Exact Relativistic Precession 

The relativistic prediction for orbital precession of a small test particle in Schwarzschild spacetime can be evaluated by several different methods, but most involve some level of approximation without explicitly establishing bounds on the accuracy of the result. Also, many derivations rely on the assumption of vanishingly small eccentricity (i.e., nearly circular orbits), even though the resulting formula is then applied to orbits with high eccentricity, such as the orbit of the asteroid Icarus, which has an eccentricity of 0.827. It would be preferable to have a derivation that can be extended to give arbitrary precision, and that does not rely on small eccentricity. Actually the original method employed by Einstein in 1915 satisfies these requirements, although Einstein didn’t carry through the calculation beyond the lowest order, at which the eccentricity doesn’t appear. Here we present the general derivation. 

Before we derive the relativistic prediction, it’s useful to review the corresponding Newtonian calculation. By direct integration of the Newtonian inversesquare force law equated to the acceleration, we find that the Newtonian kinetic energy of a test particle of unit mass is v^{2}/2 and the potential energy is –m/r where m is the mass of the central gravitating body (in geometrical units so G = c = 1). The sum of these is the constant energy E of the orbit. Thus, noting that v^{2} = (dr/dt)^{2} + (wr)^{2}, where w = df/dt is the angular velocity of the particle, we have the Newtonian equation of motion 

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Recalling that the angular momentum h = wr^{2} is constant, this can be written as 

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It’s convenient to rewrite this equation in terms of the new variable u(r) = 2m/r. Now, multiplying together the two relations du/df = (2m/r^{2})dr/df and h = (df/dt)r^{2} we have dr/dt = (h/(2m))du/df, and so the Newtonian equation of motion in terms of the variable u gives 

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Letting u_{1} and u_{2} denote the values of u at the perigee and apogee respectively, we know du/df = 0 at both of these values, so they are the roots of the right hand side of this equation. Therefore the right hand side can be written as –(uu_{1})(uu_{2}). Taking the square root of both sides and rearranging, we get 

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The angular travel between perigee and apogee is given by integrating this from u_{1} to u_{2}, which gives 

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To integrate this, it’s convenient to make a change of variables, defining a new variable a by the relationship 

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Clearly u goes from u_{1} to u_{2} as a goes from –p/2 to p/2. We also have 

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Making these substitutions for u and du into the preceding integral, we get 

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This shows that, in the Newtonian analysis, the angular travel between perigee and apogee is exactly p, so there is no precession. 

Now we will derive the relativistic prediction for the precession of a (nearly) elliptical orbit in a spherically symmetrical field. We will work with the Schwarzschild metric in the single plane q = p/2, so of course dq/dt and all higher derivatives also vanish, and we have sin(q) = 1. Thus the term involving q in the Schwarzschild metric drops out, leaving just 

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The Christoffel symbols and the equations of geodesic motion for this metric were already given in Section 5.5 of Reflections on Relativity. Taking the parameter l equal to the proper time t, those equations are 

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We can immediately integrate equations (5) and (7) to give 

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where k and h are constants of integration, determined by the initial conditions of the orbit. We can now substitute for these derivatives into the basic Schwarzschild metric divided by (dt)^{2} to give 

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Solving for (dr/dt)^{2}, we have 

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If we identify the constant (k^{2} – 1)/2 with the energy E, this equation differs from the Newtonian equation (1) only by the first term on the right hand side. (Of course, the derivative here is with respect to the proper time t, rather than Newton’s absolute coordinate time t.) We again make the substitution u = 2m/r to arrive at the relativistic equation 

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which is identical with the Newtonian equation (2) except for the term u^{3} on the right hand side. Just as in the Newtonian case we can let u_{1} and u_{2} denote the values of u at the perigee and apogee respectively, and now we let u_{3} denote the third root of the cubic on the right hand side. Recalling that the negative coefficient of the second highest degree term of a monic polynomial equals the sum of the roots, we have u_{3} = 1 – (u_{1} + u_{2}), so u_{3} is nearly equal to 1 for any realistic orbit in our solar system. Expressing the right hand side of the above equation as (uu_{1})(uu_{2})(u_{3}u), we can take the square root of both sides of that equation and rearrange terms to give the angular travel between perigee and apogee as 

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where we’ve used the binomial theorem to expand the square root factor involving u_{3} into a power series. Remember that u_{3} is very close to unity, and u = 2m/r is extremely small for realistic orbits in our solar system, because the mass m of the sun in geometrical units is only about 1475 meters, whereas the radius of the Sun itself is (6.95)10^{8} meters. Therefore, the secondorder relativistic correction will be five orders of magnitude smaller than the firstorder correction (which itself is extremely small), so only the firstorder term in the numerator of the integrand will have any appreciable effect. Nevertheless, we will carry out the calculation to higher orders, to show the full result, including the dependence on the orbital eccentricity. 

It is convenient to define the parameters L and e such that r_{1} = L/(1+e) and r_{2} = L/(1e). For Newtonian orbits L and e represent the semilatus rectum and the eccentricity respectively. We also define the parameter m = m/L. Now, we’ve already seen from the Newtonian case that du over the denominator of the integrand in the final expression above becomes simply da when we make the change of variables to a given by equation (3), so the angular travel between perigee and apogee can be written as 

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where 
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Carrying out the elementary integration and expanding into a series, we get 

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The angular travel from one perigee to the next is twice this amount, and this exceeds 2p by 

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This is the amount of precession per revolution. Naturally to the lowest order of approximation it agree with the familiar result 6pm/L, independent of eccentricity. At the next order of approximation, the eccentricity does affect the result, although for realistic orbits in the solar system the quantity m^{2} is so small that this and all higher order terms are completely imperceptible. 

The lowestorder approximation, 6pm/L, is often written in the form 

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and this sometimes misleads people into thinking this approximation includes an effect due to the eccentricity. However, the eccentricity appears in this formula only to convert the semilatus rectum L of an ellipse to the semimajor axis “a” by the geometrical relation L = a(1e^{2}). Also, recalling Kepler’s third law mT^{2} = 4p^{2}a^{3} where T is the period of the orbit (in units such that G = c = 1), we see that this alternative form is equivalent to the simple expression 6pm/L. 
