Contents
Cover
Half Title page
Title page
Copyright page
Dedication
Preface
List of Examples
Introduction
Conventions
References
Chapter 1: Prologue
1.1 Tides in Newton’s Gravity
1.2 Relativity
Chapter 2: A Brief Review of General Relativity
2.1 Differential Geometry
2.2 Slow Motion in Weak Gravitational Fields
2.3 Stress-Energy Tensor
2.4 Einstein’s Field Equations
2.5 Newtonian Limit of General Relativity
2.6 Problems
References
Chapter 3: Gravitational Waves
3.1 Description of Gravitational Waves
3.2 Physical Properties of Gravitational Waves
3.3 Production of Gravitational Radiation
3.4 Demonstration: Rotating Triaxial Ellipsoid
3.5 Demonstration: Orbiting Binary System
3.6 Problems
References
Chapter 4: Beyond the Newtonian Limit
4.1 Post-Newtonian
4.2 Perturbation about Curved Backgrounds
4.3 Numerical Relativity
4.4 Problems
References
Chapter 5: Sources of Gravitational Radiation
5.1 Sources of Continuous Gravitational Waves
5.2 Sources of Gravitational-Wave Bursts
5.3 Sources of a Stochastic Gravitational-Wave Background
5.4 Problems
References
Chapter 6: Gravitational-Wave Detectors
6.1 Ground-Based Laser Interferometer Detectors
6.2 Space-Based Detectors
6.3 Pulsar Timing Experiments
6.4 Resonant Mass Detectors
6.5 Problems
References
Chapter 7: Gravitational-Wave Data Analysis
7.1 Random Processes
7.2 Optimal Detection Statistic
7.3 Parameter Estimation
7.4 Detection Statistics for Poorly Modelled Signals
7.5 Detection in Non-Gaussian Noise
7.6 Networks of Gravitational-Wave Detectors
7.7 Data Analysis Methods for Continuous-Wave Sources
7.8 Data Analysis Methods for Gravitational-Wave Bursts
7.9 Data Analysis Methods for Stochastic Sources
7.10 Problems
References
Chapter 8: Epilogue: Gravitational-Wave Astronomy and Astrophysics
8.1 Fundamental Physics
8.2 Astrophysics
References
Appendix A: Gravitational-Wave Detector Data
A.1 Gravitational-Wave Detector Site Data
A.2 Idealized Initial LIGO Model
References
Appendix B: Post-Newtonian Binary Inspiral Waveform
B.1 TaylorT1 Orbital Evolution
B.2 TaylorT2 Orbital Evolution
B.3 TaylorT3 Orbital Evolution
B.4 TaylorT4 Orbital Evolution
B.5 TaylorF2 Stationary Phase
References
Index
Jolien D. E. Creighton and
Warren G. Anderson
Gravitational-Wave Physics
and Astronomy
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The Authors
Dr. Jolien D. E. Creighton
University of WisconsinMilwaukee
Department of Physics
P.O. Box 413
Milwaukee,WI 53201
USA
jolien@uwm.edu
Dr. Warren G. Anderson
University of WisconsinMilwaukee
Department of Physics
P.O. Box 413
Milwaukee,WI 53201
USA
warren@gravity.phys.uwm.edu
Cover
Post-Newtonian apples created by Teviet Creighton. Hubble ultra-deep field image
(NASA, ESA, S. Beckwith STScl and the HUDF Team).
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JDEC: To my grandmother.
WGA: To my parents, who never asked me to stop asking why, although they did stop answering after a while, and to family, Lynda, Ethan and Jacob, who give me the space I need to continue asking.
Preface
During the writing of this book we often had to escape the office for week-long mini-sabbaticals. We would like to thank the Max-Planck-Institut fr Gravitationsphysik (Albert-Einstein-Institute) in Hannover, Germany, for hosting us for one of these sabbaticals, theWarren G. Anderson Office of GravitationalWave Research in Calgary, Alberta for hosting a second one, the University of Minnesota for hosting a third and the University of Cardiff for our final retreat.
We thank, in no particular order (other than alphabetic), Bruce Allen, Patrick Brady, Teviet Creighton, Stephen Fairhurst, John Friedman, Judy Giannakopoulou, Brennan Hughey, Luca Santamara Lara, Vuk Mandic, Chris Messenger, Evan Ochsner, Larry Price, Jocelyn Read, Richard OShaughnessy, Bangalore Sathyaprakash, Peter Saulson, Xavier Siemens, Amber Stuver, Patrick Sutton, Ruslan Vaulin, Alan Weinstein, Madeline White and Alan Wiseman for a great deal of assistance. This work was supported by the National Science Foundation grants PHY- 0701817, PHY-0600953 and PHY-0970074.
Calgary, June 2011
J.D.E.C.
List of Examples
Example 1.1 Coordinate acceleration in non-inertial frames of reference
Example 1.2 Tidal acceleration
Example 2.1 Transformation to polar coordinates
Example 2.2 Volume element
Example 2.3 How are directional derivatives like vectors?
Example 2.4 Flat-space connection in polar coordinates
Example 2.5 Flat-space connection in polar coordinates (again)
Example 2.6 Equation of continuity
Example 2.7 Vector commutation
Example 2.8 Lie derivative
Example 2.9 Curvature
Example 2.10 Riemann tensor in a locally inertial frame
Example 2.11 Geodesic deviation in the weak-field slow-motion limit
Example 2.12 The Euler equations
Example 2.13 Equations of motion for a point particle
Example 2.14 Harmonic coordinates
Example 3.1 Transformation from TT coordinates to a locally inertial frame
Example 3.2 Wave equation for the Riemann tensor
Example 3.3 Attenuation of gravitational waves
Example 3.4 Degrees of freedom of a plane gravitational wave
Example 3.5 Plus- and cross-polarization tensors
Example 3.6 A resonant mass detector
Example 3.7 Order of magnitude estimates of gravitational-wave amplitude
Example 3.8 Fourier solution for the gravitational wave
Example 3.9 Order ofmagnitude estimates of gravitational-wave luminosity
Example 3.10 Gravitational-wave spectrum
Example 3.11 Cross-section of a resonant mass detector
Example 3.12 Point particle in rotating reference frame
Example 3.13 The Crab pulsar
Example 3.14 Newtonian chirp
Example 3.15 The HulseTaylor binary pulsar
Example 4.1 Effective stress-energy tensor
Example 4.2 Amplification of gravitational waves by inflation
Example 4.3 Black hole ringdown radiation
Example 4.4 Analogy with electromagnetism
Example 4.5 The BSSN formulation
Example 5.1 Blandfords argument
Example 5.2 Rate of binary neutron star coalescences in the Galaxy
Example 5.3 Chandrasekhar mass
Example 6.1 Stokes relations
Example 6.2 Dielectric mirror
Example 6.3 Anti-resonant FabryProt cavity
Example 6.4 Michelson interferometer gravitational-wave detector
Example 6.5 Radio-frequency readout
Example 6.6 Standard quantum limit
Example 6.7 Derivation of the fluctuationdissipation theorem
Example 6.8 Coupled oscillators
Example 7.1 Shot noise
Example 7.2 Unknown amplitude
Example 7.3 Sensitivity of a matched filter gravitational-wave search
Example 7.4 Unknown phase
Example 7.5 Measurement accuracy of signal amplitude and phase
Example 7.6 Systematic error in estimate of signal amplitude
Example 7.7 Frequentist upper limits
Example 7.8 Time-frequency excess-power statistic
Example 7.9 Nullspace of two co-aligned, co-located detectors
Example 7.10 Nullspace of three non-aligned detectors
Example 7.11 Sensitivity of the known-pulsar search
Example 7.12 Horizon distance and range
Example 7.13 Overlap reduction function in the long-wavelength limit
Example 7.14 HellingsDowns curve
Example 7.15 Sensitivity of a stochastic background search
Example A.1 Antenna response beampatterns for interferometer detectors
Introduction
This work is intended both as a textbook for an introductory course on gravitational-wave astronomy and as a basic reference on most aspects in this field of research.
As part of the syllabus of a course on gravitational waves, this book could be used to follow a course on General Relativity (in which case, the first chapter could be greatly abbreviated), or as an introductory graduate course (in which case the first chapter is required reading for what follows). Not all material would be covered in a single semester.
Within the text we include examples that elucidate a particular point described in the main text or give additional detail beyond that covered in the body. At the end of each chapter we provide a short reference section that contains suggested further reading. We have not attempted to provide a complete list of work in the field, as one might have in a review article; rather we provide references to seminal papers, to works of particular pedagogic value, and to review articles that will provide the necessary background for researchers. Each chapter also has a selection of problems.
Please see http://www.lsc-group.phys.uwm.edu/∼jolien for an errata for this book. If you find errors that are not currently noted in the errata, please notify jolien@uwm.edu.
Conventions
We use bold sans-serif letters such as T and u to represent generic tensors and spacetime vectors, and italic bold letters such as v to represent purely spatial vectors. When writing the components of such objects, we use Greek letters for the indices for tensors on spacetime, Tαβ and uα, while we use Latin letters for the indices for spatial vectors or matrices, for example vi and Mij. Spacetime indices normally run over four values, so 
, while spatial indices normally run over three values, , unless otherwise specified. We employ the Einstein summation convention where there is an implied sum over repeated indices (known as dummy indices), so that 
. In these examples, the indices α and i are not contracted and are called free indices (that is, these are actually four equations in the first case and three equations in the second case since α can have the values 0, 1, 2, or 3, while i can have the values 1, 2, or 3).
We distinguish between the covariant derivative, 
, and the three-space gradient operator 
, which is the operator 
in Cartesian coordinates. The Laplacian is 
in Cartesian coordinates, and the flat-space d’Alembertian operator is 
in Cartesian coordinates.
Our spacetime sign convention is 
so that flat spacetime in Cartesian coordinates has the line element 
. The sign conventions of common tensors follow that of Misner et al. (1973) and Wald (1984).
The Fourier transform of some time series x(t) is used to find the frequency series 
according to
(0.1) 
while
(0.2) 
is the inverse Fourier transform.
References
Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973) Gravitation, Freeman, San Francisco.
Wald, R.M. (1984) General Relativity, University of Chicago Press.
Chapter 1
Prologue
1.1 Tides in Newton’s Gravity
A brief review of Newtonian gravity is useful not only as a limit of weak-field relativistic gravity, but also as a reminder of the principles upon which general relativity was formulated. Newtonian gravity is conveniently formulated in a fixed rectilinear coordinate system in terms of an absolute time coordinate. In such coordinates as these, Newton’s laws of motion and gravitation describe the motion of a body of mass m falling freely about another body of mass M by the force
(1.1) 
where x is the position of the body with mass m, x′ is the position of the body with mass M, t is the absolute time coordinate, and 
is Newton’s gravitational constant. Famously, the quantity m cancels and
(1.2) 
If there is a continuous distribution of matter then we can sum up all contributions to the acceleration from all pieces of the distribution to obtain
(1.3) 
where ρ is the mass distribution (density) and ∇ is the gradient operator in x. Therefore, the acceleration of the body (with respect to the Newtonian system of rectilinear coordinates) is
(1.4) 
where
(1.5) 
is the Newtonian potential. The Newtonian potential satisfies the Poisson equation
(1.6) 
where we have used
(1.7) 
Because the mass of the falling body does not enter into the equations of motion, any two bodies will fall the same way. If you can only see nearby free-falling bodies, you cannot tell whether you’re falling or not. You feel the same if you are freely falling toward some massive object as you would if you were in no gravitational field whatsoever. The gravitational acceleration describes the motion of the falling body with respect to the absolute Newtonian coordinates – but is there any way for a freely falling observer to know if they are accelerating or not?
Einstein codified the observation that freely falling objects fall together as a principle known as the equivalence principle: a freely falling observer could always set up a local (freely falling) frame in which all the laws of physics are the same as they would be if that observer were not in a gravitational field. The coordinate acceleration a does not have any physical importance (as it does in Newtonian gravity) because one can always choose a frame of reference – freely falling with the observer – in which the observer is at rest.
Example 1.1 Coordinate acceleration in non-inertial frames of reference
An inertial frame of reference in Newtonian mechanics is any frame of reference that can be related to the absolute Newtonian frame of reference by a uniform velocity and a constant translation of position. That is, if x is the location of a particle in one inertial frame of reference, then another inertial frame of reference will have 
for some constant vectors 
and v. Inertial frames preserve the form of Newton’s second law since 
.
In non-Cartesian coordinates, however, the form of the coordinate acceleration is different. For example, for a two-dimensional system we could express the location of a particle in polar coordinates 
and Φ = arctan(y/x). In these coordinates, the coordinate velocity of a particle is given by dr/dt = v ⋅ er and 
where er and eΦ are unit vectors in the r- and Φ-directions, and the equations of motion for the particle are 
and 
. Even when there is no force on the particle, F = 0, there is still a coordinate acceleration in that d2r/dt2 and d2Φ/dt2 do not vanish except for purely radial motion. This merely arises because of the choice of non-Cartesian coordinates – the geometrical form of Newton’s second law, F = ma still holds.
A non-inertial frame is a frame that is accelerating relative to an inertial frame. A common example is a uniformly rotating reference frame with angular velocity vector ω. In such a reference frame, Newton’s second law has the form F = ma + mω × (ω × r) + 2mω × v where the two additional terms, the centrifugal force, mω × (ω × r) and the Coriolis force, 2 mω × v, arise because the frame of reference is non-inertial. These are known as fictitious forces.
A freely falling frame of reference in Newtonian theory is a non-inertial frame of reference because it is accelerating relative to the absolute set of Newtonian coordinates. The following coordinate transformation relates a freely falling frame of reference (primed coordinates) at point x0 with the absolute Newtonian coordinates (unprimed): 
, where 
is a constant. It is straightforward to see that 
which vanishes at point x0.
In fact, there is a way to tell if you are falling. If there is another object that is some small distance away from you then its acceleration will be slightly different. Suppose ζ is the vector pointing from you to the other object. The acceleration of that object is
(1.8) 
and so the relative acceleration or tidal acceleration is
(1.9) 
where
(1.10) 
is known as the tidal tensor field. The tidal acceleration is not really local since it depends on the separation ζ between falling bodies. The tidal field, however, is a local quantity, and it encodes the presence of the gravitational field. We will see later that in General Relativity, the tidal field is a measure of the spacetime curvature.
In the above expressions, the indices i and j run over the three spatial coordinates {x1,x2,x3} or equivalently {x,y,z} and ζi is the ith component of the vector ζ. (The three components of the vector are ζ1, ζ2 and ζ3 so we would write ζi = [ζ1, ζ2, ζ3].) The tidal field is a rank-2 tensor having nine components: 
and ε33. It is symmetric: ε12 = ε21, ε13 = ε31 and ε23 = ε32, or, more concisely, εij = εji. Einstein’s summation convention is being used here: there is an implicit summation over repeated indices. That is, the expression
is short-hand for
For example, if two objects are separated in the x3 - or z-direction, so that ζ1 and ζ2 both vanish, then the three components of the tidal acceleration are
Example 1.2 Tidal acceleration
Consider a body falling toward the Earth. The Newtonian potential is
(1.11) 
The tidal field component ε11 is
(1.12) 
the tidal field component ε12 is
(1.13) 
and so forth. The components can be written concisely as
(1.14) 
where 
and δij is the Kronecker delta
(1.15) 
and so xi = δij xj.
Suppose that a reference body is on the z-axis at a distance r = z from the centre of the Earth. Then the tidal tensor is
(1.16) 
Consider a nearby second body that is also on the z-axis, a distance δz farther fromthe centre of the Earth. The relative tidal acceleration of this body is
(1.17) 
The only non-vanishing component is the z-component:
(1.18) 
(1.19) 
and the only non-vanishing component is the x-component:
(1.20) 
Notice that a collection of freely falling objects will be pulled apart along the direction in which they are falling while being squeezed together in the orthogonal directions.
Unlike the coordinate acceleration, the tidal acceleration has intrinsic physical meaning. We witness ocean tides caused by the Moon and the Sun. These tides dissipate energy on the Earth. That is, tidal forces can do work. To compute the work, consider an extended body (say, the Earth) moving within a tidal field produced by another body (say, the Moon). An element of the extended body, located at a position x and having mass ρ(x)d3x, experiences a tidal force
(1.21) 
If the element is moving through the tidal field with velocity v then there is an amount Fivi of work per unit time done on that element. Summing over all elements that comprise the body yields the total amount of tidal work:
(1.22) 
where
(1.23) 
is the quadrupole tensor. Note that this tensor is closely related to the moment of inertia tensor
(1.24) 
and also to the (traceless) reduced quadrupole tensor
(1.25) 
Here 
.
Tidal work can also be performed by a dynamical system with a time-changing tidal field εij(t). The work performed by such a system on another body with a quadrupole tensor Iij is found by integrating Eq. (1.22) by parts:
(1.26) 
The first term is bounded, while the second term secularly increases with time and represents a transfer of energy from the dynamical system that is producing the time-changing tidal field to the other body. For example, the source of the time-changing tidal field might be a rotating dumbbell or a binary system of two stars in orbit about each other. Over a long time (large T) the secularly growing term will dominate, and we can write the work done by the dynamical source on the body with moment of inertia tensor Iij as
(1.27) 
1.2 Relativity
The special theory of relativity postulates that there is no preferred inertial frame: local measurements of physical quantities are the same no matter which inertial frame the measurement is made in. This is the principle of relativity. In particular, measurements of the speed of light in any inertial frame will always yield the same value, c: = 299 792 458 m s−1. The consequence of this is that the Newtonian separation of space and time must be abandoned. Consider a spaceship travelling at a constant speed v in the x-direction relative to the Earth (see Figure 1.1). Within the spaceship, an experimental determination of the speed of light is made in which a photon is emitted from a source in the y-direction, reflected by a mirror a distance 
away from the source, and received back at the source. The time-of-flight Δτ is measured and the speed of light c = Δy/Δτ is computed. For an observer on the Earth, however, the distance travelled by the photon is 
, where Δx = vΔt and Δt is the amount of time the observer on the Earth determines it takes the photon to travel from the emitter to the receiver. Since the observer on Earth must measure the same speed of light, 
, we see that
(1.28) 
where we have used Δy = cΔτ, and so
(1.29) 
The usual time dilation formula Δt = γΔτ, where 
is the Lorentz factor, follows by setting Δx = vΔt. This relationship between how time is measured within the moving frame of the spaceship to how time is measured on Earth is not particular to the experiment with the photon: time really does move differently in the different inertial frames of reference.
Equation (1.29) relates the amount of time Δτ between two events, as recorded in an inertial frame in which the two events occur at the same spatial position (which is known as the proper time between the two events), to the amount of time Δt between the same two events as seen in an inertial frame in which the two events are separated by a spatial distance Δx. Since the notion of an absolute time is lost in special relativity, we understand time to simply be a new coordinate which, along with the three spatial coordinates, depends on the frame of reference. Together, the time and space coordinates are used to identify points (or events) on a four-dimensional spacetime. For rectilinear coordinates in an inertial frame, we define an invariant interval (Δs)2 between two points in spacetime, (t, x, y, z) and (t + Δt, x + Δx, y + Δy, z + Δz), by
(1.30) 
which has the same form as the Pythagorean theorem except for the factor of –c2 in front of the square of the time interval. This equation is just a generalization of Eq. (1.29) with (Δs)2: = –c2(Δτ)2.
Special relativity is incompatible with Newtonian gravity because Newton’s law of gravitation defines a force between two distant bodies in terms of their separation at a given instant in time. However, in special relativity, there is no unique notion of simultaneity. In addition, different frames of reference will make different measurements of the Newtonian gravitational force, a result that is at odds with the principle of relativity.
The general theory of relativity provides a description of gravity in terms of a curved spacetime. This is discussed in Chapter 2. In general relativity, the inertial frames of reference are freely falling frames, and the principle of relativity is then taken to hold in such frames of reference. Tidal acceleration is the physical manifestation of gravitation, but measurement of a tidal field requires a somewhat extended apparatus.
Of course, Newtonian gravity must be recovered in some limit of general relativity: this limit is when 
and 
where M is the characteristic mass of the system, R is the characteristic size of the system, and v is the characteristic speed of bodies in the system. And since in Newtonian gravity a changing tidal field is capable of producing work on distant bodies, this must be true in general relativity as well. This means that in order to ensure that energy is conserved, energy must be radiated from the gravitating system that is producing the changing tidal field to the rest of the universe, because there is no way that the bodies on which the work is done can create an instantaneous reactive force on the gravitating system – this would be incompatible with relativity. The radiation is called gravitational radiation.
, reflects off of the mirror and returns to the source after a time 