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 inverse-square force law equated to the acceleration, we find that the Newtonian kinetic energy of a test particle of unit mass is v2/2 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 v2 = (dr/dt)2 + (wr)2, where w = df/dt is the angular velocity of the particle, we have the Newtonian equation of motion

 

 

Recalling that the angular momentum h = wr2 is constant, this can be written as

 

 

It’s convenient to re-write this equation in terms of the new variable u(r) = 2m/r. Now, multiplying together the two relations du/df = -(2m/r2)dr/df and h = (df/dt)r2 we have dr/dt = -(h/(2m))du/df, and so the Newtonian equation of motion in terms of the variable u gives

 

 

Letting u1 and u2 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 –(u-u1)(u-u2).  Taking the square root of both sides and re-arranging, we get

 

 

The angular travel between perigee and apogee is given by integrating this from u1 to u2, which gives

 

 

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

 

 

Clearly u goes from u1 to u2 as a goes from –p/2 to p/2. We also have

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

Solving for (dr/dt)2, we have

 

 

If we identify the constant (k2 – 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

 

 

which is identical with the Newtonian equation (2) except for the term u3 on the right hand side. Just as in the Newtonian case we can let u1 and u2 denote the values of u at the perigee and apogee respectively, and now we let u3 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 u3 = 1 – (u1 + u2), so u3 is nearly equal to 1 for any realistic orbit in our solar system. Expressing the right hand side of the above equation as -(u-u1)(u-u2)(u3-u), we can take the square root of both sides of that equation and re-arrange terms to give the angular travel between perigee and apogee as

 

 

where we’ve used the binomial theorem to expand the square root factor involving u3 into a power series. Remember that u3 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)108 meters. Therefore, the second-order relativistic correction will be five orders of magnitude smaller than the first-order correction (which itself is extremely small), so only the first-order 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 r1 = L/(1+e) and r2 = L/(1-e). For Newtonian orbits L and e represent the semi-latus 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

 

 

where

 

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

 

 

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

 

 

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 m2 is so small that this and all higher order terms are completely imperceptible.

 

The lowest-order approximation, 6pm/L, is often written in the form

 

 

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 semi-latus rectum L of an ellipse to the semi-major axis “a” by the geometrical relation L = a(1-e2). Also, recalling Kepler’s third law mT2 = 4p2a3 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.

 

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