Table Of ContentPearson New International Edition
Gravity: An Introduction
to Einstein's General Relativity
James B. Hartle
First Edition
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ISBN 13: 978-1-292-03914-5
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Chapter 1. Gravitational Physics
James B. Hartle 1
Chapter 2. Geometry as Physics
James B. Hartle 13
Chapter 3. Space, Time, and Gravity in Newtonian Physics
James B. Hartle 31
Chapter 4. Principles of Special Relativity
James B. Hartle 47
Chapter 5. Special Relativistic Mechanics
James B. Hartle 77
Chapter 6. Gravity as Geometry
James B. Hartle 105
Chapter 7. The Description of Curved Spacetime
James B. Hartle 135
Chapter 8. Geodesics
James B. Hartle 169
Chapter 9. The Geometry Outside a Spherical Star
James B. Hartle 187
Chapter 10. Solar System Tests of General Relativity
James B. Hartle 221
Chapter 11. Relativistic Gravity in Action
James B. Hartle 236
Chapter 12. Gravitational Collapse and Black Holes
James B. Hartle 257
Chapter 13. Astrophysical Black Holes
James B. Hartle 283
I
233334444455913460147914823982973373
Chapter 14. A Little Rotation
James B. Hartle 298
Chapter 15. Rotating Black Holes
James B. Hartle 312
Chapter 16. Gravitational Waves
James B. Hartle 333
Chapter 17. The Universe Observed
James B. Hartle 349
Chapter 18. Cosmological Models
James B. Hartle 368
Chapter 19. Which Universe and Why?
James B. Hartle 402
Chapter 20. A Little More Math
James B. Hartle 419
Chapter 21. Curvature and the Einstein Equation
James B. Hartle 447
Chapter 22. The Source of Curvature
James B. Hartle 473
Chapter 23. Gravitational Wave Emission
James B. Hartle 493
Chapter 24. Relativistic Stars
James B. Hartle 517
Index 543
II
PART
I
Space and Time in Newtonian Physics
and Special Relativity
The major phenomena of gravitationalphysics are briefly
described and the idea that the geometry of space and time
is a physical question isintroduced. Essential elementsof
Newtonian physics and special relativityare reviewed. Tools
for describing the geometry of spacetime are developed.
From Part I and Chapter 1 of Gravity: An Introduction to Einstein's General Relativity, First Edition. James B. Hartle.
Copyright © 2003 by Pearson Education, Inc. All rights reserved.
1
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C H A P T E R
1
Gravitational Physics
Gravity is one of the four fundamental interactions. The classical theory of
gravity—Einstein’s general relativity—is the subject of this book. General rel-
ativity is central to the understanding of frontier astrophysical phenomena such
as black holes, pulsars, quasars, the final destiny of stars, the big bang, and the
universe itself. General relativity is also concerned with the minute departures
of the orbits of the planets from the laws of Newton and is a necessary ingre-
dient in the operation of the Global Positioning System used every day. As one
of the fundamental forces, gravity is central to the quest for a unified theory of
all interactions; many of the ideas for these “final theories” originate in general
relativity.
Gravitationalphysicsisthusatwo-frontierscience.Itsimportantapplications
lieatboththelargestandsmallestdistancesconsideredincontemporaryphysics.
Onthelargestscales,gravitationalphysicsislinkedtoastrophysicsandcosmol-
ogy.Onthesmallestscalesitistiedtoquantumandelementaryparticlephysics.
Thesetwofrontiersbecomeoneatthebigbang,wherethewholeoftheobservable
universetodayiscompressedintothesmallestpossiblevolume.Thisintroductory
texttreatsonlytheclassical(nonquantum)theoryofgravitywhosedirectappli-
cationsaremostlyonlargedistancescales,buttheideasandmethodsdeveloped
herereemergeindifferentguisesatthefrontieroftheverysmall.Thisintroduc-
tion gives a brief survey of some of the phenomena for which classical general
relativityisimportant.
Theoriginsofgeneralrelativitycanbetracedtotheconceptualrevolutionthat
followedEinstein’sintroductionofspecialrelativityin1905.Newton’scenturies-
old gravitational force law is inconsistent with special relativity. According to
Newton’slaw,twobodiesofmassm andm attractoneanotherwithagravita-
1 2
tionalforcewhosemagnitudeis
Gm m
F = 1 2, (1.1)
grav r2
12
where r is the distance between them, and G is Newton’s gravitational con-
12
stant6.67×10−8dyn·cm2/g2.TheNewtoniangravitationalforceactsinstanta-
neously.Theforceononemassdependsonthepositionofthesecondatthesame
time.However,instantaneousinteractionisprohibitedinspecialrelativitywhere
3
3
4 Chapter1 GravitationalPhysics
nosignalcantravelfasterthanthespeedoflight.Newtoniangravitycantherefore
onlybeanapproximationtoayetmorefundamentaltheory.
In 1915, Einstein’s quest for a relativistic theory of gravity resulted not in a
new force law or a new theory of a relativistic gravitational field, but in a pro-
foundconceptualrevolutioninourviewsofspaceandtime.Einsteinsawthatthe
experimentalfactthatallbodiesfallwiththesameaccelerationinagravitational
fieldlednaturallytoanunderstandingofgravityintermsofthecurvatureofthe
four-dimensionalunionofspaceandtime—spacetime.Masscurvesspacetimein
itsvicinity,andthetrajectoriesalongwhichallmassesfallarethestraightpaths
inthiscurvedspacetime.InNewtoniantheorytheSunexertsagravitationalforce
on the Earth and the Earth moves around the Sun in response to that force. In
generalrelativitythemassoftheSuncurvesthesurroundingspacetime,andthe
Earthmovesonastraightpathinthatcurvedspacetime.Gravityisgeometry.
Theremainderofthischapterbrieflyintroducessomephenomenaintheuni-
verse for whose understanding general relativity is important. A few properties
ofthegravitationalinteractionthathelptoexplainwhengravityisimportantcan
alreadybeseenfromtheNewtoniangravitationalforcelaw(1.1):
• Gravityis a universal interactionin Newtoniantheorybetween all mass, and,
since E =mc2,inrelativisticgravitybetweenallformsofenergy.
• Gravity is unscreened. There are no negative gravitational charges to cancel
positiveones,andthereforeitisnotpossibletoshield(screen)thegravitational
interaction.Gravityisalwaysattractive.
• Gravityisalong-rangeinteraction.TheNewtonianforcelawisa1/r2interac-
tion.Thereisnolengthscalethatsetsarangeforgravitationalinteractionsas
thereareforstrongandweakinteractions.
• Gravity is the weakest of the four fundamental interactions acting between
individual elementary particles at accessible energy scales. The ratio of the
gravitational attraction to the electromagnetic repulsion between two protons
separatedbyadistancer is
F Gm2/r2 Gm2
grav = p = p ∼10−36, (1.2)
F e2/(4π(cid:6) r2) (e2/4π(cid:6) )
elec 0 0
wherem isthemassoftheprotonandeisitscharge.
p
These four facts explain a great deal about the role gravity plays in physical
phenomena. They explain, for example, why, although it is the weakest force,
gravitygovernstheorganizationoftheuniverseonthelargestdistancescalesof
astrophysicsandcosmology.Thesedistancescalesarefarbeyondthesubatomic
rangesofthestrongandweakinteractions.Electromagneticinteractionscouldbe
long range were there any large-scale objects with net electric charge. But the
universeiselectricallyneutral,andelectromagneticforcesareso muchstronger
than gravitational forces that any large-scale net charge is quickly neutralized.
Gravityislefttogovernthestructureoftheuniverseonthelargestscales.
4
Chapter1 GravitationalPhysics 5
1060
present universe
1050
universe at helium fusion
1040 black hole in Milky Way
MilkyWay
black hole in X-ray source
Sun
1030
GPS orbit
n grams 1020 inside black holes neutron star
s i primordial black hole
mas 1010 evaporating today
human
universe at the
1 end of inflation laboratorymeasurement
of Newton’s G
universe at the
quantum gravity scale
10(cid:1)10
strand of DNA
10(cid:1)20 probed by best
accelerators hydrogen atom
10(cid:1)30
10(cid:1)30 10(cid:1)20 10(cid:1)10 1 1010 1020 1030
distance in cm
FIGURE1.1 Gravitationalphysicsdealswithphenomenaonscalesofdistanceandmass
rangingfromthemicroscopictothecosmic—thelargestrangeofscalesconsideredincon-
temporaryphysics.Therearephenomenaforwhichgravityisimportantoverthiswhole
rangeofscalesthatareshownonthisplotofcharacteristicmass M vs.characteristicdis-
tance R.Representativeonesareindicatedbycircles.Otherillustrativephenomenawhere
gravitationplayslittleroleareshownbysquares.Phenomenaabovethediagonallineare
unobservable,becausetheytakeplaceinsideblackholes.Phenomenaclosetothediago-
nalline2GM = c2R aretheonesforwhichrelativisticgravityisimportant.Thelargest
scalesarethefrontierofastrophysics;thesmallestarethoseofelementaryparticlephysics.
Thesmallestdistanceshown(∼10−33cm)isthePlancklengthmarkingtheboundarybe-
tweenclassicalandquantumgravity.Scalesreferringtotheuniverseatvariousmoments
initshistorydenotethesizeofthevolumethatlightcouldtravelacrosssincethebigbang
andthemassinsidethatvolumeiftheuniversealwayshadtheexpansionrateithadatthat
moment.
5
