1) The document summarizes the 100-year history of gravitational wave detection, from Einstein's theory of general relativity predicting their existence to the recent direct detection by the LIGO experiment. 2) It describes how LIGO uses laser interferometry to extremely sensitively measure tiny distortions in spacetime caused by passing gravitational waves. 3) The first detected gravitational waves in 2015 matched predictions for the inspiral and merger of two black holes, with the signal analyzed to determine properties of the black holes such as their masses.

1. Gravitational Waves: A 100 Year Success Story
James G. O’Brien
Department of Sciences
Wentworth Institute of Technology
Boston, MA 02115, USA
1
obrienj10@wit.edu
Abstract
Over the course of the few years, the world of Physics has been alive with news of groundbreaking
discoveries. Many of these recent advances have been crucial to the confirmation of results that
are at the cornerstone of modern physics. These triumphs are a culmination of the efforts of
theorists and experimentalists alike. The spoils of recent physics however has in many cases been a
testament to the patience and persistence of science. For example, in 2013, after being postulated
over fifty years ago by Peter Higgs and company [1], the Higgs Boson was detected at the Large
Hardron Collider (LHC) [2]. In 2008, Gravity Probe B returned to Earth with results to further
verify the geodetic curvature effect of Einstein, as well as provide a first measurement of the mind
blowing gravitational frame dragging effect [3]. Both of these ideas were postulated nearly 100
years ago, and the first idea for using a space-craft to measure the curvature and frame dragging
effect originated over 60 years ago by the great Leonard Schiff. Today, physicists find ourselves once
again celebrating victory, with the first detection of Gravitational Waves at the Laser Interferometer
Gravitational-Wave Observatory, LIGO. This was yet another discovery, a hundred years in the
making. In this article, we will review briefly the history of Gravitational Waves, the methods used
at LIGO for detection and take a glimpse to the future of gravitational wave physics.
1

2. I. GRAVITATIONAL WAVES
In order to appreciate the magnitude of the recent discovery, we should highlight the basics
of gravitational waves. When Einstein revolutionized our understanding of gravity in 1915
with General Relativity (GR), he also forced us to expand our comprehension of the geometry
of space and time itself. GR was the first metric theory of gravity, a complete description of
gravitational phenomena resulting from the presence of mass, occupying a region of space-
time, which in turn causes the surrounding geometry to become non-Euclidian. This is easily
visualized by a small marble resting on a rubber sheet causing it to curve. In the years that
followed, this curvature effect was proven due to the deflection of star light experiment.
Since light follows the geodesics of the system in which it propagates, the straight lines
around a large object such as the sun would be curved according to GR. Hence, light should
be deflected from its original path. When this was confirmed, the Einstein description was
celebrated as the correct and complete interpretation. This was a huge deviation from the
Newtonian description of gravity since his famous equation, Fg = GM1M2
r2 ,should not couple
to massless light. Moreover, one of the motivating factors of Einstein’s special relativity
was the preservation of causality. Since it was a known shortcoming of Newton’s theory
that gravity required an instantaneous force, this is immediately at odds with causality. As
it turns out, Einstein had the answer to this conundrum as well, namely that a geometric
theory of gravity admits wave like solutions which travel at the speed of light. These
theorized waves, would become known as the elusive gravitational waves, and for Einstein,
the similarities between electromagnetism and gravity have come full circle.
Although the idea of gravity communicating via a wave is a simple analogous effect to
electrodynamics communicating via waves, the physical implications are quite different. In
order to quantify the physicality of such objects, let us turn to some basic GR. As a metric
theory of gravity, GR relates the gravitational field to a set of space-time dependent (xα
)
co-efficient functions gµν(xα
), called the metric tensor. This set of 16 quantities (as the
indices are dummies that can range from 0 to 3) fully describe the point by point geometry
of a given system in a four dimensional space-time. Since the metric tensor coefficients are
functions, they also hold the information for if and how the geometry of space time changes
in a system. Thus, according to Einstein, knowing the metric tensor of the system completely
describes the geometry and thus describes what Newton previously called gravity. So let us
2

3. assume that we have a gravitational source in space time. This source will produce a metric
to describe the gravity. As we do in the Maxwell equations, let us then look at the far field
effect of the gravity, namely the metric tensor far from the source. In this regime, the metric
tensor can take the form gµν = ηµν + hµν, where ηµν is the background geometry and hµν is
the weak perturbation to the geometry due to the source. The Einstein equations so solve
for the metric tensor are given by Rµν − 1
2
gµνRα
α = 8πG
c2 Tµν, where Rµν is the Ricci Tensor
(a quantity which contains linear combinations of first and second order derivatives of the
metric tensor), Rα
α is the Ricci scalar of the system (the trace of the Ricci Tensor) and Tµν
is the Energy Momentum tensor of the system. Upon using the expanded version of the
metric tensor in the far field, the left hand side of Einstein’s equations simplify to:
Rµν ≈
1
2
∂λ∂λ
hµν −
∂2
∂xλ∂xµ
hλ
ν −
∂2
∂xλ∂xν
hλ
µ +
∂2
∂xµ∂xν
hλ
λ (1)
We can make this simpler as well by eliminating the Ricci scalar from the system by writing
Vµν = Tµν − 1
2
ηµνTλ
λ . In GR, we maintain energy and momentum conservation in the
convenient form that ∂µTµ
ν = 0. Similiarly, if we are working in the weak field, then we get
a similar relation for the derivatives of h (to first order), viz: ∂µhµ
ν = 1
2
∂νhµ
ν . Using these in
the original formula for the Einstein equations in the far field, we obtain:
∂λ∂λ
hµν = −
16πG
c2
Vµν (2)
which if we then work in the source free zone, Vµν = 0, the Einstein equations are ∂λ∂λ
hµν =
0. Upon remembering the Einstein summation convention, we realize that we staring at the
wave equation for hµν,
∂λ∂λ
hµν =
1
c2
∂2
∂t2
− 2
hµν = 0 (3)
For further elaboration on these derivations, see for example Weinberg [4]. This overlap
with electromagnetic theory is not only remarkable, but also gives us a definitive value of
the speed of the gravitational radiation. Further, it allows us to postulate on the divergence
of the correlation with electro-magnetics, in that the metric tensor itself hµν is the quantity
which behaves like a wave. Coupling this with our earlier statement that the metric tensor
describes the actual space-time structure of the system, then we are stating that under the
right circumstances, wave-like disturbances in the space-time fabric are allowed in GR, and
can propagate far from the source. These are the true manifestations of gravitational waves,
and it was one of these disturbances in space-time that was recently discovered at LIGO. It
3

4. should be noted however that long before LIGO was operational, there was indirect evidence
of gravitational waves. in 1993, the Nobel Prize in Physics was awarded to Russell Hulse for
the indirect detection of gravitational waves by observing a pulsar binary system [5]. The
observations showed a decaying orbit of the pulsar which can be measured due to doppler
shift. Using the equations described above, one can calculate the energy carried away by
the gravitational radiation, and thus predict the orbital decay. This number was measured
within .2 percent of the prediction from GR. Now more than 30 years later, the direct
detection of such radiation has been captured. Although the above derivation is correct, we
have made the oversimplification that we set the source to zero. In order to make predictions
about gravitational waves and relating them to their source in the far field, we would need
to go back and set the weak Einstein equations to a non zero energy momentum tensor and
carry out the calculations. In this framework, the metric tensor coefficients hµν are related
to the amplitudes of the gravitational waves ¯h, and follow wavelike solutions to the wave
equation. These types of waves however, are only produced in the quadrapole moment of the
source and hence are extremely weak waves (in contrast with electromagnetic waves which
possess a dipole moment). We can get a rough estimate for the amplitudes ¯h), by assuming
we are looking at a binary system, which orbits in a circular fashion (which is in-spiraling)
at a distance R from center to center, with period T. The quadrapole moment is given by
P ≈ 2MR2
, where M is the total mass of the system. Since we can relate this back to the
Schwarzschild radius rs, then the approximation for the perturbation amplitude becomes
¯h, =
rs
r
v2
c2
. (4)
This can be further simplified by using the virial theorem for the system, which allows the
final simplification that ¯h ≈ r2
s
Rr
. The beauty of this simplification is that we can get a good
estimate for the amplitudes from simple algebra. For example, if we assume that the objects
in the binary motion are of the order of solar masses, and the event is occurring in a spatial
distance from earth (r) on the order of kpc, then ¯h ≈ 5 ∗ 10−18
m. We will see below that
this number is of significant importance to both LIGO and the signal detected a few weeks
ago.
4

5. FIG. 1: A Cartoon picture of the LIGO apparatus. In the top drawing, the two beams are not
interacting with a gravitational wave. In the Lower, the presence of the gravitational wave (yellow)
causes a small shift in space-time and thus an interference pattern after recombination.
II. LIGO
The Large Interferometer Gravitational Wave Observatory (LIGO) was first hypothesized
in the 1960’s with the advent of interferometry. The idea of using laser interferometry to
attempt to measure gravitational waves was studied by GR master KipThorne and then
revisited in the 1980’s when the first 40 meter prototype was built at Caltech. Fast forward
to the modern day, and LIGO is one of the most ambitious and long going collaborations
between powerhouse universities (MIT and Caltech to name a few), and remains the single
largest collaboration ever funded by NSF. Since its inception in the early 80’s, LIGO
has underwent many updates and renovations which culminated in the announcement on
Februaruy 11, 2016 that the first detection of a gravitational wave was found in the LIGO
array.
5

6. As being a member of the physics community, one cannot escape the irony of the detection
method used. The laser interferometer array used in the LIGO facilities is an experiment
very similar to the famous Michaelson-Morely (MM) experiment, the failed experiment that
originally paved the way for the success of special relativity over 100 years ago. The main
difference now, is that we understand that there is no ether, but instead, we are searching
for wiggles in the space time continuum. Figure 1 shows a simple illustration of this. The
beams of coherent light are fine tuned and sent into a beam splitter across two different
paths. The light bounces off mirrors down the 4000m arms over 250 times and recombine
for no interference pattern (the original MM result). This is achieved by keeping the beams
completely out of phase so that no gravity waves implies a null result. However, if a grav-
itational wave were to traverse through the continuous beam, the small deviation in space
time would cause an extremely small deviation in one of the paths, resulting in an inter-
ference pattern after recombination. It should be noted that the signals sought in LIGO
are extremely small (spatial distortions of about 10−18
m), and the systematic noise in the
system is overwhelming [6]. Many false positives of the LIGO apparatus were due to factors
such as minuscule seismic activity and simple nearby automobile traffic.
Since GR predicts gravitational waves, one can then directly correlate the amplitude of such
waves directly back to information about the source. As an experimental apparatus, we
know the sensitivity of LIGO, and therefore can figure out what types of exotic objects can
put out gravitational waves strong enough to be picked up by the interferometer. This is
how the LIGO group are sure that the gravitational wave detected, was the in-spiral of two
black holes. The signal, dubbed GW150914 was short but sweet. The GR prediction calls
for this type of merger to radiate much of the mass away and have a wavelength that changes
as the merge evolves. Over the .2 seconds, the wave began at 35 Hz, maximizing at 150
Hz. This same signal was then seen 10m/s later at the seconds LIGO site, for a combined
signal to noise ratio of 24, with 90 percent certainty. [7]. Figure 2 shows the actual signal
measured from the two detectors.
6

7. FIG. 2: The measured signal over measured at LIGO Hanford site (left) and LIGO Linvingston
sight (right).
The frequencies measured and change in frequencies can then be used to find the masses
involved via:
M =
(m1m2)
3
5
(m1 + m2)
1
5
=
c3
G
5
96
π−8
3 ν−11
3
dν
dt
3
5
. (5)
With this equation, we can estimate the sum of the masses as being on the order of 70 solar
masses. Since we relate the maximum frequency to be half the orbital frequency of the
masses in -spiraling to be 75 Hz, one can compute the Schwarzchild bound 2GM
c2 ≈ 250km .
Further analysis concludes the specifics of the two masses were about 25 and 39 solar masses
plus or minus 4. With the inferred masses and observed frequencies, this immediately
7

8. implies the two objects had to be black holes and not compact Neutron stars. Perhaps as
icing on the cake, analysis can also be performed in the fall off of the signal. As can be seen
in Figure 2, the fall off was measured, and is consistent with a damped waveform similar to
those found in a Kerr solution for a single rotating black hole, signifying that a merger and
completion was observed [8].
III. FUTURE OF LIGO
With this recent observation, the future of LIGO is promising. The signal was found
in the arrays only a few short months after the recent upgrades to Advanced LIGO [6].
This timely event shows the importance of persistence and commitment to pure scientific
research. Although LIGO may have had some funding issues in the past, it is more than
likely that with the current excitement, that LIGO has not seen its first and last signal.
LIGO India has now only a few weeks after the confirmed signal been confirmed to be new
region which will build a LIGO apparatus to give more of a baseline for future signals [9].
Other sites including Australia have also been discussed but not confirmed [10]. Perhaps
one of the outcomes of this signal, would be a re-affirmed commitment to LISA, the Laser
Interferometer Space Array. The bounds on LISA as compared to LIGO can be seen in
Figure 3, and would basically remove much of the noise and false positives found here on
earth. It would be the hope that this may prove to be a similar advancement to gravitational
wave observations as the Hubble Space telescope was to luminous observations.
8

9. FIG. 3: The parameter space as measured by LIGO, advanced LIGO (aLIGO), LISA and other
gravitational wave detectors.
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