PreprinttypesetinJHEPstyle-PAPERVERSIONGRAVITATIONALWAVEPHYSICSKostasD.KokkotasDepartment of Physics, Aristotle University of ThessalonikiThessaloniki 54124, Greece.Abstract: Gravitational waves are propagating uctuations of gravitational elds,thatis, ripplesinspacetime, generatedmainlybymovingmassivebodies. Thesedistortions of spacetime travel with the speed of light. Every body in the path of sucha wave feels a tidal gravitational force that acts perpendicular to the waves directionofpropagation;theseforceschangethedistancebetweenpoints,andthesizeofthechanges is proportional to the distance between the points. Gravitational waves canbedetectedbydeviceswhichmeasuretheinducedlengthchanges. Thefrequenciesandtheamplitudesof thewavesarerelatedtothemotionof themassesinvolved.Thus, theanalysisofgravitational waveformsallowsustolearnabouttheirsourceand, if therearemorethantwodetectorsinvolvedinobservation, toestimatethedistanceandpositionoftheirsourceonthesky.Article for the Encyclopedia of Physical Science and Technology, 3rd Edition, Volume 7 Aca-demic Press, (2002)Contents1. INTRODUCTION 11.1 Whyaregravitationalwavesinteresting? 21.2 Howwewilldetectthem? 22. THEORYOFGRAVITATIONALWAVES 32.1 Linearizedtheory 52.2 Propertiesofgravitationalwaves 112.3 Energyuxcarriedbygravitationalwaves 122.4 Generationofgravitationalwaves 132.5 Rotatingbinarysystem 163. DETECTIONOFGRAVITATIONALWAVES 183.1 Resonantdetectors 193.2 Beamdetectors 223.2.1 LaserInterferometers 223.2.2 SpaceDetectors 263.2.3 SatelliteTracking 273.2.4 PulsarTiming 284. ASTRONOMICALSOURCESOFGRAV.WAVES 284.1 Radiationfromgravitationalcollapse 294.2 Radiationfrombinarysystems 304.3 Radiationfromspinningneutronstars 314.4 Cosmologicalgravitationalwaves 311. INTRODUCTIONEinstein rst postulated the existence of gravitational waves in 1916 as a consequenceofhistheoryofGeneral Relativity, butnodirectdetectionofsuchwaveshasbeenmade yet. The best evidence thus far for their existence is due tothe workof1993Nobel laureates JosephTaylor andRussell Hulse. Theyobserved, in1974,twoneutronstarsorbitingfasterandfasteraroundeachother,exactlywhatwouldbe expectedif the binaryneutronstar was losingenergyinthe formof emittedgravitational waves. The predicted rate of orbital acceleration caused by gravitational1radiationemissionaccordingtogeneral relativitywasveriedobservationally, withhighprecision.Cosmicgravitational waves, uponarrivingonearth, aremuchweakerthanthecorrespondingelectromagneticwaves. Thereasonisthatstronggravitationalwavesare emittedbyvery massive compact sources undergoingvery violent dynamics.Thesekindsofsourcesarenotverycommonandsothecorrespondinggravitationalwavescomefromlargeastronomical distances. Ontheotherhand, thewavesthusproduced propagate essentially unscathed through space, without being scattered orabsorbedfrominterveningmatter.1.1Whyaregravitationalwavesinteresting?Detection of gravitational waves is important for two reasons: First, their detection isexpected to open up a new window for observational astronomy since the informationcarriedbygravitationalwavesisverydierentfromthatcarriedbyelectromagneticwaves. This newwindowontouniversewill complement our viewof thecosmosandwill helpusunveil thefabricofspacetimearoundblack-holes, observedirectlytheformationof blackholesorthemergingof binarysystemsconsistingof blackholesorneutronstars, searchforrapidlyspinningneutronstars, digdeepintotheveryearlymoments of theoriginof theuniverse, andlookat theverycenter ofthegalaxieswheresupermassiveblackholesweightingmillionsof solarmassesarehidden. Theseareonlyafewof thegreatscienticdiscoveriesthatscientistswillwitnessduringtherstdecadeofthe21stcentury. Second, detectinggravitationalwavesisimportantforourunderstandingof thefundamental lawsof physics; theproofthatgravitationalwavesexistwillverifyafundamental85-year-oldpredictionofgeneralrelativity. Also,bycomparingthearrivaltimesoflightandgravitationalwaves, from, e.g., supernovae, Einsteins prediction that light and gravitational wavestravel atthesamespeedcouldbechecked. Finally, wecouldverifythattheyhavethepolarizationpredictedbygeneralrelativity.1.2Howwewilldetectthem?Up to now,the only indication of the existence of gravitational waves is the indirectevidencethattheorbital energyintheHulse-Taylorbinarypulsarisdrainedawayatarateconsistentwiththepredictionofgeneralrelativity. Thegravitationalwaveis a signal, the shape of which depends upon the changes in the gravitational eld ofitssource. Asithasbeenmentionedearlier,anybodyinthepathofthewavewillfeelanoscillatingtidalgravitationalforcethatactsinaplaneperpendiculartothewavesdirectionof propagation. Thismeansthatagroupof freelymovingmasses,placedonaplaneperpendiculartothedirectionof propagationof thewave, willoscillateaslongasthewavepassesthroughthem, andthedistancebetweenthemwill varyasafunctionoftimeasinFigure1. Thus, thedetectionofgravitationalwavescanbeaccomplishedbymonitoringthetinychangesinthedistancebetween2freely moving test masses. These changes are extremely small; for example, when theHulse-Taylor binary system nally merges,the strong gravitational wave signal thatwill be emitted will induce changes in the distance of two particles on earth, that are1 km apart much smaller than the diameter of the atomic nucleus! To measure suchmotionsof macroscopicobjectsisatremendouschallengeforexperimentalists. Asearlyasthemid-1960s, JosephWeberdesignedandconstructedheavymetal bars,seismicallyisolated, towhichasetof piezoelectricstraintransducerswerebondedinsuchawaythattheycoulddetectvibrationsof thebarif ithadbeenexcitedbyagravitational wave. Today, thereareanumberofsuchapparatusesoperatingaroundtheworldwhichhaveachievedunprecedentedsensitivities,buttheystillarenotsensitiveenoughtodetectgravitational waves. Anotherformof gravitationalwavedetectorthatismorepromisinguseslaserbeamstomeasurethedistancebe-tweentwowell-separatedmasses. Suchdevicesarebasicallykilometer sizedlaserinterferometersconsistingofthreemassesplacedinanL-shapedconguration. Thelaserbeamsarereectedbackandforthbetweenthemirrorsattachedtothethreemasses, themirrorslyingseveral kilometersawayfromeachother. Agravitationalwave passing by will cause the lengths of the two arms to oscillate with time. Whenone arm contracts, the other expands, and this pattern alternates. The result is thattheinterferencepatternofthetwolaserbeamschangeswithtime. Withthistech-nique, higher sensitivities could be achieved than are possible with the bar detectors.Itisexpectedthatlaserinterferometricdetectorsaretheonesthatwill provideuswiththerstdirectdetectionofgravitationalwaves.2. THEORYOFGRAVITATIONALWAVESNewtonstheoryofgravityhasenjoyedgreatsuccessindescribingmanyaspectsofour every-day life and additionally explains most of the motions of celestial bodies intheuniverse. GeneralrelativitycorrectedNewtonstheoryandisrecognizedasoneofthemostingeniouscreationsofthehumanmind. Thelawsofgeneral relativity,though,inthecaseofslowlymovingbodiesandweakgravitationaleldsreducetothestandardlawsof Newtoniantheory. Nevertheless, general relativityisconcep-tuallydierentfromNewtonstheoryasitintroducesthenotionof spacetimeanditsgeometry. Oneof thebasicdierencesof thetwotheoriesconcernsthespeedof propagationof anychangeinagravitational eld. Astheapplefallsfromthetree, wehavearearrangementof thedistributionof massof theearth, thegravi-tational eldchanges, andadistantobserverwithahigh-precisioninstrumentwilldetect this change. According to Newton, the changes of the eld are instantaneous,i.e., theypropagatewithinnitespeed; if thisweretrue, howevertheprincipleofcausalitywouldbreakdown. Noinformationcantravel faster thanthespeedoflight. In Einsteins theory there is no such ambiguity; the information of the varyinggravitationaleldpropagateswithnitespeed,thespeedoflight,asarippleinthe3Figure 1: Theeects of agravitational wavetravellingperpendicular theplaneof acircularringofparticles, issketchedasaseriesofsnapshots. Thedeformationsduethetwo polarizations are shown.fabric of spacetime. These are the gravitational waves. The existence of gravitationalwavesisanimmediateconsequenceof anyrelativistictheoryof gravity. However,thestrengthandtheformof thewavesdependonthedetailsof thegravitationaltheory. This means that the detection of gravitational waves will also serve as a testofbasicgravitationaltheory.The fundamental geometrical framework of relativistic metric theories of gravityis spacetime, which mathematically can be described as a four-dimensional manifoldwhosepointsarecalledevents. Everyeventislabeledbyfourcoordinatesx(=0, 1, 2, 3);thethreecoordinatesxi(i = 1, 2, 3)givethespatialpositionoftheevent,whilex0isrelatedtothecoordinatetimet(x0=ct, wherecisthespeedoflight,whichunlessotherwisestatedwill besetequal to1). Thechoiceofthecoordinatesystemisquitearbitraryandcoordinatetransformationsof theform x=f(x)areallowed. Themotionofa testparticleisdescribedby acurveinspacetime. Thedistancedsbetweentwoneighboringevents,onewithcoordinatesxandtheotherwithcoordinatesx+ dx, canbeexpressedasafunctionofthecoordinatesviaasymmetrictensorg(x) = g(x),i.e.,ds2= gdxdx(2.1)ThisisageneralizationofthestandardmeasureofdistancebetweentwopointsinEuclidianspace. FortheMinkowskispacetime(thespacetimeofspecialrelativity),4g=diag(1, 1, 1, 1). The symmetric tensor is calledthe metric tensororsimplythemetricof thespacetime. Ingeneral relativitythegravitational eldis describedbythemetrictensor alone, but inmanyother theories oneor moresupplementaryeldsmaybeneededaswell. Inwhatfollows,wewillconsideronlythe general relativistic description of gravitational elds, since most of the alternativetheoriesfailtopasstheexperimentaltests.The information about the degree of curvature (i.e., the deviation from atness)of aspacetime is encodedinthe metric of the spacetime. Accordingtogeneralrelativity, anydistributionof mass bends the spacetime fabric andthe RiemanntensorR(thatisafunctionofthemetrictensorgandofitsrstandsecondderivatives)isameasureof thespacetimecurvature. TheRiemanntensorhas20independentcomponents. Whenitvanishesthecorrespondingspacetimeisat.Inthefollowingpresentation,wewillconsidermassdistributions,whichwewilldescribebythestress-energytensorT(x). Foraperfectuid(auidorgaswithisotropicpressurebutwithoutviscosityorshearstresses)thestress-energytensorisgivenbythefollowingexpressionT(x) = ( + p)uu+ pg, (2.2)wherep(x)isthelocalpressure,(x)isthelocalenergydensityandu(x)isthefourvelocityoftheinnitesimaluidelementcharacterizedbytheeventx.Einsteinsgravitationaleldequationsconnectthecurvaturetensor(seebelow)andthestress-energytensorthroughthefundamentalrelationG R12gR = kT(2.3)Thismeansthatthegravitationaleld,whichisdirectlyconnectedtothegeometryofspacetime, isrelatedtothedistributionofmatterandradiationintheuniverse.By solving the eld equations, both the gravitational eld (the g) and the motion ofmatter is determined. Ris the so- called Ricci tensor and comes from a contractionof theRiemanntensor(R=gR), Risthescalarcurvature(R=gR),whileGistheso-calledEinsteintensor, k=8G/c4isthecouplingconstantofthetheoryandisthegravitationalconstant, which, unlessotherwisestatedwill beconsideredequalto1. ThevanishingoftheRiccitensorcorrespondstoaspacetimefree of any matter distribution. However, this does not imply that the Riemann tensoriszero. Asaconsequence,intheemptyspacefarfromanymatterdistribution,theRicci tensor will vanish while the Riemann tensor can be nonzero; this means that theeectsofapropagatinggravitationalwaveinanemptyspacetimewillbedescribedviatheRiemanntensor.2.1LinearizedtheoryNow lets assume that an observer is far away from a given static matter distribution,andthespacetimeinwhichheorshelivesisdescribedbyametricg. Anychange5in the matter distribution, i.e., in T, will induce a change in the gravitational eld,whichwillberecordedasachangeinmetric. Thenewmetricwillbe g= g + h(2.4)where his a tensor describing the variations induced in the spacetime metric. As wewilldescribeanalyticallylater,thisnewtensordescribesthepropagationofripplesinspacetimecurvature, i.e., thegravitational waves. Inordertocalculatethenewtensor we have to solve Einsteins equations for the varying matter distribution. Thisisnotaneasytaskingeneral. However, thereisaconvenient, yetpowerful, waytoproceed, namelytoassumethathissmall ([h[ 1), sothatweneedonlykeeptermslinearinhinourcalculations. Inmakingthisapproximationweareeectively assuming that the disturbances produced in spacetime are not huge. Thislinearizationapproachhas provedextremelyuseful for calculations, andfor weakeldsatleast, givesaccurateresultsforthegenerationof thewavesandfortheirpropagation.TherstattempttoprovethatingeneralrelativitygravitationalperturbationspropagateaswaveswiththespeedoflightisduetoEinsteinhimself. Shortlyafterthe formulation of his theory - the year after - he proved that by assuming linearizedperturbationsaroundaatmetric,i.e.,g= ,thenthetensorh h12h(2.5)is governed by a wave equation, which admits plane wave solutions similar to the onesof electromagnetism; here his the metric perturbation and his the gravitationaleld(orthetracereverseof h). Thenthelineareldequationsinvacuumhavetheform_2t2+2_h h= 0 (2.6)(kk/x), whichisthethree-dimensional waveequation. Toobtaintheabovesimpliedform, theconditionh=0, knownasHilbertsgaugecondition(equivalent to the Lorentz gauge condition of electromagnetism), has been assumed.Agaugetransformationisasuitablechangeofcoordinatesdenedbyx x+ (2.7)whichinducesaredenitionofthegravitationaleldtensorh

h + . (2.8)Itcanbeeasilyprovedthatmustsatisfythecondition= 0, (2.9)6so that the new gravitational eld is in agreement with the Hilberts gauge condition.Thesolutionof equation(2.9)denesthefourcomponentsof sothatthenewtensorh

isalsoasolutionof thewaveequation(2.6), andthus, havethesamephysical meaning as h. In general, gauge transformations correspond to symmetriesof the eld equations, which means that the eld equations are invariant under suchtransformations. This implies that theeldequations donot determinetheelduniquely;however,thisambiguityindeterminingtheeldisdevoidofanyphysicalmeaning.Thesimplestsolutiontothewaveequation(2.6)isaplanewavesolutionoftheformh= Aeikx, (2.10)whereAisaconstantsymmetrictensor, thepolarizationtensor, inwhichinfor-mationabouttheamplitudeandthepolarizationofthewavesisencoded,whilekisaconstantvector, thewavevector, thatdeterminesthepropagationdirectionofthewaveanditsfrequency. Inphysical applicationswewill useonlythereal partof theabovewavesolution. ByapplyingtheHilbertgaugeconditionontheplanewave solution we obtain the relation Ak= 0, the geometrical meaning of which isthatAandkareorthogonal. Thisrelationcanbewrittenasfourequationsthatimpose four conditions on A, and this is the rst step in reducing the number of itsindependentcomponents. Asaconsequence,A,insteadofhaving10independentcomponents (as has every symmetric second rank tensor in a four dimensional space),hasonly6independentones. Furthersubstitutionoftheplanewavesolutioninthewave equation leads to the important equation kk= k20(k2x+k2y +k2z) = 0, whichmeans that kis a lightlike or null vector, i.e., the wave propagates on the light-cone.Thismeansthatthespeedof thewaveis1, i.e., equal tothespeedof light. Thefrequencyofthewaveis= k0.UptothispointithasbeenproventhatAhassixarbitrarycomponents,butduetothegaugefreedom,i.e., thefreedominchoosingthefourcomponentsofthevector , the actual number of its independent components can be reduced to two,inasuitablechosengauge. Thetransversetraceless or TTgaugeis anexampleof suchagauge. Inthis gauge, onlythespatial components ofharenon-zero(henceh0=0), whichmeansthatthewaveistransversetoitsowndirectionofpropagation, and, additionally, thesumof thediagonal componentsiszero(h=h00+h11+h22+h33 h = h = 0) (traceless). Due to this last property and equation (5),inthisgaugethereisnodierencebetweenh(theperturbationofthemetric)andh(the gravitational eld). It is customary to write the gravitational wave solutionintheTTgaugeashTT . ThatAhasonlytwoindependentcomponentsmeansthatagravitationalwaveiscompletelydescribedbytwodimensionlessamplitudes,h+and h, say. If, for example, we assume a wave propagating along the z-direction,7thentheamplitudeAcanbewrittenasA= h+

++ h

(2.11)where+andaretheso-calledunitpolarizationtensorsdenedby

+_____00 0 001 0 000 1000 0 0_____

_____0000001001000000_____(2.12)As mentioned earlier, the Riemann tensor is a measure of the curvature of spacetime.A gravitational wave, propagating in a at spacetime, generates periodic distortions,whichcanbedescribedintermsof theRiemanntensor. InlinearizedtheorytheRiemanntensortakesthefollowinggauge-independentform:R=12 (h + hhh) , (2.13)whichisconsiderablysimpliedbychoosingtheTTgauge:RTTj0k0= 122t2hTTjk, j, k = 1, 2, 3. (2.14)Furthermore,intheNewtonianlimitRTTj0k0 2xjxk, (2.15)where describes the gravitational potential in Newtonian theory. Earlier we denedthe Riemann tensor as a geometrical object, but this tensor has a simple physical in-terpretation: it is the tidal force eld and describes the relative acceleration betweentwoparticlesinfreefall. If weassumetwoparticlesmovingfreelyalonggeodesicsof acurvedspacetimewithcoordinatesx()andx() + ()(foragivenvalueofthepropertime,()isthedisplacementvectorconnectingthetwoevents)itcanbeshownthat,inthecaseofslowlymovingparticles,d2kdt2 Rk0j0TTj. (2.16)This is a simplied form of the equation of geodesicdeviation. Hence, the tidal forceactingonaparticleis:fk mRk0j0j, (2.17)wheremisthemassof theparticle. Equation(2.17)correspondstothestandardNewtonianrelationforthetidalforceactingonaparticleinaeld.8Keepingthisinmind,wewilltrytovisualizetheeectofagravitationalwave.Let us rst consider two freely falling particles hit by a gravitational wave travellingalongthez-direction,withthe(+)polarizationpresentonly,i.e.,h= h+

cos[(t z)]. (2.18)Then, themeasureddistancexbetweenthetwoparticles, originallyatadistancex0,alongthexdirection,willbexx0= 1 12h+ cos[(t z)] or x= xx0= 12h+ cos[(t z)]x0(2.19)whichimpliesthattherelativedistancexbetweenthetwoparticleswill oscillatewith frequency . This does not mean that the particles coordinate positions change;insteadtheyremainatrestrelativetothecoordinates,butthecoordinatedistanceoscillates. If the particles were placed originally along the y-direction the coordinatedistancewouldoscillateaccordingtoyy0= 1 +12h+ cos[(t z)] or y= yy0=12h+ cos[(t z)]y0(2.20)Inotherwords, thecoordinatedistancesalongthetwoaxesoscillateoutof phase,thatis, whenthedistancebetweentwoparticlesalongthexdirectionismaximumthe distance of two other particles along the ydirection is minimum, and after half aperiod, it is the other way around. The eects are similar for the other polarization,wheretheaxesalongwhichtheoscillationsareoutofphaseareatanangleof45owith respect to the rst ones. This can be visualized in Figure 1,where the eect ofapassinggravitationalwaveonaringofparticlesisshownasaseriesofsnapshotscloselyseparatedintime.Anotherwayofunderstandingtheeectsofgravitationalwavesistostudythetidalforceeldlines. IntheTTgaugetheequationofthegeodesicdeviation(2.16)takesthesimpleformd2kdt212d2hTTjkdt2j(2.21)andthecorrespondingtidalforceisfkm2d2hTTjkdt2j(2.22)For the wave given by equation (2.18) the two nonzero components of the tidal forcearefxm2 h+2cos[(t z)]x0, and fy m2 h+2cos[(t z)]y0. (2.23)It canbeeasilyprovedthat thedivergenceof thetidal forceis zero(fx/x0+fy/y0= 0). ItcanthereforeberepresentedgraphicallybyeldlinesasinFigure2.9Figure 2: The tidal eld lines of force for a gravitational wave with polarization (+) (leftpanel) and () (right panel). The orientation of the eld lines changes every half periodproducingthedeformationsasseeninFigure1. Anypointacceleratesinthedirectionsof thearrows, andthedenserarethelines, thestrongestistheacceleration. Sincetheaccelerationisproportional tothedistancefromthecenterof mass, theforcelinesgetdenser as one moves away from the origin. For the polarization () the force lines undergoa 450rotation.Letusnowreturnbacktothetwopolarizationstatesrepresentedbythetwomatrices+and . It is impossibletoconstruct the(+) patternfromthe()patternandviceversa; theyareorthogonal polarizationstates. Byanalogywithelectromagneticwaves, thetwopolarizationscouldbeaddedwithphasedierence(/2) to obtain circularly polarized waves. The eect of circularly polarized waveson a ring of particles is to deform the ring into a rotating ellipse with either positiveornegativehelicity. Theparticlesthemselvesdonotrotate; theyonlyoscillateinand out around their initial positions. These circularly polarized waves carry angularmomentumtheamountofwhichis(2/)timestheenergycarriedbythewave. Ifwe consider the gravitational eld, then, accordingtoquantumeldtheory, thewaves are associatedwithfundamental particles responsible for the gravitationalinteraction, andaquantumof theeldwill haveenergy andconsequentlyspin2. Thismeansthatthequantaof thegravitational eld, thegravitons, arespin-2, massless particles (sincetheytravel withthespeedof light). Another wayofexplainingwhyagravitonshouldbeaspin-2particlecomesfromobservingFigure1. One can see that a gravitational wave is invariant under rotations of 1800about itsdirection of propagation; the pattern repeats itself after half period. For comparison,we mention that electromagnetic waves are invariant under rotations of 3600. In thequantum mechanical description of massless particles, the wavefunction of a particle10isinvariantunderrotationsof3600/s,wheresisthespinoftheparticle. Thusthephotonisaspin-1particleandthegravitonisaspin-2particle. Inotherrelativistictheoriesofgravity,thewaveeldhasothersymmetriesandthereforetheyattributedierentspinstothegravitons.2.2PropertiesofgravitationalwavesGravitational waves, once they are generated, propagate almost unimpeded. Indeed,ithasbeenproventhattheyareevenhardertostopthanneutrinos! Theonlysig-nicant change they suer as they propagate is the decrease in amplitude while theytravelawayfromtheirsource,andtheredshifttheyfeel(cosmological,gravitationalorDoppler),asisthecaseforelectromagneticwaves.Thereareothereectsthatmarginallyinuencethegravitational waveforms,forinstance, absorptionbyinterstellarorintergalacticmatterinterveningbetweentheobserverandthesource,whichisextremelyweak(actually,theextremelyweakcouplingof gravitational waveswithmatter isthemainreasonthat gravitationalwaves havenot beenobserved). Scatteringanddispersionof gravitational wavesarealsopracticallyunimportant, althoughtheymayhavebeenimportant duringtheearlyphasesoftheuniverse(thisisalsotruefortheabsorption). Gravitationalwaves can be focused by strong gravitational elds and also can be diracted, exactlyasithappenswiththeelectromagneticwaves.Therearealsoanumberof exoticeectsthatgravitational wavescanexpe-rience, thatareduetothenonlinearnatureofEinsteinsequations(purelygeneral-relativisticeects), suchas: scatteringbythebackgroundcurvature, theexistenceoftailsofthewavesthatinteractwiththewavesthemselves, parametricamplica-tionbythebackgroundcurvature, nonlinearcouplingof thewaveswiththemselves(creation of geons, that is, bundles of gravitational waves held together by their ownself- generated curvature) and even formationofsingularities by colliding waves (forsuch exotic phenomena see the extensive review by Thorne[4]). These aspects of non-linearityaectthemajorityof thegravitational wavesourcesandfromthispointofviewourunderstandingofgravitational-wavegenerationisbasedonapproxima-tions. However, itisexpectedthattheerrorintheseapproximationsformostoftheprocessesthatgenerategravitational wavesisquitesmall. Powerful numericalcodes,usingstate-of-the-artcomputersoftwareandhardware,havebeendeveloped(andcontinuetobedeveloped) for minimizingall possiblesources of error inor-dertohaveasaccurateaspossibleanunderstandingoftheprocessesthatgenerategravitationalwavesandofthewaveformsproduced.For most of the properties mentioned above there is a correspondence with elec-tromagneticwaves. Gravitationalwavesarefundamentallydierent, however, eventhough they share similar wave properties away from the source. Gravitational wavesare emitted by coherent bulk motions of matter (for example, by the implosion of thecoreofastarduringasupernovaexplosion)orbycoherentoscillationsofspacetime11curvature, andthustheyserveasaprobeofsuchphenomena. Bycontrast, cosmicelectromagneticwaves aremainlytheresult of incoherent radiationbyindividualatomsorchargedparticles. Asaconsequence, fromthecosmicelectromagneticra-diationwemainlylearnabouttheformofmatterinvariousregionsoftheuniverse,especiallyabout its temperatureanddensity, or about theexistenceof magneticelds. Strong gravitational waves, are emitted from regions of spacetime where grav-ity is very strong and the velocities of the bulk motions of matter are near the speedof light. Sincemostof thetimetheseareasareeithersurroundedbythicklayersof matter that absorbelectromagneticradiationor theydonot emit anyelectro-magneticradiationatall (blackholes), theonlywaytostudytheseregionsof theuniverseisviagravitationalwaves.2.3EnergyuxcarriedbygravitationalwavesGravitational wavescarry energy andcause a deformation ofspacetime. The stress-energycarriedbygravitational wavescannotbelocalizedwithinawavelength. In-stead, onecansaythatacertainamountof stress-energyiscontainedinaregionof thespacewhichextendsoverseveral wavelengths. ItcanbeproventhatintheTTgaugeof linearizedtheorythestress-energytensorof agravitational wave(inanalogywiththestress-energytensorofaperfectuidthatwehavedenedearlier)isgivenbytGW=132_hTTij_ _hTTij_). (2.24)wheretheangularbracketsareusedtoindicateaveragingoverseveralwavelengths.Forthespecial caseofaplanewavepropagatinginthezdirection, whichwecon-sideredearlier, thestress-energytensorhasonlythreenon-zerocomponents, whichtakethesimpleformtGW00=tGWzzc2= tGW0zc=132c2G2_h2+ + h2_, (2.25)wheretGW00istheenergydensity, tGWzzisthemomentumuxandtGW0ztheenergyowalongthe z directionper unit areaandunit time (for practical reasons wehaverestoredthenormal units). Theenergyuxhasall thepropertiesonewouldanticipate by analogy with electromagnetic waves: (a) it is conserved (the amplitudedies out as 1/r, the uxas 1/r2), (b) it canbe absorbedbydetectors, and(c)itcangeneratecurvaturelikeanyotherenergysourceinEinsteinsformulationofrelativity. Asanexample, byusingtheaboverelation, wewillestimatetheenergyuxingravitational wavesfromthecollapseofthecoreofasupernovatocreatea10M

blackholeatadistanceof50-million-light-years(15Mpc)fromtheearth(at thedistanceof theVirgocluster of galaxies). Aconservativeestimateof theamplitude of the waves on earth (as we will show later) is of the order of 1022(at afrequencyofabout1kHz). Thiscorrespondstoauxofabout3ergs/cm2sec. This12isanenormousamountofenergyuxandisabouttenordersofmagnitudelargerthantheobservedenergyuxinelectromagneticwaves! Thebasicdierenceisthedurationof thetwosignals; gravitational wavesignal will lastafewmilliseconds,whereasanelectromagneticsignallastsmanydays. Thisexampleprovidesuswithausefulnumericalformulafortheenergyux:F= 3_f1kHz_2_h1022_2ergscm2sec, (2.26)from which one can easily estimate the ux on earth, given the amplitude (on earth)andthefrequencyofthewaves.2.4GenerationofgravitationalwavesAs we have mentionedearlier, whenthe gravitational eldis strongthere are anumberonnonlineareectsthatinuencethegenerationandpropagationofgravi-tationalwaves. Forexample,nonlineareectsaresignicantduringthelastphasesof blackholeformation. Theanalyticdescriptionof suchadynamicallychangingspacetimeisimpossible,anduntilnumericalrelativityprovidesuswithaccuratees-timates of the dynamics of gravitational elds under such extreme conditions we havetobecontentwithorderofmagnitudeestimates. Furthermore,therearedierencesinthepredictionsofvariousrelativistictheoriesofgravityinthecaseofhighcon-centrationsofrapidlyvaryingenergydistributions. However, all metrictheoriesofgravity,aslongastheyadmitthecorrectNewtonianlimit,makesimilar predictionsfor the total amount of gravitational radiation emitted by weak gravitational wavesources, thatis, sourceswheretheenergycontentissmall enoughtoproduceonlysmall deformationsof theatspacetimeandwhereall motionsareslowcomparedtothevelocityoflight.Letusnowtrytounderstandthenatureofgravitationalradiation,bystartingfrom the production of electromagnetic radiation. Electromagnetic radiation emittedby slowly varying charge distributions can be decomposed into a series of multipoles,wheretheamplitudeofthe2

-pole( = 0, 1, 2, ...)containsasmallfactora

,withaequaltotheratioofthediameterofthesourcetothetypicalwavelength,namely,anumbertypicallymuchsmallerthan1. Fromthispointofviewthestrongestelec-tromagneticradiationwouldbeexpectedformonopolarradiation(=0), butthisis completely absent, because the electromagnetic monopole moment is proportionaltothetotal charge, whichdoesnotchangewithtime(itisaconservedquantity).Therefore,electromagneticradiationconsistsonlyof 1multipoles,thestrongestbeingtheelectricdipoleradiation(=1), followedbytheweakermagneticdipoleand electric quadrupole radiation ( = 2). One could proceed with a similar analysisforgravitational wavesandbyfollowingthesameargumentsshowthatmasscon-servation (which is equivalent to charge conservation in electromagnetic theory) willexcludemonopoleradiation. Also,therateofchangeofthemassdipolemomentis13proportionaltothelinearmomentumofthesystem,whichisaconservedquantity,andthereforetherecannotbeanymassdipoleradiationinEinsteinsrelativitythe-ory. The next strongest form of electromagnetic radiation is the magnetic dipole. Forthecaseofgravity,thechangeofthemagneticdipoleisproportionaltotheangularmomentumof thesystem, whichisalsoaconservedquantityandthusthereisnodipolargravitationalradiationofanysort. Itfollowsthatgravitationalradiationisofquadrupolarorhighernatureandisdirectlylinkedtothequadrupolemomentofthemassdistribution.As early as 1918, Einstein derived the quadrupole formula for gravitational radi-ation. This formula states that the wave amplitude hijis proportional to the secondtimederivativeofthequadrupolemomentofthesource:hij=2rGc4QTTij_t rc_(2.27)whereQTTij(x) =__xixj13ijr2_d3x (2.28)isthequadrupolemomentintheTTgauge,evaluatedattheretardedtimet r/cand is the matter density in a volume element d3x at the position xi. This result isquiteaccurateforallsources,aslongasthereducedwavelength=/2ismuchlonger than the source size R. It should be pointed out that the above result can bederived via a quite cumbersome calculation in which we solve the wave equation (2.6)withasourcetermTontheright-handside. Inthecourseof suchaderivation,anumberofassumptionsmustbeused. Inparticular,theobservermustbelocatedatadistancer ,fargreaterthanthereducedwavelength(inwhatiscalledtheradiationzone)andTmustnotchangeveryquickly.Usingthe formulae (2.24) and(2.25) for the energycarriedbygravitationalwaves, one canderive the luminosityingravitational waves as afunctionof thethird-order time derivative of the quadrupole moment tensor. This is the quadrupoleformulaLGW= dEdt=15Gc53Qijt33Qijt3). (2.29)Basedonthisformula, wederivesomeadditional formulas, whichprovideorderofmagnitude estimates for the amplitude of the gravitational waves andthe corre-spondingpoweroutputof asource. First, thequadrupolemomentof asystemisapproximatelyequaltothemassMofthepartofthesystemthatmoves,timesthesquareofthesizeRofthesystem. Thismeansthatthethird-ordertimederivativeofthequadrupolemomentis3Qijt3MR2T3Mv2TEnsT, (2.30)wherevisthemeanvelocityof themovingparts, Ensisthekineticenergyof thecomponentofthesourcesinternalmotionwhichisnonspherical,andTisthetime14scale for a mass to move from one side of the system to the other. The time scale (orperiod) is actually proportional to the inverse of the square root of the mean densityofthesystemT _R3/GM. (2.31)Thisrelationprovidesaroughestimateof thecharacteristicfrequencyof thesys-temf =2/T. Then, theluminosityof gravitational waves of agivensourceisapproximatelyLGW Gc5_MR_5Gc5_MR_2v6c5G_RSchR_2 _vc_6(2.32)where RSch=2GM/c2is the Schwarzschildradius of the source. It is obviousthatthemaximumvalueoftheluminosityingravitationalwavescanbeachievedifthesourcesdimensionsareoftheorderofitsSchwarzschildradiusandthetypicalvelocities of the components of the system are of the order of the speed of light. Thisexplainswhyweexpectthebestgravitationalwavesourcestobehighlyrelativisticcompactobjects. Theaboveformulasetsalsoanupperlimitonthepoweremittedbyasource,whichforR RSchandv cisLGW c5/G = 3.6 1059ergs/sec. (2.33)Thisisanimmensepower,oftencalledtheluminosityoftheuniverse.Usingtheaboveorder-of-magnitudeestimates, wecangetaroughestimateoftheamplitudeofgravitationalwavesatadistancerfromthesource:h Gc4EnsrGc4Ekinr(2.34)whereEkin(with0 1),isthefractionofkineticenergyofthesourcethatisabletoproducegravitationalwaves. Thefactorisameasureoftheasymmetryofthe source and implies that only a time varying quadrupole moment will emit gravi-tational waves. For example, even if a huge amount of kinetic energy is involved in agiven explosion and/or implosion, if the event takes place in a spherically symmetricmanner,therewillbenogravitationalradiation.Another formula for the amplitude of gravitational waves relation can be derivedfromthe uxformula(2.26). If, for example, we consider anevent (perhaps asupernovaeexplosion) at theVirgocluster duringwhichtheenergyequivalent of104M

isreleasedingravitationalwavesatafrequencyof1kHz, andwithsignaldurationoftheorderof1msec, theamplitudeofthegravitationalwavesonEarthwillbeh 1022_EGW104M

_1/2_f1kHz_1 _1msec_1/2_r15Mpc_1. (2.35)15For a detector with arm length of 4 km we are looking for changes in the arm lengthoftheorderof = h = 1022 4km = 4 1017!!!Thesenumbersshowswhyexperimentersaretryingsohardtobuildultra-sensitivedetectorsandexplainswhyalldetectioneortstilltodaywerenotsuccessful.Finally, itisuseful toknowthedampingtime, thatis, thetimeittakesforasource to transform a fraction 1/e of its energy into gravitational radiation. One canobtainaroughestimatefromthefollowingformula=EkinLGW1cR_RRSch_3. (2.36)For example, for a non-radially oscillating neutron star with a mass of roughly 1.4M

andaradiusof12Km,thedampingtimewillbeoftheorderof 50msec. Also,byusingformula(2.31), weget anestimatefor thefrequencyof oscillationwhichisdirectlyrelatedtothefrequencyof theemittedgravitational waves, roughly2kHzfortheabovecase.2.5RotatingbinarysystemAmongthemost interestingsources of gravitational waves arebinaries. Thein-spirallingof suchsystems, consistingof blackholesorneutronstars, is, aswewilldiscuss later, the most promising source for the gravitational wave detectors. Binarysystemsarealsothesourcesofgravitational waveswhosedynamicsweunderstandthe best. If we assume that the two bodies making up the binary lie in the xy planeand their orbits are circular (see Figure 3), then the only non-vanishing componentsofthequadrupoletensorareQxx= Qyy=12a2cos 2t, and Qxy= Qyx=12a2sin 2t, (2.37)whereistheorbital angularvelocity, =M1M2/Misthereducedmassof thesystemandM= M1 + M2itstotalmass.According to equation (2.29) the gravitational radiation luminosity of the systemisLGW=325Gc52a46=325G4c5M32a5, (2.38)where, inordertoobtainthelastpartoftherelation, wehaveusedKeplersthirdlaw, 2= GM/a3. As the gravitating system loses energy by emitting radiation, thedistancebetweenthetwobodiesshrinksataratedadt= 645G3c5M2a3, (2.39)16Figure 3:A system of two bodies orbiting around their common center of gravity. Binarysystems are the best sources of gravitational waves.and the orbital frequency increases accordingly ( T/T= 1.5 a/a). If, for example, thepresent separation of the two stars is a0, then the binary system will coalesce after atime=5256c5G3a40M4(2.40)Finally,theamplitudeofthegravitationalwavesish = 5 1022_M2.8M

_2/3_0.7M

__f100Hz_2/3_15Mpcr_. (2.41)Inall theseformulaewehaveassumedthattheorbitsarecircular. Ingeneral, theorbits of the twobodies are approximatelyellipses, but it has beenshownthatlongbeforethecoalescenceof thetwobodies, theorbitsbecomecircular, atleastforlong-livedbinaries, duetogravitational radiation. Also, theamplitudeof theemittedgravitationalwavesdependsontheanglebetweenthelineofsightandtheaxisof angularmomentum; formula(2.41)referstoanobserveralongtheaxisoftheorbital angularmomentum. Thecompleteformulafortheamplitudecontainsangular factors of order 1. The relative strength of the two polarizations depends onthatangleaswell.If threeormoredetectorsobservethesamesignal itispossibletoreconstructthefullwaveformanddeducemanydetailsoftheorbitofthebinarysystem.As anexample, we will provide some details of the well-studiedpulsar PSR1913+16(theHulse-Taylorpulsar), whichisexpectedtocoalesceafter 3.5108years. Thebinarysystemisroughly5kpcawayfromEarth, themassesofthetwoneutron stars are estimated to be 1.4M

each, and the present period of the systemis 7hand45min. ThepredictedrateofperiodchangeisT= 2.4 1012sec/sec,while the corresponding observed value is in excellent agreement with the predictions,i.e.,T= (2.300.22) 1012sec/sec; nally the present amplitude of gravitationalwavesisoftheorderofh 1023atafrequencyof 7 105Hz.17Figure4: A pair of masses joined by a spring can be viewed as the simplest conceivabledetector.3. DETECTIONOFGRAVITATIONALWAVESThe rst attempt to detect gravitational waves was undertaken by the pioneer JosephWeber during the early 1960s. He developed the rst resonant mass detector and in-spired many other physicists to build new detectors and to explore from a theoreticalviewpointpossiblecosmicsourcesofgravitationalradiation.Apairof massesjoinedbyaspringcanbeviewedasthesimplestconceivabledetector; see Figure 4. In practice, a cylindrical massive metal bar or even a massivesphereisusedinsteadofthissimplesystem. Whenagravitationalwavehitssuchadevice, it causes the bar to vibrate. By monitoring this vibration, we can reconstructthetruewaveform. Thenextstep, followingtheideaof resonantmassdetectors,wasthereplacementofthespringbypendulums. Inthisnewdetectorthemotionsinduced by a passing-by gravitational wave would be detected by monitoring, via laserinterferometry, therelativechangeinthedistanceof twofreelysuspendedbodies.The use of interferometry is probably the most decisive step in our attempt to detectgravitationalwavesignals. Inwhatfollows, wewilldiscussbothresonantbarsandlaserinterferometricdetectors.Althoughthebasicprincipleofsuchdetectorsisverysimple, thesensitivityofdetectorsislimitedbyvarioussourcesofnoise. Theinternal noiseofthedetectorscanbeGaussianornon-Gaussian. Thenon-Gaussiannoisemayoccurseveraltimesperdaysuchasstrainreleasesinthesuspensionsystemswhichisolatethedetectorfrom any environmental mechanical source of noise, and the only way to remove thistypeof noiseisviacomparisonsof thedatastreamsfromvariousdetectors. Theso-called Gaussian noise obeys the probability distribution of Gaussian statistics andcanbecharacterizedbyaspectral densitySn(f). Theobservedsignalattheoutputof adetector consistsof thetruegravitational wavestrainhandGaussiannoise.The optimal methodtodetect agravitational wave signal leads tothe followingsignal-to-noiseratio:_SN_2opt= 2_0[h(f)[2Sn(f)df, (3.1)whereh(f) istheFourier transformof thesignal waveform. It isclear fromthisexpressionthatthesensitivityofgravitationalwavedetectorsislimitedbynoise.183.1ResonantdetectorsSuppose that a gravitational wave propagating along the zaxis with pure (+) polar-izationimpingesonanidealizeddetector, twomassesjoinedbyaspringalongthexaxisasinFigure4. Wewillassumethathdescribesthestrainproducedinthespacetime by the passing wave. We will try to calculate the amplitude of the oscilla-tions induced on the spring detector by the wave and the amount of energy absorbedby this detector. The tidal force induced on the detector is given by equation (2.23),andthemasseswillmoveaccordingtothefollowingequationofmotion: +/+ 20= 122Lh+eit, (3.2)where0isthenatural vibrationfrequencyofourdetector, isthedampingtimeoftheoscillatorduetofrictionalforces,Listheseparationbetweenthetwomassesand istherelativechangeinthedistanceof thetwomasses. Thegravitationalwaveplaystheroleofthedrivingforceforourideal oscillator, andthesolutiontotheaboveequationis=122Lh+eit202+ i/(3.3)If thefrequencyof theimpingingwaveisnear thenatural frequency0of theoscillator(nearresonance)thedetectorisexcitedintolarge-amplitudemotionsanditringslikeabell. Actually,inthecaseof= 0,wegetthemaximumamplitudemax=0LH+/2. Since the size of our detector Landthe amplitude of thegravitational wavesh+arexed, large-amplitudemotionscanbeachievedonlybyincreasingthequalityfactorQ=0 ofthedetector. Inpractice,thefrequencyofthedetectorisxedbyitssizeandtheonlyimprovementwecangetisbychoosingthetypeofmaterialsothatlongrelaxationtimesareachieved.Thecrosssectionisameasureoftheinterceptionabilityofadetector. Forthespecial caseof resonance, theaveragecrosssectionof ourtestdetector, assuminganypossibledirectionofthewave,is=3215Gc30QML2. (3.4)This formula is general;it applies even if we replace our toy detector with a massivemetal cylinder, forexampleWebersrstdetector. Thatdetectorhadthefollow-ingcharacteristics: MassM=1410kg, lengthL=1.5m, diameter66cm, resonantfrequency0=1660Hz, andqualityfactor Q=0 =2105. For thesevaluesthecalculatedcrosssectionisroughly3 1019cm2,whichisquitesmall,andevenworse, it can be reached only when the frequency of the impinging wave is very closetoresonancefrequency(thetypicalresonancewidthisusuallyoftheorderof0.1-1Hz).Inreality, the eciencyof aresonant bar detector depends onseveral otherparameters. Here, we will discuss only the more fundamental ones. Assuming perfect19Figure5: A graph of NAUTILUS in Frascati near Rome. Nautilus is probably the mostsensitive resonant detector available.isolationoftheresonantbardetectorfromanyexternalsourceofnoise(acoustical,seismic,electromagnetic),thethermalnoiseistheonlyfactorlimitingourabilitytodetectgravitational waves. Thus, inordertodetectasignal, theenergydepositedbythegravitationalwaveevery secondsshouldbelargerthantheenergykTduetothermaluctuations. Thisleadstoaformulafortheminimumdetectableenergyuxofgravitationalwaves, which, followingequation(2.25), leadsintoaminimumdetectablestrainamplitudehmin 10LQ_15kTM(3.5)ForWebersdetector,atroomtemperaturethisyieldsaminimumdetectablestrainof the order of 1020. However, this estimate of the minimum sensitivity applies onlytogravitationalwaveswhosedurationisatleastaslongasthedampingtimeofthebars vibrations and whose frequency perfectly matches the resonant frequency of thedetector. Forburstsorforperiodicsignalswhichareo-resonance(withregardtothedetector)thesensitivityofaresonantbardetectordecreasesfurtherbyseveralordersofmagnitude.In reality, modern resonant bar detectors are quite complicated devices, consist-ingof asolidmetalliccylinderweighingafewtonsandsuspendedinvacuobyacable that is wrapped under its center of gravity (Figure 5). This suspension systemprotectstheantennafromexternalmechanicalshocks. Thewholesystemiscooleddowntotemperaturesof afewkelvinsorevenmillikelvins. Tomonitorthevibra-tionsofthebar,piezoelectrictransducers(orthemoremoderncapacitiveones)are20attachedtothebar. Thetransducersconvertthebarsmechanicalenergyintoelec-trical energy. Thesignal isampliedbyanultra-low-frequencyamplier, byusinga device called a SQUID (Super-conducting QUantum Interference Device) before itbecomesavailablefordataanalysis. Transducersandampliersofelectronicsignalsrequirecarefuldesigntoachievelownoisecombinedwithadequatesignaltransfer.The above descriptionof the resonant bar detectors shows that, inorder toachievehighsensitivity,onehasto:1. Createmoremassiveantennas. Today, mostof theantennasareabout50%moremassivethantheearlyWebersantenna. Therearestudiesandresearchplansforfutureconstructionof spherical antennaeweighinguptotenthsoftons.2. Obtain higher quality factor Q. Modern antennas generally use aluminum alloy5056(Q 4 107);althoughniobium(whichisusedintheNIOBEdetector)isevenbetter(Q 108)butmuchmoreexpensive. Siliconorsapphirebarswouldenhancethequalityfactorevenmore,butexperimentersmustrstndawaytoproducelargesinglepiecesofthesecrystals.3. Lower the temperature of the antenna as much as possible. Advanced cryogenictechniques havebeenusedandtheresonant bar detectors areprobablythecoolest places inthe Universe. Typical cooling temperatures for the mostadvancedantennaearebelowthetemperatureofliquidhelium.4. Achievestrongcouplingbetweentheantennaandtheelectronicsandlowelec-tricalnoise. The bar detectors include the best available technology related totransducersandintegratethemostrecentadvancesinSQUIDtechnology.Sincetheearly1990s, anumberof resonantbardetectorshavebeeninnearlycontinuousoperationinseveralplacesaroundtheworld. Theyhaveachievedsensi-tivities of a few times 1021, but still there has been no clear evidence of gravitationalwave detection. As we will discuss later, they will have a good chance of detecting agravitational wave signal from a supernova explosion in our galaxy,although,this isaratherrareevent(2-5percentury).Thetechnicalspecicationsofthemostsensitivecryogenicbardetectorsinop-erationareasfollows:ALLEGRO (Baton Rouge, USA) Mass 2296 Kg (Aluminium 5056), length 3 m,bartemperature4.2K,modefrequency896Hz.AURIGA (Legrano, Italy) Mass 2230 Kg (Aluminium 5056), length 2.9 m, bartemperature0.2K,modefrequency913Hz.EXPLORER (CERN, Switzerland) Mass 2270 Kg (Aluminium 5056), length 3m,bartemperature2.6K,modefrequency906Hz.21NAUTILUS (Frascati, Italy) Mass 2260 Kg (Aluminium 5056), length 3 m, bartemperature0.1K,modefrequency908Hz.NIOBE(Perth, Australia)Mass1500Kg(Niobium), length1.5m, bartem-perature5K,modefrequency695Hz.Also, there are plans for construction of massive spherical resonant detectors, theadvantages of which will be their high mass, their broader sensitivity (up to 100-200Hz)andtheiromnidirectional sensitivity. Inaspherical detector, vemodesatatimewill beexcited, whichisequivalenttoveindependentdetectorsorientedindierentways. Thisoerstheopportunity,inthecaseofdetection,toobtaindirectinformationaboutthepolarizationof thewaveandthedirectiontothesource. Aprototypespherical detectorisalreadyinoperationinLeiden, Netherlands(mini-GRAIL)[7]with1mdiameterandmodefrequency 3.2kHz. TwootherdetectorsofthesamesizeareunderconstructioninItaly(Sfera)andBrazil(Shenberg).3.2Beamdetectors3.2.1LaserInterferometersAlaserinterferometerisanalternativegravitational wavedetectorthatoersthepossibility of very high sensitivities over a broad frequency band. Originally, the ideawastoconstructanewtypeofresonantdetectorwithmuchlargerdimensions. Asone can realize from the relations (3.4) and (3.5), the longer the resonant detector isthe more sensitive it becomes. One could then try to measure the relative change inthedistanceoftwowell-separatedmassesbymonitoringtheirseparationviaalaserbeamthat continuouslybounces backandforthbetweenthem. (This techniqueisactuallyusedinsearchingforgravitational wavesbyusingtheso-calledDopplertracking technique, where a distant interplanetary spacecraft is monitored from earththroughamicrowavetrackinglink; theearthandspacecraftactasfreeparticles.)Soon, it was realizedthat it is mucheasier touselaser light tomeasurerelativechanges in the lengths of two perpendicular arms;see Figure 6. Gravitational wavesthat are propagating perpendicular to the plane of the interferometer will increase thelength of one arm of the interferometer,and at the same time will shorten the otherarm,andviceversa. ThistechniqueofmonitoringthewavesisbasedonMichelsoninterferometry. L-shapedinterferometersareparticularlysuitedtothedetectionofgravitationalwavesduetotheirquadrupolarnature.Figure 6 shows a schematic design of a Michelson interferometer; the three massesM0, M1andM2arefreelysuspended. Notethattheresonantfrequenciesof thesependulumsshouldbemuchsmallerthanthefrequenciesof thewavesthatwearesupposedtodetect sincethependulums aresupposedtobehavelikefreemasses.MirrorsareattachedtoM1andM2andthemirrorattachedonmassM0splitsthelight(beamsplitter)intotwoperpendiculardirections. Thelightisreectedonthe22Figure6: A schematic design of a Michelson interferometer.twocornermirrorsandreturnsbacktothebeamsplitter. Thesplitternowhalf-transmitsandhalf-reectseachoneofthebeams. Onepartofeachbeamgoesbacktothelaser, whiletheotherpartsarecombinedtoreachthephotodetectorwherethefringepatternismonitored. Ifagravitationalwaveslightlychangesthelengthsof the two arms, the fringe pattern will change, and so by monitoring the changes ofthe fringe pattern one can measure the changes in the arm lengths and consequentlymonitortheincominggravitationalradiation1.Letusconsideranimpinginggravitationalwavewithamplitudehand(+)po-larization, propagatingperpendiculartotheplaneofthedetector. Wewill furtherassume that the frequency is much higher than the resonant frequency of the pendu-lumsandthewavelengthismuchlongerthanthearmlengthofourdetector. Sucha wave will generate a change of L hL/2 in the arm length along the x-directionandanoppositechangeinthearmlengthalongthey-direction,accordingtoequa-tions(2.19)and(2.20). Thetotal dierenceinlengthbetweenthetwoarmswillbeLL h. (3.6)Foragravitational wavewithamplitudeh 1021anddetectorarm-length4km(suchasLIGO), thiswill induceachangeinthearm-lengthofaboutL 1016.Inthegeneralcase,whenagravitationalwavewitharbitrarypolarizationimpingeson the detector from a random direction, the above formula will be modied by someangularcoecientsoforder1.1In practice,thingsare arranged so that when there isnoactual changein the arms,all lightthatreturnsonthebeamsplitterfromthecornermirrorsissentbackintothelaser,andonlyifthere is some motion of the masses there is an output at the photodetector23Ifthelightbouncesafewtimesbetweenthemirrorsbeforeitiscollectedinthephotodiode, the eective arm length of the detector is increased considerably, and themeasured variations of the arm lengths will be increased accordingly. This is a quiteecientprocedureformakingthearmlengthlonger. Forexample, agravitationalwaveatafrequencyof100Hzhasawavelengthof3000Km,andifweassume100bouncesof thelaserbeaminthearmsof thedetector, theeectivearm-lengthofthedetectoris100timeslargerthantheactualarm-length,butstillthisis10timessmaller than the wavelength of the incoming wave. The optical cavity that is createdbetweenthemirrorsofthedetectorisknownasaFabry-Perotcavityandisusedinmoderninterferometers.In the remainder of this paragraph we will focus on the Gaussian sources of noiseandtheirexpectedinuenceonthesensitivityoflaserintereferometers.a. Photonshot noise. Whenagravitational waveproducesachangeLinthe arm-length, the phase dierence betweenthe twolight beams changes byanamount =2bL/, whereis the reducedwavelengthof the laser light( 108cm)andbisthenumberofbouncesofthelightineacharm. Itisexpectedthat a detectable gravitational wave will produce a phase shift of the order of 109rad.The precision of the measurements, though, is ultimately restricted by uctuations inthe fringe pattern due to uctuations in the number of detected photons. The numberofphotonsthatreachthedetectorisproportionaltotheintensityofthelaserbeamand can be estimated via the relation N= N0 sin2(/2), where N0 is the number ofphotonsthatthelasersuppliesandNisthenumberofdetectedphotons. InversionofthisequationleadstoanestimationoftherelativechangeofthearmlengthsLby measuring the number of the emerging photons N. However, there are statisticaluctuationsinthepopulationofphotons,whichareproportionaltothesquarerootof thenumberof photons. Thisimpliesanuncertaintyinthemeasurementof thearmlength(L) 2bN0(3.7)Thus,theminimumgravitationalwaveamplitudethatwecanmeasureishmin=(L)L=LLbLN1/201bL_cI0_1/2, (3.8)whereI0istheintensityof thelaserlight(5-10W)and isthedurationof themeasurement. This limitation in the detectors sensitivity due to the photon countinguncertaintyisknownasphotonshot noise. For atypical laser interferometer thephotonshot noise is the dominant source of noise for frequencies above 200Hz,whileitspowerspectral densitySn(f)forfrequencies100-200Hzisof theorderof 3 1023Hz.24b. Radiationpressurenoise: Accordingtoformula(3.8), thesensitivityof adetector canbeincreasedbyincreasingtheintensityof thelaser. However,averypowerful laserproducesalargeradiationpressureonthemirrors. Thenanuncertaintyinthemeasurementof themomentumdepositedonthemirrorsleadstoaproportional uncertaintyinthepositionofthemirrorsor, equivalently, inthemeasured change in the arm-lengths. Then, the minimum detectable strain is limitedbyhmin mbL_I0c_1/2, (3.9)wheremisthemassofthemirrors.Aswehaveseen, thephotonshotnoisedecreasesasthelaserpowerincreases,whiletheinverseistrueforthenoiseduetoradiationpressureuctuations. If wetrytominimizethesetwotypesof noisewithrespecttothelaserpower, wegetaminimumdetectablestrainfortheoptimalpowerviatheverysimplerelationhmin 1L_m_1/2(3.10)which for the LIGO detector (where the mass of the mirrors is 100 kg and the armlengthis4km),forobservationtimeof1ms,giveshmin 1023.c. Quantumlimit. Anadditionalsourceofuncertaintyinthemeasurementsis set by Heisenbergs principle, which says that the knowledge of the position and themomentum of a body is restricted from the relation x p . For an observationthatlastssometime, thesmallestmeasurabledisplacementof amirrorof massmisL; assumingthatthemomentumuncertaintyisp mL/, wegetaminimumdetectablestrainduetoquantumuncertaintieshmin=LL1L_m_1/2. (3.11)Surprisingly, thisisidentical totheoptimal limitthatwecalculatedearlierfortheothertwotypesofnoise. Thestandardquantumlimitdoessetafundamentallimitonthesensitivityofbeamdetectors. Aninterestingfeatureofthequantumlimitisthatitdependsonlyonasingleparameter,themassofthemirrors.d. Seismic noise. At frequencies below 60 Hz, the noise in the interferometersisdominatedbyseismicnoise. Thisnoiseisduetogeologicalactivityoftheearth,and human sources, e.g., trac and explosions. The vibrations of the ground coupletothemirrorsviathewiresuspensionswhichsupportthem. Thiseectisstronglysuppressedbyproperlydesignedsuspensionsystems. Still, seismic noise is verydiculttoeliminateatfrequenciesbelow5-10Hz.e. Residualgas-phasenoise. The statistical uctuations of the residual gasdensity induce a uctuation of the refraction index and consequently of the monitored25phaseshift. Hence, theresidual gaspressurethroughwhichthelaserbeamstravelshouldbe extremelylow. For this reasonthe laser beams are enclosedinpipesovertheirentirelength. Insidethepipesahighvacuumof theorderof 109Torrguaranteeseliminationofthistypeofnoise.PrototypelaserinterferometricdetectorshavebeenalreadyconstructedintheUSA, GermanyandUKmore than15years ago. These detectors have anarmlengthof afewtensof metersandtheyhaveachievedsensitivitiesof theorderofh 1019. The construction of the new generation of laser interferometric detectorsis near completionandsomeof themhavealreadycollecteddata. It is expectedthatinayearorsotheywill beinfull operation. TheUSprojectnamedLIGO[8](LaserInterferometerGravitational Observatory)consistsof twodetectorswitharm length of 4 Km, one in Hanford, Washington, and one in Livingston, Louisiana.ThedetectorinHanfordincludes, inthesamevacuumsystem, aseconddetectorwitharmlengthof 2km. Thedetectors arealreadyinoperationandtheyhaverecentlyachievedthedesignedsensitivity.TheItalian/FrenchEGO(VIRGO)detector[9] ofarm-length3kmatCascinanear Pisa is designedtohave better sensitivityat lower frequencies is about tocommenceoperationnextyear. GEO600istheGerman/BritishdetectorbuildinHannover[10]. Ithas600marmlengthandiscollectingdataincollaborationwithLIGO[11]. TheTAMA300[12] detectorinTokyohasarmlengthof 300manditwastherstmajorinterferometricdetectorinoperation. Therearealreadyplansfor improving the sensitivities of all the above detectors and the construction of newinterferometersinthenearfuture.3.2.2SpaceDetectorsBothbars andlaser interferometers arehigh-frequencydetectors, but thereareanumberof interestinggravitational wavessourceswhichemitsignalsatlowerfre-quencies. Theseismicnoiseprovidesaninsurmountableobstacleinanyearth-basedexperimentandtheonlywaytosurpassthisbarrieristoyalaserinterferometerinspace. LISA[13] (LaserInterferometerSpaceAntenna)issuchasystem. IthasbeenproposedbyEuropeanandAmericanscientistsandhasbeenadoptedbyESA(European Space Agency) as a cornerstone mission, while recently NASA joined theeortandthelaunchdateisexpectedtobearound2011.LISAwill consistofthreeidentical drag-freespacecraftsforminganequilateraltrianglewithonespacecraftsateachvertex(Figure7). Thedistancebetweenthetwovertices (thearmlength) is 5106km. Thespacecrafts will beplacedintothesameheliocentricorbitasearth, butabout20obehindearth. Theequilateraltriangle will be inclined at an angle of 60owith respect to Earths orbital plane. Thethreespacecraftwouldtrackeachotheropticallybyusinglaserbeams. Becauseofthediractionlossesitisnotfeasibletoreectthebeamsbackandforthasisdone26Figure7: A schematic design of the space interferometer LISA.with LIGO. Instead, each spacecraft will have its own laser. The lasers will be phaselocked to each other, achieving the same kind of phase coherence as LIGO does withmirrors. The conguration will function as three, partially independent and partiallyredundant,gravitationalwaveinterferometers.Atfrequenciesf 103,LISAsnoiseismainlyduetophotonshotnoise. Thesensitivitycurve steepens at f 3 102Hz because at larger frequencies thegravitational wavesperiodisshorterthantheround-triplighttravel timeineacharm. Forf 102Hz, thenoiseisduetobueting-inducedrandommotionsofthespacecraft, andcannotberemovedbythedrag-compensationsystem. LISAssensitivityisroughlythesameasthatof LIGO, butat105timeslowerfrequency.Sincethegravitational wavesenergyuxscalesasFf2h2, thiscorrespondsto1010timesbetterenergysensitivitythanLIGO.3.2.3SatelliteTrackingTheDopplerdelayofcommunicationsignalsbetweentheearth-basedstationsandspacecraft underlies another type of gravitational wave detector. A radio signal of fre-quency f0is transmitted to a spacecraft and is coherently transported back to earth,where it is received and its frequency is measured with a highly stable clock (typicallyahydrogenmaser). Therelativechangef/f0asfunctionoftimeismonitored. Agravitationalwavepropagatingthroughthesolarsystemcausessmallperturbationsinf/f0. Therelativeshiftinthefrequencyof thesignalsisproportional totheamplitude of gravitational waves. With this technique, broad-band searches are pos-sibleinthemillihertzfrequencyband,andthankstotheverystableatomicclocks,it is possible to achieve sensitivities of order hmin 10131015. Noise sources thataectthesensitivityofDopplertrackingexperimentscanbedividedintotwobroad27classesa)instrumental andb)relatedtopropagation. Atthehigh-frequencyendof thebandaccessibletoDopplertracking, thermal noisedominatesoverall othernoisesources, typicallyatabout0.1Hz. Amongall othersourcesof instrumentalnoise(transmitterandreceiver,mechanicalstabilityoftheantenna,stabilityofthespacecraftetc), clocknoisehasbeenshowntobethemostimportantinstrumentalsourceof frequencyuctuations. Thepropagationnoiseis duetouctuations intheindexofrefractionofthetroposphere, ionosphere, andtheinterplanetarysolarplasma. BothNASAandESAhaveperformedsuchmeasurementsandthereisacontinuedeortinthisdirection.3.2.4PulsarTimingPulsars are extremely stable clocks and by measuring irregularities in their pulses weexpecttosetupperlimitsonthebackgroundgravitationalwaves(seenextsection).Ifanobservermonitorssimultaneouslytwoormorepulsars,thecorrelationoftheirsignalscouldbeusedtodetectgravitationalwaves. Sincesuchobservationrequirestimescalesoftheorderof1year,thismeansthatthewaveshavetobeofextremelylowfrequencies.4. ASTRONOMICALSOURCESOFGRAV.WAVESThe new generation of gravitational wave detectors (LIGO, VIRGO) have very goodchances of detecting gravitational waves, but until these expectations are fullled, wecanonlymakeeducatedguessesastothepossibleastronomical sourcesofgravita-tionalwaves. Thedetectabilityofthesesourcesdependsonthreeparameters: theirintrinsicgravitationalwaveluminosity,theireventrate,andtheirdistancefromtheEarth. Theluminositycanbeapproximatelyestimatedviathequadrupoleformulathatwediscussedearlier. Eventhoughtherearecertainrestrictionsinitsapplica-bility (weak eld, slow motion), it provides a very good order-of-magnitude estimatefor the expected gravitational wave ux on Earth. The rate, at which various eventswith high luminosity in gravitational waves take place is extrapolated from astronom-ical observations in the electromagnetic spectrum. Still,there might be a number ofgravitationallyluminoussources,forexamplebinaryblackholes,forwhichwehavenodirectobservationsintheelectromagneticspectrum. Finally, theamplitudeofgravitational wavesignalsdecreasesasoneoverthedistancetothesource. Thus,asignal fromasupernovaexplosionmightbeclearlydetectableif theeventtakesplaceinourgalaxy(2-3eventspercentury),butitishighlyunlikelytobedetectedifthesupernovaexplosionoccursatfargreaterdistances, oforder100Mpc, wheretheeventrateishighandatleastafeweventsperdaytakeplace. Allthreefactorshavetobetakenintoaccountwhendiscussingsourcesofgravitationalwaves.It was mentioned earlier that the frequency of gravitational waves is proportionalto the square root of the mean density of the emitting system;this is approximately28truefor anygravitatingsystem. For example, neutronstars usuallyhavemassesaround1.4M

andradiiintheorderof10km; thusifweusethesenumbersintherelation f _GM/R3we nd that an oscillating neutron star will emit gravitationalwaves primarilyat frequencies of 2-3kHz. Byanalogy, ablack-holewithamass 100M

, will have a radius of 300 km and the natural oscillation frequency will be 100 Hz. Finally, for a binary system Keplers law (see section 2.5) provides a directand accurate estimation of the frequency of the emitted gravitational waves. For two1.4M

neutronstarsorbitingaroundeachotheratadistanceof 160Km, Keplerslawpredictsanorbitalfrequencyof50Hz,whichleadstoanobservedgravitationalwavefrequencyof100Hz.4.1RadiationfromgravitationalcollapseTypeIIsupernovaeareassociatedwiththecorecollapseofamassivestartogetherwithashock-drivenexpansionof aluminous shell whichleaves behindarapidlyrotating neutron star or a black hole, if the core has mass of > 23M

. The typicalsignalfromsuchanexplosionisbroadbandandpeakedataround1kHz. Detectionof suchasignal wasthegoal of detectordevelopmentoverthelastthreedecades.However, westill knowlittleabouttheeciencywithwhichthisprocessproducesgravitational waves. For example, an exactly spherical collapse will not produce anygravitationalradiationatall. Thekeyquestioniswhatisthekineticenergyofthenonsphericalmotions,sincethegravitationalwaveamplitudeisproportionaltothis[equation(2.30). After30yearsof theoretical andnumerical attemptstosimulategravitationalcollapse,thereisstillnogreatprogressinunderstandingtheeciencyofthisprocessinproducinggravitationalwaves. Foraconservativeestimateoftheenergy in non-spherical motions during the collapse, relation (2.32) leads to events ofanamplitudedetectableinourgalaxy, evenbybardetectors. ThenextgenerationoflaserinterferometerswouldbeabletodetectsuchsignalsfromVirgoclusteratarateofafeweventspermonth.The mainsource for deviations fromspherical or axial symmetryduringthecollapseistheangularmomentum. Duringthecontractionphase, theangularmo-mentumis conserved, andthestar spins uptorotational periods of theorder of1msec. Inthiscase, consequentprocesseswithlargeluminositymighttakeplaceinthis newlybornneutronstar. Anumber of instabilities, suchas theso-calledbar mode instabilityandther-mode instability, mayoccur whichradiatecopiousamounts of gravitational radiation immediately after the initial burst. Gravitationalwavesignalsfromtheserotationallyinducedstellarinstabilitiesaredetectablefromsourcesinourgalaxyandaremarginallydetectableif theeventtakesplaceinthenearbycluster of about 2500galaxies, the Virgocluster, 15Mpc awayfromtheEarth. Additionally,therewillbeweakerbutextremelyusefulsignalsduetosubse-quent oscillations of the neutron star; f, p and w modes are some of the main patternsof oscillations(normal modes)of theneutronstarthatobserversmightsearchfor.29These modes have been studied in detail and once detected in the signal, they wouldprovide a sensitive probe of the neutron star structure and its supranuclear equationofstate. DetectorswithhighsensitivityinthekHzbandwillbeneededinordertofullydevelopthisso-calledgravitationalwaveasteroseismology.If the collapsing central core is unable to drive o its surrounding envelope, thenthecollapsecontinuesandnallyablackholeforms. Inthiscasetheinstabilitiesand oscillations that we discussed above are absent and the newly formed black holeradiates away, within a few milliseconds, any deviations from axisymmetry and endsupasarotatingorKerrblackhole. Thecharacteristicoscillationsof blackholes(normal modes) are well studied, and this unique ringing down of a black hole couldbeusedasadirectprobeoftheirexistence. Thefrequencyofthesignalisinverselyproportional totheblackholemass. Forexample, ithasbeenstatedearlierthata100M

blackholewill oscillateat afrequencyof 100Hz(anideal sourceforLIGO), whileasupermassiveonewithmass107M

, whichmightbeexcitedbyaninfalling star, will ring down at a frequency of 103Hz (an ideal source for LISA). Theanalysisofsuchasignalshouldrevealdirectlythetwoparametersthatcharacterizeany(uncharged)blackhole;namelyitsmassandangularmomentum.4.2RadiationfrombinarysystemsBinary systems are the best sources of gravitational waves because they emit copiousamountsorgravitationalradiation,andforagivensystemweknowexactlywhatisthe amplitude and frequency of the gravitational waves in terms of the masses of thetwo bodies and their separation (see section 2.5). If a binary system emits detectablegravitationalradiationinthebandwidthofourdetectors,wecaneasilyidentifytheparametersof thesystem. Accordingtotheformulasof section2.5, theobservedfrequencychangewillbef f11/3/5/3, (4.1)andthecorrespondingamplitudewillbeh /5/3f2/3r=frf3, (4.2)where /5/3= M2/3isacombinationofthetotalandreducedmassofthesystem,calledchirpmass. Sincebothfrequencyfanditsrateofchangefaremeasurablequantities, we can immediately compute the chirp mass (from the rst relation), thusobtainingameasureof themassesinvolved. Thesecondrelationprovidesadirectestimateofthedistanceofthesource. TheserelationshavebeenderivedusingtheNewtoniantheorytodescribetheorbitof thesystemandthequadrupoleformulafor the emission of gravitational waves. Post-Newtonian theory inclusion of the mostimportantrelativisticcorrectionsinthedescriptionof theorbitcanprovidemoreaccurate estimates of the individual masses of the components of the binary system.30Whenanalyzingthedataof periodicsignalstheeectiveamplitudeisnottheamplitude of the signal alone but hc=n h, where n is the number of cycles of thesignal within the frequency range where the detector is sensitive. A system consistingoftwotypical neutronstarswill bedetectablebyLIGOwhenthefrequencyofthegravitational waves is 10Hz until the nal coalescence around 1000Hz. This processwill last for about 15 min and the total number of observed cycles will be of the orderof104, whichleadstoanenhancementofthedetectabilitybyafactor100. Binaryneutron star systems and binary black hole systems with masses of the order of 50M

are the primary sources for LIGO. Given the anticipated sensitivity of LIGO, binaryblack hole systems are the most promising sources and could be detected as far as 200Mpc away. The event rate with the present estimated sensitivity of LIGO is probablyafeweventsperyear, butfutureimprovementofdetectorsensitivity(theLIGOIIphase)couldleadtothedetectionof atleastoneeventpermonth. Supermassiveblackholesystemsof afewmillionsolarmassesaretheprimarysourceforLISA.Thesebinarysystemsarerare,butduetothehugeamountofenergyreleased,theyshouldbedetectablefromasfarastheboundariesoftheobservableuniverse.4.3RadiationfromspinningneutronstarsAperfectlyaxisymmetricrotatingbodydoesnotemitanygravitational radiation.Neutron stars are axisymmetric congurations, but small deviations cannot be ruledout. Irregularitiesinthecrust(perhapsimprintedatthetimeofcrustformation),strains that have built upas the stars have spundown, o-axis magnetic elds,and/oraccretioncoulddistorttheaxisymmetry. Abumpthatmightbecreatedatthesurfaceof aneutronstarspinningwithfrequencyf will producegravitationalwaves at a frequency of 2fand such a neutron star will be a weak but continuous andalmostmonochromaticsourceofgravitationalwaves. Theradiatedenergycomesatthe expense of the rotational energy of the star, which leads to a spin down of the star.If gravitational wave emission contributes considerably to the observed spin down ofpulsars,then we can estimate the amount of the emitted energy. The correspondingamplitude of gravitational waves from nearby pulsars (a few kpc away) is of the orderof h 10251026, which is extremely small. If we accumulate data for sucientlylongtime,e.g.,1month,thentheeectiveamplitude,whichincreasesasthesquarerootof thenumberof cycles, couldeasilygouptotheorderof hc 1022. Wemustadmitthatatpresentweareextremelyignorantof thedegreeof asymmetryinrotatingneutronstars,andtheseestimatesareprobablyveryoptimistic. Ontheotherhand,ifwedonotobservegravitationalradiationfromagivenpulsarwecanplaceaconstraintontheamountofnonaxisymmetryofthestar.4.4CosmologicalgravitationalwavesOne of the strongest pieces of evidence in favor of the Big Bang scenario is the 2.7 Kcosmicmicrowavebackgroundradiation. This thermal radiationrst bathedthe31universearound380,000yearsaftertheBigBang. Bycontrast, thegravitationalradiationbackgroundanticipatedbytheoristswasproducedatPlancktimes, i.e.,at1032secorearlieraftertheBigBang. Suchgravitational waveshavetravelledalmostunimpededthrough theuniversesincethey weregenerated. Theobservationofcosmologicalgravitationalwaveswillbeoneofthemostimportantcontributionsofgravitationalwaveastronomy. Theseprimordialgravitationalwaveswillbe,inasense, anothersourceof noiseforourdetectorsandsotheywill havetobemuchstrongerthananyotherinternaldetectornoiseinordertobedetected. Otherwise,condence in detecting such primordial gravitational waves could be gained by using asystem of two detectors and cross-correlating their outputs. The two LIGO detectorsarewellplacedforsuchacorrelation.AcknowledgementsI am grateful to I. D. Jones, Th. Apostolatos, J. Ruo, N. Stergioulas and J. Agtzidisforusefulsuggestionswhichimprovedtheoriginalmanuscript.References[1] Fundamentals of Interferometric Gravitational Wave Detectors, P.R. Saulson,World Scientic (1994).[2] The Detection of Gravitational Waves, D.G. Blair, Cambridge University Press(1991).[3] Relativistic Gravitation and Gravitational Radiation, Cambridge University Press,Editors J-A. Marck and J-P. Lasota (1997).[4] Gravitational Radiation, K.S. Thorne, in S.W. Hawking and W. Israel (eds), 300Years of Gravitation, (Cambridge University Press, Cambridge).[5] Gravitational Wave Physics, K.D. Kokkotas, Encyclopedia of Physical Science andTechnology, 3rd Edition, Volume 7 Academic Press, (2002)[6] An Overview of Gravitational-Wave Sources C. Cutler and K.P. Thornegr-qc/0204090 April 30th 2002.[7] mini-GRAIL http://www.minigrail.nl[8] LIGO http://www.ligo.caltech.edu[9] VIRGO http://www.pi.infn.it/virgo/virgoHome.html[10] GEO http://www.geo600.uni-hannover.de32[11] Detector Description and Performance for the First Coincidence Observationsbetween LIGO and GEO The LIGO Scientic Collaboration, gr-qc/0308043, August14th, 2003.[12] TAMA http://tamago.mtk.nao.ac.jp[13] LISA http://lisa.jpl.nasa.gov33