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4 Motion in the Schwarzschild Field |
yet at small distances it dominates and makes dramatic changes in our predictions. It is responsible for qualitatively new effects, in particular the phenomenon of periastron advance in planetary orbits. This is a tiny but measurable effect in the solar system that can become impressively large in narrow binary systems. Historically it provided one of the three tests of General Relativity proposed by Einstein himself: the calculation of the measured and unexplained perihelion advance in the orbit of Mercury. Another test proposed by Einstein was the deflection of light-rays by a gravitational field. This has to do with the light-like geodesics whose calculation we address in Sect. 4.5.
In the Newtonian case the general integral of the orbit equation can be given in closed analytical form. For the case of Schwarzschild geometry this is also possible but it involves the use of higher transcendental functions. We will come back to this later on (see Chap. 3 of Volume 2 where we discuss rotating black holes). Here, for pedagogical purposes we present numerical solutions of the orbit equation that are produced by a short computer package in MATHEMATICA (presented in Appendix B.1) that also plots them graphically. Alternatively we can use perturbation theory and calculate the first order corrections to Keplerian orbits. This is the first example of the post-Newtonian expansion and produces the celebrated formulae for the periastron advance and for the light-ray deflection angle.
4.2 Keplerian Motions in Newtonian Mechanics
As anticipated in the introductory section it is convenient to start with a review of Kepler’s problem in classical Newtonian mechanics. The first thing to do is to fix our conventions for kinematics. We use polar coordinates and we label the points of a 2-sphere by means of two angular coordinates as described in Fig. 4.3.
In Newton’s theory the orbit equations are deduced from the equations of energy and angular momentum conservation that, in the polar coordinates we have adopted, take the following form:
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angular momentum
Following quite standard conventions one sets:
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4.2 Keplerian Motions in Newtonian Mechanics |
161 |
Fig. 4.3 Our conventions for the angular coordinates on the S2 sphere are as follows: the azimuthal angle φ takes the values in the range [0, 2π ], while the ascension angle θ runs from 0 (the North Pole) to π (the South Pole). The metric ds2 = dθ 2 + sin2 θ dφ2 is singular at θ = 0 and
θ = π . These are coordinate singularities that can be removed by redefining θ
so that (4.2.1) becomes:
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Introducing a new variable
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u 2 + u2 = C0 + 2Cu
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Taking a further dφd derivative of (4.2.6) one obtains:
(4.2.4)
(4.2.5)
(4.2.6)
(4.2.7)
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There are two kinds of solutions of (4.2.8), those where u = 0 and those where:
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(4.2.9) |
The first kind of solutions correspond to the circular orbits for which r = const, while the second are the non-circular ones. Equation (4.2.9) is immediately solved by a change of variable u = y + C1 which reduces it to the familiar equation
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of harmonic motion whose general solution is y = b cos(φ − φ0) and contains two integration constants b and φ0. Therefore the general solution of (4.2.9) can be written as follows: