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

[Hugh Logan <hloganPHY@CFL.RR.COM>]

Brian Whatcott wrote:

> A somewhat similar problem is sometimes offered: [Phys 4222 UFlorida]
>
> "Suppose that the sun were surrounded by a dust cloud of uniform
> density which extended at least as far as the orbital radius of the
> Earth. The effect of the dust cloud is to modify the gravitational
> force experienced by the Earth, so that the potential energy of the
> Earth is (neglecting the effects of the planets) U(r) = -GMm/r + 1/2
> kr^2 where M is the mass of the sun, m is the mass of the Earth, G is
> the gravitational constant, and k = 4 pi rho mG/3 (note that k > 0, so
> this additional term is attractive). The effect of the dust cloud is
> to cause elliptical orbits about the sun to precess slowly."
>
> Does this provide any traction, I wonder?

Bernard Cleyet wrote:

> My first thought was somewhere I'd heard one could expand and use the
> first and second? terms. Thereby, adding a cubic term to the force.
> (quad potential, naturellement). Some theorem, I think, that only the
> linear and quad central force result in no precession. That's why some
> early on suggested gravity was not exactly inverse square.
>
> bc, still searching his shelves.
According to Dennis W. Sciama, in his high school Science Study Series
book, _The Physical Foundations of  General Relativity_,  Anchor Books,
Doubleday, 1969 (p. 75),  there is no precession
(according to Newtonian mechanics) in a Newtonian orbit (presumably in a
pure Newtonian inverse square gravitational force fieled) -- "it exactly
repeats itself each time around."  According to Sciama,  the only other
force field to share this property is one in which the force is
proportional to the distance. In a footnote, apparently not intended for
high school students, he writes: "Those familiar with normal modes will
recognize that this is because the frequencies of the two normal modes
that represent the motion of the planet are equal (degenerate) for the
special laws of force. Thus the two modes remain in phase, the planet
returns to its starting point after one period., and the orbit is
closed. A small deviation from the Newtonian law splits the degeneracy,
the two modes get out of phase, and the orbit does not close up."

I think the scenario that Brian describes is a description that might
attempt to explain the results of general relativity in somewhat
Newtonian terms. Sciama states that "Einstein's equations are so
constructed that to a first aproximation they coincide with Newton's
equation." According to his interpretation, the sun is the source of an
inverse-square field. However, this field possesses potential energy,
which, in turn, can act as a source of gravitation. The field, in the
case of a spherically symetric non-rotating mass, is given by the
Schwarzschild metric. Sciama claims that, because of the weakness of
the sun's gravitational field, the most one can hope observe is the
first-order non-linearity of Einstein's general relativity. Sciama asks,
"How will this non-linearity show up in the orbit of a planet?" He
answers, "The crucial point is that the source of the additional
gravitational field is distributed throughout space, and is not located
at a single point far from the planet." He goes on to say that the
deviation from a pure Newtonian field could not be inferred from a
planet in a circular orbit. An increase, for example, could be
attributed to an increase in the mass of the sun. However, a deviation
of sufficient magnitude could be  detected with a planet in an elliptic
orbit  -- one in which the distance from the sun varies.

It is ironic that Brian's model involves both of the exceptional force
fields, which, acting separately, would not result in precession of the
perihelion of a planet in orbit. I suspect that the author of the
problem had in mind something like the situation that Sciama was describing.

I am not sure what Bernard was expanding to get terms in the force that
would lead to precession according to Newtonian mechanics. It seems
certain that a pure Newtonian inverse square force would not do this. It
seems to me that the point of the Goldstein problem is to show that
special relativity introduces some precession (but not enough) that does
not appear when the the same inverse square force
law is treated according to Newtonian mechanics.

In my previous posting, I referred to Wolfgang Rindler's treatment of
precession in a Coulomb field and the possibility of treating precession
in a gravitational field in the same way -- only replacing the Coulomb
const K with G*M. Rindler comments (on p. 110 of _Special Relativity_,
Oliver and Boyd, 1960), "Although we have not explicitly stated a
gravitational theory which would justify our procedure, suffice it to
say that such theories have evolved within special relativity. But the
above-mentioned disagreement with observation must be held against
them."  I must admit that I am not familiar with gravitational theories
within special relativity special to that theory, and I am only slightly
aware of them.

Hugh Logan
Retired physics teacher
Retired physics teacher
_______________________________________________
Phys-L mailing list
Phys-L@electron.physics.buffalo.edu
https://www.physics.buffalo.edu/mailman/listinfo/phys-l