The standard General Relativity results for precession of particle orbits and for bending of null rays are derived as special cases of perturbation of a quantity that is conserved in Newtonian physics, the Runge–Lenz vector. First, this method is applied to give a derivation of these General Relativity effects for the case of the spherically symmetric Schwarzschild geometry. Then the lowest order correction due to an angular momentum of the central body is considered. The results obtained are well known, but the method used is rather more efficient than that found in the standard texts, and it provides a good occasion to use the Runge–Lenz vector beyond its standard applications in Newtonian physics.

## REFERENCES

1.

Although our treatment is not confined to the solar system, we use this term to denote the point of closest approach to the central body because it seems more familiar (and more etymologically consistent) than the more correct term, pericenter.

2.

For a history of the Laplace–Runge–Lenz vector, see

H.

Goldstein

, “Prehistory of the Runge–Lenz vector

,” Am. J. Phys.

43

(8

), 737

–738

(1975

) andH.

Goldstein

, “More on the prehistory of the Laplace or Runge–Lenz vector

,” Am. J. Phys.

44

, 1123

–1124

(1976

).3.

See

C. E.

Aguiar

and M. F.

Banoso

, “The Runge–Lenz Vector and Perturbed Rutherford Scattering

,” Am. J. Phys.

64

(8

), 1042

–1048

(1996

), and the references cited therein.4.

For example, $4\pi r2$ is the area of the sphere $r=const,$ $t=const,$ and $\u2202/\u2202t$ is the time-like Killing vector that has unit length at infinity. See C. W. Misner, K. S. Thorne, and J. A. Wheeler,

*Gravitation*(Freeman, San Francisco, 1973).5.

As in Newtonian physics, it is customary in general relativity to derive the equations of motion in the Schwarzschild geometry by using all the conservation laws and identifying an effective potential in a radial energy conservation equation. For the effects we want to calculate we need the radial acceleration $d2r/d\tau 2,$ so it would be a little more straightforward to use the radial component of the geodesic equation and the conservation of angular momentum. But we follow the equivalent, customary route of finding the radial acceleration from the gradient of the effective potential.

6.

For the case of light, the affine parameter τ is defined only up to scale transformations. The quantities

*E*and*L*are therefore similarly defined only up to such rescaling in this case. The final, physical results will contain only ratios of such quantities, and are therefore independent of rescaling.7.

For a derivation see, for example, Bernard F. Schutz,

*A First Course in General Relativity*(Cambridge U.P., Cambridge, 1985), p. 275 or the reference of footnote 4 on p. 656,or Sec. IV of the present paper.

8.

The Runge–Lenz vector has been defined with various factors of

*m*by various authors. Our**A**is $(mc)\u22122\xd7$ that of Ref. 10, and has the advantage that it gives a finite value for particles of finite rest mass*m*as well as for light $(m=0).$9.

This means $M/r\u226a1,$ where

*r*is a typical orbit radius; it follows if we assume $M2/L2\u226a1.$10.

See, for example, H. Goldstein,

*Classical Mechanics*(Addison–Wesley, Reading, MA, 1980), 2nd ed.11.

We work only to first order in

*J*; more precisely, we assume $J\u2272ML$ and, as before, $M/r\u223cM2/L2\u223c\epsilon \u226a1,$ so that $J/r2\u223c\epsilon 3/2.$12.

J.

Lense

and H.

Thirring

, “Über den Einfluss der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie

,” Phys. Z.

19

, 156

–162

(1918

);J.

Lense

and H.

Thirring

, English translation in Gen. Relativ. Gravit.

16

, 711

–750

(1984

).Also see

D. R.

Brill

and J. M.

Cohen

, “Rotating Masses and Their Effects on Inertial Frames

,” Phys. Rev.

143

, 1011

–1015

(1966

).13.

For the geometrical reason for the conservation of $Q2$ see

D.

Bocaletti

and G.

Pucacco

, “Killing Equations in Classical Mechanics

,” Nuovo Cimento B

122

, 181

–212

(1997

).14.

For a summary of all the relativistic effects on orbits see I. Ciufolini and J. A. Wheeler,

*Gravitation and Inertia*(Princeton, U.P., Princeton, 1995).One of the aims of the Lense–Thirring paper cited in Ref. 12 was to integrate the equations of motion for orbits of general orientation.

15.

For a treatment using the Runge–Lenz vector, see L. D. Landau and E. M. Lifshitz,

*The Classical Theory of Fields*(Pergamon, New York, 1975), p. 336;S. Weinberg,

*Gravitation and Cosmology*(Wiley, New York, 1972), p. 230.16.

With our assumptions as spelled out in Ref. 11 the Newtonian terms of the effective potential are of order ε, both non-Newtonian terms are of order $\epsilon 2,$ and typical terms that are neglected are $J2E2/r4\u223cJEML/r4\u223c\epsilon 3,$ $J2L2/r6\u223c\epsilon 4,$ etc.

17.

Formally, this follows from the $M2\u226aL2$ assumption and requiring bound orbits.

18.

The last term in Eq. (4.11) changes sign if

**L**is antiparallel to**J**. Both terms are of order ε.19.

For nonequatorial orbits the first contribution is a precession about

**J**, whereas the second contribution is a precession about**L**, proportional to**J⋅L**.20.

However, the relative contribution of

*J*to light bending is less: We have $M/b\u223c\epsilon ,$ but $J/b2\u223c\epsilon 3/2.$
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