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



RAUBER, JOEL wrote:

Hugh wrote in part:

________________________

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."
_______________________________________________



Sciama was obviously discussing an idealized situation in which the sun
is modeled as essentially non-rotating and having a perfectly spherical
mass distribution to which the Schwarzschils solution applies exactly.
He states on p. 74 of _The Physical Foundations of General Relativity_
that the Schwarzschild solution "completely determines the motion of
light and of material bodies acted on by a spherical mass, so long as we
can neglect the gravitational action of the light and the material
bodies on the spherical mass. In practice this back action for the
planets is sufficiently well represented by the Newtonian approximation."

If memory serves, Newtonian physics does predict a precession of the
wrong value. Considering the perturbative influences of the other
planets and perhaps the mass quadrupole moment of the Sun serve to make
the gravitational field slightly different form inverse square; which is
how such a thing is possible. An earlier Goldstein problem (in the
central force section) has you calculate the precession that an inverse
cube term produces. IIRC
________________________
Joel Rauber
Department of Physics - SDSU



There is considerable observational information about the precession of
the perihelion of Mercury's orbit
on pp. 538-539 of _Classical Mechanics, 3rd ed._ by Goldstein, Poole,
and Safko (in the chapter on perturbation theory). The total observed
secular precession is 5599.74+/-0.41"/century. Of this, about
531.54"/century is attributed to the perturbation of Mercury by other
planets. Presumably, this is accounted for by Newtonian mechanics as you
pointed out -- the Newtonian approximation that Sciama seems to be
referring to. Goldstein states that most of the discrepancy results
from the precession of the equinoxes. The value to be compared with
general relativity is obtained by subtracting these two sets of effects
from the total secular precession. Goldstein, Poole, and Safko state
that the currently accepted observational value that can be attributed
to general relativity is 43.1"+/-0.5"/century. This is to be compared
with the predicted value from general relativity, 42.98"/century --
within experimental error. The contribution from general relativity is
less than a hundreth of the total secular precession. It seems to me
that it is not so much that Newtonian mechanics gives the wrong answer,
but rather only part of the answer -- to which a small correction from
general relativity must be added. The general relativity correction is
only large in comparison with the incorrect correction that special
relativity would have provided.

The problem you referred to is #21 on p. 130 of the 3rd ed. The authors
were not explicit about whether or not the effect of the mass quadrupole
moment of the sun entered into the value of the precession to be
compared with the general relativistic contribution. Your message
reminded me that, back in 1966 or 1967, that Robert Dicke, H. M.
Goldenberg, and others were performing solar oblateness experiments to
determine the quadrupole moment of the sun. It was my impression that
the purpose of doing this experiment was to get a more accurate value of
the rate of precession of the perihelion of Mercury's orbit for
comparison with relativity to see if a modified form of general
relativity, a scalar-tensor theory (the Dicke-Brans theory, I believe),
could improve on Einstein's tensor theory. I seem to recall a session of
a joint APS-AAPT meeting in 1967 in which John A. Wheeler defended
Einstein's version of general relativity, while Robert Dicke still held
out hope that his scalar-tensor theory might better fit the observed
data. As far as I know, the latter theory has been pretty much
abandoned. I searched the web for "solar oblateness." It appears that
more recent experiments have been done, not only to measure solar
oblateness, but also the the hexadecapole shape
factor.<http://www.noao.edu/noao/noaonews/dec97/node2.html>. >From the
little I could find, the interest was mainly in relation to models of
the internal structure of the sun rather than celestial mechanics.

Hugh Logan
Retired physics teacher
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