Special Relativity in General Frames: From Particles to Astrophysics

800 Pages·2013·10.7 MB·English

by  Éric Gourgoulhon (auth.)

Graduate Texts in Physics
Éric Gourgoulhon
Special
Relativity in
General Frames
From Particles to Astrophysics
Graduate Texts in Physics
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Graduate Texts in Physics
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´
Eric Gourgoulhon
Special Relativity
in General Frames
From Particles to Astrophysics
123
E´ricGourgoulhon
LaboratoireUniversetThe´ories
ObservatoiredeParis,CNRS,Universite´ParisDiderot
Meudon,France
TranslationfromtheFrenchlanguageeditionof:Relativite´restreinte:Desparticulesa`
l’astrophysique,(cid:2)c 2010EDPSciences,CNRSEdition,France.
ISSN1868-4513 ISSN1868-4521(electronic)
GraduateTextsinPhysics
ISBN978-3-642-37275-9 ISBN978-3-642-37276-6(eBook)
DOI10.1007/978-3-642-37276-6
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013942463
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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To Vale´rieandMaxime

Foreword
The theory of special relativity holds a distinctive place within physics. Rather
thanbeingaspecificphysicaltheory,itis(similartothermodynamicsoranalytical
mechanics)ageneraltheoreticalframework withinwhichvariousdynamicaltheo-
ries can be formulated.In this respect, a modern presentation of special relativity
must put forward its essential structures before illustrating them by concrete
applicationstospecificdynamicalproblems.Suchisthechallenge(sosuccessfully
met!)ofthebeautifulbookbyE´ricGourgoulhon.
Contrary to most textbooks on special relativity, which mix the presentation
of this theory with that of its historical development and which sometimes write
the specific form of “Lorentz transformations” before indicating that they leave a
certainquadraticforminvariant,thebookbyE´ricGourgoulhoniscentred,fromthe
very beginning, on the essential structure of the theory, i.e. the chrono-geometric
structure of the four-dimensional Poincare´–Minkowski spacetime. The aim is to
train the reader to formulate any relativity question in terms of four-dimensional
geometry. The word geometry has here the meaning of “synthetic geometry” (a`
la Euclid) in contrastwith “analytic geometry”(a` la Descartes). Under the expert
guidanceofE´ricGourgoulhon,thereaderwilllearntoset,andtosolve,anyproblem
ofrelativitybydrawingspacetimediagrams,madeofcurves,straightlines,planes,
hyperplanes,conesand vectors. He will get accustomedto visualizing the motion
ofaparticleasalineinspacetime,tothinkaboutthetwinparadoxasanapplication
ofthe“spacetimetriangleinequality”,toexpressthelocalframeofanobserveras
a four-dimensionalgeneralization of the Serret–Frenet triad, to compute a spatial
distanceasageometricmeanoftimeintervals(viathehyperbolicgeneralizationof
thepowerofapointwithrespecttoacircle)ortounderstandtheSagnaceffectby
consideringtwohelicesinspacetimewoundinoppositedirections.
Besides the pedagogicalcharacteristic of being centred on a geometric formu-
lation, the book by E´ric Gourgoulhon is remarkable in many other ways. First
of all, it is fully up to date and very complete in its coverage of the notions and
resultswherespecialrelativityplaysanimportantrole:fromThomasprecessionto
thefoundationsofgeneralrelativity,includingtensorcalculus,exteriordifferential
calculus,classicalelectrodynamics,thegeneralnotionofenergy–momentumtensor
vii
viii Foreword
and a noteworthy chapter on relativistic hydrodynamics. In addition, this book is
sprinkled with enlightening historical notes, in which the author summarizes in
a condensed, albeit very informative way the (sometimes very recent) results by
historians of science. Finally, the book is richly laden with many examples of
applications of special relativity to concrete physical problems. The reader will
learn the role of special relativity in various domains of modern astrophysics
(supernovanebulae,relativisticjets,micro-quasars)inthedescriptionofthequark-
gluonplasmaproducedinheavyioncollisions,aswellasinmanyhigh-technology
experiments:from laser gyrometersto the LHC, including modernreplicationsof
theMichelson–Morleyexperiment,matterwaveinterferometers,synchrotronsand
theirradiation,andthecomparisonofatomicclocksembarkedonplanes,satellites
ortheInternationalSpaceStation.
IamsurethattheremarkablyrichbookbyE´ricGourgoulhonwillattractthekeen
interestofmanyreadersandwillenablethemtounderstandandmasteroneofthe
fundamentalpillars(withgeneralrelativityandquantumtheory)ofmodernphysics.
Bures-sur-Yvette,France ThibaultDamour
Preface
Thisbookpresentsageometricalintroductiontospecialrelativity.Bygeometrical,
it is meant that the adopted point of view is four dimensional from the very
beginning. The mathematical framework is indeed, from the first chapter, that of
Minkowskispacetime,andthebasicobjectsarethevectorsinthisspace(oftencalled
4-vectors). Physical laws are translated in terms of geometricaloperations(scalar
product,orthogonalprojection,etc.)onobjectsofMinkowskispacetime(4-vectors,
worldlines,etc.).
Many relativity textbooks start rather by a three-dimensional approach, using
space+timedecompositionsbasedoninertialobservers.Onlyinthesecondstage
theyintroduce4-vectorsandMinkowskispacetime.Inthisrespect,theyarefaithful
to the historical developmentof relativity. A more axiomatic approach is adopted
here, setting from the very beginning the full mathematical framework as one of
thepostulatesofthetheory.Fromthispointofview,thechosenapproachissimilar
to that adopted in classical mechanics or quantum mechanics, where usually the
exposition does not follow the history of the theory. The history of relativity is
undoubtedlyrich and fascinating, but the objective of this book is the learning of
special relativity within a consistent and operationalsetting, from the bases up to
advancedtopics.Thetextis,however,enrichedwithhistoricalnotes,whichinclude
referencestotheoriginalworksandtothestudiesbyhistoriansofscience.
Usually, the geometric approach is reserved for general relativity, i.e. for the
incorporation of the gravitational field in relativity theory.1 We employ it here
for special relativity, taking into accounta geometric structure much simpler than
that of generalrelativity: while the latter is based on the conceptof differentiable
manifold,specialrelativityreliesentirelyontheconceptofaffinespace,whichcan
be identified with the space R4. Consequently, the mathematical prerequisites are
relatively limited; they are mostly linear algebra at the level of the first two years
ofuniversity.Themathematicsusedhereisactuallythesame asthoseofa course
1TwonotableexceptionsarethemonographsbyCostadeBeauregard(1949)andSynge(1956).
ix