Table Of ContentGraduate Texts in Physics
Ronald J. Adler
General Relativity
and Cosmology
A First Encounter
Graduate Texts in Physics
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Ronald J. Adler
General Relativity
and Cosmology
A First Encounter
123
RonaldJ. Adler
Department ofPhysics andAstronomy
SanFrancisco State University
SanFrancisco, CA, USA
GravityProbe BMission
Hansen Experimental Physics Laboratory
StanfordUniversity
Stanford, CA,USA
ISSN 1868-4513 ISSN 1868-4521 (electronic)
Graduate Textsin Physics
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Preface
Four truly profound questions have always permeated science and served as its
basis: what is matter, what is the universe, what is life, and what is thought. The
twentieth century has been extraordinary in that three of these have been at least
partially answered. The answers in brief and broad outline are: matter is made of
quarks and leptons and the quantum gauge fields that hold them together; the
universe is an isotropic and homogeneous expanding curved spacetime that is
dominated on the cosmological scale by gravity; life is a mechanism whereby the
molecular polymer deoxyribonucleic acid (DNA) makes more DNA from its
environment.
We have not been so successful concerning the nature of thought; indeed some
people are of the opinion that almost no real progress has been made. One rather
entertainingviewisthatthoughtisanillusionandwedonotreallythinkatall.That
istosaywemerelythinkthatwethink.Perhapsitisgoodthatthereremainssucha
field with so much to be explored in our future mental adventures.
While the nature of matter and life are part of the undergraduate curriculum in
physicsandbiology,thenatureoftheuniverseisoftenleftforagraduatecourseon
general relativity, and cosmology is often treated only briefly at the end of the
course.Someuniversitiesofferanundergraduatecourseoncosmologynotbasedon
generalrelativity, butofcoursethisisnosubstituteforamorecompletetreatment.
Thereisnogoodreasonthatgeneralrelativityandcosmologyshouldnotbestudied
by undergraduates as well as beginning graduate students. This book is thus
directedprimarilyatbeginninggraduatestudentsbutalsoatadvancedandconfident
undergraduatestudents.Themathematicsneededisonlyashortstepbeyondvector
and matrix analysis, the physical concepts are simpler than those of quantum
mechanics, and the subjects have become very mainstream. Such topics as black
holes, dark matter, the shape of the universe, the big bang, the primordial fireball,
and the ultimate fate of the universe can be appreciated and understood by almost
anyonewithanundergraduatephysicsbackground.Someoftheappendicesshould
help undergraduates and others with gaps in their background.
vii
viii Preface
General relativity theory began in the early twentieth century in the borderland
betweenphysicsandmathematics.Aftertheinitialconfrontationofthetheorywith
the three classic tests (red shift, Mercury perihelion shift, deflection of starlight),
there was little contact between theory and observation until the last half of the
century. But then the discovery of the cosmic microwave background radiation
madeitclearthatthetheoryhadmuchtoofferfordescribingtheevolvinguniverse.
Since then the field of observational cosmology has blossomed, using many dif-
ferent approaches to measuring the properties of the universe on a large scale.
Theoretical cosmology has naturally blossomed with it and the combination of
observationandtheoryhasresultedinthepresentstandardmodelofcosmology,the
lambdacolddarkmatterorLCDMmodel.Itisfairtosaythatthereisnownomore
active area in fundamental physics than cosmology.
Butwemustnotunderestimate theprogressinrelativitytheoryandobservation
for other basic systems, notably neutron stars and black holes. The agreement
betweenblackholetheorybasedontheKerrmetricanddiverseobservationsisone
of the most impressive successes in physics. This is most relevant now that it has
becomeapparenthowimportantsupermassiveblackholesareforthestructureand
evolution of the universe.
Another truly extraordinary prediction of general relativity has been verified
with the observation of gravitational waves. The first waves detected were gener-
ated by binary black hole and neutron star mergers, using the LIGO and Virgo
detectors.Thedetectionrequiredacenturyofthoughtanddecadesofexperimental
effort. Certainly, the connection of the two extraordinary predictions of relativity
theory,blackholesandgravitationalwaves,ismostimpressiveandgratifying.The
futurepromisestobeevenmoreinterestingsincegravitationalwavesareanentirely
newobservationalwindowonthecosmos,andthereisnowaytopredictwhatthey
might reveal.
Clearly,thefrontieroffundamentalphysicsresearchhasnowshiftedtothelarge
end of the distance scale, the universe. But our understanding of the universe
requires also an understandingof thesmallend of thedistance scale,most notably
in our study of the early universe. The thriving field now called particle astro-
physics and cosmology (PAC) did not even exist until almost the twenty-first
century but is now the center of much frontier research.
A remarkable fact concerning the detection of neutron star mergers and the
gravitational waves they emit is worth noting here; the kilonovas that are the end
resultofthemergersarethesourceofmuchoftheheavierelementsweobservein
theuniverse,includingthematterthatmakesupourplanetandnotably—ourselves.
The purpose of this book is to introduce the reader to general relativity theory
andallthatitcantellusabouttheuniverse.Itisintendedtobeasclear,simple,and
brief as possible, and as rigorous as reasonable. It is divided into four somewhat
independent parts that might be considered separate volumes.
Part I is a brief review of special relativity; most physics students will have
studied specialrelativityinother coursesandmay skimeasilyoverthispart, butit
can serve as a brief introduction for others.
Preface ix
Part II provides mathematical background regarding Riemann space and the
vectorsandtensorsthatinhabitit.Itusestheideasandnotationofthecomponentor
classic approach to tensors but also includes discussion of the more modern ideas
and notation of the intrinsic abstract view of tensors.
Part III gives a view of basic general relativity as a theory of gravity and the
geometric ideas that underlie it; it includes chapters on gravitational waves and
black holes and includes a brief heuristic discussion on Hawking radiation.
PartIVisasurveyofrelativitytheoryasusedincosmology;thisbookismostly
about theory and its mathematical basis, but Part IV includes, of necessity, a fair
amountofmaterialontheexperimentalandobservationalworkbeingdoneorbeing
planned. Also of necessity, the material on the observational work is far from
complete, but is intended to provide a start and references for anyone interested in
pursuing it further and perhaps becoming a researcher.
My prime target readers are early graduate students or advanced and confident
undergraduatephysicsstudents.Astudentwiththeusualmathematicalbackground
in calculus and vectors and matrices and a physics background in mechanics and
electromagnetism should be able to handle the material without much outside
reading. Nominally, the entire book should be readable in a two semester or a two
or three quarter course.
Thanksareduetomanypeoplewhohelpedinthewritingofthisbook.Muchof
its content is based on my teaching as an adjunct professor at San Francisco State
University and work done on the Gravity Probe B mission at Stanford University.
MySFSUrelativityclasseshavegivenmemuchfeedbackandcorrectednumerous
errors. At Stanford and at SLAC National Accelerator Laboratory my colleagues
Robert Wagoner, James Bjorken, Francis Everitt, Alex Silbergleit, Pisin Chen,
David Santiago, and John Berberian have provided many interesting ideas and
discussions. Fred Martin patiently proof-read, criticized, and improved the early
notes. James Overduin encouraged me to revise and expand the notes for publi-
cation and provided thought-provoking comments.
San Francisco/Stanford, USA Ronald J. Adler
email: [email protected]
Contents
Part I Special Relativity in Review
1 A Brief Stroll in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 The Trouble with Absolute Time. . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Simplest Lorentz Transformation. . . . . . . . . . . . . . . . . . . . 5
1.3 Some Elementary Properties and Applications . . . . . . . . . . . . . 8
2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 The Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Four-Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 The Motion of Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Energy and Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Accelerated Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Curves and Arc Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Part II Vectors and Tensors
4 Riemann Spaces and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Riemann Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Vectors, Component View. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 Vectors and 1-Forms, Abstract View . . . . . . . . . . . . . . . . . . . . 40
4.4 Tensors, Component View . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.5 Tensors, Abstract View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Tetrads and n-Trads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.7 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Affine Connections and Geodesics. . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Affine Connections, Component View . . . . . . . . . . . . . . . . . . . 59
5.2 Transformation of the Affine Connections . . . . . . . . . . . . . . . . 61
5.3 Parallel Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
xi
