Calculus, Planets, and General RelativityAuthor(s): Frank MorganSource: SIAM Review, Vol. 34, No. 2 (Jun., 1992), pp. 295-299Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2132856 .

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SIAM REVIEW (?) 1992 Society for Industrial and Applied Mathematics Vol. 34, No. 2, pp. 295-299, June 1992 004

CLASSROOM NOTES EDITED BY MURRAY S. KLAMKIN

This section contains brief notes which are essentially self-contained applications of mathematics that can be used in the classroom. New applications are preferred, but exemplary applications not well known or readily available are accepted.

Both "modem "and "classical" applications are welcome, especially modem applications to current real world problems.

Notes should be submitted to M. S. Klamkin, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1.

CALCULUS, PLANETS, AND GENERAL RELATIVITY* FRANK MORGANt

Abstract. In explaining the motions of the planets, Newton invented the calculus, John Couch Adams pre- dicted Neptune, and Einstein developed general relativity. (The full story now includes a surprise appearance by Galileo.) This article includes a very simplified explanation of general relativity and Mercury's precession.

Key words. calculus, general relativity, planets, precession, Mercury, John Couch Adams, Einstein, Galileo, Kepler, Newton

AMS(MOS) subject classifications. 01A45-60, 83-03, 83C

The history of humanity is the intellectual drama of mankind finding its place in the universe. In this historical drama, calculus has played a central role.

Poring over the tomes of data that lycho Brahe (1546-1601) had collected over a lifetime of planetary observations, Johannes Kepler (1571-1630) noticed certain pat- terns. Figuring out how the orbits would look from above, Kepler deduced that the planets were moving in ellipses. Then Isaac Newton (1642-1727) invented the calculus and related Kepler's Laws to his own F = ma and law of gravitation.

When minor corrections were made for the effects of the planets on each other, the theory at first seemed to work perfectly. But by 1820 the accumulated error in Uranus's position had reached about a full degree. Around 1844, John Couch Adams, a senior at St. John's College, Cambridge, pursued in his thesis the hypothesis that the error was due to an eighth, undiscovered planet. He predicted the position of the planet we now call Neptune (as did Leverrier).

The search for Neptune was difficult. Fortunately, star charts were being prepared at the Berlin Observatory, where Neptune was discovered by Galle in 1846. Ironically, Neptune had been observed fifty years before by Lalande in Paris, but he had concluded that the unexpected image was erroneous.

There is still more to the story. Nowadays computers often sit idle at night. In 1979 Steve Albers put them to work and computed all 21 occultations of one planet by another from 1557-2230. He noticed that Jupiter occulted Neptune during the time of Galileo's observations in 1612-1613. Sure enough, a sketch was found in Galileo's notebook showing Neptune, with a notation about its having seemed to move. It may just have been the bad weather of that period which prevented Galileo from pursuing the question and discovering the planet Neptune himself.

*Received by the editors August 23, 1991; accepted for publication October 5, 1991. This work is partially supported by a grant from the National Science Foundation.

tDepartment of Mathematics, Williams College, Williamstown, Massachusetts 01267.

295

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296 CLASSROOM NOTE

By the turn of the last century the data on the slow rotation or precession of the planetary orbits looked like this:

Planet Predicted precession Observed

Saturn 46'/century check

Jupiter 432"/century check

Mercury 532"/century 575"/century

The error in Mercury's orbit, although a mere 43 seconds of arc (about one hun- dredth of a degree) per century, was larger than could be attributed to observational error. It was to be explained by Einstein's theory of general relativity.

General relativity seems to have become the scientific standard of our age, the great intellectual achievement against which all other genius must be measured. I hope that the following discussion will help in understanding something of general relativity and impressing your students.

Nowadays most students seem to take special relativity rather casually. It builds on the principle that objects in free space move in straight lines at constant velocity, in other words, in straight lines in four-dimensional space-time. The principle of special relativity says that the laws of physics look the same in all coordinate systems moving at constant velocity, i.e., in all nonaccelerating coordinate systems. Of course, things look funny in twisted, accelerating coordinate systems. In an accelerating car, which is changing speed or direction, combs leap from the dashboard and we feel thrown side to side by mysterious forces.

General relativity tackles gravity. At base, the theory is a single simple idea, called the Principle of Equivalence. If you feel yourself pressed against the floor of a closed elevator, you do not know whether it is because the elevator is sitting on a large planet or because the elevator is accelerating upward. Locally, gravity is equivalent to acceleration, a twisted coordinate system in space-time. In such a coordinate system, distance is no longer given by the familiar formula ds2 = dx2 + dy2, but by a more general formula or metric.

Metrics. The standard distance or metric on the plane R2 in R3 is given by

ds2 = dx2 + dy2

In space-time, it takes the form

ds2 =-(dx2 + dy2) + c2dt2

where c is the speed of light. For a particle moving at the speed of light, the spatial distance dx2 + dy2 would equal c2dt2, and ds2 = 0 (no aging). Below the speed of light, dx2 + dy2 < c2dt2, and ds2 is positive.

In polar coordinates, this metric becomes

ds2 = -dr2 _ r2d02 + c2dt2.

If we add a point mass (the sun) at the origin, the right metric turns out to be

(1) ds2 = -(1 - 2GMc-2r-1)-ldr2 - r2d02 + (1 - 2GMc-2r-1)c2dt2,

where M is the sun's mass and G is the gravitational constant. This is the famous so- called "Schwarzschild Metric." Since 2GMc-2 z 3 kilometers, for r large the new factor

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CLASSROOM NOTE 297

(1 - 2GMc-2r-1) is close to one, and the Schwarzschild metric is close to the standard metric. But for r smaller (nearer the sun), the new factor slightly distorts the metric. How will this distortion affect Mercury's orbit?

Planets follow "straight lines" or geodesics (the analogues of shortest paths) in this metric, from some initial point (ri, 01, t1) in space-time to some later point (r2, 02, t2) in space-time. We know that these orbits are almost perfect ellipses in space. See Fig. 1. We will see that over long time periods the elliptical orbit shape will rotate slowly or "precess."

FIG. 1. Planets follow geodesics in the Schwarzschild metric, which are almost perfect ellipses.

The total space-time distance computed in the Schwarzschild metric is

(2) 8= Jlds = JF(r, 0, t)1/2ds,

where by (1),

(3) 1 = F(r, 0, t) = -(1- 2GMc-2r-1)-1 dr2 _2 _d2 + (1-2GMc-2rr-1c2 dt2 ds2 d s 2 ' ds2~

Consider changing just the speed an infinitesimal amount, perhaps moving faster at the beginning and slower near the end, to still arrive at the same point (r2, 02, t2) in space- time, i.e., to still arrive at the same destination at the same time. Then t as a function t(s) of s is replaced by t(s) + Et(s), where Et(s) denotes the infinitesimal change in the function t(s). (In the calculus of variations, 6 is used for the infinitesimal change 6f in a function f, just as in single-variable calculus d is used for the infinitesimal change dx in a variable x.) Since the path is a geodesic, the corresponding infinitesimal change 6s in the total space-time distance traveled is zero. Compute this infinitesimal change 6s by calculus:

(4) 0 = Es =| F-6=- 2GMc2r-1c2 dt /6 dt s I ds 6\ds/

Notice how differentiation under the integral sign repeatedly used the chain rule. The infinitesimal change b(F1/2) is 1F-1/26F. Since the functions r(s) and 0(s) are held constant (only t(s) is changing), 6F = (1 - 2GMc-2r-1 )c2 . 2(dt/ds) * b(dt/ds).

Integration by parts, f u dv = uv] - f v du, with

u -F 1/2(1 -2GMc2r-1 )c2 2 (dt) = (1 - 2GMc-2r)c2 (dt)

(since F = 1), and

dv = 6-ds = 6dt, v = Et ds

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298 CLASSROOM NOTE

yields

0= 0- ((1 - 2GMcG2r ) c2 ) 6tds.

(The first term is zero because Et must be zero at the fixed endpoints.) Since this integral vanishes for any Et, the rest of the integrand must be zero, and hence

(1- 2GMc-2r-1) dt = Cl

is constant. Similarly, considering changes 60 in 0(s) yields

2dO r -= C2 ds

With these two formulas for dt/ds and d0/ds, equation (1) can be solved for dO/dr. With the helpful substitution u = 1/r, the answer becomes

d -u (1 - 2GMc-2(u + u1 + u2))-1/2 du A(lu(-2

l+GMc-2(u+ui +U2)

O(u - u)(u - U2)

where u1 and U2 are new constants (see Fig. 2).

'V~~~~U

FIG. 2. The constants ul, U2, and 1 depend on the shape of the elliptical orbit.

Now recall that to first order the orbit is an ellipse. Integrate dO/du around the ellipse u = ul cos2 o?u2 sin2 oe(O < ca < ir) to get

(5) AS 2ir + 37rGMc-2(ui + U2) = 2ir + 67rGM/lc2

per revolution, where 1 is the "semilatus rectum" of the ellipse of Fig. 2, 1 = 2/(ul + u2). In one revolution, 0 has increased a bit more than 2ir, and the ellipse has indeed precessed! Moreover, the rate of precession is given in terms of a few big constants. Since 1 is rather large, this precession will be quite small. To compute the amount of precession, use the values

G = 6.67 x 10-11m3/kgsec2

M = 1.99 x 1030 kg,

c = 3.00 x 108 m/ sec2, I = 5.52 x 1010 m,

T = 88.0 days/revolution.

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CLASSROOM NOTE 299

Plugging these huge numbers into (5) yields a precession of 43.1"/century, in excellent agreement with observation.

Acknowledgments. Paul Davis encouraged me to write up this portion of a talk I gave at the Boston Workshop for Mathematics Faculty in the summer of 1991. I found the story of John Couch Adams in the Encyclopedia Britannica. Herman Karcher told me the story of Galileo, which appears in [DK]. I first learned the derivation of Mercury's precession from Spain [S, Chap. VIII] and Weinberg [W, Chap. 9] with the help of my friend Ira Wasserman (see [M, Chap. 7]). The short derivation given here is based on a talk by my student Phat Vu at a mathematics colloquium at Williams College, in turn based on Jeffery [J]. My colleague David Park made some helpful suggestions.

REFERENCES

[DK] S. DRAKE AND C. T KowAL, Galileo's sighting of Neptune, Sci. Amer., December, 1980, pp. 7481. [J] G. B. JEFFERY, Relativity for Physics Students, Dutton, London, 1924. [M] F. MORGAN, Riemannian Geometry: A Beginner's Guide, Jones and Bartlett, Boston, MA, 1992. [S] B. SPAIN, Tensor Calculus, Wiley, New York, 1953. [W] S. WEINBERG, Gravitation and Cosmology, Wiley, New York, 1972.

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