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C H A P T E R

Gravitational Waves

Edmund Bertschinger & Edwin F. Taylor

A modern physicist is a quantum theorist on Monday,

Wednesday, and Friday, and a student of gravitational

relativity theory on Tuesday, Thursday, and Saturday. On

Sunday the physicist is neither, but is praying to his God that

someone, preferably himself, will find the reconciliation

between these two views.

—Norbert Wiener

I ask you to look both ways. For the road to a knowledge of the

stars leads through the atom; and important knowledge of the

atom has been reached through the stars.”

—Arthur Eddington

INTRODUCTION

Gravity wave: a tidal force that propagates through spacetime.

General relativity differs from Newtonian gravity in several important ways.

One way is in the behavior of light and matter in strong gravitational fields,

especially near black holes. The black hole was predicted by Michell and

Laplace on the basis of Newtonian gravity more than a century before

Schwarzschild discovered his famous metric. However, the event horizon,

singularity, and no-hair theorems are all consequences of general relativity that

could not have been predicted from Newtonian physics.

Gravitational radiation is another phenomenon that has no counterpart in

Newton: Gravity
propagates
instantaneously.

Newtonian physics. According to Newton, the gravitational interaction

propagates instantaneously: When the Earth moves around the Sun, the

Earth’s gravitational field changes all at once throughout space, according to

Newton.

When Einstein formulated special relativity and recognized its

Einstein: No signal
propagates faster
than light.

requirement that no information can travel faster than the speed of light, he

Draft of Second Edition of

Exploring Black Holes: Introduction to General Relativity

2010 Edmund Bertschinger, Edwin F. Taylor, & John Archibald Wheeler.

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Chapter 17

Gravitational Waves

FIGURE 1

Computed emission of gravity waves. The tiny dot at the center of this image is

two black holes churning spacetime as they combine into one. The swirling patterns represent
distortions of spacetime that propagate outward as gravity waves. Close to the coalescing black
holes, the gravity waves—essentially nothing but traveling tidal forces—are lethal. In contrast,
we expect that gravity waves that could be detected on Earth are extremely small.

realized that Newtonian gravity would have to be modified. Not only would

static gravitational fields differ from the Newtonian prediction in the vicinity

of compact masses, but also time-varying fields would have to propagate. He

showed that these fields would move with the speed of light, so gravity could

not be used to send information faster than the speed of light, which would

have destroyed the fundamental basis of all relativity.

Einstein had a conceptual prototype for gravity waves: electromagnetic

radiation. James Clerk Maxwell predicted electromagnetic radiation in 1873

and Heinrich Hertz demonstrated it experimentally in 1888. (Einstein was

Gravity waves like
electromagnetic
waves?

born in 1879.) When he grew up, Einstein quickly realized that a general

relativity theory based on curved spacetime would not look like Maxwell’s

electromagnetic theory. After his theory was completed, Einstein and others

were able to compute how gravitational fields propagate.

Gravity waves are essentially tidal forces that vary with time and position;

that is all they are. As a gravity wave passes over you, you are alternately

stretched and compressed in ways that depend on the particular form of the

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Gravity wave metric

wave. In principle there is no limit to the size of gravity waves. Figure 1

pictures the calculated result of two black holes emitting gravity waves as they

combine into one. In the vicinity of the coalescence, gravity-wave-induced tidal

forces would be dangerous to life.

We predict that gravity waves from various sources are continually

sweeping over us on Earth’s surface. Sections 3 and 7 describe some of these

sources. Basically we hope to observe these waves by detecting changes in

separation between two test masses suspended near to one another—changes

in separation caused by the traveling tidal force that constitutes a gravity

wave. We expect this change in separation to be

extremely

small for gravity

Gravity wave on Earth:
An extremely small
traveling tidal force.

waves detectable on Earth.

Current gravity wave detectors on Earth are interferometers in which light

is reflected back and forth between free test masses along two perpendicular

directions, and the time difference measured between round-trip times in the

two directions. The “free” test masses are hung from wires that are in turn

supported with elaborate shock-absorbers so as to minimize the vibrations due

to passing trucks and even waves crashing on a distant shore. But the

back-and-forth pendulum-like motions of these test masses are free enough to

permit measurement of their change in separation due to tidal effects resulting

from a passing gravity wave, caused by some gigantic distant gravitational

event, for example the coalescence of two black holes modeled in Figure 1.

Does the change in separation induced by gravity waves affect everything,

for example a meter stick or the concrete slab on which a gravity wave

detector rests? Answer: Only by an amount that is entirely negligible. The

structure of meter sticks and concrete slabs is determined by electromagnetic

forces mediated by quantum mechanics. The two ends of a meter stick are not

freely-floating test masses. The tidal force of a passing gravity wave is much

weaker than the internal forces that maintain the shape of solids. The meter

stick—or the concrete slab underlying the vacuum chamber and detectors of a

gravitational-wave observatory—is stiff enough to be negligibly affected by a

passing gravity wave.

GRAVITY WAVE METRIC

Tiny but significant departure from the inertial metric

Our analysis uses a particular gravity wave: a certain kind of plane wave

arriving from a very distant source and moving in the

-direction. This wave

(and almost all of the gravity waves we discuss in this chapter) represents a

very small perturbation of flat spacetime. Here is the timelike metric for such

a particular wave that propagates along the

-axis.

Gravity wave
metric

dτ

−

(1 +

)

−

)

−

(

(1)

In this metric

is a dimensionless function of time and space.

Numerically, h

is a fractional deviation from the flat-spacetime coefficient of

in the

metric

. Another name for fractional deviation of length is

strain

, so

is also

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Chapter 17

Gravitational Waves

FIGURE 2

Progressive improvements in sensitivity of LIGO interferometers. On the

vertical axis 1e-19, for example, means a fractional change in separation of

−

between

test masses. The bottom solid line is the current goal. Spikes occur at frequencies of electrical
or acoustical noise. To be detectable, gravity wave signals must cause greater displacement
than what is represented by these noise curves. As of mid-2009, gravity wave signals have yet
to be detected. FIND MOST RECENT UPDATE THAT SHOWS EVOLUTION OF SENSITIVITY:
SCOTT HUGHES?

called the

gravity wave strain

. The wave leading to (1) is a transverse wave,

gravity

wave strain

since

describes a perturbation of space only in the

and

directions

transverse to the

-direction of propagation. The strain

varies with both

position and time. Its maximum value is very much less than one. Let two free

test masses be at rest a distance

apart in the

direction. When a

-directed gravity wave passes over them, the change in their separation,

called the

displacement

, equals

, which follows directly from the

definition of

as a “fractional deviation.”

FOLLOWING IS NUMERICALLY INCONSISTENT. GET LATEST

LIGO PARAMETERS AND SENSITIVITIES. One can use Einstein’s field

equations to make predictions about the magnitude of the function

equation (1) for various kinds of astronomical phenomena. Currently, gravity

wave detectors use laser interferometry and go by the full name

Laser

LIGO gravity
wave detector

Interferometer Gravitational Wave Observatory

LIGO

for short.

The first-generation LIGO, called Initial LIGO, was able to detect waves with

100

(approximately)

h >

−

for frequencies within a range of about 100 hertz.

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Gravity wave metric

(Abbreviation: Hz. Recall that one hertz is one cycle per second.) The

102

second-generation LIGO, called Advanced LIGO, is about 10 times more

103

sensitive; it is planned to be operational around 2014. Advanced LIGO can

104

also be tuned in frequency to achieve higher sensitivity in frequency bands of

105

interest.

106

Figure 2 compares the gradually-improving sensitivities of LIGO over

107

time. The displacement sensitivity is expressed in the units of meter/(hertz)

108

because the amount of noise limiting the measurement grows with the

109

frequency range being sampled. Note that the instruments are designed to be

LIGO sensitivity

110

most sensitive near 150 hertz. This frequency is determined by the different

111

kinds of noise faced by experimenters: Quantum noise limits the sensitivity at

112

high frequencies, while seismic noise is the largest problem at low frequencies.

113

If the range of sampled frequencies—

bandwidth

—is 100 hertz, then the best

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sensitivity is about 10

−

100

= 10

−

. This means that along a length of

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4 kilometers = 4

meters, the change in length is approximately

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−

= 4

−

meters, which is approximately one thousand times

117

smaller than a proton, or a hundred million times smaller than a single atom!

118

Hold on! Your gravity wave detector sits on Earth’s surface, but equation (1)

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says nothing about curved spacetime described, for example, by the

120

Schwarzschild metric. The expression

M/r

measures departure from

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flatness in the Schwarzschild metric. At Earth’s surface,

122

M/r

≈

−

, which is

—

ten million million!

—times greater

123

than the corresponding gravity wave factor

∼

−

. Why doesn’t the

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quantity

M/r

—which is much larger than

—appear in (1)?

125

First, the factor

M/r

is essentially constant over the size of LIGO, so we

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can ignore it. Second—and more important—the LIGO detector is “tuned”

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to detect a time-varying gravity wave of frequency near 150 hertz. LIGO is

128

totally insensitive to the small

static

curvature introduced by the factor

129

M/r

at Earth’s surface. For this reason, we simply omit static curvature

130

factors from equation (1), effectively describing gravity waves “in free

131

space” as well as for the predicted

132

In free space and for small values of

, Einstein’s field equations actually

Einstein’s equations
become a
wave equation.

133

reduce to a wave equation for

. For the most general case, this wave has the

134

form

(

x, y, z, t

). When

x, y, z,

and

are all in geometric units (for

135

example meters), this wave equation takes the form:

136

∂

∂x

∂

∂y

∂

∂z

∂

∂t

(free space and

(2)

For simplicity, think of a wave moving along the

-axis. The most general

137

solution to the wave equation under these circumstances is

138

(

−

) +

−

(

)

(3)

The expression

(

−

) means a function

of the variable

−

and

not

139

some constant

times

the quantity (

−

). The function

(

−

) describes

Assume gravity
wave moves
in

direction.

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a wave moving in the positive

-direction and the function

−

(

) describes

141

a wave moving in the negative

-direction. In this chapter we deal only with a

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gravity wave propagating in the positive

-direction and hereafter use

143

≡

(

−

)

≡

(

−

)

(wave moves in +

direction)

(4)

The argument

−

means that

is a function of

only

the combined variable

144

−

. Indeed,

can be

any function whatsoever

of the variable (

−

). The

145

form of this variable tells us that, whatever the profile of the gravity wave at

146

any given time; as time passes, that profile displaces itself in the positive

147

-direction with the speed of light (one in our units) .

148

Figure 2 shows that the LIGO gravity wave detector has maximum

149

sensitivity to gravity waves of frequencies between 75 and 500 hertz, with a

LIGO sensitive
75 to 500 hertz

150

peak sensitivity at around 150 hertz. Even at 500 hertz, the wavelength of the

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gravity wave is very much longer than the overall 4-kilometer dimensions of

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the LIGO detector. Therefore

we can assume in the following that at any given

153

time the value of

is spatially uniform over the entire LIGO detector.

154

155

QUERY 1. Uniform

156

Using numerical values, verify the claim in the preceding paragraph that

is effectively uniform over

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the LIGO detector.

158

159

It is important to understand the meaning of the coordinates in metric

Draw global
map coordinates
on rubber sheet.

160

(1). These are global

map

coordinates; global coordinates are always fictions

161

that we choose to reveal aspects of a spacetime we cannot visualize. For

= 0,

162

these global coordinates are invariably distorted. Think of the three mutually

163

perpendicular planes formed by pairs of space coordinates (

x, y

), (

y, z

), and

164

(

z, x

). Draw a grid of lines on a rubber sheet lying in each corresponding

165

plane. The gravity wave distorts the rubber sheet as it passes through it.

166

Glue map clocks to the intersections of these grid lines on the rubber sheet

167

so that they move as the rubber sheet distorts. A gravitational wave moving in

168

the +

direction (Figure 5) passes through a rubber sheet lying in the

169

plane, so that the grid ruled on the rubber sheet stretches and contracts in

170

different directions within the plane of the sheet (Figures 3 and 4). The map

171

clocks, glued at intersections of map coordinate grid lines, ride along with the

172

grid as the sheet distorts,so that the map coordinates of any clock does not

173

change.

174

Think of two ticks on a single map clock. Between ticks the map

Map time

read on clocks
glued to the
rubber sheet.

175

coordinates of the clock do not change:

= 0. Therefore metric (1)

176

tells us that the wristwatch time

dτ

between two ticks is also map time

177

between ticks. Therefore map time

corresponds to the time measured on the

178

clocks glued to the rubber sheet, even when the strain

varies at their

179

locations.

180

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Gravity wave metric

FIGURE 3

Change in shape (greatly exaggerated!) of the map coordinate grid at four

times as a periodic wave passes through in the

-direction (perpendicular to the page). NOTE

carefully!: The

-axis is stretched while the

-axis is compressed and vice versa. The areas of

the panels remain the same.

FIGURE 4

Effects of a periodic gravity wave with polarization “orthogonal” to

that of Figure 3 on the map grid in the

plane. Note that the axes of compression

and expansion are at 45 degrees from the

and

axes. All grids stay in the

plane

as they distort. As in Figure 3, the areas of the panels are all the same.

Figure 3 represents the map time variation of the space distortion of the

181

rubber sheet at a given location due to a particular polarization of the gravity

182

wave. Although gravity waves are transverse like electromagnetic waves, the

183

polarization forms of gravity waves are different from those of electromagnetic

184

waves. Figure 4 shows the distortion caused by the wave “orthogonal” to that

185

shown in Figure 3.

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SOURCES OF GRAVITY WAVES

187

Many sources; only one with clear prediction

188

Sources of gravity waves include collapsing stars, exploding stars, stars in orbit

189

around one another, and the big bang. Neither electromagnetic waves nor

190

gravity waves result from a spherically symmetric distribution of charge (for

191

electromagnetic waves) or matter (for gravitational waves), even when that

192

spherical distribution pulses symmetrically in and out. Therefore,

symmetric

193

collapses or explosions emit no waves, either electromagnetic or gravitational.

194

The most efficient source of electromagnetic radiation is oscillating pairs of

195

electric charges of opposite sign, the result technically called

dipole

196

radiation

. But mass has only one “polarity;” there is no gravity dipole

197

radiation from masses that oscillate back and forth along a line. Emission of

198

gravity waves requires

asymmetric

movement or oscillation; the technical name

199

for the simplest result is

quadrupole radiation

. Happily, most collapses and

200

explosions are asymmetric; even the motion in a binary system is sufficiently

201

asymmetric to emit gravitational waves.

202

We study here gravity waves emitted by a binary system consisting of two

203

neutron stars—or a neutron star and a black hole—orbiting about one another

204

(Section 6). All such pairs that we have detected are too far away to see

205

directly; at least one neutron star needs to be a

pulsar

that emits a steady

Binary system
emits gravity
waves . . .

206

stream of pulses that we can receive at a great distance. Pulsars turn out to be

207

extremely stable clocks. As the two objects orbit, they also emit gravity waves

208

that cause the binary system to lose energy, so that the orbiting objects

209

gradually spiral in toward one another. These orbits are fairly well described

210

by Newtonian mechanics until about one millisecond before the two objects

211

coalesce.

212

Emitted gravity waves are nearly periodic during the Newtonian phase of

. . . whose
amplitude is
predictable.

213

orbital motion. As a result, these particular gravity waves are easy to predict

214

and therefore easy to search for. When the two objects coalesce, they emit a

215

burst of gravity waves (Figure 11). After coalescence the resulting structure

216

vibrates (“rings down”), emitting more gravity waves as it settles into its final

217

state as a black hole. Initial LIGO has already completed its efforts and would

218

have been sensitive enough (Figure 2) to detect binary neutron star systems

219

coalescing at a distance of about 26 million light years. Unfortunately, no such

220

coalescences were detected during more than one year of observation.

221

Advanced LIGO extends the detection radius to 200 Megaparsecs

≈

650

222

million light years. The volume of space increases approximately as the cube of

223

the distance, so the improved sensitivity will vastly increase the number of

224

galaxies that can be “seen” by LIGO from about one thousand to millions,

225

increasing the odds of success thousands of times.

226

227

QUERY 2. Increased volume of detection

228

Use numerical values given in the preceding paragraph to calculate to two significant figures the

229

increased “odds of success” of Advanced LIGO compared with Initial LIGO.

230

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Motion of Light in Map Coordinates

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Binary coalescence is the only source for which we can currently make a

232

clear prediction of the signal (and therefore of the detection distance limit).

233

Other conceivable sources include supernovae and the collapse of a massive

From other sources:
hard to predict.

234

star to form a black hole—the event that triggers so called

gamma-ray

235

bursts

. But we have only speculations about how far away any of these can be

236

and still be detectable by either Initial LIGO or Advanced LIGO.

237

DETECTORS DO NOT AFFECT GRAVITY WAVES

238

We are used to the fact that metal structures can distort or reduce the

239

amplitude of electromagnetic waves passing across them. Even the

240

presence of a receiving antenna can distort an electromagnetic wave in

241

its vicinity. The same is not true of gravity waves, whose generation or

242

modification requires massive moving structures. Gravity wave detectors

243

have negligible effect on the waves that they are designed to detect.

244

Indeed, it is the smallness of the influence that gravity waves have on

245

mechanical structures that makes gravity waves so difficult to detect.

246

247

QUERY 3. Electromagnetic waves

vs.

gravity waves. Discussion.

248

What property of electromagnetic waves makes their interaction with conductors so huge compared

249

with the interaction of gravity waves with matter of any kind?

250

251

MOTION OF LIGHT IN MAP COORDINATES

252

Light reflected back and forth between mirrored test masses.

253

The LIGO detector is an

interferometer

that employs mirrors mounted on

254

“test masses” suspended at rest at the ends of an L-shaped vacuum cavity.

255

The length of each leg of the L is 4 kilometers. Detection of the gravity wave is

256

accomplished by measuring the relative round-trip

time delay

between light

LIGO is an
interferometer.

257

sent down one leg of the detector and light sent down the other, perpendicular

258

leg.

259

Suppose that a gravity wave of the polarization illustrated in Figure 3

260

moves in the

-direction as shown in Figure 5 and that one leg of the detector

261

lies along the

-direction and the other leg along the

-direction. In order to

262

analyze the operation of LIGO, we need to know (a) how light propagates

263

along the

and

legs of the interferometer and (b) how the test masses at the

264

ends of the legs move when the

-directed gravity wave passes over them. In

265

the present section we analyze the motion of light in map coordinates; Section

266

5 begins the description of the motion of test masses in map coordinates.

267

With what map speed does light move in the

-direction in the presence of

Motion of light in
map coordinates.

268

a gravity wave implied by metric (1)? To answer this question, set

= 0

269

in that equation, yielding

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FIGURE 5

Perspective drawing of the relative orientation of legs of the LIGO

interferometer lying in the

and

directions on the surface of Earth and the

direction of the incident gravity wave descending vertically. [Illustrator: Rotate lower
plate and contents CCW 90 degrees, so corner box is above the origin of the coordinate
system. Same for Figure 10.]

dτ

−

(1 +

)

(5)

As always, the proper time is zero between two adjacent events on the

271

worldline of a light pulse. Set

dτ

= 0 to find the speed of light in the

272

-direction.

273

(1 +

)

−

(light moving in

direction)

(6)

The plus and minus signs correspond to a pulse traveling in the positive or

274

negative

-direction, respectively—that is, in the plane of LIGO in Figure 5.

275

Remember that the magnitude of

is very much smaller than one, so we use

276

the approximation inside the front cover. To first order:

277

(1 +

)

≈

1 +

1 and

(7)

Apply this approximation to (6) to obtain

278

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Motion of Ligo Test Masses in Map Coordinates

≈ ±

−

)

(light moving in

direction)

(8)

In words, the map speed of light is changed (slightly!) by the presence of our

279

gravity wave. Since

is a function of time as well as position, the map speed of

Gravity wave
modifies map
speed of light.

280

light in the

-direction is not constant, but varies as the wave passes through.

281

(Should we worry that the speed in (8) does not have the standard value one?

282

No! This is a

map speed

—a mythical beast—measured directly by no one.)

283

By similar arguments, the map speeds of light in the

and

directions for

284

the wave described by the metric (1) are:

285

≈ ±

(1 +

)

( light moving in

direction)

(9)

( light moving in

direction)

(10)

MOTION OF LIGO TEST MASSES IN MAP COORDINATES

286

“Obey the Principle of Maximal Aging!”

287

Consider two test masses with mirrors suspended at opposite ends of the

-leg

288

of the detector. The signal of the interferometer due to the motion of light

289

along this leg will be influenced only by the

-motion of the test masses due to

290

the gravity wave. In this case the metric is the same as (5).

291

How does a test mass move as the gravity wave passes over it? As always,

292

we answer this question with the Principle of Maximal Aging, maximizing the

293

wristwatch time of the test mass across two adjoining segments of its worldline

294

between fixed end-events. In what follows we verify the surprising result

295

anticipated in Section 2 above, namely that a test mass initially at rest in map

296

coordinates rides with the expanding and contracting map coordinates drawn

297

on the distorting rubber sheet, so this test mass does not move with respect to

298

map coordinates as a gravity wave passes over it. This result comes from

299

showing that an out-and-back jog in the vertical worldline in map coordinates

300

leads to smaller aging and therefore does not occur for a free test mass

301

Figure 6 pictures this idealized case: an incremental linear deviation from

302

a vertical worldline from origin 0 to the event at

= 2

. Along Segment A the

303

displacement

increases linearly with time:

, where the speed

is a

304

constant. Along segment B the displacement returns to zero at the same

Idealized case:
Linear jogs
out and back.

305

constant rate. The strain

has average values ¯

and ¯

along segments A

306

and B respectively. We use the Principle of Maximal Aging to find the value of

307

the speed

that maximizes the wristwatch time along this worldline. We will

308

find that

= 0. In other words, the free test mass initially at rest in map

309

coordinates stays at rest in map coordinates; it does not deviate from the

310

vertical worldline in Figure 6. Now for the details.

311

Write the metric (5) in approximate form for one of the segments:

312

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FIGURE 6

Trial worldline for a test mass; incremental departure from vertical line

of a particle at rest. Segments A and B are very short.

∆

≈

∆

−

(1 + ¯

)∆

(11)

where ¯

is an average value of the strain

across that segment. Now we apply

313

(11) first to Segment A in Figure 6, then to Segment B. We are going to take

314

derivatives of these expressions, which will look awkward applied to ∆ symbols.

315

Therefore we temporarily ignore the ∆ symbols in (12) and let

stand for ∆

316

for ∆

, and

for ∆

, holding in mind that these symbols actually represent

317

increments, so equations in which they appear are approximations.

318

With these substitutions, equation (11) becomes, for the two adjoining

319

worldline segments:

320

≈

2
0

−

1 + ¯

(

)

Segment A

(12)

≈

2
0

−

1 + ¯

(

)

Segment B

so that the total wristwatch time along the bent worldline from

= 0 to

321

= 2

is the sum of the right sides of equations (12).

322

We want to know what value of

(the out-and-back speed of the test

323

mass) will lead to a maximal value of the total wristwatch time. To find this,

324

take the derivative with respect to

of the sum of individual proper times

325

and set the result equal to zero.

326

dτ

≈ −

(1 + ¯

)

−

(1 + ¯

)

= 0

(13)

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Motion of Ligo Test Masses in Map Coordinates

In middle expression of (13), all quantities are fixed except for

. The only

Initially at rest
in map coordinates?
Then stays at rest
in map coordinates.

327

way that (13) can be satisfied is if

= 0.

The test mass initially at rest does

328

not change its map x-coordinate as the gravity wave passes over.

329

Our result seems rather specialized in two senses: First, it treats only the

330

vertical worldline traced out by a test mass at rest. Second, it deals only with

331

a very short segment of the worldline, along which ¯

is considered to be nearly

332

constant. Concerning the second point, you can think of (13) as a tiny

333

out-and-back “jog”

anywhere

on a much longer vertical worldline. Then our

334

result implies that

any

jog in the vertical worldline does not lead to an

335

increased value of the wristwatch time, even if

varies a lot over a longer

336

stretch of the worldline.

337

The first specialization, the vertical worldline, is important: The gravity

338

wave does not cause a kink in a

vertical

map worldline. The same is typically

339

not

true for a particle that is moving in map coordinates before the gravity

340

wave arrives. (We say “typically” because the kink may not appear for some

341

directions of motion of the test mass and for some polarization forms and

342

directions of propagation of the gravity wave.) In this more general case, a

Not at rest in map
coordinates? Maybe
kink in map worldline.

343

kink in the worldline corresponds to a change of velocity. In other words, a

344

passing gravity wave can change the map velocity of a moving particle just as

345

if it were a velocity-dependent force. If the particle velocity is zero, then the

346

force is zero: a particle at rest in map coordinates remains at rest.

347

348

QUERY 4. Disproof of relativity? (optional)

349

“AHA!” declares Kristin Burgess. “Now I can disprove relativity once and for all. If the test mass

350

moves

, a passing gravity wave can cause a kink in the worldline of the test mass as observed in the

351

local inertial Earth frame. No kink appears in its worldline if the test mass is at rest. But if a worldline

352

has a kink in it as observed in one inertial frame, it will have a kink in it as observed in all overlapping

353

relatively-moving inertial frames. An observer in any such frame can detect this kink. So the

absence

354

a kink tells me

and every other inertial observer

that the test mass is ‘at rest’ ? We have found a way to

355

determine absolute rest using a local experiment. Goodbye relativity!” Is Kristin right? (A detailed

356

answer is beyond the scope of this book, but you can use some relevant generalizations drawn from

357

what we already know to think about this paradox. As an analogy from flat-spacetime

358

electromagnetism, think of a charged particle at rest in a purely magnetic field: The particle

359

experiences no magnetic force. In contrast, when the same charged particle moves in the same frame, it

360

may experience a magnetic force for some directions of motion.)

361

362

In this book we make every measurement in a local inertial frame, not

363

using differences in global map coordinates. So of what possible use is our

364

result that a particle at rest in global coordinates does not move in those

At rest in map
coordinates?
Still can move
in Earth coordinates.

365

coordinates when a gravity wave passes over it? Answer: Just because

366

something is at rest in map coordinates does not mean that it is at rest in

367

local inertial Earth coordinates. In the following section we find that a gravity

368

wave

does

move a test mass as observed in the Earth coordinates.

369

LIGO—attached to the Earth—can detect gravity waves!

370

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Chapter 17

Gravitational Waves

DETECTION OF A GRAVITY WAVE BY LIGO

371

Make measurement in the local Earth frame.

372

Suppose that the gravity wave that satisfies metric (1) passes over the LIGO

373

detector oriented as in Figure 5. We know how the test masses at the two ends

374

of the legs of the detector respond to the gravity wave: they remain at rest in

375

map coordinates (Section 5). We know how light propagates along both legs:

376

as the gravity wave passes through, the map speed of light varies slightly from

377

the value one, as given by equations (8) through (10) in Section 4.

378

The trouble with map coordinates is that they are arbitrary and need not

Earth frame
tied to LIGO slab

379

correspond to what an observer measures. Recall that we require all

380

measurements to take place in a local inertial frame. So think of a local

381

reference frame anchored to the concrete slab on which LIGO rests. As

382

explained in the Introduction (Section 1), the gravity wave has essentially no

383

effect on this slab. Call the coordinates in the resulting local coordinate

384

system

Earth coordinates

. Earth coordinates are analogous to shell

385

coordinates for the Schwarzschild black hole; useful only locally but yielding

386

the numbers that predict results of measurements. The metric for the local

387

inertial frame then has the form:

388

∆

≈

∆

2
Earth

−

∆

2
Earth

−

∆

Earth

−

∆

Earth

(14)

Compare this with the approximate version of (1):

389

∆

≈

∆

−

(1 +

)∆

−

)∆

−

∆

(

(15)

Legalistically, in order to make the coefficients in (15) constants we should use

390

the symbol ¯

, with a bar over the

, to indicate the average value of the

391

gravity wave amplitude over the detector. However, in Query 1 you showed

392

that for the frequencies at which LIGO is sensitive, the wavelength is very

Earth frame
coordinate
differences

393

much greater than the dimensions of the detector, so the amplitude

of the

394

gravity wave is effectively uniform across the LIGO detector. Therefore it is

395

not necessary to take an average, and we use the symbol

without a

396

superscript bar.

397

Comparing (14) with (15) yields:

398

∆

Earth

= ∆

(16)

∆

Earth

= (1 +

)

∆

≈

(1 +

)∆

∆

Earth

= (1

−

)

∆

≈

−

)∆

∆

Earth

= ∆

where we use approximation (7). Notice, first, that Earth time lapse ∆

Earth

399

between two events is identical to their map time lapse ∆

and the

400

component of their space separation in Earth coordinates, ∆

Earth

, is identical

401

to the

component of their separation in map coordinates, ∆

402

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Detection of a gravity wave by LIGO

Now for the differences! Let ∆

be the map

-coordinate separation

403

between the pair of mirrors in the

-leg of the LIGO interferometer and ∆

404

the map separation between the corresponding pair of mirrors in the

-leg. As

405

the

-directed wave passes through the LIGO detector, the test masses at rest

406

at the ends of the legs stay at rest in map coordinates, as Section 5 showed.

407

Therefore the value of ∆

remains the same during this passage, as does the

Test masses move
in Earth coordinates.

408

value of ∆

. But the presence of the time-varying strain

(

) in (16) tells us

409

that these test masses move when observed in Earth coordinates. More: when

410

the distance between test masses increases (say) along the Earth

-axis, it

411

decreases along the perpendicular Earth

-axis; and vice versa. Perfect for

412

detection of a gravity wave by an interferometer!

413

Earth metric (14) is that of an inertial frame in which the speed of light

Light speed

= 1

in Earth frame.

414

has the value one in whatever direction it moves. With light we have the

415

opposite weirdness to that of the motion of test masses initially at rest: In

416

map coordinates light moves at speeds different from one in the presence of

417

this gravity wave—equations (8) through (10)—but in Earth coordinates light

418

moves with speed one. This is reminiscent of the corresponding case near a

419

Schwarzschild black hole: Light moves at speeds different from one in

420

Schwarzschild map coordinates but at speed one in shell coordinates.

421

In summary

the situation is this: As the gravity wave passes over the

422

LIGO detector, the speed of light propagating down the two legs of the

423

detector has the usual value one as measured by the Earth observer. However,

Different Earth
times along
different legs

424

for the Earth observer the separations between the test masses along the

-leg

425

and the

-leg change: one increases while the other decreases, as given by

426

equations (16). The result is a difference in the round-trip times of light along

427

the two legs. It is this difference that LIGO is designed to measure and

428

thereby to detect the gravity wave.

429

What will be the value of this difference in round-trip times between light

430

propagation along the two legs? Let D be the length of each leg in the absence

431

of the gravity wave. The round-trip time is twice this length divided by the

432

speed of light, which has the value one in Earth coordinates. From equations

433

(16) we find that the difference in round-trip times between light propagated

434

along the two legs is

435

∆

Earth

= 2

(one round trip of light)

(17)

Using the latest interferometer techniques, LIGO reflects the light back

Time difference
after

round trips.

436

and forth down each leg approximately

= 140 times. That is, light executes

437

approximately 140 round trips, which multiplies the detected time shift,

438

increasing the sensitivity of the detector by the same factor. Equation (17)

439

becomes

440

∆

Earth

= 2

N Dh

(

round trips of light)

(18)

Quantities

and

have no units, so the unit of time in (18) is the same as

441

the unit of D, for example meters.

442

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Chapter 17

Gravitational Waves

443

QUERY 5. LIGO fast enough?

444

Do the 140 round trips of light take place in a time small compared with one period of the gravity wave

445

being detected? (If it does not, then LIGO detection is not fast enough to track the change in gravity

446

strain.)

447

448

449

QUERY 6. Application to LIGO.

450

Each leg of the LIGO interferometer is of length

= 4 kilometers. Assume that the laser emits light of

451

wavelength 1000 nanometer = 10

−

meter (infrared light from a NdYAG laser). Suppose that we want

452

LIGO to reach a sensitivity of

= 10

−

. For

= 140, find the corresponding value of ∆

Earth

453

Express your answer as a decimal fraction of the period

of the laser light used in the experiment.

454

(For background see

http://www.ligo.caltech.edu/LIGO_web/about/

)

455

456

457

QUERY 7. Faster derivation?

458

In this book we insist that global map coordinates are arbitrary human choices, so that we cannot

459

depend on map coordinate differences to be directly measurable quantities. However, the value of

460

(1) is so small that the metric differs only slightly from an inertial metric. This once, therefore, we treat

461

map coordinates as directly measurable and ask you to redo the derivation of equations (17) and (18)

462

using only map coordinates.

463

Remember that test masses initially at rest in map coordinates do not change their coordinates as

464

the gravity wave passes over them (Section 4), but the gravity wave alters the map speeds of light, and

465

differently in the

-direction, equation (8), than in the

-direction, equation (9). Assume that each leg

466

of the interferometer has the length

map

in map coordinates.

467

A. Find an expression for the difference ∆

in map time between the two legs for one round trip of

468

the light.

469

B, How great do you expect the difference to be between times ∆

and ∆

Earth

and the difference

470

between distances

(in Earth coordinates) and

map

? Taken together, will these differences be

471

great enough so that the result of your prediction and that of equation (18) could be

472

distinguished experimentally?

473

474

475

QUERY 8. Different directions of propagation of the gravity wave

476

Thus far we have assumed that the gravitational plane wave of the polarization described by equation

477

(1) descends vertically onto the LIGO detector, as shown in Figure 5. Of course the observers cannot

478

prearrange in what direction an incident gravity wave will move. Suppose that the wave described by

479

(1) propagates along the direction of, say, the

-leg of the interferometer, while the

-direction lies

480

along the other leg, as before. What is the equation that replaces (18) in this case?

481

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Binary System as a Source of Gravity Waves

FIGURE 7

A binary system with each object in a circular path.

482

483

QUERY 9. LIGO fails to detect a gravity wave?

484

Think of various directions of propagation of the gravity wave pictured in Figure 3, together with

485

different directions of

and

in equation (1) with respect to the LIGO detector. Give the name

486

orientation

to a given set of directions

and

—the transverse directions in (1)—plus

(the direction

487

of propagation) in (1) relative to the LIGO detector. How many orientations are there for which LIGO

488

will detect

no signal whatever

, even when its sensitivity is 10 times better than that needed to detect

489

the wave arriving in the orientation shown in Figure 5? Are there zero such orientations? one? two?

490

three? some other number less than 10? an infinite number?

491

492

BINARY SYSTEM AS A SOURCE OF GRAVITY WAVES

493

Proof of the existence of gravity waves?

494

Now we consider in more detail gravity waves generated by a binary system

495

consisting of two neutron stars, each in circular orbit around their center of

496

mass. The binary system is the only known example of a stellar system for

Unequal masses,
each in circular
orbit.

497

which we can explicitly calculate the emitted gravity waves. Suppose that the

498

stars of the binary system have masses

and

and are assumed to orbit

499

at a constant distance

apart, as shown in Figure 7.

500

The basic parameters of the orbit are adequately computed using

Energy of the system.

501

Newtonian mechanics, according to which the energy of the system in

502

conventional units is given by the expression:

503

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Chapter 17

Gravitational Waves

conv

−

(Newton)

(19)

As these neutron stars orbit, they generate gravity waves. General

Rate of
energy loss . . .

504

relativity predicts the rate at which the orbital energy is lost to this radiation.

505

In conventional units, this rate is:

506

conv

−

(

)

(

)

(Newtonian circular orbits)

(20)

Equation (20) assumes that the two stars are separated by much more than

507

their Schwarzschild radii and that they are moving at nonrelativistic speeds.

508

Deriving equation (20) involves a lengthy and difficult calculation starting

. . . derived from
Einstein’s equations.

509

from Einstein’s field equations. The same is true of the derivation of the metric

510

(1) for a gravity wave. These are two of only three equations in this chapter

511

that we simply quote from a more advanced treatment of general relativity.

512

513

QUERY 10. Energy and rate of energy loss in geometric units

514

Convert equations (19) and (20) to geometric units to be consistent with our notation and to get rid of

515

the constants

and

. Use the sloppy professional shortcut, “Let

= 1.”

516

A. Show that (19) and (20) become:

517

−

(Newton: geometric units,

in units of length)

(21)

−

(

)

(

)

(geometric units,

in units of length)

(22)

B. Verify that in both of these equations

has the unit of length.

518

C. Suppose you are given the value of

in meters. Show how you would convert this value first to

519

kilograms and then to joules.

520

521

522

QUERY 11. Rate of change of radius

523

Derive an expression for the rate at which the radius changes as a result of this energy loss. Show that

524

the result is:

525

−

(

)

(Newton: circular orbits)

(23)

526

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Binary Pulsar PSR1913+16

BINARY PULSAR PSR1913+16

527

Proof of gravity waves?

528

On July 2, 1974 Russell A. Hulse was carrying out observations at the world’s

Hulse and Taylor
discover binary.

529

largest radio telescope at Arecibo, Puerto Rico. Hulse—a graduate student

530

working under the direction of Joseph H. Taylor, then at the University of

531

Massachusetts, Amherst—detected signals from a pulsar later named

532

PSR1913+16. (PSR stands for “pulsar” and the numbers denote its celestial

533

coordinates.) Here is an account of the discovery, excerpted from the Nobel

534

Foundation website (which also has wonderful illustrations)

535

http://www.nobel.se/physics/laureates/1993/illpres/discovery.html

536

(Copyright c

2001 The Nobel Foundation)

537

THE DISCOVERY OF THE BINARY PULSAR

538

During 1974 Joseph Taylor and Russell Hulse were searching for new

539

pulsars with the Arecibo telescope. They discovered 40, one of which was

540

to be very important.

541

When Hulse was observing the new pulsar, which has been named

542

PSR1913+16, he found that the pulses arrived sometimes more often

543

and sometimes less. The simplest interpretation was that the pulsar was

544

orbiting another star very closely and at high velocity: Here one ”pulsar

545

year” is only about eight hours.

546

By observing the shift in the pulses, Hulse and Taylor found that the

547

stars were equally heavy, each weighing about 1.4 times as much as the

548

Sun. Since they were not visible on any photographs either, it was

549

concluded that the other body, somewhat unexpectedly, was also a

550

neutron star. Seen from Earth, however, it does not show up as a pulsar.

551

. . . .

552

MEASURING gravity waves

553

Since the two neutron stars in PSR1913+16 are moving so fast and close

554

together they should, according to General Relativity, emit large

555

amounts of gravity waves. This makes them lose energy: Their orbits will

556

therefore shrink and their orbiting period will shorten.

557

Indirect evidence:

The binary pulsar has been observed continuously

558

since its discovery, and the orbiting period has in fact decreased.

559

Agreement with the prediction of General Relativity is better than

560

1/2%. This is considered to prove that gravity waves really exists. In

561

turn, this result is currently one of our strongest supports for the

562

validity of the General Theory of Relativity.

563

The signal from the pulsar constituted a very stable clock, stable to 10

564

significant figures. As a result, Hulse and Taylor were able to use general

565

relativity to analyze the motion of the system in detail, verifying many general

Data from
binary pulsar

566

relativity predictions, some of which allowed them to determine the individual

567

orbiting masses

and

(given below), which Newtonian mechanics does

568

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Chapter 17

Gravitational Waves

not reveal. Their results show that the binary system PSR1913+16 has the

569

following parameters:

570

= (1

442

003)

Sun

(pulsar)

(24)

= (1

386

003)

Sun

(companion)

= 2

3418

0001

light seconds

(Semi-major axis of both)

= 0

617127

000003

(Eccentricity of both)

Orbital period,

≈

7.75 hours

571

Rate of advance of the periastron

≈

4.2 degrees per Earth-year

572

Distance from Earth

≈

7 kiloparsecs or about 20 000 light years.

573

(This distance is quite uncertain.)

574

Each neutron star follows its own elliptical path about the center of mass.

575

The semi-major axis of the elliptical orbit for a neutron star—label it

—is

576

half of the major axis, the longest distance from one side of its orbit to the

577

other. The semi-minor axis—label it

—is half of the minor axis. Then the

578

eccentricity

≡

(

−

)

. The word

periastron

refers to the point of

Meaning of

periastron

579

closest approach of these “astron”omical objects (just as the word

perihelion

580

refers to the point of closest approach of an orbiting object to our Sun: Greek,

581

“Helios”). Note how large the rate of this periastron advance is compared with

582

43 arcseconds of advance of the perihelion of the planet Mercury

per

583

Earth-century

584

The non-zero eccentricity in equation (24) tells us that the neutron stars

585

in PSR1913+16 are

not

in circular orbits. General relativity predicts that

586

when a binary system has non-circular orbits it will radiate gravity waves at a

587

greater rate than when the orbits are circular. Nevertheless, in the following

588

QUERIES we assume for simplicity that the orbits are effectively circular, as

589

in Figure 7. That is, we assume a binary system in which each companion is in

We assume
circular orbits.

590

a circular orbit with constant radial separation

equal to the major axis,

591

twice the value of the semi-major axis given in (24). This is equivalent to

592

setting to zero the eccentricity of each neutron star orbit.

593

594

QUERY 12. Shrinkage of

per orbit

595

For a single orbit, the separation

between the orbiting neutron stars (assumed to be in circular

596

orbits) does not change much, but it does change a little. For one orbit, what is the approximate value

597

of the change in this separation

? Express your answer in millimeters. (

Hint:

No integration is needed

598

for an approximate calculation of this incremental change.)

599

600

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Binary Pulsar PSR1913+16

FIGURE 8

Decrease in the period in seconds (vertical axis) over the years 1975 to 1998

(horizontal axis) of binary system PSR1913+16. Agreement with the prediction of general
relativity, assuming the change is due to emission of gravity waves, is now within 0.3 percent.
This agreement appears to eliminate any other possible explanation for the change in orbits.
From a paper (and Copyright c

2000) by J.H. Taylor and J. M. Weisberg.

601

QUERY 13. Energy radiated by idealized binary PSR1913+16

602

A. What is the power currently being radiated in gravity waves? Express your answer as a unitless

603

measure (energy in meters divided by time in meters) and also in watts (joules per second).

604

B. Use equation (19) or (21) to calculate how much total energy in joules will be radiated in

605

gravity waves from the present year to the future time when the two companions are separated

606

= 20 kilometers (approximately the sum of their radii)? This total energy corresponds to

607

how many kilograms of mass converted entirely to energy?

608

C. How long a time in years will it be before the two neutrons stars in PSR1913+16 are separated

609

by only

= 20 kilometers, so that coalescence is imminent? (Only in the last millisecond or so

610

before coalescence does the Newtonian description become completely useless.)

611

612

ADD QUERY ABOUT RADIATION RATE OF SUN-MERCURY BINARY

613

SYSTEM AND LENGTH OF TIME TO COALESCE DUE TO GRAV

614

RADIATION.

615

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Chapter 17

Gravitational Waves

FIGURE 9

Figure 7 augmented to show the center of mass and orbit radii of individual

components of PSR1913+16.

GRAVITY WAVE AT EARTH DUE TO DISTANT BINARY SYSTEM

616

How far away from a binary system can we detect gravity waves?

617

Can LIGO on Earth’s surface detect the gravity waves emitted by the distant

618

binary system PSR1913+16 (idealized as one in which the neutron stars move

619

in circular orbits as shown in Figure 7)? To answer this question we need to

620

calculate the magnitude of

in the metric of equation (1).

621

Here is the third and final result of general relativity quoted without proof

Gravity waveform . . .

622

in this chapter. The function

(

z, t

) is given by the equation (in conventional

623

units)

624

(

z, t

) =

−

cos

πf

(

−

)

(conventional units)

(25)

where

is the frequency of the binary orbit,

is the (constant!) distance

625

between orbiters in Figures 7 and 9, and

is the distance from source to

626

detector. Convert (25) to geometric units by setting

= 1. Note that

627

(

z, t

) is a function of

and

628

Figure 10 schematically displays the notation of equation (25), along with

629

relative orientations and relative magnitudes assumed in the equation. This

630

equation makes the Newtonian assumptions that (a) the two stars are

631

separated by a distance

much larger than their Schwarzschild radii and (b)

632

they move at nonrelativistic speeds. Additional assumptions are:

633

between the binary system and Earth is very much greater

634

than a wavelength of the gravity wave. This assumption assures that the

635

radiation at Earth constitutes the so-called “far radiation field” where it

636

assumes the form of a plane wave given in equation (4).

637

(d) The binary stars are orbiting in the

plane, so that from Earth the

638

orbits would appear as circles if we could see them (which we cannot, because

639

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gravity wave at Earth Due to Distant Binary System

FIGURE 10

Schematic diagram,

not to scale

, showing notation and relative magnitudes

for equation (25). The binary system and the LIGO detector lie in parallel planes.[Illustrator:
See note in caption to Figure 5.]

they are too far away). Unfortunately this assumption is not true of the plane

640

of the orbit of binary PSR1913+16, as we know from Doppler shifts of signals

641

from the orbiting pulsar.

642

Equation (25) describes only one linear polarization at Earth, the one

643

generated by metric (1) and shown in Figure 3. The orthogonal polarization

. . . for one case

644

shown in Figure 4 is also transverse and equally strong, with components

645

proportional to (1

). The formula for the magnitude of

in that

646

orthogonally polarized wave is identical to (25) with a sine function replacing

647

the cosine function. We have not displayed the metric for that orthogonal

648

polarization.

649

In order for LIGO to detect a gravity wave, two conditions must be met:

650

(a) the amplitude

of the gravity wave must be sufficiently large, and (b) the

Detection
requirements

651

frequency of the wave must be in the range in which LIGO is most sensitive

652

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Chapter 17

Gravitational Waves

(100 to 400 hertz). QUERY 13 deals with the amplitude of the wave. The

653

frequency of gravity waves, discussed in QUERY 14, contains a surprise.

654

655

QUERY 14. Amplitude of gravity wave from PSR1913+16 at Earth

656

A. Use (25) to calculate the maximum amplitude of

at Earth due to the radiation from the

657

“idealized circular-orbit” binary system PSR1913+16. Consider this amplitude to be positive.

658

B. Can either Initial LIGO or Advanced LIGO detect the gravity waves whose amplitude is given

659

in part A?

660

C. What is the maximum amplitude of

at Earth just before coalescence of PSR1913+16, when

661

the neutron stars are separated by a distance

= 20 kilometers (but with orbits still described

662

approximately by Newtonian mechanics)?

663

664

665

QUERY 15. Frequency of gravity waves emitted from PSR1913+16

666

A. In order for either Initial LIGO or Advanced LIGO to detect the gravity waves whose amplitude

667

is given in Query 13, the frequency of the gravity wave must be in the range 100 to 400 hertz. In

668

Figure 9 the point C. M. is the stationary center of mass of the pulsar system. Using the

669

symbols in Figure 9, fill in the steps to complete the following derivation.

670

(for

, Newton, conventional units)

(26)

(for

, Newton, conventional units)

(27)

(center-of-mass condition)

(28)

orbit

≡

orbit

πr

(common orbital frequency)

(29)

where

orbit

and

orbit

are the frequency and period of the orbit, respectively. From these

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equations, show that for

≡

the frequency of the orbit is

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orbit

(

)

(30)

B. Here is a surprise: The frequency

of the gravity wave generated by this binary pair and

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appearing in (25) is twice the orbital frequency.

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gravity wave

= 2

orbit

(31)

Why this doubling? Essentially it is because gravity waves are waves of tides. Just as there are

675

two high tides and two low tides per day caused by the moon’s gravity acting on the Earth,

676

there are two peaks and two troughs of gravity waves generated per binary orbit.

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July 7, 2010 08:06

GravWaves100707V2

Sheet number 25 Page number 25

AW Physics Macros

References

FIGURE 11

The “chirp” of one polarization of gravity wave as the two components of a binary

system coalesce. (The plot for the other polarization is indistinguishable from this

one.)

C. Approximate the average of the component masses in (24) by the value

= 1

Sun

. Find the

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distance

between the binary stars when the orbital frequency is 75 hertz, so that the frequency

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of the gravity wave is 150 hertz. [ANS: Approximately 100 km.]

680

D. Using results quoted earlier in this chapter, estimate the time for the binary system to decay

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from the current radial separation to the radial separation calculated in part C.

682

ANS:

−

= 5(

−

)

(256

), everything in unit meter.

683

684

Newtonian mechanics predicts the motion of the binary system

“Chirp” at
coalescence

685

surprisingly accurately until the two components touch, a few milliseconds

686

before they coalescence. As this happens, the gravity wave sweeps upward in

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both frequency and amplitude in what is called a

chirp

. Figure 11 is a

688

predicted wave form for such a chirp.

689

To hear an audio simulation of the chirp, visit one of the following

690

websites:

691

http://www.lsc-group.phys.uwm.edu/~patrick/work/talks/itp/chirp.002.au

692

http://www.lsc-group.phys.uwm.edu/~patrick/work/talks/itp/chirp.002.wav

693

Detection of such a waveform sweeping through the frequencies for which

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LIGO is sensitive would be a “smoking gun” for the coalescence of a binary

695

source. Although LIGO cannot detect emission from PSR1913+16, we expect

696

that many other binary systems are close to provide a detectable signal for

697

Advanced LIGO.

698

REFERENCES

699

First initial quote: Norbert Wiener,

I am a Mathematician

, MIT Press,

700

Cambridge, MA, 1956, page 109

701

Second initial quote: Arthur Eddington,

Stars and Atoms

(1928), Lecture 1

702

LIGO sensitivity, Figure 2, at

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http://www.ligo.caltech.edu/advLIGO/scripts/summary.shtml

704