Re: Newton's 3rd law? was Re: inertial forces (definition)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Cliff Parker's questions:

> OK, lets see if I am thinking correctly.  Are you saying that the fact that
> forces work at a distance means that during the time the initiating force (from
> body one)  travels to a second body the interaction of the fields through this
> transmittance process changes the initial force and the reacting force so that
> they do not exactly match?

I'm not certain I understand you question, but I think I would say, no.
It's not the mediating field's interaction that "changes the initial
force and the reacting force so that they do not exactly match".  Rather
it is simply that the force body A experiences at time t due to body B is
due to the state of affairs at body B at an earlier time, say t' where
t-t' is the time required for the information about the state of body B
to reach body A so body A can be told at time t how to be pushed.  During
this time delay body B could be doing something so wild (due to other
forces on it not due to body A) so that the force at A is 'obsolete' by
the time t.

For example let's consider a frame in which body A is at rest and (and
is nailed down by other forces or by its huge sluggish mass).  In this
frame consider body B venturing across the static field of body A.  At
each instant of time body B feels a force locally determined by the
local field strength at the current location of B in A's field.  Since
A's field is static the force that B experiences is always directed to
the central point A of A's field.  But now let's look at the force A
feels due to the moving body B.  As B moves along at time t A can only
know about where B was and what it was doing doing back at time t'.  A
gets obsolete info about B, but B has access to the current info about A
because of A's pre-established static field.  If A cannot properly guess
what B is doing during the time interval between t' and t it will not be
able to know how to have the force that it feels point equal and opposite
to the force that B experiences at time t.  Since A doesn't even know
where B is at time t so that the force it (A) feels can point along its
current line connecting it with B.  In order for Newton's 3nd law to hold
A (or, more precisely, B's field at the *location* of A) would always
have to have perfect foreknowlege of B's future behavior so it can make
the proper extrapolations.

> Does this process also indicate that the action
> reaction forces do not occur instantaneously but rather "dance" with each other
> over a period of time?

I guess you could say that.

> I do not understand the significance of 1/c^2 in your statement.  It would seem
> that the "dance" of interacting forces would tend to in large part cancel each
> other out.  Why would it work out this way more for gravitational force than fo
r
> electromagnetic or other forces?

See my other post about this whose subject title was "Re: Impact of
Spheroidicity".

> Yes that is correct.  I had not thought calling space-time a field before.  Now
> what is this about stress counting as a source of gravitation?  That is a new
> idea for me.

In GR gravitational effects are due to the curvature of spacetime.  The
sources that produce such a curvature are energy, momentum, and stress.
Einstein's field equations spell out how it works in detail.  In Newton's
theory of gravity only mass produces gravitational effects.  In
Einstein's theory mass is just one of many things that produce
gravitational effects.

An example of how stress can add to a gravitational field in GR is in
evaluating the internal pressure inside a spherical distribution of a
fluid mass (e.g. a star or planet).  In Newton's theory the pressure at
a given radius depends on the weight of the *mass* outside that radius.
This weight depends on the value of g and the mass distribution outside
this radius.  But the value of g at any radius depends *only* on the mass
inside that radius (by Gauss' law).  For a finite amount of mass in a
planet or star the value of g is finite as well, (it even tends to
decrease with decreasing radius near the center, and vanishes at the
center).  Since both g and the mass are finite, this means the pressure
everywhere inside the object is finite, but the pressure is greatest at
the center.

Now in a GR analysis some things are the same and some are different.  In
GR the pressure at some radius is also due to the weight of the overlying
mass. But in this case the value of g at a given radius depends on *both*
the mass inside that radius *and* the pressure profile inside that radius
as well.  Recall that pressure *is* a form (an isotropic form to be
precise) of stress.  The pressure itself increases the value of g over
the value it would have had if pressure did not contribute to
gravitation.  This means that g at a given radius in GR is greater than
its corresponding Newtonian value at the same radius for the same mass
distribution.  Since an increased value of g makes for a higher
pressure, and a higher pressure makes for a higher g value, we have a
case of positive feedback for the value of the pressure.  This feedback
can be so severe in GR that a finite distribution of mass (finite mass
with everywhere a finite mass density) can result in the central pressure
diverging toward infinity an spite of the finiteness of the mass.  In
fact, it can be shown that any finite spherically symmetric mass
distribution will develop an infinite central pressure if that matter is
just compressed enough.  (Actually the core must collapse before the
infinite pressure develops.)  The precise value of the critical density
for this depends on things like the matter's compressibility as a funcion
of pressure/density.  In any event, a rigorous lower bound on the
critical outer radius for the matter occurs when the matter has a uniform
interior radial density profile (i.e. constant interior density), and in
this bounding case the central pressure diverged toward infinity when the
outer radius (in standard coordinates) is squeezed down to 9/8 of the
gravitational radius of the total mass.  Thus, if any spherical mass
distribution is squeezed down enough so its outside radius is *no less
than* 9/8 of the radius of the Schwarzschild radius that mass would have
if it was a black hole, then the central pressure inside will diverge to
infinity and the matter in the center *must* be crushed as no kind of
matter has an infinite bulk modulus.  Once the center begins to collapse
the rest of the outer parts of the matter fall inward to fill the hole
resulting in the matter distribution spontaneously and unstably
collapsing inward to form a black hole.  Thus any stable (spherical)
astronomical object must have an outer radius greater than 9/8 of its
gravitational radius.  If it wasn't for the fact that the pressure in the
star as well as the star's mass contributes to the value of g this
unstable positive-feedback collapse would be averted.

It can be shown that the gravitational redshift is exactly 2 for light
escaping a spherical gravitational well starting from 9/8 of the
gravitational radius of the mass that forms the well.  This means that
we cannot observe any gravitational redshifts greater than 2 from
stable (non-black hole) astronomical objects assuming the light coming
from them was emitted at the surface of the object.  The redshift could
be greater if the outer layers of the object were transparent to light
emitted from deeper inside.

David Bowman
David_Bowman@georgetowncollege.edu