Newton: Sphere Theorem, Continued Response to DavisBJ

Easy and simple:

(1) "geometry" , i.e., Euclidean properties.


Lets measure the angle of the ellipse to the test particle in the following
manner:

Lets take the angle between the ellipse and the GC line to the particle,
to define the angle of the ellipse to the particle.

To illustrate, when the angle between the ellipse plane and the GC line
is 90 degrees, we will say the ellipse is perpendicular to the particle,
and we can say the ellipse is "head-on" to the particle,
(i.e., not at an angle in ordinary parlance). Call this the 'normal position'.

In this 'normal position', if the cone is circular, so is the ellipse,
and also the GC line and the Cone-axis are identical (superimposed).
This can be called the 'normal', 'start', or 'root' or 'base' position.

In that case, we are letting the circular cone define the shape of the ellipse,
by intersection.

Now, any 'angle' of the ellipse will be measured "from" this base position.



For instance, we could talk of the ellipse being 45 degrees from base.

Suppose then that the plane was tilted in just this way: 45 degrees away
from its base position which is perpendicular or 90 degrees from the
Center of Gravity (CG) line to the particle.

We choose still to use the CG line as the 'standard', because it keeps the thought experiments simple, and makes the calculations and arguments simple too. Before you reject this choice, keep in mind two things here: (a) Using the axis through the center of a circle or ellipse will keep the averaged or effective distance of the object constant when we rotate it, for purposes of the thought experiments to test the Center of Mass theorem/idea. (b) Dividing the circle or ellipse at its center keeps the calculations for each half simple, by making the mass exactly half, and also makes comparisons and logic regarding relative strength of force easy.

If we let the intersection of the plane with a circular cone define
the shape of the ellipse, then we will get a moderate ellipse as in the
diagrams we have been using.

In this case, as we observed above, the center-line of axis for the cone
drifts off from the GC line.

Now we are ready to talk about the "Geometry" part of this thought experiment.

First let us define what we are going to hold constant:
Namely

(a) the shape (circular) and the breadth (edge to axis angle)
of the CONE, and
(b) the distance of the GC of the ellipse to the test particle, and

(c) the orientation of our view of the GC and particle, by fixing
the GC line as a horizontal line on the page, for convenience.

Starting at the base position, with a perpendicular circle,
and the GC-line and Cone-axis congruent, we note that
the Cone-Axis drifts upward as we rotate the PLANE at the GC intersection.

And here is the "Euclidean Geometry" part of the observation:

The ANGLE of the GC-line to the Cone-Axis is entirely determined
by the ANGLE of the ellipse, as measured and defined from the
GC-line, irrespective of scale.

We can now say that the cone-axis angle is absolutely FIXED by
the plane-angle for the ellipse, irrespective of the scale or size.

____________________________

Suppose now instead of a changing shape, defined by the cone
cutting the plane, we choose again for simplicity in discussion,
a fixed moderate ellipse, as we both want to explore.

It is this fixed moderate ellipse that I'd like to consider for a moment,
to clarify another idea:

We rotate now this fixed moderate ellipse on the bisecting axis,
through its GC as we have been doing previously, but keep its GC
fixed in space as we also fix the test-particle.

This is the experiment with which we can test the Center of Mass idea
in the simplest manner possible, because we can keep the MASS of the
Ellipse CONSTANT (=1), and divide the near and far half exactly (1/2),
and we can observe whether the EFFECTIVE Center of Mass moves off
from the Geometric Center (GC) of the ellipse, which it obviously does.

In fact, there are TWO observations to be made here:

(a) The DIRECTION of the Force Vector on the particle moves (upward).

(b) The MAGNITUDE of the Force Vector on the particle changes (increases).

(This note should clarify the imprecise words I used in the previous posts
concerning the Force Vector as well. I certainly don't want to muddy
the waters. My point was that the Force Vector must be examined
as a 'two-part' entity, having both direction and magnitude.
We are interested not just in the fact that the vector changes, but also
HOW it changes, specifically in the first instance, the DIRECTION.
Later, when comparing the forces from opposing disks/ellipses,
the MAGNITUDE will also be of interest, because they are supposed
balance, not just in direction but in magnitude resulting in NET ZERO
FORCE.)

These profound results show that the standard "Center of Mass" idea
as normally calculated (i.e., weighted average of mass and position),
FAILS for nearby particles, because the actual EFFECTIVE 'Center of Mass'
MOVES as the object is rotated on its GC in space.

This result is significant for the Sphere Theorem also, because
Newton used circular disks in his own analogical argument for
the Sphere Theorem.

We have ourselves pointed out elsewhere (check all the gravity threads)
that just because Newton's analogy was wrong, doesn't mean
the Sphere Theorem is wrong. However, because his arguments
fail in the manner we have shown, in fact the Sphere Theorem fails.

We should also point out that not only does the Center of Mass concept fail,
and only remain valid as an approximation when the object is at a great
enough distance so that it effectively acts as a 'point particle',

But there is a further concept that comes out of that failure:

We have substituted the "Center of Mass" with a more flexible and
useful concept, namely the "Equivalent Point-Mass" (EPM), defined as
the position one would place a point-mass of the same mass as
an object with extension in space to give the same Force Vector
on the test particle.

As it happens, the EPM moves around in the space around an object
which is not radially symmetrical like a sphere. It is totally dependent
upon the position of the test-particle.

As a result, we were able to show that Gravity Waves are a natural
result of Newtonian Gravitational theory, and that General Relativity
is not needed in order to have Gravity Waves.

Newton's Gravity Waves