A short introduction to the one of the nicest bits of physical reasoning ever, which led to Albert Einstein's General Relativity, gravitational waves and our research on gravitational wave sources. Designed by Joseph John Fernandez for LJMU FET Research Week.

1. The EEP is a very powerful statement: it implies that if we know the laws of physics in the
absence of gravity, then we also know their description in the presence of gravity. Because free-
fall is equivalent to motion in an accelerated reference frame, and acceleration can be removed
with an adequate change of reference frame, this means we can describe gravity by means of a
coordinate transformation. Conversely, what we know as the gravitational field is a change of
the geometry, or curvature, of the space-time, due to the presence of mass. This is embodied in
a mathematical object known as the metric tensor, which tells us how to measure distances in
curved space [3]:
This is how gravity is understood in GR: the presence of mass changes the geometry of space-
time. Gravity is a fictitious force which bodies feel as they free-fall through curved space-
time. One of the strong points of this description is that it is local, a feature embodied in
gravitational waves (GWs).
During the early years of the 20th century, it became clear that Newton's universal law of
gravitation required modification. It gave predictions for the precession of Mercury's perihelion
which did not coincide with observations. From a theoretical perspective, its explicit non-locality
made it incompatible with special relativity (SR).
To a great degree of accuracy, the inertial mass (the opposition of a body to change its state of
motion) and gravitational mass (a measure of how much a particle is affected by a gravitational
field) are equal. A direct consequence of this is the universality of free fall, or the weak
equivalence principle[1]:
"All bodies, independently of mass or composition,accelerate equally under the influence of a
gravitational field."
From this it follows that a free-falling observer would not be able to tell the difference between
being immersed in a gravitational field or in an accelerated reference frame: the dynamics
would be identical.
Einstein took this principle a step further, proposing what is now known as the Einstein
equivalence principle(EEP) [2]:
"The laws of physics in free-fall are locally equivalentto those in the absence of gravity"
Joseph John Fernandez* & Shiho Kobayashi
AstrophysicsResearch Institute,Liverpool John Moores University
Liverpool Science Park IC2, 146 Brownlow Hill, Liverpool L3 5RF, UK
General Relativity and gravitational waves: a primer
Abstract
The LIGO/Virgo observatories recently detected gravitational waves (GWs) from black hole (BH) mergers. These avoided detection for a century after their prediction by Albert Einstein, who
thought such a feat would never be possible. The observation of these waves has allowed, for the first time since the formulation of General Relativity (GR), for the direct observation of black
holes and the study of gravity in the strong, dynamical regime. We give a brief overview of the modern description of gravity, gravitational waves and their origin. On a closing note, a dynamical
formation channel for BH binaries, tidal encounters of these systems with galactic centre massive BHs, will be discussed.
Introduction: the equivalence principle
Gravitational waves and black hole binaries
Black hole binaries: where did they come from?
A dynamical source of merging black hole binaries in the galactic centre
Waves are ubiquitous in physics. They are how information regarding changes in the local
properties of a system reach spatially separated regions. Examples include sound waves, which
transport pressure and density changes in a fluid, or electromagnetic waves such as light. In GR,
small perturbations of the gravitational field are transported by GWs. That is, perturbations to
the metric of the form [3]
propagate obeying a wave equation, and at a finite speed. Many of their properties are
analogous to those of EM waves (see table 1). Gravitational waves are the physical manifestation
of the locality of General Relativity. Newtonian gravity does not present this feature. In this
theory changes in mass distributions, and hence the gravitational field propagates
instantaneouslyto all points in space.
Systems which give rise to time-varying gravitational fields, such as compact binaries, radiate
GWs which carry away energy and momentum from them. As GWs propagate through space-
time, they locally perturb the metric at each point, changing the proper distance between
particles as they pass. Like EM waves, they have two independent polarization states. The effect
of these on the relative positions of a set of particles is illustrated in figure 1.
Compact binary systems, such as BH binaries, are prime sources of GWs. GW emission causes
these systems to inspiral: their orbits circularize (if they are eccentric) and tighten, until they
eventually coalesce. As they inspiral, the frequency of their orbit and of their GW
radiation increases, giving rise to a chirp like signal (see figure 2).
As of 2018, the joint aLIGO/VIRGO GW observatory network has reported a total of five
confirmed (and one tentative) BH and one neutron star merger. It is expected that at design
sensitivity these facilities will detect tens to hundreds of merging binary systems, allowing for
unprecedented verificationof the predictions of GR in the dynamical regime [3].
EM waves Gravitational waves
Type of wave Transverse Transverse
Polarizationstates 2: horizontal and vertical 2: h+ and hx (see fig. 1)
Propagationspeed c (in vacuum) c (confirmed within error by LIGO/VIRGO)
Lowest order source Time varying dipole Time varying quadrupole
Table 1. Comparison of some properties of gravitational andelectromagneticwaves.
Figure 1. Illustration of the effect of the independent
polarization states of GW waves, the h+ and hx states, on
the relative positions of a set of test particles. Source: [4]
(1)
General relativity
(2)
Figure 2. Top: depiction of the inspiral and merger. Middle:
Reconstructed waveform compared to numerical relativity
prediction. Source: [5] (modified)
Before the 2015 LIGO detections black holes had never been observed directly. These
observations also proved the existence of stellar-mass binary black holes. Many more are
necessary to determine the origin of these objects [5]. Several formation channels have
been proposed, and can be divided into two broad categories [6]:
· Isolated field binary scenarios: These consider massive field binaries which
evolve independently. Examples of these models include homogeneous chemical
evolution and the massive overcontact binary model [7,8].
· Dynamical formation scenarios: These consider that interaction with other compact
objects (scattering, N-body exchanges, the Kozai-Lidov mechanism...) in dense
environments are what eventually lead to binary coalescence [9,10].
We study a dynamical formation process of
GWS. We consider a close encounter with a
MBH can alter the orbits of BH binaries [6].
· We find that if the binary survives the
encounter, its GW merger time can be
reduced, in some cases by several orders of
magnitude.
· We have also characterized the
expected effective spin and eccentricity
distributions expected from this process.
Figure 3. Example trajectory of the secondary component in
the primary comoving frame. The tidal encounter makes the
binary hard and eccentric.
Figure 4. Cumulative distribution of the post-encounter GW
merger time. The merger time is reduced by a factor of 100
or more in 10% of cases, and by a factor of 100000 or more
in 1% of cases.
Figure 5. Post-encounter effective spins for three initial BH
spin configurations. Around 5-7% of the mergers produced
by the mechanismhave negative effectivespins.
References:
[1] Landau L.D., Lifshitz,E.M. (1975). The classical theory of fields. Butterworth-Heinemann.Oxford.
[2] Bertotti, B.,Grishchuck, L.P., The strong equivalence principle, Class.Quantum Grav 7(1990)
[3] Schutz, B. (2009). A first course in general relativity.Cambridge UniversityPress.Cambridge
[4] https://commons.wikimedia.org/wiki/File:Pol.jpg
[5] B.P. Abbot et al., Observation of Gravitational waves from a Binary Black Hole merger,Phys. Rev. Lett 116, 061102, 2016
[6] Fernandez.,J.J, Kobayashi, S.,Black hole mergers induced by tidal encounters with a galactic centre black hole, in prep, 2018
[7] Marchant, P. et al.,A new route towards merging massive black holes, A&A 588, 2016
[8] Mandel, I., de Mink, S. E., Merging binary black holes formed through chemically homogenous evolution in short period stellar binaries, MNRAS 458, 2016
[9] VanLandingham,J.H., et al.,The Role of the Kozai--Lidov Mechanism in Black Hole Binary Mergers in Galactic Centers, Astrophys.J,828, 77016, 2016
[10] Rodriguez, C.L., et al., Binary Black Hole Mergers from Globular Clusters:Implications for Advanced LIGO, Phys. Rev. Lett. 115, 051101
@joefdez
linkedin.com/in/jospehfdez/
J.J.Fernandez@2017.ljmu.ac.uk*