Conquering the Perihelion

On 18 November, 1915, shortly before arriving at the final field equations of general relativity, Einstein published a derivation of Mercury’s orbital precession based on the vacuum field equations, which turned out to carry over unchanged in the final theory. As early as 1907 he had written to Conrad Habicht that he was working in a theory of gravitation that he hoped would account for the anomalous precession of Mercury. (See An Anomaly in Translation.) Now, eight years later, he was finally was able to derive this result. He told a friend that he was beside himself with excitement for several days after establishing this agreement between theory and observation. The derivation he published in 1915 is mathematically interesting, not just for how he inferred the equation of motion from the vacuum field equations (without the benefit of the Schwarzschild metric), but also for his method of inferring the amount of precession from this equation. In reference to this derivation, the great mathematician David Hilbert, who at the time was working on a unified field theory based in part on Einstein’s nascent gravitational theory, wrote enviously to Einstein

… congratulations on conquering perihelion motion. If I could calculate as rapidly as you, in my equations the electron would correspondingly have to capitulate, and simultaneously the hydrogen atom would have to produce its note of apology about why it does not radiate.

Hilbert may not have been aware of it, but Einstein had an advantage in “conquering” the perihelion calculation so rapidly, because he had performed the same calculation previously (together with his friend Michele Besso) based on earlier versions of his theory. From the theoretical standpoint the important part of this work was obviously deriving the equation of motion, but from a purely mathematical standpoint, in order to quantitatively compare the results with observation, the determination of the implied perihelion precession rate was also important. This step introduced no novel concepts, but it was not an entirely trivial exercise. The “quadrature” approach taken by Einstein is not followed by most modern texts (an exception being Weinberg, 1972), so it’s interesting to review the paper of 18 Novomber 1915 paper to see exactly how he did it. His explanation is rather terse (and there are a couple of typos in the published paper), so it takes a bit of effort to reconstruct his reasoning.

First it should be remembered that Einstein did not arrive at the final form of the field equations (with the “trace” term) until November 25th, but the perihelion motion depends only on the vacuum solution, which is unaffected by the trace term, so its absence didn’t invalidate the November 18 results on Mercury’s precession. Second, not only was Einstein not in possession of the full field equations, he didn’t yet know the exact spherically symmetrical vacuum solution, something which was found by Schwarzschild less than a month later (working at his post on the Russian front). For this reason, Einstein worked with just an approximation to the spherically symmetrical solution of the (vacuum) field equations. He gave this metric in terms of a Cartesian coordinate system, but essentially his approximate metric can be written in polar coordinates as

Schwarzschild soon showed that the coefficient of (dr)² should really be (1 - 2m/r)^-1, which agrees with Einstein’s approximation only to the first order in m/r. Given the high degree of symmetry in this case, it actually isn’t difficult to determine the exact solution from the field equations (or even from Kepler’s third law), but Einstein hadn’t expected any simple exact solution to exist, so he hadn’t looked very hard. (He replied to Schwarzschild’s letter “I would not have thought that the strict treatment of the mass-point problem was so simple”.) His approximate metric coefficient g_rr is related to the exact Schwarzschild g_rr by

Thus the coefficients differ in the second order in 2m/r. Using his approximate metric for the spherically symmetrical vacuum field, Einstein evaluated the Christoffel symbols to determine the geodesic equations of motion, and arrived (just as in modern derivations) at the equation

where x = 1/r is the inverse of the radial distance from the Sun, f is the angular coordinate in the orbital plane, the symbols A and B and constants of integration (B is the angular momentum and A is related to the energy), and a = 2m where m is the Sun’s mass in geometrical units. If we use the exact Schwarzschild metric, this equation is exact with q = 1, but with Einstein’s approximate metric the value of q should actually be 1 – a²x². Dividing through by q, or, what amounts to nearly the same thing, multiplying through by 1 + a²x², the actual equation (1) based on Einstein’s approximate metric would be

Fortunately Einstein recognized that he could take q = 1 without affecting the lowest-order non-Newtonian effect, so he proceeded to use equation (1) with q = 1, which happens to be exactly correct, even though he thought it was an approximation.

From this point most modern derivations differentiate equation (1) again with respect to f, leading to a second order “harmonic” equation with a small relativistic correction term, from which the perihelion precession can be inferred. (See for example the derivation in Anomalous Precessions.) However, this is not how Einstein proceeded. Instead, he took the square root of the reciprocal of both sides of the above equation, giving the elliptic integral for the angular travel between the two extremal inverse radial parameters x₁ and x₂

(Incidentally, if we integrated over r instead of x, we would get a factor of r² in the denominator, due to the fact that dx = -dr/r².) Determining the explicit expression for an elliptic integral in terms of elementary functions is not generally possible, so this approach may seem unpromising, but Einstein was able to approximate the integral with the necessary degree of accuracy. To do this, he made use of the fact that the limits of integration x₁ and x₂ represent the reciprocals of the apogee and perigee distances, at which the derivative of r with respect to f vanishes. Hence we need to integrate between two roots of the cubic under the square root sign. As in Einstein’s paper, let a₁ and a₂ denote these two roots. We will also let a₃ denote the third root, so the polynomial under the square root can be written as

Also, since the coefficient of x² in the monic polynomial on the left side is -1/a, we have

Consequently the product of a and (x – a₃) can be written as

Furthermore, noting that all the quantities ax, aa₁, and aa₂ are all extremely small compared with 1 (because each of them is roughly twice the Sun’s mass in geometrical units, which is less than 1.5 km, divided by the radius of Mercury’s orbit, which is over 55 million km), we see that the denominator 1 – ax in the second factor on the right hand side represents a correction on the order (ax)² to the overall factor, so it is negligible. Hence with sufficient accuracy we can write

and therefore the elliptic integral can be written as

Now, making use of the approximation (1-z)^-1/2  ≈  1 + z/2 for small z, we can bring the constant factor outside the integral, and raise the final factor, so the equation can be written in the form

The definite integral can be evaluated in closed form, giving the result

This is the arc length from the apogee to the next perigee, and equivalently from the perigee to the next apogee, so the total arc length for one “cycle” from one perihelion to the next is twice this amount, and if we subtract 2p we get the precession per cycle. The third term is negligible, so we have the result

where L is the semi-latus rectum of the orbital ellipse. Inserting the values for the Sun’s mass in geometrical units (1.475 km) and the semi-latus rectum of Mercury’s orbit (55.4430 million km) gives 0.1034 arc seconds per revolution, and since Mercury completes 414.9378 revolutions per century, we get 42.9195 arc seconds per century, which agrees very closely with the observed value.

This derivation might seem to rely on knowledge of the indefinite integral

but of course the right-hand expression simplifies considerably upon substitution of either b or c for the variable x. For either of these arguments the second term on the right side vanishes, and the first term reduces to

Hence the definite integral from x = b to c is simply

In the case a = 0 the integral is simply p, for any values of b and c. This is such a nice result that it might have been part of the standard mathematics curriculum at the end of the 19th century, so it’s possible Einstein (or Besso or Grossmann) might have known this definite integral, even without needing to evaluate it. On the other hand, it isn’t too difficult to evaluate this integral directly, especially if we convert to a variable w defined by the relation

The variable x ranges from b to c as the variable w ranges from –p/2 to +p/2. Also, we have

Substituting into (4) then gives

The integral of the sine term is a cosine term, which evaluates to equal values for w = ±p/2, so those drop out of the definite integral, and we are left with equation (4).

At the conclusion of his letter of 22 December 1915 informing Einstein of the exact spherically symmetrical metric, which he had been prompted to seek while studying Einstein’s paper on Mercury’s precession, Schwarzschild wrote

It is a wonderful thing that the explanation for the Mercury anomaly emerges so convincingly from such an abstract idea.

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