Applied General Relativity - Theory and Applications in Astronomy, Celestial Mechanics and Metrology PDF
Preview Applied General Relativity - Theory and Applications in Astronomy, Celestial Mechanics and Metrology
Astronomy and Astrophysics Library Michael H. Soff el Wen-Biao Han Applied General Relativity Theory and Applications in Astronomy, Celestial Mechanics and Metrology SeriesEditors: MartinA.Barstow,UniversityofLeicester,Leicester,UK AndreasBurkert,UniversityObservatoryMunich,Munich, Germany AthenaCoustenis,Paris-MeudonObservatory,Meudon, France RobertoGilmozzi,EuropeanSouthernObservatory(ESO), Garching,Germany GeorgesMeynet,GenevaObservatory,Versoix,Switzerland ShinMineshige,DepartmentofAstronomy,Sakyo-ku,Japan IanRobson,TheUKAstronomyTechnologyCentre, Edinburgh,UK PeterSchneider,Argelander-InstitutfürAstronomie,Bonn, Germany VirginiaTrimble,UniversityofCalifornia,Irvine,CA,USA DerekWard-Thompson,UniversityofCentralLancashire, Preston,UK Moreinformationaboutthisseriesathttp://www.springer.com/series/848 Michael H. Soffel (cid:129) Wen-Biao Han Applied General Relativity Theory and Applications in Astronomy, Celestial Mechanics and Metrology 123 MichaelH.Soffel Wen-BiaoHan Instituteofplanetarygeodesy ShanghaiAstronomicalObservatory Lohrmann-Observatory ChineseAcademyofSciences Dresden,Germany Shanghai,China ISSN0941-7834 ISSN2196-9698 (electronic) AstronomyandAstrophysicsLibrary ISBN978-3-030-19672-1 ISBN978-3-030-19673-8 (eBook) https://doi.org/10.1007/978-3-030-19673-8 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Cover illustration: ‘Space-Time curvature’ (2019) by M. Soffel and W.-B. Han with the St. Marien (Rostock)astronomicalclockoverlaid(photographbyM.L.Preis). ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface At present, there is a vast number of textbooks on Einstein’s theory of gravity (general relativity, GR) that are available for different kinds of readers and at differentlevelsoftechnicalcomplexity.Thereisaseriesofclassicaltreatments,e.g., Eddington (1922), Tolman (1934), Bergmann (1942), Weyl (1950), Pauli (1958), Fock (1959), Synge (1960), Adler et al. (1965), Fokker (1965), Rindler (1969), Carmelietal.(1970),LandauandLifshitz(1971),Weinberg(1972),Møller(1972), Misner et al. (1973), and Hawking and Ellis (1973), but also many more modern books, like Geroch (1978), Wald (1984), Schutz (1985), Woodhouse (2007) or Carroll (2013) to name just a few. In addition, there are the Living Reviews in Relativity, such as e.g. Will (2006) or Blanchet (2014), that can be downloaded fromthewebforfree. ManybooksdealwithtestsofGR;thestandardreferenceisWill(1993,2006), but only a few deal with specific applications, e.g. in the important field of metrology.Inthefieldof‘AppliedGeneralRelativity’itisessentiallythebooksby Soffel (1989) and Kopeikin et al. (2011) where the reader can learn how general relativistic effects enter such fields as the realization of time scales, practical clock synchronization, satellite- and lunar laser ranging or very long baseline interferometry. Now, the first of these books is completely obsolete, whereas the secondoneisnotreallyatextbook,writteninahomogenousstylewherethereader shouldbeabletounderstandtheargumentsstep-by-step. In some sense, this book presents an improvement, extension and actualization of my old Springer book (Soffel 1989). This is especially true with respect to the selection of subjects treated in this book: the main emphasis lies on relativity in astrometry, celestial mechanics and metrology, thus on certain aspects of applied science. We have borrowed heavily from that book; some parts that we think are stilluptodateweretakenalmostliterally(wehavealsoborrowedseveralpartsfrom Soffel and Langhans 2013). This book is clearly not a textbook on all aspects of Einstein’stheoryofgravity(‘generalrelativity’,GR).Thoughsomeaspectsrelated withexactsolutionsoftheEinsteinfieldequationsaretreated,thephysicsofobjects with strong gravitational fields, like black holes, neutron stars or white dwarfs or gravitationalwaves,willnotbediscussedhere. v vi Preface In another sense, this is a completely new book. After Soffel (1989) came out, both, the theoretical relativistic formalisms and the observational techniques, have drastically been improved so that large parts of Soffel (1989) became obsolete. A good example for theoretical improvements is the Brumberg-Kopeikin Damour- Soffel-Xu (BK-DSX) formalism for relativistic celestial mechanics. For the first timeinhistory,anew formalismfortreatingtherelativisticcelestialmechanics of systems of N arbitrarily composed and shaped, weakly self-gravitating, rotating, deformablebodieswasintroduced.Thisformalismisaimedatyieldingacomplete description, at the first post-Newtonian approximation level, of (1) the global dynamicsofsuchN-bodysystems(‘externalproblem’),(2)thelocalgravitational structure of each body (‘internal problem’), and (3) the way the external and the internal problems fit together (‘theory of reference systems’) (Damour et al. 1991; DSX-I). This BK-DSX formalism is based on the first post-Newtonian approximation of Einstein’s theory of gravity, and an extension to higher orders will be difficult. Nevertheless, it is sufficient for many applications at the present levelofaccuracies. The multipolar post-Minkowskian (MPM) formalism that has been worked out by Blanchet, Damour and Iyer (see Blanchet 2014 for an overview) is another exampleforthat.Thoughimportantpapersonthatsubjectdatebacktothesecond halfofthe1980s,ithasonlybeeninrecentyearsthattheMPMformalismhasbeen worked out completely. The MPM formalism is able to describe the gravitational fieldofweak-fieldsourcesinsideofsomecompactregionbasicallytoallordersof GM/(c2R)inasinglecoordinatesystemandhasbeenemployedverysuccessfully totheemissionofgravitationalwavesfrombinarysystems. AlsothecharacterofthebookisverydifferentfromSoffel(1989).Forexample, alotofworkhasbeenspendondidacticalaspects,likealargenumberof(partially solved) exercises have been included. The title of the book, Applied General Relativity (AGR), points to two aspects: applied science on the one side and theoreticalframeworkofGRontheotherside.Itisnotdifficulttorealizethatthese twoaspectsusuallyarerepresentedbytwodifferentexpertgroups.Itisonegoalof thebooktoillustratetooneofthesegroupsthedisciplineoftheother.Thefieldof AGRhasadvancedtoamultidisciplinarystagesothatbothgroupsshouldfertilize eachother. Chapter 2 deals with the language of relativity: differential geometry, in which the reasons for that will be discussed later. This treatment is fairly standard. For moredetails,thereaderisreferredtothestandardliterature(e.g.BeyerandGostiaux 1988;Pressley2010;Bär2011;KobayashiandNomizu2014). Chapter 3 introduces Newtonian celestial mechanics. It starts with the Weak EquivalencePrinciple(universalityoffree-fall)andNewton’stheoryofgravity.Of specialinterestistheexteriorgravitationalfieldofsomematterdistribution(body) anditsdescriptionwithmultipolemoments.Here,inadditiontotheusualspherical moments that are based upon an expansion of the exterior Newtonian potentials in terms of spherical harmonics, the expansion in terms of Cartesian symmetric and trace-free (STF) tensors is introduced. Whereas the spherical moments are very well known, e.g., for geodesists under the name of ‘potential coefficients’, Preface vii this is still not the case for the STF moments. For experts in relativity, they play a crucial role, e.g. because Lorentz transformations usually are formulated in Cartesian coordinates. As was, e.g., nicely demonstrated by Hartmann et al. (1994) the use of STF moments can be employed very efficiently, like for the derivationoftranslationalandrotationalequationsofmotionintheN-bodyproblem orforarepresentationofthetidalpotentialinNewtoniancelestialmechanics.The Newtonian two-body problem of two mass monopoles moving under their mutual gravitational attractive forces is treated exhaustively since it serves as basis for a description of the relativistic (post-Newtonian) two-body problem. The usual classical celestial mechanical first-order perturbation theory is outlined. There are special topics in classical celestial mechanics that are of interest for the central subject of the book, e.g. the anomalous perihelion shift of planetary orbits due to theactionofotherplanets,whichcanbetreatedwithsuchafirst-orderperturbation theory.Later,itwillalsobeusedinconnectionwithrelativisticdynamicalproblems suchasplanetaryorsatellitemotion. Chapter4isdevotedtoMaxwell’stheoryofelectromagnetismasanintroduction to the field of relativity. Here, a metric tensor is introduced from a physical point of view for the first time, and the Lorentz transformation is derived. The electro- magneticfield(Liénard-Wiechertpotentials)ofamovingpointchargeisdiscussed in some detail. The problem of the ‘speed of propagation’ in electromagnetism is exhaustively treated here. The attentive reader will ask for a reason for this. First, thisproblemisgenerallynotunderstoodverywell,thoughalldetailscanbefoundin theliterature.Oneoftenhearsthatrelativitymeansnothingmovesfasterthanlight invacuum,whichisabsolutelynotthecase.Thoughcausalityisalwaysassuredby the use of retarded potentials, physics is full of superluminal speeds as explained in the text. To many readers, it might not be clear which velocity is restricted to thevacuumspeedoflightifthepropagationofanelectromagneticwavethrougha dielectric medium is concerned. Another common error is that the use of retarded potentialsimpliesretardedpropagatingactionwiththevacuumspeedoflightinall cases.Inanycase,thispartshedslightontheproblemofthe‘speedofgravity’that hasbeenthesubjectofmanycontroversialdiscussionsinthepast. Chapter5introducesEinstein’stheoryofgravity.Thefieldequationsarederived, andtheproblemofcoordinates,thegaugeprobleminGR,isdiscussed.Observables have to be coordinate-independent quantities (as measured objects, they cannot depend upon a certain coordinate system that a theoretician employs for his calculations) and, therefore, have to be described as scalars. This implies that observers and parts of the measuring devices (e.g. in form of tetrad-vectors) have tobeintroducedexplicitlyintotheformalism.ForAGR,thereareonlythreetypes of observables that play a central role besides proper time (a time that might be read-off some idealized clock): (1) the ranging observable as measurable time interval between emission and reception of some electromagnetic signal (as in laser ranging) (2) the spectroscopic observable presenting measurable frequencies ofsomeincidentlightrayand(3)theastrometricobservableastheanglebetween twoincidentlightraysasseenbysomeobserver. viii Preface Chapter 5 also deals with the Landau-Lifshitz formulation of Einstein’s theory whichpresentsthebasisfortheMPMformalism,aperturbationtheorythatformally can be extended to any order of corresponding small parameters. The Landau- Lifshitzformulationchoos√estheharmonicgaugefromthebeginningandworkswith the‘gothicmetric’gμν = −ggμν insteadoftheusualone,gμν (g =detg ). μν Chapter 6 presents some exact solution of Einstein’s field equations in the vacuumthatmightbeofsomeuseforthefieldofAGR.Iamnotconvincedthatfrom a methodological point of view, it is preferable to start from some exact solutions beforeonedealswithapproximations:Einstein’stheoryofgravitymightbeviolated at some level of accuracy; to deal with exact models, a real problem has to be over-simplifiedandsoon.Nevertheless,exactmodelssometimesmightbeofhelp how toextend acertain approximative framework tohigher orders.Inthischapter also,cosmologicallyrelevantspacetimesareintroducedinrelationtothefollowing question:Howdoestheglobalexpansionoftheuniverseinfluencethegravitational physicsinthesolarsystem? Chapter 6 also introduces multipole moments for stationary gravitational fields obeyingthevacuumfieldequationsandposesthepropertiesofasymptoticflatness. DuetothedefinitionsofGeroch(1970)andHansen(1974),suchfieldmomentscan be defined rigorously. Later, they will be related with body moments as integrals overtheenergy-momentumtensorofthebody.Thornein1981hasintroducedfield moments differently by the structure of the metric tensor in special coordinates systems (asymptotically Cartesian and mass-centred coordinates) that were later showntobeequivalenttotheGeroch-Hansenmoments(Gürsel1983). Chapter 7 introduces the post-Newtonian approximation of GRT as slow- motion,weakfieldapproximationandthemultipolarpost-Minkowskianformalism. Acanonicalformofthemetrictensorisdiscussedinthefirstpost-Newtonian(PN) approximation,wherethegravitationalfieldisentirelydescribedwithtwopotentials only:ascalarpotentialw thatgeneralizestheNewtoniangravitationalpotentialU andavectorpotentialwthatdescribesgravito-magnetic-typegravityresultingfrom mattercurrents(movingorrotatingmasses).Thecorrespondingfieldequationsare verysimilartoMaxwell’sequationsofelectromagnetism. ThischapteralsodealswiththeexteriorfieldofabodytofirstPN-order,where post-Newtonian multipole moments (Blanchet-Damour moments) come into play, andthelastpartisdevotedtothemultipolarpost-Minkowskian(MPM)formalism. Firstapplications,whichdonotrequireaformalismforthegravitationalN-body problem,arediscussedinChap.8onthebasisofthefirstPNframework.Discussed arethegravitationalfieldoftheEarth,equipotentialsurfaces,clocksandtimescales (TCG, TAI, TT, UTC) in the vicinity of the Earth and in the barycentric system (TCB,TDB),clocksynchronization,thegravitationallightdeflectionandtimedelay in the field of a single central body, the PN motion of torque-free gyroscopes and thesatellitemotioninthefieldofarotatingEarthtoPN-order. Chapter9isdevotedtotheBK-DSXframeworkofrelativisticcelestialmechan- ics which is based upon a total of N + 1 different coordinate systems in the gravitational N-body problem: one global system with coordinates (ct,x) that, neglecting all matter outside the system of N bodies and assuming asymptotic Preface ix flatness, extends to (spatial) infinity and is used to describe the overall motion of the N bodies and N local systems and one for each body A with coordinates (cT ,X )thatisco-movingwithbodyAandusedforadescriptionofphysics(e.g., A A geophysics)inthelocalA-system.Formanyapplications,theBK-DSXformalism isthebestone,providinghighestaccuracyatpresent. Chapter10dealswiththepost-NewtoniangravitationalN-bodyproblem.Laws andequationsofmotionforthetranslationalandrotationalcasesarediscussedhere. ForasystemofPuremassmonopoles,thefamousLorentz-DrosteEinstein-Infeld- Hofmannequations(inharmonicgauge)fortranslationalmotionarederived.They form the basis for any modern high-precision numerical ephemeris, such as one fromtheDE,INPOPorEPMseries(seee.g.SoffelandLanghans2013). Chapter 11 is devoted to relativistic astrometry while Chap. 12 to relativistic metrology, where many techniques like pulsar timing, navigation by means of GNNS, satellite and lunar laser ranging (SLR and LLR), very long baseline interferometry (VLBI), etc. are theoretically described in a consistent relativistic framework. Of course, there are other books that deal with Applied General Relativity in one way or another. We would like to mention especially the book by Kopeikin et al. (2011), where many of the subjects are in common with those of this book. Acomparison,however,revealsthatthesebooksareofverydifferentcharacter. Itisthehopethatthisbookfillstheobviousgapinthefieldof‘AppliedGeneral Relativity’.Itisatextbookwithaclearredthread,containingmanyexerciseswith solutions in most cases. We hope that we have not forgotten a field of application that really is of practical interest. It is a pleasure for us to thank all those people whohavecontributedtothisbookinonewayoranother:AndreasBauch,Francisco Frutos, Franz Hofmann, Enrico Gerlach, Sergei Klioner, Sergei Kopeikin, Jürgen Müller,GerhardSchäfer,MaximilianSchanner,HaraldSchuh,IrinaTupikovaand SvenZschocke,tonamejustafew.Clearly,forallthemistakes,onlytheauthorsare responsible. Dresden,Germany MichaelH.Soffel Shanghai,China Wen-BiaoHan April2019
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