Co- authors Wanted for Journal Paper (related to "Jello-O... " thread)

MacM, I am going through your site. It is pretty confusing.

With Newtonian gravity it is sufficient to consider two point masses. Everything else can be described by integrating the two point masses over a distributed volume.

I think with UniKEF gravity that we will need to consider three point masses since an intervening mass will (according to my understanding) attenuate the UniKEF "graviton". I think this may be the mathematical source of all of the geometric effects you describe, like the CoS.

I need to think some more on it, but I believe that if we can calculate the forces on 3 co-linear point masses then we should be able to generalize to arbitrary geometries.

-Dale

 

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Agreed but I have never considered simplicity to be a requirement for truth.

I don't follow your arguement for three point masses but I am sure you will be more explicit if and when you present that view.

 

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I was refering to the presentation, not the ideas.

OK, I have gone over the material and think I have a plan for integration. The thing that makes Newtonian gravity easy to integrate is that the forces between any two differential masses are independent of any other masses. With UniKEF it appears that the "flux" between two differential masses can be "attenuated" by intervening mass. I think this is your "CoS" effect.

So, my plan is to start with a spherical mass of uniform density and a point mass at an arbitrary distance. I will integrate the UniKEF force over each line drawn through the point and the sphere. That will give me the formula for the force on a point mass from a sphere. To find the force on a second sphere I will integrate that formula over all of the points the second sphere. However, the force on a point mass on the far side of the sphere will be affected by the "attenuation" of intervening lines of mass.

Before I do that, I want to make sure that I have everything straight in my mind. I think I know the answer to these questions, but I want to verify before proceeding.

In all of my questions I am refering to point masses which are infinitesimally small and line masses which are infinitessimally skinny. All distances are small relative to the size of the universe. All line masses are "pointing" straight at the relevant point mass and all point masses are colinear. Here are my questions: A) are the forces on two point masses affected by the distance between the masses? B) is there any difference between the forces on two point masses and the forces on a point mass and a line mass of the same mass as the second point. C) what are the formula for the forces on each of two point masses and on each of three colinear point masses?

-Thanks
Dale

PS I had a drawing illustrating my questions, but I am having trouble uploading the images. Any hints?

 

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That sounds like a good planto me, but perhaps it can be even simplier, For example:

1)after you have the resulting force on the point expression, vary the separation from the center of the sphere. (To look for the inverse square law, which if not found, means there is little point in doing the second part.)

Reasons I am suggesting (2) below is you can not fill a sphere with line masses - either they will over lap near the center of the sphere or there will be space between them near the outter part of the sphere, if I understand your proposal correctly.

By edit: My old (2) used “hexagonal cones” but they will not “close pack” to fill the sphere either. Also it did not properly address the ‘shadow calculation“ . Thus, I have replaced old (2) with:
(2) Consider thin disk of radius R with field point on line perpendicular from its center a separation S > R away. Behind it is thin disk n =2 of radius r slightly less than R etc for many disks N > > 1. And the radius of each is as large as possible to be contained in side a hemisphere of radius R. Like wise there are N disk in front of the first one I asked you to consider that complete the approximation of the sphere. Unfortunately, I don not think you can do the one disk problem and use those results as id the force of gravity originated in the matter of the disks (That is Newton’s idea, not MacM’s) You will need to consider the whole sphere when doing the problem.

Thus I think we need to do the problem along the lines of MacM’s CoS. I also note that my discussion below is closely related to Dale‘s, but my “field point” is inside the sphere and Dale’s is outside I think mine may be easier to do. That is getting MacM’s differential flux and then the force on a point inside the sphere may be relatively simple compared to one outside.

As I understood you prior idea about 3 co-linear masses, it was one would be the "field or test mass" to feel the force, the most distant from it, to be the "source of the force" and the intervening one to be the the "absorber of the force" with the idea that any shape could be construced by a set of "most distant ones" and any absorber could be construced by a sum of the "intermediate ones". MacM did not comment and how and said he did not understand, so I will venture a comment for him, and ask him my own question, very closel related to yours:

I don't think Dale quite has MacM's CoS concept correct yet (Perhaps I do not either, but I think I do) you can not separate an object into source and absorber compontent (as I think you were trying to do) but the object is feeling the force due to differential absorption of flux and also contributing to the making of the flux inhomogenious.

I too do not know how to place drawings here so bear with me as I use words to explain (and also ask MacM if I have it correct):

Consider a small volume r from the center of an isolated uniform density sphere of radius R and the line Rr. Also consider the plane, P, passing thru that small volume which is perpendicular to line Rr. P divides the sphere into two unequal parts and the bigger one I call +H and the smaller -H. In the limit as r approaches zero, +H & -H approach equal [h]H[/b]emispheres (if r = 0 line Rr is ill defined.) Got the picture? OK

For any r not zero there is a net force on the small volume towards the center because every ray that contibutes to that force passing thru th -H part is stronger than the ray along the same line passing thru the +H part.

For example if r = 2R/3 and we consider the opposing rays along the Rr line, the one passin thru the -H part travels R/3 thru sphere matter to hit the "small volume" and the one traveling thru the +H part travels thru (R + 2R/3) = 5R/3 matter to hit the "small volume." Thus, this pair of rays produces a force proportional to (5/3-1/3)R = 4R/3 towards the center. If one considers the very small volume just below the surface. For example, if r = R -R/1000, the force on the small volume is tending to 2R proportionality.

Thus I am also asking MacM if this is a correct quantative description of how the force on a "small volume" depends upon the location of that small volume inside the sphere?

 

DaleSpam,

I do not follow your suggestion for use of point masses. All lines of force do not pass through such a point. The flux consists of beams of parallel rays at all possible angles within the CoS.

Billy T,

I do not want to get into calculating small volumes within a sphereical mass. The inverse square finding is based on the integration as described.

**************************************************

Fig 6 I think is the easiest way to explain it.

http://www.unikef-gravity.com/UniKV2/page2.htm

This chart is a representation of beams (if you will) which vary in cross-section as angle of penetration changes for all angles within the CoS. The CoS is (Cones of Sources). That is the volume of the universe where a source could project a straight beam which would penetrate some part of both masses (scattering ignored for reasons explained).

The table shows to seperation distances and the affect on the volumes so penetrated.

I should probably find a better acronym than CoS since it is often confused with the Cos or Cosine of an angle.

 

To Dale:
Can you try this integration***, for volume normalization of:

Eggoid 1 (z-axis symmetric but “un-normalized”): (z/a)^2 + 2(rf/b)^2 = B.
Where:
(1) “B” is a volume normalization parameter to be evaluated by integration of this eggoid’s volume and setting the results equal to V, the volume of radius R sphere.
(2) “r” is the distance of interior point, (x, y, z), from the z-axis: r = +sqrt (x^2 +y^2).
(3) “f” is a simple, near unity function, f(z), which if it were unity and “a” is not equal to “b“, makes the equation describe an ellipsoid. When a & b are not equal, we assume a > b, so that the ellipsoid’s major axis is a segment of the z-axis. (We have assumed the z-axis is an axis of cylindrical symmetry elsewhere, e.g. in the use of “r.”) Note also that if f = 1 and a = b = R then the above equation describes a sphere.

Two examples of function “f(z)” of interest are:
(a)……“Linear f”: f(z) = 1 + Dz.
Where here and elsewhere, the “Distortion constant, D,” is a very small positive value. I.e. 0 < D < < < 1.
(b)……“Step f”: f = 1 + D for z > 0 and f = 1 - D for z < 0 and again f = 1 at z = 0.

This discontinuous “step f” function (but a constant inside each hemisphere) is probably more convenient if integrations over the eggoid are done separately for the two hemispheres (+z and -z hemispheres). More discussion as to why some surface slopes of the “step f” or “tangents” being discontinuous at z = 0 is unimportant are found in the first post this thread. The fact that at z = 0 the max value of “r” is discontinuous is more difficult. I think we simply define a surface for the eggoid at z = 0 to be the annulus between two circles. (One being the limit of r max as +z approaches zero and the other being the limit as -z approaches zero.)
(Note that when we do the virtual work or potential calculation, some of the rays incident upon a point in the more constricted "hemisphere" can pass thru part of the "hemisphere" with the greater r for z nearly zero, then re-enter vacuum before entering the "smaller waist hemisphere" to strike the point being evaluated, if the "step f" is used.) I hope you can integrate the “linear f” case, so this complexity can be avoided. (See end of this post for eggoid 3 equation, which is equally good, I think.)

I suspect that the “step f” and two separate integrations is the easier case to integrate. If the integration can be facilitated by some other, near unity f(z) which is slightly greater than unity in the z > 0 space and slightly less than unity in the z < 0 space and equal to unity when z = 0 feel completely free to describe and use it.

I.e. I want you to adjust “B” to make the volume of this eggoid equal to that of a sphere of radius R, or volume V. That is find, B(R or V, f, a, b), --- express B as function of these four variables.

Briefly, the another integration I will want is for you to integrate the dot product of force * “distortion-movement” vectors for all differential volume elements of the ellipsoid that are not identical with a differential volume element of the sphere it can evolve from.
E.g. where r exceeds R, (mainly along the extremes of z) there is work being done against the external flux pressure by the ellipsoid to expand and we need to calculate it. Likewise, where the ellipsoid surface has moved closer to its center of mass than R, the external flux field is doing work on it that must be calculated.
I am not entirely sure it is necessary, if you separately integrate over the entire volumes of the sphere and the eggoid some expression for the local element's potential**, instead of just the region of space where they differ, but these “not identical” sectors (see the light tan and blue regions of the 2D drawing of the first post) will depend upon how we imagine that the sphere and ellipsoid are “centered” or “superposed” one on the other. Thus I strongly suspect that this energy or net virtual work done calculation should proceed only after we (read that as Dale) have “shifted” the center of mass of the eggoid back to the origin. (See original post discussion about the evaluation of C, which is used in the coordinate transform of z to z’ where z’ = z - C.)

For now, please see if you can find the value of the volume normalization parameter B.

Eggoid 3 (z-axis symmetric “un-shifted, un-normalized): {(1/f)(z/a)}^2 + 2(r/b)^2 = F

Where “F” is evaluated, as “B” was, from the volume normalization requirement. I doubt if this alternative form helps with the integration, but wanted to mention it.

__________________________________________
**How to write this expression for the potential energy stored in each volume element is less clear to me than how to compute the work done during the deformation.
***Even I can reduce this to a single integration over dz by the construction of a "stack of differentially thin disks" with each having radius "max r" possible for that value of z. - The differential volumes of each disk being &pi; ({r max of z}^2) dz and the r^2 being repalced with its value by solving the eggoid equation for r^2 max as function of z and a,b,f and V or R of the sphere. B is not a function of the integration variable z, for fixed values of a,b,f & V.

PS - There is a little notational problem. In the first post I used capital Z and R to refer to points on the surface of eggoid 1 etc. and lower case r & z for interior points, and I still like this convention, but it makes for confusion between the constant R of the sphere and the varying R of the eggoid.
What do you think of returning to this "surface is cap, interior is lower case" convention with this confusion removed by Rs is constant radius of the sphere and Re(z) is the varying surface radius of the eggoid?

 

Ok - I will assume that the method I outlined in recent post is consistent with you intentions. I did note in your paragraph [025] discussing fig.7 the following:

"The length of each penetration line represents some force which also has a trigometric function that limits the effective force ..."

This is my approach also. I.e. if I want to know the differential force INSIDE an isolated sphere of radius R on a small volume element at r from the center, due to fact more flux shadow is being created on one side of it than the other (except when r = 0 ) then the max force is along the max length line passing thru both the center and thru the small volume element and is proportional to (R+r)-(R-r) OR 2r, which does go to zero as r does and has the greatest force on the volume element when it is at the surface of the sphere with full cosmic flux pushing towards the center and the force on the interior side of the volume element is reduced the max amount possile by passing thru the length 2R.
That is I, like you, will assume that the absorption of flux is linear with the distance it travels thru matter, assuming uniform mass density and neglecting other parameters (Iron vs Hydrogen etc.). If you think this is wrong for the isolated star's internal forces, please say so.

For other rays (those passing only thru the small volume, not the center also) at angle &theta; I will include a reduction in their force (which was not as great as the max force described above) by cos(&theta; ), where &theta; is the angle between the max force or center passing ray and the lesser force (because the total length of star transversed is less) ray. I think your reference to "trigometric function" is my cos(&theta; ) is it not?

 

This finally sounds correct although I need to look closer at your formulas. But it is supper right now. The issue of Fe or He, etc is moot in that if you assume a homogeneous sphere for simplicity the force is only a matter of multiplying the integration times UG * density.

 

It is just two different ways to break up the problem. They way you are describing you look at a given direction and then find all of the corresponding point masses (that is the approach on the website). The way I am describing is to look at a point and find all of the corresponding directions. Mathematically the two approaches are the same.

Whenever I try to solve a difficult problem I like to try to have at least two different approaches in mind. That way I can solve it both ways and then I have a built-in check, both answers should agree. If they don't then something is wrong. Also, sometimes one way has an analytical solution but the other is only numerical.

So, I could really use answers to these questions. It will help my understanding of your theory in general and help me to set up the pointwise integration approach.

In all of my questions I am refering to point masses which are infinitesimally small and line masses which are infinitessimally skinny. All distances are small relative to the size of the universe. All line masses are "pointing" straight at the relevant point mass and all point masses are colinear. Here are my questions: A) are the forces on two point masses affected by the distance between the masses? B) is there any difference between the forces on two point masses and the forces on a point mass and a line mass of the same mass as the second point. C) what are the formula for the forces on each of two point masses and on each of three colinear point masses?

-Dale

 

I can try. (Jedi Master Yoda would be very disappointed with me for trying rather than just doing

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)

-Dale

 

No, it is not a test mass. Lets say the we have three colinear point masses, m1, m2, and m3 with m2 between m1 and m3. And lets say that we want to find the force on m3 due to m1. For Newtonian gravity it is independent of m2, however, for UniKEF some of the flux going from m1 to m3 is absorbed by m2. This is the fundamental source of the CoS and the reason why UniKEF is difficult to integrate. You have to consider all mass on a given geometric line. I just want to understand from "first principles" and I believe that for UniKEF the first principles require 3 point masses as opposed to the 2 point masses required for Newton.

I hope that is clear.

-Dale

 

DaleSpam,

It appears you and I had similar thoughts on how to approach the integration. However, I think you've gone astray with defining any "point masses". Point masses are possible in Newtonian gravity because of the independent effect of gravity as generalized by Newton.

This is all true, but it does not apply to the situation of the gravity between two spheres. The integration MacM has described is rather straight forward and appears to give the same prediction as Newton's law of gravity. However, a better integration method would be to integrate over all points of the sphere and calculate the net force on that point. Is that what similar to what you are suggesting? It doesn't involve point masses, but it is an integration method that would work for all geometries.

I gave up on it though when I realized MacM's integration method is completely different from this.

 

Sorry without a sketch I can't be sure you are doing what UniKEF requires. If you can't post a sketch here then feel free to send one to lmccoin@elp.rr.com.

I concur.

By being infinitesimally small it suggests that it is the equivelent of two softballs a light year apart. In such case the CoS is virtually zero degrees and distance would not be a factor. But any point with a finite diameter, however small, the distance will be effective at changing the CoS.

This gets tricky. In the preferred view "~" varies with "U" making gravity a function of total mass and not mass squared. In that case there is a decrease in force between a series of points and the change is not a linear function. The work done so far assumes there is a uniform statistical decrease of force with linear penetration of mass and gravity is still a function of the product of PM1 * PM2.

I can't be sure of this answer but it seems such point masses should have a CoS of zero (trig function = 1.000) and the volume is equal to the PV. But how does a point mass have a volume? Without volume how does it have mass and hence how would you calculate any penetration losses?

 

Verbally what you wrote sounds correct. I'm not sure where you think it is different than what is done in UniKEF.

 

The integration method you've described requires you to calculate the volume of intersection between two given spheres over all orientations, correct?

The integration method I've described requires you to calculate the force at each point in a sphere over all orientations. There is no calculation of the "Pseudo-volume" with this method.

 

How are you deriving a force without first having a volume, density and "G" factor.

Each parallel flux ray penetrates numerous colinear points over all possible angles within the CoS. It is that integration which I call the Pseudo Volume. It sounds as though you are integrating that volume in terms of a direct force but I fail to see how that is possible.

 

The force at a point for a given orientation would be dependent on the length of the line that goes through each sphere.

For example:

--o---............---

Image o as the given point and the "-" lines as the line through the spheres. The forces on &middot; will be:

<--o--->--->

to give a net force of:

o---->

for that particular point.

 

Again, it is just two different ways of approaching the problem. The CoS geometry is still there, but now it is implicit in the integration and we are explicitly moving over the volume. With the approach shown on the website the CoS geometry is explicit and you are implicitly moving over the volume.

It is like this, if you are trying to calculate the volume of a sphere of radius R you can integrate the surface area of spherical shells from radius 0 to R. Alternatively, you can integrate the area of circular cross sections from radius 0 to R and back to 0. You had better get the same answer both ways because the mathematical descriptions are equally valid.

-Dale

 

Exactly, the appeal of the method I am proposing is that it is easier to generalize to arbitrary geometries. I guess everyone is getting confused by the term "point mass". I mean "masses contained in an infinitessimally small volume", not really "masses contained in zero volume". I need to know the forces on a differential element in order to integrate.

-Dale

 

Exactly, that is what I mean. And that is what I had thought you were describing.

OK, it sounds like we need some work on your first principles then. See my comments below.

Think of the softballs a light year apart example. That is what I mean by point mass. I have to establish the function you want to integrate over an infinitessimally small volume. It isn't that the softball has zero volume, it is that it has an infinitessimally small volume compared to the dimensions of the problem. That is what I am looking for.

Since the line mass is already "tricky" here is what we need to do. Give me the formula you want to use for two point masses and then I will try to integrate it over a line. From there we can proceed to higher dimensions and different geometries.

I don't know how much calculus you know, so I apologize if I sound pedantic. What I need is to calculate the effect on a differential element. A differential element is an element with an infinitessimal volume, not a zero volume. That is what I am describing as a point mass.

-Dale