>To my regret I will make one contribution here before unsubscribing again.
>Consider the exponent in the inverse square laws of electromagnetism and
>gravity. One can calculate the odds or approximate the odds that this
>value would be chosen at random in a truly randomly constructed universe.
>The fact is that if the exponent is less than 2 or greater than 2 the earth
>will either spiral into the sun or away from the sun. (I am too tired to
>night to figure out which goes with which).
Unless the exponent of the inverse power law force is at least 3 (Glenn's
epsilon1 >= 1) the orbits will remain stable in that there will be no
spiraling either into or out from the center of force (i.e. the "Sun").
If the exponent *is* greater than 3 then the orbit will either spiral
into the "Sun" if the initial distance is close enough since then the
attractive force will always overpower the repulsive 1/r^3 centrifugal
force (as seen in a frame rotating with the revolving orbiting body and
coming from the effective potential generated by the conserved orbital
angular momentum's associated kinetic energy), or if the initial distance
from the "Sun" is far enough then the orbit will spiral away from the
"Sun" since in this case the attractive force is always overpowered by
the centrifugal force. For such a force law there is a unique radius
where an *unstable* circular orbit is conceptually possible where the
centrifugal force exactly balances the attractive force. Orbits closer
than this radius will spiral into the "Sun" and those farther out than
this radius will spiral out from the "Sun".
If the exponent is less than 3 but not exactly 2 (or -1) then fixed
elliptical orbits become impossible and then the orbits will precess such
that the location of the perihelion and the aphelion will move around the
"Sun" with time and the orbits will not be simple closed figures. In the
general situation the orbits will trace out a pattern not all that
different from that made by a Spiro-Graph (TM) with the perihelion and
aphelion distances from the "Sun" remaining fixed and only the angular
location of where they occur varying.
It is only for exponents of exactly 2 and -1 (i.e. Kepler/Coulomb and
simple harmonic oscillator force laws respectively) that
fixed-orientation elliptical orbits are possible for the bound motions.
In general, fixed-radius perihelion and aphelion distances will occur for
*any* attractive central force law that causes the effective radial
potential (for the 2-body problem) to develop a local minimum as a
function of radial distance. The effective radial potential here is the
sum of the actual attractive potential energy function (that gives the
actual force law in question) and the repulsive centrifugal potential
(i.e. L^2/(2*m*R^2) where L is the relative orbital angular momentum
about the center of mass, m is the reduced mass of the 2-body system, and
R is the radial distance between the attracting bodies). If this
effective potential has a local minimum then stable bound orbits are
possible for distances sufficiently close to the radius of the minimum.
A circular orbit is possible at the radius of the minimum in the
effective potential. Note that different amounts of orbital angular
momentum will yield different effective potential functions each with its
own radius for its minimum. Such a minimum of the effective potential
will occur for any attractive power law central force function whose
(inverse power) exponent is less than 3. If this exponent is negative
then (we don't have an inverse power law force any more, but rather have
a direct power law force and) then *all* motions are bound since then the
potential energy diverges at an infinite separation distance, and the
system then has an infinite escape energy.
<snip Glenn's stuff about epsilon1 and epsilon2, the habitable range of
Earth's orbit, and his a priori probability comments>
> .... Obviously the way
>out of this argument is that 2 is chosen for the exponent by some logically
>apriori set of physical laws. But if one choses that out, then I think it
>is incumbent upon them to demonstrate such a theory.
Actually the exponent for long range interactions such as those of EM and
gravitation is exactly 2 as an *automatic consequence* of deeper physical
laws than classical Newtonian mechanics. From relativistic quantum field
theory we understand physical interactions between objects as being due
to the exchange of virtual quanta of the boson field which ties the
interacting particles together in the interaction. In the case of the
electromagnetic interaction the interaction is between electric charges
and the intermediate boson field is the Maxwell field, and its quanta are
the photons. In the case of gravitation the intermediate boson field is
the metric of spacetime and its quanta are the so-called gravitons. If
the quanta of the intermediate boson field have *zero* mass (which *does*
seem to be the case for photons and gravitons) then the force law for the
corresponding interaction (assuming that its intrinsic coupling strength
is sufficiently weak to be treated perturbatively) *must* be an inverse
square law-type force with no adjustment of the exponent possible. If
the mass of the mediating bosons is not zero then the interaction's force
law is *not* another power law with a different exponent, but rather it
is multiplied by an *exponential* factor exp(-R/a) where the constant a
is the reduced Compton wavelength for the mediating boson's mass M (i.e.
a = h_bar/(M*c)). In the limit that the boson mass vanishes then a
becomes infinite making the exponential factor just the number 1 and the
force law boils down to the usual inverse square law form. If the mass M
is instead nonzero then the interaction is intrinsically a short range
force where the force between 2 objects interacting via the exchange of
the virtual bosons becomes negligible at a distance of a small multiple
of (say about 5 times) the e-folding distance a. This is the case for
the Weak interaction where the mass of the intermediate bosons (W+, W-,
& Z0 particles) is about 80-90 GeV/c^2 and this makes the range of the
interaction a ~= 2 x 10^(-18) m.
BTW, if the intrinsic coupling of the interaction is too great to
adequately treat perturbatively (such as for the Strong interaction)
then the force between the interacting objects is not dominated by the
exchange of single virtual mediating bosons, and the interaction force
tends to become too complicated to figure out as a simple low energy long
distance formula. In the case of the Strong interaction the Strong
coupling strength (combined with the self-interacting nature of the
mediating gluons) ends up causing the force between colored (i.e.
Strong-type "charge") objects to *grow* with distance, and this causes
colored quarks to be forever confined in color-neutral clusters we call
hadrons (such as neutrons and protons).
It should be noted that in the case of gravity the gravitational force is
intrinsically so weak that we have yet to actually directly observe any
gravitational radiation at all, let alone its quantum particles (quanta),
the gravitons. So the claim that the 1/R^2 force law for gravity is an
automatic consequence from quantum field theory is a little premature.
However, it should be noted that the weak-gravity, slow-speed limit of
(classical) General Relativity also automatically gives the Newtonian
1/R^2 force law between masses. The inverse square law for gravitating
masses is still an *automatic consequence* of Einstein's equations of GR
which describe how matter curves the spacetime in which the matter
resides.
It should also be noted that if (as recent measurements of distant
Type Ia supernovas seem to indicate) the Cosmological Constant is nonzero
then the Newtonian 1/R^2 force law is *incorrect* as it stands and must
be modified by the addition of a long-range repulsive term. In this case
the attractive gravitational field strength g due to a mass M at a
distance R from the mass has the form: g = G*M/R^2 - b*R where the
repulsive last term is a consequence of the extra curvature of spacetime
caused by the Cosmological constant giving the background vacuum an
effective residual energy density. The constant b above is *very tiny*
and directly proportional to the correspondingly tiny Cosmological
constant. This modified force law says that 2 masses attract each other
at sufficiently short distances like Newton claimed, but they tend to
repel at sufficiently large distances with the strength of the repulsion
*growing* with further distance. We don't have to worry about the - b*R
term above causing any problems for us on distances a small as the solar
system. This term tends to only become significant at Cosmologically
great (i.e. multi-billion light year) distances.
>So why shouldn't a randomly chosen universe have a gravitational force with
>an exponent of 10?
Because that then would not be a gravitational-type force. It would be
something else. Gravitation has to do the effects and the means by which
matter curves spacetime and spacetime's curved geometry influences the
behavior of the matter in it. The effects of curved spacetime do not and
cannot produce an exponent of 10.
Regarding the business of attempting to calculate the a priori
probabilities for the various constants of nature, I agree with Howard's
comments on that score.
David Bowman
dbowman@georgetowncollege.edu
David Bowman
dbowman@georgetowncollege.edu