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# A critique of the mathematics of Newton’s “law” of gravity

5 posters ## A critique of the mathematics of Newton’s “law” of gravity

The silent war for world-view dominance
The heliocentric model won over the geocentric Ptolemaic and Tychian (of Tycho Brahe) models in no small part because of the work of Tycho Brahe’s masonic disciple Johannes Kepler. By carefully studying the celestial objects (on the firmament) under extended periods, Tycho Brahe had a passion for exact facts and owned data that his young apprentice Kepler wanted to use for his purpose to verify the Copernican heliocentric model. Still, Brahe would not separate himself from it and argued that the data lacked the required parallax of the stars to support the heliocentric model. Then Brahe mysteriously fell very ill and died (likely poisoned), and Kepler used the data to claim verification of the Copernican model and refusal of the Tychian and Ptolemaic models.[ref #1]

By the time Newton came up with the idea of “gravity” as a force, most if not all of the theoretical and doctrinal groundwork was therefore al-ready in place to allow for the authoritative cultural revolution we know as history. The first piece was the already established idea of heliocentricity and the consequence that there had to be something holding the earth in its orbit around the sun, and “A God-given path” was not a desirable answer.

If “gravity” was to be proven as a force acting at a distance, the force had to extend beyond the objects themselves. Gauss divergence theorem was revamped to give the foundation needed. Since the force was verifiably not a function of speed dX/dt  (where X is the position vector), it had to be an expression of the second time-derivative, acceleration (denoted “a”), d2X/dt2. And so, Newton's first "law" (F=m*a) was born, and "mass" (or “m”) consequently gained the unit of [F]/[a] = [Newton]*[seconds2/meter].

By doing this, Newton and his society effectively established a new paradigm where “mass” became a dead-end for inquiries on the true nature of “gravity” since “mass” was/is established as the only source and variable for it (apart from distance). It also brought separation from electromagnetism with a completely different “field” of acceleration such that at each point, you would experience a different acceleration depending on the distance from the “mass” generating it, tied to space and time (X,t) through d2X/dt2, immeasurable using anything else than another mass, yet affecting electromagnetism (as later shown) as in frequency change in light (red-shift). Only one-way apparently. No one has ever really explained exactly what “mass” is - except to use the circular reasoning of referring to gravity or make Einstein’s claim that it is inherent energy divided by the speed of light squared or referring to the quantum physics straw-man that it is due to an exchange of “gravitons.” Nowhere has it been clarified and proved what in "mass" creates "gravity" or why “mass” and not just any energy gets to have this property of being the source of “gravity.”

Gauss Theorem
Gauss’s theorem is fundamentally a continuity equation. An expression of the postulate that energy cannot be created, only transformed. It is used as a foundation for much of both practical and theoretical physics., It can be stated as follows.

Let (V) be a volume with boundary surface (S) and let there be a flow F across this surface, i.e., flow into, out from, or through the volume. Gauss theorem then makes theoretical equality between (one hand) summing-up flows through the surface area (S) and (on the other hand) the sum of flows coming from sources and sinks inside the tiny volume-elements dV that make up the entire volume (V). The total sum of flows through the closed surface S is coming from or being swallowed by the sources/sinks inside the volume.

The strengths of the sources/sinks inside the volume are counted by the amount of deflection (or "divergence") they cause on the flow (F) in the point where they are. If the divergence in a point in the volume is positive, then there is (assumed to be) a generation of flow (F) in that point. And if the divergence is negative, there is assumed to be a sink of flow within the point. How ever that generation or swallowing of flow is happening. Note the small surface area element dS and the normal vector (n) pointing perpendicular out from that surface element. This is a construction used to summarize the flow crossing the whole surface (S). More on that below. Gauss theorem then says that Here, the left-hand side (LHS) of the equality really means that we (in each small part of the surface, dS) only count the part of the flow (F) that is pointing straight out of the surface (volume), i.e., is aligned with the normal vector (n) at that crossing-point. The rest of the LHS expression, just mean that we add up all those small flow-contributions from each surface area-elements (dS) into a total cross-surface flow over the whole surface (S).

Now to the right-hand side (RHS) of (Eq.1) i.e. . As mentioned initially, this summarizes in 3 dimensions over the whole volume (V) the contributions of sources of the flow (F) residing within each tiny volume-element (or point) dV.

The strength of each tiny source/sink is expressed in terms of the spreading (“divergence”) of the flow (F) that it creates in each point, written as .

Physically, the value of is the sum of change of (F) in respective dimensions in that point. The value of is positive if the source is outpouring and (F) exhibits an outward spreading, and it is negative if
it acts as a sink and (F) exhibits a contraction of force or flow. In standard orthogonal (x,y,z) coordinates, the "divergence" is formulated as . Naturally, then, in a flow where there are no sources or sinks, all these derivatives are zero, and in those cases =0  everywhere, the flow (F) will not be diverging in any way but be parallel. Therefore, if we sit on top of a flow volume element (dV), the "divergence" can also be interpreted as measuring the (spatial = lengths dimensions) rate of expansion (or contraction) of the volume element as it flows along (F).

Note that the source is a scalar, and thus it only defines an amount of the flow medium created or added to the reality in that point (or unit volume dV). The source has no defined direction, and so for most cases, it is presumed that the added or removed flow simply fills or vacates all spatial dimensions equally, i.e., “isotropically” i.e. that dF/dx = dF/dy = dF/dz.

Note also that the source of (F) is (must be) generating spatial derivatives of (F). Consequently, if (F) is an acceleration-"field" (which is the case claimed in Newton's "law") having the applied unit m/s2, then the source of that "field" will have to be generating "substance" of unit 1/s2 in relevant points of the volume in consideration i.e., be proportional to frequency squared.

The creation of Newtons "law" of gravity
Newton’s second “law” of “gravity” was clearly a premeditated theoretical construction from a number of presumptions. It starts with simply cutting away the RHS of Gauss Theorem (Eq.1). and then, this new total “gravity” flow or “field” of acceleration (g) on the LHS was matched with a new RHS (Eq.2) and assumed to be proportional to the “mass” (M) and the basic idea of (F) being a flow was replaced with (g) being a "field". The proportionality constant in front of (M) was defined as -4*Pi*(G). Gone was Gauss RHS integral (summation) of sources over the volume (V). The unit of the “gravitational constant,” (G), then consequently became set to [meters / (Kilograms * seconds2) ], since if (g) is position-dependent acceleration i.e. meters/second2,the unit of "mass" (M) is in kilograms and Pi is dimensionless, then the unit of (G) follows.

By applying then this proposal to a spherical volume (a ball), Newton presumed that the “gravitational field” (g) would be spherically symmetric. This is a special case since Gauss theorem makes no such prerequisite on the “field” (F) or the distribution or shape of the sources included in the RHS volume (V) of (Eq.1).

Newtons then used his own “shell theorem” to show that (in terms of "gravity") a spherically symmetric body affects external objects as though all of the “mass” of the sphere was concentrated in a point at its center. This is why the RHS of (Eq.2) expresses proportionality to the total "mass2 (M) as being in a point. The shell theorem also states that the sum of all “gravitic” forces experienced from a material sphere by an external body would be acting in a straight line between the central point of the sphere and the “center of mass” of the outer body. With these presumptions, Newtons stated that “the gravity field” could be theoretically simplified and written as the multiplication of a scalar field g(r) with (er) being the radial base vector of length one pointing in the direction of the external body, so (er) is a normal vector of the surface (S): A scalar field has a strength or number assigned to each point in the realm. Under these conditions, (Eq.2) becomes Due to the assumption of spherical symmetry of the “mass” (M), the strength of the “gravity,” i.e., g(r), will be the same (constant) anywhere on the surface of any sphere of radius r from the center, including the larger sphere (dashed line above) that is intersecting the external body. So, g(r) and can therefore be moved outside of the integral. And the remaining LHS integral summarizes to the area of the surface of the sphere of radius r: Here we see why Newton added that strange 4*Pi factor to the RHS in (Eq.2). It was NOT to have that baked into the “gravitational constant” (G). Instead, that 4*Pi cancels out, and we are left with the expression for the “gravitational” strength at distance r from “mass” (M): Accordingly, the sought-for vector "field" variant of “gravity” from (M) then, according to (Eq.3) will be And since then two point “masses” (M) and (m) will be affecting each other, the total attractive force F between them becomes - not to confuse the force F with the flow (F) in Gauss Theorem. (Eq.5) is called “Newtons Law,” but again, it is NOT a law but a theory (or rather a mathematical construction).

Critique of the creation of Newtons (second) "law"
As we have already seen, it builds on a number of presumptions and at least two errors. I will simply outline them here in no particular order.

1. Frankenstein mathematics is not science
Taking the Gauss theorem (Eq.1), chopping it in half, and creating a new equation from its LHS and mathematical stretch it beyond the realm of its original validity, is not scientific. It could be argued that the LHS of Gauss theorem was selected as basis for Newton’s proposition because it provides an opportunity to “prove” an already made-up idea. But the original balance, limit of validity, and physical meaning of the Gauss theorem are entirely gone in Newton’s application.

2. Presumptive logic is not science
Straight-up presuming “mass” (alone) is the only driver of an entirely new kind of “field” separate from electromagnetics and buoyancy and straight up, assuming a linear relationship as postulated in (Eq.2), is at best narrow-minded and certainly not scientific. In a situation like this, scientists consider existing science and “antecedents” (variables suspected to be driving or shaping the effect) to be brought into the situation and candidate function, not so in this case.

And this would be less of a problem had the theory not had such immense consequences and not the scientific community been so quick and decisive in accepting and proclaiming the idea as a “law,” a title given only to natural effects that have been long-time verified and are really not disputed at all. This is certainly not the case for “gravity.” But this did not stop this piece of math from becoming the “law” and motivation for the entire world becoming indoctrinated with an in-finite punctured vacuum “universe” and earth evolution into a spinning ball whizzing around the sun.
Immense religious propaganda and ridicule towards protest around a worldly perception should always be a wake-up call to anyone that still has not been turned into blind acceptance.

3. Breaking the boundaries of validity, physics and logic
An inverse square “law” cannot be presumed to exist or be valid outside of the volume in which the sources of the flows are. – There is a reason why in the Gauss theorem, the closed surface (S) is completely surrounding the volume (V), and no part of the volume is sticking out and away from the limiting surface.

The inverse square law is predicated on a central point source and a uniformly spreading flow from there. This, in turn, means that the flow sources must be uniformly distributed throughout the volume (V), or the flow field would not be spreading in such a uniform way that the inverse square property would be fulfilled. But if the shell (S) and the flow across it (LHS of Eq. 1) is presumed to implode to a central point, then so must also all those tiny “masses” that are the sources or sinks of it (RHS of Eq. 1). Because the volume (V) cannot be assumed to become larger than the surface (S), or vice-versa. If this happens (as Newton postulates it), the central physicality of the Gauss theorem fails completely. - The shape and strength of the "field" outside of that central point becomes just as unknown as it is outside of the original volume (V) and the bounding surface (S) in Gauss Theorem.

Fact 1: The permissible realm of Gauss theorem (and likewise integral theorems, including Newtons idea) is only at or inside the surface of the volume - not anywhere outside of it.

Fact 2: An “implosion” can only be allowed is to change both volume and surface together, never either one alone.

So essentially, this idea of imploding the surface and presume the matter is in a point yet assume the sources of the "field" are still spread far and wide to accommodate an inverse square "law", is a dream. It has NO foundation in physics or real mathematical logic.

Furthermore, the theoretical implosion is in itself not physically possible, not even in a “black hole” (presuming the very “gravity” theory that is questioned). Black holes are likely as much constructed science-fiction as Newton's second "law".

4. The deceptive shell theorem switch
As mentioned earlier, Newton’s shell theorem, which motivates the idea of imploding the “mass” to a central point, is built upon the predicate of “gravity” being ruled by the very same inverse square “law” that Newton’s “law” then results in. (presumptive or circular reasoning!). The theorem is twofold in its statement:
A. “A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center.” This statement claims action on a distance without intermediate transfer, which violated the physical reality of the volume as the source and extent of “gravity.”
B. “If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object’s location within the shell.” This is the same violation as in (a.), even as it occurs inside a shell and relies on that “gravity” has vectorial superposition.

Newton also adds that“inside a solid sphere of constant density, the gravitational force within the object varies linearly with distance from the center, becoming zero by symmetry at the center of mass.” And thereby verifying that implosion to a point is congruent with requiring a homogeneous density in the now larger volume. In its construction, the shell theorem is a preparation for the “action on a distance” that Newton abhorred but which the masonic society pushed to acceptance.

The only reason an inverse square relationship is obtained by Newton is that it numerically matches between the imploded LHS flow with the RHS (summation of sources/sinks) of a volume (V) holding uniformly distributed tiny sources/sinks. But by replacing the RHS with a “mass” (M) and a constant (-4*Pi*G), the range of validity of the original theorem to the volume (V) is removed from the reasoning, leaving it fully open to then expand the flow-from an imploded point to infinity by division by r2 (Eq.4). This may look mathematically correct but is neither scientific nor physically accurate.

The shell theorem builds on the same idea as the main theory relies on and "proves", this is one very intelligent piece of deception. The shell theorem moves the critical "magic" away from the scenery of the main theorem. It becomes just a footnote, albeit a very crucial one. A true mark of masonic "handywork".

6. Half of the physics is not accounted for
Gauss theorem does not include any solenoidal part of the “field” (F) i.e. flow or "field" that does not flow through the surface (S) but stays rotating inside the volume (V). Seeing that Newton clearly presumed that the exact same kind of flow moving out of the “mass” was the very “gravity” he was theorizing, it is very odd that Newton makes no effort in understanding the likely relation between this inner rotating part of the flow/"field" and his “gravity”. For this reason alone, his theory is as incomplete as it is invalid and mathematically impossible.

Conclusions

Newtons’s “law” is indeed a very intelligently created theoretical physics made by butchering of Gauss integral theorem under clear presumptions and goals, and consequently, the theory violates both logic and the fundamental domain of validity.

What is most concerning is that most modern physics is firmly based on this inverse square idea and likewise prearranged equations. Electrostatics i.e Columb’s “law” being the prominent example, and thereby also electrodynamics and (so-called ) quantum mechanics/dynamics/physics.

It would be a good idea if someone could make a formulate a new theory that stays within the realm of validity and reality.
The idea that "gravity" simply is buoyancy seems intuitively correct. However, boyancy build on there being a "gravity"-like force pushing one of the densities into the other, or the two different objects/media would be separated. - There is more to the story to be uncovered.

References

Last edited by PacMan on Tue Sep 14, 2021 12:56 am; edited 1 time in total

PacMan

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## Re: A critique of the mathematics of Newton’s “law” of gravity

“Newton's first "law" (F=m*a) was born, and "mass" (or “m”) consequently gained the unit of [F]/[a] = [Newton]*[seconds2/meter].”

Just to clarify Newton’s First Law of Motion is Inertia and his Second Law of Motion is the F = m*a and the unit of force 1 N = 1 kg ⋅ m/s2

As far as my research has taken me his Law’s of Motion do not conflict with the Geocentric model. No issues.

However his Universal Law of Gravitation is a problem. It’s one big assumption. If anything it corrupts his laws of motion.

Awesome find. Thank you! tycho_brahe

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## Re: A critique of the mathematics of Newton’s “law” of gravity

Yes. Agree.

Whatever the true nature of "gravity" the force is real and likely really boyancy driven by differences in local material strain of the respective media my current idea of the next step in detailing/adjusting Eric's overall math of boyancy/relative density (inner stress/strain tensor being relative to density).

Again I agree. The reason Earth is not globe is found in other factors than "gravity", such as existence of Polonium Halo's in rocks, lack of curvature, inconsistent topography in the southern hemisphere and other proofs.

But I have to say that with "gravity" very likely not being a factor of "mass" alone, "mass" being unexplained as source of "gravity" and also the force unit of "newton" being defined in this, the 1st "law" (F=ma) is also really suspicious and seem just a constructed. It seems to be no coincidence that the unit of force contain the unit of mass (with all the consequences to all other related physical units). including inertia, which is equally based on density i.e. "mass" per unit volume (of insignificant size). We need to get away from this Newtonian paradigm.

PacMan

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## Re: A critique of the mathematics of Newton’s “law” of gravity

Terms like mass, point, volume, field, weight isn't well defined in consistent manner.
Brian Greene doesn't know what space is tuOhm18T3CA etc.
Correct Prism Model - Johann Goethe xJ8BJVkEsSI
Secrets of Magnetism & the Magneto-Dielectric foundation of the Cosmos
lncd-K4MqB0
First Time Ever Seen - Secret of Light - 140 Year old mystery solved! Crookes Radiometer CCrnDGOl2xA

Crystals (1958) kw84ZH_kXr8 and Water Crystals In Motion - Messages From Water - Dr Masaru Emoto and the different attitude the static compared with running electricity has, as well the double slit experiment.
All these indicates that such paper laws does not explain the mechanism in which nature work, but only describe the situation we observe, many times with contradicting explanations. If solid crystals can grow and analog liquid devices predict weather why we had to assume that only we had life and wisdom? Maybe life isn't only attributed to carbon compounds.
"Black holes are where God divided by zero" or rather we are joking because we are not able to understand or comprehend infinite wisdom and so we keep invent words and assign to them arbitrary theorems.

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## Re: A critique of the mathematics of Newton’s “law” of gravity

Any chance someone could explain this in like 2-3 short sentences for the elementary reader? I consider myself pretty bright but this is way over my head. I am an FEer / and I don't believe in Gravity// thanks!

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## Re: A critique of the mathematics of Newton’s “law” of gravity

Wow great study! I will come back and read this more in depth later (although I’m not great at math I can usually see the points made). Nicely layed out and written. Tree

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## A step towards a better formulation of "gravity" - The math of soap-bubbles and balloons

As a step towards a better (non-"mass"-based, non-Newtonian) formulation of "gravity", let's consider the math of basic soap-bubbles and balloons.

Fundamentally, atoms/molecules in a medium/material are drawn to each other or bound by some form of binding force (dipole-binding, electron-binding, covalent binding etc.). This is essentially the force holding all substances together and this force is small to none and even negative (repelling) in a gas whereas it in a fluid is conclusively positive while being large to extremely large in solid materials.

Considering a fluid as example, inside the medium, one atom/molecule experiences a zero net force of movements from being equally pulled in all directions or pushed equally from all directions, keeping the atoms/molecules i.e. the substance together while allowing it to adjust to external forces.

The molecules at the surface, on the other hand, do not have neighboring molecules ‘above the surface’ and will therefore to be drawn towards the molecules/atoms inside the fluid instead. This increased tension on the surface creates an (additional) internal pressure and the locally higher energy of the surface makes the surface of the material strive to become minimal.

The geometric shape that provides the minimum surface to volume ratio is a sphere (soap bubbles, balloons,…). Note that in the case of a balloon or soap bubble, the surface is made up of some other stretchy material than the volume substance, for instance rubber. This allows for the volume substance to even be a gas that would otherwise fly apart.    This energy-minimum relationship of a material bubble is described by the Laplace equation, and as a step toward a new formulation of “gravity” outside of the Newtonian paradigm, this equation is what I aim to derive here and also to see what we might learn form this step. The case of a soap bubble (with air inside), is mathematically interesting, not only because it shows the fundamental steady state of this bubble configuration that then we will likely see take part in other cases. The soap being a liquid shell, also provides necessary basic understanding before we might consider a liquid only scenario. It also gives an interesting case of where a less dense material (air) is trapped by a more dense one (soap). Another example of this would be the rising CO2- bubble in a glass of beer (I know you have been looking at them ...) or the rise of a hot-air or helium-mix balloon in the atmosphere. These cases are all physically similar and so they all potentially come from the same or similar physical effects and driving-forces.

If we look at the energy (E) of the initial atoms/molecules pictures, a molecule in contact with a neighboring molecule has a lower energy than if isolated. The reason is that a molecule (or atom) in the bulk of a solution has as many neighboring atoms/molecules and on average has a lesser energy difference relative these (the rest of the substance) than a molecule that is set apart or locally lumped or tied harder together. Being at rest under larger forces, molecules at the surface lack neighbors and will therefore have a higher energy relative to the solution overall since any movement will then require or release more energy. Consequently, for the solution to totally minimize its energy, the number of surface molecules (i.e. the surface area) needs to be minimized.

The SI-units of surface tension are N/m or J/m2, representing two possible understandings of surface tension:
1. As the force, F, required to stretch out a film of a substance a tiny distance, dx, i.e. (force/length) in N/m.
2. The free energy (or surface energy), dE, required to enlarge a surface area, dA, of a substance i.e. energy/area (J/m2)

Looking at item (2.), at equilibrium, this change in free energy is equal to the increase in surface energy, dE, such that
dE = γ·dA                               (Eq.1)

where the stretching coefficient, γ (gamma), is the proportionality constant that express how much surface energy is stored per unit area of the soap film, so γ=dE/dA. It can also be interpreted as how much energy is needed to change the area of the bubble area one tiny unit.

To understand and experimentally determine γ, consider a soap film with area A residing on a wireframe is stretched one-dimensionally (using a piece of wire that is crossing the frame) a tiny length dx by a force F. By this action, the free energy, dE, generated/added to the soap film is

dE = F·dx                              (Eq.2) This apparatus is of course very simple and there are way more precise ways to do these measurements, but for the sake of this case, let's accept this way of doing it. We see that in this specific case then
dA =2·L·dx                              (Eq.3)

where the “2” comes from that fact that the film in this case has two surfaces, one facing towards the reader and one facing away from the reader (into the screen). Double surfaces is indeed the case when we look at soap-bubbles also, one facing inwards into the bubble and one facing outwards, but in general this need not be so. One such different case is for solid bubbles of fluids (like water) in zero “gravity”. Another is fluid-filled balloons, where the “surface” material is made of different and a solid material (rubber). In those cases, there is no fluid surface and without a surface tension on the fluid, the rubber pose the main controlling force. In case there is a gas in the balloon, the rubber is definitely the only thing keeping the status-quo going.

Note also that in our fluid film example here, we are assuming that the inner volume of the fluid of  the film requires no extra force to stretch, i.e. that the inner bulk fluid molecules/atoms are very easy-flowing, at least compared to the atoms in the surfaces, so that the force, F, required is (approximately) only attributable to overcoming the tensions of the surface(s).

Putting together Eqs. 1-3, we get
F·dx = dE = γ·dA = γ·2·L·dx

From this, we see that for this case
F = γ·2·L

and so
γ = F/2L                              (Eq.4)

It is interesting to reflect on the meaning and source of γ as it is clearly determined by the stress-energy and stress-force relationship of the substance in the film. It is not a force dependent on the “mass” nor the volume of the substance.

As we can see in our initial model pictures of the bonding forces between the molecules/atoms of the substance, it is clear that γ is a combined macro-level kind-of spring-constant of the surfaces, valid only as a first linear approximation related to very small stretching of the film, dx. It is also a first concept to expand into a 2-dimensional stretching of the surfaces of a soap-film making up a soap-bubble at/near equilibrium.

If we turn from the force-perspective to the thermodynamic or energy-perspective and consider the situation for a bubble that undergoes a tiny change in size i.e. a tiny change in radius, dr, we note the basic area, volume and derivatives of those for a sphere are:

The energy equation of this tiny change will look something like this:
dE = dA γ + ΔW                              (Eq.5)

Here, the reduction of energy in the soap film covering the bubble is due to change of area, dA and this is opposed by mechanical work, ΔW, inside the bubble and relative to the surroundings. As we can see in the picture above, dA= -8πr·dr is already clear as we are considering a shrinking sphere (dr is negative). We also know how to obtain, γ for the substance by the previous experiment. ΔW is a bit trickier but considering that this work will mainly (only) come from an increased pressure, ΔP, inside the bubble, it can be expressed as
ΔW = - ΔP·dV = - ΔP·A·dr = - ΔP·4πr2dr                              (Eq.6)

Here, we have to assume that higher-order terms of dr (like dr2 and dr3) are relatively negligible because since the surface is bent, such higher-order terms will appear if we consider all effects on pressure. Putting (Eq. 5) and (Eq. 6) together, we now get the expression for dE as:
dE = γ (-8πr·dr) + ΔP·4πr2·dr

which at equilibrium (dE=0 & dE/dr=0) gives
0 = γ (-8πr·dr) + ΔP·4πr2·dr
So
γ·8πr = ΔP·4πr2
And this gives us the Laplace equation for a soap bubble

ΔP = 2γ/r                             (Eq.7)

This equation means that the pressure inside a soap-bubble is greater than the ambient (outside) pressure and that the pressure difference is due to the radius and the surface tension of the soap film. It also means that for a fixed radius, the pressure inside the bubble cannot be allowed to grow larger than the ambient pressure than 2γ/r because this is the limit of the tension that the soap surface. Any larger pressure difference and the bubble bursts. This especially becomes a factor when the surrounding pressure drops. One example of this is the bursting of helium balloons at high enough altitudes. As with the rubber surrounding the helium, there will not be enough soap to cover an ever expanding soap-bubble and γ will likewise eventually not be strong enough to keep the bubble together. Note also that a smaller bubble will have a larger pressure difference i.e. measured in the same ambient pressure, the pressure inside a small bubble is greater than in a larger bubble.

Some notes:

1. This mathematics regarding one soap-film could be mathematically integrated (with some adjustments, shells upon shells becoming a solid) to account for a fluid-bubble in zero-“gravity”.

2. The analysis especially show how the inherent material stickiness and in some cases also surface tension plays a key role in how internal pressure is set up inside a substance/material.  Consequently "density" and "mass" as defined in the Newtonian paradigm likely misses the point and the true nature of the driving force behind "gravity" as it builds on total/external "heaviness" rather than internal stickiness. But then again, the purpose of Newton & C/o was not to create real science but to win the war of world-views.

3. The stickiness (or stress tensor) of materials, not "mass" (or "weight"), is most likely the driver behind buoyancy and the illusion of "gravity" as well as the key to modelling every case correctly. The math here outlined will likely also be a major factor in understanding situations like bubbles of lower density being forced to the surface of a denser fluid and likely also how a denser material will “fall” through a less dense fluid/gas to join with a material of the same or even more dense nature.

4. It can also be clearly seen that the inner stickiness of the material sets up an internal pressure gradient regardless of any external "gravity".

5. There is a relationship between the internal stickiness and the density of the substance and this is the reason for why Newton's triad of concepts are working at all to partially explain reality. It is also the reason why relative density shows up as the moderating factor of the Newtonian "gravity" in Eric's work on buoyancy. This avenue will not be further pursued since it will be building on and thereby help keep rather than entirely expel the deceptive Newtonian concepts of "mass", "density" and "Newtons". Instead a "Newton-free" stress-tensor kind of formulations should be pursued.

6. On a large scale, the surface tensions likely plays a less to no role compared to the total amount of ordinary (non-surface) material stickiness. If we consider a boat floating in water, this buoyancy is determined in very little degree by surface tension (a fraction of a ppm?) but rather entirely by the reluctance of the bulk sub-surface volume of the water to divide itself (the bulk sum of stickiness).

7. Considering that all material bonding ultimately is electromagnetic in nature and residing in a "space" that is also basically electromagnetic, this avenue of continued thought and math, looks promising.

PacMan

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## Re: A critique of the mathematics of Newton’s “law” of gravity

Well, I tried but I don’t think I can understand this math. Not saying it’s wrong or right it’s just not something I can evaluate. What exactly does Gauss mean by ‘flow’?

Glad you could make sense of the math though and find some counter arguments to the mainstream dogma. Tree

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## In short

kirastar wrote:Any chance someone could explain this in like 2-3 short sentences for the elementary reader? I consider myself pretty bright but this is way over my head. I am an FEer / and I don't believe in Gravity// thanks!

It seems the concept and "law" of "gravity" was created by butchering an older physics equation that has its limit of validity. They then used a deceptive trick to fool people that the "gravity" in the new Frankenstein equation ultimately could extend beyond the "mass" generating it, utterly violating all physical validity.

In short, both the "gravity", "mass" and force in "newtons" were total inventions aimed at high-jacking what likely is simple buoyancy to motivate the existence of "gravity" as a completely separate force that could motivate the heliocentric model. - Genius but still, a complete mathematical hoax.

We all know there is a force there in some cases, but it is clear that it is NOT due to a "mass"-driven field extending beyond the surface of the source. Consequently the range of validity and capability to explain the "universe" is seriously questioned. This also includes the ideas of "gravity" as modeled in the theory of "relativity" as well as "quantum gravity" as those build on these invalid concepts. Together they may explain the reality but they also hinder a true science to be developed.

Sadly, these Newtonian concepts and units have long since completely saturated all of our science and everyday lives.
We are going the have to go back to the drawing-board and redefine the true source of the illusion of "gravity".
Toward that end, I have just now uploaded a small contribution.

PacMan

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## Re: A critique of the mathematics of Newton’s “law” of gravity 