Substituting back this result into the original Einstein equations we realize that they can be written as

Transcription

1 CHAPTER 8 THE CHWARZCHILD-DROTE OLUTION As you see, the wa teated me kindly enough, in spite of the heavy gunfie, to allow me to get away fom it all and take this walk in the land of you ideas chwazschild s lette to Einstein duing Wold Wa I Most of the wok done till now has been elated to weak-field solutions of the Einstein equations. In this Chapte, we go a step fowad an look fo exact solutions. Given the non-lineaity of the field equations and the associated difficulty in finding analytical solutions fo abitay matte distibutions, we will estict ouselves to vacuum solutions. To detemine ou stating point, let me ewite the Einstein equations G µν + Λg µν = κ 2 T M µν in a much moe convenient fom. Multiplying by the invese metic and taking the tace we obtain a elation between the Ricci scala, the cosmological constant and the tace T M g µν T M µν of the locally conseved enegy-momentum tenso T M µν, namely R µ µ 1 2 Rδµ µ + Λδ µ µ = κ 2 T M R = κ 2 T M + 4Λ. 8.1) ubstituting back this esult into the oiginal Einstein equations we ealize that they can be witten as R µν = κ T 2 µν 1 ) 2 g µνt + Λg µν. 8.2) Vacuum solutions T M µν = Λ = 0) coespond then to solutions of the equation athe that to solutions of G µν =0. R µν = 0, 8.3) Vacuum solutions ae not necessaily flat Eq. 8.3) does not imply the vanishing of the Riemann tenso R µ νρσ, which contains exta components.

2 8.1 A spheically symmetic ansatz 110 The poblem of finding a solution of this equation is futhe simplified in those cases in which the poblem is highly symmetic. In what follows, we will look fo spheically symmetic solutions. 8.1 A spheically symmetic ansatz Conside the spacetime outside a spheically symmetic mass distibution, which can be static o not. A spacetime is said to posses a paticula symmety if the functional fom of the metic unde the action of such a symmety is maintained. In paticula, a spheically symmetic spacetime is a spacetime whose line element is invaiant unde otations o, if you want. a spacetime with the symmeties of the sphee ). The only otational invaiants of the spacelike coodinates x = x i and thei diffeential ae x x 2, dx dx, x dx. 8.4) The most geneal spatially isotopic metic that can be constucted with these elements takes the fom ds 2 = at, )dt 2 2bt, )dt x dx) + ct, ) x dx) 2 + dt, )dx dx, 8.5) with a, b, c and d some abitay functions of t and. The equied invaiance unde otations suggests the use of spheical coodinates {, θ, φ}. Pefoming the change of vaiables we ealize that all the angula dependence in 8.5) is isolated in the dx dx pat x x = 2, x dx = d, dx dx = d dθ sin 2 θdφ ) ubstituting these expessions into 8.5) we aive to the equivalent fom ds 2 = at, )dt 2 2bt, )dtd + ct, ) 2 d 2 + dt, ) d dω 2), 8.7) whee we have defined dω 2 abitay, functions = dθ 2 + sin 2 θdφ 2. Collecting tems togethe and defining some, still At, ) at, ), Bt, ) bt, ), Ct, ) 2 ct, ) + dt, ), Dt, ) 2 dt, ), to take into account the exta factos of in Eq. 8.7), we ae left with ds 2 = At, )dt 2 2Bt, )dtd + Ct, )d 2 + Dt, )dω ) The esulting metic can be futhe simplified by using the feedom in the choice of coodinates. Fo instance, we can define a new adial coodinate 2 Dt, ) and eliminate and d in tems of, t, d and dt. This gives ise to a big mess that changes the explicit fom of the coefficients A, B, C to some new, but still abitay, coefficients A, B, C ds 2 = A t, )dt 2 2B t, )dtd + C t, )d dω ) The next thing we can do is to find some new coodinate time tt, ) to get id of the nasty tem dtd. To do that, let me define this new time as d t = µt, ) [A t, )dt + B t, )d ] = t Ψt, )dt + Ψt, )d, 8.10) whee the new unknown integating facto µ is detemined by the condition that the second equality holds fo some Ψ. In othe wods, we equie µt, ) [A t, )dt + B t, )d ] to be a total diffeential, so that the fist equality makes sense. quaing Eq.8.10) d t 2 = µ 2 A 2 dt 2 + 2A B dtd + B 2 d 2) 8.11)

3 8.2 pheical symmety and staticity 111 and isolating the tems elated to dt 2 and dtd, we get A dt 2 + 2B dtd = 1 A µ 2 d t 2 B 2 A d ) In tems of the new tempoal coodinate t the coss tem disappeas and the antsatz 8.9) becomes diagonal ds 2 = 1 ) A µ 2 d t 2 + C + B 2 d dω ) A ince the functions of t and in this expession ae abitay we can collect them into some abitay new functions 1 e 2α 1 A µ 2, e2β C + B 2 A, 8.14) and wite ds 2 = e 2α t, ) d t 2 + e 2β t, ) d dω ) Dopping the bas to maintain the notation as light as possible, we aive to ou fist impotant esult ds 2 = e 2αt,) dt 2 + e 2βt,) d dω ) Just by using spheical symmety and ou feedom to change coodinates, we have been able to educe the 10 functions in g µν to two functions of only two vaiables! Rathe impessive. 8.2 pheical symmety and staticity The unknown functions α and β can be detemined by inseting the antsatz 8.16) into the vacuum Einstein equations 8.3). The fist step in this pocedue is to compute the metic connection Γ µ νσ. The job is conceptually staightfowad but athe tedious. Whateve the way you do it 2, you should obtain 12 non-vanishing components out of 40, namely Γ t tt = t α, Γ t t = Γ t t = α, Γ t = e 2β α) t β, Γ tt = e 2α β) α, Γ t = Γ t = t β, Γ = β, Γ θθ = e 2β, Γ φφ = sin 2 θ Γ θθ, Γθ θ = Γ θ θ = 1/, 8.17) Γ θ φφ = sin θ cos θ, Γ φ θφ = Γ φ θφ = cot θ, Γφ φ = Γ φ φ = 1/. The non-vanishing components of the Riemann tenso associated to these Chistoffel symbols ae given by R t t = e 2β α) [ 2 t β + t β) 2 t α t β] + [ α β 2 α α) 2], R t θtθ = e 2β α, R t φtφ = e 2β sin 2 θ α, R t θθ = e 2α t β, R t φφ = e 2α sin 2 θ t β, R θθ = e 2β β, R φφ = e 2β sin 2 θ β, 8.18) R θ φθφ = 1 e 2β ) sin 2 θ.. 1 This exponential fom is specially useful fo witing compact expessions fo the components of the metic connections and the Riemann tenso. 2 The quicke way to get Γ µ νσ is by using the Lagangian pocedue fo geodesics, but you can also use the bute foce method and compute them via Eq. 4.62).

4 8.2 pheical symmety and staticity 112 which, contacted, povide us with the non-vanishing components of the Ricci tenso R tt = [ 2 t β + t β) 2 t α t β ] + [ 2 α + α) 2 α β + 2 α ] e 2α β), R = [ 2 t β + t β) 2 t α t β ] e 2β α) [ 2 α + α) 2 α β 2 β ], 8.19) R t = 2 tβ, R θθ = 1 + e 2β [ β α) 1], R φφ = R θθ sin 2 θ. Undestanding the esult The esult 8.19) can be easily undestood fom simple symmety consideations. Conside fo instance the R θ component and note that the metic 8.16) is invaiant unde eflections in the θ and φ coodinates, i.e. θ θ and φ φ. When θ θ, the sign of R θ changes and we ae foce to have R θ = 0. The same kind of agument can be applied to many components to get R θ = R φ = R tθ = R tφ = R θφ = ) The elation between R φφ and R θθ can be also deived without pefoming the explicit computation. To see this, conside the coodinate tansfomation θ, φ) θ, φ) and wite the expession fo the angula pat of the line element in both coodinate systems [ ) 2 ) ] 2 θ φ dθ 2 + sin 2 θdφ 2 = θ + sin 2 θ θ dθ ) The invaiance of the line element unde otations implies the equality ) 2 ) 2 θ φ θ + sin 2 θ θ = ) ubstituting this into the tansfomation law fo the R θθ component ) 2 ) 2 θ φ R θ θ = θ R θθ + θ R φφ R θθ = 1 sin 2 θ ) ) 2 φ θ + and demanding R θ θ = R θθ, we get the sought-fo elation R φφ = sin 2 R θθ. ) 2 φ θ R φφ The empty-space field equations ae obtained by setting each of the components 8.19) equal to zeo. These gives ise to 5 equations among which only 4 ae useful since the R φφ component simply epeats the infomation of the R θθ component. Among these 4 equations, the simplest one is that associated to R t. A simple inspection of this equation eveals a vey inteesting popety: the function β must be independent of time R t = 0 t β = 0 β = β). 8.23) Taking into account this esult and pefoming the time deivative of the vacuum equation R θθ = 0, we get t R θθ = 0 t α = 0 α = γ) + κt). 8.24) The coefficient e 2α,t) can be then splited into two pieces e 2α,t) = e 2γ) e 2κt). This allows us to pefom an exta coodinate edefinition dt e κt) dt, γ) α), 8.25)

5 8.3 The chwazschild-doste solution 113 in Eq. 8.16) to obtain a much simple line element ds 2 = e 2α) dt 2 + e 2β) d dω 2, 8.26) specified by only two time-independent functions α) and β). The esulting metic is static 3 even though we did not impose any equiement on the souce apat fom being spheically symmetic. The souce could be as dynamical as a collapsing o a pulsating sta and the metic outside the matte distibution would still take the fom 8.26), as long as the collapse is symmetic. This esult is in pefect ageement with ou discussion on gavitational waves: if a spheically symmetic body undegoes pue adial pulsations, thee is no quadupole and thee is no emission of gavitational waves. All vacuum solutions of the Einstein equations with O3) symmety ae necessaily static. 8.3 The chwazschild-doste solution Thanks to symmety, we ae left with 3 equations of a single vaiable fo two unknows α and β. Let me ewite them as ) R tt = + α + α 2 α β + 2α e 2α β) = 0, 8.27) ) R = α + α 2 α β 2β = 0, 8.28) R θθ = 1 e 2β 1 + α β ) = 0, 8.29) with the pime denoting deivatives with espect to. Note that the fist two equations ae athe simila. Multiplying the fist one by e 2α β) and adding it to the second we get e 2β α) R tt + R = 2 α + β ) = 0 α + β = 0 α) + β) = constant. 8.30) The integation constant appeaing in the pevious expession can be always set to zeo by simply pefoming a coodinate edefinition, allowing us to set α = β. Inseting this esult into Eq. 8.29) we get R θθ = α ) e 2α = 1 e 2α) = 1, 8.31) which can be easily integated to obtain e 2α = + C e 2α = e 2β = 1 + C, 8.32) o equivalently ds 2 = 1 + C ) dt C ) 1 d dω ) 3 A static spacetime is one in which i) The components of g µν ae independent of the timelike component x 0. ii) The line element is invaiant unde the tansfomation x 0 x 0. If the second condition is not satisfied the spacetime is athe said to be stationay. A paticula example of stationay metic is the one geneated by a otating sta, whee the change x 0 x 0 changes the sense of otation.

6 8.3 The chwazschild-doste solution 114 The obtained metic is asymptotically flat: it tends to the Minkowski metic when. Bikhoff s theoem Any solution of the vacuum Einstein equations with O3) symmety must be static and asymptotically flat. The only thing left is to associate the constant C to some physical paamete. The most impotant use of a spheically symmetic vacuum solution is to epesent the spacetime outside stas o planets. In that case, we would expect to ecove the Newtonian limit g 00 = 1 2GM ), g = 1 + 2GM ), 8.34) at lage values. Compaing 8.34) with the limit of the metic 8.32) g 00 = 1 + C ), g = 1 C ), 8.35) we get C = 2GM, which allows us to wite the final and taditional expession fo the so-called chwazchild-doste metic 4 ds 2 = 1 R ) dt R ) 1 d dω ) with the chwazschild adius 5. R 2GM c 2 ) M 3km, 8.37) M Execise Veify that 8.36) satisfies Eq. 8.29). Explain why this is guaanteed to happen even though we initially had thee equations fo two unknowns. how that the chwazschild metic can be witten in a fom that makes explicit its isotopic chaacte, namely ) 2 1 R ds 2 4ρ = ) 2 dt R 4ρ 1 + R ) 4 dx 2 + dy 2 + dz 2), 8.38) 4ρ with ρ = 1 2 GM + ) 2 2GM. 8.39) 4 Kal chwazschild found this exact solution in 1915 while seving in the Geman amy on the Russian font duing the Wold Wa I and died a yea late fom pemphigus, a painful autoimmune disease. An altenative deivation of this solution based on the Weyl method was pesented by Doste aound the same time but fo some eason the physics community completely ignoed it. 5 We have momentaily estoed the factos of c.

7 8.4 Measuing distances and times Measuing distances and times Which is the physical meaning of the coodinates t,, θ, φ) appeaing in the chwazschild-doste solution? Although they povide a global efeence fame fo an obseve making measuements at an infinite distance fom the souce asymptotic flatness), not all of them epesent physical quantities measued by abitay obseves. While θ and φ have the same intepetation than the spheical angula coodinates in flat spacetime, the coodinate adius and the coodinate time t cannot be geneically intepeted as the physical adius o the physical time measued by a clock. Physical quantities must be computed fom the metic. The physical inteval in the adial diection measued by an abitay local obseve is given by the pope distance dt = dθ = dφ) ds = 1 R ) 1/2 d, 8.40) while the time measued by an stationay clock at d = dθ = dφ = 0) is given by the pope time inteval dτ = 1 R ) 1/2 dt. 8.41) Undestanding the esult In the chwazschild metic, space is foliated by sphees 2 of aea 4π 2 sepaated by a pope distance 1 R ) 1/2 d. 8.5 Visualizing chwazschild spacetime A mental image of the chwazschild-doste spacetime can be obtained by embedding a subset of it into a highe dimensional spacetime 6. ince ou solution is static and spheically symmetic, we can, without loss of geneality, fix t =constant and θ = π/2. This leaves us with a 2-dimensional suface dx 2 = 1 R ) 1 d dφ 2 = f) 1 d dφ 2, 8.42) which can be easily embeded into the odinay 3-dimensional Euclidean space [ ) ] 2 dz) dx 2 = dz 2 + d 2 + dφ 2 = 1 + d dφ ) d The function z) can be obtained by simply compaing 8.42) and 8.43) ) 2 dz) 1 + = f) 1, 8.44) d and pefoming a tivial integation z) = 0 d 1 f ) f = 2 R R ) + constant. 8.45) ) The esulting embedding diagam is the Flamm s paaboloid shown in Fig The distances between cicles on this suface ae lage than just, in clea ageement with the discussion pesented in the pevious section. 6 Remembe that such embedding diagams can be misleading. Fo instance, a 2-dimensional cylinde embedded in 3-dimensional Euclidean space can seem to be cuved even though it is intinsically flat, K = κ 1 κ 2 = 0.

8 8.6 Appaent singulaity 116 Figue 8.1: Embedding diagam fo the chwazschild φ) plane: Flamm s paaboloid 8.6 Appaent singulaity The line element 8.36) appeas to contain two singulaities, one at = 0 coming fom the g tt component blue dashed line) and anothe at = R coming fom g ed line). Is this a poblem? Not necessaily. In most of the astophysical applications the typical size R of the souce is much lage than the chwazschild adius 8.37) R R 10 9 R, R 10 6 R, R ) N This fact makes the singulaities at = 0 and = R s completely ielevant in most of the cases, since they lie in the inteio of objects whee the exteio chwazschild solution does not apply. Indeed, the poblem disappeas when one conside ealistic inteio solutions of the Einstein equations ds 2 = 1 2GM) ) dt GM) ) 1 d dω 2, 8.47) since the function M) deceases faste than and effectively kills all the above singulaities.

9 8.7 Geodesics in chwazschild metic 117 We should woy and speculate about the singulaities only in those cases in which the size of the object is such that the chwazschild-doste solution applies all the way down to = R. This kind of objects ae called black holes. Even in that case the two singulaities descibed above ae not on equal footing. The metic coefficients in the line element 8.36) depend on the choice of a paticula coodinate system and you should not extact any conclusion fom them alone. Let me pesent an illustative example. A woked-out examples: Coodinates should not be tusted Conside the completely egula and singulaity-fee Euclidean space in two dimensions dx 2 = dx 2 + dy 2, 8.48) and pefom a geneal coodinate tansfomation to a new vaiable ρ defined though x = 2 ρ to get ds 2 = 1 ρ 2 dρ2 + dy ) The metic appeas to blow up at ρ = 0 even though we know that ou space is, by constuction, flat and fee of singulaities. The appaent singulaity is a beakdown of ou coodinate system at the point in which ρ becomes negative. It has nothing to do with a beakdown of the undelaying manifold! In ode to detemine if we ae dealing with some atifice of ou coodinate system o with a tue physical singulaity, we cannot neithe look to the cuvatue tensos alone, since thei components ae coodinate-dependent 7. We should athe constuct scalas out of the cuvatue tensos. If any the scala blows up in a paticula coodinate system, it will do in all of them. The simplest possibility would be to conside the Ricci scala, R but we can also constuct highe ode scalas such as R µν R µν R µνρσ R µνρσ. Fo the paticula case of the chwazschild-doste metic, the fist two quantities ae not useful since ae identically equal to zeo. We ae foced then to conside the squae of the Riemann tenso, the so-called Ketschmann scala. Taking into account the non-vanishing components of the Riemann tenso 8.18), we obtain K = R µνρσ R µνρσ = 12R2 6, 8.50) which is a pefectly egula quantity at the chwazschild adius, but becomes infinity at = 0. This last point is a eal physical singulaity! The singulaity at = R is, on the othe hand, just a pathology of the specific coodinate system used. 8.7 Geodesics in chwazschild metic Let us study the motion of pointlike objects in ou ecently found chwazschild solution. To do that, let me conside the epaametization invaiant action 3.27) = L dσ = 1 dσ e 1 dx µ dx ν ) σ)g µν 2 dσ dσ m2 eσ), 8.51) 7 They can catch singulaities when going fom one coodinate system to anothe though the tansfomation matices x µ / x ν.

10 8.7 Geodesics in chwazschild metic 118 in the massive eσ) = 1/m) and massless eσ) = 1, m 0) cases 8 massive = 1 2 m dx dσ g µ dx ν ) µν dσ dσ 1, massless = 1 2 dx dσ g µ µν dσ dx ν ). 8.52) dσ The geodesic equations fo both actions can be diectly witten by taking into account the nonvanishing Chistoffel symbols 8.17). Let s denote the deivatives with espect to the affine paamete σ by a dot. The explicit fom 9 obtained by following this pocedue tuns out to be not vey useful since the esulting equations ae coupled R ẗ = R )ṙṫ, 8.53) = R R ) 2 3 ṫ 2 R + R ) θ + sin 2 θ 2 R, 8.54) θ = 2 θṙ + sin θ cos θ φ 2, 8.55) φ = 2 φṙ 2 cot θ θ φ. 8.56) Fotunately, ou task can be geatly simplified by consideing the symmeties of the chwazschild- Doste metic. ince 8.36) does not depend on the coodinates t and θ they ae cyclic coodinates in 8.52)), we have two consevation laws t L = 0 E = 1 R ) ṫ = constant, 8.57) φ L = 0 h = 2 sin 2 θ φ = constant, 8.58) with a clea physical intepetation. In the massless case, E and h ae the elativistic enegy and angula momentum that the paticle would have at =. In the massive case, they ae the elativistic enegy and angula momentum pe unit mass. Execise Check this by taking the non-elativistic limit of 8.57) and 8.58) at the equatoial plane θ = π/2. Consevation of angula momentum means that the paticle moves in a plane, which we can set to be the equatoial plane θ = π 2 without loss of geneality. Indeed, a simple inspection of Eq. 8.55) shows that if we conside a geodesic passing though a point on the equato θ = π 2 and tangent to the equatoial plane θ = 0, we will always have θ = 0 and θ = 0. On top of the above symmeties, we have still a geneic consevation law associated to the invaiance of the action 8.51) unde epaametizations of the path σ σ = fσ) cf. ection 3.5.1). This eads d gµν u µ u ν) = 0 g µν u µ u ν = ɛ, 8.59) dσ with ɛ = 1 and 0 fo massive and massless paticles espectively. Expanding this equation 10 8 Remembe that σ = τ in the massive case. 9 Up to a global facto m in the massive case. 10 Remembe that θ = π/2. 1 R ) ṫ R ) 1 ṙ φ2 = ɛ 8.60)

11 8.7 Geodesics in chwazschild metic 119 and plugging 8.57) and 8.58) we obtain a single equation fo σ) with V ) ɛ GM playing the ole of an exact effective potential and 1 2 ) 2 d + V ) = E, 8.61) dσ E h2 2 2 GMh ) E 2 ɛ ). 8.63) Eq. 8.61) is stuctually equivalent to that of a paticle of unit mass and enegy 11 E moving in an effective potential V ). It is inteesting to compae the obtained potential with the Newtonian esult V N ) = GM + h ) The fist two tems in Eq. 8.62) ae just the univesal gavitational attaction and the centifugal baie that wee aleady pesent in Newton s theoy of gavity. The thid tem is new. At sufficiently long distances, the exta contibution is athe small and does not significantly modify the Newtonian effective potential 12 cf. Fig. 8.4). The situation is completely diffeent at shot distances. The new tem eventually dominates ove the centifugal baie fo small and dives the potential to 13. Let me analyze the massive and massless case sepaately. Massive paticles, ɛ = 1, σ = τ: We can distinguish two cases: If h 2 > 3R 2, the potential displays both a maximum and a minimum at dv ) d = 0 max,min = h2 1 ± 1 3 ɛ=1 h R R ) 2, 8.65) We have then fou possibilities depending of the elation between the effective enegy of the paticle and the potential cf. Fig.??): 1. Cicula obits: If E = V max ) o E = V min ) the paticle descibes an unstable o stable obit espectively. 2. Bound pecessing obits: If 0 > E > V min ) the paticle is tapped into the potential and descibes an elliptical obit with shifting peihelion see below). 3. catteing obits: If V max ) > E > 0 the paticle bumps in the potential and eteats back to infinity. 4. Plunging obits: If E > V max ) the paticle sails ove the top of the potential to finally spial into the black hole. 11 The tue enegy pe unit mass in E but the effective potential fo athe esponds to E. 12 The small coection will play howeve a cental ole! ee next section. 13 Note that the potential is always zeo at = R.

12 8.7 Geodesics in chwazschild metic VN V Figue 8.2: Effective potentials in Newtonian gavity and Geneal Relativity fo massive paticles. Diffeent lines coespond to h 2 /R 2 = 0, 1, 3, 5, 7, 9 fom bown to blue). Note the change in the potential at the citical value h 2 /R 2 = 3. Figue 8.3: Obits fo massive paticles in chwazschild-doste geomety

13 8.7 Geodesics in chwazschild metic 121 Execise What happens with max and min when h 0? And when h deceases? Which is the minimal value of h and allowing fo a stable cicula obit? If h 2 < 3R 2 the centifugal baie disappeas and the paticle has no othe option but to spial into the singulaity. Conside fo claity the limiting case h = 0 in which the paticle follows a adial tajectoy. In this case, the adial equation of motion 8.61) becomes 14 d dτ = ± R ) 1/2 Integating the pevious equation we get d 1/2 = R dτ. 8.66) τ) = 2 3 R 3/2 0 3/2), 8.67) with 0 > an integation constant fixing the initial value of the pope time to zeo. The paticle eaches the chwazschild adius in a finite pope time τ. The inteval measued by an obseve at est at spatial infinite is howeve quite diffeent. Indeed, it is infinite, as can be easily seen by evaluating d dt = dτ d dt dτ = 1 R ) ) 1/2 R dt = R 1/2 3/2 d R 8.68) at = R. Fo the obseve at infinite the paticle appeas to appoach but neve quite coss the hoizon! This is just anothe indication that the chwazschild coodinates ae flawed nea R = R. Execise What happens with t when the obseve cosses the hoizon? Massless paticles, ɛ = 0: The potential 8.62) with ɛ = 0 displays a unique maximum fo all values of h at Thus, the motion of massless paticles can be divided into thee cases: max = 3 2 R. 8.69) 1. Cicula obit: If E = V max ) the paticle descibes an unstable cicula obit. 2. catteing obits: If V max ) > E the paticle bumps in the potential and eteats back to infinity deflection of light). 3. Plunging obits: If E > V max ) the paticle sails ove the top of the potential to finally spial into the black hole. 14 Among the two signs in the squae oot we take the negative one, in such a way that we fall towad 0

14 8.7 Geodesics in chwazschild metic VN V Figue 8.4: Effective potentials in Newtonian gavity and Geneal Relativity fo massless paticles. Diffeent lines coespond to h 2 /R 2 = 0, 1, 3, 5, 7, 9 fom bown to blue). Figue 8.5: Obits fo massless paticles in chwazschild-doste geomety

15 8.8 olving the adial equation olving the adial equation Let us detemine the equation fo the obits descibed in the pevious section. Fo doing that, we make use of Eq. 8.58) with θ = π/2 and change the deivatives with espect to the affine paamete in Eq. 8.61) to deivatives with espect the angula vaiable φ to obtain d dσ = d dφ dφ dτ = h d 2 dφ, 8.70) ) 2 h d 2 + h2 dφ 2 = ɛ2gm + 2GMh E. 8.71) The ticks to solve this kind of equation ae well known. Let s pefom a change of vaiable u 1/ in 8.71) ) 2 du + u 2 = ɛ 2GMu dφ h 2 + 2GMu 3 + 2E h 2, 8.72) and deive the esult with espect to φ. This gives ise to a second ode diffeential equation of the fom d 2 u dφ 2 + u = ɛgm h 2 + 3GMu ) The massive case: Peihelion advance of Mecuy In the massive case ɛ = 1 and 8.73) becomes d 2 u dφ 2 + u = GM h 2 + 3GMu ) The esulting equation is extemely simila to the Newtonian equation of motion of a paticle of mass m in the equatoial plane d 2 u 0 dφ 2 + u 0 = GM h ) even though the intepetation of the adial vaiable is completely diffeent 15. As you pobably emembe fom you Classical Mechanics couse, the geneal solution of 8.75) is a conic u 0 = GM h e cos φ) 0 = a1 e2 ) 1 + e cos φ) 8.76) with a1 e 2 ) = h2 GM. 8.77) 15 In Newtonian gavity is the adial distance fom the mass while in the elativistic it is just a adial coodinate that can be only elated to a distance though the metic.

16 8.8 olving the adial equation 124 Obital Mumbo Jumbo a=semi-majo axis: 1/2 of the long axis of the ellipse. b=semi-mino axis: 1/2 of the shot axis of the ellipse. e=eccenticity : It chaacteizes the deviation of the ellipse fom cicula. When e = 0 the ellipse is a cicle, when e = 1 the ellipse is a paabola. It is defined in tems of the semi majo and semi mino axes a and b as e = 1 ) 2 b. 8.78) a f=focus: The point ove the semi-majo axis at a distance f = ae fom the geometic cente of the ellipse. l=semi-latus ectum: The distance l = b2 a paallel to the semi-mino axis. fom the focus to the ellipse along a line p =peiapsis: The distance p = a1 e) fom the focus to the neaest point of appoach of the ellipse. a =apoapsis: The distance a = a1+e) fom the focus to the futhest point of appoach of the ellipse. The equation of the obit: It gives the distance to the obiting body fom the focus of the obit as a function of the pola angle θ θ) = a1 e2 ) 1 + e cos θ. 8.79) If the gavitational field is sufficiently weak, Newtonian gavity alone is expected to povide a good appoximation to the motion of massive paticles in Geneal Relativity. This suggest to teat to exta tem 3GMu 2 as a petubation of top of the solution of Eq. 8.75). The petubative solution of Eq. 8.74) can be detemined by consideing the antsatz u = u 0 + u, 8.80)

17 8.8 olving the adial equation 125 with u 0 given by 8.76). Inseting this into 8.74) we get d 2 ) u [1 dφ 2 + u = A + e2 + 2e cos φ + 1 ] 2 2 e2 cos 2φ with 8.81) A = 3 GM)3 h ) a tiny paamete. To solve this equation let me notice two identities d 2 φ φ sin φ) + φ sin φ = 2 cos φ, 8.83) dφ2 d 2 cos 2φ) + cos 2φ = 3 cos 2φ. 8.84) dφ2 A diect compaison of 8.83) and 8.84) with 8.81) suggests the solution ) u = A [1 + e2 16 ] 2 cos 2φ + eφ sin φ, 8.85) which can be checked by diect diffeentiation. The thee tems in the squae backet ae athe diffeent. The fist and the second one ae just a constant and an oscillatoy tem aound zeo, both of them vey small due to the constant A in font. The thid one is diffeent since it accumulates ove successive obits and gadually gows with time. Retaining only this last tem we get which can be witten in a moe enlightening way by taking into account that fo u = GM [1 + e cos φ + αφ sin φ)] 8.86) h2 u GM [1 + e cos 1 α) φ] 8.87) h2 cos [φ 1 α)] = cos φ cos αφ + sin φ sin αφ cos φ + αφ sin φ 8.88) α 3 GM)2 h ) The solution 8.87) shows that the obit is still peiodic, but with a peiod that is not longe 2π, but athe 2π1 α). The values of epeats on cycles lage than 2π and the obit pecesses.

18 8.8 olving the adial equation 126 The advance of the peihelion in one evolution is φ = 2π 1 α 2π 2φα = 6πG2 M 2 h 2, 8.90) which taking into account 8.77) can be witten as 16 note that we estoe the c factos) φ = 6G2 M 2 h 2 c 2 = 6πGM a1 e 2 )c ) Because it is a small effect, let s accumulate this ove 100 yeas to get the obsevable quantity φ 100 φ T 100 yeas centuy, 8.92) with T the peiod of the obit in yeas. In tems of obsevable obits within the sola system, Mecuy is the closest planet to the un, and so it should have the lagest pecession. Object Mass Mean Equatoial Radius Peiod emimajo axis Eccenticity kg) 10 3 km) days) 10 8 km) Mecuy Venus Eath Mas Jupite atun Uanus Neptune Taking into account the values fo Mecuy s obit, we obtain φ ) The majo axis of Mecuy pecesses at a ate of 43 acsecs pe centuy. The obsevational esults ae in excellent ageement with Geneal Relativity Planet Obseved esidual GR pediction Mecuy ± 0.45) Venus 8.4 ± 4.8) 8.6 Eath 5.0 ± 1.2) The use of the expessions fo the unpetubed solution is justified by the fact that we ae looking to a vey small quantity.

19 8.9 The massless case: Gavitational deflection of light acseconds and the end of the Newtonian empie Newton s theoy had been a vey successful theoy, extensively used by astonomes fo centuies. It had pedicted the etun of comet Halley 1758) with an eo of 33 days, the elliptical chaacte of the ecently discoveed Uanus 1781) and even moe supisingly the location, mass and obit paametes of Neptune, even befoe it was diectly obseved 1846). Leveie discoveed it just with the point of his pen! a ; clealy an amazing poof of the univesality of the gavitational inteaction. Nevetheless, at the end of the 19th centuy thee wee still some caveats elated to Mecuy s obit. As you should know the 1/ 2 dependence of the Newton s foce gives ise to elliptical tajectoies on a plane, and the coesponding peihelion is a pioi a fixed point b. Howeve, diffeent petubations due fo instance to the pesence of othe massive objects in the ola system, such as Jupite, o to the quadupole moment of the un), give ise to a peihelion advance, and theefoe to an ellipse tuning on the plane. Even when all those effects wee taken into account thee was a esidual contibution to the shift. As pointed out by Leveie and Newcomb at the end of the 19th, Mecuy s peihelion pecesses at a ate of 575 pe centuy, but only 532 can be explained by the petubations associated to the othe planets. The emaining 43 pe centuy could not be accounted fo by the Newtonian theoy even when eos wee taken into account. The obsevational poblem was basically closed fo eveyone apat fom Leveie c ), but the theoetical poblem would emain open till the intoduction of Geneal Relativity in a Fancois Aago b The Laplace-Runge-Lenz vecto is conseved. c He died believing that the histoy of the discovey of Neptune would epeat and a new planet with a mass enough to account fo the 43 pe centuy would be encounteed between the un and Mecuy. 8.9 The massless case: Gavitational deflection of light Let us conside now the massless case whee ɛ = 0 and 8.61) becomes d 2 u dφ 2 + u = 3GMu ) In the absence of the tem 3GMu 2, the pevious solution educes to the simple hamonic oscillato equation d 2 u 0 dφ 2 + u 0 = 0, 8.95) whose solutions u 0 = sin φ, 8.96) b can be intepeted as staight lines passing at a distance b fom the oigin. Following a simila pocedue to the one used in the pevious section, we look fo petubative solutions of the fom u = u 0 + u = sin φ b + u 8.97) with u 0 given by 8.96). ubstituting 8.97) into 8.94) we get a linea equation in u whose solution is given by d 2 u dφ 2 u = 3GM 2b 2 3GM + u = b 2 sin 2 φ, 8.98) ) cos 2φ). 8.99)

20 8.10 The post-newtonian fomalism 128 Adding this to the unpetubed solution we get u = sin φ b + 3GM 2b 2 which in the limit, u 0 and fo small φ gives ise to The total deflection angle is twice this value ) cos 2φ), 8.100) φ 2GM bc ) φ = 2R s b Fo ays coming fom a distant stas and gazing the suface of the un 17 we get = 4GM bc ) b R = km M = Kg 8.103) φ = 4GM c 2 R = ) Light paths so close to the un ae of couse not visible by day, but they become visible at the time of a total eclipse. Thei position elative to the othe backgound stas duing the total eclipse appeas shifted elative to the position in the usual night sky. This pediction of Geneal Relativity was veified in 1919 just a few yeas late the fomulation of the theoy. Two sepaate goups led by Athu Eddington and Andew Commelin moved to Guinea and Bazil to obseve the total eclipse of May 29, They epoted deflections of 1.61 ± 0.40) and 1.98 ± 0.16), in easonable ageement with Einstein s pediction 8.104) The post-newtonian fomalism Nowdays, the ageement between theoy and obsevation is at the level of a few pats in a thousand. The deviations fom the Geneal Relativity ae usually paametized in tems of the so-called post- Newtonian paametes β and γ measuing espectively the non-lineaity in the supeposition law fo gavity and the spatial cuvatue poduced by unit est mass ds 2 = 1 2GM + 2 β γ) G2 M 2 ) 2 dt γGM ) 1 d dω ) When this paametes ae taken into account Eqs. 8.91) and 8.102) become espectively ) ) 2 β + 2γ 6πGM 1 + γ 4GM φ = 3 a1 e 2 )c 2, φ = 2 bc ) The Geneal Relativity limit coesponds to γ = β = 1. Recent measuements povide values γ = ± and 2γ β 2 < , in excellent ageement with GR. 17 In this case, the effect is maximized and easie to obseve.