part 1 - mass and gravity

7/31/2019 Part 1 - Mass and Gravity

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The Spacetime ModelVersion 3.0616 July 2009

Jacky JEROME Ingnieur Europen EUR-ING

Ingnieur DPE (Diplm Par l'Etat)Ingnieur IPF

Ingnieur ITP-ECIEmail: [email protected]

ISBN 97829531234-0-1Editions Arts et Culture 42

4, square Kennedy42120 LE COTEAU (France)

Cover: On the left of the photo of Einstein: Maxwell, Feynman, Max Planck, SchrdingerOn the right: Pauli, Niels Bohr, Marie Curie, De Brogglie, Dirac, Heisenberg

A step toward the Theory of Everything

Part 1

Mass and Gravity


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The Spacetime Model - Part 1 - - Introduction II

Patent Rights

This theory, the Spacetime Model, was registered atINPI, the French Patent Institute, under the followingreferences:

238268, 238633, 244221, 05 13355-2 895 559,248427, 258796, 261255, 268327, 297706,297751, 297811, 297928, 298079, 298080,329638, 332647, 335152, 335153, 339797 .

This list is not exhaustive and some recent registrations atINPI are not mentioned. The Spacetime Model was alsoregistered in other legal forms for Copyright.

First deposit date at INPI: May 5 th , 2005 Major deposit date at INPI: December 27 th , 2005

In 2006, the two versions of this document, English andFrench, were addressed to more than 7000 physicistsworldwide by e-mail. Several paper copies were sent inOctober 2006 to the most important Academics of Scienceand Committees of Foundations for Research.

The Spacetime Model was also published on November30, 2006, on 31 different web sites. It is also referenced onmany sites like Google Books, Yahoo, DMOZ...

The Spacetime Model is the intellectual property of its

author, Jacky JEROME, and any illicit appropriation of the theory will be subject to prosecution.


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The Spacetime Model - Part 1 - - Introduction III

Abstract

The Spacetime ModelThe Spacetime Model is a new theory based on spacetime.

Instead of limiting spacetime to relativity, the author has extended it to all elements of theuniverse: matter, charge, electromagnetism, leptons, quarks, antimatter... In combination andrelying on experimentation, this new theory explains, in a perfectly logical and rational way,more than 50 enigmas of quantum mechanics.

Moreover, the Spacetime Model unifies the three basic forces (gravity, electroweak andstrong nuclear force) in two generic forces: the Coulomb Force and the Hooke Force.

To improve readability, the Spacetime Model has been divided into five parts:

Part 1 .......... Mass and gravity (this part) (www.spacetime-model.com/files/mass.pdf) Part 2 ..........Constitution of matter (www.spacetime-model.com/ files/duality.pdf)Part 3 ..........Quarks and antimatter (www.spacetime-model.com/files/quarks.pdf)Part 4 ..........Electromagnetism (www.spacetime-model.com/files/electromagnetism.pdf)Part 5 .......... Forces, the Universe (www.spacetime-model.com/files/universe.pdf)

This document is the first part of the Spacetime Model. It offers the solution to the mass andgravity enigmas. Here are the main lines of this new theory.

The curvature of spacetimeLet's fill up a container with water. We drop a billiard ball into thecontainer. The volume of the ball produces a displacement of water.

The same phenomenon applies to spacetime. Contrary to generallyaccepted ideas, it is not mass which deforms spacetime, but volume,more exactly "closed" volume.

Mass = "Closed" volume?In our world, mass and volume seem to be two different quantities because in atoms, the massis not proportional to the volume. So, we have a large range of atoms with different massesand volumes. However, at the particle level, mass = volume (with some reservationsexplained in chapter 1 of this document).

In reality, we have two main classes of volumes:! Closed volumes (fig. A): These volumes make

a displacement of spacetime, which produces apressure on the surface of the volume. Amass effect appears, i.e. an effect that has allcharacteristics of mass. Nuclei and electronsare examples of closed volumes. Only closedvolumes produce a mass effect and,therefore, have a mass.

A B


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The Spacetime Model - Part 1 - - Introduction IV

Open volume

Closed volumes

! Open volumes (fig. B on the previous page):These volumes exist but do not produce anydisplacement of spacetime. They are porousregarding spacetime. If there is no curvature, there

is no mass effect either. So, open volumes aremassless. Orbitals of electrons in atoms areexamples of open volumes.

Each atom has a particular proportion of open and closedvolume. This is why mass and volume give us the illusionof being two different quantities.

What is Gravity?Two closed volumes inserted into spacetime curve it. Since spacetime is elastic, its curvatureproduces pressures on these two volumes. This tends to bringthem closer to each other. So, contrary to what we think:

Gravity is not an attractive force betweenmasses but a pressure force exerted byspacetime on closed volumes .

Since a pressure force is the opposite of an attractive force, andconcave and convex curvatures are in opposition too, there is nodifference between the Newton-Einstein Theory and this newexplanation of mass and gravity:

Attractive force + Pressure force +Concave curvature of spacetime Convex curvature of spacetime

Validation by mathematics and experimentation Usually, the Schwarzschild Metric and Newton Law are calculated from the Einstein FieldEquations (EFE). Chapters 4 to 6 of this document propose a new and innovative calculationof these two formulas, based on the Hooke Law (elasticity laws).

The suggested theory is also in perfect accordance with the 1921s Von Laue Diagram.

Moreover, this document proposes a simple and low cost ($5,00) experimentation, whichproves that the curvature of spacetime produces a pressure force, not an attractive force. Thisexperiment also has a curiosity: it highlights a black hole behavior when R = Rs.

The Higgs BosonThe Higgs Theory doesn't propose to the Physicists Community a simple explanation, a lowcost experimentation, and a full mathematical validation (Schwarzschild-Newton-Einstein) asthose explained in this document. The conclusion is immediate: the Spacetime Model ismuch more credible than the Higgs Boson Theory .

=


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The Spacetime Model - Part 1 - - 1 - Open and Closed Volumes 1

1 Open and Closed Volumes

Mass, gravity and spacetime are so linked that it is not possible to explain one without the other. To understand these phenomena, it is necessary to study two new concepts.

An understanding of these solves the mass and gravity enigma.

This is why, for teaching purposes, this subject is divided into two chapters: Open and Closed Volumes, and Gravity. Each chapter covers half of the solution.

1.1 Starting points

The following three arguments has been used as starting points:

1. Enigma of electron and positron massWe know that the mass of positrons is exactly equal to that of electrons 1. In addition,since the positron is the antiparticle of the electron, according to the symmetry theories(Richard Feynman, Nobel Prize 1965), their volumes must be identical (fig. 1-1).

If electrons and positrons have the same mass (511 KeV) and the same volume, they

also have the same mass per volume unit. Therefore, the following problem arises:

How can, two particles, having strictly the same mass per volumeunit, also have diametrically opposed internal constitutions 2?

1 The mass of the electron, 510.998918 +/- 0.00004 keV, is equal, or very close, to that of the positron. Theaccuracy of measurement is: |me+ - me-| /m < 8.10 -9, with a CL of 90%. For the following demonstration, thisaccuracy is significant enough to allow us to assume that electrons and positrons have the same mass.2 It is possible to consider that there is a probability that two particles with the same mass per volume unit canalso have opposed internal constitutions. Such a proposal would not be illogical, but the measurement is soaccurate (510,998918 KeV + 0,000044) that the probability of such a possibility would be infinitesimal. For example, on Earth, let us try to find two objects having the same mass per volume unit with an accuracy of 0,0000086% and different internal constitutions. Obviously, this challenge seems impossible.

Fig. 1-1

SameVolume

Same mass :510,998918 KeV


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This enigma allows for only one solution: mass and volume are two differentaspects of a unique and identical entity. In other words, mass = k.volume.

2. Inconsistencies in EFE (Einstein Field Equations)This subject is covered in paragraph 2-3.

3. The fifth dimension

Is mass function of spacetime? We are facing to the following two possibilities:

Mass is independent of the four known dimensions. In this case, it is not possibleto express the mass in the form m = f (x,y,z,t) . The only solution to this problem isto consider a fifth dimension like the Higgs field, which expresses all objects of the universe by a law of the form f (m,x,y,z,t) .

Mass is the function of four known dimensions: m = f (x,y,z,t) . In this case, allobjects of the universe can be expressed with the variables x, y, z and t.

According to Einstein, the universe has 4 dimensions. His field equations wereinitially formulated in the context of a four-dimensional theory. Thus, it would beadvisable to begin our research following the theory this great scientist proposed. Thissupposes that mass is function of spacetime.

There is no objection to envisaging new dimensions, like the Higgs Field, but only asa last resort, after having exhausted all the resources of the traditional spacetime. Aswe will see in this theory, spacetime is far from having revealed all of its secrets to

us In other words, it would be logical to begin our research in priority in 4D,excluding extra dimensions.

1.2 Basic concept

According to these three arguments, the following question arises:

is the mass nothing more than volume?

The answer is both, YES and NO.

! YES, because, as we'll see further, the Einstein Equations prove that mass and volumeare the same concept. We'll also see that this concept explains most of the greatestenigmas of physics: What is a quark? Where is antimatter? E=mc? 1 etc

! NO, because this definition is incomplete and requires some complements which aregiven in the following paragraphs.

1 Parts 2 to 5 of the "Spacetime Model" cover this topic.


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1.3 Current theory of spacetime curvature

Einstein's equations connect mass to the spacetime curvature. Consequently, we'll try tounderstand mass in an indirect manner. Indeed, the solution to this enigma arises from the

spacetime curvature.

Scientific authors often represent the spacetime curvature by a drawing like that of figure 1-2.This representation is very informative but does not answer the question how is it possible

for a mass to curve spacetime?

1.4 Explanation of spacetime deformation

Let's fill up a container with water. We drop a billiard ball into the container. Thevolume of the ball produces a displacement of water.

The same phenomenon applies to spacetime. Contrary to generally accepted ideas, it is notmass which deforms spacetime, but volume (fig. 1-3).

Fig. 1-2

This figure was simplified to two dimensions for teaching purposes.

In addition, our example does not correspondexactly to spacetime since water is not elastic. Itwould be more exact to compare spacetime to akind of deformable crystal or EPP (expanded

polypropylene). Since spacetime is elastic, if weremove the central object, the curvature mustdisappear.

Fig. 1-3


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This deformation isn't concave but convex. However, we will see in chapter 2 that the natureof gravity (a pressure instead of an attractive force) exactly compensates this modification of curvature. So,

It must be noted that the word volume may not be exact since the universe has four dimensions and not three but, whatever the dimension of the space is, 1D, 2D, 3D or 4D, thereasoning is the same. For example, let's imagine a simple line (1D). A small segment insertedinto the middle of the line will push out its two ends to make room. Similarly, in 2D, a smallsurface inserted into a larger one will push out the surrounding surface to make room.

Spacetime shares the same principle. Any volume inserted in spacetime pushes outsurrounding spacetime to "get room". Thus, whatever the dimension of the space, 1D, 2D, 3Dor 4D nD, we always have the same phenomenon.

Note: Considering that spacetime is present everywhere, more particularly in volumes, we could think that any volume can't curve spacetime. In reality, particles are hermetic regarding spacetime. This

problem is covered in Part 2 of the Spacetime Model: "Constitution of Matter" (please download thePDF document "Part 2: Constitution of Matter" from the Web site www.spacetime-model.com).

1.5 Mass-volume equivalence

In the Einstein Equations, we should note two curvatures of spacetime:

A first curvature, produced by mass. The mass variable, "m", is present in theSchwarzschild Solution for example.

A second curvature, produced by volume. The volume variable, "v", must be present

in the Einstein Equations. This second curvature should have the form C = f (v).The formulation of the spacetime curvature should therefore be C = F (m, v) , which is thecombination of two curvatures, one from Einstein's theory produced by masses, C = f (m),which does not need to be verified, and another one produced by volume C = f (v). This secondcurvature is necessary, based on logical reasoning and good sense.

However, on close examination of Einstein Field Equations and their solutions, we find onlyone curvature, the one produced by mass. We don't see any trace of volume curvature.

We naturally deduce from this that the curvature of Einstein's theory due to mass and the

curvature due to volume are the same phenomenon. In other words, this means that mass andvolume are the same entity. but things are not quite so simple We'll study this subjectthoroughly in paragraph 1.7.

Any volume inserted in spacetimenecessarily produces a curvature of it


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1.6 Einstein Field Equations

Note: Readers who are not familiar with tensors may bypass this section.

In this section, we will try to solve the same problem with Einstein Field Equations.These equations, usually called Einstein Field Equations 1, or EFE, emphasize an identity between the properties of matter, mass-energy-momentum, and the geometry of spacetime,which is a variety of Riemann Spaces. Einstein discovered equivalence between the stress-energy tensor T jk and the geometrical tensor of curvature R jk - (1/2) g jk R. This mathematicalequivalence can be expressed as follows:

Geometry of spacetime Matter and energy

Hiding the cosmological constant , the EFE, which must be read from right to left, becomes:

R jk - (1/2) g jk R = (8 G/c4) T jk

The member of left is a geometrical tensor verifying a mathematical property of Lorentzian conservation. R jk is the tensor of Ricci and R the scalar of Ricci. Themember of left describes a representation of the geometry of spacetime.

The member of right is the T jk stress-energy tensor. Since energy comes from mass(special relativity), it gives the representation of the energy-momentum.

The simplest solution of Einstein's equations is the Schwarzschild Solution. Its metricdescribes the deformation of spacetime produced by a static object with a spherical symmetry.

Setting down (x 0, x1, x2, x3) = (ct, r, , ), this metric is written as follows:

The expression of mass M exists in the elements T 00 and T 11 , but we do not see any trace of the expression of volume 2 V. This expression should exist since, like in our example of a

billiard ball into water,

1 Sometimes called "Eintein-Hilbert Field Equations"2 We are looking for an expression of volume that produces the spatial component of spacetime deformation.This volume should not be confused with the spherical coordinates R, , and of the point of measurement.

0 0 0

0 0 0

0 0 r 2 0

0 0 0 r 2sin2

(g) =

21

rcGM +

211

rcGM

Any volume inserted in spacetime mustcurve it. It is an absolute necessity.


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The Schwarzschild Metric has been verified many times. So, the only means by which tosolve this enigma is to consider that mass and volume are connected by an equation of theform m = f (v).

It should be noted that the same reasoning could be applied to other metrics1

but, since theSchwarzschild Metric reflects an anomaly, it is not necessary to go further.

1.7 Classes of volumes

Let's replace the billiard ball used in the example in paragraph 1.4 by a balloon having thesame volume (fig. 1-4 A). This does not change anything.

However, if we make some holes in the balloon (fig. 1-4 B), water will go inside it and the

displacement of water will disappear.

The same phenomenon also exists in spacetime. In reality, we have two main classes of volumes:

Closed Volumes (fig. 1-4 A): These volumes make a displacement of spacetime,which produces a pressure on the surface of the volume. A mass effect appears, i.e.an effect that has all characteristics of mass (see chapter 2 and 4). Protons, neutrons,

electrons, positrons, muons, taus, quarks, mesons are examples of closed volumes.Only closed volumes produce a mass effect and, therefore, have a mass.

Open Volumes (fig. 1-4 B): These volumes exist but they are porous regardingspacetime. Therefore, they do not produce any displacement of spacetime. Since thereis no curvature, there is no mass effect either . Orbitals of atoms are examples of open volumes. Open volumes are massless.

Empty space between atoms or molecules is considered as an open volume. This empty space,of course, is massless.

All atoms of the universe, and therefore all objects, are combinations of these two classes of

volumes: closed volumes (with mass), and open volumes (massless).

1 Kerr, Reissner-Nordstrm, Robertson-Walker etc

A B

Fig. 1-4


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1.8 Example: Atoms

In atoms, orbitals are crossed on both sides by spacetime (fig. 1-5). It is a normal situationsince orbitals are made of a vacuum. As a result, orbitals do not produce any spacetimecurvature. The volume of orbitals is massless.

On the other hand, the nuclei and electrons are closed volumes and, therefore, have a mass.

1.9 Conclusions

Calculations are not modified by this equivalence mass-volume. However, we must alwaysremember that "mass" means "closed volume" and conversely (see chapter 4 for conversionformulas). We must also take note of the difference between open volumes, which aremassless, and closed volumes, which has mass.

Fig. 1-5

Open volume (orbital)

Closed volumes:Electron(s) and

nucleus

Spacetime

Experimentation confirms that atoms are made of:

1/ Open volumes: massless orbitals (a vacuum)2/ Closed volumes: nucleons and electrons, with mass.

This is why we have the illusion, on Earth, thatmass and volume are two different concepts.


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So:

To understand what is mass, we must replace

Open and/or closed volume

Closed volume

Volume by

Mass by

Fig. 1-6Note: As indicated, complex elements like atoms

have a combination of open and closed volumes.


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The Spacetime Model - Part 1 - - 2 - Mass and Gravity 9

2 Mass and Gravity

It seems obvious that it is the volume (closed volume), and not the mass, whichdeforms spacetime. Now the challenge is to understand how such a spacetimecurvature by a volume can produce the mass effect and gravity.

2.1 Principle of gravity

Two closed volumes inserted into spacetime curve it (fig. 2-1). Since spacetime is elastic, itscurvature produces pressures on these two volumes. This tends to bring them closer to eachother.

So, contrary to what we think,

Gravity is not an attractive forcebetween masses but a pressure force

exerted by spacetime on closed volumes

Note: This figure, in 1D, isfor teaching purposes only.

Fig. 2-1


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This leads to the following remarks:

! In fluid mechanics, constraint tensors are always built on volumes, not onmasses . Indeed, it is the volume and not the mass that produces the viscosityand pressure in fluid mechanics, as in the proposed theory.

! Since the stress-energy tensor is built from the constraint tensor, why has massreplaced the volume?

! Viscosity, in the Einstein's tensor, is the same as the one in the constraint tensor.If the viscosity is the same, why would the trace be different? Why has anattractive force replaced the initial pressure force?

These inconsistencies probably have the following explanation.

Since Newton's time, physicists have supposed that gravity is an attractive force. This is onlyan assumption. We must keep in mind that, in Newton's time, spacetime was unknown.

Today, we have a better knowledge of spacetime and the context is different. Therefore, wemust reconsider this question because Newton's point of view concerning gravity as an

attractive force is wrong.

Be that as it may, these inconsistencies lead to the following deduction:

T00 T01 T02 T03

T10 T11 T12 T13

T20 T21 T22 T23

T30 T31 T32 T33

Gravity = Attractive force

Fig. 2-3

Pressure

T00 T01 T02

T10 T11 T12

T20 T21 T22

Constraint tensorin fluid mechanics

Stress-energy tensorin General Relativity

In the original constrainttensor, the significance of the

trace is a pressure.

Einstein built his tensor from theconstraint tensor. Curiously, he

replaced the initial pressureforce (the trace) by an attractive

force. Why ???


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This conducts to the following equivalence:

Gravity + concave curvature = Pressure force + convex curvature

Figures 2-4 and 2-5, on the following page, show the difference between the two concepts.

2.4 Synthesis

In this synthesis, we will only consider closed volumes since open volumes do not produce amass effect. The whole phenomenon is summarized as follows:

1. Closed volumes curve spacetime.2. Since spacetime is elastic, a pressure force (Hooke Force) appears on the surface

of these closed volumes.3. This pressure force produces a mass effect.4. The same pressure force on the surface of two or more volumes tends to bring

them closer to each other. This effect is gravity.It should be noted that volume exists physically, but mass doesn't. Mass is only an effect dueto pressure from the curvature of spacetime produced by closed volume.

Note: The term effect, used on several occasions here, doesn't mean gravity is a kind of "virtualeffect". Since the spacetime curvature is quite real, the mass effect is real too. In other words, this"mass effect" is, of course, a reality. The reader must avoid misinterpretations of the word effect

2.5 Wheeler's Intuition

Let's note the intuition of the great physicist John Archibald Wheeler:

Mass and energy tell spacetime how to curve itself and the spacetime curvature tellsmatter how to behave.

If we replace mass by closed volume, Wheeler's expression describes, word for word ,the theory presented here:

Closed volumes tell spacetime how to curve itself, and the

spacetime curvature tells matter how to behave.

The connection between the constraint tensor and the stress-energy tensor clearly shows that:

! Spacetime is curved by closed volumes (not by masses)! Gravity is a pressure force (not an attractive force)


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Fig. 2-5

Closed volumescurve spacetime

Spacetimeelasticity

Closedvolume

Object

Proposed Theory (close to Wheeler's intuition)

Masseffect

Spacetime produces a

pressure force onclosed volumes

Gravity

A mass effectdue to the

pressure of spacetime is

associated witheach closed

volume.

Only volume physically exists. Mass does not exist per se. Mass is aneffect produced by the pressure of spacetime on closed volume.

1

2 3

4

Fig. 2-4

Gravity ??? (*)

Mass ???

(*)Volume

Curvature of spacetime ??? (*)

Current Theory = No Explanation

(*) No explanation

Object


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2.6 Example of mass and gravity

In this example, imagine that you hold a very light polystyrene ball in your hand (fig. 2-6). Aflexible EPP (expanded polystyrene) foam, which is curved by the ball, replaces spacetime.

The curvature of the foam (spacetime) produces a pressure on the top of the ball, towardEarth. If you close yours eyes, you will sense that the ball is heavier. It is not the mass of the ball, which increases, but the pressure on the top of it. The result is the same.

This is what is called a "mass effect".

As we see in this example, mass is directly related to the pressure of the flexible polystyrenefoam on the ball. It is the volume of the ball that deforms the foam, not its mass. Transposedto spacetime, this example clearly shows that mass physically doesn't exist. Mass is a simpleeffect due to the pressure of spacetime exerted on closed volumes.

This example also means that gravity is a pressure force produced by volume, instead of anattractive force made by mass.

2.7 Equivalence principle

The proposed theory fully explains the equivalence of gravitational and inertial mass.

Lets consider an object on Earth (Fig. 2-7). The volume of this object produces a curvature of spacetime which exerts a gravity force on it. The acceleration due to gravity is g = 9.81 m.s -2 on the surface of Earth.

Fig. 2-6

Note: For teaching purposes, in thisfigure, Earth has

been omitted.

EPP


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Lets now consider the same object which is accelerated out of gravity (fig. 2-8A). Theacceleration, , is supposed identical to g, i.e. = 9.81 m.s -2.

Without any reference, a local observer cant say if the acceleration comes from the object or from the curvature of spacetime. In fact, figures 2-8A and 2-8B are identical and depend onwhere the observer stands, as described in Spacial Relativity. Since

g = 9.81 m.s -2 (fig. 2-7) is identical to = 9.81 m.s -2 (fig. 2-8). These examples uses the same object. Therefore, the curvature of spacetime produced

by the volume (more exactly, by the closed volume) of the object is identical. So, the mass effect produced by these curvatures will be the same.

According to Einstein, we deduce that the gravitational mass effect (fig. 2-7) is identical tothe inertial mass effect (fig. 2-8B). In other words:

Gravitational mass = Inertial mass = Spacetime effect

Fig. 2-7

g Note: For teaching purposes, in thisfigure, Earth has

been omitted.

Fig. 2-8

A B

This figure isidentical to fig.2-7 since g =


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2.8 Relativistic particles

The mass of a particle increases when its speed approaches the speed of light, c. This well-known phenomenon is defined in Special Relativity but no one can explain it with logic. The

proposed theory solves this enigma with a great simplicity (fig. 2-9).

As the velocity increases toward the speed of light c, the curvature of spacetime produced bythe volume of the particle grows too. This phenomenon is identical to the opposition force

produced by the pressure of air on cars which increases with speed (F=kv).

At relativistic speed, spacetime is compressed. The pressure of spacetime on the volume of the particle increases. This increase of pressure explains the apparent growth of the mass.

Therefore, contrary to a preconceived idea:

At relativistic speed, the mass of a particleremains unchanged. It is its mass effect due to

the compression of spacetime that increases.

Current Theory Proposed theory

Fig. 2-9

V = 0

m 0

Relativistic speed V

m 0

1 v 2/c 2m =

The speed of the particle produces acompression of spacetime, which givesus the illusion that the mass increases.

The particle curves spacetimeDirection of movement

Why ? No one can explain


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2.9 Crystals

We know that the mass of a particle moving inside a crystal increases (fig. 2-10).

No one can explain this strange phenomenon. However, the solution of this enigma is verysimple, considering that mass is an effect of the spacetime curvature. In this example, lets takethe cubic system, which is the simplest structure of a crystal.

We can consider the lattice of a crystal as an array of tunnels. The particle is moving insideone of these tunnels.

Closed volumes inside each atom of the tunnel (the nuclei and electrons) curve the spacetimelocated on the path of the particle. Therefore, the density of spacetime will be higher insidethe tunnel than outside the crystal. Since a high density (or curvature) of spacetime producesan opposition force, the apparent mass, more exactly the mass effect, of a particle crossing acrystal increases too.

It is a necessity.

This explanation is exactly in accordance with experimentation.

Note 1: The volume of the particle remains unchanged. The only thing that increases is the spacetime densityinside the tunnel of the crystal, therefore the mass effect.

Note 2: The same phenomenon also exists inside matter different than crystals. However, it is impossible, or at least very difficult, to measure exactly the mass effect inside matter.

Note 3: It would be interesting to calculate new solutions of the EFE with different kinds of Bravais Lattice:cubic, orthorhombic, monoclinic, triclinic... structures. These metrics would be useful to predict the increase of the mass effect of a particle moving inside a particular crystal.

Fig. 2-10

m (out of any mass)

M

M > m

Note: For teaching purposes, in thisfigure, the mass of the

particle has beenincreased. In reality, itis its "mass effect"which increases.


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2.10 Example of mass-volume equivalence 1

Fig. 2-11

1 In this example, the mass of electrons, the binding energy etc has been ignored. Moreover, we consider thatthe volume of protons is identical to that of neutrons.

This simple thought experiment demonstrates that what wecall mass is, in reality, a mass-effect, which is the

pressure exerted by spacetime on closed volumes.

On our left, we have an eraser and on our right, a pen thatweighs two times more.

x2

If we remove the 99.999% of vacuum existing inside the atomsof the two objects, we obtain two"heaps" of nucleons. We willhave two times more nucleons for

the pen than for the eraser because all the mass is practicallyconcentrated in nuclei.

Since all nucleons have the samevolume, the pen nucleons willhave a total volume two timessuperior to those of the eraser.

x2

Therefore, the pen will producetwice more curvature of spacetime than that produced bythe eraser.

Since spacetime curvature =ressure , the spacetime will

produce a pressure two timesstronger over the pen than over the eraser in the pen/eraser-Earthcontext. Therefore, the pen will

be two times heavier than theeraser.

x2

x2Pp


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2.11 The mass excess

No one knows exactly the origin of the mass excess, why it is negative in some nuclides, andnull or positive in others. The Bethe-Weizscker formula is accurate but doesnt explain this

well-known but strange phenomenon.

The proposed theory gives a rational and consistent explanation of this enigma.

Now, we know that the mass effect is function of the volume which produces the curvatureof spacetime. The shape of the surface must also be taken into account because spacetimeexerts a pressure on the surface of the volume.

Before going on, we must note that two objects, like a cube and a sphere, can have identicalvolumes with different surfaces. The mass excess phenomenon is based on this ratiovolume/surface.

In the following example (figure 2-12), we consider a nuclide with 19 nucleons.

! Independent nucleons (A) If the volume of a nucleon is V and its surface S, thetotal volume is 19V and the total surface 19S.

! Atom (B) The 19 nucleons are linked to make an atom. The grey surface representsthe enclosed volume between nucleons. In reality, this closed volume curvesspacetime.

! Atom (C) From an external view, the figure (B) looks like (C). The global volume

and surface are different than those calculated from the 19 independent nucleons (A).Therefore, the mass effect of figure (C) is different from that of figure (A).

A B C

Fig. 2-12


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In all cases, we have 19 nucleons, but the volume and surface are different, depending of thearrangement of these nucleons. The calculation isnt a simple multiplication followed by anaddition.

So, it is obvious that, when two or more nucleons are linked, the overall volume and surfacehighlight minor differences. These differences are nothing but the mass excess (positive or negative) since mass = closed volume .

Note:More examples of the equivalence Mass = Closed Volume are described in Part 4, chapter 4.

2.12 Conclusion

Current Theory Proposed Theory

What is mass? No one knows

Gravity is an attractiveforce between mass

Why ???

Closed volumecurves spacetime

Gravity is a pressureforce on volumes

produced by spacetime

Logical andconsistent

explanation of Massand Gravity

No explanation ofthese phenomena

Mass = closed volume

Mass curves spacetimeWhy ???


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The Spacetime Model - Part 1 - - 3 - Validation by Experimentation 21

3 Validation by Experimentation

The following experimentation can be repeated many times, under the same conditions. It always produces the same results. Therefore, this experiment can be considered as avalid scientific simulation of mass and gravity.

3.1 Introduction

Einstein demonstrated that spacetime has an elasticity behaviour and can be calculated fromFluid Mechanics.

Therefore, it is possible to simulatespacetime with an EPP (expanded

polypropylene) foam, since spacetime andEPP follow the same rules. Indeed, elasticitylaws can be applied in both cases and the

basic principle is identical.

The purpose of this experiment is todemonstrate that gravity isnt an attractiveforce between masses, but a simple pressureforce produced by spacetime on closedvolumes (fig. 3-1).

3.2 Basic material

To conduct this experiment, we need:

A piece of expanded polypropylene foam (EPP) measuring 30x21 cm, 2 cm thick A drawing, on the EPP foam, of a set of lines spaced 5 mm apart. Two cylinders with a 2 cm in diameter Two Force Sensing Resistors (FSR) - see the following paragraph - Some basic tools such as a soldering iron, a power supply, a multimeter, a cutter... A few basic components such as wire, resistors, trimmers...

Fig. 3-1


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3.3 The FSR

A Force Sensing Resistor (FSR, fig. 3-2) is a polymer thick film (PTF)device which exhibits a decrease in resistance with an increase in the force

applied to the active surface.The two FSRs used in these experiments are manufactured by theInterlink Company, part # SS-U-N-S-00015 (price = $1.00 each). The

pressure ranges from 0.007 to 7 bars, and the resistance decreases from 10M to 1 K with an increase of force.

3.4 Experiment #1

Experiment #1 shows that far from any mass, the resistance of a FSR is 10M (open circuit,fig. 3-3). When a mass is placed on a FSR (fig. 3-4), its resistance decreases to 35 K .

Since the weight of a mass is directly related to gravity (second Newton Law, Weight=mg),this experiment shows the following simple but important deduction:

Fig. 3-2

Decrease of FSR resistance = Presence of gravity

Fig. 3-3 Fig. 3-4


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3.5 Experiment #2

Two FSRs measure the pressure produced by an expanded polypropylene (EPP) foam oneach side of a volume V1. Figures 3-5 and 3-7 represent the first step of this experiment.

In order to get an accurate measurement and to use only one galvanometer in this experiment,the two FSRs are inserted in a Wheatstone Bridge. This setup is frequently used in straingauge measurements. Figure 3-6 represents the circuit diagram.

The volume V1 is inserted in the EPP foam and the Wheatstone Bridge is adjusted by VR1 toobtain a zero voltage between the two midpoints A and B (fig. 3-6). No current flows throughthe galvanometer Vg. The pressure of the EPP foam (or spacetime) on both sides of thevolume V1 is identical.

A

FSR2 FSR1

R1 VR1

Vg B

C

D

Fig. 3-6Fig. 3-5

FSR2 V1 FSR1

Pressure of the EPP foam(= spacetime) on the two FSRs

V1

Fig. 3-7

FSR1

FSR2

CA

D

B


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Without changing anything, a second volume, V2, is inserted near the first volume V1 (fig. 3-8). We note a deviation on the galvanometer (fig. 3-10), which indicates an additional

pressure on FSR1.

To summarize, with one volume, the galvanometer indicates no voltage (fig. 3-9). With twovolumes, it indicates a voltage proportional to the pressure produced by the second volume onFRS1 (fig. 3-8 and 3-10). This force beetween V1 and V2 can be identified to gravity since"voltage on FSR = Presence of gravity" (paragraph 3.4).

Fig. 3-9 Fig. 3-10

Fig. 3-8

V2

FSR1

V1

FSR2

CA

D

B


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3.6 Conclusions! Out of any volume (or mass), the voltage in a Wheatstone Bridge is 0 V (fig. 3-9).! When a second volume is inserted into the EPP foam the voltage of FSR1 increases

(fig. 3-10).! Experiment #1 (fig. 3-3 and 3-4) proves that a voltage on a FSR 1 indicates a

"presence of gravity" .! Deduction: The voltage on FSR1 clearly shows that between V1 and V2 a force

identical to gravity has appeared.

At last, this experimentation also confirms that gravity is a pressure force exerted byvolumes (not by masses) since mass of the cylinders is irrelevant. Only the volume of thecylinder is taken into account.

3.7 The light deflection

Every physicist is familiar with the observation of the light deflection by the sun made by Sir Arthur Eddington and his collaborators during a total solar eclipse in 1919.

Replacing the spacetime elasticity with that of an EPP foam (fig. 3-11), this simpleexperiment explains with consistency this strange phenomenon.

3.8 The Von Laue Diagram 2

The following thought experiment demonstrates that the proposed theory is in accordancewith the Von Laue Diagram. The Author, due to practical limitations, can't actually performthis simple experiment 3.

1 More exactly, the FSR resistance decreases and, therefore, the voltage increases.2 Von Laue, 1921, page 226, reported by Jean Eisenstaedt "Einstein and General Relativity", page 247.3 This very simple experiment is not accurate with traditionnal EPP foam. To increase accuracy, this experimentmust be conducted by a laboratory and the scientific community. The Author is a physics hobbyst and does notwork in an institutional establishment. Therefore, he has not the financial possibility to conduct this experimentthat requires some funds.

Fig. 3-11

Light


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A set of concentric circles, spaced 5 mm apart, has been drawn on EPP foam (fig. 3-12A). Avolume is inserted in the middle (fig. 3-12B). Therefore, the lines are displaced since the EPPfoam is elastic.

These concentric circles have been duplicated in fig. 3-12C. The Von Laue Geodesics has been drawn over these circles. We see that the Von Laue Geodesics match EXACTLY theconcentric circles. So, the Von Laue Diagram and the proposed theory conduct to the sameresults. It means that the two theories are based on the same principle.

In other words,

It seems that in 1921, Von Laue predicted the proposed theory.

B

Fig. 3-12

A

2GM/c

3GM/c

3 3 GM/c

C

Von Laue Geodesics


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The Spacetime Model - Part 1 - - 4 - Mass-Volume Conversion 27

4 Mass-Volume ConversionThe purpose of chapters 4 to 6 is to validate the entire theory described in this document using mathematics. This chapter studies the mass-volume conversion equation.

4.1 Abbreviations used in formulas

F = Force, in Newtons [ML/T]P = Pressure, in Pascals or N/m [M/LT] = Conversion constant: 1,5704459.10 28 m3/kg.s for the proton [L 3/MT] = Strain x/x 0 [1]G = Universal Constant of Gravity: 6,67428.10 -11 [L3/MT]

4.2 Conventional mass-volume conversion

The volume-mass equivalence described in the preceding chapters may be expressed by anequation like (values for the proton):

m = k.v k = 5.9239 x 10 17 kg/m 3 (CODATA 2006)or v = K.m K = 1.6881 x 10 -18 m3/kg

The constants k and K have respectively the dimensional quantity of [M/L 3] and [L 3/M].These two constants are simple constants of conversion. On Earth, we often use similar constants as mass per unit volume". However, things are not so simple.

4.3 Starting pointIn physics, we use a constant, G, which has the dimensional quantity of [L 3/MT]. Letsexamine these two expressions, K, the traditional volume-mass ratio, and G, the universalconstant of gravity:

K = [L 3/M]G = [L 3/MT]

We remark that K and G are two constants very close to each other. The only difference isthat G has an additionnal term, [1/T].


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A basic constant is a ratio between two or more quantities. For example, c, the speed of lightin a vacuum, is the ratio between x and t, which leads to a constant, c = 300 000 km/s. A basicconstant cant have only one value, as [1/T]. If such is the case, it is not a basic constant butthe value of a particular object, as 511 KeV for the mass of electron 1.

If G = [L 3/MT] and K = [L 3/M] are two basic constants, the ratio G/K must be another constant having the dimensional quantity of:

If we consider that the mass per volume unit K is a constant, we are faced a problem since thesecond member of this equation, [1/T], can't be a basic constant 2.

4.4 The "mass effect"

Lets examine figure 4-1, which is identical to figure 2-5.

1 Except with Planck Units t p, m p .... In this case, these units become references and replace G, h, c.... However,this point of view doesnt change the reasoning stated above.2 This point of view, considering that [L 3/M] could not be constant, must be taken with great care since manycombinations of basic constants are possible. It is offered only as a suggestion, a possible way of research for anew theory, nothing more.

Fig. 4-1

Closed volumes

curve spacetime

Spacetimeelasticity

Closedvolume

Object

Masseffect

Spacetime produces a

pressure force onclosed volumes

Gravity

A mass effectdue to the

pressure of spacetime is

associated with

each closedvolume.

1

2 3

4


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On one hand, the "mass effect" M must be proportional to the volume V that produces thespacetime displacement. On the other hand, the "mass effect" must be inversely proportionalto the surface S since the pressure exerted by spacetime on the object follows the rule P = F/S(the "mass effect" acts as a pressure). Combining these two considerations, the mass effect

would be proportional to:

This preliminary result is interesting.

Let's return to the above equation (4-1). This formula can be written as:

or:

with K = volume per mass unit [L 3/M] 1. The dimensional quantity of (4-5) becomes:

or:

This equation reflects reality since it includes the surface component but it isn't homogenous.To render it homogenous, we must add an unknown term (constant or variable) having the

dimensional quantity of [1/L] :

We can transform [1/L] in [1/T] without changing anything, using the well-known formula x= ct, or x = ct. Equation (4-8) becomes:

1 K already includes a basic conversion constant without dimensional quantity

(4-8)

(4-9)

(4-1)

(4-2)

(4-3)

(4-4)

(4-5)

(4-6)

(4-7)


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Rearranging the right member:

This is a second interesting result.

4.5 Mass-volume conversion

Deduction of paragraph 4.4, and more precisely equation 4-10, strongly suggest that thesimplest form of an equation matching the process of figure 4-1 and including the surfacecomponent (4 R = [L]) could be:

Where (the following values concern the proton and may differ with other particles):V = Volume (volume of the proton = 2.8235 x 10 -45 m3) = [L 3 /MT] ( for the proton = 1.57045 x 10 28 m3 /kg.s)

R = Radius (radius of the proton = 8.768 x 10 -16 m) M = Mass (mass of the proton = 1.672622 x 10 -27 kg)

And:4 R = Surface (surface of the proton, assuming it is spherical = 9.6607 x 10 -30 m) 4 R/c = Volume-mass conversion factor [L 3 /M] = 1.68808 x 10 -18 m3 /kg

Since V = 4/3 R 3, assuming the proton spherical, the formula (4-11) may also be written asfollows:

Simplifying:

or:

As indicated in equation 4-3, the relation between R and M is linear.

(4-10)

(4-11)

(4-13)

(4-12)

(4-14)


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4.6 Confirmation by experimentation

The following table calculates some characteristics of protons and deuterons.

On Earth, for a given element, the mass per unit volume is constant. For example, that of ironis 7,874 g.cm -3. However, the problem seems more complex for nuclear matter.

Proton Deuteron Ratio deuteron/proton

Mass M 1.67262E-27 3.34358E-27 1,999

Radius R 8.76800E-16 2.14020E-15 2,441

Volume V = 4/3 R3 2.82352E-45 4.10631E-44 14,543

R/M (Radius/Mass) 5.24207E+11 6.40092E+11 1,221

Considering that the mass is proportional to the volume, as on Earth, the volume of thedeuteron must be twice that of the proton since the deuteron/proton mass ratio is 1.999.

Instead of x2 , we have x14.5 !!!.

On the other hand, the last line of the table calculates the ratio, R/M. This value isn't 1.00, aswe could expect, but 1.221 1. It confirms our previous result (equation 4-3, paragraph 4.4),which is summarized as:

This table, from CODATA 2006, clearly shows that the Mass/Volume conversion isn't linear,as we could think.

Moreover, this conclusion exactly matches equation (4-13) that we obtained in the previous paragraph. So, CODATA values based on experimentation provide a first validation of the proposed theory.

4.7 Calculation of the curvature of spacetime

The variable R in equation 4-13 must not be interpreted as a radius but as a displacement of spacetime. As we know, a simple displacement doesn't produce a force. In reality, theelasticity of spacetime produces a force (or a pressure) on closed volumes. Therefore, wemust take into account R, an "elastic displacement", not "R" which is a simple displacement.

R is calculated from the coefficient of elasticity of spacetime, , a constant withoutdimension. Since Einstein built his EFE from the Fluid Mechanics, we can use, as Einstein,the Fluid Mechanics to define R by the well-known formula:

1 perhaps because the deuteron isn't spherical

(4-15)


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Since R


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The Spacetime Model - Part 1 - - 5 - The Schwarzschild Metric 33

5 The Schwarzschild Metric

The purpose of this chapter is to validate the theory described in this document byretrieving the Schwarzschild Metric from the mass-volume formula (4-21 and 4-22).

Calculations are made on the assumption of a static non-rotating spherical symmetrymass with a weak fields approximation: x


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Experimentation with EPP foam

r Distance measured between the base and the point of measurement r of the curvature of

spacetime (see fig. 5-1 on the previous page).

y(r exp) The curvature of spacetime y r is calculatedfrom the distance r measured in thisexperiment. The radius component of theSchwarzschild formula has been normalizedto:

y(r exp) is the experimental calculation.

The following values are contracted in 2.2 with:R = 22mm (the radius of the cap of the milk bottle) = 0.1, an arbitrary EPP foam coefficient(see 4.7)R = R = 0.1 x 22mm = 2.2 mm.

Note: The dimensional quantity of 2.2/r is [1] since 2.2is a length , not a mass.

r y (r) 22

24,4 1,1026,8 1,09

29,4 1,0832,2 1,0735,2 1,0738,3 1,0641,8 1,0645,9 1,0550,4 1,0555,1 1,0459,9 1,0464,6 1,0469,3 1,03

74,4 1,0379,3 1,0384,4 1,0389,5 1,0394,5 1,0299,5 1,02

104,2 1,02108,8 1,02113,6 1,02118,4 1,02123,2 1,02

128 1,02132,9 1,02

(5-1)

0,96

0,98

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 3 5 7 9 11 13 15 17 19 21 23 25Fig. 5-3

Table of figure 5-3


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Theoretical calculation

Now, we are going to calculate thedata on a mathematical basis, usingthe same formula as the precedentone (5-1). The values are:

! Col. 1: "n" is the rank of each line! Col. 2: Lines "r 0" are spaced 5 mm.

apart, out of gravity, with an offsetof 22 (22 mm. is the radius of theclosed volume).

! Col. 3: r, is calculated using theelasticity of spacetime, , supposedto be 0.1 in this simulation.

!

Col. 4: Finally, the distance between the base and the point of measurement of the spacetimecurvature, r, is computed from R and r 0.

! Col. 5: Result y (r theor.)

As in our previous example, thecurve of function y (r theor.) is:

Col. 1 Col. 2 Col. 3 Col. 4 Col. 5

n r 0 = 5n+22 r = 0.1 r 0 r = r 0 - r y(r) 1 27 2,70 24,30 1,10

2 32 3,20 28,80 1,083 37 3,70 33,30 1,074 42 4,20 37,80 1,065 47 4,70 42,30 1,056 52 5,20 46,80 1,057 57 5,70 51,30 1,048 62 6,20 55,80 1,049 67 6,70 60,30 1,04

10 72 7,20 64,80 1,0411 77 7,70 69,30 1,0312 82 8,20 73,80 1,0313 87 8,70 78,30 1,0314 92 9,20 82,80 1,0315 97 9,70 87,30 1,0316 102 10,20 91,80 1,0217 107 10,70 96,30 1,0218 112 11,20 100,80 1,0219 117 11,70 105,30 1,0220 122 12,20 109,80 1,0221 127 12,70 114,30 1,0222 132 13,20 118,80 1,0223 137 13,70 123,30 1,0224 142 14,20 127,80 1,0225 147 14,70 132,30 1,0226 152 15,20 136,80 1,02

0,96

0,98

1,00

1,02

1,04

1,06

1,08

1,10

1,12

1 3 5 7 9 1 1

1 3

1 5

1 7

1 9

2 1

2 3

2 5

Fig. 5-4

Table of figure 5-4

(5-2)


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Conclusions

As we see, the two tables and their associated graphics 5-3 and 5-4 are very close to eachother. In particular, their last column (in bold) are practically identical. It is only a questionof a coefficient. This leads to two important conclusions:

1. The curve calculated from the Schwarzschild Metric (fig. 5-4), is very closeto that plotted from experimentation using a simple EPP foam (fig. 5-3).

2. In these two tables, volume (more exactly radius) replaces mass

5.2 The Schwarzschild Radius Rs

Lets return to our experiment (fig. 5-1) and to its table andcurve (fig. 5-3). In formula 5-1, when the length of thecurvature of spacetime R is increased to 40 mm., a singularityappears (fig. 5-5).

We get the behaviour of the Schwarzschild Equation around Rs,the Schwarzschild Radius. The table and the curve (fig. 5-5)show an asymptote when r = 40 mm.. This value, 40 mm., isnothing but the Schwarzschild Radius Rs.

We can also remark that the signature is changed from + to -, asinside a black hole.

This experiment is very interesting because it proves, one moretime, that the curvature of spacetime is produced by the volume,not by the mass . Indeed, during all these explanations, onlylengths have been considered. Masses have been totally ignored.It is the radius, not the mass, which has been increased to 40mm. to calculate a black hole behaviour.

(5-3)

-30,00

-20,00

-10,00

0,00

10,00

20,00

30,00

1 3 5 7 9 1 1

1 3

1 5

1 7

1 9

2 1

2 3

2 5

Fig. 5-5

r = Rs

-

+

r y = f (r) 22

24,4 -1,5626,8 -2,0329,4 -2,7732,2 -4,1335,2 -7,3338,3 -22,5341,8 23,2245,9 7,7850,4 4,8555,1 3,6559,9 3,0164,6 2,63

69,3 2,3774,4 2,1679,3 2,0284,4 1,9089,5 1,8194,5 1,7399,5 1,67

104,2 1,62108,8 1,58113,6 1,54118,4 1,51

123,2 1,48128 1,45132,9 1,43

Table of figure 5-5


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5.3 Discussion concerning Rs

Theoretically, the radius Rs should exist. In reality, this is not certain....

From a mathematical point of view, it is possible to have r < R, R beeing the radius of the particle or object. From a physical point of view, this point of view isn't obvious. The debateis ongoing, but logically, r can't be less than R.

What happens if an electromagnetic wave comes at a distance r = R? The Von LaueDiagram (fig. 3-12) partially gives the solution to this enigma:

! Far from the volume that makes the displacement of spacetime (fig. 5-6 A), the lightis only deviated. It follows the geodesic of spacetime.

! Near the volume (fig. 5-6 B), the light is captured. In this case, the light turns aroundthe object and doesn't have the possibility of escaping.

! In front of the volume, the light comes in collision with the object (Compton Effect).In other words, the "famous" black hole capture could be nothing but an ordinaryCompton effect

At last, we must note the three following remarks (these remarks are only suggestions thatare not proven):! Since a particle (electron, proton) is a closed volume, its behaviour could be

identical to that of a black hole.! If the light comes in r = R (fig. 5-6 B), it is possible that a resonance takes place if

the circumference of the particle is a multiple of its wavelength.! In that case, it is possible that particles of groups 2 and 3 of the Standard Model

could be nothing but particles of group 1 in resonance. For example, the muon could be an electron in a "level 1 resonance". In the same manner, the tau could be anelectron in a "level 2 resonance". This resonance increases the volume (the mass) of the particles but keep their charge unchanged (-1 in this example). That is in perfectaccordance with the proposed theory. For this reason, this possibility must becarrefully studied.

A

B

Fig. 5-6


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5.4 The Minkowski Metric

The expression of the Minkowski Metric, in spherical coordinates, is:

Using this formula, which comes from special relativity, we will calculate theSchwarzschild Metric using results from the proposed theory.

The Schwarzschild Metric is nothing but a Minkowski Metric, expressed in a sphericalcoordinates, with two unknown functions: A(r) and B(r) 1:

To determine A(r) and B(r), the Einstein Vacuum Field Equations are employed andsimplified. The details of calculations are described in many books concerning GeneralRelativity 2. These two functions A(r) and B(r) have the form 3:

5.5 The Schwarzschild Metric

Lets come back to our preceding simulation of (fig. 5-1) that becomes figure 5-7.

1 The term r(d + sin d) doesnt need an unknown function as A(r) or B(r) since the Schwarzschild Metrichas a spherical symmetry2 For example, Notes on General Relativity - S. Carroll -.3 Some authors prefer writing A(r)B(r) = K with K=c. In that case, the term c must be excluded from theMinkowski Metric (5-4). However, in both cases, the result is the same. Note: in order to simplify equations,some Authors also replace c and G by 1. In this document, we don't follow this rule because a simple number (1 in this case) 1 doesn't have a dimensional quantity like c [L/T] or G [L 3/MT].

(5-4)

(5-5)

(5-6)

Fig. 5-7

Curvature


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As everyone knows, spacetime is curved near a mass (more exactly near a closed volume inthe proposed theory). On the contrary, spacetime is flat far from any mass.

This phenomenon is exactly what figure 5-7 shows, replacing spacetime by an EPP foam, and

the mass by a volume. The two lines in bold on figure 5-7 simulate with a great realism thecurvature of spacetime, out and in a gravity field.

What is the relation between C out, the curvature of spacetime out of gravity (far from anyclosed volume), and C in, the curvature of spacetime near a closed volume?

As we can see on figure 5-7, if the upper line moves toward the central volume, its curvatureincreases. The relation between two adjacent curvatures C out and C in is:

or

where:! Cout is the curvature of spacetime out of gravity (in a flat space)! is an elementary increase of spacetime curvature ( = Cout/Cout)! Cin is the curvature of spacetime toward the volume.

Therefore, an elementary radius dr out approaching the central volume will be subject to the

same increase of spacetime curvature:

Since


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This result is nothing but the coefficient of dr in formula (5-5), i.e. expression of A(r). Thecalculation of B(r) is immediate, taking into account that A(r)B(r) = 1 (formula 5-6). Hence:

So, equation (5-5) becomes:

The curvature of spacetime, , is given by formula 4-22 which is re-written as follows:

Finally, replacing by GM/rc in equation 5-15 leads to the well-known SchwarzschildMetric:

As we see, the proposed theory demonstrates that the Schwarzschild Metric can be easilyobtained

! With few logical deductions,! Conducting a simple experiment which costs less than $5.00,! And with the help of few elementary mathematical manipulations.

This demonstration also confirms Einsteins Point of view: The universe is very simple .

(5-13)

(5-16)

(5-15)

(5-17)

(5-14)


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The Spacetime Model - Part 1 - - 6 - The Newton Law 41

6 The Newton Law

This chapter calculates the first Newton Law from the conversion formula of chapter 4.

6.1 The Potential

Equation (4-20) indicates the relationship between mass and the curvature of spacetime:

In quantum mechanics, we have the following equivalence, with E having the dimensionalquantity of [ML/T] :

Formula (6-1) is constructed on the same principle:

The only difference is that U(r) has the dimensional quantity of [L/T] instead of E =[ML/T]. Finally, equation (6-3) may be written as:

This formula, which is a solution of the Poissons equation, is the "gravitational potential"since:

! U(r) is a continuous function,! If r , U(r) 0, i.e. its expression varies as "1/r",! It can be derivated,! And its dimensional quantity is that of a gravitational potential: [L/T].

(6-1)

(6-2)

(6-3)

(6-4)


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The Spacetime Model - Part 1 - - 6 - The Newton Law 42

6.2 The Newton Formula

Derivating formula (6-4) gives an gravitational field which is actually equal to thegravitational acceleration.

Since we are in a spherical symmetry, the nabla operator is reduced to:

The gravitational field then becomes :

Finally, applying the Newton Second Law (F = mg) to equation (6-7) gives the well-known Newton Formula:

with, as in formula (4-21):

For the proton, the values for and are: = 1,57045.10 28 m3 /kg.s= [L 3 /MT] =1,41664.10 -39 = [1]

Note:The product x is constant but each term isn't. these variables depend of many parameters,as proven by the table of paragraph 4-6 (proton-deuteron ratio). Parameters which enters inthe construction of and are mainly the volume, the surface and the radius (remember that,for two objects having different shapes, the volume isn't proportional to the surface).

(6-5)

(6-7)

(6-8)

(6-9)

(6-6)


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The Spacetime Model - Part 1 - - Complements I

Complements

Predictions

The gravity formulas (Schwarzschild Metric, Newton Laws ...) are not modified by the proposed theory and results of experimentations concerning this new theory and generalrelativity are identical. So, it is impossible, or at least very difficult, to conduct anexperimentation verifying the proposed theory.

Replacing spacetime by an EPP foam

Following the first publication of his theory in 2005, the author predicted that an EPPfoam could have the same behaviour than spacetime. Therefore, he devised the twoexperimentations described in chapters 3 and 5. These two experimentations prove that

when we replace spacetime by an EPP foam, we obtain the same results.

1. The first experimentation in chapter 3 proves that the curvature of the spacetime isnot produced by masses but by volumes,

2. and that the pressure force measured with the two FSR's is identical to gravity.

3. The curves of chapter 5 are identical to those of the Schwarzschild Metric,

4. and also prove that the displacement of spacetime is produced by a volume (thecap of the milk bottle), not by a mass.

5. Finally, a thought experimentation with r = r s, the Schwarzschild Radius, has beendescribed using an EPP foam. Results also prove that it is the volume, not the mass,which produces the spacetime curvature.

Particle behaviour near a surface

The proposed theory demonstrates that the spacetime density increases near closedvolumes. So, when a particle is moving near atoms or molecules, its masstheoretically must increase since the density of spacetime have an influence on themass effect.

The proposed experimentation could highlight a very light difference of the mass of a particle moving near and far a massive surface.


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The Spacetime Model - Part 1 - - Complements II

The moon influence

The curvature of spacetime produced by the moon is added or subtracted to that produced by Earth. Therefore, the measurement of the "mass effect" of a particledepends of the position of the moon during the experimentation. So, if twoexperimentations measuring the mass of a particle are conducted a) when the moon isat zenith and b) 12 hours after, we should note different data.

If this prediction is verified, all the particles must be re-caculated to eliminate theeffect of the spacetime curvature due to Earth. The only real mass of a particle must bethat calculated out of any gravitational field, i.e. far from any mass which could

perturb the measurement: Moon, Earth, Sun etc. 1

Partitioning the theory

The five parts of the Spacetime Model can be downloaded at the following URL addresses:Part 1 .......Mass and gravity.................www.spacetime-model.com\mass.pdf Part 2 .......Constitution of Matter ........www.spacetime-model.com\matter.pdf Part 3 .......Quarks and Antimatter .......www.spacetime-model.com\quarks.pdf Part 4 .......Electromagnetism...............www.spacetime-model.com\electromagnetism.pdf Part 5 .......Forces, the Universe ...........www.spacetime-model.com\forces.pdf

Note: Some informations included into parts 3 to 5 are nor proven and must be taken with reservation.

1 However, it is not impossible that the curvature of spacetime could produce an infinitesimal perturbation of EMfield.

Before understanding the constitution of matter, the author had to solve three enigmas:

1. How to explain the wave-particle duality from a scientific point of view.2. Why electromagnetic waves have a constant speed of 300 000 km/s.3. How an e+e- pair can be transformed into two gammas of 511 KeV, i.e. howmatter is transformed into waves and the converse.

For example, a drop of water can be transformed into awave. The same phenomenon exists in physics: Wave-

particle duality. This enigma is solved in only one case,if the particle and the wave have same constitution, likewater/water in our example.

The solving of this enigma conducts to the knowledge of the constitution of matter and EM waves. This newtheory is confirmed by experimentations.

Part 2 - Constitution of Matter


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QuarksThis part demonstrates that we need two positrons to make three u quarks. A uquark with an electron becomes a d quark (please note that the rule of addition of fermions is covered in Part 4). This deduction, from the wave-particle duality andspacetime, has been extended to all particles. Finally, u quarks, d quarks,antiquarks, muons, antimuons, taus, mesons, baryons etc... can be made with onlytwo basic particles: electrons and positrons.

AntimatterFrom this discovery, we can deduce that antimatter is not located at the bottom of the universe but right before our eyes, embedded in u and d quarks.

A simple calculation demonstrates that any atom is made up of an equal number of electrons and positrons, exactly 2A, with A = atomic number. For example, the C12is made of 24 electrons and 24 positrons, the latter being embedded in quarks.

The calculation is fully explained in this Part and is 100% accurate for all 2930

known isotopes .

Part 3 Quarks and Antimatter

The mystery of the wave-particle duality solved in Part 2 leads to a full knowledgeof electromagnetism. This phenomenon is quite simple to understand.

In short, when a charged particle is motionless, its electric field has a sphericalsymmetry. When it moves, it becomes a wave and its spherical symmetrydisappears. Its 1D space is transformed into a 2D/3D space. A magnetic component(2D/3D) is added to the electric field (1D) of the particle.

This phenomenon is exactly what experimentation proves ( q/ t).

Part 4 - Electromagnetism


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ContactYou can contact the author 1 by email at:

[email protected]

or writing to:

M. Jacky JEROMERsidence Le Kennedy4 square Kennedy42120 LE COTEAU(France)

1 The author is a physics hobbyist and does not work in an institutional establishment. The writing of theSpacetime Model has been done entirely on his own money and time, with no help from the scientificcommunity. If you find some error in this document, please let him know.

Nuclear forceElectrons or positrons, which surround other particles as a spacetime wave, produce arecall force toward the center of the particle, like a rubber band. This force is nothing butthe "strong nuclear force".

Unification of forcesThis part unifies the three basic forces (gravity, electroweak and strong nuclear force)in two generic forces: the Coulomb Force and the Hooke Force.

The UniverseA suggestion regarding the creation of the universe is proposed. In reality, the Big-Bang Theory does not explain the electron mystery" discussed in Part 5.

Part 5 offers two suggestions, much more credible than the Big-Bang, regarding thecreation of the universe.

Part 5 - Forces, the Universe


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Table of content

Introduction .................................................................................................. I IV

1. Open and Closed Volumes1.1 Starting points............................................................................................11.2 Basic concept.............................................................................................21.3 Current theory of spacetime curvature ......................................................31.4 Explanation of spacetime deformation......................................................31.5 Mass-volume equivalence.........................................................................41.6 Einstein Field Equations............................................................................51.7 Classes of volumes ....................................................................................61.8 Example: Atoms ........................................................................................71.9 Conclusions ...............................................................................................7

2. Mass and Gravity2.1 Principle of gravity....................................................................................92.2 Principle of split ........................................................................................102.3 Stress-energy tensor...................................................................................102.4 Synthesis....................................................................................................122.5 Wheeler's Intuition.....................................................................................122.6 Example of mass and gravity.....................................................................142.7 Equivalence principle ................................................................................142.8 Relativistic particles ..................................................................................162.9 Crystals ......................................................................................................172.10 Example of mass-volume equivalence ......................................................182.11 The mass excess ........................................................................................19

2.12 Conclusions ...............................................................................................20

3. Validation by Experimentation3.1 Introduction ...............................................................................................213.2 Basic material ............................................................................................213.3 The FSR.....................................................................................................223.4 Experiment #1 ...........................................................................................223.5 Experiment #2 ...........................................................................................233.6 Conclusions ...............................................................................................253.7 The light deflection....................................................................................253.8 The Von Laue Diagram .............................................................................25


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4. Mass-Volume Conversion

4.1 Abbreviations used in formulas.................................................................274.2 Conventional mass-volume conversion.....................................................27

4.3 Starting point .............................................................................................274.4 The "mass effect".......................................................................................284.5 Mass-volume conversion...........................................................................304.6 Confirmation by experimentation..............................................................314.7 Calculation of the curvature of spacetime .................................................31

5. The Schwarzschild Metric5.1 Experimentation ........................................................................................335.2 The Schwarzschild Radius Rs ................................................................... 365.3 Discussion concerning Rs..........................................................................375.4 The Minkowski Metric ..............................................................................385.5 The Schwarzschild Metric .........................................................................38

6. The Newton Law6.1 The potential..............................................................................................416.2 The Newton Formula.................................................................................42

Complements .............................................................................................. I - IV

part 1 - mass and gravity

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