Re: Universal Nonconservation.

[Hugh Logan <hlogan@ix.netcom.com>]

> I am beginning to appreciate that there are then three (at
> least) manners, or maybe better levels, at which we and the universe exist.
> That is on one level, locally (supply your own definition), is for all
> intents and purpose Newtonian, with conservation, inertia, almost
> mechanical. Then possibly next is relativistic in which the Newtonian laws
> begin to breakdown in favor of variances found in extremis, though
> conservation is primarily retained intact.

Are you referring to special relativity?*

> Third then is Hubble space in
> which finally the laws of conservation begin to fail, the universe is
> largely homogenous and recognized as a unit.

I think the expression "locally" pertains as much to special relativity
as to Newtonian physics. Special relativity holds in flat spacetime.
However, it also holds true in locally flat regions of curved spacetime;
i.e. regions so small that the effects of the curvature of spacetime is
negligible. In particular, a free-falling space capsule (assumed to be 
small enough that tidal forces are negligible) provides a locally flat
reference frame. Experiments in such a free-float reference frame will
be in agreement with special relativity. Whether or not Newtonian
physics is adequate in such a frame depends on velocities relative to
that frame -- whether v/c is small enough. Except for gravitational
interactions, the results of special relativistic mechanics should
correspond to those of Newtonian mechanics when v/c approaches zero.
For example relativistic kinetic energy reduces to the classical value
(1/2)*m*v^2 (where m is the (rest) mass) as v/c --> 0. But the
relativistic expression is correct when v/c is small -- just
inconvenient to use when the classical formula is good enough.

*I can think of one case where a general relativistic result corresponds
to Newtonian mechanics -- that of radial free fall of a small particle
in the presence of a spherically symmetric object of mass M. Using the
Schwarzschild solution, one can obtain

		(d^2 r)/(d tau)^2 = - (a*c^2)/(2*r^2)  ,

where tau is the proper time. Using Newtonian mechanics with m being the
mass of the small object, m*a= F , where F = -G*(M*m)/r^2, or

		m*(d^2 r)/(dt)^2 = -G*(M*m)/r^2 .

If the motion of the small particle is slow enough, the difference
between t and tau is negligible, so that the GR equation reduces to

		(d^2 r)/(d t)^2 = - (a*c^2)/(2*r^2)  .

Comparing the reduced GR and Newtonian equations, we see that they are
the same if

		a = 2*(G/c^2)*M   .

a turns out to be the Schwarschild radius, and we see that it can be
expressed in terms of the universal gravitational constant of Newtonian
physics. This part of the discussion is based on Sec. 4.3 on pp. 55-56
of General Relativity, A First Course for Physicists (Revised Ed.)_ by
J. L. Martin, Prentice Hall, 1996.

These are just little pieces of a much more global discussion.

Hugh Logan