...need to rewrite!! ...
??? show Figure of "capped" sphere ???
For Chapter 4, Lucas chooses an observer Frame of Reference (RFo) that is of the same [scale, orientation] as the particle Frome of Reference (RFp), and that is initially coincident with it (t=0). Furthermore, the particle's velocity within RFo is constant
This section deals , whereas all other points in space (or Points Of Interest (POI) including the observer, are fixed with resepct to RFo. This vastly simplifies the geometry and formulations.
Note that within RFp, given symmetry and constant observer velocity, Et,E0,Ei, and Bt,B0,Bi are only functions of [rp,Op,t] within the PARTICLE reference frame (eg Et(rp(t),Op(t)). The time variance is useful for following changes in [B,E,F, etc] of a fixed "Point of interest" in the particle & observer reference frames.
??? show Figure of plane defined by the particle center and Lo ???
Refering to Figure ??? "capped" sphere ??? :
Figure "Chapter 4 reference frames"
This sub-section contains a "partially modified symbol" version of my initial Sep2015 analysis, which includes a mix of Cartesian, Spherical] coordinate expression, and which is the basis for later "General reference frames" work. It should all be updated to my current symbol definitions, but it's actually easier to redo the derivations rather than modify the old.
For Chapter 4, changes in [E,B,...] occur in RFo, not RFp, so it is important to derive expressions for a Point Of Interest (POI) that is fixed in the observer reference frame (RFo).
For this case, particle reference frame (RFp) calculations of [E,B,...] will vary with time for the POI, unlike the previous section, whereas observer coordinates to the POI are now constant.
/*********************
>>>>>>>>> Galilean transformation of the (observer, particle) reference frames, RFp <=> RFo
For Chapter 4, Lucas chooses a Reference Frame for the observer (RFo) that is of the same [scale, orientation] as the Reference Frame for the particle (RFp), and that is initially coincident with it (t=0). Furthermore, the particle's velocity within RFo is constant
For Chapter 4, symmetry around Roch, and a constant particle velocity along a coordinate system axis, makes Pp irrelevant to most POIp measures and calculations here.
+-----+
27Mar2018 WRONG!!
As the POIp moves with the particle :
[Rpcs,ROpcs,ROPI2pcs,Opc,Ppc,Poc,Ep,Bp] are NOT functions of time.
[Rocs,ROocs,ROPI2ocs,Ooc, Eo,Bo] are functions of time.
+-----+
For illustrations of the geometries and symbols, see the illustrations and definitions in "Howell - Symbols for Bill Lucas, Universal Force.odt".
OBJECTIVES :
Where convenient, express variables as functions of FIXED variables :
(POIp) -> -Vodv(PART),t,Rpcv(POIp),Aθpc(POIp),Aφpc(POIp)
(POIo) -> Vodv(PART),t,Rocv(POIo),Aθoc(POIo),Aφoc(POIo)
To "translate" (transform) RFo to RFp coordinates, the Galilean transformationapplies. See the Figure "Galilean transformation" below for details. That the [observer, particle] reference frames are of the SAME [rotation, scale], and are exactly coincident at time t=0, makes [analysis, formulae] MUCH easier !!!!
Figure "Galilean transformation"
Click to see http://www.BillHowell.ca/
Click to see file:///media/bill/SWAPPER/Lucas - Universal Force/Images/Howell - Chapter 4 - Galilean transformation - cropped.png
/*********************
>>>>>>>>> Generalized ether reference frames
While the [observer, particle] reference frames seem clear enough, for general reference there may be a need to define a "Generalized ether reference frame" that will likely be completely independent of the [observer, particle] frames.
At this stage of my Chapter 4 work (20Dec2017), I have not yet elaborated on systems of reference frames for "generalized ether frameworks". As such, my current checks on derivations for Chapter 4 simply follow Lucas's own interpretations. Lucas does incorporate the Lorentz-Poincare relativistic correction factor (adopted by Einstein, but which interferometer data clearly contradict[?2012 Rob Johnson?]), even though he uses Thomas Barnes approach to deriving the factor via classical physics rather than relativity theory.
It seems that essentially all physicists seem "conceptually stuck" with one simple ether concept from the start of the discussions about the apparently anomalous speed of light, very, very long ago, It seems that only rare physicists are [interested, willing, able] to consider the vastly different concepts, or to realize that General Relativity itself can be interpreted s incorporating an ether concept.
As a quick, inaccurate list, of ether fields from simplest to more powerful (albeit not necessarily more correct) :
1. there is no ether, as the ether concept was a mistake of applying to strictly [solid, liquid, gas, plasma] wave physics to light
2. material
3. electromagnetic
4. gravitational
5. General relativity - While initially Einstein and colleagues made great ado about abolishing ether, it seems that they later realized that General Relativity DOES incorporate ether-like concepts, but they decided to avoid the use of the word "ether" and the great about-face they had done? [many references – list??].
6. Zero point energy
7. PLUS a seemingly endless series of other suggestions for ether
8. Any combinations of the above
9. Note that for some concepts, some of the concepts above are a manifestation of other concepts, such as :
Satellite atomic clocks from a conventional point of view – clock rates are [solar activity, altitude]-dependent,
General relativity
Zero-point energy
Lucas's "Universal force" :
gravity is simply the net attractive 4th order electromagnetic force between neutral vibrating dipoles (similar to Van der Waal's force in chemistry)
material (mass-related) is a manifestation of electromagnetic energy, involving Mach's principle
10. Note that [Ether - particle – electromagnetic] interactions are not discussed here
It is very important to specify assumed motions of an ether with respect to one or more of the [Earth, Sun, local region of Milky Way, Milky Way center, Center of Universe] :
1. non-rotational movement at a constant rectilinear velocity
2. rotating at a constant angular velocity
3. accelerating [translational (recti-linear), rotational, perhaps even sub-atomic spin] - I don't remember any examples of this beyond GR, but I haven't checked.
4. Point 3 for systems that are multi-scalar [periodic, quasi-periodic, chaotic] and/or [coupled, synchronous, entangled...]
For disciples of the great science [fashion-cum-cult-cum-religion] of the Big Bang Theory and its expansion of the universe would seem to scream for combinations of perhaps all of the concepts above.
/*********************
>>>>>>>>> Euclidean versus Riemannian geometries
https://en.wikipedia.org/wiki/Non-Euclidean_geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.
The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ.
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:
In Euclidean geometry the lines remain at a constant distance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
In elliptic geometry the lines "curve toward" each other and intersect.
https://www.quora.com/How-do-I-explain-the-difference-between-Euclidean-geometry-and-Riemannian-geometry-to-a-curious-and-intelligent-15-year-old
How do I explain the difference between Euclidean geometry and Riemannian geometry to a curious and intelligent 15 year old?
Rahul Anand, Work smart, Live hard and Enjoy easy
Written May 25, 2016
Briefly speaking Euclidean Geometry is the study of flat spaces. In case you have noticed all the axioms and the postulates are mainly dedicated to 2-dimensional. There are a few exceptions.
Now, Riemannian Geometry is an example of the non-euclidean geometry( There are forms of geometry that contain a postulate (axiom) which is equivalent to the negation of the Euclidean parallel postulate. )
Simply saying Riemannian Geometry is the study of curved surfaces.
Hope you understood it finally. Cheers
http://mathforum.org/library/drmath/view/64459.html
Euclidean and Riemann Geometry
Date: 12/29/2003 at 01:33:36
From: Ming
Subject: Riemann Geometry
I'm a bit confused about the basic premises of Riemann Geometry. I
want to write a paper comparing it to the basic premises of Euclidean
Geometry and do a comparison on how they differ, and how that
subsequently affects our understanding of geometry and the natural
world.
What I've done so far is discuss the first 4 axioms of Euclidean
Geometry, then talked about why the fifth axiom is more a statement of
fact than it is an axiom, and why people have tried to prove it but
failed. Then I move on to Riemann's work. Here is where the problem
arises--I understand that he changed it, saying that parallel lines
always meet. But how did he come up with this, and why is it true?
Date: 12/30/2003 at 07:46:56
From: Doctor Edwin
Subject: Re: Riemann Geometry
Hi, Ming.
This is a fun topic. I like it because it illustrates a lot about how
math relates to the world.
Really, geometry isn't about the world. It's about geometry. It's a
nice, closed system. But of course the entities in geometry are a lot
like things in the real world, aren't they? And that's what makes it
useful.
Euclid built a closed system that was similar to the way things behave
in the real world. But as you pointed out, there were lots of things
that were true in the real world that couldn't be derived in geometry
with just the four postulates. So Euclid took a thing that seemed
true in the real world and added it to his system as a fifth
postulate. Then he was able to have a pretty complete model of how
shapes worked in the real world.
Many people (including, I seem to recall reading, Euclid) were unhappy
with the fifth postulate and tried to get rid of it. If you could
derive it from the other four postulates, you could have it as a
theorem, and be back to the original four.
One way to prove the fifth postulate would be by negation. If I
assume the opposite of the fifth postulate, and add that to the
system, and I can find a way that it contradicts one of the other four
postulates, then I have proven that the fifth postulate is true, using
the other four.
There are two ways to negate the fifth postulate. You can either
assume that there are NO lines through a point not on line AB that are
parallel to AB, or that there are an infinite number of them.
Riemann's geometry assumes that there are no parallel lines--that
all lines must intersect.
However, when this was done, no contradiction was found. You could
generate theorems using the negation of the fifth postulate along with
the other four from now until the cows come home, and you'd have
nothing but a perfectly self-consistent closed system that is about
itself. You'd have an alternate geometry.
Okay, here's the cool part. Just like Euclid's geometry models the
way shapes work on a plane, Riemann's geometry models the way shapes
work in a space that curves back on itself, like on the surface of a
sphere.
Now here's the really cool part. Einstein said that the universe
actually fits Riemann's geometry--that the three-dimensional universe
we perceive actually curves back on itself in four dimensions like the
two-dimensional surface of a sphere does in three dimensions.
Does that help answer your question?