The Precession of Mercury s Perihelion

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1 The Pecession of Mecuy s Peihelion Owen Biesel Januay 25, 2008 Contents 1 Intoduction 2 2 The Classical olution 2 3 Classical Calculation of the Peiod 4 4 The Relativistic olution 5 5 Remaks 9 1

2 1 Intoduction In this pape, we will attempt to give a demonstation that Geneal Relativity pedicts a ate of peihelion pecession equal to that of Mecuy s obit aound the un when the influences due to othe planets have aleady all been accounted fo). Fist, we will use classical physics to seve a twofold pupose: to demonstate that classical obits ae closed) ellipses, and also to illustate the methods involved in the elativistic solution. econd, we will apply these methods to a geneal elativistic teatment of geodesics in the chwazschild metic, and show that an obit matching Mecuy s specifications can be expected to shift by appoximately 43 acseconds pe centuy. 2 The Classical olution We will begin with thee-dimensional pola coodinates, whee the metic is ds 2 d dω 2 with dω 2 dθ 2 + sin 2 θdφ 2. In these coodinates, we can expess the Lagangian as L 1 2 mẋ2 V x) 1 2 m ṙ θ2 + 2 sin 2 θ φ 2] + GMm whee we have substituted the gavitational potential V x) GMm x. The equations of motion fo a paticle ae then given by the Eule-Lagange equations L d L x i dt. These become, with x i, θ, φ espectively: ẋ i m θ2 + sin 2 θ φ 2] GMm 2 d mṙ) m dt m 2 sin θ cos θ φ 2 d ) m 2 θ dt 0 d m 2 sin 2 θ dt φ ) Notice that these equations ae invaiant unde θ π θ, unde which sin θ sin θ, cos θ cos θ, and θ θ. Then fo an initial value poblem with θ0) π 2, θ0) 0 that is, the motion of the paticle begins in the equatoial plane), any solution with t), θt), φt)) immediately gives Page 2 of 10

3 us anothe solution t), π θt), φt)), which contadicts local uniqueness of the solution to the initial value poblem unless θt) π 2. ince any initial value poblem can be otated into one of this fom, we will now assume that θ π 2, educing the Eule Lagange equations whee we have also canceled m) to: φ 2 GM 2 0 d dt ) 2 φ The second equation says that L 2 φ is a constant of the motion; if it is zeo we find that GM < 0 and hence is concave down. ince concave 2 down functions ae unbounded below, we would find that fo some t, 0, which descibes the uninteesting event in which the object cashes into the sun. Hence, we will estict ou attention to L 0. In that case, φ is eithe always negative o always positive, in which case φt) is monotone and we can wite t tφ). Hee we ae letting φ ange though R, and consideing the fact that φ and φ+2π descibe the same point only as a cuiosity.) Then t φ φ L 2 φ. Hence we may ewite the othe equation of motion as φ 2 GM 2 L2 3 GM 2 L 2 φ ) L 2. φ To make this equation moe eadily solvable, we make a change of vaiables to u 1/. Then denoting diffeentiation with espect to φ by a pime, we have u / 2, and so the diffeential equation becomes L 2 u 3 GMu 2 L 2 u 2 u u + u GM L 2 We can easily solve this equation as uφ) A cosφ φ 0 )+ GM. By suitably L 2 tanslating φ, we can choose φ 0 0 and A 0, in which case we can ewite this as uφ) GM 1 e cosφ)) 1) L2 with e AL2 GM 0. It is well-known 1] that Equation 1 descibes an ellipse of eccenticity e. Page 3 of 10

4 3 Classical Calculation of the Peiod As an altenate demonstation that φ) is peiodic with peiod 2π, conside that the binding enegy pe unit mass) of the system is anothe constant of the motion: E 1 ) ṙ φ2 GM 2, whee we have used E instead of E so that E > 0. Then we can solve this fo ṙ: ṙ 2 2E + 2GM ) L 2 2 2E + 2GM ) 2 2E L GM 2 1 ± R + 1 R + L2 2 L2 2 L ) ) ) 1 R ) ) 1 R whee we have intoduced the notation R ± fo the nonzeo oots of the quatic polynomial in 2); since these ae the only places whee 0 and 0, we may identify them as the aphelion and the peihelion of a closed obit. ince φ 1 φ/, we can find the amount of φ equied to pass fom R to R + by integating: R+ d φ + φ ) ) R 1 R + R 1 R + ) R ) + 2 R + R actan 2 R + ) R )R R + actan+ ] actan ] π 2 + π 2 π ] R+ Hence the paticle will tavel fom R to R + and back evey time φ φ+2π, so the obit φ) is peiodic with peiod 2π, and so closed. R Page 4 of 10

5 4 The Relativistic olution In the geneal elativistic case, we assume that the paticle is a test paticle taveling along a geodesic though spacetime. Geodesics can also be descibed as stationay points of the integal I ẋ, ẋ dτ, which is the fomulation of the geodesics we will use. Assume that the metic fo the sola system is spheically symmetic, static, and asymptotically flat, so that it can be epesented as follows: ds 2 e 2αR) dt 2 + e 2βR) dr 2 + e 2γR) dω 2, 3) whee the dω 2 dθ 2 +sin 2 θdφ 2 tem comes fom spheical symmety and T is the coodinate poduced by the timelike Killing vecto field, of which the metic components ae all independent. We would like to change coodinates fom R to, whee coesponds to physical measuements of adius. If we define the adius of a sphee as the squae oot of its aea divided by 4π, then the coefficient of dω 2 is fixed as 2, and so we can eexpess 3) as ds 2 e 2A) dt 2 + e 2B) d dω 2, 4) whee we define A) αr) α γ 1 ln ) and similaly fo B). If we assume that the obit of Mecuy is a geodesic in a vacuum, this futhe constains ds 2 to satisfy the vanishing of the Ricci Tenso: R µν 0. We can compute the nonvanishing Chistoffel symbols Γ λ µν 1 2 gλρ gρµ x ν fo 4) as 2]: + gρν x µ ) gµν x ρ Γ B ) Γ φφ sin2 θe 2B) Γ θ θ Γθ θ 1 Γ φ φ Γφ φ 1 Γ θθ e 2B) Γ tt A )e 2A) 2B) Γ θ φφ sin θ cos θ Γ φ φθ Γφ θφ sin θ cos θ Γ t t Γ t t A ) Then we can compute the Ricci tenso components 2]: R µν Γλ µλ x ν Γλ µν x λ + Γη µλ Γλ νη Γ η µνγ λ λη as Page 5 of 10

6 R A + 2A ) 2 A A + B ) 2 B R θθ 1 + e 2B A B ) + e 2B R φφ sin 2 θr θθ R tt A + 2A ) 2 ]e 2A 2B + A e 2A 2B A + B ) 2 A e 2A 2B R µν 0 µ ν Note that if we take the combination R +R tt e 2B 2A, we obtain 2 A +B ). Then the vacuum equiement R µν 0 implies that A +B 0, i.e. A+B const. ince A, B 0 as by asymptotic flatness, we must have A B. Then the vacuum conditions become: R θθ 1 + 2A e 2A + e 2A 0 R A + 2A ) A 1 2e 2A R θθ 0 ince the second condition follows fom the fist, we need only choose A so that 1 2A e 2A + e 2A e 2A ). The geneal solution to this is e 2A const., i.e. e 2A 1 const.. It is known that in a gavitational field that esembles Newtonian gavity, we must have g 1 2Φ, whee Φ GM is the Newtonian potential. Then the obsevation that ou gavitational field appoximates Newtonian gavity gives us the chwazschild metic: ds 2 1 R ] dt R ] 1 d dω 2, 5) whee R 2GM is the chwazschild adius of the sun. Now if we paameteize a cuve xτ) T τ), τ), θτ), φτ)) by pope time, then we find that letting L ẋ, ẋ whee the dot efes to diffeentiation with espect to pope time), L is both a constant of the motion 1, in fact) and also satisfies the Eule-Lagange equations so that I L dτ is stationay. By exactly the same easoning as in the classical case, we may estict ou attention to motion in the equatoial plane and assume that θτ) π/2, so that the Lagangian becomes L 1 R ] T R ] 1 ṙ φ2 6) Page 6 of 10

7 Then the Eule-Lagange equations fo φ and T ead: 0 d ) 2 2 φ dτ 0 d dτ 2 1 R ) ) T This implies that L 2 φ and E T R / 1) ae two constants of the motion. Then the elation L 1 gives us: 1 1 R ] T 2 1 R E 2 1 R / ṙ 2 1 R / L2 2, ] 1 ṙ 2 2 φ2 ṙ) 2 E 2 1) + R L2 2 + R L 2 3 Once again, assuming L 0 allows us to invet φ φτ), so we may obtain as a function of φ with ṙ L, and hence we have 2 i.e. ) 2 E2 1 L R L R Now the equiement that of a closed obit with ) 2 0 imposes some constaints on L, E, and R ; we need a connected component of { : 0} to be a compact subset of R +. This means thee exist at least two values R + and R whee 0, i.e. aphelion and peihelion. Then the angle shift fom R to R + is given, as in the classical case, by φ + φ R+ R d. 7) E 2 1 L 4 + R 2 L R 2 Given that R + ) and R ) ae factos of E R L R L 2, we can solve fo E 2 1 and L 2 in tems of R ± and R : which give E 2 1)R L 2 ) R R R + ) R R 3 + E 2 1)R 4 + L 2 ) R 2 + R R ) R R 3 E 2 1 L 2 R + R R + R + + R )R 2 R + R R + + R + R ) R + + R ) 2 R R+R 2 R 2 R + R R + + R + R ) R + + R ) 2 R Page 7 of 10

8 It is convenient to intoduce the combination D R +R, R + + R which has units of distance. Then the above expessions fo E 2 1 and L 2 become: E 2 1 R /R + R ) + R 2 /DR +R ) 1/D + R /R + R ) R /D 2 ) L 2 R 1/D + R /R + R ) R /D 2 ) We would like an expession fo ε, the thid nonzeo oot of E R L R 0. We know that the sum of the thee nonzeo oots is L 2 3 R E 2 1 the coefficient of 3 with the polynomial in standad fom); using the above expessions we can swiftly obtain: ε R 1 R /D Now we can appoximate 7), by witing E 2 1 L R L R 1 E2 L 2 R + ) R ) ε). We obtain: L 2 R+ 1 φ + φ 1 E 2 R R+ ) R ) ε) d L 2 R+ 1 1 E 2 R 1 ε ) 1/2 d R + ) R ) Now use the Taylo seies expansion 1 ε/) 1/2 1 + ε/2, with an eo E bounded by E ε/) 5/2 ε/) ε/r +) 5/2 ε/r ) 2, which poduces: L 2 1 E 2 R+ R 1 + E R + ) R ) + ε/2 2 R + ) R ) d We aleady evaluated the integal of the fist tem in the classical case; it is just π1 + E)/ R + R. The second integal is tickie, but can be evaluated in closed fom: R+ ε/2 R 2 R + ) R ) d πε/2 2 R + R R + + R R + R Page 8 of 10 1 πε R+ R 4D.

9 Then if we ecognize that L2 /R + R 1 1 E 2 1 R /D, we find that L φ + φ π1 + E) 2 /R + R L 1 E /R + R πε 1 E 2 4D π ) R /D π + 1 R /D 4 1 R /D 1 R /D E. Using the obseved values R km, R km fom which we obtain D km), and R 2GM/c km, we find that the second tem is bounded above by π ε/r +) 5/2 ε/r ) 2 / 1 R /D π, making the fist tem R /D 4 1 R /D π a tustwothy estimate of φ + φ half a evolution, in adians). ince Mecuy completes evolutions each centuy, and thee ae /2π acseconds pe adian, we find that Mecuy s peihelion advances by ) ) acseconds pe centuy. π 5 Remaks R /D In the geneal elativity solution, we opted to estimate a single integal, athe than attempt a sot of fist-ode appoximation to a diffeential equation. The eason fo this is that such appoximations ae typically not well justified, and neglect cetain tems as small without poviding estimates fo the neglected eo. On the othe hand, we made some assumptions in ou teatment as well. Apat fom the standad assumptions that the sola system is spheically symmetic which it is not), Mecuy is a test paticle and tavels along a geodesic of this backgound spacetime which it is not and does not), and that spacetime is asymptotically flat who knows?), etc., we also assumed that thee even existed a geodesic coesponding to the specifications we gave fo Mecuy s obit. In addition, in ou deivation of the chwazschild metic, we used a few aguments that stand on somewhat shaky gound such as the use of A R )/4π, with A R ) the aea of the foliating sphee of adius R in ou oiginal coodinates, as a smooth coodinate), though the use of the chwazschild metic is also standad. In any case, we need to match ou physical obsevations to theoy at some point, and we have demonstated that the assumption of Mecuy taveling along a geodesic of the chwazschild metic models ou obsevations well. ) Page 9 of 10

10 Refeences 1] Etgen, Hille, alas. Calculus: One and eveal Vaiables. Wiley, ] Weinbeg, teven. Gavitation and Cosmology: Pinciples and Applications of the Geneal Theoy of Relativity. John Wiley and ons, Page 10 of 10