Theoretical Sciences - Rhodes University · Please give a brief description of the Program. It would help if you could include in your description : (i) names of possible lecturers,

Approximation methods in general relativity for gravitational-wave astrophysics

P. Ajith International Center for Theoretical Sciences, Bangalore, India

Lecture 1

24th Chris Engelbrecht Summer School Rhodes University, Grahamstown, South Africa. 15 to 24 January 2013

Motivation of this course: Gravitational-wave astronomy

2LIGO (Livingston)

LIGO (Hanford)

LIGO (Livingston)Virgo (Pisa)

GEO600 (Hannover)

KAGRA (Japan)

LIGO-India

?

A worldwide network of advanced GW observatories will be operational in the near future, opening a new observational window to the Universe.

[Also, see Levin’s lectures on PTAs]

Motivation of this course: Gravitational-wave astronomy

• In order to make the first detections, to interpret observations, and to understand the physics and astrophysics of the sources, we need reliable models of the expected GW signals.

• This is a “theory for observers” course!

• Plan “From the bottom up”. Start from Newtonian gravity, classical mechanics and electrodynamics. Starting from approximate descriptions to more complex descriptions.

3

Overview of the lectures

• Lecture 1 Introduction to GWs, order-of-magnitude calculation of GW amplitudes, overview of the promising sources, coalescing compact binaries, need of accurate modeling of the sources.

• Lecture 2 Two-body problem in Newtonian gravity, radiation reaction -- examples from electrodynamics, GWs from a two-body system -- leading order calculation in GR.

• Lecture 3 Adiabatic approximation, calculation of GW phasing in the adiabatic approximation, inclusion of post-Newtonian (PN) effects, inclusion of spin effects.

• Lecture 4 [On black board] Different adiabatic PN approximants for computing GW signals from inspiralling compact binaries, beyond-adiabatic approximation, effective one body approach, issues with PN approximation, comparison with numerical relativity (NR).

• Lecture 5 Need for combining PN and NR, Matching analytical and NR waveforms, hybrid waveforms, EOB-NR, other phenomenological approaches for constructing inspiral-merger-ringdown waveforms.

4

Gravitational waves

• The existence of gravitational waves (GWs) is one of the most intriguing predictions of General Relativity.

• GWs are freely propagating oscillations in the geometry of spacetime − ripples in the fabric of spacetime.

5

Gravitational waves

• The existence of gravitational waves (GWs) is one of the most intriguing predictions of General Relativity.

• GWs are freely propagating oscillations in the geometry of spacetime − ripples in the fabric of spacetime.

6

accelerating charges(time-varying dipole moment)

accelerating masses(time-varying quadrupole moment)

electromagnetic waves

gravitational waves

• Poisson’s equation ...

... has the solution:

Newtonian gravity does not have GWs

7

r2� = 4⇡G⇢

Newtonian potential mass density of the source

�N(x, t) = �GZ⇢(y, t)

rd3y

r = |x � y|

• Poisson’s equation ...

... has the solution:

• Change in ϕ produced by a change in ρ (e.g. due to two stars orbiting each other) is instantaneous → ϕ does not satisfy a wave equation. Newtonian gravity has no GWs!

Newtonian gravity does not have GWs

8

r2� = 4⇡G⇢

Newtonian potential mass density of the source

�N(x, t) = �GZ⇢(y, t)

rd3y

time is the same on both sides

Newtonian gravity + special relativity

• Special Relativity No information can travel faster than c, the speed of light.

• The simplest way to make gravity consistent with SR: insert a delay (retardation) between the potential and density.

• This potential ϕR satisfies the scalar wave equation:

9

�R(x, t) = �GZ⇢(y, t � r/c)

rd3y

retarded time

Waves that propagate with speed c !

r2�R �1c2@2�R

@t2 = 4⇡G⇢

Newtonian gravity + special relativity

• Newtonian gravity + Special relativity → waves that propagate with speed c. But essentially different from GWs in GR.

10

Newtonian gravity + special relativity General relativity

Scalar field Tensor field

Longitudinal waves Transverse waves

• Linearized gravity is an adequate approximation to GR when

Gravitational waves in linearized GR

11

5 Institute of Physics !DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

is valid in linearized theory. (As we will discuss in section 2, this is allowable because, inlinearized theory, the position of a spatial index is immaterial in Cartesian coordinates.) A quantitythat is symmetrized on pairs of indices is written as

A(ab) = 12(Aab + Aba).

Throughout most of this paper, we use ‘relativist’s units’, in which G = 1 = c; mass, space andtime have the same units in this system. The following conversion factors are often useful forconverting to ‘normal’ units:

1second = 299 792 458 m ! 3 " 108 m1M! = 1476.63 m ! 1.5 km

= 4.92549 " 10#6 seconds ! 5 µseconds.

(1M! is one solar mass.) We occasionally restore factors of G and c to write certain formulaein normal units.

Section 2 provides an introduction to linearized gravity, deriving the most basic propertiesof GWs. Our treatment in this section is mostly standard. One aspect of our treatment that isslightly unusual is that we introduce a gauge-invariant formalism that fully characterizes thelinearized gravity’s degrees of freedom. We demonstrate that the linearized Einstein equationscan be written as five Poisson-type equations for certain combinations of the spacetime metric,plus a wave equation for the transverse-traceless components of the metric perturbation. Thisanalysis helps to clarify which degrees of freedom in general relativity are radiative and whichare not, a useful exercise for understanding spacetime dynamics.

Section 3 analyses the interaction of GWs with detectors whose sizes are small comparedto the wavelength of the GWs. This includes ground-based interferometric and resonant-massdetectors, but excludes space-based interferometric detectors. The analysis is carried out in twodifferent gauges; identical results are obtained from both analyses. Section 4 derives the leading-order formula for radiation from slowly moving, weakly self-gravitating sources, the quadrupoleformula discussed above.

In section 5, we develop linearized theory on a curved background spacetime. Many of theresults of ‘basic linearized theory’ (section 2) carry over with slight modification. We introducethe ‘geometric optics’ limit in this section, and sketch the derivation of the Isaacson stress–energytensor, demonstrating how GWs carry energy and curve spacetime. Section 6 provides a verybrief synopsis of GW astronomy, leading the reader through a quick tour of the relevant frequencybands and anticipated sources. We conclude by discussing very briefly some topics that we couldnot cover in this paper, with pointers to good reviews.

2. The basic basics: gravitational waves in linearized gravity

The most natural starting point for any discussion of GWs is ‘linearized gravity’. Linearizedgravity is an adequate approximation to general relativity when the spacetime metric, gab, maybe treated as deviating only slightly from a flat metric, !ab:

gab = !ab + hab, $hab$ % 1. (2.1)

Here !ab is defined to be diag(#1, 1, 1, 1) and $hab$ means ‘the magnitude of a typical non-zerocomponent of hab’. Note that the condition $hab$ % 1 requires both the gravitational field to

New Journal of Physics 7 (2005) 204 (http://www.njp.org/)

spacetime metric(analogous to potential ϕ

in Newtonian theory) “flat” metric

diag(-1, 1, 1, 1)

small perturbation

[More in GR lectures by Tiglio]



12





1second = 299 792 458 m ! 3 " 108 m1M! = 1476.63 m ! 1.5 km








gab = !ab + hab, $hab$ % 1. (2.1)



spacetime metric(analogous to potential ϕ

in Newtonian theory) “flat” metric

diag(-1, 1, 1, 1)

small perturbation


On the surface of the Earth, where we aim to detect GWs, linearized gravity is an adequate

approximation to GR.


• In an appropriate coordinate system (Lorentz gauge), the linearized Einstein’s equations can be cast as a wave equation.


13





1second = 299 792 458 m ! 3 " 108 m1M! = 1476.63 m ! 1.5 km








gab = !ab + hab, $hab$ % 1. (2.1)




so that the trace-reversed metric becomes

h!ab = h!

ab " 12!abh

! = hab " 2"(b#a) + !ab"c#c. (2.9)

A class of gauges that are commonly used in studies of radiation are those satisfying the Lorentzgauge condition

"ahab = 0. (2.10)

(Note the close analogy to Lorentz gauge3 in electromagnetic theory, "aAa = 0, where Aa is thepotential vector.)

Suppose that our metric perturbation is not in Lorentz gauge. What properties must #a satisfyin order to impose Lorentz gauge? Our goal is to find a new metric h!

ab such that "ah!ab = 0:

"ah!ab = "ahab " "a"b#a " !#b + "b"

c#c (2.11)

= "ahab " !#b. (2.12)

Any metric perturbation hab can therefore be put into a Lorentz gauge by making an infinitesimalcoordinate transformation that satisfies

!#b = "ahab. (2.13)

One can always find solutions to the wave equation (2.13), thus achieving Lorentz gauge.The amount of gauge freedom has now been reduced from four freely specifiable functionsof four variables to four functions of four variables that satisfy the homogeneous wave equation!#b = 0, or, equivalently, to eight freely specifiable functions of three variables on an initial datahypersurface.

Applying the Lorentz gauge condition (2.10) to the expression (2.7) for the Einstein tensor,we find that all but one term vanishes:

Gab = " 12!hab. (2.14)

Thus, in Lorentz gauges, the Einstein tensor simply reduces to the wave operator acting on thetrace-reversed metric perturbation (up to a factor "1/2). The linearized Einstein equation istherefore

!hab = "16$Tab; (2.15)

in vacuum, this reduces to

!hab = 0. (2.16)

Just as in electromagnetism, the equation (2.15) admits a class of homogeneous solutions whichare superpositions of plane waves:

hab(x, t) = Re!

d3kAab(k)ei(k·x"%t). (2.17)

Here, % = |k|. The complex coefficients Aab(k) depend on the wavevector k but are independentof x and t. They are subject to the constraint kaAab = 0 (which follows from the Lorentz gaugecondition), with ka = (%, k), but are otherwise arbitrary. These solutions are gravitational waves.3 Fairly recently, it has become widely recognized that this gauge was in fact invented by Ludwig Lorenz, ratherthan by Hendrik Lorentz. The inclusion of the ‘t’ seems most likely due to confusion between the similar names; see[38] for a detailed discussion. Following the practice of Griffiths ([39], p 421), we bow to the weight of historicalusage in order to avoid any possible confusion.



be weak, and in addition constrains the coordinate system to be approximately Cartesian. Wewill refer to hab as the metric perturbation; as we will see, it encapsulates GWs, but containsadditional, non-radiative degrees of freedom as well. In linearized gravity, the smallness of theperturbation means that we only keep terms which are linear in hab—higher order terms arediscarded. As a consequence, indices are raised and lowered using the flat metric !ab. The metricperturbation hab transforms as a tensor under Lorentz transformations, but not under generalcoordinate transformations.

We now compute all the quantities which are needed to describe linearized gravity. Thecomponents of the affine connection (Christoffel coefficients) are given by

"abc = 1

2!ad(#chdb + #bhdc ! #dhbc) = 1

2(#chab + #bh

ac ! #ahbc). (2.2)

Here #a means the partial derivative #/#xa. Since we use !ab to raise and lower indices, spatialindices can be written either in the ‘up’position or the ‘down’position without changing the valueof a quantity: f x = fx. Raising or lowering a time index, by contrast, switches sign: f t = !ft.The Riemann tensor we construct in linearized theory is then given by

Rabcd = #c"

abd ! #d"

abc = 1

2(#c#bhad + #d#

ahbc ! #c#ahbd ! #d#bh

ac). (2.3)

From this, we construct the Ricci tensor

Rab = Rcacb = 1

2(#c#bhca + #c#ahbc ! !hab ! #a#bh), (2.4)

where h = haa is the trace of the metric perturbation and ! = #c#

c = "2 ! #2t is the wave

operator. Contracting once more, we find the curvature scalar:

R = Raa = (#c#

ahca ! !h) (2.5)

and finally build the Einstein tensor:

Gab = Rab ! 12!abR = 1

2(#c#bhca + #c#ahbc ! !hab ! #a#bh ! !ab#c#

dhcd + !ab!h). (2.6)

This expression is a bit unwieldy. Somewhat remarkably, it can be cleaned up significantlyby changing the notation: rather than working with the metric perturbation hab, we use thetrace-reversed perturbation hab = hab ! 1

2!abh. (Notice that haa = !h, hence the name ‘trace

reversed’.) Replacing hab with hab + 12!abh in equation (2.6) and expanding, we find that all terms

with the trace h are cancelled. What remains is

Gab = 12(#c#bh

ca + #c#ahbc ! !hab ! !ab#c#

dhcd). (2.7)

This expression can be simplified further by choosing an appropriate coordinate system,or gauge. Gauge transformations in general relativity are just coordinate transformations. Ageneral infinitesimal coordinate transformation can be written as xa# = xa + $a, where $a(xb) isan arbitrary infinitesimal vector field. This transformation changes the metric via

h#ab = hab ! 2#(a$b), (2.8)


“trace-reversed” metric perturbation stress-energy tensor

(analogous to density ρin Newtonian theory)


a, b = 0...3G = c = 1

Greek letters in Tiglio’s lectures

h jk(t) ⇠ 2

d¨Qjk ⇠

✓Gc4

◆4µ

d

✓GMR

◆cos 2!t


14

• In transverse-traceless gauge, the metric perturbation can be related to the quadrupole-moment of the source.

source

distan

ce, d

observer

second time-derivative of the

quadrupole moment

distance to the source

GW amplitude

j, k = 1...3


m2

m1

ωt

R

M = m1 + m2

Most promising sources are relativistic phenomena involving massive and compact

objects

15

frequency of the source

“compactness” of the source

reduced-mass of the

source

Keplerian v210−45!


h jk(t) ⇠ 2

d¨Qjk ⇠

✓Gc4

◆4µ

d

✓GMR

◆cos 2!t


m2

m1

ωt

R

M = m1 + m2

Most promising sources are relativistic phenomena involving massive and compact

objects

16

10−45!


! =

rGmR3

h jk(t) ⇠ 2

d¨Qjk ⇠

✓Gc4

◆4µ

d

✓GMR

◆cos 2!t

Exercise: Typical GW amplitudes and frequencies

• A “GW generator” Two masses with 103 kg separated by 10m orbiting each other at a frequency of 10 Hz (d ~ 104 km).

• Planets A Jupiter-sized planet orbiting a Sun-like star in our own galaxy (d ~1 kpc).

• Binary stars A main-sequence star binary in our own galaxy (d ~1 kpc).

• White-dwarf binaries A white-dwarf binary in our own galaxy (d ~ 1 kpc).

• Binary pulsars A binary pulsar in our own galaxy (e.g. Hulse Taylor binary; d ~ 6.4 kpc).

• Compact binary coalescence Binary black-hole(BH)/neutron-star coalescence in Virgo cluster. (d ~ 16.5 Mpc)

• Supermassive BH binaries Merger of two galaxies each hosting a supermassive BHs at 1Gpc.

17

Exercise: Typical GW amplitudes and frequencies

• A “GW generator” Two masses with 103 kg separated by 10m orbiting each other at a frequency of 10 Hz (d ~ 104 km). [h ~ 10-43, FGW ~ 20 Hz, Hopelessly small to detect!]

• Planets A Jupiter-sized planet orbiting a Sun-like star in our own galaxy (d ~1 kpc).

• Binary stars A main-sequence star binary in our own galaxy (d ~1 kpc).

• White-dwarf binaries A white-dwarf binary in our own galaxy (d ~ 1 kpc).

• Binary pulsars A binary pulsar in our own galaxy (e.g. Hulse Taylor binary; d ~ 6.4 kpc).

• Compact binary coalescence Binary black-hole(BH)/neutron-star coalescence in Virgo cluster. (d ~ 16.5 Mpc)

• Supermassive BH binaries Merger of two galaxies each hosting a supermassive BHs at 1Gpc.

18

Expected sources of gravitational waves

19

00.10.20.30.40.50.60.70.80.91

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

00.10.20.30.40.50.60.70.80.91

Frequency 10−16 Hz 10−9 −10−6 Hz 10−5 −10−1 Hz 10−1 −1 Hz 1 −104 HzWavelength 1021 km 1014 −1011 km 1010 −106 km 106 −105 km 10 5 −10 kmDetection CMB Polarization Pulsar timing eLISA/NGO BBO/DECIGO LIGO/Virgo/KAGRA/ET

!! !"#$ !" !%#$ % %#$ " "#$ !!!

!"#$

!"

!%#$

%

%#$

"

"#$

!

Coalescing binaries of stellar- & intermediate- mass BHs / neutron stars

Spinning neutron starsN

ASA

/CX

O/S

AO

Core-collapse supernovae, gamma-ray bursts, soft gamma-ray repeaters ...N

ASA

/HST

NA

SA/S

wift

/M P

at,

H-K

eith

, J Jo

nes

!! !"#$ !" !%#$ % %#$ " "#$ !!!

!"#$

!"

!%#$

%

%#$

"

"#$

!

Coalescing SMBH binaries

!! !"#$ !" !%#$ % %#$ " "#$ !!!

!"#$

!"

!%#$

%

%#$

"

"#$

!

White-dwarf binaries

Energetic processes in the early universe

U M

aryl

and

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Neutron stars / stellar-mass BHs inspiralling into SMBHs


• Burst sources Collapse of massive stellar cores can produce a burst of GWs.

may also produce a supernova/long GRB

leaves behind a compact object

(black hole or neutron star)

Class. Quantum Grav. 26 (2009) 063001 Topical Review

-10 0 10 20 30 40 50-200

-150

-100

-50

0

50

100

t - tbounce (ms)

h+D

(cm

)

Type I

Type II

Type III

Figure 1. Axisymmetric GW burst signal (h+D in units of cm, where D is the distance of thesource) as a function of time after core bounce for models A1B3G3 (type I), A2B4G1 (type II)and A1B3G5 (type III) of the 2002 GR study using polytropic initial models and a simple analytichybrid polytropic/ideal-fluid EOS carried out by Dimmelmeier et al [112]. The waveforms wereobtained from [126].

4. Rotating core collapse and bounce

Rapid precollapse rotation, in combination with angular momentum conservation duringcollapse, leads to significant asphericity in the form of a GW emitting, rapidly time-varying! = 2 oblate (quadrupole) deformation of the collapsing and bouncing core.

Rotating collapse and core bounce is the most extensively studied GW emission processin the massive star collapse context. In 1982, Muller published in [127] the first GW signalsfrom rotating core collapse and bounce that were based on the axisymmetric (2D) Newtoniansimulations of Muller and Hillebrandt [128]. A large number of 2D studies followed withvarying degrees of microphysical detail, inclusion of GR, and precollapse model sets (see,e.g., [39–41, 86, 112, 116, 129] and references therein). These computational parameterstudies demonstrated that the general analytic picture of stellar core collapse derived byGoldreich and Weber [130] and Yahil [131] for spherically symmetric collapse also holds togood approximation for rotating cores: From the beginning of collapse, the collapsing coreseparates into a subsonically homologously (v ! r) contracting inner core and a supersonicallyinfalling outer core. The mass of the inner core at core bounce sets the mass cut for the matterthat is dynamically relevant in bounce (see, e.g., [4]) and responsible for the GW burst. It alsodetermines the initial size of the PNS.

Furthermore, these studies identified at least three GW signal ‘types’ that can be associatedwith distinctly different types of collapse and bounce dynamics (figure 1 displays representativeexamples). Type I models undergo core bounce governed by the stiffening of the nuclear EOSat nuclear density and ‘ring down’ quickly into postbounce equilibrium. Their waveformsexhibit one pronounced large spike at bounce and then show a gradually damped ringdown. Type II models, on the other hand, are affected significantly by rotation and undergocore bounce dominated by centrifugal forces at densities below nuclear. Their dynamicsexhibits multiple slow harmonic-oscillator-like damped bounce–re-expansion–collapse cycles(‘multiple bounces’), which is reflected in the waveform by distinct signal peaks associatedwith every bounce. It is interesting to note that type-II models are related to fizzlers, proposed

8

[Dimmelmeier et al (2002)]

EGW = 10�12 � 10�4 M� c2

20


• Burst sources Collapse of massive stellar cores can produce a burst of GWs

• Continuous sources Spinning neutron stars with non-axisymmetric deformations.

Gravitational collapse can leave the newly born neutron star highly spinning.

h(t)will get Doppler modulated by the motion and

spin or the earth.

some of the NSs are observed as pulsars (e.g. Crab)

21




• Compact binary coalescences driven by GW emission.

merger (involving NS) might also produce a short GRB

h(t)!! !"#$ !" !%#$ % %#$ " "#$ !!!

!"#$

!"

!%#$

%

%#$

"

"#$

!

EGW ' 0.01 � 0.15 Mc2

22




• Compact binary coalescences driven by GW emission.

• Stochastic GW background Produced by superposition of a number of astrophysical sources or by energetic processes in the Early Universe.

analogous to CMBR

0 10 20 30 40 50 60 70 80 90 100−4

−2

0

2

4

6

Detector 1Detector 2

h(t)

Correlation is exaggerated

23

Coalescing compact binaries

Coalescing binaries of compact objects (black holes or neutron stars) are arguably the most promising sources for the first

detection of gravitational waves .

Number of astrophysical channels exist to form such systems. Once formed, they “inspiral” and coalesce by GW emission.

24

!! !"#$ !" !%#$ % %#$ " "#$ !!!

!"#$

!"

!%#$

%

%#$

"

"#$

!

Coalescing compact binaries: The most promising GW sources

•Highly “efficient” sources ~1−15% of the rest mass is radiated as GWs. Can be observed up to large distances.

25

most of the energy is radiated over the late-inspiral &

merger (time scale ~ 1000 M)

LGW ⇠ 0.1Mc2

1000GM/c3⇠ 1022L�

larger than the total luminosity of the

observable EM universe!



• Strong observational evidence A dozen galactic binary neutron stars have been observed. Binary quasars are expected to be supermassive black hole binaries.

26

December 16, 2010 1:9 WSPC/INSTRUCTION FILE gebfest˙jl

4 James M Lattimer and Madappa Prakash

Fig. 1. Measured neutron star masses. References in parenthesis following source numbers areidentified in Table 1.

December 16, 2010 1:9 WSPC/INSTRUCTION FILE gebfest˙jl

4 James M Lattimer and Madappa Prakash

Fig. 1. Measured neutron star masses. References in parenthesis following source numbers areidentified in Table 1.

[Lattimer & Prakash (2010)]



• Strong observational evidence A dozen galactic binary neutron stars have been observed. Binary quasars are expected to be supermassive black hole binaries.

•“Clean systems” Expected signals can be accurately modelled using GR. Enable us to use the most sensitive search techniques.

27

Search for GWs from CBCs: Matched filtering

• Signals are rare, weak, and buried in the noise. Need sophisticated data analysis techniques.

28


• Signals are rare, weak, and buried in the noise. Need sophisticated data analysis techniques.

29

h(�

)

t t ⌧

d

Cross-correlationSignal template Data

source parameterssignal

template

dataSNR

⇢ ⌘ max�

hd ? ˆh(�)

i

d?h(�

)

[More in DA lectures by Sintes]


30

h(�

)

t t ⌧

d

Cross-correlationSignal template Data

d?h(�

)

Detection requires accurate models of the expected GW signals, as computed in GR.

Also, understanding the physics & astrophysics of sources requires accurate theoretical models of the source.

Summary of the lecture

• GWs can be thought of as an essential consequence of the finite speed of propagation of gravitational interaction. Thus, Newtonian gravity does not have GWs.

• Loudest sources of GWs are relativistic phenomena involving massive and compact astrophysical objects (e.g. black holes, neutron stars etc.). Among them, the most promising ones for the first detection of GWs are likely to be binaries of compact objects.

• GW signals from even these “loud” sources are extremely weak; need accurate models of the expected signals in order to detect them and to extract the physics and astrophysics. Topic of the next lectures.

• Further reading

B. F. Schutz, "Gravitational waves on the back of an envelope", American Journal of Physics 52, 5 (1984)

B. S. Sathyaprakash and B. F. Schutz, "Physics, astrophysics and cosmology with gravitational waves", Living Reviews in Relativity 12, 2 (2009).

31

Theoretical Sciences - Rhodes University · Please give a brief description of the Program. It would help if you could include in your description : (i) names of possible lecturers,

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