100 Years of Relativity: Space-time Structure Einstein And Beyond

527 Pages·2006·6.68 MB·English

by  Abhay Ashtekar

100 Years
Relativity
Space-Time Structure:
Einstein and Beyond
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KMears
Relativity
Space-Time Structure:
Einstein and Beyond
editor
Abhay Ashtekar
Institute for Gravitational Physics and Geometry,
Pennsylvania State University, USA
"X Vfe World Scientific
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100 YEARS OF RELATIVITY
Space-Time Structure: Einstein and Beyond
Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
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October12,2005 14:40 WSPC/TrimSize:9inx6inforReviewVolume preface
PREFACE
Thegoalofthisvolumeistodescribehowourunderstandingof space-time
structure has evolved since Einstein’s path-breaking1905 paper on special
relativity and how it might further evolve in the next century.
Preoccupation with notions of Space (the Heavens) and Time (the
Beginning, the Change and the End) can be traced back to at least 2500
years ago. Early thinkers from Lao Tsu in China and Gautama Buddha
in India to Aristotle in Greece discussed these issues at length. Over cen-
turies, the essence of Aristotle’s commentaries crystallized in the Western
consciousness, providing us with mental images that we commonly use.
We think of space as a three-dimensional continuum which envelops us.
We think of time as (cid:13)owing serenely, all by itself, una(cid:11)ected by forces in
thephysicaluniverse.Together,theyprovideastageonwhichthedramaof
interactionsunfolds.Theactorsareeverythingelse|starsandplanets,ra-
diationandmatter,youandme.InNewton’shands,theseideasevolvedand
acquired a mathematically precise form. In his masterpiece, the Principia,
Newtonspelledoutpropertiesofspaceandtheabsolutenatureoftime.The
Principia proved to be an intellectual tour de force that advanced Science
inanunprecedentedfashion.Becauseofitsmagni(cid:12)centsuccess,thenotions
ofspaceandtimewhichlieatitsfoundationsweresoontakentobeobvious
and self-evident. They constituted the pillars on which physics rested for
over two centuries.
It was young Einstein who overturned those notions in his paper on
specialrelativity,publishedon26thSeptember1905inAnnalender Physik
(Leipzig).LorentzandPoincar(cid:19)ehaddiscoveredmanyoftheessentialmath-
ematical underpinnings of that theory. However, it was Einstein, and Ein-
stein alone, who discovered the key physical fact | the Newtonian notion
of absolute simultaneity is physically incorrect. Lorentz transformations
of time intervals and spatial lengths are not just convenient mathematical
waysofreconcilingexperimental(cid:12)ndings.Theyarephysical facts.Newton’s
assertion that the time interval between events is absolute and observer-
independentfailsintherealworld.TheGalileanformulafortransformation
ofspatialdistancesbetweentwoeventsisphysicallyincorrect.Whatseemed
obvious and self-evident for over two centuries is neither. Thus, the model
v
October12,2005 14:40 WSPC/TrimSize:9inx6inforReviewVolume preface
vi A. Ashtekar
of space-time that emerged from special relativity was very di(cid:11)erent from
that proposed in the Principia.
Space-timestructureofspecialrelativityhasnumerousradicalrami(cid:12)ca-
tions | such as the twin \paradox" and the celebrated relation E =Mc2
between mass and energy | that are now well known to all physicists.
However, as in the Principia, space-time continues to remain a backdrop,
an inert arena for dynamics, which cannot be acted upon. This view was
toppled in 1915, again by Einstein, through his discovery of general rela-
tivity. The development of non-Euclidean geometries had led to the idea,
expounded most eloquently by BernhardRiemann in 1854, that the geom-
etry of physical space may not obey Euclid’s axioms | it may be curved
duetothepresenceofmatterintheuniverse.KarlSchwarzschildeventried
to measure this curvature as early as 1900. But these ideas had remained
speculative. General relativity provided both a comprehensive conceptual
foundation and a precise mathematical framework for their realization.
In general relativity, gravitational force is not like any other force of
Nature; it is encoded in the very geometry of space-time. Curvature of
space-time provides a direct measure of its strength. Consequently, space-
timeisnotaninertentity.Itactsonmatterandcanbeactedupon.AsJohn
Wheeler puts it: Matter tells space-time how to bend and space-time tells
matterhowtomove.Therearenolongeranyspectatorsinthecosmicdance,
nor a backdrop on which things happen. The stage itself joins the troupe
of actors. This is a profound paradigm shift. Since all physical systems
reside in space and time, this shift shook the very foundations of natural
philosophy. It has taken decades for physicists to come to grips with the
numerousrami(cid:12)cationsofthisshiftandphilosopherstocometotermswith
the new vision of reality that grew out of it.
Einstein was motivated by two seemingly simple observations.First, as
Galileodemonstratedthroughhisfamousexperimentsatthe leaningtower
of Pisa, the e(cid:11)ect of gravity is universal: all bodies fall the same way if
the onlyforce actingon them is gravitational.This is a direct consequence
of the equality of the inertial and gravitational mass. Second, gravity is
always attractive. This is in striking contrast with, say, the electric force
whereunlikechargesattractwhilelikechargesrepel.As aresult,whileone
can easily create regions in which the electric (cid:12)eld vanishes, one cannot
build gravity shields. Thus, gravity is omnipresent and nondiscriminating;
it is everywhere and acts on everything the same way. These two facts
make gravity unlike any other fundamental force and suggest that gravity
is a manifestation of something deeper and universal. Since space-time is
October12,2005 14:40 WSPC/TrimSize:9inx6inforReviewVolume preface
Preface vii
alsoomnipresentandthe sameforallphysicalsystems,Einsteinwasledto
regard gravity not as a force but a manifestation of space-time geometry.
Space-timeofgeneralrelativityissupple,depictedinthepopularliterature
as a rubber sheet, bent by massive bodies. The sun, for example, bends
space-timenontrivially.Planetslikeearthmoveinthiscurvedgeometry.In
a precise mathematical sense, they follow the simplest trajectories called
geodesics | generalizations of straight lines of the (cid:13)at geometry of Euclid
to the curved geometry of Riemann. Therefore, the space-time trajectory
of earth is as \straight" as it could be. When projected into spatial sec-
tions de(cid:12)ned by sun’s rest frame, it appears elliptical from the (cid:13)at space
perspective of Euclid and Newton.
The magic of general relativity is that, through elegant mathematics,
it transforms these conceptually simple ideas into concrete equations and
usesthem tomakeastonishingpredictionsabout the nature of physicalre-
ality.It predictsthatclocksshouldtickfasteronMontBlancthanin Nice.
Galacticnucleishouldactasgiantgravitationallensesandprovidespectac-
ular,multipleimagesofdistantquasars.Twoneutronstarsorbitingaround
each other must lose energy through ripples in the curvature of space-time
caused by their motion and spiral inward in an ever tightening embrace.
Over the last thirty years, precise measurements have been performed to
test if these andother evenmoreexotic predictionsarecorrect.Eachtime,
general relativity has triumphed. The accuracy of some of these observa-
tionsexceedsthatofthe legendarytestsof quantumelectrodynamics.This
combinationofconceptualdepth,mathematicaleleganceandobservational
success is unprecedented. This is why general relativity is widely regarded
as the most sublime of all scienti(cid:12)c creations.
Perhaps the most dramatic rami(cid:12)cations of the dynamical nature of
a
geometry are gravitationalwaves, black holes and the big-bang.
aHistoryhas a tinge of irony | Einstein had di(cid:14)culty with all three! Like all his con-
temporaries,hebelievedinastaticuniverse.Becausehecouldnot(cid:12)ndastaticsolution
tohisoriginal(cid:12)eldequations,in1917heintroducedacosmologicalconstanttermwhose
repulsivee(cid:11)ect could balance the gravitational attraction, leading to alarge scale time
independence.HisbeliefinthestaticuniversewassostrongthatwhenAlexanderFried-
manndiscoveredin1922that theoriginal(cid:12)eldequations admitacosmologicalsolution
depictinganexpandinguniverse,forawholeyear,EinsteinthoughtFriedmannhadmade
an error. It was only in 1931, after Hubble’s discovery that the universe is in fact dy-
namicalandexpanding thatEinsteinabandoned theextratermandfullyembracedthe
expanding universe. The story of gravitational waves is even more surprising. In 1936
EinsteinandNathanRosensubmittedapapertoPhysicalReviewinwhichtheyargued
thatEinstein’s1917weak(cid:12)eldanalysiswasmisleadingandinfactgravitationalwavesdo
notexistinfullgeneralrelativity.ItisnowknownthatthepaperwasrefereedbyPercy
October12,2005 14:40 WSPC/TrimSize:9inx6inforReviewVolume preface
viii A. Ashtekar
Gravitationalwavesareripplesinthecurvatureofspace-time.Justasan
oscillating electric dipole produces electromagnetic waves, using the weak-
(cid:12)eld approximation of general relativity, Einstein showed already in 1917
thatatime-changingmassquadrupoleproducestheseripples.Inthe1960’s
HermannBondi,RainerSachs,RogerPenroseandothersextendedthethe-
ory to full nonlinear general relativity. They (cid:12)rmly established that these
ripples are not \coordinate-gauge e(cid:11)ects" but have direct physical signi(cid:12)-
cance. They carry energy; as Bondi famously put it, one could boil water
with them. Through careful observations of a binary pulsar, now spanning
three decades, Russell Hulse and Joseph Taylor showed that its orbit is
changingpreciselyin the manner predicted bygeneralrelativity.Thus, the
realityof gravitationalwavesis now (cid:12)rmly established. Gravitationalwave
observatorieshavebeen constructedtoreceivethem directlyonearth.It is
widely expected that they will open a new window on the universe.
Generalrelativityushered-intheeraofmoderncosmology.Atverylarge
scales, the universe around us appears to be spatially homogeneous and
isotropic. This is the grandest realization of the Copernican principle: our
universehasnopreferredplacenorfavoreddirection.UsingEinstein’sequa-
tionsin1922,AlexanderFriedmannshowedthatsuchauniversecannotbe
static. It must expand or contract. In 1929, Edwin Hubble found that the
universe is indeed expanding. This in turn implies that it must have had
a beginning where the density of matter and curvature of space-time were
in(cid:12)nite.Thisisthebig-bang.Carefulobservations,particularlyoverthelast
decade,haveshownthatthiseventmusthaveoccurredsome14billionyears
ago. Since then, galaxies are moving apart, the average matter content is
becoming dilute. By combining our knowledge of general relativity with
laboratory physics, we can make a number of detailed predictions. For in-
stance,wecancalculatethe relativeabundancesof lightchemicalelements
whose nuclei were created in the (cid:12)rst three minutes after the big-bang; we
can predict the existence and properties of a primal glow | the cosmic
Robertson, a renowned relativist, who pointed out the error in the argument. Einstein
withdrew the paper saying that he had not authorized the Editors\to show the paper
tospecialistsbeforeitwasprinted"and\hesawnoreasontoaddressthe|inanycase
erroneous | comments of your anonymous expert". Finally, he did not believe in the
existence ofblackholes.Aslateas1939,hepublishedapaper intheAnnals of Mathe-
maticsarguingthat black holescouldnot beformed through the gravitational collapse
of a star. The calculation is correct but the conclusion is an artifact of a nonrealistic
assumption.Justafewmonthslater,RobertOppenheimerandHartlandSnyder|also
members of the Princeton Institute of Advanced Study | published their now classic
paperestablishingthatblackholesdoinfactresult.
October12,2005 14:40 WSPC/TrimSize:9inx6inforReviewVolume preface
Preface ix
microwave background | that was emitted when the universe was some
400,000 years old; and we can deduce that the (cid:12)rst galaxies formed when
the universe was a billion years old. What an astonishing range of scales
and variety of phenomena! In addition, general relativity also changed the
philosophical paradigm to phrase questions about the Beginning. It is not
just matter but the space-time itself that is born at the big-bang. In a pre-
cise sense, the big-bang is a boundary, a frontier, where space-time ends.
General relativity declares that physics stops there; it does not permit us
to look beyond.
Through black holes, general relativity opened up the third unforeseen
vista. The (cid:12)rst black hole solution to Einstein’s equation was discovered
already in 1916 by Schwarzschild while serving on the front lines during
the First World War. However, acceptance of its physical meaning came
very slowly. Black holes are regions in which the space-time curvature is
so strong that even light cannot escape. Therefore, according to general
relativity,theyappearpitchblacktooutsideobservers.Inthe rubbersheet
analogy, the bending of space-time is so extreme inside a black hole that
space-time is torn-apart, forming a singularity. As at the big-bang, curva-
ture becomes in(cid:12)nite. Space-time developsa (cid:12)nal boundary and physics of
general relativity simply stops.
And yet, black holes appear to be mundanely common in the universe.
General relativity, combined with our knowledge of stellar evolution, pre-
dicts that there should be plenty of black holes with 10 to 50solar masses,
the end products of lives of large stars. Indeed, black holes are prominent
players in modern astronomy. They provide the powerful engines for the
most energetic phenomena in the universe such as the celebrated gamma
ray bursts in which an explosion spews out, in a few blinding seconds, as
much energy as from a 1000 suns in their entire life time. One such burst
is seen every day. Centers of elliptical galaxies appear to contain a huge
black hole of millions of solarmasses.Our own galaxy,the Milky Way, has
a black hole of some 3 million solar masses at its center.
Generalrelativityisthebesttheoryofgravitationandspace-timestruc-
turewehavetoday.Itcanaccountforatrulyimpressivearrayofphenomena
ranging from the grand cosmic expansion to the functioning of the more
mundane global positioning system on earth. But it is incomplete because
it ignores quantum e(cid:11)ects that governthe subatomic world. Moreover,the
two theories are dramatically di(cid:11)erent. The world of general relativity has
geometric precision, it is deterministic; the world of quantum physics is
dictated by fundamental uncertainties, it is probabilistic. We maintain a