Table Of ContentAstronomy 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
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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
