Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. atI:Itouto eea eaiiy(6lcue)4 lectures) (16 Relativity General Introduction II: Part 1 Contents cp n aiiyo h hoy6 Theory Newtonian of Review 3 Theory the of Validity and Scope 2 ttcadsainr erc,PudRbaEprmn 16 Experiment Pound-Rebka metrics, stationary and Static 6 Spacetime Curved 5 Relativity Special of Review 4 1.3 1.2 1.1 1.4 1.5 . xml 8 ...... 5 ...... 5 ...... Example . . . . 2.1 ...... books . Appropriate conventions miscellaneous 1.7 Other 1.6 . ute xmls...... 20 . 19 . . . . . 18 ...... 17 . . . . . 18 ...... 12 ...... Examples . . rays Further . . Light . . of . . . . Motion 6.5 . . . . . Limit . Newtonian . 6.4 . Spacetimes The Static . . in . . motion 6.3 Particle . Redshift . Gravitational 6.2 . The . . 6.1 ...... Example 3.1 Units Schedule The Pre-requisites intr Convention Signature uvtr Conventions Curvature .. xml:Edpitvrain n oetmcnevto 21 conservation momentum and variations point End Example: 6.5.3 .. h cwrshl erc...... 20 ...... The Metric Schwarzschild 6.5.2 The 6.5.1 4 ...... atI eea Relativity General II Part k oeto-akrMti 20 ...... Metric Robertson-Walker 0 = 4 ...... 4 ...... abig B 0WA, CB3 Cambridge abig University, Cambridge ibroc Road, Wilberforce eray1 2006 1, February Gibbons W G D.A.M.T.P., 4 ...... 5 ...... U.K. 1 15 12 8 Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 0Prle transport Parallel 10 1Curvature 11 egh,Age n ofra eclns21 Analysis Tensor Rescalings 8 Conformal and Angles Lengths, 7 ieetaigTensors Differentiating 9 . h eiCvt oncin...... 30 ...... 29 ...... Connection . Levi-Civita The . . 9.2 . . . . Connections Affine Symmetric 9.1 05*rjcieEuvlne 39 ...... 36 ...... 33 . Equivalence* . *Projective 32 . . 10.5 ...... 35 ...... Principle . Fermat’s metrics . and . related metrics . conformally . Static of . . 10.4 connections . Levi-Civita . The . . 10.3 ...... Force and . Acceleration . . 10.2 curves Autoparallel 10.1 13Cneune ftemti rsrigPoet 44 ...... 41 Property Preserving metric . the . of Consequences . 11.3 ...... of . justification . coordinates, inertial . local . of . significance Physical . 11.2 . . Coordinates Inertial Local 11.1 . GahclNtto*...... 27 . . 27 27 ...... 26 ...... Notation* . . *Graphical . Shuffling . Index . . for 8.4 . Rules . properties tensorial Theorem 8.3 their Quotient preserving tensors on 8.2 Operations 8.1 .. xml:TeNjnusBakt...... 30 30 ...... 31 ...... torsion . . with . connections . metric-preserving . . Example: . . . . 9.2.1 Bracket . Nijenhuis . The . Example: Derivative Exterior 9.1.2 Example: 9.1.1 042Eape iclrnl edsc 37 . . . . same . the having . connections preserving . metric *Example: . . 10.5.1 . 34 . 34 closed . in . geodesics . null . . projection, . Stereographic . . *Example: geodesics . . null 10.4.3 . Circular . . Example: . . . 10.4.2 Schwarzschild . the . for metric . . 34 optical and . . coordinates Isotropic . . . . 10.4.1 ...... Rockets . Relativistic . rest Example: . at Particles particle 10.2.3 Charged a Example: of acceleration 10.2.2 The Example: 10.2.1 131*xml:Tecraueo h pee 46 . 44 . 43 42 ...... 43 . . . 43 . . sphere* . 42 . the . . . of . . . curvature . . . The . . 41 . . . *Example: ...... 11.3.1 ...... Curvature* . . . . . Projective . . . . Example:* . . . . . 41 Connections* 11.2.5 . *Weyl . . . contraction* . Example: possible . . . . other . . . 11.2.4 . The . . . . 40 *Example: . Identity . Bianchi . . . 11.2.3 The . . . . . Consequences: Identity . . . . Cyclic 11.2.2 . . The . . Consequences: ...... 11.2.1 . . . postulate . clock . . the . . Coordinates . Inertial . Local . . of . Existence . . . 11.1.1 . . . tensor Ricci co-vectors The for Identity 11.0.3 Ricci The 11.0.2 edsc*...... 40 ...... 37 ...... geodesics* . 38 . . lens* . eye . fish Maxwell’s . and . universes Friedman-Lemaitre . . . . . delay time Shapiro solution: 2 28 31 40 23 Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 2TeEnti Equations Einstein The 12 3TeEnti il qain ihMte 52 equations field Einstein The 14 Matter with Equations Field Einstein The 13 6Goeisi h cwrshl erc61 59 metric Schwarzschild the in Geodesics 16 metrics vacuum Symmetric Spherically 15 14Cnrce inh dniis...... 46 . . . . . 47 ...... Tensor . Riemann the . of . Properties the . of . Summary Identities 11.5 Bianchi Contracted 11.4 25Eape...... 51 . . 54 52 ...... 48 ...... 50 ...... spacetime . . . curved . to . . . 48 . Generalization . . Tensor . 13.2 Momentum . . . Energy . . . The . . . . 13.1 ...... vector . connecting Example . of . 12.5 48 . choice . convenient . A . . . 12.4 . Deviation . . Geodesic . . 12.3 . . symmetry* theorem Conformal Lovelock’s and equations: *Dilatation Einstein 12.2 the of Uniqueness 12.1 61Tesaeo h ri 61 ...... 58 . . . . . orbit . the of . shape . The . 16.1 . 58 . 55 ...... metrics . Friedman-Lemaitre-Robertson-Walker Mistake Example Greatest . Einstein’s 14.3 . Example: . postulate 14.2 geodesic the of derivation and Example 14.1 221*h elCnomlCraueTno*...... 48 ...... Tensor* Curvature Conformal Weyl *The 12.2.1 612Apiain2 ih edn 62 . . 62 63 ...... perihelion . . the . of . Precession . . 3: . . Application bending light 16.1.3 2: limit Application Newtonian 1: 16.1.2 Application 16.1.1 56 ...... pressure* with fluid *perfect Example: 14.1.1 3 47 54 Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. nodrntt lte pfrua,frtems at nt ilb sdin used be will units part, most the for light, formulae, of up velocity clutter the to which not order In 1.3 matter, of presence the in equations Field [2] motion. identities. of Bianchi equations energy. field [5] holes. black collapse, periheli gravitational deflection, horizon, light Event red-shift, gravitational vance. orbits, and Rays tensors, solution. Ricci and Riemann geodesics. paral [5] curve, Curvature. derivatives, deviation. of covariant geodesics. geodesic duration and as angles, autoparallels absolute transport, magnitudes, symbols, tensor, Christoffel Metric Connection, tensors. of ntesbetcnuig h etsrtg st utvt h blt oswit to ability the cultivate to is strategy best The confusing. abound which subject conventions the notational in various remembering find often Beginners elementary using 1.4 restored be will, and may, they required analysis. When dimensional unity. to set ek xeiet nrdcint eea eaiiy nepeaino the spacetimes, of Static interpretation principles. [4] equivalence relativity: limit. geodesics, Newtonian general hypothesis, Pound- to clock the Introduction gravitation, metric, and relativity experiment. Special Rebka spaces. Riemannian and Curved which schedule, the in 1.2 given *aster- order in the enclosed in is lectured material follows. This be as will it. reads techniques of material the end The using the accessible by isks*. is which mastered sc but have the lectured, should in be included you not not will material which illustrative and additional ules, of amount small a and aisi eial.Yuaepriual die orvs atsa tensor Cartesian ‘index-shuffling’. revise of practice to the advised and Classical particularly convention II summation are Part Einstein You and the essential desirable. are Relativity is Special namics and Methods IB Part (16 Relativity 1.1 General Introduction II: Part 1 hr ilb he xml sheets. example three be will There gravitational of non-localisability coupling, minimal principles, Equivalence Schwarzschild the spacetimes, symmetric Spherically equations. field Vacuum oain n otaain esr,tno aiuain ata deriv partial manipulation, tensor tensors, contravariant and Covariant h nun oe r eindt cover to designed are notes ensuing The lectures) Units intr Convention Signature h Schedule The Pre-requisites c n etnscntn fGravitation, of constant Newton’s and , 4 almost l h aeili h course the in material the all G nad- on atives are , hed- Dy- lel ch s, Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. rceswl eue odnt ymtiainadsur rcest denote to brackets square and symmetrization denote thus to anti-symmetrization, used be will brackets en d rsbrc h aeepeso with expression same the subtract or add means h symbol The rn ftetno n eoe ata eiaie eiclnfloe ya by followed semi-colon A derivative. partial tensor denotes a and after subscript tensor the of front om olwdb subscript a by followed comma conventions A miscellaneous Other 1.6 h uvtr n ic esrcnetos(hs enn ilb explained be will are meaning course) (whose the conventions in tensor later Ricci and curvature The 1.5 intr ovnini wthdt h poieoe epn h curvature symbols the Christoffel keeping the one, then opposite unchanged, conventions the tensor to Ricci switched and is convention signature R ffiecneto opnnsΓ components connection affine rs 1985 Press 2001 Press −− h intr ovnin freapet osl etokwihue the uses which metric textbook the a replace to consult suffices to will it example indices one), Space (for opposite Cyrillic conventions Greek, alphabet. signature say the the than of by (rather beginning denoted letters the be latin from taken case lower usually by Hebrew) denoted be will oee o r die o od oi h ideo oml.Tesignature The formula. a of below middle arbitrary given the (+++ such in are is so conventions on use do will changing to depend I convention not facilitate can advised to are statement you hints physical however Some no as conventions. specially desired, as ahmtclSceySuetTxsn.5 abig nvriyPes1990 Press University Cambridge 5, no. (+ Texts Student Society Mathematical print) schedules. the in listed are following The books Appropriate 1.7 db −− − ..Sht is orei eea eaiiy abig University Cambridge Relativity. General in Course First A Schutz B.F. University Oxford Cosmological and General Special, Relativity: Rindler W. ..Hgso n ..TdA nrdcint eea eaiiy London Relativity. General to Introduction An of Tod K.P. (out and Wesley Hughston 1979 Addison L.P. Gravity Arnold Hartle, Edward B J Relativity. General Elementary Clarke, C. † )( = .dIvroItouigEnti’ eaiiy lrno rs 92(+ 1992 Press Clarendon Relativity. Einstein’s Introducing d’Inverno R. R R uvtr Conventions Curvature ) c )( bcd R ) ± r nhne.TeRicci-scalar The unchanged. are ( a ,j k j, i, ↔ b fe esra xrsincnann h ne pair index the containing expression tensorial a after ) n iltk ausfo o3 fyune ochange to need you If 3. to 1 from values take will and ; a ∇ or a S ∇ ( ∇ ab − b a V ) n pctm nie wihrnoe values) 4 over run (which indices spacetime and ) nfoto esrdntscvratderivative. covariant denotes tensor a of front in = a c b ∇ − 1 2 c , ¡ a uvtr tensor curvature , S b ab fe esrmastesm as same the means tensor a after ∇ 5 a + V S c ab = R ¢ g ab R and = a c by g dab and ab A − V R [ ab g R d ab b ab ] , c necagd Round interchanged. = . R hne sign. changes dab db 2 1 ¡ n ic tensor Ricci and A = ab R − c dcb A ab fthe If . © ¢ a ∂ . b a c ab or ª in − ; , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 90( + + (+ 1990 • taottelvlo h rsn ore hysol l eaalbei college in available be ( all should They course. present the of libraries. level the about at o)(+ mon) n aeapcs ucsflter hudalwadsrpino ih both light of tation description a particle know allow both waves. we should has as Since theory and matter, consist particles successful all field. as fully a indeed gravitational a aspects, and a wave give light, in and that moving must Mechanics light it Quantum of particular Newtonian from motion In compatible, the of making Relativity. account or Special Theory ‘unifying’, and the from Gravity of results Validity Relativity and General Scope 2 AdsnWesley) (Addison carr it However course. the is the text carries in It undergraduate and done further. is an up-to-date much as as completely applications far the designed as is almost is it far, It but quite mathematics GR. students physics of American side Greek. physical for are the they for that ommend means (G) useful Roman, textbook a known are contains better indices the book spactime of some first means in (R) The used conventions various Relativity. the General of summary develops then and course eea,frua nGnrlRltvt nov both. • involve Relativity General in formulae general, −− radius − rvt siprati h yia velocities typical the if important is Gravity eaiiyi motn if important is Relativity ++)( + .Sehn eea eaiiy2deiin abig nvriyPress University Cambridge edition. 2nd Relativity General Stephani H. .W inr ..Ton n ..Welr rvtto.WH Freeman W.H. Gravitation. Wheller, J.A. and Thorne K.S. Misner, W. C. material useful much contain books advanced more following the addition, In .Wibr,GaiainadCsooy(ie)( (Wiley) Cosmology and Gravitation Weinberg, S. ..Lna n ..Lfht,TeCasclTer fFed (Perga- Fields of Theory Classical The Lifshitz, E.M. and Landau L.D. ..Seat dacdGnrlRltvt CmrdeUiest rs)(+ Press) University (Cambridge Relativity General Advanced Stewart, J.M. etnsLw fGaiyaeepesduigNwos osato Gravi- of constant Newtons’ using expressed are Gravity of Laws Newton’s .B ate rvt nItouto oEnti’ eea Relativity General Einstein’s to Introduction An : Gravity Hartle, B. J. rec- strongly I which out came book textbook new outstanding II an Recently Part the for suitable level a at Electrodynamics both covers third The ..Wl,GnrlRltvt CiaoUiest Press)( University (Chicago Relativity General Wald, R.M. )( G G R ) −− − satisfy n fcus pca eaiiyitoue h eoiyo light of velocity the introduces Relativity Special course of and G − ) )( )( R R ). ) £ 35.99 v 2 v ≈ 2 ≈ 6 GM R c 2 . v nue yamass a by induced , − ++)( + − G ++)( + ) M niea inside R c ) In . ent (1) ies s. − Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. h ytmw need we system the ~ hudsffradeflection a suffer should n ae ae pb alc.Tenm BakHl’a ondb John by coined is gravity Hole’was of ‘Black absence the name in The ’in 1960’s. Laplace. 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This effects relativistic important. general large how of measure a thus is number dimensionless The sw hl e ae ntecus,teeatase s o matparameters impact for is, large answer as exact twice the radius, course, Schwarzshcild the the with in compared later large see shall we as parameter impact with scattered when nohrwrsif words other In asenergy mass xml faglrfrequency angular of example n ec,b lnksrlto nenergy an relation Planck’s by hence, and ω eea eaiiyi motn if important is Relativity General codn oteBlitcTer,lgti aeu fprilswoespeed whose particles of up made is light Theory, Ballistic the to According phenomenon a Hole, Black a is actually or to close is body the words other In eea eaiiybek onwe eaiitcqatmeet become effects quantum relativistic when down breaks Relativity General oprbewt h etms energy mass rest the with comparable R S scle h cwrshl aiso h oy fterdu fabody a of radius the If body. the of radius Schwarzschild the called is Mc etms energy mass rest 2 ftesse ilb eddif needed be will system the of saevelocity escape c codn oNwoinmcais uhparticles such mechanics, Newtonian to According . ω δ R R ≈ = n wavelength and ≈ 2 ≈ δ rvttoa oeta energy potential gravitational 2 c GM 4 2 GM 2 R GM c = c λ R R π 2 2 S b C 2 7 ≤ b = ≈ GM c = nfc codn oEnti’ theory, Einstein’s to according fact In . = ≈ 2 = R, b 2 ih eoiy(4) velocity light ~ R 1 Mc GM R Mc c ω 2 ~ S , . 2 R b tlatcmaal ihterest the with comparable least at S 2 , , λ . eaeeaiig olocalize To examining. are we R ene naon fenergy of amount an need we ssalrthan smaller is R . ihlgtfor light with 1784 (10) (5) (3) (8) (7) (6) (2) (9) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. disafnaetlui feeti hreadtemnwoivne the invented who man using the units of and system charge fundamental or electric absolute nature of an that ‘electron’constructed recognize unit name to fundamental person first a the admits Stoney, Johnstone Planck, before Long Example 2.1 unu ehnc by mechanics quantum otc ihi ae ok reyfligpril a h qaino mo make of can equation the we has which particle system, form falling coordinate a freely inertial in A an theory work. in Newtonian later review in with to contact useful Theory prove Newtonian will It of Review 3 units? rvttoa as codn oeprmnso aie,Nwo n E¨o and Newton Galileo, of experiments have to we According mass. gravitational and units eaemr iea eeetr atce’hnamcocpcbd.Tu the Thus body. by macroscopic bounded is a gravity ’than will designed Newtonian body particle classical is our of ‘elementary which because realm theory an particles field like of number quantum more indefinite use behave an to with have systems we describe to point this At particles. where than apn hr esalsyn oeaoti nti oreecp opitout point to except course this time in and it length an mass what about are about more it known with no is associated say little that shall very Since we Gravity. there Quantum happens of domain the is which where n thus and 1 oeie alda nrilrfrneframe. reference inertial an called sometimes c hrceiigterlvn cl.Te okott be to out work They scale. relevant the characterizing , R u o using not but R g ≈ C qaiyo nriladPsieGaiainlMass Gravitational Passive and Inertial of Equality ste“rvttoa field, “gravitational the is scle h opo ais fw r olclz atcet better to particle a localize to try we If radius. Compton the called is R lnkMass Planck c ene omc nryta ernters fcetn more creating of risk the run we that energy much so need we lnkLength Planck lnkTime Planck ~ needn faymnmd ovnin,called conventions, man-made any of independent , o i ed t o r i nt eae oPlanck to related units his are How it? do he did How . R > R M boueo udmna ytmo hscluisof units physical of system fundamental or absolute m C P i hs w elsitreta h lnkscale Planck the at intersect realms two These . d d dt dt 1 = L 2 2 T x 2 x 2 m P P ¡ m c = i = G = = ~ = i 8 m ¢ g ¡ ¡ h nrilms and mass inertial the 2 1 m ( G p c G x c 3 g ≈ ~ p 5 t , ~ ( ¢ x ¢ 2 ) 2 1 , 2 1 t , × ≈ ≈ ) , 10 1 4 . R > R − 6 × 5 × g 10 ≈ 10 − S − 10 44 33 n non-relativistic and s. 19 i.e. , cm GeV m p h passive the Planck tion, tv¨os (13) (11) (14) (15) (16) (12) G , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hc mle the implies which hsauiomgaiainlfil ie n o which for one (i.e. field gravitational uniform a Thus )B hoig( choosing By i) cf. ( that observe We suosral:i a lasb lmntdb asn oasial frame. suitable a field to gravitational passing by The eliminated ii) be always can it unobservable: is ceeaini netra rvttoa ed hsalw st ast new a to pass to us allows This system field. coordinate gravitational (non-inertial) external an in acceleration agrsmer ru i atifiiedmninl hnjs h Galilei the just than dimensional) infinite fact a (in motion of group equations group. symmetry Newton’s because larger variable physical a a not is gravity) nwhich in rtco rsol,adwowoeaon 0 A.D. 500 around wrote who and Aristotle, of critic initial same the have they if path, same ity. the follow interactions self tional Principle the Equivalence of Weak terms in situation this describe to u ihtesm ceeainwt rcso foepr namlinmil million a in part one of precision a the with test the acceleration (10 towards to same falls method earth the different on this with of everything sun used that weights on in have showing two by torque part others suspended principle equivalence and periodic are one weak Dicke which a than by from exert Experiments whose balance better not torsion materials. to pendula a does equal of simple sun arm E¨otv¨os are two the the Baron materials more showed of different a million. periods made from a the Newton made that Pisa. are showing of bobs Tower by Leaning check the qualitative from balls dropping repute, 12 olwn ik’ enmn fEnti’ rgnlaayi ti customar is it analysis original Einstein’s of refinement Dicke’s Following hsie svr l,adge aka es oJh hlpns passionate a Philoponus, John to least at back goes and old, very is idea This aie hce hsb iigabl oln ona nlndpaead by and, plane inclined an down rolling ball a timing by this checked Galileo .Teeaecretypasb AAadEAt yada-resatellite drag-free a fly to ESA and NASA by plans currently are There ). ahohr ri hyd,i ilb ya mecpil amount, between one. imperceptible difference to an two of by ratio kind the be from that in all will have differed at not but it differ them, did do, not weights they will the times if were, although one the or if half, but other, less. other measure, each much wide the a is and by times double, differ the not say ratio did between the weights difference with the if correspond the Thus the not but that does weights, see motion their will their wide you of of very height, times a same the by the of other from ratio each them from drop differing and weights measure, two take you if For t utbyw a set can we suitably ) nvraiyo reFall Free of Universality isensLift Einstein’s d dt 2 x 2 ˜ g l reyfligbde ihngiil gravita- negligible with bodies falling freely All : = ie h oa au fteaclrto u to due acceleration the of value local the (i.e. g ˜ ( x ˜ x = t , ) = ) x g ˜ 9 + ln h aho igeparticle. single a of path the along 0 = g b ( ..alprilsfl ihtesame the with fall particles all i.e. , ( x t t , ) , ) − b ¨ ( t ) . g ( x t , sindependent is ) veloc- dmit (18) (17) lion x y ) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. ttemto ftonihorn atce ihpositions with particles neighbouring two of motion the at ore sueteursrce aiiyo h ekeuvlneprinciple. equivalence weak the of validity responsible unrestricted field the range assume long course, hypothetical there purely gravitational) far a so fam and for and are nuclear additional we an forces strong be four nuclear, may the weak to addition (electro-magnetic, in with that be would explanation possible thus ie rmtoei h nlgu qain nelectro-stat in equations analogous the in those from differ the ewieti as this write We oesr htteinrstliei nfe al)uigti h rpsr plan proposers the this using 10 in fall.) part free rock one in and to is sensors claim satellite has Philoponus’ check inner which to the satellite that evacuated ensure larger to a inside satellite (one oa est fatv rvttoa asmatter mass gravitational active of density local potential Newtonian the of Hessian the be to seen is tensor tidal the whence curl conservative, is field gravitational the Now n so and rt h oetorder lowest the to or 2 ycnrs o-nfr rvttoa fields gravitational non-uniform contrast By Now o hudcekta o nesadtesgsi 2)ad(28 and (27) in signs the understand you that check should You etna Potential Newtonian fhforce fifth oso’ equation Poisson’s ic hr sn vdnefrsc oc esal nthis in shall, we force a such for evidence no is there Since . d dt 2 N 2 i E + d by dt ij g 2 E N 2 rGussLwrltstegaiainlfil othe to field gravitational the relates Law Gauss’s or = = ij N E − ( = − d div ∇ j d ij dt dt grad ∂ 2 d 2 dt 0 = N j 2 N x 2 = ˜ 2 2 g g U x 2 i i . = . grad) ∂ = g U . ( = 4 = = i ∂ 10 g − j g ( ∂ U πGρ 4 x ( j ia Tensor Tidal g πGρ x edscDeviation Geodesic g + = t , i ( + i ) 17 a N N ) E = . , a . 2 O j (20) ) ij , are g − . If )( . ρ ,ads emyintroduce may we so and 0, = ∂ ics. N i li al tti ee,a level, this at fails claim his a i bevbe osehw look how, see To observable. U : 2 (21) ) n o n h they why and how and ) x and x ˜ = x + N (28) (27) (22) (26) (25) (24) (23) (19) iliar ets . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. is rvttoa edvector field gravitational asw a rptesbcitadwiePisn’ qainas equation Poissons’s write and subscript the drop can we mass where eteo aso h at onsse ooclaewt h ua period. the lunar expect the would with one oscillate to hold system not seen. moon is did earth effect centimetre principle the a such Equivalence of than Strong mass better to of the known centre If is distance moon so. earth or the orbit the back get etna hoyti olw rmNwo’ hr a htato n reac- force and The action opposite that and Law equal Third be Newton’s should self-interact from gravitational tion follows significant this with theory bodies Newtonian for even hold will ciple Principle Equivalence Strong ehv en t asv rvttoa as hsfc ssmtmsknow sometimes is fact This of mass. principal gravitational passive its seen, have we n hudlo pn(2 san as (32) upon look should One because Finally, rvttoa aso oy2 o etnsTidLaw, Third Newton’s Now 2. body of mass gravitational us etfo at otemo n nooeo hs onr srflce back laser reflected is A corners these corner angles. of some one right into behind at and left moon mutually the visit meeting in to to earth mirrors astronauts from plane sent last pulse three The i.e. moon. reflectors, the hold. of necessarily motion not Third the would the if laws holds, conservation that converse these such the hold extent function some not potential To did a hold. Law of all existence will conserved the is be of energy must and w momentum masses angular bodies, passive tum, of and pairs active possible that all sensibly, for units true out equal. is choosing equation by this that, that require we If rtdb oydepends body a by erated 3 precisely ie h dniyo nril ciegaiainladpsiegravi passive and gravitational active inertial, of identity the Given n ipewyt hc h togEuvlnePicpei yloigat looking by is Principle Equivalence Strong the check to way simple One momen- of conservation of law the that follows then it theory Newtonian In o xeietrvastermral atta h rvttoa edgen- field gravitational the that fact remarkable the reveals experiment Now o hudb bet rv this prove to able be should You m p (2) dniyo cieadPsieGaiainlMass Gravitational Passive and Active of Identity stepsiegaiainlms fbd and 2 body of mass gravitational passive the is h poiedirection opposite the E ij = − F E g ∂ (21) ii i only j iial n hudrgr h ymtycondition symmetry the regard should one Similarly . g 4 = i ic tipista h ekEuvlnePrin- Equivalence Weak the that implies it since , = ehave we nisttlieta aswihi uneul,as equals, turn in which mass inertial total its on πGρ Gm m m nerblt condition integrability E 3 p (1) a (1) p (2) i n ytmn o ogtepletksto takes pulse the long how timing by and [ j,k m = ∂ 11 ] k a (1) il Equation Field 0 = E m m ij | a (2) p (2) r r . (1) F = (1) . (21) ∂ − − i E r r xre ybd nbd 2 body on 1 body by exerted (2) kj (2) | or 3 . , F o h xsec fthe of existence the for (21) rsmtmsthe sometimes or , m = a (1) − F steactive the is (12) requires tational os In ions. sthe as n see e (29) (32) (31) (30) No Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. n 4) and 2)a nitgaiiycniinfrteeitneo etna potenti Newtonian a of existence the for condition integrability an as (26) eiwo pca Relativity adopt usually Special will I of Review 4 densities tides mean that the Given about sun? say earth. the you the and on can in moon and tide what degree) the sun same gradients, of a gravity the The half by approximately produced (about star. raise are angle symmetric also same spherically They the a approximately sky. to subtend due moon tensor the tidal the Calculate Example 3.1 uhitgaiiycniin r fe called often are conditions integrability Such oriae,btsmtmsIwl call will I sometimes but coordinates, etna Gravity Newtonian ovnin ihu ute omn.Nt htfo o nie ncoordi- on The indices now is “upstairs”. points from be that always Note will nates comment. further without conventions paig sa noopimwhile endomorphism an is Λ speaking, ovnin h nevli nain ne oet transformations Lorentz under invariant is interval The convention. with ri arxnotation matrix in or where where ne aesrw n h eodlwridxlbl oun,adte hudbe should they vectors and column columns, on labels acting upper, index as first, lower of The second thought conventions. the and usual rows the labels with index consistent are but unfamiliar, look n a ht3) that has One eaenwi oiint umrz h ai qain n tutr of structure and equations basic the summarize to position a in now are We η ab t ∇ ⇒ eoe arxtasoe h ne oiin ntemtie may matrices the on positions index The transpose. matrix denotes diag(1 = 2 U 1) 4 = , 1 πGρ , 4) x ⇒ 3) 1 2) a , d − dt 2 . ( = E N ) nohrwrsw s h mil ls signature plus” “mainly the use we words other In 1). ds 2 jk i x 2 E E E + i = = ii x , j ij [ E x i,k 4 = η − d 4 = ij a ab x ), ] ∂g N → 2 0 = E Λ πGρ Λ i j − j a ji t 1 = )ad4) and 3) , x ˜ 0 = η c dt a x interval Λ = Λ η 12 x 4 Λ = 2 b , saqartcfr,adfrta reason, that for and form, quadratic a is a 2 d x = hr h ne aesrw.Strictly rows. labels index the where , inh Identity Bianchi , inh Identity Bianchi 0 = η, 3 il Equation Field a η a , n aeteatnetcagsof changes attendent the make and ab edscDeviation Geodesic b η inh Identities Bianchi x cd ewe egbuigspacetime neighbouring between dx b 1 = , ⇒ a dx , g 2 i b , , = 3 , sieta spacetime inertial as 4 − . ∂ i U ⇒ . E ij = ∂ al i (33) (35) (34) (36) ∂ j U U . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. with functional I Method ways: tion where ychoosing by i Choosing ii) ecall We For )teato functional action the i) that Note The oho t nie r oee.Frti esni ae ai needn sense independent basis makes symmetric, it is reason it this that For say lowered. to are indices its of both h edscPostulate Geodesic The λ by S remto a edsrbduiga using described be can motion Free esythat say We oethat Note ih rays light [ lc Postulate Clock x λ λ ˜ a u = oeprmtraogtecremaue lpe rprtime proper elapsed measures curve the along parameter some ( a λ λ f sacntn etr ftewrdln is line world the If vector. constant a is ]= )] an ( λ ok nyfrtmlk o pclk)cre.W ayteaction the vary We curves. spacelike) (or timelike for only works λ λ , ) ffieparameter affine hscno edn and done be cannot this τ − ob rprtime proper be to = sidpneto h hieo aaee eas fw replace we if because parameter of choice the of independent is m τ dx dλ t is Z a = r = λ eaaerzto invariant reparametrization ttsta lc oigaogawrdline world a along moving clock a that states x r − 4 = f − eget we , S ttsta reprilsmv nsrih lines straight on move particles free that states η ′ d dλ τ dx η ab 2 d srprmtiaininvariant, reparametrization is ab x S λ ˜ ⇒ a 2 λ dx a η dλ with , τ dx = dλ ab → . 0 = a dx = dλ − a dx = dλ aλ a dx Z m λ ⇔ η b dx dλ f b Z ba + dλ p ′ dλ b x . ,b a, b = − η dt a 13 λ = ab η = r = sabtayu oa ffietransforma- affine an to up arbitrary is d dλ ab s λ ˜ = − aitoa Principle Variational 1 x x ˙ − but a mτ − a η 0 + (0) x η ab ˙ ¡ ab b ∈ u dλ. d = dt . dx a x d R u timelike λ − ˜ ¢ λu b a . 2 m = dx d a λ ˜ Z , − b d 1 p λ ˜ . ecnnormalize can we − η ab x na es two least at in ˙ a x ˙ x b dλ. a = x a (39) (41) (40) (42) (43) (38) (37) ( u λ a ) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. n ob ote’ theorem Noether’s by so and with h qaino oinbecomes motion of equation The piece additive irrelevant an to up velocities, small at coincides which h diiecntn in constant additive the h tnadnnrltvsi xrsin eueteascae Euler-Lagrange associated the use We equations expression. non-relativistic standard the fapril,btti eoe ess foerflcsta rma btatpoint abstract an from that reflects one if so less becomes this but convenient. particle, more a usually of is II Method practice, In spacetimes. a curved which particles. equations massless obtain other we or zero, is rays constant light the for If valid curves. timelike for result II Method L ftecntn sngtv ecos tt be to it choose we negative is constant the If Now ue-arneeutosare equations Euler-Lagrange rtadscn hoe epciey u hsterminology sh this they but clarity respectively, For theorem shortly. second using does o and be Lagrangian first depend the will when not we other the which does and variable, here Lagrangian using the are we when which one theorem, Noether’s = 5 4 tmyse tag htteei ouiu cinpicpefrtemotion the for principle action unique to no extended is be there readily that can strange II seem Method may and It I Method both that out turns It hc hs o hudas hc htyuudrtn htth that understand you that check also should irrelevant. is You it why explain this. to Check able be should You η L ab = dx dλ − a p dx dλ srte uce.W aea cinfunctional action as take We quicker. rather is b − oeta hsato is action this that Note . η ab x ˙ a x ˙ b ⇒ ∂ dλ τ ∂L d x ˙ othat so ∂x a ¡ ∂L η η S ab a = 5 ab = dx .Now 0. = dλ dλ dλ − d dx d dτ Z a L ¡ 1 ¡ b λ η dx ∂ ∂L ¢ η ∂λ dλ ∂L η ab = x ab ab ˙ 0 = b a 14 dx dλ dx τ dx 0 = dλ ¢ constant = dλ b = nti a ercvrorprevious our recover we way this In . not b ⇒ a ¢ , = dx 0 = dλ ∂x ∂L d dτ eaaerzto nain.The invariant. reparametrization − 2 b a − x dλ, . η 2 a n n htw a choose can we that find and 1 ab 0 = . sntuniversal. not is dx udprasb aldNoether’s called be perhaps ould dτ b . o eeduo dependent a upon depend not h needn variable independent the n r r w ae of cases two are ere 4 (49) (44) (45) (46) (47) (50) (48) to , re Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. g oi sa trcieie oacietecraueo h ah ffel fal freely of spacetime paths constant the the that of assume curvature We the spacetime. ascribe metric of to curvature the idea to attractive particles an is it so g r eaeln um indices, dummy relabelling or, eas fthe of Because Spacetime Curved 5 r since or, The same the get we that verify to differ many like take tha may of we fact you if point the motion point stationary with of a the equations familiar home be may rub are To point We functiona given functions. variables. a a of many dimensions, point infinitely finite stationary in of as function motion a the characterizing of are we view of where in are tions Using edfiethe define We re-written maybe This ab ab ( Thus . x lc n edscPostulates Geodesic and Clock uhta h nevli ie by given is interval the that such ) ehdI Method f η )i almost is () ab L if 1 = srpae yagnrlsaeadtm eedn uvdmetric curved dependent time and space general a by replaced is nvraiyo reFall Free of Universality nes metric inverse g ab g ab λ ehv set have we d dτ − = 2 d any dτ x 2 2 dλ b x τ ds d 2 , b dτ + d 2 ¡ ucino t argument. its of function + L 1 = ¡ 2 1 g ∂g ³ g ∂x g τ ab ab ∂g by dτ ab g L ∂x dλ ab = dx c L ab dx ( ad = x c dx Z dτ b = b ( = ) ¢ f dx ¢ + c o eda eoebtwith but before as read now p p ( = = dx η g dτ a ∂g ∂x ab − − − dx 15 2 1 − b 1 g ac g dx h oini needn fms,and mass, of independent is motion the ¡ d ) dλ b − ab 2 ab ab ∂g 1 g , ∂x L − a x x ˙ ˙ 2 1 ¡ = cd a a dx a dλ ∂g ∂g ∂g x x ∂x ∂x ∂x ˙ ˙ ¢ g b b b dλ. dx cd n h ue arneequa- Lagrange Euler the and , cd ba cd ) dτ a a a ab , ´ ¢ c dx dτ = dx dx dx dλ dτ dτ c g c c d dx ba dτ dx dx . dλ dτ . d d d 0 = , 0 = , . η ab elcdby replaced ,i.e l, (56) (54) (52) (51) (58) (53) (57) (55) ling ent t, Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. sw hl e shortly, see shall we As g otato f(57) of Contraction where gi ehave we Again a nrdc rvlgdtm coordinate time privileged a introduce may erci called is Pound-Rebka metric A metrics, stationary and Static 6 examples. particles. massless for works also II Method spacetime, with that so hsget thus time-reversal hudjs etogto sa ra ffntos n nfc,t anticipate to fact, in and functions, of are array t they an being, as follows, time of the thought For be shortly. just properties should mathematical their explore shall We aclto using calculation o asv atce,tecntn sngtv n ecos tt be to it choose we and negative is constant the particles, massive For 4 i n httemti a ecs nteform the in cast be may metric the that and 0 = ntenx eto esalapyteeutosw aedvlpdt some to developed have we equations the apply shall we section next the In ecudhv rcee using proceeded have could We h ahrsrnecleto fobjects of collection strange rather The erci called is metric A Experiment δ b a λ steKoekrdlaadeult if 1 to equal and delta Kronecker the is = ..ivratudra involution an under invariant i.e. , τ enwoti u rvoseutos oee,a nflat in as However, equations. previous our obtain now We . ∂L ∂λ ds not L 2 stationary n ote’ hoe yields theorem Noether’s and 0 = = = g n h opnnso esrfield. tensor a of components the ea g g static c ab 44 ds e U n relabelling and d ( dx ( d d 2 dλ x o dλ dτ x 2 2 a ) = g ly h oeo h etna potential. Newtonian the of role the plays ) x x = dt fi ssainr n nadto nain under invariant addition in and stationary is it if 2 2 ab dx a a dλ fi sidpneto ie hsmasta one that means This time. of independent is it if h 2 b 2 1 dx + + ij dλ + g yields ( ea n n g a x g ac c c ) dx ij ³ dλ dx a a ehdII Method g ( ∂g ∂x cb b d d x 16 i o o ad ) dx e c constant = = dx dx dx dλ dτ → + j n δ i c c − dx b a c t a ∂g ∂x dx dx , dλ a dτ = e j o yields now ac d 2 d d d 2 + U o x lgtysotranalogous shorter slightly A . T − ( 4 0 = 0 = x r called are . ) g such : ∂g ∂x dt 4 a t . , i cd ( 2 a − → = x . ´ ) ∂t . dtdx ∂ b ( t n eootherwise. zero and g hitfflsymbols Christoffel hsipisthat implies This . ab i 0 = ) . . Thus − and 1 what (65) (59) (62) (61) (60) (64) (63) hey . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. sn h rprtm,so time, proper the using ∆ time nate y1+ 1 by n iial o h bevdfrequency observed the for similarly and aldagaiainlrdhf n sqatfidi em faquantity a of terms in quantified is and redshift gravitational a called vdnl fteeitri talwrvleo h rvttoa potenti gravitational the of value lower a at is emitter the if Evidently . h rvttoa Redshift Suppose Gravitational The 6.1 U n ec h ratio the hence and )Ahuitcdrvto ftegaiainlrdhf a lob ie us given be also can redshift gravitational the of relation derivation Planck’s heuristic A i) i h rvttoa esitis second redshift the gravitational of The machine gravitational ii) motion the perpetual up photon a impossible. the climbing construct is unless to which energy able photon, kind of be absorbed re- would amount the of then one predicted of because well, is energy the process, photon the lowering precisely the the to loses during due and weight obtained point extra absorb be starting is the can potential, the energy higher Since to a to lowered emitted. lower a slowly from absorber sent is the photon a which in process ocibu h rvttoa oeta eli eue.Tu codn othe speeds different to have according will Thus sources have reduced. different particles’which is from ‘light well coming the light potential of Theory, according gravitational speed Ballistic Theory the The Ballistic up the conserved. climb using also to problem is the energy analyze which to to interesting clocks) is by It mesured v) as places. different (i.e. the at time rates as that different shows to at redshift referred gravitational sometimes The iv) is statement This clocks, curved. physical argument by is measured as really that time shows above. redshift gravitational given The impos lines the iii) the using along proved machines be also motion could perpetual this Again of particles. massless all for ( x ermr that remark We o ,tercie rqec ilb oe hnteeitdfeuny hsis This frequency. emitted the than lower be will frequency received the ), z = n . ussaesn rma mte at emitter an from sent are pulses ν ν t o e h mte rqec il ytecokpsuae eobatined be postulate, clock the by will, frequency emitted The . . E ν ν = o e hν = s n isensformula Einstein’s and g g 44 44 ν ν ( ( o e x x universal = = 0 e ) ) ∆ ∆ exp = t t p p 17 − − h esiteprecdi the is experienced redshift the , n n g g ¡ 44 44 U ( ( ( x x x x e o e 0 ) ) ) oa bevrat observer an to E − = U mc ( x e 2 ) n esu cyclic a up sets One . ¢ . x al, o ncoordi- in U z sibility ( space- Schild x given same runs e (66) (68) (67) ) ing ed, < . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. etr nt n so and units restore hs oprn ih60 with comparing Thus, The . atcemto nSai Spacetimes Static in have We motion Particle source. its the of contradicts independent 6.2 prediction and speed universal the his is that But Relativity) earth Special on up here it. setting received check light when of use to we prism (which suggested fact a and observed happen with would this experiment precisely predicted an Michell John 1784 in fact In nteaoew aeset have we above Limit the In Newtonian The 6.3 Now ntetmlk aew a choose may we case timelike the In hr h constant the where needneof Independence o ecnnwepn h erci nes oesof powers inverse in metric the expand now can we Now x i qaino oinis motion of equation e 2 U ¡ t dτ d dτ dt ie,fo ote’ theorem, Noether’s from gives, ¡ E n t h ¢ j 2 → ij steeeg.Now energy. the is i − dx k dτ ct o h c j h quantity The . ij = ¢ .T nesadti prxmto eshould we approximation this understand To 1. = dx e L dτ + n 2 1 h 2 c g 4 U 2 = i ij h dτ e 44 d i 2 dx n is e dτ + 1 = 4 U = e 2 ¡ 4 = ³ o 2 U ∂ j e 4 U λ ∂h δ i 2 ∂x = − i = 1 = U ij t U dt ˙ dτ o = 2 18 c sk ¡ dτ − + j dt 2 − 2 = dτ τ c dt h = U + 2 The . + O ¢ g h ij U ∂ ab ¢ ij O E, ( 0 = i + E ∂ 2 ∂h U ∂x c srpae by replaced is x x 1 j ˙ ˙ 1 (77) (1) 2 2 = . . . . e i a e sj ) x . 2 k x ˙ − ˙ x U . j 1 2 b 2 . 4 − ( U = ∂ qaino oinis motion of equation i ∂h − ∂x − h jk h (with 1 jk s ij ˙ ) x ´ dx dτ j . x ˙ c U i 2 k c dx : . . dτ λ j = . τ gives ) (79) (71) (70) (76) (78) (69) (75) (72) (73) (74) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. eobtain we hsjsie u dnicto ftequantity the of identification our justifies This λ λ n h atthat fact the and fabd fmass of body a of o h on-ek experiment Pound-Rebka the For lal,ti steeuto feeg osrainfrannrltvsi pa mass non-relativistic unit a for per conservation energy energy of of equation the is this Clearly, put we order, non-trivial lowest the to Thus, . oino ih rays Light gives of theorem Noether’s Motion 6.4 τ osbet mrv 8)s hti sacrt oorder to accurate is it that so (80) improve to possible potential. ai feege swl defined. well is energies of ratio ≈ → → ecnnwgv h rvttoa esitsffrdb htni h field the in photon a by suffered redshift gravitational the give now can We fw,i diin sueta h atcei oigsol,w a set may we slowly, moving is particle the that assume addition, in we, If eas h ffieparameter affine the Because nfc,uigteEnti edeutos hc ehv o e e,i is it met, yet not have we which equations, field Einstein the using fact, In h nepeaino 8)wl egvnltri h course. the in later given be will (87) of interpretation The t aλ aλ , E otoi h ‘Energy’ the is too so , , E + 1 = → E a E hsmasta,i ueypril hoyo ih,ol the only light, of theory particle purely a in that, means This . efidfo (76) from find We . g d M ds dλ ab 2 ds x x 2 ˙ 2 nteNwoinapproximation Newtonian the in i a 2 − ≈ x + ˙ − ≈ b E hn ie c.76) (cf. gives 0 = c 2 oigi etna rvttoa potential gravitational Newtonian a in moving c j dt E 2 E ν ν i dt o e 2 2 k h qain r nain ne h rescaling the under invariant are equations The . 1+ (1 o 2 = = 2 1 1+ (1 e r + λ v 2 o δν e GM U ν 2 2 = h sdfie nyu oa vrl multiple, overall an to up only defined is 2 U + dλ c dt jk U = 2 19 2 h r c ( U U e h 2 ij + ) r = + 1 gh is c e + ) dx = dλ 2 ∂ h E, − . d s where , i E U x ¡ dx . r dλ 1 1 2 i o ¡ − dx j ) dλ + 1 . , 1 2 2 j c ln( U 2 h dx O dλ ¢ stehih ftetower, the of height the is − ( k d c c g 1 x 1 4 2 0 = 44 2 ) . ¢ steNewtonian the as ) . . rticle (81) (82) (86) (85) (83) (80) (87) (84) U . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. magnitude erae.Telclymaue s-ald‘euir)velocity ‘peculiar’) (so-called measured locally The decreases. htni mte ttime at emitted is photon pta etosaecurved. are sections spatial where hsi h nvreepnsadtesaefactor scale the and expands universe the if Thus is metric the hole, black if or true star be the will than which massive geometry, less movi spacetime much particles the or very of the on is Provided motion effect it star. the negligible a study a or to has hole black particle position static good symmetric spherically a a in around be now should You Metric Schwarzschild The 6.5.1 Examples Further 6.5 ie by given is ewl aeamr ealdlo tti ae ntecourse. The the in later this 6.5.2 at look detailed more a take will We lodcessecp ntelmto eomass zero of limit the in except decreases also u nvrei o ttc u ahri sepnig oago approximation good by a given To is expanding. universe is our it of rather metric but the static, not is universe Our ote’ hoe implies theorem Noether’s not 6 h function The ncsooyoesuisamr eea ls fRobertson-W of class general more a studies one cosmology In osre.I atuigtenraiaino h -eoiyoegets one 4-velocity the of normalization the using fact In conserved. p sacntn etr u eas ftetm-eedne energy time-dependence, the of because but vector, constant a is | v ds | k 2 .Teeeg famsls atcedecreases particle massless a of energy The 1. = 0 = = a 1 ( oeto-akrMetric Robertson-Walker t − scle h cl atr o atceo mass of particle a For factor. scale the called is ) dr 2 2 M r t oetmconservation momentum + e ds E E n eevda time at received and v r E ( ( 2 2 t t = ( e e = ma dθ = ) ) v p − + 1 = E 2 s 2 = dt ( sin + p = t m 2 ) 2 6 a 20 d dλ 2 + m ( + x t + z p ) 2 m dτ dt a d = dt = θdφ 2 a x 2 ( p 2 a p t a a ( 2 ) 2 m t 2 , ( ( d ( ) ) t t t x o e . ) when − ) ) 2 t . . o µ a ( 1 h esiti ie by given is redshift the t nrae h energy the increases ) − v scntn n funit of and constant is le erc nwihthe which in metrics alker 2 M r ¶ dt ∝ 2 . a ( 1 t ) hsi a if thus , biting (95) (90) (92) (93) (91) (88) (94) (89) m ng E , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. rrvsat arrrives hr h points the where from h oa cinaddmn hti vanish: it that demand and action total the nsaeie ecnuetemetric the use can we spacetime, in nifiieia etrdisplacements vector infinitesimal on V erigthat learning ewe hmi ie by given is them between ftetovcosare vectors two the If fteaction the of where fteedpit ftewrdln from line world conservation the momentum of and points variations end the point If End Example: 6.5.3 eaiiysol od hsi ewr nasalnibuho fapoint a of Speci neigbourhood of small laws a the in locally work we that if is Thus Relativity hold. General Rescalings should of Relativity Conformal aspect and important An Angles Lengths, 7 o osdra2pril olsnat collison 2-particle a consider Now ftetovcosare vectors two the If h aito at variation the hr h aoia momentum canonical the where a by A atcetowt mass with two particle , C S ihmass with AX δS ( δS ,A X, X ( ,B ,D C, B, A, , ( ,X C, ,A X, BX htmmnu scnevda h collision: the at conserved is momentum that is ) spacelike + ) , timelike = ) m XC 3 δS Z n atce4arvsat arrives 4 particle and r edfie but fixed held are A and ( cosh | X ,X D, V p the , cos n ohaeftr ietd the directed, future are both and , a 1 p δx a p a i | + m XD a θ a θ = p = + ) g ¡ 2 p a sdfie by defined is = = angle ab g a 2 ∂x q X m ∂L = from ab V − V utb edsc,w lodsoe from discover also we geodesics, be must = 21 δS g A | | i a U g nwihpril ihmass with 1 particle which in g ab a ∂ a g odfiea ne product inner an define to ∂L | ab p ab U , U U − to ab ( x a U ˙ a 3 θ ,A X, || V a a b dx B U a + dτ dp V a X ewe hmi ie by given is them between || . dλ a V a V b b V p n en h egho vector a of length the define and n fe h olso atce3 particle collision the after and a V b | r loe ovr,tevariation the vary, to allowed are . + ) a 4 X b ¢ b . b | , | dλ . . salwdt ay swl as well As vary. to allowed is δS + D ( £ ,B X, p ihmass with a δx 0 = ) a ¤ A X , , m 4 rapidity evary We . m 1 starts (101) (102) (100) (103) (104) (98) (99) (97) (96) x al θ a Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hc sotie yacnomlrsaigo h a n ttcMinkowski static and flat the of coordinate rescaling The conformal bracket. the a inside by metric obtained is which u nlsadrpdte r unchanged are rapidities and angles but where size o hsrao elrsaig r localled also are Weyl-rescalings reason this For efidta l egh rescale lengths all that find we if otn xml spoie ytetm-eedn oeto-akrmetric Robertson-Walker time-dependent universe the expanding by an provided of is example portant rae,a esol xeti a pctm,adthen distances and de- greater spacetime, first flat and at in greater size expect at angular should apparent we size, the as intrinsic creases, redshifts, same greater and the greater of equivalently all galaxies, of called asi hstp fepniguies r h aea hywudb nflat in be would light they of as system same by the made are universe angles expanding the of that spacetime. type follows this It in rays coincide. metrics related htteaprn nua iei es taredshift a at least is size angular apparent the that n Ω( and nl asmdvr ml)a time at small) very (assumed angle a ( h oml 10 a nauigcneune fw osdrafamily a consider we If consequence. amusing an has (110) formula The fw hnetemti ymliligb oiiefnto aprocess (a function positive a by multiplying by metric the change we If o xml,aglx mtiglgto nrni rprsize proper intrinsic of light emitting galaxy a example, For t a ) ( d Weyl-rescaling t ∝ x e ) = ) t p ntecnomlyrltdMnosimti n hssbed an subtends thus and metric Minkowski related conformally the in ih0 with , a ( t .A esalseltr h ulgoeiso w conformally two of geodesics null the later, see shall we As ). ds < p < 2 ): = − dt hr aclto,wihyusol hc,shows check, should you which calculation, short a 1 2 g ∆ + η ab o θ + 1 a | − → = 2 V η ( η t a Ω z = a ) e → | t θ d ( o 2 = = t x Z → ( e 22 ie by given x 2 )( ¡ Z Ω ) = a p 1 θ. η g t d dt | e ( t ab o ¢ V η t o a ) 1 − a 2 − sdfie by defined is a p ˜ = , | ( dt p ( , η t t . ) e g ) ofra Rescalings Conformal ³ ) ab . , − dη + increases d x 2 ´ d o example, For . ttime at nim- An . t (108) (106) (110) (107) (105) (111) (112) (109) e or , has Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. ytm˜ system utteeoealwteueo rirr oriaesses hsdsr is desire This Covariance General systems. of coordinate Principle arbitrary of formalism the Our use in spacetime. the formalized curved allow general a therefore in must exist to understanding Physics. cease Mathematical coordinates good inertial of a but part than given, standard relevant a be less become course much has of are level. what can they of unsophisticated manifolds purposes relatively differentiable practical a field of for mathematically course Einstein’s accounts is introductory deeper what to this i.e. In Much at and equation, geometry. progress proceed Poisson’s more further shall of some make we develop analogue To to the need geodesics. down we just equations, write using to far able so be get only can One Analysis Tensor physics. 8 for relevant become they did discover universe Einstein Hubble’s the with before of 1920’s long expansion the century, the were in nineteenth notice of later would the Only we Relativity. of whether General end and formulated the overnight towards size frequent in doubled quite universe it. i the measure metric if which Minkowski pen would instruments flat of she gett the also set had that are a she maintain made provide is if to should she wishes but one which really of ‘correct’one, smaller, is atoms one universe getting the If the that are agree that smaller. who to have, we had would possi- have is always also Alice have it Roberston-Walker is imagines it but the one course expanding by as Of ordinary not given rather expands. of maintain, as universe say to the lengths them ble built measure to instruments, relative they and measuring that metric, conventional find use we we atoms, If ments. onteeutoso hsc nawywihi ai nalcodnt syst coordinate all be in to valid said is are which way equations Such a in physics of equations the down a,w ol iet a owtotitouiggoa nrilcoordinates. inertial that. just global do introducing to without us allows so Analysis say Tensor or to were Calculus really like described Tensor spacetime would be if Thus principle we in systems. flat, can cordinate particular spacetimes to such reference that without requires simply It reference. oriaesseswihmyb xrml sfli rcie ete osit does Neither practice. in particular useful of t extremely use important out be the rule is preclude may It not which does systems systems. Covariance coordinate coordinate General of all Principle in the form that same the take should u nlsso h ekEuvlnePicpesosta rvlgdglobal privileged that shows Principle Equivalence Weak the of analysis Our hap- would what about speculations philosophical that striking perhaps is It instru- measuring our upon depends ‘correct’metric the is metric Which ie n oriaesystem coordinate one Given x priori a a ˜ = x a ( x h osblt fpiiee ytm fcodntso rmsof frames or coordinates of systems privileged of possibility the b n aclt the calculate and ) covariant hc ttsta n hudb bet write to able be should one that states which Λ x a a b ecnawy ast e coordinate new a to pass always can we = aoinmatrix Jacobian 23 rsometimes or ∂ ∂x x ˜ a b . form-invariant ic they since realize o (113) the s ems. ing y Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. tion) x o einr,o hneggdi opiae acltos hsntto can notation this calculations, However complicated read. because in to here difficult engaged useful. usage formulae when that be printed adopt make or to not beginners, tends shall for and I i.e. eye type, the tilded. same strains both the it of or Einstein be un-tilded the must both case, indices either that repeated but In holds differentiated. convention being posi- summation are index variables what the remembering convention. nicely summation matri how Einstein and the Note differentiation course, partial of of matrix. rules and the multiplication, constant to a themselves accommodate is tions matrix Jacobian the sn h hi rule chain the Using transformat Lorentz a as such transformation, linear a for example, For The hsfruamtvtstedfiiino hti alda called is what of definition field the motivates formula This n thus and ehave we n ncodnts˜ coordinates in and a = enwconsider now We to aid further a as indices the on tildes of use the encounters sometimes One oevr ftetransformation the if Moreover, hi rule chain ssto quantities of set as x a ( λ nsaeie t agn etri the in vector tangent Its spacetime. in ) mle htif that implies T ˜ a x a = etrFields Vector tis it , ∂ ∂x T x x ˜ ˜ a a a b ( x T Λ = rnfrigudracodnt hnea (121). as change coordinate a under transforming ) b , x ˜ ˜ ∂ ∂x a d a det dλ ∂x ∂ x ˜ ˜ x ˜ = b x ˜ T a b T a omtvt h ento osdracurve a consider definition the motivate To . x ˜ a ³ b b a x otaain etrfield vector contravariant x a ˜ ˜ = = , ∂x ∂ ∂ ∂x a a = = x x ˜ (˜ ˜ 24 ∂ ∂ → ∂ ∂x x c b a b x d ˜ x ˜ x dx ˜ ˜ c dλ ´ dλ = x satidcodnt ytm then system, coordinate third a is ) a ˜ c a b x ˜ ∂ ∂x a a 0 6= ∂ ∂x a dx δ dλ x ˜ , . x ˜ c a sinvertible is a b c b b . , , Λ = . x a a b , oriae s(ydefini- (by is coordinates otaain vector contravariant . ion, (121) (117) (116) (114) (119) (120) (115) (118) x Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. sivrat o since, Now invariant. is pca eaiiy ydmnigta h interval the that demanding by Relativity, Special ehave we fafunction a of a omthe form may h otato sivrat hti,i sa is it is, that invariant, is contraction the r if or, etrfil.Ti stu,adte r called are they and true, is This field. vector rnfrsas transforms paig hytasomi h poiewy neapei the is example An way. opposite the in transform they speaking, ecndecompose can We field tensor covariant rank second symmetric) necessarily (not general A Q ˜ g ˜ h rnfrainpoete fthe of properties transformation The ie otaain etrfield vector contravariant a Given transformation coordinate the If of kind other some is there that suggests contravariant epithet strange The cd cd F ˜ = = a = g Q ab ab ∂ ∂f x ˜ ∂x ∂ ∂x ∂ f a x ˜ contraction x , ( ˜ a c x a c ∂ ∂x .Tecanrl gives rule chain The ). ∂ ∂x x ˜ F x ˜ ˜ d b d a b F , T , ˜ a a = ds = F ∂ ∂x ymti eodrn oain esrfield tensor covariant rank second symmetric 2 ymti eodrn oain esrfield tensor covariant rank second symmetric ∂ ∂x x a ˜ = T x a ˜ b Q a b F a ˜ g F cd ∂x ∂ b ∂f ∂x dx F = ab ˜ a , x ˜ a = a dx = a a e T = T = = Q a x a ∂ F b dx a ∂ ∂x a ( 25 oain etrfield vector covariant F T ∂x cd ∂ a ∂x ∂ f x ˜ → e Because . x b ˜ x a ˜ ) a b = a b = a c ˜ = + F erctno field tensor metric n oain etrfield vector covariant a and x ˜ clrfield scalar d ∂ b ∂f ∂x a δ x Q g ˜ . , , ∂f x ˜ oain etrfields vector covariant b e cd c b sivril,w lohave also we invertible, is [ T , a dc d , b ] x ˜ F , c e d = x ˜ d . T a F a , olw uta in as just follow, gradient n roughly and . . F a (127) (128) (122) (125) (124) (126) (123) (130) (129) (131) one , Q ab Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. (ii) (i) prop- tensorial their preserving tensors on Operations 8.1 (iii) nie n h otato ilb esrfil ftype of field tensor a be will contraction the and indices xhne o qaebrackets. square for exchanged into and gi h ymti n nismercprstasomit themselves transformations. into transform coordinate parts general anti-symmetric and symmetric the Again the hsdcmoiini nain ne eea oriaetransformation. coordinate general a under example invariant for Thus, is decomposition This Addition n a en otaain eodrn esr analogously tensors rank second contravariant define may bracketsOne round with part anti-symmetric the for holds argument identical An ngnrl n a lascontract always may one general, In h rnfrainrl o has now rule transformation The o xml a example For ngnrl n a osdrabtaytno ed ftype of fields tensor arbitrary consider may one general, In h ptisaddwsar nie a ecnrce oyedasaa called scalar a yield to contracted be can indices downstairs and upstairs The Multiplication trace ue rTno Products Tensor or Outer Q erties q p [ cd nie ontis ..covariant i.e. downstairs, indices nie ptis ..contravariant i.e. upstairs, indices Q ] = ( cd Q o aetype) same (of ) 2 1 ( cd ∂ ∂x ¡ x Q ) ˜ ¡ a c = cd M 1 1 ˜ ∂x ∂ ¢ yscalars by 1 2 − x a ˜ esrfil rnfrsas transforms field tensor d c ¡ a Q Q = = dc cd Q V ¢ ∂ ∂x + M a x cd ˜ ˜ = P W ˜ a c Q a ∂x ab ∂ − ∂ ∂x b b dc x ˜ x Q ˜ = ( = a ¢ e.g. , c a d [ dc = M ∂x P ∂ ∂ ∂x p ] sa is x x ˜ ˜ cd 26 Q c d atr of factors a c c ) r d ( ∂ ∂x ∂x dc ∂ ¡ = = otaain nie with indices contravariant x ˜ x 1 1 ˜ ) t anti its a c ¢ d b Q δ ∂x ∂ tensor M c d cd x M ˜ c c b ∂ t ymti at(132) part symmetric its ∂x d c . x ˜ ∂ ∂x − d . x ( ˜ . c a ymti part symmetric = and ∂ ∂x M x ˜ b d ) d ¡ q p q d = − − atr of factors . r r ¡ ¢ Q ˜ , ¡ ¢ r q p ( ab ¢ ≤ . ) with . min( r ∂x ∂ covariant x ˜ . ,q p, under (138) (133) (134) (137) (136) (135) ). Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. mte.Teovosaaoywt hmcldarm xliswyatensor a why explains diagrams chemical with type analogy of obvious The omitted. ymti rttlyat-ymti esr,ti sntapolmsnet since totally problem for a course Of not is order. cyclic this the tensors, of anti-symmetric track totally keep to or need symmetric the is notation this o xml,if example, For Theorem Quotient 8.2 . ue o ne Shuffling Index example For for Rules 8.3 v otatoscrepn ojiiga non n noton ro,no delta arrow, Kronecker outgoing The an (vi) vertex. and same ingoing the an to unimportant attached joining is necessarily to node correspond in outgoing course Contractions attached Of and (v) are ingoing node. arrows an each the of around indices anti-clockwise, order the example relative of for the order order, the cyclic of definite track arrow. keep a ingoing to an order by In by (iv) represented is index covariant outgoi A an (iii) with edge an attaching it by on represented arrow is index contravariant vertex, A a (ii) by represented is this tensor Each In (i) century. nineteenth the firs tensors in for Sylvester ‘chemical’notation and notation or Cayley graphical Clifford, a by adopt introduced to manipulations, convenient index be complicated may involving it those especially purposes, some For Notation* *Graphical 8.4 (v) (iv) and then (vi) foehsadsigihdmti rb-ierfr,te h rosmybe may arrows the then form, bi-linear or metric distinguished a has one If T ne Interchange Index Contractions ymtiainadAnti-symmetrization and Symmetrization [ W abc ab ] ¡ q p = sacvrattno.Ti euti oeie called sometimes is result This tensor. covariant a is ¢ 3! 1 ssmtmssi ohave to said sometimes is ³ T V abc ab + W T ab cab T sasaa o nabtaycnrvrattensor contravariant arbitrary an for scalar a is ab δ a + b satensor a is V srpeetdb oeesarrow. nodeless a by represented is T ( bca ab V ) W − ( ab ab T ) W bac ⇔ = 27 [ valence ab − V T ] ba ab T 0 = cba W satensor a is . ( ab − p ) + T , acb q o example For . h anda-akof draw-back main The . ´ . satensor a is esrDetection Tensor . (139) (140) (142) (141) V hen ab ng t t . , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. (iii) i)Atn navco field, vector a on Acting (iv) o n arof pair any for (ii) (i) hitfflsymbols Christoffel want is eiaieoperator derivative Γ h rttr is term first The hscneto snti nvra s,ads n hudtak symme should to attention one restrict so unnecessary. will and ind becomes we use, first distinction shortly the universal However, that in convention not books. the is using convention am I This indices. lower the pt in hscnb sdt iegahclpof ftno identities tensor of proofs tens of of graphical number set the give tenso a counting to same to from analogous the used constructed is invariants to be problem possible rise latter can all gives The of This type enumeration same the sign. or the a of to another over up arrow one lifting ehv enthat seen have We Tensors Differentiating 9 molecule. a not 7 b ∇ edmn of demand We nodrt osrc esrfilsw utitoueaso-called a introduce must we fields tensor construct to order In oain eiaieoeao sas aldan called also is operator derivative covariant A o eea ffiecneto,teei osmer ihresp with symmetry no is there connection, affine general a For ∇ c ∇ a ∇ r t components its are φ a eodrn oain esrfil.Frsmlrraos ete r the are neither reasons, similar For field. tensor covariant rank second a sLeibnizian: is a = omtswt contractions, with commutes ohv rprisa ls spsil otoeof those to possible as close as properties have to ∂ a φ , ¡ q p good ¢ ∇ and ∂x ∇ © ∂φ = a a a a u h eodi clearly is second the but that = b ∂x hc maps which ∂ ¡ ∇ sac-etrfil u htaotteHessian? the about what but field co-vector a is c ∂x q p x ˜ ª ∂x ∂ ′ a ′ 7 a c ¢ . ( x ∂φ ∇ ˜ h opnnso esrfil ftype of field tensor a of components the a UV ∂ ∂x a c esrfields tensor ∂x a x ˜ ∂ ∂x V d b b x ˜ ( = ) b ∂ d b = x = ˜ ∂ ∂ c 2 x ˜ ∂x ∂ ∂ ∂ ∂ ¡ φ ∇ c x x q p ˜ a ˜ 2 ∂ a ¢ V φ a d c 28 x ˜ U ∂ esr to tensors b + d ) ∂ U x ˜ Γ + V + c ∂x ∂ and ³ x ˜ + ∂x a a ∂ c ∂x ∂ U b x bad ˜ ∂ 2 a c V d b ∂ x ( x ˜ ∂x ˜ V ∇ 2 c ∂ d , ∂φ ¡ nohrwrs h Hessian the words, other In . x ∂x ˜ c x b a ˜ . d q p V d rcan oncin n this and connections affine tric ∂φ x +1 ˜ xi h ‘differentiating’index. the is ex b ffieconnection affine ´ d ) aewe osligtext- consulting when care e ∂ , ∂φ ¢ . x ˜ esr oevrwe Moreover . tensors d c oitrhneo the of interchange to ect isomers ∂ a . facertain a of ¡ 2 1 covariant ¢ . n the and (146) (145) (144) (143) (147) ors. r Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hc mle that implies which lasasm httetrinvnse,ietecmoet fteconnection the of components the i.e vanishes, symmetric torsion are the that assume always ednt that note field satno ftype of tensor a is operator sn ne hffln n h qaiyo ie atasw euethat deduce we partials mixed of equality the and shuffling Connections index Using Affine Symmetric 9.1 field. tensor a of components the are o hudcekta o nesadwytewr eddaoesosthat systems. shows coordinate above all did in we holds work which the statement why a understand is you this that check should You aetransformation nate Similarly, rpris()(i,ii,i)dtrieteato ftecovariant the of action the determine (i),(ii),(iii),(iv) Properties tflosthat follows It ti sfleecs ocekthat check to exercise useful a is It field. vector a for (iv) expression the with compared sign minus the Note are They ( = ∂ ∇ a W a naytno ed o xml,t e how see to example, For field. tensor any on not b ∇ ) Γ V a a b ( b h opnnso a of components the W ( + c ∇ → b ¡ a V ∂ 2 1 W ¢ a b Γ ˜ V ( = ) T cb a aldthe called a Γ ˜ b b ∇ = ) b [ c a W c a = ∇ = = b ∂ W b Γ = a c a ∂x ] ∂ ∇ ( = W b ∂x ∂x ∂ ∂ W Γ = x ˜ x x = ˜ ˜ a a cb a e b b e e b ∇ b ∇ ) b ∂x ∂ b ∂ ∂x ∂ − ∂ ∂x ∂x V ∂ oso tensor torsion b c c x a ˜ x a ˜ φ x ˜ x ˜ W b e b W − Γ = a g Γ a [ g 29 © a g + = ∂ a ∂x ∂x ∂ b 2 1 b ∂x ∂ Γ ∂x x ˜ ) ª ∂ x e ˜ W − x ∇ c V ˜ c [ g d x c c ˜ a d c b esrfil.I at ne coordi- a under fact, In field. tensor b d W a b c b Γ Γ ∂ Γ ] a ∂x a ( ∂ + [ Γ a ∇ x eb g g ˜ b . = g c φ c d e e W a ] − b e d d − V Γ W ] d b g + Γ T b ( . nwa olw,w shall we follows, what In . c = ) ∂ e a c . a ∂x ∂ d b e V x ˜ a b ∂ e b W b a ∂ Γ + ce ( ∇ ∂ x ˜ W a 2 a . x ∂ b a cso covector a on acts V e x ˜ b c b c (149) ) V . c ) , derivative (150) (157) (154) (148) (156) (152) (151) (153) (158) (155) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. sas a also is iial,uigtesm ehiu,oemypoeta ie two given that prove may one technique, same the Bracket using Nijenhuis Similarly, The Example: 9.1.2 h ih adsd satno n hrfr h ethn iei esr The tensor. a is side hand left the therefore and tensor a operator is side hand right The eakbeproperty remarkable fields S theorem. iaetasomto n e httebdtrscne,o oeexpeditiously, more coor- or a cancel, under terms behaviour for bad the that the out show that write either see can and we transformation field dinate tensor a is this prove To nat-ymti otno scle a called is co-tensor anti-symmetric Derivative An Exterior Example: 9.1.1 curl (i) that fact striking a Connection is It Levi-Civita The 9.2 (i) (ii) h udmna hoe fDffrnilGeometry Differential of Theorem Fundamental The a 8 b ecl hscneto the connection this call We h eiCvt connection Levi-Civita The true is statement stronger a fact, In n a osrc ( a construct may One The © sn utprildffrnito.W define We differentiation. partial just using , c b = A a p a c A 1fco scnetoa n ae o ipictosi th in simplifications for makes and conventional is factor +1 ¡ ª b d 2 1 e and ¢ transform odfie cigon acting defined so a ∂ esrfil hc satsmercin antisymmetric is which field tensor e n symmetric any B B b a c b − ∇ then , A a precisely b Γ g bc e b ∂ a p = a c 1 om aldthe called form, +1) dω B Γ = ∂ e ∂ abc... a ffieconnection affine c [ g sasmercan connection. affine symmetric a as a − © bc © c p ω b A b a eiCvt connection Levi-Civita bc... − om snilpotent: is forms a a ( = e b c , c c Γ ª ª ] ∂ a p e = 30 steuiu ffiecneto s.t. connection affine unique the is = B e 1) + p-form b ∇ b © g i . a [ ec c e a + . ∂ a ω ssmerc(164) symmetric is − [ A b a bc... ª ω b . Γ xeirderivative exterior bc... e a ] ∂ . 8 e c ] a B c . g e and d eb a 2 + 0 = thstefollowing the has It . 0. = B c . omlto fStokes’s of formulation e e . a ∂ e A b or c − generalized ¡ B 1 1 ¢ b e tensor ∂ (161) (160) (159) (163) (162) a A e c − B e c ∂ e A b a + B b e ∂ c A e a Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. emtn h he nie n hntk utbelna obnto.The combination. linear suitable a take then Γ of and symmetry indices three the permuting Proof bouederivative absolute hc gives which If transport Parallel 10 the called sometimes is (ii) eotndenote often We eetteaoeeecs o oncinwt oso ofidta h connec- the that find by to given torsion now with are connection coefficients a tion for torsion exercise above with the connections Repeat metric-preserving Example: 9.2.1 γ h diinlterm additional The o ae( take Now sacregvnby given curve a is ewieottecvratcntnycniintretms cyclically times, three condition constancy covariant the out write We : Γ b e c = i a ( ( ∇ ) ( iii 2 1 ii b i − ∇ a ) g c ) ) g es ( b favector a of ed ocancellations: to leads cb ii V ¡ 2 Γ ) ∂ g 0 = a b ∇ b ea − ∇ ∇ by g otrintensor contorsion e a Γ sc x c ( c , b K g g iii b a g V bc = ab + ( b ca e λ n s h ymtyo h oncint get to connection the of symmetry the use and ) a e c = ∂ and ) 1 2 = = c ; + c V b DV g h erci oainl constant covariantly is metric the g DV ∂ Dλ = ∂ es Dλ n so and ∂ a ∂ sb a c b a g ¡ T g along g a − g bc ∂ ab a b ca T bc b = e ∂ g − = a − − c − sc 31 s T g − = Γ V Γ Γ γ bc ∂ + b a . c a ∇ c b T ¢ e dx by g e ∂ dλ ; e bc b b + ab b a c c V a T g g g e g ec T sb − b eb a ea t agn etrw en the define we vector tangent its − . b . − − ∂ − e − T b c cb Γ g ∂ Γ Γ − ac a s c e b g e e T bc e 0 = c b bc a g ¢ g g eb ea . e ec , − 0 = 0 = 0 = T cb . . . . e . (166) (165) (167) (169) (172) (171) (168) (173) (174) (170) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hs r uvsaogwihtetnetvector tangent the which along curves are These curves Autoparallel 10.1 ehave we nta au ftevector, the of value initial ilhv ieetvectors different have will along n o ysetting by so, and fa of htis That that say We or along r infinitesimally, or, tdpnson depends it aallcreas aesnefran oncin hc ar which connections affine for sense make also curve parallel 9 oeta ecudhv eaddteaprnl ekrcondition weaker apparently the demanded have could we that Note o h eiCvt oncin Γ connection, Levi-Civita the For lhuhw o’ s hsfc nti ore h oin of notions the course, this in fact this use won’t we Although geodesic γ γ hsi ierode along o.d.e. linear a is This . o oefunction some for . V γ a n w curves two and is g g aall rnpre along transported parallely ˙ = f eget we , V f V d dλ dV ( 2 a dλ gU a ˙ λ x 0.However (0). dV 2 ,btif but ), tteedpoints. end the at a a γ a DV Γ + ( + a and Dλ + = V DV DU DT g T Dλ Dλ a Dλ b a − DU γ b a a Dλ = = Γ ′ Γ 32 a a c b a onn h aetoeet nspacetime in events two same the joining b b dx γ c f gU dλ a 0 = 0 = 0 = a a ( = n a nqeslto ie the given solution unique a has and c λ = e c aalltasoti ahdependent path is transport parallel a © V ) ) dx . . dλ , V V a fgU e γ b c b dx a c 0 = 0 = ª b if a T , ercvroroddefinition old our recover we a , . o eesrl symmetric. necessarily not e sprleytransported parallely is aalltasotadauto- and transport parallel (181) (179) (178) (180) (175) (183) (182) (177) (176) 9 , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. where − n ecnnwueorpeiu eakt set to remark previous our use now can we and efind we where nti aeNwo’ a od nteform the in holds law Newton’s case this In o some for oainl constant, covariantly where n egtbc oorpeiu odto.Sc hieo aaee scalled is parameter of choice a Such condition. previous our an to back get we and oc utb rhgnlt h 4-velocity the to orthogonal be must force hste4vlct n ceeainvco r rhgnl npriua the particular in orthogonal, The are spacelike. vector is acceleration vector and acceleration 4-velocity the Thus 02Aclrto n Force and Let Acceleration 10.2 λ by ˜ 1 uhthat such . ffieparameter affine Differentiating oeta ecudhv eaddteaprnl ekrcniinthat condition weaker apparently the demanded have could we that Note τ epoe ieaogatmlk curve timelike a along time proper be U m a f sters aso h atce hc ewl suei constant. a is assume will we which particle, the of mass rest the is = ( λ .Hwvri ecag parameter change we if However ). dx dτ a sthe is g ab n ti nqeu oa ffietransformation affine an to up unique is it and Dg U Dλ 4-velocity a ab U b λ ,w n that find we 0, = ˜ g along 1 = → = DT a d dλ d λ Dλ F λ ˜ n snraie othat so normalized is and λ ( ˜ a 2 → T mU a D + dτ a U , a D g a T = ab a 4-force ˜ ,a b a, b, λ ˜ = λ ˜ = a a a 33 U a = ) DU γ f g Dτ 0 = b T 0 = = ( ˜ T λ ˜ ˜ λ n eebrn httemti is metric the that remembering and 0 = a ( a a ) F γ λ , = , T F F . , hnthe then , ) a . a , a a , dx , d g g ∈ ˙ = sdfie by defined is λ ˜ = a R ma , . f a n n e parameter new a find and n xrsinfrthe for expression Any . 4-acceleration U a U a = g ab sdefined is U a (189) (186) (187) (185) (192) (184) (190) (188) (193) (191) U b = Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. h erci the in metric the hc sago oa oe o h eaiu ftemti eraKlighorizon Killing a near If metric later. the of see behaviour shall the we for model as local good a is which inis tion equation x eaiitcrceshv aiberest-mass, variable have rockets Rockets Relativistic Relativistic Example: 10.2.3 4-velocity. the to orthogonal indeed is stescn akatsmerccovariant antisymmetric rank second the is oc eddt eptepril rmfligrdal nad,dvrea the at diverge inwards, radially falling from particle the keep horizon to needed force o atceo charge of particle Particles a For Charged Example: 10.2.2 nteShazcidmti,ti equals this metric, Schwarzschild the In where ntecs f2dmninlMnosispace Minkowski 2-dimensional of case the in atcea eti ttcmti has metric rest static at a particle in rest a at of particle acceleration A The Example: 10.2.1 hste4aclrto atcea eti ietdradially directed is rest at particle a 4-acceleration the Thus antd qae sgvnby given is squared magnitude 1 direction. Curves ocekta hsi esnbe ti ot okn hog hsexample this through working worth is it reasonable, is this that check To r 2 = ρ osathv acceleration have constant = M . ide wedge Rindler e x U F oigi neetoantcfil ehv the have we field electromagnetic an in moving ˙ 0 ab a = U ˙ = a ds ρ m U = sinh g ˙ 2 ac DU a x DmU Dλ = 4( 1 = F Dτ ,x t, g > − c a 1 − 44 b ρ | = ( 34 ) x a 2 = ∂ 2 dt 2 0 U 1 eF i h = | g − − g M 2 a 44 ij r is 44 4 2 F J 1 2 + a ρ 1 = ∂ M r aaa tensor Faraday ba a , m b = i n hsi lal ntepositive the in clearly is this and dρ g 0) U , h ceeain n ec the hence and acceleration, The . √ 44 E = ρ = b . 2 | , ∂ 1 1 g cosh . F , 44 j m 1 g [ | ab 44 ( naclrtn coordinates accelerating in δ τ ] 4 a t, , . .Tereuto fmo- of equation Their ). n 4-acceleration a and h oet 4-force Lorentz the , outward Lorentz n its and (197) (196) (199) (195) (198) (194) (200) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. h ttoayosre hsaqiigavelocity a acquiring thus observer stationary the utb timelike, be must optr h eodosre,iiilya etwt epc otefis observer first the a to using respect primes with two rest of at v initially sum observer, the second is The number computer. even every that conjecture Goldbach’s where edsco h metric the of geodesic hr aclto eel that reveals calculation short A uoew aetocnomlyrltdmtissc that such metrics related conformally related two have conformally we Supose of connections Levi-Civita The 10.3 expended. fuel the of mass or where eti rprtm eursalwrbudo h oa aso h ulused fuel the a of over mass total set-up, the Twin-Paradox on the bound in lower a as requires acceleration, time certain proper a certain obtain to Thus (1+ ulgoei ftemetric the of geodesic null a d dλ initial 2 x 2 osdrtoosres n fwo sa etadeggdi checking in engaged and rest at is whom of one observers, two Consider o,gvnacurve a given Now, o,i eea,if general, in Now, ntodmninlMnosispacetime Minkowski dimensional two In a z + .Teices ntert fgi fifraini one yteenergy the by bounded is information of gain of rate the in increase The ). ,dcdst s iedlto ofidotfse yaccelerating by faster out find to dilation time use to decides 0, = θ J © a sthe is metrics b a stert feiso f4mmnu fteeet.Physically ejecta. the of 4-momentum of emission of rate the is c ª ′ rapidity dx dλ © b b m a J m dx dλ a c initial final ª J c ′ a g U efind We . g = ab = γ < x ab ′ a , a < dx © oee hr sa xeto.Spoigthat Supposing exception. an is there However . (cosh = d dλ x ,wihlast h inequality the to leads which 0, ( dλ b 2 λ a r ln( a a x ( sagoei ftemetric the of geodesic a is ) 2 g λ a dx c dλ ab 1 + 1 ª ehave we ) m + m − b + hnw have we then initial © g final 0 = θ, v v ab b ′ δ m m initial initial ˙ b a a sinh Ω = ∂ c > ) ⇒ Ω ª c 35 Ω < dx | dλ r a θ g 2 + Z ) a g ab ′ b + 1 1 ab | | ⇒ dx . δ dλ dx | − dλ c . a E a c ∂ a a v v 1 a Ω | b +2 , final final dτ. a dx 1 Ω dλ | v = b − dx final dλ = 0 = g dτ dθ a as + 1 Ω Ω ˙ n a and , , ∂ 1 − Ω s g Ω z g ab g as . bc leshift blue twill it ∂ Ω s Ω g ab not dx dλ towards a factor (204) (206) (207) (202) (208) (205) (203) (201) ea be dx dλ γ J b is a . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. u now But adsd hti sas ulgoei ftecnomlyrltdmetric related conformally the of right the geodesic of null vanishing a the also from is deduce it we and that vanishes side side hand hand left the then hsi,i at pca aeo u rvoseapewt = Ω with example previous our of case special a fact, in is, this 3-metric n oif so and tflosthat follows It Thus o nrdc e pta metric spatial new a introduce Now efudta h qaino oino htnis photon a of motion of equation the that found we metric related so and 04Sai erc n emtsPrinciple Fermat’s and If metrics Static 10.4 rays. light with solely made measurements not. of is parametrized affinely being statement, invariant conformally a n has One Since hsclytocnomlyrltdmtiscno edsigihdb means by distinguished be cannot metrics related conformally two Physically d γ dλ 2 λ x x etil a ultnetvco ihrsett h confomally the to respect with vector tangent null a has certainly 2 a a sn ogra ffieparameter affine an longer no is ( λ + sanl edsco h metric the of geodesic null a is ) x g © © ab λ ′ b j sannanl amtie edsco the of geodesic parmetrized non-affinely a is Now . a i d dλ c k 2 d dλ ª ª x 2 2 ′ ′ x i ds dx 2 dλ = i + ds o 2 + b © h ds = 4 2 dx j © dλ © = 2 i j j e c k i − i = e ª k = k 2 2 ª ª h U − U − ′ ij ′ d h + h e dλ dx δ 2 dλ ij 2 − = j i x U h dx 2 ∂ j a is dt dt k e dx 36 dλ + U h i − 2 2 dx jk 2 k + © − + U ∂ j b hswiebiganl edscis geodesic null a being while Thus . 2 = h h s δ h a = ij U k i ij ij ′ c ∂ i . ª dx h dx dx j dλ g dx ij ′ U dx dλ dλ ab i i i dx dx dx j ∂ + b with k dx i dx j dλ j dλ h dx U i is k , c dx j dλ ∂ . 2 + 0 = λ s k Uh nan parameter, affine an . dx . dλ jk pia rFermat or optical a . Ω Ω e ˙ U , . (216) (209) (215) (214) (213) (211) (210) (212) g ab ′ . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 374-376. eto eea eaiiyuigrdolnswt h Cassi the with links radio using relativity general of test A h icmeec facrl fcodnt radius coordinate of circle a of geodesics circumference null The Circular Example: 10.4.2 rmltri h course. the in later orem and eeattm ea is delay time relevant n peial ymti -erci ofral flat conformally is 3-metric symmetric spherically Any Schwarzsc the for metric optical and coordinates Isotropic 10.4.1 h ffcierfatv index refractive effective The at)t n ati ude thousand t hundred opposition or a in waves in was part radio it one when with satellite to compared Cassini earth) the down the (using called slowed recently is verifed thus effect been are This away. hole keep black which light or star a near hc maps which tagtln niorpccodntsjiigteerhadtesatellite. the and earth the radii joining at coordinates are isotropic in line straight an duces n prahstehrznat horizon the approaches one h a pc eutmyb bandfo h ieitga 2 integral line with the compared from delay obtained time be the may accuracy, result relevant space the flat To the back. and satellite the ercbecomes metric 11 10 h -ercisd h rc sflt hsteotclmti so h form the of is metric optical the Thus flat. is brace the inside 3-metric The hs neetdcnlo pteatceo h e BBertotti (B web the on article the up look can interested Those o hudb bet rv hswe o aeudrto h p the understood have you when this prove to able be should You n ds ( x 2 sa ffciesaedpnetrfatv ne ie by given index refractive space-dependent effective an is ) = ouin hpr iedelay time Shapiro solution: isotropic − r ∞ 1 ³ and + 1 1 > ρ > − coordinate r M M 2 2 2 ρ ρ ´ epciey and respectively, M 2 2 dt to 2 + ∞ ³ n n ρ 4 r + 1 ( ( > r > ds GM r by x x c = 3 nrae rmuiya nnt oifiiyas infinity to infinity at unity from increases ) = ) 2 = o 2 ρ M 2 2 = µ πn ρ ln ,ρ M, ¡ n + 1 ´ 2 ¡ 37 b + 1 µ ( 2 1 M 4 ρ ( stedsac fnaetapoc,the approach, nearest of distance the is n 4 − x ) 11 dρ r ρ. M 2 b ) na11wy hnteSchwarzschild the then way, 1-1 a in = 1 2 ρ 2 d 2 M ai aei etfo at to earth from sent is wave radio A . | r M 2 | x x ¶ x 2 | M + 2 | 2 ¶ ¢ hpr iedlyeffect delay time Shapiro 2 ¢ , 3 ih rrdowvspassing waves radio or Light . , ρ . . 2 ¡ ρ ispacecraft, ni dθ ete nteoii is origin the on centred 2 10 sin + nfc,i n intro- one if fact, In . esadPTorora, P and Iess L , ofo iko’ The- Birkhoff’s of roof 2 R γ θdφ Nature ndl 2 ¢ where o 425 n has and fthese If (2003) γ (221) (219) (222) (218) (220) (217) the o hild sa is Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hc vr a trigfrom starting ray every which thstermral rpry(biu rmtedsrpini em fthe of terms in description the point’ from ‘object every (obvious for property that remarkable sphere) the has It h ihdge fsymmetry of degree high the naln hog h rgnwihcuts which origin the through line a on n nfc,eeygoei a etogto steitreto fteshr with sphere the of intersection the of as origin of the thought through be 2-plane can a geodesic every fact, In ntecs fteShazcidmti h icmeec a iiu value minimum a has circumference the metric Schwarzschild at the of case the In 043*xml:Seegahcpoeto,nl edsc in An geodesics null projection, Stereographic *Example: spacetime. in null curve the 10.4.3 closed although a geodesic, not null course circular of a is context itself this geodesic in called metric, optical S projection stereographic osrie osatisfy to constrained i aea nigtelgtry namdu frfatv index refractive of medium a in rays light the finding as same where with hc ok u ob ie by given be to out works which medns alit call embeddings, 1 = n iesoa ulda pc hs coordinates whose space Euclidean dimensional 1 + ρ If oeta hr sa is there that Note h ercisd h rc sfltadtemptaking map the and flat is brace the inside metric The is n = , n -sphere n ofral flat conformally 2 d i n , . . . , M n Ω ,a pia eieo hssr scle a called is sort this of device optical an 3, = i remnLmir nvre n awl’ s y lens* eye fish Maxwell’s and universes Friedman-Lemaitre n 2 1+ (1 ,i.e. 1, = − 1 S emyset may We . stemti on metric the is q n 3 4 fui aismyb endb t medn into embedding its by defined be may radius unit of ,i.e. ), d n Ω h rbe ffidn h edsc on geodesics the finding of problem The . i ds n 2 iclrgeodesic circular ∈ sgvnby given is , n scerycnoml nohrwrstemti on metric the words other In conformal. clearly is and X 2 r S = d 3 = 0 d n Ω cos = Ω − SO 1+ (1 n 2 ( M 1 n 2 S x X = e n n E ( = n 0 and ( 4 0 − n hscrepnst iclrgoei fthe of geodesic circular a to corresponds This ( dχ passes. ρ n ) ,X χ, x 1 2 +1), +1 2 dX fw set we If . = ) x ) 2 ( + 2 e vr itntpi f2pae intersect 2-planes of pair distinct Every . sin + © 38 0 ≤ hr sasnl iaepoint’ ‘image single a is there dρ X ) + 1 S every flnt 2 length of 2 χ n i 2 ( + ) 2 2 i 2 + tatpdlpoints antipodal at ≤ x χd = 1 = 2 dX ρ edsci lsdwt egh2 length with closed is geodesic ρ π . Ω n 2 d tan = i n 2 i , Temti nue rmthe from induced metric .The Ω ) sin − 2 n 2 π 1 , − , χ, at 2 ª χ 2 awl ihEeLens Eye Fish Maxwell X | efidthat find we , . x 0 | ,n χ, X , ,adhnefrom hence and 1, = 1 i X , to X S α 2 ,n ρ, n X , . . . , and x stu the thus is o i scalled is through − closed E X (228) (227) (223) (226) (224) (225) n n α +1 are π . . , . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. tagtlnst l tagtlnsbtw edt atrb h cinof action the by factor to need we but lines straight all to lines straight coordinates X h rgnwt h yepaeΠ cigwith Acting Π. hyperplane the with origin the by a,wl aesrih ie osrih ie.Hwvr some However, lines. straight to lines straight take will way, etvlyequivalent jectiveley banamr ymtia itr,oead xr onsa nnt oΠ One Π. to infinity at points extra all adds al almost one at while intersect picture, defines not Moreover symmetrical do more ‘infinit pararallel Π. a are at which obtain hyperplane some lines the once, intersect straight intersect lines straight not to do lines which straight 2-planes take will transformations where bu h parametrization. the about scalled is Universe ihtesm ffieprmtr tflosthat follows It parameter. affine same the with aallpaths. parallel where hscneti iia ota fcnomleuvlneecp htw focus we that Γ except connections equivalence affine conformal linear two of say that We to auto-parallels. similar on is concept This Equivalence* *Projective Lens. Eye Fish Maxwell 10.5 a in would they as precisely behave universe the o oec-etrfield co-vector some for n l asicdn rhgnlyo hspaa ufc ilb ousdont focussed be will surface planar this at on axis orthogonally the incident rays all and oml 28 nietehmshr.Tepaa aeo h esi at is lens the of face planar The hemisphere. the inside (228) formula t daie om eset we form, idealized its 12 α X lblyoesol consider should one Globally A h ai xml scntutdfo tagtlnsin lines straight from constructed is example basic The o osdrthe consider Now eae yeo es sdfrsm aa ytm,i the is systems, radar some for used lens, of type related A ti ovnett en aha h mg facre ..to i.e. curve, a of image the as path a define to convenient is It dnie uhthat such identified 0 rjcietransformation projective a .Srih ie orsodt h nescin f2pae through 2-planes of intersections the to correspond lines Straight 1. = RP ( t ofra time conformal sthe is ) n x X 3 stesto ie hog h rgnin origin the through lines of set the as α . 1 = , α cl factor scale 0 = fte hr h aeatprle paths autoparallel same the share they if k , remnLmir metric Friedman-Lemaitre 1 = 1 Γ A hsfo u okaoew e htlgtry in rays light that see we above work our from Thus . n , . . . , ( X n b b ds . c α usd h hemisphere the outside 1 = ds d 2 ) hsmti scnomlt h isenStatic Einstein the to conformal is metric This . ≡ 2 = = ecntikof think can We . or = RP λX a Γ ˜ 2 collineation − ( dη ( b t α n dt c ) odsrb hs nrdc homogeneous introduce this, describe To . , © = d 2 39 − ) λ + + a dη dt .Clearly 0. 6= ( a t δ 2 2 ) b c ( A + t ) ae uoprle ah oauto- to paths auto-parallel takes d d d + Ω Ω R 3 2 3 2 SL δ , ª n d c , A stehprln given Π hyperplane the as ( b R n , n x ho wyteinformation the away throw GL 1 + b +1 3 c > d ( ..( i.e. , R , n R and 12 n 0, ueuglens Luneburg 1 + . nteobvious the in ) o necessarily not , | x Γ ˜ SL | , n b R n use and 1 = 1)-tuples + c ’ ..to i.e. y’, ae all takes ) ( d n are R 1 + x 3 \ (230) (231) (229) (232) .To l. pro- to 0 0 = In . , R o ) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 051*xml:mti rsrigcnetoshvn h sam the having connections preserving metric *Example: 10.5.1 ercpeevn oncinwt torsion with connection preserving metric A n ae.Teetn owihi al od oi esrdb the by measured is so do path to fails the it upon which depends to it Tensor extent Curvature non-commutative; The is takes. transport one parallel general, In Curvature 11 h eiCvt oncini n nyi h oso sttlyantisymm totally is torsion the if only and if connection Levi-Civita the ne hs2 this Under edsc on geodesics in sn h xrso o h uvtr esradtedfiiino h covariant the of definition that the check may and one tensor co-vectors curvature for the derivative for co-vectors expresion for the Identity Using Ricci The 11.0.2 rgn oee,at-oa onson points anti-podal However, origin. hoe oso that show to Theorem nismercin anti-symmetric eiaie of derivatives itntsrih ie neetoc n nyonce. only and once intesect lines straight of distinct action effective an get esethat see We con- by field tensor a is identity Ricci that the fact of the side using righthand Now the struction. is point The the as known sometimes formula a efidthat find we RP Clearly, ene ocekthat check to need We n sw aese,b osdrn awl’ es vr itntpi of pair distinct every lens, Maxwell’s considering by seen, have we As . geodesics* S − R S n apn ra ice,ie edsc on geodesics i.e. circles, great mapping 1 V n c ⊂ dab a nesc wc.On twice. intersect ti ierin linear is It . hc a edfie by defined be may which , R a R n eed nyo Γ on only depends and c +1 dab ( R ∇ ( a empe notesto ietostruhthe through directions of set the onto mapped be may ∇ c b SL P a = : dab ∇ a R ∇ ∇ ∂ b c a a b ∇ − dab V side tensor. a indeed is ( Γ R n ∇ − c b c 1 + c T = sa is dab b d abc ∇ V b ic identity Ricci ∂ , Γ + ∇ RP a d a R = ¡ ) = V a 40 W a 3 1 .Oemyas hc hteeypi of pair every that check also may One ). S n hrfr ecnueteQuotient the use can we therefore and ) − ¢ a c n V b T n c c Γ + c R esr htvri s ti obviously is it is, it Whatever tensor. [ c hs w nescin r identified. are intersections two these abc = utb dnie since identifed be must e c n t rtdrvtvs u o on not but derivatives, first its and = Γ ftecneto ssymmetric is connection the if dba ] − b a T . R e c R a d c . e b d V dab − c . cab e spoetvl qiaetto equivalent projectively is ( , V a W d ↔ d , S . n b ) a osrih lines straight to map . X α Riemann − ≡ etric (233) (238) (237) (235) (234) (236) then X α e . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. (ii) and are (i) hs r uhta taycoe on nspacetime, in point chosen any at that such Coordinates are These Inertial Local 11.1 loca introducing by seen easily most are coordinates. which inertial symmetries extra imposes This satis which connection In Levi-Civita the not. in or interested symmetric mainly connection, are we affine any course for this work above given definitions The ycnrcinw a en the define can we contraction tensor By Ricci The 11.0.3 r equivalently, or, eodrn oain tensor covariant rank second a hl n ene ofi nytefis n eoddrvtvsof derivatives second and first the only fix to need we find shall to eesr oriaetasomtostkn st ˜ to us taking transformations coordinate necessary esato nacodnt system coordinate a in Coordinates off Inertial start Local We of Existence 11.1.1 x hr sa nlgu xrsinfra rirr esro type of tensor arbitrary an for expression analogous an is There ne oriaetasomto ehave we transformation coordinate a Under oa nrilcodntsaeas called also are coordinates inertial Local p r qiaety of equivalently, or, em ihpstv inand sign positive with terms ( ∇ a ∇ Γ ˜ a b g ab ∇ − b x c 0 = (0) = ihrsett ˜ to respect with b R ∇ ∂x ∂ db x a ˜ η g ˜ ) e b Q = ab ab Q h ∂g Γ cd ∂x cd ∂ ∂x ic tensor Ricci R ∇ diag(+1 = = a q x n has one , ˜ ab a c c a = em ihngtv in npriua,for particular, In sign. negative with terms a g g dab x g b ¯ ¯ ¯ cd ∂x ∂ x 0 0 = (0) bc 41 a − =0 x ˜ ∂ ∂x R hc sntieta n ov o the for solve and inertial not is which = d 0 = c x ˜ Γ x e 0 = a c − g cab tteoii 0. origin the at . ∂x ∂ imn omlcoordinates normal Riemann , R e . +1 x , ˜ d Q a d b + dba ed , +1 ∂ − . ∂ x ˜ x , R a 2 a − x ∂ e x )(242) 1) e x which ˜ dab a c say, 0 = i Q ce are . x ˜ nril We inertial. ihrespect with ¡ q p ¢ there : . (245) (241) (240) (239) (243) (246) (244) fies l Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. nyi nrilcodnts sn Γ Using coordinates. inertial in only coordinates nlclieta oriae tapitw have we point a at coordinates inertial Identity local Cyclic In The Consequences: 11.2.1 u hscicdswt rprtm ln ieiegoei with geodesic timelike a along time proper with coincides this But and g pctm.Tu foewie onteeutoso unu ehnc de- mechanics quantum of of elapse equations the then the rest, down at clock writes atomic one an scribing if Thus spacetime. n thus and aoihtelcleeto rvt ypsigt reyfligfae I frame. falling freely to becomes a one equations to allow geodesic passing they the by that coordinates, gravity is inertial of coordinates effect inertial local local ‘abolish’the of significance physical j The coordinates, inertial local of significance Physical derivatives. second the 11.2 for equation linear a solving involves then second The hc a lasb oepoie h metric the provided done be always can which n ogoeisaesrih ie olws re,js si ikwk spa geodesic. Minkowski timelike in any as just order, lowest to lines straight time. are geodesics so and ˜ ab 0 = (0) ehv nrdcdtenotation the introduced have We o etitto restrict Now oeta oa hsc nlclieta oriae stesm si Minko in as same the is coordinates inertial local in physics local along that coordinates Note inertial local introduce may one that showed Fermi fact, In o ecnpick can we Now Γ ˜ a icto ftecokpostulate clock the of tification b η c ab 0 = (0) and Γ ∂x ∂ ˜ x a ˜ x e b d R b ( ( dτ ( iii ¯ ¯ ¯ .W have We 0. = 2 ii c i d x ∂x ∂ x 0 0 = (0) ˜ ) =0 ) 2 x ˜ cab ) g ˜ a d b ab x ˜ + ¯ ¯ ¯ h x a 0 = (0) + ∂ © ∂x =0 ( x R τ b ˜ R R R ˜ = ) a a g . d d and d d ¯ ¯ ¯ c h rtcniinivle diagonalizing involves condition first The cab x bca ª bca abc g =0 x d cd dτ ′′ a x ˜ ′′ ′′ + ∂x ∂ (0) ∂ 0 + (0) b = x ∂ = ˜ x ˜ = d ′′ dτt a R 2 d ′′ c a 42 ′′ x ∂ x ∂ ∂x ˜ ′′ ¯ ¯ ¯ d = e x ˜ b c x x ˜ ∂ ∂ c abc ∂ c =0 ′′ a c a U = 0 = ¯ ¯ ¯ c b Γ x ¯ ¯ ¯ Γ Γ = Γ oidct hta qaini true is equation an that indicate to a x Γ =0 b = 0 = a (0) =0 c g d d d e c sw ih ed ot make to so do We wish. we as b ∂x ∂ τ a c d − − x d b ˜ g 0 + (0) − + dτ x d 2 b ab a ∂ 4 R ∂ x ˜ ¯ ¯ ¯ ∂ . . . b 2 x a eddc hti inertial in that deduce we a swa h lc measures. clock the what is c d Γ a h orc signature. correct the has =0 Γ Γ [ a + abc c b ∂ d d d ∂ x ˜ . . . c ] b a 2 a . x ∂ e x ˜ c ¯ ¯ ¯ x =0 x i i constant. = local n g (253) (252) (251) (254) (247) (248) (249) (250) ab us- wski (0) ce- Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. o oeco-vector some for hs rsreage ne aalltasotbtnttemti.Thus metric. the not but transport parallel under angles preserve Connections* These *Weyl Example: 11.2.4 o sn h qaiyo ie atasw get we partials mixed of equality the using Now o oeone-form some for hs locally, thus, elcneto fta yei ahrtiilbcuei a eeiiae by trivi contraction eliminated second of be the measure can is gauge-invariant it sense A because this trivial rather ‘gauge’transformation. is conformal transpo type a parallel that under of invariant is connection which Weyl metric A related conformally a find can h eodBaciiett gives identity Bianchi second The dniytlsu htti sntindependent: not is this that us tells identity o eea ymti oncinteRcitno a opriua symme- particular no has tensor Ricci Let the connection tries. contraction* symmetric general possible a other For The *Example: 11.2.3 the drop can we Now oeta,if that, Note nlclieta oriae tapitw lohave also we point a at coordinates inertial Identity local system Bianchi In coordinate The all in Consequences: true is it 11.2.2 system, coordinate one in true is it if hence, S ab g ab = Γ ( Ω = R − ( iiii ( b ii A i S d S c ) ) b a cab ab d ) 2 . Thus . g ˜ = ′′ ; ab S = e db © then , + R = R R b R d ′′ = d R c e d cab d cea d eab eas h qaini esra eain and relation, tensorial a is equation the because cbe R ª S cea S ; bd a ; e − A ∇ b ; bd a adb ′′ eteohrpsil otato.Tecyclic The contraction. possible other the be ′′ ; a a = b ′′ S 1 2 = g = = + 2 = [ ¡ = bc bd ∂ = ′′ δ ′′ R b R ′′ b c ; 43 = c S ∂ A n bd ∂ ∂ ] d d ∂ e a Ω ¡ b 0 = d A cbe ∂ a Ω ∂ ∂ − − + a ∂ d a e ovrey if Conversely, . Γ ; b Γ A g ∂ R a , δ Γ bc b a d b d db c = 0 = e S d d , A − d . b c c b , c − ∂ − − − b A ∂ ∂ g R ∂ e b d db ∂ a ∂ ¢ d b ∂ g a . c Γ e Γ ce [ ab Γ a e A b d ; A d e e d c ] c a ¢ . c . 2 = ∂ a Ω Ω hnwe then , lt in ality (261) (264) (257) (262) (260) (259) (256) (255) (258) (263) rt. s. Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. h ylciett n h inh dniyhl for hold identity Property Bianchi Preserving the and metric identity the cyclic The of Consequences 11.3 .hsi ossetwt u icsinof discussion our with consistent is 0.This where u hti twr o-rva,te h esrn oswudntrtr to return not would rods measuring the to returning then on non-trivial, size were original it their if that out now ic esr.Hwvr sn omlcodntsoemyso httefol- the that W show may one coordinates and normal tensor Riemann using lowing different However, have connections tensors. equivalent Ricci projectively two general Curvature* In believed Projective now Example:* is It sense. this Ψ. 11.2.5 in function gauge-invariant wave exactly its are of physics phase of the equations of the change that a in results now R r aefo tm n tm aeqiedfiiesizes. definite quite have atoms and atoms from made are icvrdadi a elzdta ntepeec fa lcrmgei field, electromagnetic an of presence was the the mechanics length, replace in quantum should of that when one units realized was after of it soon independent and of resurfaced discovered gauge-invariance sense of the in idea gauge-invariant conformal be a should by eliminated be would could metric which the change hence rescaling. conformal and radius, trivial Bohr a the undergo transport parallel under position, h rjciecrauesmlfisto simplifies curvature projective the connections. both for same pctm ihwa enwcl elcneto nwihmauigrods measuring which in connection Weyl a call now from moved we what with spacetime eto.T aefrhrpors ems s h ercpeevn condition. preserving metric the use must we progress further make To nection. a a abd vni h hreo h electron the on charge the if Even radius Bohr the from constructed are time and length of units atomic fact, In lhuhWy aiiyaadndhster htteeutoso physics of equations the that theory his abandoned rapidily Weyl Although ntecs fteLv-iiacneto,teRcitno ssmercand symmetric is tensor Ricci the connection, Levi-Civita the of case the In hsasaeo osatcrauewith curvature constant of space a Thus elsgetdta n ih eeaieEnti’ hoyb endowing by theory Einstein’s generalize might one that suggested Weyl bcd A b | = b = steeetomgei etrptnil aalltasoto nelectron an of transport Parallel potential. vector electro-magnetic the is | S en htteindex the that means R bd a A bcd aihs hscnrdcsteosre atta esrn rods measuring that fact observed the contradicts This vanishes. to + ( B n ∂ 1)( + a ln path a along by Ψ W 2 n a bcd ¡ − ∂ a 1) = b − R δ soitdfo h nismerzto,sthe anti-symmetrization,is the from omitted is R [ a B d i γ A R a ˜ e = e A ue aalltasot isenpointed Einstein transport. parallel suffer bcd c ysm te uv nestecurvature the unless curve other some by ] b rtems fteelectron the of mass the or 44 a 4 + ´ m + πǫ nteSh¨dne qain where Schr¨odinger equation, the in Ψ ( e n 0 n S e ˜ 2 n 1)( + − 2 2 R . and 1 2 a δ n [ a bcd n d RP R − c = ] any 1) b n . . δ K δ [ a d ymti ffiecon- affine symmetric R [ a d | b g | c c ] ] b − m has n e 1 + aidwith varied 2 W δ a b a bcd R (265) (266) (267) [ cd = ] , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. h ih adsd aihsadhence and vanishes side hand right The n ntescn aro nie ooti h result. the obtain to indices of pair second the on and pairs and n ple h h ic dniyfrc-esr 29 otemetric the to (239) co-tensors for identity Ricci the the applies One works. it that check whence eoddrvtvsdtrieadaedtrie ytecrauetno.One tensor. curvature the by determined are has and determine derivatives second foede o iht s nrilcodntsoemypoeea follows. as procede may one coordinates inertial and use (268) to of wish side not hand does left the one into If (273) substitute to exercise useful a is It n o ae ( takes now One oetbihthat establish To nlclietlcodntstefis eiaie ftemti aihadthe and vanish metric the of derivatives first the coordinates inertal local In tensor Ricci the that follows It that deduce We that note to is proceeding of way quickest The ab and R ef abcd ∇ n a aefu oiso h ylcidentity cyclic the of copies four take may one e ′′ ∇ i = f ( + ) g g R ′′ ( ( ( ab ab ( iii iv ii abfe i ∂ 1 2 ) ) ′′ e ) ∇ − ) ii ¡ ∂ = ∂ ) f b g ssmercwt epc osapn h index the swapping to respect with symmetric is ′′ − ∂ f ab R R c η R R ∇ R ( g ab abcd abfe iii R = badc ad dbac R e cbda g abcd − abcd ) R − ab + R − + ab + + 1 3 + 3 1 ab + ∂ = R ¡ = ( R 45 a R = = R R iv R R ssymmetric is ∂ − acbd adbc bafe bdca R dacb d cabd aebf n h kwsmer ntefirst the on skew-symmetry the and ) R R R g [ cdab ba ab bc g x + ] aef + c 0 = + + cd − + x R R d R R ∂ R g acdb . + b bcad gb dcba cdab afbe ∂ d O − g (276) 0 = ( ac (277) 0 = ´ (279) 0 = (278) 0 = R x . 3 g − ) bef , ∂ a g ∂ ag c g . bd ´ (274) (273) (275) (271) (270) (269) (268) (272) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. is n aeue h ymtiso h imn esrt give to tensor Riemann the of symmetries the used have and n ec 21 utb reeeyhr on everywhere true be must (281) hence and u,by But, trorpi oriae nteui n-sphere unit sphere* the the on of coordinates curvature Stereographic The *Example: 11.3.1 rdfiigteEnti esrby tensor Einstein the defining or otatn h inh identity Identities Bianchi Bianchi the Contracting Contracted 11.4 . r imn omlcodntscnrdo h origin. the on centred coordinates normal Riemann are hr edfiethe define we where h ercis metric the on a d ecntd pti xrsint give to expression this up tidy can We rmti n eue hta h origin the at that deduces one this From o otato with on contract Now then and SO a gives ( n )ivrac,teoii sntapiiee on ntesphere the on point privileged a not is origin the invariance, 1) + R ic scalar Ricci = R R g d g R ab cab abcd ce R R cb R ³ G abcd c ; ; ds oget to e e ab R b ab R = ; + − 2 ab c = by a − 4 = = = R R a G − b 1 g 2 d ce R = g R ab ab ¡ 1 2 cea ac ; ; g 1+ (1 b ab b R g 46 R ; ac b g + + ab ; e dx a b bd − g 0 = aeb R R bd + cbe R x i − 1 2 d ´ dx d k R − . g g ; = cbe x g b b S ab ce d ; i ad k d g 0 = n g cbe ; ) . ad R, d S (285) 0 = g 2 ab ihui ais fteradius the If radius. unit with bc (284) 0 = n g ; a g bc . cd nrdcderir nwhich in earlier, introduced (283) 0 = ¢ R . acbd (287) (286) (281) (289) (290) (288) (282) (280) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. Λ wihse ob ceeaigaa rmu)ado h omcmicrowave cosmic the of but length and small are us) very Λ constant from is cosmological away Λ that accelerating indicate be background to seem (which rpossibly or oeta fΛ if that Note ehv o eeoe ucettno nlsst banteEnti equa- Einstein the obtain to analysis fact tensor In sufficient tions. developed now Equations have We Einstein The 12 u since but hc mle that implies which 15Smayo h rpriso h imn Tensor Riemann the of Properties the of Summary 11.5 ihΛthe Λ with ,te pctm antb flat. be cannot spacetime then 0, 6= eakbyvr eetosrain ftercsino itn galaxies distant of recession the of observations recent very Remarkably Thus g ab omlgclConstant Cosmological g nvacuo in ab ,w antput cannot we 0, 6= = ( δ nadto ( addition in i ( a a ) iii ,cnrcino 26 with (296) of contraction 4, = ( ) ii (Λ ) g R ab abcd R ) ; R abcd b iii Λ Λ = abcd = Λ R ) ≈ ; ; ab R a h sΛacntn?W nwthat know We constant? a Λ is Why . e − − ⇒ R ab + ; (10 = + ; R a 2 g b ab ab ³ abdc = R n roughly and Λ = = ∂ R 47 27 R a 0 = abec acdb = 0 = Λ 1 2 1 2 cm) ab (4Λ) g = ∂ η , ab ; − a d ab + R, − − + 1 2 ic then since , 2 . R R ; positive a . g R bacd adcb ab 2Λ = abde R ´ = g 0 = ; c ; ab ; b h iesoso the of dimensions The . a R 0 = 0 = . gives cdab . R . ab R .Ta sif is That 0. = 4Λ. = (300) (293) (295) (291) (299) (294) (297) (298) (292) (296) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. C 221*h elCnomlCraueTensor* = Curvature Ω Let Conformal Weyl *The 12.2.1 eea ofra ymty one symmetry: conformal general the by broken is symmetry the course term. Of cosmological name. its hence spacetime, Minkowski mass h time the odmninu uniy olnt o xml.A osqec o every for consequence contain a equations As vacuum example. Einstein’s metric for with term, length solution cosmological no quantity, a dimensionful of no absence symmetry* the Conformal In and *Dilatation 12.2 allowed. are coefficients constant yafactor a by in ehave we where aiyse htmtpyn h ercb factor a by metric the mutiplying that sees easily hscno apn fw etitt tensors to restrict we If happen. cannot this n h ercwt osatcecet.I particular In coefficients. constant with metric the and hc sconserved, is which nhge iesosteeeittensors exist there dimensions higher In rmtemetric the from oeso eoddrvtvso h metric the of derivatives second of powers olwn rgam ntae yWy n atn oeokpoe the proved Lovelock Cartan, Theorem and Weyl by initiated following programme a the- Following Lovelock’s equations: Einstein the of Uniqueness 12.1 nGnrlRltvt n opr twt etna hoyfrwih swe as which, for theory Newtonian with it seen compare have and Relativity the consider General we in equations Einstein the motivate To Deviation Geodesic vanish. contractions its of 12.3 all Moreover, metrics. two the for same the is ab ∂ oeta ngnrl vni ,teEnti qain ontadmit not do equations Einstein the 0, = Λ if even general, in that Note a cd ∂ M b = λ g cd R orem sa rirr oiiecntn.Ti sbcueudrsc scaling a such under because is This constant. positive arbitrary an is e R → t Φ hnol iercmiain fteEnti esradmti with metric and tensor Einstein the of combinations linear only then , ab ′ ab nfu pctm iesos h nytensor only the dimensions, spacetime four In f˜ If . n ailcoordinates radial and λ λM = cd oseti ndti,cnie h cwrshl erc One metric. Schwarzschild the consider detail, in this see To . − R g g ab h ymtytu eeaie h iaainsmer of symmetry dilatation the generalizes thus symmetry The . ab n ab l egh,tmsadmse ntemetric the in masses and times lengths, All . Ω = − 1 n t rtadscn ata derivatives partial second and first its and V 1 g ab ¡ 2 ab R g ; b ab c a hr saohrslto ihmetric with solution another is there δ 0 = hnteWy tensor Weyl the then , d b − d R salna obnto fteEnti tensor Einstein the of combination linear a is , dt 2 d a N 2 δ i c cannot b t − + R → E 48 c b δ ij λt d a ∂ N allow + V a , j ∂ R ab r V 0 = b d b g → ab δ cd hc oti ihrta first than higher contain which c a λ . ¢ nfu pctm dimensions spacetime four In . hc r omr hnlinear than more no are which λr + odpn pnposition. upon depend to λ edscdvainequation deviation Geodesic ( 2 n rvddoersae the rescales one provided seuvln orescaling to equivalent is − V 1)( V ab 1 ab n ssymmetric. is ∂ osrce solely constructed − a g g 2) ab bc ′ g R ab ′ and r rescaled are ¡ δ = c a δ ∂ d b a λ − (302) (301) ∂ 2 b δ g g d a ab cd δ , c b ¢ Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. emytikof think may We ebgnb osdrn a considering by begin We a em ntecodnt rnfraincne.I u ae esyta the that say we case, our In cancel. transformation vector tangent coordinate the in terms bad sas etrfil,aldthe field,called vector a also is a o fixed for w oecoordinates more two (i.e. stetnetvco otegoei aeldby labelled geodesic the to vector tangent the is lbln h attocodntsby coordinates two last the (labeling and Thus Thus Now oncigvector connecting nfc o n w etrfils ti h aethat case the is it fields, vector two any for fact In enwcamta if that claim now We T o ymti connection symmetric a for a stetneto edsc and geodesic) a of tangent the is σ . T a σ n h oncigvector connecting the and Thus . and x 1 ∂T ( N T h x , ∂σ ii ,N T, a λ a 2 ) a ; aiyo geodesics of family ; T b T T b a ehv,i h oriaesse ( system coordinate the in have, we say sasaeadtm oriae fw introduce we If coordinate. time and space a as N = T a a a i ( i b b ; a ,b b commutator = ) ∂N T N = = N ∂λ = N T a ehave we δ b b N T 4 a a T a ; a b − ,b N , − a ,b N a = a = T = σ N ∂ T N 49 N b a a ∂x and b b ∂x ∂σ∂λ ∂λ − ∂ ; N a ∂σ a b Γ + Γ + 2 T ; ,b a b b a N x T b T ¯ ¯ ¯ − ¯ ¯ ¯ a a λ or σ a λ b b 0 = b b ) N = ; N a a = 0 = b 0 = bracket uhthat such T a c c δ a , T N σ ∂λ∂σ 3 a b ∂ ∂ commute . ecall We . . c . a 2 0 = c N . T x T a b b b syumycek the check, may you As . , , . , . x . a ( ,σ λ, sageodesic a is ) x 1 x , 2 ,λ σ, , (310) (313) (305) (311) (304) (312) (303) (308) (309) (306) (307) ), Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. hsi osbebcueo h following Lemma the of because possible is This T But uigtegoei qain(ii)) equation geodesic the (using h acltosgvnaoewr ftegeodesic the vector if work connecting above of given calculations choice The convenient A 12.4 (ii)). commutivity the (by otatn with Contracting b omtvt,(i) xadn ehave we Expanding (ii)). commutivity, (by ie ieieo ul ti fe ovnnett choose to convenenient often is It null. or timelike like, Proof where (i.e. fgoei eito,smtmscle the called sometimes deviation, geodesic of a , o sn ( using Now T a h ic dniy ..tedfiiino uvtr is curvature of definition the i.e. identity, Ricci The and If T N a N a ⋆ omt,te h eea eaiitcvrino h equation the of version relativistic general the then commute, ) a = eget we ) D D Dτ T 0 = N ⋆ Dτ 2 T 2 a N D N N a 2 Dτ D 2 a 2 ; toepitof point one at Dτ = a b b N 2 N + 2 N gives + T = a T 2 b b E a a a E ¡ ; = T c a T ; ; T = b a b a b ¡ N a ; E c b c N ; N T b N ; ( + − b b a N b ; a a ; c b b c = T T b ; ; T T b b edscDeviation Geodesic 0 = T = ; ; a c T c a c c T a T N − ; N b R − c ; a γ ¢ ; 50 c c a b b a ; T then , T ; c T − T b = dbc 0 = T N a c a c ) T c ; ; − ; c b + T b c ; aoieuto,is equation, Jacobi ⁀ a = . ; b N R N c d T T ; ¡ T c T T a c T b T c a c a c ; ¢ dbc − b a c ; . N + b T = ; ; N b γ b T a T b T N N − b a d 0 = ihparameter with 0 = a b ; . b ; E c c ; ¢ T c T ; a ; N c . b c nalpit of points all on c T b T . a N ; c b c N N ob rhgnlto orthogonal be to b b b λ sspace- is γ (324) (317) (322) (323) (316) (318) (321) (319) (320) (315) (314) . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. cu’ Lemma Schur’s pc fcntn uvtr hshas thus curvature constant of space A V -ln ssaeiew a choose may we spacelike is 2-plane Proof position. of K becomes identity. Bianchi contracted the by oeta,b h ymtiso h imn esr if tensor, Riemann the of symmetries the by that, Note b commutivity) (by uvtr fta ln sdfie by defined is plane that of curvature Proof h atse sstefc h h erci oainl osatadthat and constant covariantly is metric the the fact the uses g step last The (because fi szr o n au of value one for zero is it If o hudcekta hsepeso a l h eurdsymmetries. required the all has expression this that check should You pctm fcntn etoa uvtr a ydefinition by has curvature sectional constant of spacetime A Example 12.5 ab a V → T ycnrcinwith contraction By nasaeo osatscinlcraue h qaino edscdeviation geodesic of equation the curvature, sectional constant of space a In Thus nagnrlsaeie if spacetime, general a In a a T = a 2 b K, − osataogageodesic. a along constant = T 1 a W , fte2paeΠis Π 2-plane the If stnett geodesic) a to tangent is nasaeo osatscinlcurvature sectional constant of space a In a ( W D T Dλ a a 2 N N K g +1 = 2 ac a R a ( ¢ ,W V, abcd ; = + λ n finds one c T 3 V , twl ezr o l ausof values all for zero be will it K 1 2 K c W Dλ timelike = ( = ) D ¡ = g R T a constant = Λ = = ab K bd c V Dλ ¡ and T D T T R N ¡ a 3 = c c a g a V abcd a N W 51 T bd ¡ T a T K T a b V a g c emychoose may we Kg 1 = a ; ) a ac V c − ¢ ; .Ti xstesaeof scale the fixes This 0. = a osato l -lns Moreover 2-planes. all on constant = N N c a N 0 = bd − T pnatopaeΠ h sectional the Π, two-plane a span W , a W c = ) c c , g N b 0 = . ad V c , T g T c bc . W a a N ¢ ¢ a d W 0 = a . ; c a T V +1 = . c a V and λ a K . V , → sindependent is W aV a uhthat such a K c W + fthe If . c (331) (332) (333) (330) (326) (327) (325) (328) (329) bW 0. = a , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. domain od-itrsaeie hntmlk edsc eaaeepnnilyas exponentially separate geodesics timelike then spacetime) de-Sitter to ntetmlk aew a set may we case timelike the In o epsuaeta h nyfre c hog h boundary the through act forces only the that postulate we Now -ouesc htsc httemomentum the that such that such 3-volume D where oprn h eea eaiitcgoei qainwt hti Newtonian in that equation with Poisson’s equation Matter recal have geodesic with we relativistic theory Equations general Field the Comparing Einstein The 13 osdrater nfltsaeie ecndfiea nrydensity energy an define can We spacetime. flat in theory Tensor a Momentum Consider Energy The 13.1 roughly eset We n a lodfiea define also may One hssget htterltvsi nlgesodb oehn like something be shold analogue relativistic the that suggests This ihuiseeg e ntsailvlm n an and volume spatial unit per energy units with o nteNwoinlimit Newtonian the in Now nt nryprui raprui ie osraino nrynwreads now energy of Conservation time. unit per area unit per energy units ontis uta n osi atB hi oiin ilb will positions Their PartIB. in does one as just downstairs, where hnte silt ihpro 2 period with oscillate they then 13 in o h etfweutosw eett atsa esrnota tensor Cartesian to revert we equations few next the For E τ T 3 T ab D orc,btt e httecretase sw edt introduce to need we is answer correct the what see to but correct, spoetm.I Λ If propertime. is is a N sasial esrdtrie ytemte itiuin hsis This distribution. matter the by determined tensor suitable a is a n obtain and 0 = oetmdensity momentum dp dt i = D < ( Dλ E i Z 2 ) R ii wihcrepnst nid-ie spactime) Anti-de-Siter to corresponds (which 0 N D 2 p ab = π a T ∂t π i E q T R i = + a E ii d a T ab ∂ ∂t − 3 i T Z KT a H 4 = 3 x i Λ D 52 b ∝ = ≈ . = ≈ + π c T πGρ T i R − − ab ∂ d 4 π c i πGρ. 3 i n fΛ if and 1 i N , Z s . x 4 ihuiso oetmprunit per momentum of units with i i ∂D a 4 π 0 = = (334) 0 = natm-needn domain time-independent a in T nryflxvector flux energy ij R dσ 44 dutdshortly. adjusted e htis that , j > , inadpaealindices all place and tion wihcorresponds (which 0 s ∂D i a with say H fthe of q (339) (337) (338) (340) (336) (335) Λ 13 3 say τ , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. element nain ne oet rnfrain utetn h ymtyproperty symmetry the extend must T transformations Lorentz under invariant h ethn iei h aeo hneof change of rate the is side hand left The where i.e. domains oetmisd h domain the inside momentum Proof hr h oc xetdb h aeilisd h domain the inside material the by exterted force the where hsipisarlto ewe nryflux energy between relation a implies This set we provided htis that nodrt edrtecnevto faglrmmnu condition momentum angular of conservation the render to order In ij dt d fti qaini refraldmisw uthave must we domains all for true is equation this If u w udmna qain a ecmie nasnl equation single a in combined be may equations fundamental two Our nertn eget we Integrating and o hudcekta l opnnsof components all that check should You o osraino nua oetmipsstesmer condition symmetry the imposes momentum angular of conservation Now = Z D T G ¡ ij ∂ x i dσ D a k oterltvsial nain condition invariant relativistically the to si h xenltru ntesse.I hs r ob qa o all for equal be to are these If system. the on torque external the is is π ( = i then i sgvnby given is T − ∂ 44 x ∂t i ∂ ∂ , i = T π ¡ ij 4 k x H with ) ¢ k ∂t ∂ d utb symmetric. be must π 3 i ¡ = x x − = k − x T π x ∂ ( i − i 4 i π j ii 0 − ¡ = k Z ) = x T ¢ ∂D D x k ct + ab k T s h rttr ntergtadside righthand the on term first The . c π F ¡ . i ij T ∂ ∂ x i = T , T i cπ ¢ a ∂π j ab k − ∂t ij = ¡ T T = µ x i i ab x ij = = 53 T k = H s − i + c T ij − i T 0 = T T x ij dσ kj s ∂ 0 ji c x ba k ǫ i i j − kij cπ T ¢ ∂ i . s T j , = T T j ij . . i + ij x kj i T L ab n oetmdensity momentum and cπ ¶ i ij T 0 = j ¢ T where , ik dσ aetedimensions the have i kj + T , − ¢ . j x + = i T ∂ ki Z k T T D ki L ij kj ¡ j T − = stettlangular total the is ik T T − ik ij F T i . , ki nasurface a on ¢ d 3 T 3 x. π − ij energy ǫ volume kij = (341) (346) (350) (347) (345) (343) (342) (352) (351) (348) (344) (349) G T k ji , . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. where ihrta eoddrvtvso h erc are no metric, from the constructed of and derivatives tensorial second are than which higher equations equations, candidate field obvious The Einstein The 14 where Proof a eitrrtda etna oc ic Γ since force Newtonian a as interpreted be may oain n oeul twl lob osset hsw suethat be assume certainly we will Thus simply equation consistent. we the be comma way also this a will In containing it relativity hopefully semi-colon. and a special covariant by in comma equation the an replace see we when othat so emk hti fe eerdt sa as to referred often is spacetime what curved make We to Generalization 13.2 ihPisn qainnwsosthat shows now equation Poissons with n h eutflos h osrainlwmynwb re-written be now may law conservation The follows. result the and o o n n aaee aiyo ymti matrices symmetric of family parameter one any for Now ∆( t ealthat Recall Thus nfact In nteNwoinlmtwhen limit Newtonian the In n a b diagonalization) (by has one ) ,B C B, A, r osat.Atn with Acting constants. are T T ab ab ; Γ b ; b R b = = R a ab c ab √ ∂ − Γ = b 1 − = Trace ∆ = ∆ T a ˙ 1 2 Γ g 1 2 b ab AT g a ∂ b g T ab b a Γ + ad = g ab ¡ b R ab √ ³ det = = √ sdmntdby dominated is g = c − + bd,c 1 − b 54 1 2 gT A Bg b 8 g g T πG c cd iia coupling minimal ∇ ∂ + g = ab 4 M ac ab a ab ∂ a ¢ ¡ g T a 8 Γ + − . √ hw that shows cd,b + Γ + c πG g ab 1 4 cd − M, CRg . ˙ − c . g − 4 c a ¢ i a Λ , g d 4 d ab bc,d g T T T ≈ ab cd , 44 cd . ¢ ∂ C M , 0 = i h eodtr Γ term second the 0 = U = ( . supin htis that assumption, t . ihdeterminant with ) . 1 2 n comparison and , (360) (354) (355) (356) (353) (361) (358) (359) (357) 4 i 4 ρ Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. An ud hs agn etr r aallt to to parallel are vectors tangent whose fluid, where thus safrems aih enwddc that be- deduce is now We act this would vanish. Physically which must gradient force pressure geodesics. any a are that as lines implies pressure world vanishing the the caues that means equation This ρU h ceeaini rhgnlt h 4-velocity the to orthogonal is acceleration The rdc f(6)with (368) of product which for fluid the of frame rest local and density mass and 41Eapeaddrvto ftegoei postulate geodesic the of derivation A and Example 14.1 hsi h omlgclcntn spstv hntepesr sngtv,i negative, tension, is of pressure state a the in then is positive medium is the constant words, other cosmological the if Thus hr the where hc mle that implies which efc fluid perfect a rsuefe udi fe called often is fluid pressure-free A For qaino state of equation U b U ehave We . ˙ akenergy dark a = uein4-velocity Eulerian U ..nivsi udhas fluid inviscid i.e.an , a ; b U T P b ab T racsooia osatw have we constant cosmological a or steaclrto ftefli,ie ftewrdie fthe of worldlines the of i.e. fluid, the of acceleration the is ab sarelation a is h rsue hsi ossetwt h atta nthe in that fact the with consistent is This pressure. the T ; b U ab = strace-free is a = gives ¡ T ρU 44 ρU a ftefluid the of = T a U U ab ρc b ¡ P ¢ b ρ c dx ρU 1 2 ; dτ c = P 2 + b U + = T , T = 2 ˙ P a = b a − P = U ¢ = 55 P = = 0 = ; ρ ¡ 8 b a Λ ( dust 8 − g U Λ T πG ρ ˙ U 0 = πG c ( = ab 1 3 U ρ. c .Freapefra for example For ). a ab 4 . a ρ, ij 2 a + + . g g thas It . c, . = ab ab satisfies U U c 0 1 , 2 U δ P 0. = a a ), ˙ U ¡ a n hc satisfy which and ρU U ij a U a . b b P .Tu aigtedot the taking Thus 0. = U ¢ ¢ ; , a b n hence and 0 = (362) ) U 0 = a = , − aito gas radiation c 2 and ρ T sthe is (363) (368) (365) (370) (369) (366) (371) (367) (364) ab = n Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. u sn h atthat fact the using But oyai aibewihw a aet eteeeg density ther- energy independent the one be ‘perfect only to is homogeneous take there may A we that which property variable thermodynamics. unusual modynamic rather some the need ’has we fluid further proceed To ecncnetti oa to this convert can we hsi a fcnevto frs mass. rest of conservation of law a is This 411Eape pretfli ihpressure* with fluid If *perfect Example: 14.1.1 h rhgnlt h ieie4-velocity timelike the to orthogonal h Thus b a ab T hseuto xrse h osraino etmass. rest of conservation the expresses equation This aigteinrpoutwith product inner the Taking nrdc tensor a Introduce h aetu osrainlwaie eas ehave we because arises law conservation true name The ecnnwitgaeoe pta domain spatial a over integrate now can We rjcigotooa to orthogonal Projecting sa is ab = ( = h ba rjcinoperator projection ρ and + P ) h U b a a U ³ U dt d b ( b ρ ,btif but 0, = + Z ∂t ¡ + ∂ h D ρU g P b a recnevto law conservation true P ¡ ³ √ √ ( = ( b ) ab ρ − ρ ) U − Γ ; hc rjcsvcosit h pclk 3-plane spacelike the into vectors projects which + U b g + U a b h osrainlwgives law conservation the g ´ b = U ∂ c a P U P c ; b c U c U ie eebrn that remembering ( gives a 4 ˙ ¡ ) ∂ 4 ) U = a b ρd √ U V U b ρ U b ¡ + = ¢ a − a a 3 ρU √ ( + b ´ U + x ; U gρU δ a − 1 − ; gives 56 a b a a = a b + ∂ ρ h thstepoet that property the has It . g ¢ ρ . − i then 0 = a b + ¡ ∂ − U Γ + b + ∂ √ ¢ b ∂ a b P ¡ Z a P − 0 = P ∂ √ ∂D U P ) b a gU U . − c ˙ ρ b . c ρ g D a 0 = U i + √ ¢ ρ (378) 0 = , b h ¢ oget to − ∂ b a . 0 = a 0 = gU V P b . i 0 = . dσ = h i b a V . U ˙ a b nohrwords other in , = ρ U ˙ pressure , h a b a ) h c b = (372) (380) (373) (376) (377) (374) (375) (379) h P c a , Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. thus n hence and nrp density entropy , hsw a ewie(7)a h a fetoyconservation entropy of law the as (379) re-write may we Thus hc scalled is which Thus o o ooeeu usac,if substance, homogeneous a for Now a hc reads which law hswr o qa otelclvle h eprtr ieec ol eused be If could difference temperature. temperature the redshifted value, the local work. poten- have the do lower and to to of energy equal high location not gain at a were would energy to this photons them local The sending the could and converting tial. one photons by so thermal machine not into gravitatio were motion potential the this perpetual factor.If as a redshift increases the construct which precisely temperature by decreases a potential have must potential tional and h atrfruai oeie aldthe called sometimes is formula latter The o uda eti ttcmetric, static a in rest at fluid a For oudrtn h te qainw oethat note we equation other the understand To omnsrdhf law redshift Tolman’s s rtemperature or ∂ U i ˙ T a ln( dS T √ = U ˙ √ − ρ ¡ ds s T a ¡ s sU dP g − + ds T 0 = = 44 T ∂ , g = a neulbimi ttcgravita- static a in equilibrium in gas A . = P a 44 − U dU constant = ¢ i sw ih hyaelne ytefirst the by linked are They wish. we as 57 ρ ρ ; = ln( = ) = h a = + dρ + a b + 0 = dρ ib-ue relation Gibbs-Duhem T √ ∂ dT ρ V T P P. − b dV P − T . ∂ . . , g . i 44 S ln ) , = T ¢ sV hsgives this , tflosthat follows It . (389) (387) (385) (388) (384) (390) (386) (381) (382) (383) nal Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. o ftefli sa rest at is fluid the if Now and u oil nerto osat hc so oreΛ sn th manag he it Using Λ. the course called of universe left is static had which a he construct constant, that to integration realized he term posible earlier, constant static a followed cosmological a we a out construct lines have to the H not along unable did cosmology. which thinking was However theory He to original static. it his be with applied should model Einstein universe gravity, the of that Mistake believed theory Greatest his Einstein’s constructing After Example: 14.2 where gives sntsai,dsatglxe eercdn rmu.Enti elzdta he that realized Einstein us. from universe receding the were that galaxies announced distant b Hubble static, realized paper also not fact Lemaitre’s is a have after Λ, could Shortly without one or that Lemaitre. with out solutions pointed Friedman cosmological dependent paper, time Einstein’s after years 5 About met- Friedman-Lemaitre-Robertson-Walker Example 14.3 eevsgda meddi orfltdimensions flat four in embedded as envisaged be usiuinit h isenequations Einstein the into Substitution tflosthat follows It ti o icl oprud n ftefollowing the of one persuade to difficult not is It Thus g ij rics samti fcntn curvature constant of metric a is R ij ( X R = R ab 1 ab ) a 2 2 U 2 R = − 1 2 g ( + T ds a ikjl ij Rg µ ab 1 2 = 2 X Rg 0 0 0 and = 4 ab = = δ πGρ 2 4 a ab ) ρU = − a a 2 2 1 and R 2 2 dt ( + 8 = a µ g i ¡ = U ij 2 g i 58 − b ij X = + πGρU ¶ isenSai Universe Static Einstein 0 a a = 1 g 3 3 2 2 g kl a ) 6 ij µ 2 2 Λ = − K . dx ( + ρ 0 0 a a 3 2 R g U 0 i . = g il dx X a b ij 0 g a − kj 4 ¶ a j ¶ 1 ) 2 = ¢ 2 Λ on g = fact a 6 ab 2 a S . 2 3 h -peemay 3-sphere The . h ercis metric The . (396) (395) (394) (393) (392) (391) (397) (398) ed y e . Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. eueto reduce h case The r tl loe ocag h coordinates the change to allowed still are nrdcino stegets lne fhslf.Budro o,i has it not, the or of Blunder expansion metric the the life. consider dominate we to his his how seems of to see and To refered blunder energy, alledgedly universe. vacuum greatest and as theory, the re-surfaced his as no of Λ prediction a of this introduction make to failed had hr o h pta erci ae ob -peeo radius of 3-sphere a be to taken is metric spatial the now where al)teSai cwrshl metric Schwarzschild Static the cally) Theorem Birkhoff’s metrics establishing by vacuum start We Symmetric Spherically 15 tp1 h metric The 1: Step Proof symmetric. i.Pyial hscnb nesodi atfo h atta gravitati directio that a fact define the would from polarization on part Any field in polarizable. and understood transverse Schwarzschild be are the can waves by given this exactly exploding, Physically metric pulsating, exterior ric. a an has e.g. star source, collapsing symmetric or spherically Any collapse. ational ds tagtowr u ieoecluainsosta h isenequati Einstein the that shows calculation tiresome but straightforward A hr a eno be can There hsgnrlzsNwo’ hoe n ral ipie h td fgravi- of study the simplifies greatly and theorem Newton’s generalizes This 2 = A S k ( 2 ,t r, = u yteHiyBl hoe tcno hrfr espherically be therefore cannot it Theorem Ball Hairy the by but ds ) − 2 dt sas allowed. also is 1 = 2 2 + − ¡ g 1 B aθ r h ei a ecs nteform the in cast be may meric The − h nqeshrclysmercvcu erci (lo- is metric vacuum symmetric spherically unique The ( ds → ,t r, ( or ii 2 2 GM ) r ) ˜ ( = g r drdt i aφ ˜ = ( ) iii − ¢ r dt em with terms ˙ ) dt ( a a + ˙ ,t r, 2 2 2 2 a a ¨ C + + + ) = ( t , a ρ ,t r, a 1 59 2 k − + 2 ( − t ) dr 4 ) = dr 3˙ g a πG 2 3 a ,t r, ij 2 a GM 2 → 8 r ρ ( πGρ + = x ρ 0 = 3 t ˜ k + R ,t r, + ) = dx 2 . Λ r 3 ( + t ˜ ,t r, 2 yshrclsmer.We symmetry. spherical by ( i ¡ dx ,t r, Λ 3 dθ ) j ¡ ) 2 dθ . sin 2 sin + 2 θdφ 2 2 √ ¢ θdφ k , 2 k ¢ 0 = (404) (405) (400) (401) (399) (402) (403) met- onal ons , 1. n Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. tp2 h edequations field The 2: Step h id’ ehv hw httems eea ercta ene oconsider to need we that metric general most the that is shown have we tilde’s the ( eaeuigtenotation the using are We tp3 ro fstaticity of Proof 3: Step sn hsfedmw a define can we freedom this Using hr ehv rpe h id on tilde the dropped have we where tp4 ouino h ttcfil equation field (i) Static o.d.e. the non-linear of order first Solution 4: Step coordianate where icsinrlsotti possibility this out rules discussion iv 14 ) h eann oriaefedmmyb sdt eliminate to used be may freedom coordinate remaining The esilhv h reo ocag h iecodnt.Dfieanwtime new a Define coordinate. time the change to freedom the have still We h ercnwboe aietystatic manifestly bcomes now metric The titysekn ene oko that know to need we speaking Strictly f ( G t sa rirr ucino nerto.Tu h ercis metric the Thus integration. of function arbitrary an is ) θθ t ˜ = ds by ds sin 2 ds 2 ( = 1 i = 2 2 ( + ) e θ ( = − ( − iii G i λ e ) e φφ ) ( iii ν − r ( ) r,t λ ) ( e ∝ ( ii f f ( r ⇒ ) ˙ G ( ii ) dt ) t G e dt ≡ ) ) tt − 4 2 dt λ − rr 2 ⇒ λ ′ ∂ + ∝ + 1 2 otn acltosgive calculations Routine + + ¡ t d ∝ r ˜ ,f f, G + 2 e λ t ˜ ˙ e e ν = ν λ r rt − = e λ 0 = e e ( ′′ ′ 2 r t − r,t ( λ λ λ R +( 0 = ′ r 2 . ∝ e ( λ ) ¡ 60 ≡ ) r + ( dr 1 2 − R dr ¡ ) ,t r, ν ⇒ e f dr 1 ( rλ ∂ − ′ + 1 ( 2 ⇒ ,t r, ) 2 t − r ) 2 2 ) ( λ + f + ′ ν 14 dt. + i o culyacntn.Aseparate A constant. a actually not )is + + ν e 2 . 0 = = e r . λ − r 2 r λ 2 = ) r 2 λ ( λ ¡ 2 λ ¡ r + ν ˙ . dθ ( ¡ + − dθ ′ r dθ − (409) 0 = rλ λ ) 2 rν 2 . λ ( 2 sin ′ sin + enwne oslethe solve to need now We r ′ ¢ ′ sin + ) + ) ¢ − (408) 0 = 2 (410) 0 = θdφ ν f ′ 2 λ 2 ( θdφ ′ t 2 θdφ ¢ ) ¢ − , , 2 e 2 ¢ − 4 ¢ g ν rt ¡ Dropping . 2 λ ¨ +( λ (407) (413) (406) (411) (412) (416) (414) (415) (417) ˙ ) 2 − λ ˙ ν ˙ ¢ 0 = Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. where oget to where substitution the making by done be can This ihu oso eeaiyw a set can metric we Schwarzschild generality of the loss in Without Geodesics 16 h ouinis solution The 61Tesaeo h orbit get the we of (iii) Using shape The 16.1 etof dent where uino i)ad(i)i i gives (i) in (iii) and (ii) of tution where Since Since L h E M k ( φ sidpnetof independent is i steaglrmmnu prui asi h ieiecs) Substi- case). timelike the in mass unit (per momentum angular the is +1 = steeeg prui asi h ieiecs) Since case). timelike the in mass unit (per energy the is ) L sa rirr nerto constant. integration arbitrary an is ote’ hoe gives theorem Noether’s sidpnetof independent is L , = ¡ 0 1 , − − − ¡ o ieie ihlk rsaeiegoeisrespectively. geodesics spacelike or lightlike timelike, for 1 1 2 h r M r − ³ 2 2 ³ dλ dr ¢ 2 ( dφ dr ³ M r ii t ´ dλ dt ) 2 ´ ote’ hoe gives theorem Noether’s ¢ ( 2 ³ = ´ iii λ = 2 dλ ote’ hoe gives theorem Noether’s , dt e E ) − − ¡ E 2 ´ D 1 λ D 1 D 2 − 2 − + 1 = = − + ′ − ¡ r θ 2 1 = 61 1 2 − ¡ M 1 2 r ³ = 1 − M r − 2 − dφ dλ − ¢ M r D 1 π 2 ³ 2 1 r ³ 2 M r ´ . 2 ( dλ M dr r h arninbecomes Lagrangian The . dλ , dt r M r = ) ¢¡ ³ , ´ ´ ¢¡ 2 dλ h, dr k = − k + ´ E, + r 2 h r 2 + 2 2 ³ h r ¢ 2 2 dφ dλ r . ¢ 2 . ³ ´ dφ 2 dλ = ´ k, 2 . L sindepen- is (421) (423) (418) (426) (420) (419) (425) (424) (422) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. 612Apiain2 ih bending set light We 2: Application 16.1.2 oodrzero order To hseuto a erdcdt udaueadtease xrse nterms impact large in for expressed vaild answer method, simplest the parameter the and perhaps however quadrature functions, to elliptic reduced of be can equation This htis that hc stesadr etna eutpoie eidentify we provided result Newtonian standard the is which oget to set Now h rvttn oy h ai ftetr elce ota eando the on retained that to neglected term the is of side ratio hand left The body. gravitating the fw rptels emo h ih adsd eget we side hand right the on term last the limit drop Newtonian we 1: If Application 16.1.1 gives Order Next hc steeuto fasrih iewoedsac fnaetapoc to approach nearest of distance whose line straight a is origin of the equation the is which hshsa solution as has This ercgieti stefis nerlo h eododro.d.e. order second the of integral first the as this recognize We k oget to 0 = b st okperturbatively work to is b , usiuigtezr re ouini h ih adsd f(431) of side hand right the in solution order zero the Substituting 1 2 O ³ dφ du ( h dφ d 2 2 ´ u u 2 2 2 + u hc sesl ent be to seen easily is which ) + dφ = d u 1 2 2 dφ d u = u 2 2 1 b 2 u 2 ¡ + dφ d 3 = sin + dφ d b 2 M u 2 2 u 2 2 u u r E 2 φ 0 = h sin + sin = 2 + + 2 u u 2 − u 62 = . Mk M φ 2 h φ 3 = b ⇒ = 2 2 = 1 r = 3+cs2 cos + (3 k h Mk Mu 3 + u 2 b, h 3 2 (1 2 = M b 2 Mu 2 . − sin . (1 2 b 2 MU − φ φ O . ) , o 2 cos ¢ ( + ) v c , 2 2 ). φ Mu (434) ) M 3 . stems of mass the as (431) (432) (430) (428) (427) (435) (429) (433) Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. ers)dsac is distance nearest) 613Apiain3 rcsino h perihelion the of set now Precession We 3: Application arc. of seconds 16.1.3 1.75 about by deflected is sun the grazing oodrzero order To terms. where hc a solution has which eset We oteaypoei at is asymptote the so hr ehv hsnteitgaincntn omk the make to constant integration the chosen have we where 1 φ etrn dimensions restoring hsi nelpeo eiltsrectum semi-latus of ellipse an is This axis etorder Next circular. orsod ovanishing to corresponds ecluae u hc r o eeatfru.Ti sapeesn lis in ellipse precessing a is This us. for relevant not are which but calculated be − = hsi ocdsml amncmto ihfrequency with motion harmonic simple forced a is This ueial,if Numerically, esalasm htteecnrct ssalads h ri snearly is orbit the so and small is eccentricity the that assume shall We ofidtedflcinangle deflection the find To 3 h π 2 a M 2 . l = hst etorder next to Thus . u ′ ≈ = 1 − l l h M k e 2 2 and .easm htteobti erycircular. nearly is orbit the that assume 1.We = h pein(utet itneis distance (furthest) aphelion The . + ehave we w e b ′ n xadt oet(ier re in order (linear) lowest to expand and 1+ ≈ saslrrdu and radius solar a is l e φ e . u ∞ d dφ u r h orce ausof values corrected the are = 2 u = w n cusfrsmall for occurs and dφ 2 d ≈ 1 u l − 2 + 1+ (1 u 2 = 2 1 b M b w + ¡ dφ d δ l 1 φ 2 (1 δ ′ Tettldflcinis deflection total .The u ent htiiilaypoeo h orbit the of asymptote initial that note we e u 1+ (1 2 + = cos − = + 2 63 4 l 6 M M u h b GM φ c e M h ecnrct 0 ,eccentricity 2 2 ′ ) = 2 b cos l , + 3 + 2 M M = ) h O . 2 ωφ Mu ( oa aste ai wave radio a then mass solar a φ 3 ) φ 2 = M h , 2 ) = 4 . ¢ M h 3 , φ 2 . 1 l ∞ . − ≤ l and e eexpand We . δ < e w n h eiein( perihelion the and = u e goigquadratic ignoring − n eimajor semi and 1 ω ymti about symmetric hc a easily can which 2 = φ ∞ q n hence, and 1 − 6 h (441) (442) (438) (440) (436) (437) (439) M 2 ≈ Copyright © 2006 University of Cambridge. Not to be quoted or reproduced without permission. vrho ravneb naon e revolution respectively, per perihelion amount and an aphelion by the advance points, or nearest overshoot and furthest the which usrw e bu ere e year. per degrees 4 about get we pulsar hsteprhlo dac is advance perihelion the Thus o ecr egtaot4 eod facprcnuy o h binary the For century. per arc of seconds 43 about get we Mercury For 2 ω π − 2 π c 2 ≈ a 6 (1 πGM 6 64 πM h − 2 e 2 2 ) = . 6 πM l . (444) (443)