GENERAL RELATIVITY: THE GEODESICS OF THE SCHWARZSCHILD METRIC GILBERT WEINSTEIN 1. Intoduction Recall that the exteio Schwazschild metic g defined on the 4-manifold M = R R 3 \B 2m ) = {t,, θ, φ): > 2m} is given by: g = 1 2m ) dt 2 + 1 2m ) 1 d 2 + 2 dθ 2 + sin 2 θ dφ 2 ). It descibes the gavitational field outside a spheically symmetic body of mass M. Fou of the tests of geneal elativity ae based on this metic and its geodesics: the gavitational edshift, the bending of light, the pecession of the peihelion of Mecuy, and the time delay of ada signals. In these lectues, we will study the null and timelike geodesics of the Schwazschild metic, and descibe the fist thee tests mentioned above. The following simple esult can be viewed as a special case of Nöthe s Theoem. Poposition 1. Let M, g) be a pseudo-riemannian manifold, let ξ be a Killing vecto field of M, g), and let γ be a geodesic of M, g). Then gξ, γ) is constant. Poof. Extend γ in a neighbohood of γ to a vecto field X. Since L ξ g = 0, we have Thus: 2g ξ X, X) = ξ gx, X) ) = 2g[ξ, X], X), X gξ, X) ) gξ, X X) = g X ξ, X) = g ξ X [ξ, X], X) = 0. Since along γ, we have X = γ, and γ γ = 0, we conclude γ gξ, γ) ) = 0. The Schwazschild metic is ich in isometies; its isomety goup is G = R SO3). This, in conjunction with Poposition 1, will be used to integate explicitly the geodesic equations. The vecto field t = / t is a futue diected timelike Killing field, which we call the static Killing field. Note that g t, t ) = 1 2m/). The integal cuves of t when epaameteized by pope time ae the wold lines of static obseves, i.e. obseves which ae fixed as viewed fom infinity. 1
2 GILBERT WEINSTEIN 2. The Gavitational Redshift Light, as all electomagnetic waves, is modeled in elativity by solutions of the Maxwell equations. Howeve, fo high fequencies, the geometical optics appoximation is quite accuate. In this appoximation, light signals ae modeled by null geodesics. If a light signal is given by a null geodesic γ, then its fequency measued by an obseve β eceiving the signal at a point p M is ω = g γ, β) p. Conside now two static obseves β 1 and β 2 in the Schwazschild spacetime, and suppose that β 1 emits a light signal γ at point p 1 which is eceived by β 2 at point p 2. Fom the fact that both obseves ae static, it follows that β j = t 1 2m/, j = 1, 2. By Poposition 1, we have g t, γ) constant along γ, hence the gavitational edshift facto, defined to be one less than the atio of the fequency of the light emitted ω 1 to the fequency of the light eceived ω 2, is: ω 1 ω 2 1 = 1 2m/ 2 1 2m/ 1 1. If the light is emitted close to the cente, whee the gavitational field is stonge, and eceived futhe fom the cente, whee the gavitational field is weake, then the fequency is shifted towad the ed. This effect has been measued to 1% accuacy in laboatoy expeiments on eath. The gavitational edshift of spectal lines fom the Sun has also been obseved, though with lesse accuacy due to othe effects. We make one last inteesting obsevation elated to the edshift. One can show athe easily that fo any spheically symmetic pefect fluid body in equilibium, thee holds M/R 4/9, egadless of the equation of state, whee M is the total mass, R is the aea adius R = A/4π, and A is the suface aea of the body. It follows that the maximum edshift facto, when 2 = and 1 = R, is 2. The edshift facto fom quasas is commonly fa lage than 2. This is pesumably due to cosmological effects. 3. Integation of the Geodesic Equations Let γ be any geodesic then g γ, γ) is constant along γ. Thus if γ is not null, we can assume that it is paameteized by pope time τ, i.e. that g γ, γ) = 1. If γ is null, we assume that τ is any affine paamete along γ. Note that we still have the feedom to escale τ this case. We also make the following simple obsevation. Lemma 1. Let M, g) be a pseudo-riemannian manifold, and let ϕ be an isomety of M.g). Let Σ be the set of fixed points of ϕ: {p M : ϕp) = p}. Then any geodesic γ initially on Σ and tangent to Σ emains in Σ. In paticula, Σ is totally geodesic.
GEODESICS OF SCHWARZSCHILD METRIC 3 Poof. Indeed, suppose to the contay that γτ) Σ fo some τ. Then ϕγτ)) γτ). Howeve, ϕ γ is a geodesic with the same initial conditions as γ, since ϕγ0)) = γ0) and ϕ γ0)) = γ0). This violates the uniqueness theoem fo geodesics. Fo the Schwazschild solutions M, g), the map t,, θ, φ) t,, π θ, φ) is an isomety whose set of fixed points is the equatoial plane Σ = {t,, θ, φ): θ = π/2}. If γ is any geodesic then thee is an isomety ϕ SO3) such that ϕγ0)) Σ and ϕ γ0)) T Σ. By Lemma 1, it follows that ϕ γ emains in Σ. Theefoe, it is sufficient to conside geodesics in the equatoial plane. The same esult could have been obtained fom Poposition 1. Let γ = t,, π/2, φ) be a geodesic in the equatoial plane. Then t,, φ) is a citical point of the Lagangian: 1) E = { 1 2m ) ṫ 2 + 1 2m ) 1 ṙ 2 + 2 φ2} Using ξ = t and ξ = φ in Poposition 1, we obtain two conseved quantities: E = 1 2m ) 2) ṫ dτ. 3) L = 2 φ, which we call the enegy and the angula momentum of γ espectively. If the angula momentum is zeo, it follows that φ = constant. These geodesics, called adial geodesics, will be studied in Section 4. The last conseved quantity is obtained fom the condition g γ, γ) = constant: 1 2m ) ṫ 2 + 1 2m ) 1 4) ṙ 2 + 2 φ2 = ɛ, whee ɛ = { 0 if γ is null; 1 if γ is timelike. Substituting fom 2) and 3) into 4), we obtain: ṙ 2 + 1 2m ) ) L 2 5) 2 + ɛ = E 2. This is equivalent to the equation of motion fo a paticle of mass 1 and enegy E 2 on a one-dimensional line in the effective potential V = 1 2m/)L 2 / 2 + ɛ). The qualitative analysis of these geodesics will be done in Section 5 fo null geodesics ɛ = 0, and in Section 6 fo timelike geodesics ɛ = 1.
4 GILBERT WEINSTEIN 4. Radial Geodesics Fo adial geodesics, Equation 5) educes to: ṙ 2 + ɛ 1 2m ) 6) = E 2. Fo null geodesics ɛ = 0, we ae fee to escale the affine paamete τ so that E = 1, and we obtain ṙ = ±1. Thus, afte adjusting the oigin of the affine paamete τ, we have = ±τ. Fo outgoing null geodesics ṙ = 1, and Equation 2) now yields: t t 0 = dτ 1 2m/τ = τ + 2m logτ 2m). The outgoing null geodesics thus have the equation: 7) t t 0 = + 2m log 2m). The incoming null geodesics can be obtained similaly, and have the equation: 8) t t 0 = + 2m log 2m) ). Note that all null geodesics each = 2m within finite affine paamete even though Schwazschild time t is infinite upon thei aival. Equations 7) and 8) ae impotant when discussing extensions of the Schwazschild spacetime beyond its event hoizon = 2m. We now tun to the timelike adial geodesics. In this case, Equation 6) becomes: ṙ 2 + 1 2m ) = E 2 It is inteesting to note that the equation of motion in this case is = m/ 2 just as in the Newtonian case. The effective potential has no citical points, and theefoe the motion is vey simple. If E 2 < 1, then the obit is a cash obit, i.e. climbs to a maximum at which point it tuns aound and deceases monotonically until = 2m within finite affine paamete. If E 2 1, then depending upon the initial diection, the obit eithe escapes o cashes. We call these cash/escape obits. The equations of motion can be integated in tems of quadatues: d τ = E 2 1 + 2m/ E 2 dτ t = 1 2m/. Although, these integals can be caied out explicitly, the esulting fomulae bea no paticula inteest.
GEODESICS OF SCHWARZSCHILD METRIC 5 5. Null Geodesics and the Bending of Light Fo null geodesics, we may as befoe, escale the affine paamete to set E = 1. Equation 5) then becomes: ṙ 2 + L2 2 1 2m ) = 1. The equations of motion again can be integated in tems of quadatues: dτ d = 1 9) 1 L 2 / 2 )1 2m/) dt dτ = 1 10) 1 2m/ 11) dφ dτ = L 2. The effective potential has one citical point at = 3m. This coespond to an unstable cicula obit, a tapped ay. The potential has a maximum of L 2 /27m 2 at this citical point. Hence an obit initially in < 3m can escape if and only if L 2 < 27m 2, and convesely and obit initially in > 3m will be not get tapped if and only if L 2 > 27m 2. Conside now an obit coming in fom infinity, i.e. with as τ. Without loss of geneality, assume that L > 0. Multiplying Equation 11) by Equation 9), we obtain 12) dφ d = L 2 1 L 2 / 2 )1 2m/), fom which it follows immediately that dφ/d L/ 2, and hence φ φ 0 = constant as τ. If L 2 > 27m 2, then the obit will not get tapped, and also as τ. Thus, φ φ 1 = constant as τ +. We wish to compute the maximum deflection angle δ = φ 1 φ 0 π of the obit fom a linea obit in flat space. It is moe convenient to intoduce the vaiable u = 1/, and the paamete a = 1/L. This leads to the equation 13) du dφ ) + u 2 1 2mu) = a 2. Fom the qualitative analysis, it is easily seen that the obit will be symmetic about its peihelion whee will be at a minimum, say 0, and consequently u at a maximum u 0 = 1/ 0. We point out that u 0 is the positive oot of the cubic equation 14) a 2 u 2 + 2mu 3 = 0. In view of Equation 13), we now deduce that 15) δ + π = 2 u0 0 du a 2 u 2 + 2mu 3. This is an elliptic integal, and cannot be caied out explicitly. Howeve, we only need to compute the linea contibution of m to this integal. To
6 GILBERT WEINSTEIN this pupose, we make the substitutions v = u/u 0 and µ = u 0 m in 15). Then, noting that Equation 14) implies a 2 = u 2 0 1 2µ), we find δ = 1 0 dv 1 2µ v 2 + 2µv 3. Since clealy δ is only a function of µ, we have dδ = δ µ dµ = δ µ u 0 dm + m du 0 ). When m = 0, we find δ µ 0) = 2 1 0 1 v 3 ) 1 v 2 dv = 4, ) 3/2 and also dµ = u 0 dm. Consequently, we obtain dδ m=0 = 4u 0 dm. Thus, to fist ode in m, a null obit passing at a peihelion of 0 will expeience a deflection in its azimuthal angle of δ 4Gm c 2 0, whee G is the gavitational constant and c the speed of light. In cgs units G = 6.67 10 8 cm 3 /g sec 2, and c = 3 10 10 cm/sec. With m = 2 10 33 g and 0 = 7 10 10 cm the mass and adius of the sun espectively, one obtains fo an obit which gazes the suface of the sun: δ 0.85 10 5, o about 1.7 seconds of ac. This is within 1% of obseved data. 6. Timelike Geodesics and Peihelion Pecession We now tun to the study of the timelike geodesics. Equation 4) now has the fom ṙ 2 + 1 2m ) ) L 2 2 + 1 = E 2. Thus, as befoe, this is identical to the motion of a paticle of mass 1 on a line in the effective potential V ) = 1 2m/ + L 2 / 2 2mL 2 / 3. It is inteesting to note that the only diffeence with the Newtonian case is the elativistic coection tem 2mL 2 / 3. The qualitative analysis of this potential depends on the atio L 2 /m 2. If L 2 < 12m 2, then V has no citical points and just as in the adial case, the analysis is simple: depending whethe the enegy E 2 < 1 o E 2 1 the obit is eithe a cash o a cash/escape obit. If L 2 = 12m 2, the situation is the essentially the same, with the exception that thee is now an unstable cicula obit at = 6m, and an exceptional obit with E 2 = 8/9 which spials into this obit. If L 2 > 12m 2, thee ae two citical points 1 < 6m < 2, the oots of the equation m 2 L 2 + 3mL 2 = 0. We will not poceed futhe with this analysis, but instead point out that the cicula obit at = 2 is now a stable cicula obit. Obits with initially close to 2 and enegy E 2 slightly lage
GEODESICS OF SCHWARZSCHILD METRIC 7 than V 2 ) will have oscillating between min and max. When L 2 /m 2 is much lage than 12, than 2 will be lage, hence those obits will have lage thoughout and the elativistic coection tem small. These obit will be nealy elliptical. Conside such an obit, with peihelion at τ = 0 φ = 0, = min. The obit will each peihelion again at some pope time τ = τ 0, and φ = φ 0 We now wish to compute the peihelion pecession of such an obit, i.e. ψ = φ 0 2π. We can wite 16) max ψ = 2 min L d 2 E 2 V ) 2π. Intoduce the vaiable u = 1/, and the paametes u min = 1/ min and u max = 1/ max. Then the integal in 16) can be ewitten as 17) ψ = 2 α 1 du E 1 1)/L 2 + 2m/L 2 )u u 2 + 2mu 3 2π. Again, this is an elliptic integal, and cannot be evaluated explicitly. Howeve, as befoe, it is sufficient to compute the linea contibution of m. Fo this pupose, intoduce the paametes a and e by: min = a1 e), max = a1 + e), and the vaiable v = au. In analogy with Newtonian mechanics, we call a and e the eccenticity and semimajo axis of the obit espectively. With these substitution, the integal in 17) becomes ψ = 2 v2 v 1 dv µv0 v)v v 1 )v 2 v) 2π, whee µ = 2m/a, v 1 = 1/1 + e), v 2 = 1/1 e), and v 0 + v 1 + v 2 = 1/µ. It is clea fom this fomula that ψ is a function of e and µ. It is easy to see that when ψ = 0 when µ = 0, hence ψ e 0, e) = 0. Diffeentiating with espect to µ at µ = 0, we find: ψ µ 0, e) = v2 Theefoe, we conclude that v + v 1 + v 2 ) v 1 v v1 )v 2 v) dv = dψ µ=0 = 3π 1 e 2 dµ. 3π 1 e 2. Thus, fo an obit of ellipticity e and semimajo axis a, we have obtained to fist ode in m a peihelion pecession ate ψ 6πGm c 2 a1 e 2 ), pe evolution. Fo the obit of Mecuy aound the sun, we have e = 0.206 and a = 5.79 10 12 cm, giving a peihelion advance ψ 5 10 7 pe
8 GILBERT WEINSTEIN evolution. The peiod T of Mecuy can be obtained fom Keple s Thid law T 2 = 4π2 a 3 Gm = 5.7 1013 sec 2, i.e. a peiod of about 7.6 10 6 sec o 88 days, with about 415 evolutions pe centuy. Thus, we get a peihelion pecession ate of about 43 seconds of ac pe centuy. This is exactly the obseved value, afte taking into account the Newtonian petubations due to neaby planets. Einstein is quoted as having said that when he discoveed this esult he felt as if the univese had whispeed in his ea.