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[Phys-L] Re: Help on a problem from Goldstein



I have been off the PHYS-L list for a couple of months until today. I
don't know if the Goldstein problem is still looking for a solution.
This problem is discussed in_Special Relativity_ by Wolfgang Rindler,
Oliver and Boyd, 1960, which I have at hand. It might be in later books
by the same author. The discussion is in the section, "Orbits under a
Coulomb Force," starting on p. 107. The (incorrect) application to
planetary orbits is obtained by setting the Coulomb constant k to G*M.
(Although Rindler uses the 4-vector approach, he uses the subscript "0"
for proper mass, M_0. I will suppress the subscript.)

Leigh,

Apparently the assignment is to apply special, but not general
relativity. I suppose I'd try to solve the equation

-G M r_vec/r^2 = d(gamma v_vec)/dt

separating out the radial and azimuthal components and performing
something like the usual small amplitude radial oscillation analysis
to find the apsidal angle.

Rindler starts with two equations of motion arrived at from the 4-vector
theory. The first, with k set equal
to G*M, corresponding to the three spatial components of the 4-vector
equation, is the same as the above equation (except that the denominator
in the left member should be r^3 rather than r^2. I changed Rindler's
"u" to "v" to agree with John's equation.) It seems that the only use
made of this equation in his solution is is to obtain the special
relativistic angular momentum (per unit mass) constant,
gamma*r_vec x v_vec= _h_, the plane of the orbit being perpendicular to
_h_. With polar coordinates in that plane, gamma^r^2*(d theta/dt)=h.
Noting that gamma = dt/(d tau),
h = r^2*(d theta/d tau).

The time component of the 4-vector equation is used to get the energy
(per unit mass) constant W. This equation is d/dt (gamma*c^2) =
-G*M*(dr/dt)/(r^2). Integrating, one gets
W=gamma*c^2 - G*M/r.

The key to the solution is writing the special relativistic flat space
metric for the plane of the orbit. For the orbit, c^2*(d
tau)^2=-dr^2-r^2*(d theta)^2+c^2*dt^2. Divide through by (d tau)^2 and
make substitutions using the energy and angular momentum constants. One
gets a rather messy differential equation for the orbit in terms of r
and theta. Substituting 1/w for r, and doing some mathematical
manipulation, one can arrive at a differential equation of very
recognizable form,

(d^2 w)/(d theta)^2 + p^2*w=G*M*W/(c^2*h^2) where p^2=1-G*M^2/(c^2*h^2).

(p would be 1 in the classical case.)

Leaving out Rindler's discussion of the constant apsidal distances, the
advance of the perihelion per revolution is obviously

(delta theta)=2*pi/p - 2*pi=2*pi*[(p^-1)-1] .

Rindler's assertion that this is approximately pi*(G*M)^2/(c^2*h^2) if
the fraction is small can be seen by
writing p^-1=[1-G*M^2/(c^2*h^2)]^(-1/2) and applying the binomial theorem.

Incidentally, the application to the Coulomb case was done for an
electron in the hydrogen atom by Arnold Sommerfeld in 1916 to expain the
fine structure of the hydrogen spectrum. Unlike the gravitational case,
the approach for electrons is not invalidated by any curvature of
spacetime produced by the mass of the electron -- too small.

According to vague memory, the use of the metric in Rindler's analysis
is remniscent of the way Taylor and Wheeler arrived at the precession of
Mercury's orbit from the Scharzschild metric, substituting the
appropriate energy and angular momentum contants. There the deviation
from classical behavior was the result of the curvature of spacetime
reflected in the metric.

Hugh Logan
Retired physics teacher

But maybe that isn't saying anything you hadn't already thought of
and the stumbling block lies further along that path.


John Mallinckrodt
Cal Poly Pomona


Problem 13 from Chapter 6 of Goldstein's "Classical Mechanics" reads:

Show that the relativistic motion of a particle in an attractive >
inverse square law of force is a precessing ellipse. Compute the >
precession of the perihelion of Mercury resulting from this > effect. (
The answer, about 7" per century, is much smaller than > the actual
precession of 40" per century which can be accounted > for correctly
only by general relativity.) > >Can someone kickstart my brain with a
hint as to what approach I should >consider to solve this problem? >
Thanks, > >Leigh
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