The Physics Update Vol. 10 No.1 2006 33 General Relativity: Warping of Space and Time C. M. Kenneth Hong Department of Physics, National University of Singapore E-mail: phyhcmk@nus.edu.sg Gravityistheforcethatkeepsourfeetonthe groundandisthesourceofaccelerationthat returnsaballtoEarth.Thisarticleexploresthe scienceofgravityandhowitevolvedfromthe impressiveachievementsofNewtonslawstothe revolutionaryadvancesofEinsteinstheoryof relativity. Einsteins theory of relativity comprises oftwo:specialrelativityandgeneralrelativity. TheformeronereplacesNewtonianmechanics when the speed is close to the speed of light and is tobeconsistentwithMaxwellstheoryof electromagnetism.Generalrelativity,ontheother hand, supersedes Newtonian gravity when mass or energyisverylarge.Boththeoriescompletely revolutionizedourconceptsinspaceandtime,as well as, the way we view our universe. Special Relativity Inthenineteenthcentury,thewell-knowntheory ofelectromagnetismwasbasedonMaxwells equationswhichdescribethebehaviourof electromagnetismandelectromagneticwaves. Maxwellsequationshowedthatlightisan electromagneticwaveandrequiredthatall electromagneticwavespropagateinvacuumata fixed speed. Atthattime,itwasbelievedthatlightrequireda mediumthesocalledaetherforpropagation. Theuniversewasthusfilledwithaetherinwhich Maxwellsequationshold.Thatis,aether constitutedanabsolutereferenceframeagainst whichspeedscouldbemeasured.Aetherseemed tohavesomecontradictoryproperties:itwas sufficient elastic to support electromagnetic waves butithadnoresistancetobodiesmovingthrough it. Many experiments were devised to detect some measurableeffectsoftheaether.However,none weresuccessfulandthisledtomanyimaginative explanations to account for it. The most famous experiment is the one carried out by Albert Michelson and Edward Morley in 1887. TheyattemptedtodetermineEarthsmotion relativetotheabsolutespacethroughwhichlight supposedly moved by measuring the speed of light atdifferenttimessothattheorientationoftheir equipmentwouldchange.However,totheir surprising,theymeasuredpreciselythesame speedoflightforanyorientationoftheir apparatus!Theydemonstratedthattheobserved speedoflightisindependentoftheobservers motion through space. Figure 1A schematic setup for the Michelson-Morley experiment. 1 Albert Einstein had his first insight about relativity fromthinkingaboutelectromagnetism.In particular,attheageofsixteen,hepuzzledover the consequence of travelling at the speed of light. One of these is now known as Einsteins mirror: What would you see if you and the mirror you were looking into were both moving at the speed of light? Thespecialtheoryofrelativity(orjustspecial relativity) was proposed by Einstein in 1905 at the ageof26.Specialrelativityresultsfromtwo fundamentalpostulates:(1)Principleofrelativity: The laws of Physics are the same for all observers movingatsteadyspeedswithrespecttoeach 1 T. Hey and P. Walters, Einsteins Mirror, P40, Cambridge University Press, 1991. The Physics Update Vol. 10 No.1 2006 34 other; and (2) The constancy of the speed of light: Thespeedoflightisconstantregardlessofthe motion of the observer or of the sender. Figure 2Einsteins 1905 paper on special relativity. 2 Withthespecialrelativity,Einsteinelevatedthe speed of light to the status of a constant of nature. Herewrotethenewlawsofmechanicstoreflect this new fact and led to a deeper understanding of theuniverse.Thistheoryiscalledspecial because it applies only to restricted inertial frames inwhichtheeffectsofgravitycanbeignored. SpecialrelativityisequivalenttoNewtonian mechanicsindescribingobjectsthatmovemuch moreslowlythanthespeedoflight,butitdiffers significantly in its predictions at high speeds. Specialrelativityhasseveralimportant consequencesthatstuckmanypeopleasbizarre. Wedescribeheresomeofthoseodd consequences.Imaginethatarocketisflyingpast youcloselyatarelativespeedcomparabletothe speed of light.You begin to notice that the rocket appearstocontractinthedirectionofmotion.A one-meter ruler on board, which is identical at the launchtotheoneyoukeepinyourlaboratory,is nowshorterthanitstwin.ThisiscalledLorentz length contraction. Atthesametime,therocketsclock,which synchronized prior to launch with yours, now ticks moreslowly.Thisphenomenon,calledtime dilation,canbefoundontheroutineworkin particleaccelerators.Ofcourse,fromtheview point of the astronaut in the rocket, you are the one movingrapidly.Hence,asobservedfromthe 2 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/SpecialRel. html astronaut,youappeartobecompressedinthe direction of motion and your clock run slowly! Whenmeasuringthelengthofthemovingruler, youdosobynotingthepositionsofthetwoends atthesametimeaccordingtoyourclock. However,thosetwoeventsthetwo measurementsyoumadedonotoccuratthe sametimeasobservedbytheastronaut.Thislack of simultaneity, together with time dilation, means thattimeisrelative.Thepreviouslyaccepted Newton concept of an absolute and universal time thatwasthesameforallobservershasnowbeen abolished. Additionalcarefulexperimentationwouldreveal thatthemassoftherocketalsoincreases. Theoretically,themassoftherocketbecomes nearlyinfinitelylargeasthespeedoftherocket approachesthespeedoflight.Therefore,objects becomehardertoaccelerateastheirspeeds increase.Itbecomesimpossibletoaccelerateany objectuptoandbeyondthespeedoflight.Asa corollary,thespeedoflightbecomestheultimate speed for any physical object and signal. Finally,perhapsthebest-knownpredictionof specialrelativityisthattherocketsenergyand massareproportionaltooneanotherviathe famousformulaE=mc2.Thisisonewaytosee whyobjectswithanymasscanneverreachthe speedoflightaninfiniteamountofenergyis requiredtogetthere!Themassandenergyare now equivalent. With this concept of mass-energy equivalence,conservationofmassandenergyare combinedallowingmassandenergycanbe convertedtoeachother.Theconversionfactoris justthesquareofthespeedoflight.Bothnuclear powerplantsandatomicweaponsaretheexplicit validations of this formula. Newtonian Gravity IssacNewtondevelopedthegravitationallaw that summarizeshowgravitydependsonmassand distance.Newtonslawsaysthattheforceof gravitybetweentwoobjectsisproportionaltothe mass of each of them. The greater the gravitational attractionsbetweenthemiftheobjectsaremore massive. The law also says that the force between two objects is proportional to the inverse square of their distance. Newtonsexplanationofgravitationalinteractions mustbeconsideredoneofthemostsuccessful The Physics Update Vol. 10 No.1 2006 35 physicaltheoriesofalltimes.Itaccountsforthe motionsofalltheconstituentsofthesolarsystem withuncannyaccuracy,permitting,forinstance, the prediction of eclipses hundreds of years ahead. InNewtonstheoryofgravity,thestrengthhas nothing to do with how long the objects have been ineachotherspresence.Thismeansthatthe objectswillimmediatelyfeelachangeintheir mutualgravitationalattractioniftheirmassesor theseparationchanges.Forinstance,iftheSun weresuddenlytoexplode,theEarth,beingsome 150millionkilometresaway,wouldinstantly experiencethechangeontheirmutual gravitational pull. The knowledge that the Sun had explodedwouldbeinstantaneouslytransmittedto theEarththroughthesuddenchangeintheir mutualgravitationalpullalthoughthelightfrom theexplosionwouldtakeabouteightminutesto reach the Earth. This conclusion is in direct conflict with Einstein's specialrelativitysincenophysicalsignalcanbe transmitted faster than speed of light. Confident in the success of special relativity, Einstein began the search of a new theory of gravity compatible with specialrelativity.Almostadecadelater,in1915, hepublishedthegeneraltheoryofrelativity(or justgeneralrelativity).Generalrelativityisnow generalenoughtoincludeallpossiblereference framesinwhichtheeffectsofgravitycannotbe ignored. Mass and Acceleration TheGreekphilosopherAristotlebelievedthatall objects have a natural tendency to fall towards the center of the universe, which was considered to be the center of the Earth. According to Aristotle, the heavierobjectsfallfasterthanlighteronessince theheavieroneswerebeingpulledharderby gravity. The influence of Aristotle in the following centuriesmadeitdifficulttochallengeanyofhis pronouncements. GalileoGalileiwasthefirstmajorscientistto refuteAristotlestheories.Accordingtolegend, Galileodroppedvariousobjectsfromthetopof theLeaningTowerofPisa.Hedemonstratedthat allobjectsfallatthesamerateunderthepullof gravityifairresistanceiseliminated.This experimentwasalsocarriedoutbyApollo15 astronaut David Scott by dropping a hammer and a feathersimultaneouslyontotheMoon.Itwas observed that these two objects fell at precisely the same rate and reached the lunar surface at exactly the same time. Toexplainwhyallobjectsfallatthesamerate underthegravitationalpull,Newtonproposedthe equivalenceofinertialmassandgravitational mass. Inertial mass is the resistance of an object to any change in its state of motion. This is the mass inNewtonssecondlaw.Itisapermanent property of the object and does not change depend onitslocation.Gravitationalmass,ontheother hand,istherespondofanobjecttogravity.It determineshowstronglytwoobjectsattracteach otherbygravity.Itdependsonthelocalstrength ofgravityandappearstovaryaccordingtothe environment. For instance, an object in deep space stillhasinertialmassbutitsgravitationalmassis zero.Thereisnoreasonwhythesetwoquantities from so different origins are the same. However, it istheapparentequivalenceofthesetwotypesof masseswhichresultsintheuniformityof gravitational acceleration Galileos result that all objects fall at the same rate independent of mass. Adirecttestoftheequivalencebetweeninertial andgravitationalmassesisthecomparisonofthe accelerationoftwoobjectsofdifferent compositioningravitationalfield.Ifthis equivalenceisviolated,thentheaccelerationsof differentobjectswoulddiffer.Newtonhimself usedafixedlengthpendulumwithweightsof varyingcompositiontotestthisequivalence.The periodoftheswingofthependulumsshould dependonlyonthelengthofthependulum regardless of the compositions. Newton concluded thatthesetwotypesofmassesarethesametoan accuracy of at least 1 part in 1000. Towardstheendofthenineteenthcentury,the Hungarian physicist Roland von Etvs performed anexperimentsignificantlyimprovedupon Newtonsaccuracy.Hedevelopedadevicecalled atorsionbalance,whichconsistsofapairof objects of equal mass attached to the opposite ends of a rod suspended by a fine wire. In additional to gravitationalforceoneachobject,thereisan inertial force due to the rotation of the Earth. If the inertial and gravitational masses are not the same, therodwillrotateaboutaverticalaxisuntilitis haltedbytherestoringtorqueofthetwistedwire. Theamountoftwistcanbemeasuredbyrotating thewholeapparatusthrough180.Therodwill nowtwistintheoppositedirection.However,if there are exactly equal, there will be no movement oftherodbetweenthesetwoorientationsofthe The Physics Update Vol. 10 No.1 2006 36 apparatus.Usingthisapparatus,Etvswasable toverifythisequivalencetowithinafewpartsin 1000million.ModernversionsofEtvstype experimentshavebeencarriedouttoverifythis equivalence to even higher degrees of accuracy. Einstein's Happiest Thought GalileoandNewtonhadacceptedtheequality betweeninertialandgravitationalmassesasa happycoincidence.However,Einsteinwasnot contentandstartedtolookforanevendeeper insight behind this equality. In 1907, while pondering some issues on gravity at hisdeskinthepatentofficeinBern,Switzerland, Einsteinhadthecentralinsightleading himtohis generalrelativity.Inalecture,Einsteintoldthe story about what happened: I was sitting in a chair in the patent office atBernwhenallofasuddenathought occurred to me: If a person falls freely he willnotfeelhisownweight.Iwas startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation. Einsteinlaterreferredtothisthoughtasthe happiestthoughtofmylife.Thishappiest thoughtinfersthatspecialrelativityandallother lawsofphysicsappearasusualinaframeof referencefallingfreelyundertheinfluenceof gravity. TounderstandEinstein'sinsight,consideran observer in a rocket in the outer space without the influenceofgravity.Therocketisundergoing uniformaccelerationupwardsat9.8m/s2.Tothis observer,hewillfeeltheforceoftheflooronhis feet.Thisisthesamesituationthatyouwillfeel theforceofyourseatonyourbackifyouareon thecarbeingacceleratedforward.Einstein's realization was that the observer will not be able to distinguishthisacceleratedsituationfromone withoutaccelerationbutwithgravity.Now,ifthe rocketisstationaryontheEarthssurface,i.e. undertheinfluenceofEarthsapproximately uniformgravityg=9.8m/s2,thisobserverwill againfeelthefamiliarforceoftheflooronhis feet. Figure3Iftherewerenowindowsintheelevator, Einsteinwouldnotbeabletotellwhetherheisatrest onEarthoracceleratingataconstantratethrough space. 3 Instead,iftheobserverisinafree-fallinglift abovetheEarth,hewillbefloatingfreelyinthe lift.Heseemstofeelnogravitationalforce.All objectswillalsofloatingwithhimasifgravity wereabsent.Thisis,ofcourse,onlyan approximationasweknowthatthegravityatthe topoftheliftisalittleweakerthanthatatthe bottom.Therewillstillberesidualeffectof gravity, called tidal force, which is responsible for the ocean tides on the Earth. However, if the lift is sufficientlysmall,itwillbeclosertoazero-gravity situation. The observer would be unable to tell whether he was in free fall or in a gravity-free environmentuntilthelifthittheground!In effect, acceleration due to free fall is equivalent to agravitationalfieldthatcancelsouttheEarths gravity.Therefore,thissmallfree-fallingliftcan be treated as an inertial reference frame. Einsteincalledtheindistinguishabilitybetween acceleratedmotionswithgravitytheequivalence principle. It plays a central role in the construction ofgeneralrelativity.Thisprinciplecouldbeused to produce artificial gravity on a space station. The stationcouldbemadetospinsothatthe centripetal acceleration in the living quarters is 9.8 m/s2.Thiswouldmaketheweightofanyonein the living quarters the same as on the Earth. Non-Euclidean Geometry Euclidean geometry is the geometry we learn from schools. This approach to geometry was laid down byEuclidandhasformedthebasisforteaching thesubjectformorethan2000years.Weareall 3 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/GenRelativity .html The Physics Update Vol. 10 No.1 2006 37 educatedtobelievethatparallelstraightlines neverintersect,threeinterioranglesofatriangle adds up to 180, and so on. Euclidsgeometryisbasedonfivepostulates:(1) Alinecanbedrawnfromanypointtoanyother point; (2) A finite line segment can be extended to alineofanylength;(3)Acirclecanbedrawn withanycentreandatanydistancefromthat centre;(4)Allrightanglesareequaltoone another;and(5)Parallelpostulate:Givenaline andapointnotonthisline,thereisonlyoneline throughthispointthatisparalleltotheoriginal line.Thesepostulateswereassumedtobeself-evidentlytrue.However,thefifthpostulatewas notsoobvious.Manymathematicianstried unsuccessfully to prove it. Figure 4 A triangle on the surface of a saddle for which the sum of the three interior angles adds up to less than 180. 4 Non-Euclideangeometryarosefromtheproblem ontheEuclidsfifthpostulate.Theessential differencebetweenEuclideanandnon-Euclidean geometryisthenatureoftheparallellines.In 1829, the Russian Nikolai Lobachevski built a new geometrywithoutthefifthpostulate.In Lobachevskisgeometry,thereismorethanone line through a point which is parallel to any given line. The sum of the interior angles in a triangle is lessthan180inthisgeometry.Thisactually describesgeometryonasurfacesuchassaddle which appears concave in direction along the spine ofthehorsebutconvexintheotherdirection. Geometrywiththesepropertiesisknownas hyperbolic geometry now. In1854,BernhardRiemannfoundedanothertype ofgeometry,knownasellipticgeometrynow,by droppingthesecondEuclidspostulate.He proposedthatlinescanbefiniteinlengthbut endless.Inthisgeometry,therearenoparallel 4 http://www.physics.nus.edu.sg/einstein/ linesandtheanglesofatriangleadduptomore than180.Atypicalexampleofellipticgeometry isthesurfaceofasphere.Thestraightestline betweentwopointsonthissurfaceispartofa great circle defined such that a slice through a greatcirclecutsthesphereintotwoequalhalves. Examples of great circles are the lines of longitude andtheequator.Thisgeneralizationofthenotion ofastraightlineiscalledageodesic.Ingeneral, thegeodesicsarepathsofshortestdistanceand define the straight lines for a given surface. Figure5AtriangleonthesurfaceoftheEarthfor whichthesumofthethreeinterioranglesaddsupto more than 180. 5 InEuclideangeometry,thedistancebetweenany twopointscanbecalculatedusingthefamiliar Pythagorastheorem.Riemanndiscoveredthat distancesbetweentwopointsonanysurfacecan alsobecalculatedbyageneralizationofthe Pythagorastheorem.Riemannfurtherextendedit tospacesofanynumberofdimensions,which cannot be easily visualized, and to surfaces whose curvaturewasabletovaryfrompointtopoint. Thismathematicalapparatuswasexactlytheone neededbyEinsteininordertoconstructhisnew theory of gravity. Spacetime Geometry Spaceandtimearetobetreatedonequalfooting andcanbeinterrelatedinspecialrelativity.In 1908,HermannMinkowskidevelopedtheideaof spacetimecontinuumastheunderlying geometrybehindthespaceandtimerelationships proposedbyspecialrelativity.Minkowski proposedtreatingtimeasavariableinthesame wayas onetreatsthespacecoordinatesofa point inthree-dimensionalspace.Insteadofspaceand timeseparately,oneshouldnowthinkintermsof 5 T. Hey and P. Walters, Einsteins Mirror, P182, Cambridge University Press, 1991. The Physics Update Vol. 10 No.1 2006 38 eventsinfour-dimensionalspacetime.Thisfour-dimensionalspacetimeisaveryusefulconceptin relativity(bothspecialandgeneralrelativity)and is called Minkowski spacetime in special relativity. Thisfour-dimensionalspacetimeiseasiertobe visualized in the mean of spacetime diagrams. The conventionsusedinspacetimediagramsareas follows:thehorizontaldirectiondenotesspace, whiletheverticaldirectiondenotestime. Horizontalsectionswhicharehigherupinthe diagram give the spatial position at a later time. Thebasicelementinspacetimeisevent.Inthe usualEuclideangeometry,wecanrepresentthe positionofanypointbythreecoordinates.Now, an event is specified by four coordinates: the three coordinatesx,yandz,describingwheretheevent happened,togetherwithtime,t,oftheevent, specifyingwhenithappened.Anevent corresponds to a point in the spacetime diagrams. Figure6Aspacetimediagramofthemotionofthe Earth around the Sun. 6 Theworldlineofanobjectisthesequenceof spacetimeeventscorrespondingtothehistoryof theobject.Ittracesoutapathinthespacetime diagrams. For instance, consider the motion of the EartharoundtheSun.SincetheSunisatrest,its world line is just a vertical straight line. In the case oftheEarth,itsorbitinspaceis(nearly)circular andisrestrictedtoaplane.TheEarthswordline traces out a spiral helix in spacetime diagram. Geodesicsinspacetimearethestraightestpaths inspacetime.Accordingtowhetherthespacetime 6 T. Hey and P. Walters, Einsteins Mirror, P60, Cambridge University Press, 1991. relationshipbetweentwonearbyeventsonthe geodesicisspace-like,null(light-like),ortime-like,thegeodesicsinspacetimeareclassifiedas space-like, null (light-like) or time-like. Space-like geodesicsarepathsonthesurfaceofsimultaneity accordingtoaninertialobserver.Thepathofa lightrayinspacetimeisdescribedbyanull geodesic.Thetime-likegeodesicsisthepath traced out by an inertial observer and it is the path with the longest elapsed time. Minkowskispacetimeisaflatspacetimejustlike theEuclideanspaceisflat.Considertwoinitial observerswhichareinitiallystationarywith respecttoeachother.Theirinitialgeodesicsare twoparalleltime-likegeodesics.Accordingto specialrelativity,inertialobserverswhoinitially areatrestalwaysremainatrestwithrespectto eachother.Thus,initiallyparalleltime-like geodesicsalwaysremainparallel.Thesame conclusionappliestobothnullandspace-like geodesics. Therefore, Minkowski spacetime is flat. Gravity and Geometry Einsteinconcludedthatspacetimemustbecurved fromhisequivalenceprinciple.However,the spacetimeappearslocallyflattothefree-falling observer.Gravitythatwefeelwhenweare notin afree-fallingisamanifestationofspacetime curvature. Figure7Aschematicdiagramillustratestheideathat the curvature mimics a force acting between the ants. 7 ToillustratehowEinsteincouldmaketheleap fromhisequivalenceprincipletotheideaof curvedspacetime,imagineacolonyoftwo-dimensionalcreatures,Spherelander,livingona two-dimensionalsurfaceofthesphere, Sphereland.Spherelandersconstructasufficient largetrianglebyusingasetofverystraight aluminium rulers. They intend to measure the sum oftheinteriorangles.Accordingtotheir knowledge of Euclidean geometry, they expect the 7 J. A. Wheeler, A Journey into Gravity and Spacetime, P69, Scientific American Library (New York), 1990. The Physics Update Vol. 10 No.1 2006 39 sumtobe180.Surprisingly,theyobtainthesum tobe200.Theythenpostulatethatthereisforce actingonthealuminiumrulers.Thissuperstitious force causes the rulers bend in such a way to make the sum exceeds 180. Totesttheirpostulate,theyconstructthesame trianglebutnowusingplatinumrulers.Again, they obtain the same result exactly. After repeating thesameexperimentbyusingdifferenttypeof rulers,theyconcludethatthissuperstitiousforce affectsallrulersequally.Next,theSpherelanders constructsmallertriangleusingshorterrulersand thesumturnsouttobe190.Theyrealizethat smallertrianglesyieldsmallersum.Asthe trianglesgetsmallerandsmaller,thesumgets closer and closer to the expected value of 180. Sinceallrulersbehavedexactlythesameway, Spherelandersthensuggestthatitmighthaveless todowiththerulersthemselvesratherthanthe underlying structure of Sphereland. They come out theideathatperhapstheirworld,Sphereland,is actuallycurved.Furthermore,whentheyconfine their attention to progressively smaller regions, the worldappearsmoreandmoreequivalenttoaflat world. Einstein used a similar line of arguments from his equivalenceprincipletocurvedspacetime.All objectsfallatthesamerate.Maybe,the gravitationalforceactingonthemhaslesstodo withtheobjectsthemselvesratherthanwiththe underlyingspacetimegeometry.Justas Spherelandwascurved,ourspacetimeisalso curved.Thetrajectoriesoffallingobjectssimply reflectthedistortionofspacetime.Insufficiently smallfree-fallinglaboratories,allobjects experiencenogravity.Thespacetimeinsidethe laboratoryisapproximatelyMinkowskispacetime justlikethesmallerspaceinSpherelandwas approximately the flat space. Furthermore,Einsteinarguedthatthisshouldalso applytoalllawsofphysics.Inotherwords,all equations were to be written in the usual form they had in the Minkowski spacetime when applied to a free-fallingframe.Themotionofthefree-falling bodiesisalongthegeodesicsofthecurved spacetimejustlikethestraightestlinesin Sphereland correspond to geodesics. General Relativity Einsteinconcludedthattheeffectsofgravitycan bedescribedintermsofcurvedspacetime.To completehisformulationofgeneralrelativity,it remainstospelloutinquantitativedetailthe relationshipbetweenspacetimecurvatureandthe distributionofmatter.AlthoughEinsteinhada clearphysicalpictureoftheequivalenceprinciple andofthecurvedspacetimeasearlyas1907,it tookhimeightyearsmoretoarriveatthe equations of general relativity. Figure 8 Einsteins 1916 paper on general relativity. 8 Thecentralcontentofgeneralrelativityisan equation which has the schematic form: densityenergy - masscurvature spacetime This relation, called the Einsteins equations, is the fieldequationofgeneralrelativityinthewaythat Maxwellsequationsarethefieldequationsof electromagnetism.Maxwellsequationsrelatethe electromagneticfieldtoitssourceschargesand currents.Einsteinsequationrelatesspacetime curvature to its source the mass-energy density. Thefieldequationsallowustocalculatehow muchcurvatureisgeneratedinthepresenceof matter.Theequivalenceprinciplethentellshow matterrespondstoitfreelyfallingbodiesmove along geodesics. A famous statement by American physicistJohnWheelerillustratesthemajor principleingeneralrelativity:mattertells 8 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/GenRelativity. html The Physics Update Vol. 10 No.1 2006 40 spacetimehowtocurve,andcurvedspacetime tells matter how to move. Toobtainafeelforthisnewviewofgravity,we considerthefollowingrubbermembrane-bowling ballanalogy.Intheabsenceofanymatteror energy, the spacetime would be flat. This could be visualizedasasheetofrubbermembrane.Now, the presence of masssuggested that the spacetime istobewarped.Thisismuchlikeasituationthat weputabowlingballontherubbermembrane. Theregionoftherubbermembranearoundthe bowling ball becomes distorted. We now place a small ball bearing on the distorted rubbermembraneandsetifoffwithsomeinitial velocity.Iftherubbermembraneisflat,theball bearing will travel along a straight line. Due to the distortionontherubbermembranebybowling ball,theballbearingwillnowtravelalonga curvedpath.Furthermore,ifwesettheball bearingmovingwithjustarightspeedinjustthe right direction, it will move in an orbit around the bowling ball. Now,theSun,likethebowlingball,warpsthe spacetimearoundit,andtheEarthsmotion,like thatoftheballbearing,isdeterminedbythe warpingofthespacetime.TheEarth,liketheball bearing,willmoveinorbitaroundtheSunifits speed and orientation have certain suitable values. ThiseffectonthemotionoftheEarthiswhatwe normallyrefertoasthegravitationalinfluenceof the Sun. Figure9Therubbermembrane-bowlinganalogyto illustratethenewviewofgravityfromgeneral relativity.9 InNewtonsview,themechanismholdingthe Earthinorbitisduetothemysterious 9 T. Hey and P. Walters, Einsteins Mirror, P192, Cambridge University Press, 1991. instantaneousactionoftheSun.UnlikeNewton, Einsteinhasspecifiedthemechanismbywhich gravityistransmittedtobethewarpingofthe spacetime. That is to say the Earth is hold in orbit due to the warping of the spacetime caused by the presence of the Sun. Gravitational Red/Blue Shift Thegravitationalredshiftwasthefirstof Einstein's great predictions after he had formulated hisversionoftheequivalenceprinciple.This phenomenon is the gravitational analogue of the Doppler effect. You should have the experience of the Doppler effect in your daily life a noticeable changeinthepitchofthesirenastheambulance passesyou.Whenthesource(ambulance)is approachingyou,theeffectivewavelengthofthe soundwave(siren)isshortenedasthesource movescloserduringtheperiodofthesirenand henceahigherpitchofthesirenisdetected. Similarly,thepitchof the sirenwillbeloweredif it is receding from you. The easiest way to understand the gravitational red shiftistoconsiderthefollowingthought experiment.Thereisanemitterofawell-defined wavelengthoflightplacedontheground.A receiver is placed at the top of a high tower and is tunedtoreceivetheincomingsignalfromthe emitterontheground.Assumethatthereisan observeratrestrightnexttothereceiver.He beginstofallbacktoEarthattheinstantthe emitter sends the light upward. Accordingtothisfree-fallingobserver,gravityis absentduetoequivalenceprinciple.Hemeasures thewavelengthofthelighttobethesameasthe emittedwavelength.Asthelightmovesupward, he also begins to fall. However, the wavelength of thelightremainsunchangedasseenbyhim.He willobservethereceivermovingawayfromhim asheisfalling.Therefore,whenthereceiver receivesthelight,itisgoingtodetectalonger wavelengththanthatmeasuredbythefree-falling observerduetoDopplereffect.Inthiscase,we havegravitationalredshifteffectwhenlightis movingagainstthegravity.Ontheotherhand,if the emitter had been at the top and the receiver on theground,theshiftwouldhavebeentoward shorterwavelengthsincethereceiverwouldbe movingtowardthefree-fallingobserver. Gravitational blue shift effect occurs in this case. The Physics Update Vol. 10 No.1 2006 41 Figure10Thegravitationalredshiftexperiment carried out at Harvard tower. 10 Thegravitationalredshiftwasverifiedinthe famous Pound-Rebka experiment. This experiment is very close in concept to the one described in the abovethoughtexperiment.Itwascarriedoutat Harvard University's Jefferson laboratory in 1959. Thisresultconfirmedthepredictionsatthe10% level and was later improved to better than the 1% level by Pound and Snider. Bending of Light Einstein predicted that the deflection of light under the influence of gravity. It is quite easy to see how Einsteinsequivalenceprincipleleadstoa deflectionoflightfromthefollowingthought experiment.Consideranastronautinarocket locatedintheouterspaceundernoinfluenceof gravity. A light ray is emitted from one side of the rocket to a light detector on the opposite side. The astronautobservesthatthelightrayistravelling alongastraightlinefromonesidetotheother. Now,considerthesamerocketinfree-fallnearto theEarth.Accordingtoequivalenceprinciple,the astronautagainwillobservethatthelightray travels in a straight line across the rocket. Consider the same experiment from the view point ofastationaryobserverontheground.This stationaryobserveralsoneedstoobservethatthe lightrayisdetectedbythedetector.However,by the time the light has crossed the rocket, the rocket andthedetectorhasfallendownadistance. Therefore,thisstationaryobserverwillobserve thatthelightrayfollowsacurvedpath.Sincethe 10 http://hyperphysics.phy-astr.gsu.edu/ stationaryobserverisinagravitationalfield,he will conclude that the gravity bends light. Figure 11 A schematic diagram for the bending of light due to gravity. 11 Thisbendingoflightwasthefirstexperimental test of Einsteins predictions. The idea is to detect thechangeinpositionofstarsasthestarlight gazedtheSun.Theobservationswereperformed bySirArthurEddingtonandhiscollaborators duringthesolareclipseof1919.Theresultwas consistent with Einsteins prediction and made the frontpage of mostmajornewspapers.Itmadethe nameofEinsteinandhisgeneralrelativityworld famous. Gravitational Time Dilation Generalrelativitypredictsthatclockstickslower inastronggravitationalfield.Thiscanbe illustratedasfollowing.Fromthediscussionon gravitational red shift, we know that light is going tobered-shiftedwhenitistravellingupwards. The observer on the top is going to detect the light withlongerwavelength.Thiswillmeanthatheis goingtoreceivethelightpulseslessfrequently thantheobserverontheground.Forinstance,if theobserveronthegroundsendslightpulses at1 secondintervals(accordingtotheclockonthe ground),thentheobserveronthetopwould receivethepulsesatintervalsofgreaterthan1 second(accordingtotheclockonthetop).The observeronthetopthereforeconcludesthatthe ground clock is running slower. Alternatively,iftheobserveronthetopsentlight pulsesatonesecondintervalsdownwards,the 11 http://hyperphysics.phy-astr.gsu.edu/ The Physics Update Vol. 10 No.1 2006 42 observer on the ground would receive the pulses at intervalsoflesserthanonesecond.Theground observerwillthenconcludethatthetopclockis runningfaster.Wethereforeconcludethatclocks run slower in a stronger gravitational field and the strongerthegravitationalfieldtheslowerthe clocks! Thiseffectalsoaffectsourbiologicalclocks. Hence,itmakescertainsensethatitwouldbe better if we always stay on the basement level. We willbeyoungerthanthosestayonthehigh-floor level! However, this effect is going to be a minute one on the Earth due to the weak gravity. Imagine thatwearelivingonasurfaceofamuchmore massive object. In this case, we can simply stay on the basement in order to be younger. Although one willbeolderstayingonhigh-floorlevel,one would be smarter there since one could think faster there comparatively! Gravitationaltimedilationhasbeen experimentallymeasuredusingCesiumatomic clocks by Hafele and Keating on 1971. They made airline flights around the world in both directions, once eastward and once westward. They compared theirclockswiththeclockoftheObservatoryin Washington,D.C.whentheyreturned.To calculatetheexpectedtimes,wehavetoinclude bothgravitationaltimedilationandspecial relativistic effect. The result matched the theory to betterthan10%.Laterexperimentsinvolving rocketsandspacecraftimprovedonthisaccuracy. Nowadays,thiseffecthastobetakencareto resolve the discrepancy between the clocks on the satellitesandtheclocksonthegroundinthe GlobalPositioningSystem(GPS).Thisisa practicaldemonstrationofthetheoryofrelativity in a real-world system. Precession of the Perihelion of Mercury AlongstandingprobleminthestudyoftheSolar SystemwasthattheorbitofMercurydidnot behaveasrequiredbyNewton'stheory.Asthe MercuryorbitstheSun,ittracesoutanellipse withtheSunatonefocus.Thepointintheorbit thatistheclosesttotheSun,calledperihelion,is fixed,sothatalinedrawnfromtheSuntothe perihelion points in a fixed direction. It was found thattheperihelionofMercurydoesnotalways occuratthesamelocationbutrotatesaroundthe Sun. The ellipse is then rotates in its plane so that the actual orbit describes a kind of rosette pattern. This rotation of the orbit is called a precession. Theprecessionoftheorbitisnotpeculiarto Mercury,alltheplanetaryorbitsprecess.Infact, Newton'stheorypredictstheseeffectsasmostly beingproducedbythegravitationalattractionsof theotherplanetsononeanother.Thequestionis whetherNewton'spredictionsagreewiththe amountanorbitprecesses.Theprecessionofthe orbitsofallplanetsexceptforMercury'scan,in fact, be understood using Newton's equations. But Mercury seemed to be an exception. Figure12Artist'sversionoftheprecessionof Mercury's orbit. 12 ThetotalobservedprecessionofMercuryis5600 arc-secondspercentury(onedegreeequalsto 3600arc-seconds)withrespecttothepositionof theVernalEquinoxoftheSun.Newton's equations predict a precession of 5557 arc-seconds percentury.Thishastakenintoaccountallthe effectsfromtheotherplanets,theslight deformation of the Sun due to its rotation, as well as,thefactthattheEarthisnotaninertialframe. Thereisadiscrepancyof43arc-secondsper century. A number of ad-hoc proposals were put forward to account for the discrepancy. One was the existence of new matter (planet, dust or asteroid) in an orbit betweenMercuryandtheSun.Unfortunately,no solid observational evidence was ever found. Lots ofsuggestionsweremadetoexplainthis discrepancy,somesimple,someserious,some verycomplicated,andnoneverysuccessful.In contrast, Einstein was able to predict, without any adjustmentswhatsoever,thattheorbitofMercury shouldprecessbyanextra43secondsofarcper century should the general relativity be correct. 12 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/EinsteinTest .html The Physics Update Vol. 10 No.1 2006 43 Black Holes Blackholeisprobablyoneofthefamous predictionsofgeneralrelativity.Ablackholeisa cosmichotelinsuchawaythatyoucancheckin butyoucannotcheckout!Itisanobjectso massivethatnotevenlightcanescapeits attractions. It is surprising that the idea of a black holewasfirstanticipatedover200yearsago although general relativity is to be employed to the full description of black holes. Normally,whenyouthrowaballupintothe air,itfallsbackdowntotheEarth.However,if you throw a ball up at a speed of greater than 11.2 kilometers per second, Earth's escape speed, it will notreturn.Suchspeedisrequiredinorderforthe astronautsto gototheMoon.Escapespeedisthe keytounderstandblackholes.Bydefinition,the escape speed is the minimum speed required for an objecttoescapefromthegravitationalpullof another.Abodysescapespeedisproportionalto thesquarerootofthebodysmassdividedbythe square root of its radius. In1783,JohnMichell,aBritishgeologistand astronomer,pointedoutthatasufficientlydense objectmighthaveanescapespeedfasterthan light.Sincenotevenlightcanescapefromthis object,itwouldbeinvisible.In1796,theFrench mathematicianPierreSimonLaplacepromoted similar ideas to those of Michell. These objects are often referred to as dark stars. The phase black hole was coined by Wheeler in 1967. Considerahypotheticalexperimentinwhichthe Sunisbeingcompressed.AstheSunshrinks,its massremainsthesame,butitsescapespeed increases because the Suns radius is decreasing. If theSuncouldbecompressedtolessthan3 kilometers,theescapespeedwouldexceedthe speedoflight.Sincenothingcangofasterthan light,absolutelynothingcouldescapefromthe surface of the compressed Sun. Our Sun would be invisibleanduncommunicative.Thecompressed Suncouldbesaidtohavedisappearedfromthe universe. Only its gravitational field would remain behind betraying the presence of its mass. The connection of black holes to general relativity startswithaGermanastronomerKarl Schwarzschild.In1916,Schwarzschildfirst obtainedanexactsolutionfromEinsteinsfield equationsdescribingastaticpointmass.This solutionisnowknownastheSchwarzschild solution.Thecriticalradiusatwhichtheescape speedfromanobjectwouldequaltothespeedof lighthasbeennamedafterSchwarzschildasthe Schwarzschildradius.Everyobjecthasa Schwarzschildradiusatwhichtheobjectwould havetobecompressedforittobecomeablack hole. In other words, a black hole is an object that happenstoliewithinitsownSchwarzschild radius. Figure13BasicstructureofaSchwarzschildblack hole. 13 Thesurfaceofanimaginaryspherewithradius equal to the Schwarzschild radius and centered on thepointmassiscalledtheeventhorizon.We couldthinkoftheeventhorizonasthesurface ofablackhole.Theeventhorizonservesasa loyalcosmictrafficofficerensuringaone-way traffictoblackhole.Itdefinestheregionwithin whichnoeventcaneverbeknownbyoutside observers.Anythingfromoutsidescancrossthis surface;however,anythingthathappensinside remains forever hidden to outside observers. There is a singularity at the center of a black hole. Thisisapointatwhichbothitsdensityand gravitationalfieldbecomeinfinite.Singularities arenotphysical;rather,theyalwayssignalthe breakdownofthetheoryproducingthem.Many strangethingsmayoccuraroundsingularities.In viewofthis,RogerPenroseproposedthecosmic censorshiphypothesisstatingthatNaturealways hidesanysingularityin1969.Forinstance,the blackholesingularityisfoundinsideanevent horizon.Eventhoughphysics fails,itsbreakdown cannot affect us outside. A black hole could be formedat the final stage of stellarevolutionforahighmassstar.Astarlike ourSunispreventedfromgravitationalcollapse bytheoutwardpressuregeneratedbynuclear 13 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/BlackHole Anat.html The Physics Update Vol. 10 No.1 2006 44 fusions in the interior. This is similar as in the case of a balloon in which it is not flattened due to the pressure of air molecules interior. Toward the end ofastarslife,thenuclearfuelforthesenuclear reactionsbecomesexhaustedandthestarnow begins to contract under gravity. The final destiny of the star is governed simply by its mass. A not very massive star, such as the Sun, will shed some of its outer layers to form planetary nebula.Atthecenterofthenebularemainsthe coreofthestar,whichcoolsdowntobecomea smallbutdensewhitedwarf.Awhitedwarf consistsofnucleiwanderingaboutintheseaof electrons.Itispreventedtobefurther gravitationallycollapsedbecausetheinward gravityisbalancedbytheelectrondegeneracy pressure.Thiselectrondegeneracypressureisa consequenceofthePauliexclusionprinciplein whichtheelectronsareresistedtobesquashed together. StarsmoremassivethanourSuncometoamore dramaticend.TheIndian-bornastrophysicist, SubrahmanyanChandrasekhar,showedthatthe electrondegeneracypressureisnotstrongenough tocounterbalancethegravitationalpullifthestar ismoremassivethan1.4solarmasses Chandrasekharlimit.Theelectronsarethen absorbedbyprotonstoformneutrons,andthe final collapse of the core is very rapid. This results averyviolentstellarexplosion,whichiscalled supernovabyWalterBaadeandFritzZwickyin 1934,andaneutronstarisleftbehind.The neutrondegeneracypressurenowservesasa mechanismtopreventfurthergravitational collapse.In1939,RobertOppenheimerand HartlandSnyderfurthershowedthattheneutron degeneracypressureisinsufficienttoprevent furthercollapseifthemassoftheneutronstaris morethan3solarmasses.Thecorewillcollapse that results in a singularity and a black hole is then formed. Blackholesarenotcosmicvacuumcleaner.They do not cruise around interstellar space and suck up everythinginsight.Infact,theorbitofanobject near a black hole is the same as its orbit near a star of the same mass. Only if the object is too close to theeventhorizon,itsorbitwouldbesignificantly deviatefromthepredictionsfromNewtonian gravity.Ofcourse,iftheobjectisfallingintoa black hole, it would be unable to come out. To examine the nature of the black hole, lets orbit around a black hole with 10 solar masses at a safe distanceinaspacecraft.Thisblackholewould have a 30-kilometer event horizon. Suppose a poor observerisfallingintotheblackholetoprobeit. Everythingwouldappeartobeasusualtothe infallingobserver.Hewillrealizenothingspecial andeventuallycrossestheeventhorizonthe pointofnoreturn.Thispoorobserverisnow doomed to fall towards the singularity of the black hole. As he falls closer to the singularity, he would start to be uncomfortable.If, as he falls in, his feet are closer to the singularity than his head, his feet will be pulled more strongly inward than his head. Hewouldfindhimselfstretchedenormouslyin heightandsqueezedunmercifullylaterally.The nearinfinitespacetimecurvaturewouldstretch him like a piece of spaghetti and rip his body apart spaghettification.Theremnantofhisbody wouldbehittingtothesingularityandtheblack hole has now added the mass of this poor observer. Watchingfromasafedistanceinourorbiting spacecraft,wewouldrealizethatanincreasing amountoftimeisrequiredtoreceivethesignal fromthepoorinfallingobserverasheis approaching the event horizon gravitational time dilation.Furthermore,thesignalfromthepoor infallingobserverwouldbedetectedtohave longerwavelengthgravitationalredshift.Upon reachingtheeventhorizon,thetimeofthe infalling observer would seem to stop from us and thesignalemittedwouldbered-shiftedto infinitelylongwavelengths.Theoretically,the signalcouldstillreachusstillmovingatthe speedoflightbutwithzeroenergy.Thus,the emittedsignalwouldbered-shiftedbeyondour perception.Consequently,theimageofthe infallingobserverwouldbefrozenontheevent horizonandwewouldneveractuallywitnessthe infalling observer sink below the event horizon. Alltheobservationalevidenceforblackholesis necessarilyindirect.Asmatterfallingtowardsa blackholewillbeacceleratedtospeeds approachingthespeedoflight,weexpectblack holes to be strong sources of X-rays caused by the charged matter falling towards them. There is now agreatdealofindirectastronomicalevidencefor blackholesintwomassranges:(1)stellarmass black holes with masses ranging from 4 to 15 solar masses;and(2)supermassiveblackholeswith massesintherangeoforder105to1010solar masses. The Physics Update Vol. 10 No.1 2006 45 Manystarsinourgalaxyarebelievedtooccurin binarysystemsinwhichtwostarsareinorbit roundeachother.Somebinarysystemshave peculiarpropertiesthatthevisiblememberisin orbitaroundtheinvisiblecompanion.The invisiblememberemitslargeamountsX-raysand itsmassismeasuredasseveralsolarmasses.Itis thereforebelievedthattheinvisiblecompanionis ablackhole.Amongst,oneparticularbinary systemlyingintheconstellationCygnushas drawn much attention. The black hole candidate is anX-raysourcecalledCygnusX-1.Thevisible companionisablueB-typesupergiantandthe massoftheCygnusX-1isabout10solarmasses makingittoomassivetobeanythingbutablack hole. Some other black hole candidates include the thirdX-raysourcediscoveredintheLarge MagellanicCloudcalledLMCX-3witha mass of nearly10 solar masses, as well as, the X-raybinarysystemA0620-00whichcontainsan invisible compact object of mass 3.8 solar masses. However,thestrongestevidenceforblackholes comesfromobservationsofthecentersofmany galaxies including our own. It is currently believed that most if not all galaxies contain a supermassive blackholeattheircenters.SagittariusA*isa brightandcompactradiosourceatthecenterof our own galaxy Milky Way which is believed tobeassociatedwiththe2.6million-solar-mass supermassiveblackhole.InMay2004,30 previously-hiddensupermassiveblackholes outsideMilkyWayhavebeendiscovered.This discoverysuggeststhatthereareatleasttwiceas many of these black holes as previously thought. Additionally,thereissomeevidencefor intermediate-massblackholes,thosewithmasses ofafewhundredtoafewthousandsolarmasses. In2000,thefirstintermediate-massblackholes havebeenobservedfarfromthecenterofthe galaxyM82.Theseblackholesareprobably youngandhavemassesbetween100and1000 solarmasses.Thediscoveryofthefirst intermediate-massblackholeinourgalaxyhas beenreportedin2004.Thisblackholeof1300 solarmassesisorbitingthreelight-yearsfrom SagittariusA*.Thediscoveriesofintermediate-massblackholesprovidemissinglinkbetween stellarmassblackholesinbinariesandthe supermassive black holes in the hearts of galaxies. Expansion of the Universe Generalrelativityalsoplaysaroleontheorigin andevolutionofthewholeuniverse.Afterhis greatsuccesswithgeneralrelativity,Einstein begantothinkabouttheimplicationofhistheory fortheuniverseasawhole.In1917,hecameout the firstmathematicalmodel of the universe. This begananewfieldofphysicsrelativistic cosmology. Tohisgreatsurprise,whentheequationsare appliedtotheuniverseasawhole,hereacheda remarkableconclusion:theoverallsizeofthe universemustbechangingintime.Atthistime, theuniversewasstronglybelievedtobeavery staticplace.Forthisreason,Einsteinchoseto modify his field equations by introducing an extra termknownascosmologicalconstant.Withthis extraterm,heintroducedanewrepulsiveforce andthisallowedhimtoobtainastaticuniverse solution. In1922,RussianmeteorologistAlexander FriedmannhadusedEinsteinsoriginalfield equationsandobtainedsolutionsdescribing expandinguniverses.Thenextimportantstepin revealingthenatureoftheuniversewastakenby EdwinHubble.In1929,Hubblediscoveredthat distantgalaxiesarerecedingfromoursataspeed proportionaltotheirdistanceHubbleslaw. Hubblesobservationisdirectevidencethatthe universeisexpanding.OwingtoHubbles discovery,Einsteinlateradmittedthatthe introductionofcosmologicalconstantwasthe greatest blunder of my life. Figure 14 All stars will see all other stars moving away fromtheminanexpandinguniverse.Arisingloafof raisin bread is a good visual model: each raisin will see allotherraisinsmovingawayfromitastheloaf expands. 14 14 http://hyperphysics.phy-astr.gsu.edu/ The Physics Update Vol. 10 No.1 2006 46 In1931,GeorgesLematre,aBelgianpriestand mathematician,proposedthattheuniversemay have started with the explosion of a primeval atom containingthetotalmassoftheuniverse.The observedexpansionwascausedbytheexplosion ofthiscosmicatom.Thisexplosionwaslater calledtheBigBangandthisdescriptionofthe creationoftheuniverseisnowknownastheBig Bang model. The implications of the Big Bang model were first workedoutbyRussian-bornphysicist,George Gamow.Heconsideredindetailhownuclear reactionstakingplaceaftertheBigBangcould createtheelementswehaverightnow.Together withHansBetheandhisgraduatestudentRalph Alpher,theyrealizedthattheBigBangwouldbe expectedtocreatelighterelements,hydrogenand helium,inthecorrectproportionstoexplaintheir abundance in the early universe. The abundance of lightelementsisoneoftheprimarypiecesof evidence for the Big Bang model. TheBigBangmodelhadaseriousproblemthe Earthwasolderthantheuniverse.Partly motivatedbythisproblem,FredHoyle,Thomas Gold and Hermann Bondi developed an alternative modelSteadyStatemodeltoaccountforthe expansionoftheuniverse.Thisinvolvedthe continuouscreationofmatterandyieldeda continuouslyexpandinguniversewithaconstant averagedensity.Infact,itwasHoylewhocoined the name Big Bang for Lematres model. Thesupportofthesetwomodelswasquiteeven foranumberofyears.However,thediscoveryof the cosmic background radiation has ruled out the SteadyStatemodel.Gamow,AlpherandRobert Hermanhaveactuallypredictedtheexistenceof thecosmicbackgroundradiationin1948. However, itwasArno Penzias and Robert Wilson at the Bell Telephone Laboratories in Murray Hill, NewJersey,whoaccidentallydiscoveredthis radiationin1965.Theradiationwasactingasa sourceofexcessnoiseinaradioreceiverthey werebuilding.Thiscosmicbackgroundradiation isnowabout2.7Kcorrespondingtothe microwaveportionoftheelectromagnetism spectrum. Although it is invisible to the naked eye, itfillstheuniverseandcanbedetected everywhere we look. Thecosmicbackgroundradiationwasemitted onlyafewhundredthousandyearsaftertheBig Bang,longbeforestarsorgalaxieseverexisted. Thus,bystudyingthedetailedphysicalproperties oftheradiation,wecanlearnaboutconditionsin theuniverseonverylargescales,sincethe radiationweseetodayhastraveledoversucha largedistance,andtheuniverseatveryearly times.Observationsofthecosmicbackground radiationhavecontinued,withthemost spectacularresultscomingfromtheCosmic BackgroundExplorer(COBE)satellitelaunched in1989,aswellas,theWilkinsonMicrowave AnisotropyProbe(WMAP)satellitelaunchedin 2001. TheBigBangisaverysuccessfulmodelonthe universe. However, it fails to account some of the basic questions about the universe. For instance, it providesnoclueonwhytheuniverseisbeingso flat,whatdrives theexpansion,etc.In1981,MIT physicistAlanGuth,cameuptheideaofthe inflationaryuniverse.Heproposedthattheearly universe went through a period of extremely rapid expansion.Duringthisinflationepochlasted between 10-35 to 10-32 seconds after the Big Bang theuniverseexpandedinsizebyafactorof1050 from smaller than an atom to bigger than a galaxy. Itwasdrivenbyvastamountsofenergyreleased whenasymmetrybreakingphasetransition occurred. Figure 15 The introduction of inflationary period to the early universe. 15 Theevolutionoftheuniverseisdeterminedbya strugglebetweentherateofexpansionandthe averagedensityofmatter.Thecurrentrateof expansionismeasuredbytheHubbleconstant.If theaveragedensityofmatterislessthanthe critical density, which is proportional to the square oftheHubbleconstant,thentheuniversewill 15 http://hyperphysics.phy-astr.gsu.edu/ The Physics Update Vol. 10 No.1 2006 47 expandforeveropenuniverse.Iftheaverage densityofmatterisgreaterthanthecritical density,thenthegravitywilleventuallywinand theuniversewillcollapsebacktoitself,theso calledBigCrunch,closeduniverse.Ifthe averagedensityofmatterexactlyequalstothe criticaldensity,therateofexpansionwill asymptotically approach zero flat universe.However, recent observations of distant supernova havesuggestedthattheexpansionoftheuniverse isactuallyaccelerating,whichimpliesthe existenceofaformofmatterwithastrong negativepressure,suchasthecosmological constant.Thisstrangeformofmatterisalso sometimes referred to as the dark energy. If dark energyinfactplaysasignificantroleinthe evolution of the universe, then in all likelihood the universe will continue to expand forever. Time Travel Noideafromsciencefictionhascapturedthe humanimaginationasmuchastimetravelthe abilitytotraveltoanypointinthefutureorpast. Onecouldgotothefuturetotakeavacationor even bring back a cure for cancer. Onecould also go to the past to witness major historical events or even meet historical figures. Nowadays,thesubjectoftimetravelhasjumped fromthepagesofsciencefictionstothepagesof physicsjournalsasphysicistsexplorewhetherit mightbeallowedbythelawsofphysics.The paradoxes associated with time travel always pose achallengeandallowphysiciststotestthe boundariesofcurrentphysicallaws.InNewtons universe, time travel was inconceivable. However, ithasbecomearealpossibilityinEinsteins universe. Timedilationisthekeytotimetraveltothe future.Itisbestillustratedthroughthefamous twin paradox. Suppose there are twin brothers AlbertandBob.AlbertstaysontheEarth.Bob takes off in a spaceship at 80 percent of the speed oflighttoAlphaCentauriwhichis4light-years away.Bobstriptherewilltake5Earthyears. However,AlbertwillseeBobsclockrunning slowerthanhisby40percents,Bobwillageonly 3 years during the trip. Bob turns around when he reaches Alpha Centauri and returns to Earth again at 80 percent of the speed of light. The return also takes5Earthyears.So,Albertis10yearsolder whenBobarrivesbackhome.Onthereturntrip, Albert sees Bobs clock again running slower than his by 40 percents, so Albert sees Bob age another 3 years during his return. Bob will be 6 years older when he gets back. Bob has time-travelled 4 years into the future. Accordingtospecialrelativity,timewillslow downifonemovesfasterandfasterand approachesthespeedoflight.Ifonecouldreach thespeedoflight,timewouldstop.And,ifone couldmoveevenfasterthanspeedoflight,then one,inprinciple,couldgobacktotime. Unfortunately,thespeedoflightistheultimate speedlimitandonecannotgofasterthanit. However,ingeneralrelativity,spacetimecan become so distorted that permits shortcuts through spacetime.Thiswouldallowonetobeatalight beam and travel back to the past. In1988KipThorneandhisassociateshave proposed the idea of taking a shortcut back in time bytravelingquicklythroughawormhole.Ina nutshell,wormholesaretunnelsconnectingtwo distantregionsofspacetime.Ifonecouldtake suchashortcut,onecouldreachthedestination aheadofalightbeamacrosscurvedspace.In principle,onecouldevenbeabletogetbackin time to see oneself off. Figure16Awormholecreatesashortcutbetweenthe Earth and Alpha Centauri. 16 To illustrate the idea, suppose there is a wormhole withonemouthneartheEarthandtheothernear AlphaCentauri.YoucouldthenreachAlpha Centauribytakingtheordinaryfour-light-year routeorjumpingthorughthewormholewithlets say10kilometers.IfyoudivefromtheEarth through the wormhole in the year 2006, you might perharpsemergeatAlphaCentauriintheyear 1996. You could now travel back to the Earth at a veryhighspeedsothatyougetbacktoEarthin year 2000. You would have time-travelled 6 years backintoyourpastandcouldevenshakehands with yourself when you took off in the year 2006! 16 J. R. Gott, Time Travel in Einsteins Universe, P119, Houghton Mifflin (USA), 2001. The Physics Update Vol. 10 No.1 2006 48 Figure 17 A warp drive creates a shortcut betweenthe EarthandAlphaCentauriviaaU-shapeddistortionin spacetime. 17 Warpdriveisasisterpossibilityintimetravelto wormhole. In Star Trek, the crew of the Enterprise usedwarpdrivetoalterspacesothattheycould travel among stars at speed faster than that of light. Insimpleterms,awarpdrivecreatesaU-shaped distortioninspacetimeandhenceashortcut betweentwodistantregions.Onecouldalsouse warp drive to travel to the past just like the case in wormhole. Future Outlooks General relativity also predicts gravitational waves that move through space. Gravitational wave is the gravitationalcounterpartofanelectromagnetic wave. Gravitational radiation results from changes in the strength of a gravitational field. In principle, a gravitational wave should be emitted at the speed oflightwheneveranymassiveobjectsaccelerate. The passage of gravitational wave should produce smalldistortionsinthespacethroughwhichit passes.However,gravityisanexceedinglyweak forcecomparedwithelectromagnetism,these distortions are expected to be very small. The detection of gravitational wave would provide such strong support for the theory of relativity that scientistsareeagertosearchforthem. Gravitationalwaveshavenotyetbeendetected directly,butwebelievethattheyhavebeen detectedindirectlybyradioastronomersinthe binarypulsarsystem1913+16.Asthepulsaris orbitingarounditscompanion(every8hoursin thiscompactsystem),generalrelativitypredicts thatgravitationalwavesshouldbeproduced. Althoughthesewavesarefartoofainttobe detecteddirectly,thebinarypulsarsystemis losingenergythroughthisradiation,andthe pulsar/neutronstaranditscompanionare predictedtobeslowlyspirallingtogether.The rapidradiopulsespermitprecisetimingofthe 17 J. R. Gott, Time Travel in Einsteins Universe, P119, Houghton Mifflin (USA), 2001. pulsarorbitbyDopplershiftsofthepulseperiod as the pulsar moves toward or away from us. Since thediscoveryofthebinarypulsarin1974,timing ofthepulsarhasshownthatthestarsareindeed spiralling together just as predicted. In 300 million yearsthestarswillcoalesce-thatshouldproduce gravitational radiation that can be easily detected! Directdetectionofgravitationalwavehaslong beensought.Laserinterferometerobservatories arebeingconstructedthatwillbeabletodetect gravitationalwavesfrompossiblesources potentiallyasfarawayas100millionlightyears. In1992,LaserInterferometricGravity-wave Observatory(LIGO),acollaborationbetween CaltechandMIT,beganitssearchforcosmic gravitational waves that are theoretically created in supernova collapses of stellar cores, collisions and coalescencesofneutronstarsorblackholesand theremnantsofgravitationalradiationcreatedby thebirthoftheuniverse.Thereareatleastfive otherprojectsamongEuropean,Australian, Japanese, and Space Physicist groups constructing otherGravitational-WaveObservatories.This would open up a whole new observational window on the universe. Figure18Detectionofgravitationalwaves:Inthe presenceofgravitationalwaves,thedistancebetween the mirrors will fluctuate. 18 Ourunderstandingoftheuniversehasdeepened profoundlyduringthepastcentury.Wenowhave twofoundationalpillarsuponwhichmodern physicsrests.OneisEinsteinsgeneralrelativity whichprovidesatheoreticalframeworkfor unverstandingtheuniverseonthelargestscales: stars,galaxiesandbeyondtotheuniverseitself. Theotherisquantummechanicswhichprovides theoreticalframeworkforunderstandingthe universeonthesmallestscales:molecules,atoms andallthewaytosubatomicparticleslike electronsandquarks.Boththeorieshave tremendous success. 18 http://archive.ncsa.uiuc.edu/Cyberia/NumRel/LIGO.html The Physics Update Vol. 10 No.1 2006 49 However,generalrelativityandquantum mechanicscannotbebothrightthereare mutuallyincompatible.Sofar,physicistsstudy things that are either huge and heavy or things that aresmallandlight,butnotboth.Forthesecases, physicistsneedonlygeneralrelativityoronly quantummechanics.Undercertainextreme conditions,wherethingsareverymassiveand verysmallnearthecenteroftheblackholeor the whole universe at themoment of the big bang werequirebothgeneralrelativityandquantum mechanics.Unfortunately,whenwetryto combinegeneralrelativityandquantum mechanics,theirunionbringsviolentcatastrophe. Physicistsarenoweagertosearchforacorrect theoreticalframeworkquantumgravityto unifythesetwofoundationalpillarsforadeeper understanding. Further Readings 1.T.HeyandP.Walters,Einstein'sMirror, Cambridge University Press (1997). 2.K.S.Thorne,BlackHolesandTimeWarps: Einsteins Outrageous Legacy, W.W.Norton (New York), 1994. 3.J.R.Gott,TimeTravelinEinstein's Universe:ThephysicalPossibilitiesofTravel throughTime,HoughtonMifflin(Boston), 2001. 4. J.A.Wheeler,AJourneyintoGravityand Spacetime,ScientificAmericanBooks(New York), 1990. 5.J.Schwinger,EinsteinsLegacy:TheUnity ofSpaceandTime,ScientificAmerican Books (New York), 1986. 6.C.M.Will,WasEinsteinRight?Putting GeneralRelativitytotheTest,2ndEd.,Basic Books (New York), 1986. 7.R.M.Wald,Space,TimeandGravity:The Theory of the Big Bang and Black Holes, 2nd Ed.,UniversityofChicagoPress(Chicago), 1992. 8.S.Weinberg,TheFirstThreeMinutes:A ModernViewoftheOriginoftheUniverse, 2nd Ed., Basic Books (New York), 1993. 9.B.R.Greene,TheElegantUniverse: Superstrings,HiddenDimensions,andthe Quest for the Ultimate Theory, W. W. Norton (New York), 1999. 10.A.H.Guth,TheInflationaryUniverse:The QuestforANewTheoryofCosmicOrigins, Perseus Books (Cambridge), 1997.