈 𝝁𝝂
• This determines the “shortest distance” between two points.
• This is a symmetric tensor of order 2, that is 𝑔 𝜇𝜈 = 𝑔 𝜈𝜇
• This is taken to be nondegenerate, meaning that the determinant 𝑔 = 𝒈 𝛍𝝂
does not vanish, so this allows us to define the inverse metric 𝑔 𝜇𝜈 as
𝑔 𝜇𝜈 𝑔 𝜈𝜎 = 𝑔 𝜆𝜎 𝑔 𝜆𝜇=𝛿 𝜎
𝜇
• The symmetry of 𝑔μ𝜈 implies that 𝑔 𝜇𝜈
is also symmetric.

Line Element
𝑑𝑠2
= 𝑔 𝜇𝜈 𝑑𝑥 𝜇
𝑑𝑥 𝜈
Where 𝑑𝑥 𝜇
is an infinitesimal coordinate displacement
This is an invariant, for example in R3
o Cartesian coordinates 𝑑𝑠2
= 𝑑𝑥2
+ 𝑑𝑦2
+ 𝑑𝑧2
o Cylindrical coordinates 𝑑𝑠2 = 𝑑𝜌2 + 𝜌2 𝑑𝜑2 + 𝑑𝑧2
o Spherical coordinates 𝑑𝑠2 = 𝑑𝑟2 + 𝑟2 𝑑𝜃2 + 𝑟2 𝑠𝑖𝑛𝜃2 𝑑𝜙2
o Toroidal coordinates 𝑑𝑠2
= (𝑎 + 𝑏𝑐𝑜𝑠𝜑)2
𝑑𝜗2
+ 𝑏2
𝑑𝜑2

Curvature
It depends on the metric, which defines the geometry of a manifold.
All the ways in which curvature manifests itself depend on something called a
connection which gives us a way of relating vectors in the tangent space of nearby
points.
There is a unique connection that we can construct from the metric called the
Christoffel symbol, given by
Γ𝜇𝜈
𝜆
=
1
2
𝑔 𝜆𝜎
𝜕𝜇 𝑔 𝜈𝜎 + 𝜕𝜈 𝑔 𝜎𝜇 − 𝜕 𝜎 𝑔 𝜇𝜈
where: 𝜕 𝛼 𝑔 𝛽𝛾 =
𝜕𝑔 𝛽𝛾
𝜕𝑥 𝛼
The notation makes Γ𝜇𝜈
𝜆
look like a tensor, but it is not; this is an “object” or “symbol”.
And this is symmetric ⇒ Γ𝜇𝜈
𝜆
= Γ𝜈𝜇
𝜆

Riemann Tensor
This technical expression of curvature is contained in the Riemann tensor
R 𝜌
𝜎𝜇𝜈 = 𝜕𝜇Γ𝜈𝜎
𝜌
− 𝜕𝜈Γ𝜇𝜎
𝜌
+ Γ𝜇𝜆
𝜌
Γ𝜈𝜎
𝜆 − Γ𝜈𝜆
𝜌
Γ𝜇𝜎
𝜆
R 𝜌𝜎𝜇𝜈 = 𝑔 𝜌𝜆 R 𝜆
𝜎𝜇𝜈
R 𝜌𝜎𝜇𝜈 =
1
2
𝜕𝜇 𝜕 𝜎 𝑔 𝜌𝜈 − 𝜕𝜌 𝜕𝜇 𝑔 𝜎𝜈 − 𝜕𝜈 𝜕 𝜎 𝑔 𝜌𝜇 + 𝜕𝜌 𝜕𝜈 𝑔 𝜎𝜇
where: 𝜕𝜇 𝜕 𝜎 𝑔 𝜌𝜈 =
𝜕2 𝑔 𝜌𝜈
𝜕𝑥 𝜎 𝜕𝑥 𝜇
Everything we want to know about the curvature of a manifold is given to us by the
Riemann tensor; and it will vanish if and only if the metric is flat.

Ricci Tensor and Ricci Scalar
o If we make a contraction of indices, 𝜌 = 𝜈, we obtain 𝑹 𝝆
𝝈𝝁𝝆 = 𝑹 𝝈𝝁 which is
called Ricci tensor.
o If we rise an index, 𝒈 𝝈𝜶 𝑹 𝜶𝝁 = 𝑹 𝝁,
𝝈 we obtain a mixed system.
o We again make other contraction of indices,  = , we have 𝑹 𝝁,
𝝁
= 𝑹 which is
called Ricci scalar (or scalar curvature) and it is invariant.
Finally, with all these elements, we can present the Einstein equation.

Einstein’ Equation
where
 𝑮 𝝁𝝂  is the Einstein tensor
 𝑹 𝝁𝝂  is the Ricci tensor
 R  is the Ricci scalar
 𝒈 𝝁𝝂  is the metric
 𝑮  is the gravitational constant of Newton
 c  is the speed of light
 𝑻 𝝁𝝂  is the energy-momentum tensor
The term 8𝜋GT𝜇𝜆 generalizes the density term in the Poisson equation for the
Newtonian potential Φ = −
GM
𝑟
.
𝛻2
Φ = 4πG𝜌
𝑮 𝝁𝝂 = 𝑹 𝝁𝝂 −
𝟏
𝟐
𝒈 𝝁𝝂 𝑹 =
𝟖𝝅𝑮𝑻 𝝁𝝂
𝒄 𝟒

Minkowski Space
𝑑𝑠2
=  𝝁𝝂 𝑑𝑥 𝜇
𝑑𝑥 𝜈
where
 𝑥 𝜇
:
𝑥0 = 𝑐𝑡
𝑥1
= 𝑥
𝑥2
= 𝑦
𝑥3
= 𝑧
  𝜇𝜈 is the Minkowski metric
 𝜇𝜈 =
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
 So, we have a spatiotemporal metric
𝑑𝑠2
= −𝑐2
𝑑𝑡2
+ 𝑑𝑥2
+ 𝑑𝑦2
+ 𝑑𝑧2
 this is flat space, so 𝒈 𝝁𝝂  𝝁𝝂

 Gravitational Wave
This is a ripple in the curvature of spacetime, which propagates with the
speed of light c.
In the real universe, gravitational waves propagate on the background of a
large-scale, slowly-changing spacetime curvature, created by the universe’s
lumpy cosmological distribution of matter (stars, galaxies, clusters, …).
 What is this?

Linearized Einstein Equations for Weak Gravitational Fields
We assume weak gravitational field
The weakness of the gravitational field allows us to decompose the metric into
Minkowski metric plus a small perturbation in Cartesian coordinates:
𝑔 𝜇𝜈 = 𝜂 𝜇𝜈 + ℎ 𝜇𝜈 ℎ 𝜇𝜈 ≪ 1
This ℎ 𝜇𝜈 being so small, it allows us to ignore anything that is higher than first
order in this quantity.
Far from sources, and if r is the radial coordinate, the space must be
asymptotically flat, that is
lim
𝑟→∞
ℎ 𝜇𝜈 = 0 ∀𝜇, 𝜈

We want to find the equation of motion obeyed by the perturbations ℎ 𝜇𝜈, which
comes by examining Einstein’s equation to first order.
o We begin with the Christoffel symbols
Γ𝜇𝜈
𝜆
=
1
2
𝑔 𝜆𝜎
𝜕𝜇 𝑔 𝜈𝜎 + 𝜕𝜈 𝑔 𝜎𝜇 − 𝜕 𝜎 𝑔 𝜇𝜈
Γ𝜇𝜈
𝜆
=
1
2
(𝜂 𝜆𝜎
−ℎ 𝜆𝜎
) 𝜕𝜇(𝜂 𝜈𝜎+ℎ 𝜈𝜎) + 𝜕𝜈 𝜂 𝜎𝜇 + ℎ 𝜎𝜇 − 𝜕 𝜎(𝜂 𝜇𝜈+ℎ 𝜇𝜈)
But, 𝜂 𝜇𝜈 are all constants, and to first order in h, we have
Γ𝜇𝜈
𝜆
≈
1
2
𝜂 𝜆𝜎
𝜕𝜇ℎ 𝜈𝜎 + 𝜕𝜈ℎ 𝜎𝜇 − 𝜕 𝜎ℎ 𝜇𝜈
o The Riemann tensor becomes
R 𝜇𝜈𝜌𝜎 =
1
2
𝜕𝜌 𝜕𝜈ℎ 𝜇𝜎 + 𝜕 𝜎 𝜕𝜇ℎ 𝜈𝜌 − 𝜕 𝜎 𝜕𝜈ℎ 𝜇𝜌 − 𝜕𝜌 𝜕𝜇ℎ 𝜈𝜎

R 𝜆
𝜈𝜌𝜎 = 𝑔 𝜆𝜇
R 𝜇𝜈𝜌𝜎 ≈ 𝜂 𝜆𝜇
R 𝜇𝜈𝜌𝜎
R 𝜆
𝜈𝜌𝜎 ≈
1
2
𝜂 𝜆𝜇
𝜕𝜈 𝜕 𝜌ℎ 𝜇𝜎 − 𝜕𝜇 𝜕𝜌ℎ 𝜈𝜎 − 𝜕𝜈 𝜕 𝜎ℎ 𝜇𝜌 + 𝜕𝜇 𝜕 𝜎ℎ 𝜈𝜌
R 𝜆
𝜈𝜌𝜎 ≈
1
2
𝜂 𝜆𝜇
𝜕𝜈 𝜕𝜌ℎ 𝜇𝜎 − 𝜂 𝜆𝜇
𝜕𝜇 𝜕𝜌ℎ 𝜈𝜎 − 𝜂 𝜆𝜇
𝜕𝜈 𝜕 𝜎ℎ 𝜇𝜌 + 𝜂 𝜆𝜇
𝜕𝜇 𝜕 𝜎ℎ 𝜈𝜌
Doing the contraction  = 
R 𝜆
𝜈𝜌𝜆 = R 𝜈𝜌 ≈
1
2
𝜂 𝜆𝜇 𝜕𝜆 𝜕𝜌ℎ 𝜇𝜆 − 𝜂 𝜆𝜇 𝜕𝜇 𝜕𝜌ℎ 𝜈𝜆 − 𝜂 𝜆𝜇 𝜕𝜈 𝜕𝜆ℎ 𝜇𝜌 + 𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆
R 𝜈𝜌 ≈
1
2
𝜼 𝝀𝝁 𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 + 𝜕𝜆 𝜕𝜌ℎ 𝜇
𝜇
− 𝜕𝜇 𝜕𝜌ℎ 𝜈
𝜇
− 𝜕𝜈 𝜕𝜆ℎ 𝜌
𝜆
R 𝜈𝜌 ≈
𝟏
𝟐
𝜼 𝝀𝝁
𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 +
1
2
𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜇 𝜕𝜌ℎ 𝜈
𝜇
− 𝜕𝜈 𝜕𝜆ℎ 𝜌
𝜆
where
ℎ = ℎ 𝜇
𝜇
= 𝑖=1
4
ℎ𝑖
𝑖

2R 𝜈𝜌 ≈ 𝜼 𝝀𝝁
𝝏 𝝁 𝝏 𝝀 𝒉 𝝂𝝆 +
1
2
𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜇 𝜕𝜌ℎ 𝜈
𝜇
+
1
2
𝜕𝜆 𝜕𝜌ℎ − 𝜕𝜈 𝜕𝜆ℎ 𝜌
𝜆
2R 𝜈𝜌 ≈ 𝜼 𝝂𝝁
𝝏 𝝁 𝝏 𝝂 𝒉 𝝂𝝆 + 𝜕𝜌
1
2
𝜕𝜈ℎ − 𝜕𝜇ℎ 𝜈
𝜇
+ 𝜕𝜈
1
2
𝜕𝜌ℎ − 𝜕𝜆ℎ 𝜌
𝜆
2R 𝜈𝜌 ≈ 𝜼 𝝂𝝁
𝝏 𝝁 𝝏 𝝂 𝒉 𝝂𝝆 + 𝜕𝜌 𝜕𝜇
1
2
𝛿 𝜈
𝜇
ℎ − ℎ 𝜈
𝜇
+ 𝜕𝜈 𝜕𝜆
1
2
𝛿 𝜌
𝜆
ℎ − ℎ 𝜌
𝜆
Imposing
𝜕𝜇
1
2
𝛿 𝜈
𝜇
ℎ − ℎ 𝜈
𝜇
≡ 0
𝜕𝜆
1
2
𝛿 𝜌
𝜆
ℎ − ℎ 𝜌
𝜆
≡ 0
 Hilbert – Einstein gauge
Doing = 

o Then the Ricci tensor reduces to
where  is the d’Alembertian, and for a flat space is
 = −
1
𝑐2 𝜕𝑡
2
+ 𝜕 𝑥
2
+ 𝜕 𝑦
2
+ 𝜕𝑧
2
= −
1
𝑐2 𝜕𝑡
2
+ 𝛻2
o We calculate the Ricci scalar from the mixed tensor
R 𝜈
𝜇
= 𝑔 𝜇𝜌
R 𝜈𝜌 ≈ 𝑔 𝜇𝜌 1
2
𝜂 𝜈𝜇
𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌 ≈ 𝜂 𝜇𝜌 1
2
𝜂 𝜈𝜇
𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌
R 𝜈
𝜇
≈
1
2
𝜂 𝜈𝜇 𝜕𝜇 𝜕𝜈ℎ 𝜈𝜇 ≈
1
2
𝜕𝜇 𝜕𝜈ℎ 𝜇
𝜇
≈
1
2
𝜕𝜇 𝜕𝜈ℎ
Contracting =  we obtain
R 𝜇
𝜇
= R ≈
1
2
𝜕𝜇 𝜕𝜇ℎ =
1
2
ℎ
𝑅 𝜈𝜌 ≈
1
2
𝜂 𝜈𝜇
𝜕𝜇 𝜕𝜈ℎ 𝜈𝜌 =
1
2
ℎ 𝜇𝜎

o Putting it all together, we obtain the Einstein tensor in vacuo.
𝑅 𝜇𝜈 −
1
2
𝑔 𝜇𝜈 𝑅 = 0 ⇒ 𝑅 𝜇𝜈 −
1
2
𝜂 𝜇𝜈 + ℎ 𝜇𝜈 𝑅 = 0
⇒
1
2
ℎ 𝜇𝜈 −
1
2
𝜂 𝜇𝜈 + ℎ 𝜇𝜈
1
2
ℎ = 0
Again, to first order,
1
2
ℎ 𝜇𝜈 −
1
4
𝜂 𝜇𝜈ℎ = 0 ⇒
1
2
ℎ 𝜇𝜈 −
1
2
𝜂 𝜇𝜈ℎ = 0
o Finally, we obtain the wave equation
 𝒉 𝝁𝝂 −
𝟏
𝟐
𝜼 𝝁𝝂 𝒉 = 𝟎
Thus, the quantities 𝒉 𝝁𝝂 −
𝟏
𝟐
𝜼 𝝁𝝂 𝒉 propagate in space-time with the speed
of light in vacuo.

(a) and (b) transverse, spin 2 (tensor)
(c) transverse, spin 0 (scalar)
(d) purely longitudinal, spin 0 (scalar)
(e) and (f) longitudinal transverse, spin 1 (vector)
Polarization
This describes how spacetime is distorted in
three different spatial directions when a
gravitational wave propagates. General Relativity
predicts these waves are transversal. This means
that they stretch and compress space-time in the
plane perpendicular to the direction of
propagation.
The allowed distortions are only of two types,
called polarizations (a) "+" ("more"), and (b) "x"
("cross"). The space-time distortion occurs in the
plane perpendicular to the direction in which the
gravitational wave propagates (axis z).
(c) again shows a transverse polarization, while
(d), (e) and (f) illustrate distortions that
propagate in one direction (indicated by the
arrow) that is in the same plane as the space-
time distortion.

 LIGO and Virgo
The LIGO and Virgo detectors are giant Michelson-type laser interferometers with arms of
4 and 3 kilometers respectively, both have been designed to detect gravitational waves.
o LIGO  Laser Interferometer Gravitational-Wave Observatory
Hanford, Washington, USA

o Virgo  is named for the Virgo Cluster of about 1,500 galaxies in the Virgo
constellation.
Cascina, Pisa, Italy

The Observed Signal GW170814
GW170814 was a gravitational wave signal from two merging black
holes, detected by the LIGO and Virgo observatories on 14 August 2017
at 10:30:43 UTC. This was the fourth confirmed event
after GW150914, GW151226 and GW170104. It was the first binary
black hole merger detected by LIGO and Virgo together.
The detected gravitational waves, "wrinkles in spacetime", were
emitted during the final moments of the fusion of two black holes with
masses of approximately 30.5 𝑴⨀ and 25.3 𝑴⨀ and located 540 Mpc
(1.8x102 ly) away. The resulting spinning black hole is about 53.2 𝑴⨀.
This means that approximately 2.6 𝑴⨀ were converted into energy in
the form of gravitational waves during coalescence.

The blue area corresponds to the fast localization based only on data from
the two LIGO detectors.
Adding Virgo leads to the orange area, which is an order of magnitude
smaller, 100 𝑑𝑒𝑔2
compared to 1160 𝑑𝑒𝑔2
.
The green region is the result of the complete analysis of parameter
estimation using the three detectors: an area of 60 𝑑𝑒𝑔2
.
The right part of the figure compares the probability distributions for the
luminosity distance of the source.
Location of the source of GW170814 in the sky.

What detected each of them?
Signal-to-noise ratio as a function of time. The peak occurs
at different times in each detector because the gravitational
wave propagates at the speed of light. GW170814 arrived
first at LIGO-Livingston, 8 ms later it arrived at LIGO-
Hanford, and 6 ms later it arrived at Virgo.
Temporal series of amplitude with the best wave models
selected by the adapted filtering technique (black curves)
and search methods without models (gray bands)
superimposed
Time-frequency representation
of the amplitude of the signal.

 Sources of Gravitational Waves
In general, any acceleration that is not spherically or cylindrically symmetric
(with varying 4-pole moment) will produce a gravitational wave. For example:
• Supernovae  This explosion will produce gravitational waves if the mass is
not ejected in a spherically symmetric way.
• Spinning star  A perfectly spherical star will not produce a gravitational
wave, but a lumpy star will.
The gravitational waves that modern detectors are sensitive to would be in the
audible frequency range if they were sound waves. In that sense, these
detectors can be thought of as ‘gravitational wave radios.’

Continuous Gravitational Waves
They are produced by systems that have a constant
and well-defined frequency. Examples of these are
binary star or black hole systems orbiting each
other or a single star rotating rapidly about its axis.
The sound these gravitational waves would
produce is a continuous tone since the frequency
of the gravitational wave is nearly constant.

Inspiral Gravitational Waves
They are generated during the end-of-life stage of binary systems where the two
objects merge into one. These systems are usually two neutron stars, two black
holes, or a neutron star and a black hole whose orbits have degraded to the point
that the two masses are about to coalesce.

Burst Gravitational Waves
They come from short-duration unknown or
unanticipated sources.
• Gamma ray bursts (GRBs)  gamma-ray bursts
(GRBs) are extremely energetic explosions that have
been observed in distant galaxies.
• Supernovae (SN)  it is a dramatic and catastrophic
destruction which is marked by one final titanic
explosion.

Stochastic Gravitational Waves
They are the relic gravitational waves from the
early evolution of the universe.
Much like the Cosmic Micro-wave Background
(CMB), which is the leftover light from the Big
Bang, these gravitational waves arise from a
large number of random, independent events
combining to create a cosmic gravitational wave
background.
If these gravitational waves
truly originated in the Big
Bang, these waves will have
been stretched as the
universe expanded and
they can tell us about the
very beginning of the
universe.

Black Hole Hunter

Thankyou
foryourpatience