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New viewpoint of General Relativity that explains Massand Gravity and some Enigmas of Modern Physics

Jacky Jerome, Frederic Jerome

Sciences-Tech, France. Email: [email protected]

Using different viewpoints on mass, volume, and spacetime, we may associate well-known theories (LeSage, Higgs, and General Relativity) to solve some enigmas of modern science such as: What is themechanism behind the force of gravitation? How does mass curve Spacetime? What is the mechanismof E =mc2?. . . This paper does not intend to cast doubt on or undermine Higgs or General Relativity,but it attempts to demonstrate that it goes further by taking different viewpoints. Several mathematicaldemonstrations support this paper such as a rewriting of the Newton’s Law, Energy-Momentum Tensor,or Equivalence Principle . . . This paper also gives a very simple method to quickly obtain the Schwarzschildmetric without using the Einstein Field Equations, in just two pages instead of 20 currently. In conclusion,this article opens the way to many applications of modern physics. Several enigmas of Astrophysics will bealso more readily explained with this viewpoint.

1. Introduction

When Einstein developed General Relativity (GR) in the 1910s [1-11], knowledge in physicswas significantly less than today. The atom was analogized as a plum pudding in which electronswere represented by raisins [12,13]. The proton and neutron were discovered later, in 1919 and1932. So, GR was developed on an incomplete view of reality. The Einstein Field Equations workperfectly, but the mechanism of curvature of spacetime remains somewhat obscure [14].

The solution to this mystery lies in the mass/volume ratio of each component of the atom. Sincethe density of the plum pudding model of J.J. Thomson is uniform, in 1915 we had one, and onlyone, definition of “volume”. This is not the case today because we have two different kinds ofvolumes in atoms: some volumes have mass and curve spacetime (nucleus, electrons), while othersdo not (volumes of orbitals). This construction was ignored when GR came up.

It means that we must reconsider GR to be in line with reality. Here we propose to replace theplum pudding model (one volume) by the current atomic model (two different volumes). This leadsto a new viewpoint of GR which fully explains:

– What is the mechanism of curvature of spacetime?– What is mass? (in accordance with the Higgs Theory)– What is the mechanism of the force of gravitation?– Why does the mass of relativistic particles increase? etc . . .

Note: Replacing mass by a “special kind of volume” is a new concept that upsets preconceived ideas. Like anynovelty, this may lead to a misunderstanding of this article. Another problem is that the concept here developed is verysimple (scholar level) for a scientific paper that covers GR. So, it is possible that this excessive simplicity appearsdisconcerting on first reading, but Physicists who prefer a mathematical version can access Appendix E.

2. The Curvature of Spacetime

To date, there is no explanation of the mechanism of how mass can curve spacetime (Fig. 1a,next page). However, if we replace mass by volume (Fig. 1b), all becomes clear, but this assumptionis problematic because it is not confirmed by experiment.

;

Figure 1a - Current representation of the curvature of spacetime, such as that made by a black hole.Figure 1b - Curvature of spacetime made by a volume (after Von Laue, 1927, see Appendix F).

This raises the question:

Is spacetime curved by Mass?This explanation is validated by experiment but does not answer why

...or by Volume?Appears more logical than Mass (see explanations below)

In this paper we demonstrate that spacetime is not curved by mass as is thought, nor by volume,but by a “Special Kind of Volume”, a kind of volume which fully explains mass and gravity.

3. What is this “Special Kind of Volume”?

One tends to assume that one, and only one, definition of Volume exists, saying that a volume isthe 3D space of a body such as a ball, or a box, or a house etc. This definition must be specified inGR because all volumes are not equal regarding mass and the curvature of spacetime. Indeed, somevolumes have mass and curve spacetime (e.g. particles), while others do not (e.g. orbitals). Sincemass and spacetime are at the heart of GR, we must consider these volumes separately.

In reality, we do not have only one type of volume, as the plum pudding model suggests, butfive, described in Appendix K. The three major classes of volumes are:

3.1 Volumes with Mass, or “Matter Volumes”Our definition, “Matter Volumes”, is nothing but the contraction of “Volume of Nuclear

Matter”. These volumes, such as nuclei or electrons, have mass (Fig. 2a). Since the 1930s, weknow that Mass (M) and “Matter Volumes” (V) are connected together by the expression:

M = kV = f(x,y,z) (k= 2.3× 1017kg/m3) (1)

A more accurate formula is proposed in Appendix D:

M =c2

G0εvV

Sρs = f(V ) (2)

. . . where εv is elasticity of spacetime, and ρs is the density of spacetime relative to a flat spacetime.

It is important to see that, whichever of Eqs. (1) or (2) used, Mass is directly connected to“Matter Volume”. We always have:

M = f(MatterV olume) (3)

Therefore, we can replace Mass by “Matter Volume” in all cases where “Volume” means“Volume of Nuclear Matter” (or “Matter Volumes”). It is a simple mathematical operation.

3.2 Volumes without Mass, or “Empty Volumes”These volumes are nothing but an empty 3D space such as a vacuum or gap between atoms (Fig.

2b). They exist but are transparent regarding spacetime. So, “Empty Volumes” are massless and donot curve spacetime, even therein virtual particles would be present (in this case, if a curvature isdetected, it is produced by the virtual particles, not by empty volumes). “Empty Volumes” must beignored in GR calculus. Example: the volume of orbitals of atoms exists, but this volume does nothave mass and does not curve spacetime.

Figure 2a - “Matter Volumes”, such as nuclei or particles, have mass and curve spacetime. Figure 2b - “EmptyVolumes”, such as orbitals of atoms (excluding the nucleus and electrons), are massless and do not curve spacetime.

3.3 Combinations of Volumes, or “Apparent Volumes”Objects we use daily are made of molecules, which are combinations of “Matter Volumes”

(nuclei, electrons) with mass, and “Empty Volumes” (orbitals, gap between atoms), without mass.In daily life these combinations of volumes are simply called “Volumes”. To avoid confusion with“Matter Volumes” and “Empty Volumes”, here we call these volumes “Apparent Volumes”.

We have the feeling that mass and volume are two different quantities because in “ApparentVolumes”, the proportion of Matter Volumes/Empty Volumes varies from one atom to another,from one molecule to another, from one object to another.

Apparent Volumes =∑

Matter Volumes +∑

Empty Volumes

4. Replacing Mass by “Matter Volume”

We have shown in Section 3.1 that we can substitute mass by “Matter Volume”. So:

Mathematics of GR: We can continue to use Mass in GR calculus. We can also replace Massby “Matter Volume” without problem since these two quantities are connected together by Eqs.(1–3). The two approaches give identical results.

Explanation of phenomena: On the contrary, to understand basic phenomena such as “Whatis the mechanism of the curvature of spacetime”, we must replace Mass by “Matter Volume”. Thissubstitution also explains several enigmas of modern physics (see below and Appendix L).

5. Validation by experiment

The following flow chart (Fig. 3) demonstrates that the substitution of Mass by “Matter Volume”is not a new theory per se, but simply the adaptation of GR to our current knowledge of atoms. Thisadaptation does not need any extra validation by experiment since all these observations have beenwell known since the 1930s.

Figure 3 - This flow chart is grounded on experiment validated since the 1930s. It shows that we are facing with twokinds of volumes: “Matter Volumes”, with mass, which curve spacetime, and massless “Empty Volumes”, which do not.

This flow chart clearly demonstrates that only "Matter Volumes" curve spacetime. Therefore,we could continue by saying “Spacetime is curved by mass”, but mass is not at the origin of thecurvature of spacetime. It is a consequence (see below), the origin being "Matter Volumes". So, itis more accurate to say:

Spacetime is curved by “Matter Volumes”

6. Example of the Curvature of Spacetime

If we plunge a marble into a glass filled with water, it is the volume of the marble, not its mass,that displaces the water. Since water and spacetime obey to the same laws of Fluid Mechanics onwhich is grounded GR, this example can be transposed to spacetime as follows:

An electron of volume “V” crosses a cube of volume “1,000 V” (Fig. 4, next page). Bothvolumes have their own spacetime. The only difference is that the volume of the particle is a“Matter Volume” and has mass, whereas that of the cube is a massless “Empty Volume” sinceit is a vacuum. The internal spacetime of the particle will push the surrounding spacetime to makeroom. Since spacetime has elasticity properties (Einstein), the cube will contain 1,001 elementsof volumes inside a global volume of 1,000 V. The result is that spacetime will be curved. Thisexample shows that it is the volume of the particle, not its mass, which curves spacetime.

7. Mechanism of Creation of Mass

The aforementioned deductions fully explain the creation of mass:

– An object is placed into an empty space (Fig. 4a, next page);– As explained before, the “Matter Volume” of the object (not its mass) curves spacetime;– Einstein demonstrated in GR that spacetime has elasticity properties (Fig. 4b);– Due to this elasticity, a pressure force appears on the surface of the object (Hooke Law);– This pressure increases the resistance to movement of the object (Higgs Mechanism),– In others words, the object acquires mass.

Figure 4 - (a) An electron of volume V crosses a cube of 1,000 V.. Since the electron has mass, it is a “MatterVolume”. The cube is a vacuum and is a massless “Empty Volume”. Due to its elasticity properties, spacetime will becurved. (b) The curvature of spacetime exerts a pressure force on the surface of the electron (here zoomed).

8. The Higgs Mechanism

This scheme is that of the Higgs Mechanism, but reservations must be made:– The Higgs Theory does not explain the mechanism of curvature of spacetime (this paper does).– It is not proven that the Higgs Field permeates all space since the Big Bang epoch because,

despite the 2.8oK of the deep universe, the Big-Bang remains an assumption, not a certainty.– The Higgs Field has no potential from which it derives. This contradicts the Poisson Law.Substituting Mass by “Matter Volume” solves these three points saying:

The Higgs field does not permeate all space but is a Newtonian field in spacetimecreated by the “Matter Volume” of the particle, which acts as a potential.

9. What is the Mechanism behind the Force of Gravitation?

Gravity is usually represented by Fig. 5a, but this scheme fails in some particular situations suchas if two identical spheres (M1 =M2) are away one to the other (Fig. 5b).

Figure 5 - Current representation of gravity (a). However, despite its popularity, this scheme does not work in someparticular situations (b). This means that our current representation of the force of gravitation must be revised.

We usually consider that gravity is ascribed to the curvature of spacetime. This is true, but it doesnot explain the nature of the force of gravitation. In reality, confusion exists between the curvatureof spacetime, which is a vector (a kind of media between objects), and the force of gravitation. Thecurvature of spacetime IS NOT a force. The dimensional quantity of geodesics of spacetime are[L3], while that of a force is the product Mass x Length x 1/Time2, i.e. [ML/T 2]. Therefore, wecannot identify the force of gravitation [ML/T 2] to the curvature of spacetime [L3].

Since to date, no one can rationally explain the mechanism behind the force of gravitation, weare faced with the following alternative (Fig. 6):

1/ Is gravity an attractive force? (probability = 0.5)2/ . . . or a pressure force? (probability = 0.5)

In both cases, the results are identical: two bodies move one toward the other.

Figure 6 - An attractive force (a) gives the same results as a pressure force exerted by the curvature of spacetime(b). In both cases, each body moves one toward the other. Since these two schemes have an identical probability, 0.5, wemust not exclude the possibility that gravity could be a pressure force instead of an attractive force.

In 1748, a Swiss physicist, Le Sage, imagined “Push Gravity” [15-20]. He thought that thepressure was exerted by “Ultramundane Particles”, i.e. particles that come from another world. In1904, J.J. Thomson thought that the Le Sage Pressure came from a kind of X-ray. In the 1930s,Louis de Broglie thought that the pressure was produced by neutrinos.

Here, we demonstrate that the Le Sage principle is a good foundation, but the pressure does notcome from ultramundane particles, X-rays, or neutrinos, but simply from the curvature of spacetimecreated by “Matter Volumes”. The mechanism of pressure is fully described in the Bulk Modulusand Hooke’s Law sections of classical physics (Appendix B) and does not need validation.

Therefore, gravity may be explained by the association of three different theories:1. Le Sage “Push Gravity”;2. GR, in which the Curvature of Spacetime replaces “Ultramundane Particles”;3. The present paper showing that spacetime is curved by “Matter Volume”, not by Mass.

Figure 7 - Mass and Gravity are two similar phenomena. In both cases, the vector of mass or gravity remains thecurvature of spacetime produced by the Matter Volumes of objects. Mass: Elasticity of spacetime exerts a pressure onthe surface of the “Matter Volume” of the object. This pressure increases its resistance to movement (Higgs Mechanism).Gravity: Elasticity of spacetime exerts an external pressure force (not an attractive force) on objects that tends to bringthem closer to each other.

So:

Gravity is not an attractive force between objects,but a pressure force made by the curvature of space-

time, which tends to bring objects closer to each other.

The curvature of spacetime is producedby the “Matter Volumes” of the objects.

10. Conclusions

This paper is based on a very simple idea: All volumes are not equals regarding the curvatureof spacetime and mass. It shows that we should classify volumes in at least three basic categories:1/ "Matter Volumes", 2/ "Empty Volumes", and 3/ "Apparent Volumes". Finally, the curvature ofspacetime, mass, and gravity are explained by the association of four theories (Fig. 8):

– Le Sage Theory, which shows that gravity could be a pressure force;– Einstein GR, which demonstrates that spacetime can be curved;– Higgs Mechanism, which explains the basic principle of mass creation;– This paper (Author’s contribution), which shows that spacetime is curved by Matter Volumes.

This paper is important because it touches the weakest nerve of GR by asking “how mass curvesspacetime”? It also complements the Higgs Mechanism and Le Sage Theory to fully explain massand gravity, and indicates a solution to several enigmas of modern physics such as: (Appendix L).

– How can we explain E = mc2?– Why does the mass of relativistic particles increase? (see also Appendix G)– How can we explain the dilatation of time in GR?– What is the mystery of filaments of galaxies?– How can we explain the headlong rush of galaxies (dark matter, dark energy)? etc . . .

Figure 8 - Each object has a Matter Volume [1], which curves spacetime [2]. Spacetime exerts a pressure on the MatterVolume of the object [3]. This pressure increase the resistance to movement of the object, which acquires mass [4].

References[1] A. Einstein (1905), "Zur Elektrodynamik bewegter Korper", Annalen der Physik, 17: p. 891.[2] A. Einstein (1907), "Uber das Relativitatsprinzip", Jahrbuch der Radioaktivitat, 4: p. 411.[3] A. Einstein (1907), "On the Inertia of Energy Required by SR", Annalen der Physik, 23: p. 371-384.[4] A. Einstein (1915), "Die Feldgleichungen der Gravitation", SPA der Wissenschaften, p. 844-847.[5] A. Einstein (1916), "Der allgemeinen Relativitatstheorie", Annalen der Physik, 49.[6] A. Einstein (1916), "Die Grundlage der allgemeinen Relativitatstheorie", Annalen der Physik, 354(7): p. 769-822.[7] A. Einstein (1917), "Kosmologische und Relativitatstheorie", SPA der Wissenschaften, p. 142.[8] A. Einstein (1920), "Relativity, The Special and the General Theory", Routledge, ISBN 0-415-25384-5.[9] A. Einstein, M. Grossmann (1913), "Outline of a Generalized Theory of Relativity and of a Theory of Gravitation",

Zeitschrift fur Mathematik und Physik, 62: p. 225-261.[10] A. Einstein (1914), "Covariance Properties of the Field Equations of the Theory of Gravitation Based on the

Generalized Theory of Relativity", Zeitschrift fur Mathematik und Physik, 63: p. 215-225.[11] A. Einstein, H. A. Lorentz, H. Minkovski, H. Weyl (1923, 1952), The Principle of Relativity: a collection of original

memoirs on SR and GR, Courier Dover Publications, ISBN 0-486-60081-5.[12] G. J. Stoney (1894), "Of the Electron or Atom of Electricity", Philosophical Magazine, 5 38 (233): p. 418-420.[13] J. J. Thomson (1904), "On the Structure of the Atom", Philosophical Magazine, 6 7 (39): p. 237.[14] H. Friedrich (2005), "Is general relativity understood?", Annalen der Physik, 15 (1-2): p. 84-108.[15] N. Fatio de Duillier (1690), "Oeuvres completes de Christian Huygens", Societe Hollandaise des Sciences, 2570: p.

381-389.[16] N. Fatio de Duillier (1701), "Die wiederaufgefundene Abhandlung von Fatio de Duillier: De la cause de la

Pesanteur", Mathematics Genealogy Project.[17] N. Fatio de Duillier (1743, 1949), "De la Cause de la Pesanteur", Royal Society of London, 6 (2): p. 125-160.[18] G. L. Lesage (1748), "Lucrece Newtonien", Memoires de l’Academie Royale des Sciences et Belles Lettres de

Berlin, p. 404-432.[19] G. L. Lesage (1748), "The Le Sage Theory of Gravitation", Annual Report of the Board of Regents of the

Smithsonian Institution, p. 139-160.[20] P. Prevost (1818), "Physique Mecanique des Georges-Louis Le Sage, Deux Traites de Physique Mecanique", Editions

J.J. Paschoud, p. 1-186.

Appendices

Summary

A – Complements on GravityB – New version of Newton’s LawC – New version of the Schwarzschild MetricD – Expression of the mass effect in 4DE – Partial Rewriting of the Einstein Field Equations (EFE)F – Von Laue GeodesicsG – Explanation of the Increase of the Mass of Relativistic ParticlesH – Equivalence Principle using the Curvature of SpacetimeJ – Bethe-Weizsacker Semi-Empirical Mass Formula (SEMF)K – Other Classes of VolumesL – Applications and Simulations (with reservations)

Bibliography

Appendix A

Mass and Gravity

A-1 Building the EFEsThe heart of General Relativity (GR) is the “Einstein Field Equations” (EFEs). The EFEs are not

classical equations such as y= f(x), but an equivalence between two sets of tensors. The left sidemember defines the curvature of spacetime whereas the right side member represents the descriptionof mass-energy. Due to this equivalence of tensors, we have two possibilities:

1/ “Mass is at the origin of the curvature of spacetime”2/ “Mass is a consequence of the curvature of spacetime”

To answer this question, we must understand the whole phenomenon of the mechanism of masscreation (see Fig. A-1).

(A) Einstein grounded GR on the Fluids Mechanics of the 1850s. He assimilated spacetimeto a fluid, which exerts pressures on the surface of an object.

(B) Fluids Mechanics is based upon the Riemann curvature tensor. The Riemann Tensorbecame then the Einstein Tensor in the EFEs as follows: Riemann tensor⇒ Christoffel symbols⇒Geodesic Equation⇒ Riemann-Christoffel Tensor⇒ Ricci Tensor⇒ Einstein Tensor.

(C): The result was “Mass curves spacetime”. This assertion is not logical because GR isgrounded upon Fluid Mechanics, in which “Volume (not mass) curves spacetime”.

Figure A-1: Mechanism of the construction of EFEs

To be logical and consistent, the result of the EFEs must be “Volume curves spacetime”, not“Mass curves spacetime”.

A-2 ExampleIf we drop a marble into a glass filled with water, it is the volume of the marble, not its mass, that

displaces water. This example can be transposed to spacetime since water and spacetime obey tothe same laws of the Fluid Mechanics used by Riemann and Einstein. Therefore, figure A-1 shouldbe replaced by figure A-2.

Appendix A

Figure A-2: Since the EFEs are built from the Fluids Mechanics, spacetime must be curved by volumes, not by mass.

However, various experiments show that it is the mass of the object, not its volume, that curvesspacetime. The solution to this enigma lies in what we call “Volumes” (see below).

A-3 Definition of “Volume”In current life, we consider that there is one, and only one, kind of volumes. These volumes are

called “Apparent Volumes” below.

In reality, we have 5 different definitions of volumes. The three main kinds are:

1/ “Matter volumes”, or volumes of nuclear matter, such as electrons, protons . . .2/ “Empty volumes”, such as a vacuum, orbitals of atoms, gap between atoms . . .3/ “Apparent volumes”, which are a combination of matter and empty volumes.

In GR, we can ignore “Empty volume” since they are empty. We must take into considerationonly “Matter Volumes” (or the “Matter Volumes” part of “Apparent Volumes”).

A-4 “Matter Volumes”“Matter Volumes” is a contraction of “Volume of Nuclear Matter”. The relation between Mass

and “Matter Volumes” is well known since the 1930s:

M = k.V (k= 2.3.1017kg/m3)

Therefore, in the EFEs, we can replace Mass by “Matter Volume”. It is a simple mathematicaloperation. This does not change anything in calculus, but this substitution solves the enigma of theorigin of the curvature of spacetime. This substitution demonstrates that the above Fig. A-2 is morelogical than Fig. A-1.

Appendix A

A-5 What is the origin of Mass?Substituting Mass by “Matter Volume” solves the enigma of mass creation. The process is:

1/ The "Matter volume" of an object curves spacetime (like in Fluid Mechanics);2/ . . . Spacetime exerts a pressure on the surface of the object (Fig. A-2);3/ . . . this pressure increases the resistance to movement to the object;4/ . . . In other words, the object acquires mass (Higgs Mechanism).

A-6 ValidationThese assertions are confirmed by two ways:

1/ Mathematics: The main article is supported by several mathematical demonstrations(appendices B, C, D, E, G, H, J);

2/ Experiment: The mass creation is identical to the Higgs Mechanism (however with somereservations explained in Section 8 of the main article), which has been validated on the 4th July2012 at CERN. This discovery was followed by a Nobel Prize to Peter Higgs. However, the HiggsTheory does not answer the question: “From where does the Higgs Field come from?”. The mainpaper solves this enigma. Therefore, this paper may be considered as a complement to the HiggsTheory about the mass creation.

A-7 Conclusions about MassThis paper demonstrates that the assertion “Mass curves spacetime” must be replaced by

“(Matter) Volume curves spacetime”. This assertion is confirmed by mathematics and experiment.

To summarize, the solution to mass is the combination of three theories:

1/ Einstein’s GR, which shows that spacetime can be curved;2/ Higgs Mechanism, that explains the mass creation (with reservations);3/ Replacing Mass by Matter Volume (Author’s contribution).

A-8 Introduction to GravityIn the current life, we see that two objects are attracted one toward the other. So, our immediate

deduction is that gravity is an attractive force.The fact that 99,99% of people think that gravity is an attractive force is not a proof per se. In

reality, since no one knows the origin of the gravity force, we are faced with two possibilities:1/ Gravity is an attractive force (probability = 0.5);2/ Gravity is a force of pressure (probability = 0.5).For example, a sheet of paper (Fig. A-3, next page) can be curved either by attracting it (A), or

by pushing it (B). In both cases, the result is the same. Gravity follows the same principle.

A-9 The Einstein Field EquationsA thorough examination of the Einstein Field Equations, or “EFE”, gives the solution to the

gravity enigma. Gravity would be a pressure force, not an attractive force.

On one hand, we know that a fluid exerts a pressure on a “Matter Volume”. As explained below,extending this pressure to several volumes gives the mechanism of gravity.

Appendix A

Figure A-3: An attractive force (A) produces the same effect as a pressure force (B).

On the other hand, the trace of the energy-momentum tensor Tuv (Appendix E) represents theisostatic pressure force on the surface of a “Matter Volume”. By “Isostatic Pressure” we mean apressure force, not an attractive force. An interesting proof is that the energy-momentum tensor Tuv

is also used in fluid mechanics. So, we return to our original definition of Isostatic Pressure, where“Pressure”, means “Pressure”, nothing else (a pressure force is the opposite of an attractive force).

A-10 What is “Push Gravity”It seems that the first suggestion of gravity as a pressure force comes from a friend of Newton,

Nicolas Fatio de Duillier [29-31]. In 1690, he wrote a letter to Huygen in which he suggested thatgravity could be a pressure force. His theory is dated 1690-1743.

In 1748, Georges-Louis Le Sage completed the De Duillier’s theory [32-34]. His “PushGravity”, also known as “Shadow Gravity”, describes gravity as a pressure force. In the Le Sage’stheory, space could be filled with tiny particles called “Ultramundane Corpuscles”.

During the last century, several physicists have tried unsuccessfully to explain gravity. As weknow to date, the vector of gravity is spacetime, but its mechanism remains an enigma.

In 1904, J. J. Thomson considered a Le Sage-type model in which the ultramundane fluxconsisted of radiations such as X-rays. In the 1930s, De Broglie thought that Le Sage Corpuscleswere neutrinos. However, whatever the nature of the flux, ultramundane particles, X-rays, orneutrinos, “Push Gravity” must be reconsidered today in a spacetime context.

Here we show that gravity is a Le Sage “Push Gravity”, not a Newton Attractive Force. However,two major differences exist between the Le Sage “Push Gravity” and our explanation:

1/ The external pressure force comes from the curvature of spacetime;2/ Mass must be replaced by Matter Volume, which is identical since m= f(MatterV olume).

To summarize, Push Gravity seems much more logical than the well-known Newton-typegravity. The pressure force does not come from Le Sage hypothetical Ultramundane Corpusclesbut from the curvature of spacetime produced by Matter Volumes.

A-11 Proposed TheoryThe mechanism of gravity becomes clear and easy to understand if we think “Matter Volume”

instead of “Mass” because a mass cannot curve spacetime. A volume does.We have demonstrated in the main article that spacetime exerts a pressure on any matter volume.

Gravity is nothing but an extension of this ascertainment (Fig. A-4).

Appendix A

Gravity would not be an attractive force between masses,but a pressure force exerted by the curvature of spacetime on

“Matter Volumes”, which tends to bring them closer to each other.

Figure A-4: Mass and gravity are two similar phenomena

To summarize:

Convex vs. Concave:Appendix E shows that the curvature of spacetime is convex, not concave (see the black hole

figure).

Pressure Force vs. Attractive Force:The enregy-momentum tensor also shows that the convex curvature of spacetime exerts a

pressure force on objects, not an attractive force.

Results:As shown in the main article and Fig. A-1, the results are identical in both cases. Therefore, we

can combine these assertions into the following expression concerning spacetime:

Concave curvature + Attractive force ≡ Convex curvature + Pressure force

If we assign by convention a positive sign to a convex curvature of spacetime and to a pressureforce, the above assertion gives (−−) = (++). As we see, this does not change the EFE.

This means that gravity is a combination of three elements:- The Le Sage’s Push Gravity- The Einstein’s Curvature of Spacetime- Our explanation of “Matter Volumes” and EFE inconsistencies (Appendix E).

This combination gives a rational and consistent explanation of gravity.

Moreover, this new explanation of gravity is fully demonstrated with mathematics in AppendicesB, C and E in which we replace “Mass” by “Matter Volume”.

A-12 Curvature of Spacetime vs. PressureFig. A-5a shows the curvature of spacetime produced by the “Matter Volumes” of two spheres.

On the left sphere, spacetime has two different magnitudes of curvature:

Appendix A

– Point L (Left side): The curvatures of spacetime of the two spheres are added.– Point R (Right side): The curvature of spacetime of the right sphere is subtracted fromthat of the left one because the two curvatures are in opposition.

The difference of the curvature of spacetime on each side of the two spheres leads to a differenceof pressure (black and grey arrows on Fig. A-5b). In reality, Fig. A-5a and Fig. A-5b are twodifferent views of the same phenomenon: curvature of spacetime vs. pressure.

The same principle also applies if the two spheres are the Earth and the Moon.

At the Lagrangian point (oval on Fig. A-5a), the curvature of spacetime produced by the Earthis canceled by that produced by the Moon. The curvature of spacetime is null at this point and, as aconsequence, gravity disappears.

Figure A-5: Curvature of spacetime vs. pressure on the surface of each sphere (gravity).

A-13 Conclusions about GravityThis appendix shows that the combination “Le Sage Push Gravity + Einstein Curvature of

Spacetime + Our theory about volumes and EFE inconsistencies” solves the enigma of gravity.Therefore, the original “Push Gravity” must be modified as follows:

1/ “Ultramundane particles” must be replaced by the curvature of spacetime.2/ Spacetime is curved by “Matter volumes”, not by mass;3/ The curvature of spacetime is convex, not concave, as imagined by Von Laue (Appendix F);4/ Spacetime exerts a pressure force on objects, which is “Gravity”.

Appendix B

New Version of theNewton’s Law

B-1 IntroductionThe Newton’s Law is obtained from the Einstein Field Equations (EFE). It is a particular solution

for a spherical static symmetry object using the weak field approximation. Here we show thatNewton’s Law can be easily explained and obtained replacing mass by “Matter Volume”. Thismethod is simpler than calculating the Newton’s Law from the EFE.

This new version of Newton’s Law is very important because it demonstrates that gravity is apressure force exerted by the curvature of spacetime (see Appendix A).

B-2 Bulk ModulusThe bulk modulus KB of a substance measures the substance’s resistance to uniform

compression (Fig. B-1). It is defined as the pressure increase needed to cause a given relativedecrease in volume, Eq. (1).

Figure B-1: Bulk modulus

KB =−V ∆P

∆V(1)

Starting with the Navier-Stokes Equations of the Fluid Mechanics, Einstein, helped byGrossmann, demonstrated in the 1910s that spacetime:

• Can be identified to a Newtonian fluid,• Returns to its rest shape after having removal of the applied stress (elasticity).

Appendix B

Therefore, the Bulk Modulus, Eq. (1), which is a version of the Cauchy Tensor of the FluidMechanics, can also be applied to spacetime. It means that the curvature of spacetime made by a“Matter Volume” exerts a pressure on the surface of the latter (fig. B-1).

B-3 Elasticity LawElasticity phenomena follow the well-known logarithmic law:

ε= ln

(R−∆R

R

)(2)

where ε = coefficient of elasticity (in fact “true strain”).

The Schwarzschild Metric gives an order of magnitude of the curvature (compression) ofspacetime, which is infinitesimal. For example, the ratio curvature of spacetime/radius of theSchwarzschild Metric, or ∆R/R=GM/Rc2, is 1.4166× 10−39 for the proton, with M = 1.672×10−27 kg, R = 8.768× 10−16 m, G = 6.674× 10−11, c2 = 8.987× 1016 m/s.

Under these conditions, whatever the formula used, logarithmic or not, the curvature ofspacetime can be considered as a linear function since we are working on an infinitesimal segmentnear to the point zero. So, Eq. (2) becomes in first order approximation:

εR ≈∆R

R(3)

or, with volumes:

εV ≈∆V

V(4)

For the moment, the linear εR and volumetric εv coefficients of elasticity (in fact “strain”) ofspacetime are unknown.

B-4 Curvature of SpacetimeA matter volume V inserted into spacetime pushes it to make room (Fig. B-2). So the following

volumes are identical:

V = V1 = V2...= Vn (5)

Figure B-2: Volumes V, V1, V2, V3... are identical

Appendix B

Since the curvature of spacetime is a linear function, the coefficient of elasticity of spacetime εvcan be considered constant. So, combining Eqs. (4) and (5) gives:

∆V = ∆V1 = ∆V2...= ∆Vn (6)

However, there should not be any confusion between a simple displacement of spacetime, Vx,produced by the insertion of a matter volume into a flat spacetime, and the curvature ∆Vx = εvVxdue to the elasticity of spacetime (fig. B-3).

Figure B-3: Simple displacement (Vx) vs. curvature of spacetime (∆Vx)

B-5 Solving ∆x= f(∆R)

Since the ∆V’s are infinitesimal, the volume ∆Vx is simply the product of ∆x by the surfaceSx (fig. B-4):

∆Vx = Sx ∆x = 4πd2∆x (7)

The volume ∆VR is also the product of ∆R by the surface SR:

∆VR = SR ∆R = 4πR2∆R (8)

Figure B-4: Displacement and curvature at distances R and d

Appendix B

From Eq. (6) we have:

∆VR = ∆Vx (9)

Combining Eqs. (7), (8) and (9) gives:

4πR2∆R= 4πd2∆x (10)

Finally, we get:

∆x=R2

d2∆R (11)

Where:• R is the radius of the matter volume VR,• ∆R is the infinitesimal variation of the curvature of spacetime on the surface of the matter

volume VR,• d is the distance of the point of measurement,• ∆x is the infinitesimal variation of the curvature of spacetime at distance d.

B-6 Curvature ∆x vs Mass MAs explained in the main text, the “Mass Effect” is a combination of the volume and the surface

of a “Matter Volume”.

On one hand, the “Mass Effect” is proportional to the curvature of spacetime, i.e. to the“MatterVolume” since it is the latter that curves spacetime. On the other hand, it is inversely proportional tothe surface S since the origin of the “Mass Effect” is a pressure. If we combine these two statements,we get the following dimensional quantity:

[M ]≡ [L3][1/L2]≡ [L] (12)

At this point, we do not know the relation between ∆R and M but, referring to Einstein’s works,we have good reasons to believe that this relation is a simple linear function like:

∆R=KM (13)

. . . where K is a constant having the dimensional quantity of [L/M]. This value is in line withthe dimensional quantity of some terms of the Schwarzschild Metric: 2ε= 2∆r/r= 2GM/rc2

(sometimes, we use the Schwarzschild Radius, ε= rs/r).

The challenge, now, is to calculate K to retrieve the Newton Law.

B-7 The Newton LawPorting Eq. (13) in Eq. (11) gives:

∆x=R2

d2KM (14)

or

∆x

R2=K

M

d2(15)

Appendix B

Since x = ct, replacing R2 by c2t2 gives:

∆x

c2t2=K

M

d2(16)

or :

∆x

t2= c2K

M

d2(17)

The value ∆x/t2 has the dimensional quantity of an acceleration [L/T2]. So, replacing thisfraction by the acceleration symbol “a” (see the note below), we get:

a= c2KM

d2(18)

Notes: Here, ∆x is an infinitesimal quantity, not a differential quantity such as dx. Moreover,we are working in a linear segment of the elasticity of spacetime. In such a situation, ∆x/∆t≈ x/t.

The multiplication of a constant c2 by a second constant K gives a constant. So, we can replacethe product c2K by a new and unknown constant for the moment, G for example:

c2K = G (19)

or (the following equation is not necessary here but will be used further):

K =G

c2(20)

Porting Eq. (19) in Eq. (18) gives:

a=GM

d2(21)

To be consistent, this unknown constant G must have the same dimensional quantity of theproduct c2K, Eq. (19):

• c2 : Dimensional quantity⇒ [L2/T 2]• K : Dimensional quantity⇒ [L/M ] (see section B-6)

So,

The dimensional quantity of this new constant G is[c2K] = [L2/T2][L/M] = [L3/MT2].

On the other hand, we know that a force F is the product of an acceleration a by a mass m.Therefore, Eq. (21) can be written as follows:

F =GMm

d2(22)

Appendix B

For the moment, G is unknown but we have demonstrated that:

• G is a constant,• Its dimensional quantity is [L3/MT 2].

So, we can identify G to the constant of gravitation issued from experimentation:

G= 6, 67428× 10−11 (23)

In other words,

Eq. (22) can be identified to theNewton Law of Universal Gravitation.

This result is very important because the mathematic development in this appendix is basedexclusively on “Matter Volumes” instead of “Mass”.

Appendix C

New Version of theSchwarzschild Metric

C-1 IntroductionHere we show that the Schwarzschild Metric can be easily obtained using matter volumes

instead of masses. The following demonstration does not require the knowledge of tensor calculus.

C-2 The Minkowski MetricThe expression of the Minkowski Metric, in spherical coordinates, is:

ds2 =−c2dt2 + dr2 + r2(dθ2 + sin2 θdφ2) (1)

The Schwarzschild Metric refers to a static object with a spherical symmetry. It is built from aMinkowski Metric, in spherical coordinates, with two unknown functions: A(r) and B(r):

ds2 =−B(r)c2dt2 +A(r)dr

2 + r2(dθ2 + sin2 θdφ2) (2)

The Minkowski Equation must follow the Lorentz Invariance in Special Relativity (SR) orGeneral Relativity (GR). To get this invariance, we must set A(r) = 1/B(r). Details of calculusare described in books concerning SR and GR [11, 50-79]. So:

B(r)A(r) = 1 (3)

C-3 The Schwarzschild MetricTo calculate the Schwarzschild Metric, we can start with fig. C-1 (next page), which is issued

from the main article, where:• drout is an elementary differential radial variation outside of any “Matter Volume”, i.e. outside

of any mass;• drin is an elementary differential radial variation inside a Schwarzschild spacetime;• r is the point of measurement.

Section B-3 of Appendix B shows that the order of magnitude of the coefficient of elasticity εR(here noted ε) is infinitesimal. So, we can use the first order approximation Eq. (3) of Appendix B.Since ε is a simple coefficient, we can calculate the relation between two differential elementaryradius dr(out) and dr(in), out and in a gravitational field, closed one to the other:

drin = (1 + ε) drout (4)

Since ε� 1, Eq. (4) can be written as:

drin =1

(1 − ε)drout (5)

Appendix C

Figure C-1: Spacetime has been reduced to 2D

or, elevating in square:

dr2in =1

(1 − ε)2dr2out (6)

Developing the denominator (1 − ε)2 = 1 − 2ε+ ε2 and ignoring the last term ε2, we obtain:

dr2in =1

(1 − 2ε)dr2out (7)

This result is nothing but the radial component of the Schwarzschild Metric, i.e. the function A(r)of dr2 in Eq. (2). Then, the calculation of B(r) is immediate, taking into account that A(r)B(r) = 1from Eq. (3). So:

A(r) =1

(1 − 2ε)(8)

B(r) = (1 − 2ε) (9)

Thus, Eq. (2) becomes:

ds2 =−(1 − 2ε)c2dt2 +1

(1 − 2ε)dr2 + r2(dθ2 + sin2 θdφ2) (10)

On the other hand, Eqs. (3), (13) and (20) of Appendix B can be rewritten as:

ε=∆R

r=KM

r=

GM

rc2(11)

Finally, porting this expression in Eq. (10) gives:

ds2 =−(

1 − 2GM

rc2

)c2dt2 +

1(1 − 2GM

rc2

) dr2 + r2(dθ2 + sin2 θdφ2) (12)

Appendix C

C-4 ConclusionsThis new calculus of the Schwarzschild Metric, which is exclusively based on matter volumes

instead of mass (Fig. C-1), gives identical results than developing the EFE in the case of a staticspherical symmetry. This is due to the fact that the origin of EFE is the Fluid Mechanics, which isitself based on volumes, not on masses.

Another point: it is remarkable to see the simplicity of this 2 page demonstration vs reducingthe EFE to the case of a static sphere with a spherical symmetry which requires about 15 pages ofcomplex equations (tensors).

Appendix D

Expression of the Mass Effect in 4D

D-1 Expression of “M”In the main paper and appendix B, we have seen that the displacement of spacetime VR is equal

to that of the matter volume, V, which produces this displacement (fig. D-1):

VR = V (1)

Figure D-1: The curvature of spacetime

On the other hand, the curvature of spacetime is (Eq. (4), section B-3, Appendix B):

∆VR = εv VR (2)

Porting Eq. (1) in Eq. (2) gives:

∆VR = εv V (3)

Since the volume ∆VR is infinitesimal, we can consider that the radial curvature of spacetime,∆R, at the surface of M, is calculated dividing the volume by the surface:

∆R=∆VRS

(4)

Replacing ∆VR by Eq. (3) gives:

∆R= εvV

S(5)

Appendix D

Porting Eq. (13), Section B-6, Appendix B, in Eq. (5) gives:

KM = εvV

S(6)

orM =

εvK

V

S(7)

Porting in Eq. (7) the expression of K given in Eq. (20) of Appendix B gives the expression ofthe “Mass Effect”. Here, we have added a new coefficient, ρs (see explanation below):

M =c2

G0εvV

Sρs (8)

with:M = Mass effect (kg)V = Volume of the “Matter Volume” (m3)S = Surface of the “Matter Volume” (m2)εv = Coefficient of volumetric elasticity of spacetime. This parameter is unknown but canbe calculated from the mass/diameter of spherical particles such as some leptons or“magic” nuclei. See section D-3.c = Speed of the light (m/s)G0 = Universal constant of gravitationρs = Density of surrounding spacetime relative to a flat spacetime. This parameter is equalto 1 in a Minkowsky Spacetime.

It seems useful to differentiate εv, the coefficient of elasticity of spacetime, and ρs, the densityof surrounding spacetime. We could merge these two parameters in one common parameter sincewe are faced with two dimensionless coefficients. In both cases, the result is identical.

The coefficient of elasticity of spacetime εv is an intrinsic coefficient of spacetime. Thiscoefficient does not change. On the contrary, the density of surrounding spacetime, ρs, can changefrom one point of the universe to another. This is why the two parameters have been separated.

For example, the particle may be located in a Riemannian spacetime, such as near a black hole.In this case, the “Mass Effect” of the particle will increase because the curvature of spacetime (ρs)near a black hole is huge.

To summarize, Eq. (8) shows that the “Mass Effect” is not constant and may vary. For example,the mass of electron is 510.998928(11) KeV/c2 on Earth, but it can be for example 498 KeV/c2outside our solar system in a flat spacetime, or more than 100 GeV/c2 near a black hole.

D-2 Case of a sphereIn the particular case of a sphere, we have V = 4/3πR3 and S = 4πR2. Thus, Eq. (8) becomes:

M =εvRc

2

3 G0ρs (9)

Appendix D

D-3 NucleiGenerally nuclei are not spherical. Since we don’t know exactly their shape, it is not possible to

apply Eq. (9) to calculate their mass effect.

The radius R of a nucleus is often defined as R= 1.2 ×A1/3, A being the mass number.In reality, this equation does not mean that mass follows a A1/3 law. This rule concerns thearrangement of nucleons inside the nucleus. The result is equivalent but the significance is different.

We must also note that a surface component also exists in the Bethe-Weizsacker expression.It means that, early in 1937, Bethe and Weizsacker predicted the Eq. (8) here demonstrated, byconnecting the “Mass Effect” to the volume and surface. This topic is developed in Appendix J.

We must keep in mind that nuclei are made of empty volumes and matter volumes (seeAppendix J). However, the nucleus has a behavior of “matter volume” because the empty volumesare enclosed inside the nucleus. The space between nucleons may vary from one nucleus to another.Thus, it is necessary to know the arrangement of nucleons with accuracy before any calculations.A particular case are magic nuclei (nuclei having a null quadripolar moment) because they aresupposed to be spherical. It will be interesting to make accurate experiments on the relationvolume/surface/mass effect of magic nuclei.

On the other hand, some nuclei have a halo made of empty volumes that are not relevant in masscalculation (see section J-2, Appendix J). This is the case for example of the 11Li (3p8n), whichhas empty volumes between the 9Li and the 2n orbitals. These exceptions highlight the difficulty tomake accurate calculations of the mass effect. In all cases, before any calculation, we must knowexactly the geometry of matter volumes and empty volumes inside the nucleus. It means that thecalculus of the mass from the geometry of the nucleus, Eq. (8), is not as simple as it sounds.

As a direct consequence of the proposed theory, it could be possible that a relationship existsbetween the sphericity of particles or nuclei and the accuracy of measurements. This deductionsuggests that leptons could be spherical since their mass effect is known with an excellent accuracy.This is also the case of some particles such as the proton, neutron, or π meson. Inversely, this is notthe case of quarks. This means that quarks could have a non-spherical and complex shape.

Appendix E

Partial Rewriting of theEinstein Field Equations (EFE)

NOTE: A full development of the EFEs starts at section E-12

E-1 Original EFEThe original EFE are fully described in [11] and in several books [50-79].

Rµν −1

2gµνR=

8πG0

c4Tµν (1)

with:Rµν = Ricci Curvature Tensor;gµν = Metric Tensor;R = Scalar Curvature;8πG0/c

4 = Einstein Coefficient;Tµν = Energy-Momentum Tensor. See below and Fig. 2 of the main text.

Figure E-1: Energy-Momentum Tensor

Tµν =

T00 T01 T02 T03T10 T11 T12 T13T20 T21 T22 T23T30 T31 T32 T33

=

E/V ρmcvx ρmcvy ρmcvzρmcvx σx τxy τxzρmcvy τyx σy τyzρmcvz τzx τzy σz

(2)

Appendix E

E-2 Expression of MassEq. (8) of Appendix D shows that Mass= f(MatterV olume):

M =c2

G0εvV

Sρs (3)

with:M = Mass effect (kg)c = Speed of the light (m/s)G0 = Universal constant of gravitationεv = Coefficient of volumetric elasticity of spacetime.V = Volume of the matter volume (m3)S = Surface of the matter volume (m2)ρs = Density of surrounding spacetime relative to a flat spacetime.

E-3 Density of MatterIn the case of non-relativistic objects, the density of matter, ρm, is:

ρm =M

V(4)

Porting Eq. (3) in Eq. (4) gives, after simplification by V:

ρm =c2

S G0εv ρs (5)

E-4 Components of TµνComponents of the Tµν tensor are given in Eq. (2).

Element T00

T00 =E

V=mc2

V(6)

Since m= ρmV (ρm = density of matter):

T00 =ρmV c

2

V(7)

Porting Eq. (5) in Eq. (7), after simplification by V:

T00 =c4

S G0εv ρs (8)

Elements T01,T02,T03,T10,T20,T30

Tµν = ρm c vµν =c2

S G0εv ρs c vµν (9)

or:

Tµν =c4

S G0εv ρs ·

vµνc

(10)

Appendix E

Elements T11,T22,T33

σµν is a normal pressure and aµν is a normal acceleration:

σµν =FµνS

=maµνS

=c2

G0εvV

Sρs ·

aµνS

(11)

or:

σµν =c4

S G0εv ρs ·

V

S c2aµν (12)

Elements T12,T13,T23,T21,T31,T32

τµν is a tangential pressure and aµν is a tangential acceleration.

τµν =FµνS

=maµνS

=c2

G0εvV

Sρs ·

aµνS

(13)

or:

τµν =c4

S G0εv ρs ·

V

S c2aµν (14)

E-5 New Energy-Momentum TensorEqs. (8-10-12-14) have the following common expression:

c4

S G0εv ρs (15)

Extracting this expression from each component Eqs. (8-10-12-14) leads to a new tensor. Toavoid confusion with the traditional Energy-Momentum tensor Tµν , this new tensor is called Jµν(this symbol is different to the Bessel Symbol Jν used in some countries). We get:

Jµν =

1 v01/c v02/c v03/cv10/c (V/Sc2)a11 (V/Sc2)a12 (V/Sc2)a13v20/c (V/Sc2)a21 (V/Sc2)a22 (V/Sc2)a23v30/c (V/Sc2)a31 (V/Sc2)a32 (V/Sc2)a33

(16)

This gives the following equivalence:

Tµν =c4

S G0εv ρs Jµν (17)

Appending the Einstein Constant to Eq. (17), the full right hand side of the EFE becomes:

8πG0

c4× Tµν =

8πG0

c4× c4

S G0εv ρs Jµν (18)

After simplifying by G0 and c4 the right hand side we get the new EFE:

Rµν −1

2gµνR=

8πεvρsS

Jµν (19)

where:εv = Coefficient of volumetric elasticity of spacetime.S = Surface of the matter volume (m2)ρs = Density of surrounding spacetime relative to a flat spacetime.

Appendix E

E-6 Conclusions1 - Spacetime DensityThis new version of EFE includes a constant, ρs, which is the density of the surrounding

spacetime. Here, it is a constant but, in reality, it is a variable (a partial derivative) because it variesfrom one point of the universe to another. It means that this new version of EFE may give differentresults depending of the location of the object(s) in the universe.

For example, we could suppose that the density of spacetime ρs is greater in the center of theuniverse than on its borders since the whole spacetime of the universe exerts a pressure towardthe center. This density of spacetime, ρs, can be calculated on each point of the universe usingcomputers. The result is a new vision of the universe that could explain dark matter and dark energy.

In some sense, ρs, can be assimilated to the Cosmological Constant Λgµν that Einstein removedfrom his EFE. The only difference is that, contrary to Λgµν , ρs is fully explained (Appendix D).

2 - Using traditional EFEThis new version of EFE needs to know the volume and the surface of the matter volume which

produces the curvature of spacetime. Moreover, it also needs to know the coefficient of curvatureof spacetime, εv, and the density of surrounding spacetime relative to a flat spacetime, ρs. In thatsense, it would be more convenient to continue using the traditional Energy-Momentum Tensor onEarth. In astrophysics, we must bear in mind that ρs varies at each point of the universe. In thatparticular case, using the traditional Energy-Momentum Tensor could lead to erroneous results.

E-7 Case of a static sphereIn this case we have:

V =4

3πr3 (20)

andS = 4πr2 (21)

henceV

S=

4πr3

3

1

4πr2=r

3(22)

Porting this expression in the tensor Jµν , Eq. (16), gives, after simplification, a new tensor Kµν :

Kµν =

1 v01/c v02/c v03/cv10/c (r/3c2)a11 (r/3c2)a12 (r/3c2)a13v20/c (r/3c2)a21 (r/3c2)a22 (r/3c2)a23v30/c (r/3c2)a31 (r/3c2)a32 (r/3c2)a33

(23)

Finally, replacing “S” of the right hand side of Eq. (19) by its expression in Eq. (21) gives, aftersimplification, a reduced version of the EFE in the case of a static sphere:

Rµν −1

2gµνR=

2εvρsr2

Kµν (24)

Appendix E

Here are three inconsistencies found in the EFEs and solved in the main article and theseappendices.

E-8 Inconsistency 1: Direction of the Curvature of SpacetimeIn fluid mechanics, the curvature of the fluid made by an object is convex (Fig. E-2a). In the

scientific literature, the curvature of spacetime produced by a mass such as a black hole is concave(Fig. E-2b). Therefore, since GR is grounded on the Fluid Mechanics, the curvature of spacetime inGR must be convex, not concave, as Von Laue thought in 1927 (Appendix F).

Figure E-2. The curvature of the fluid is convex in Fluid Mechanics (a). Since GR is grounded on the Fluid Mechanics,the curvature of spacetime in GR must be convex too, not concave (b).

E-9 Inconsistency 2: Pressure or attractive force?The fluid exerts a pressure on objects (black arrows in Fig. E-2a). This is expressed by the

Cauchy Stress Tensor where each element has a pressure-like dimension (Fig. E-3a). Since theenergy–momentum tensor is grounded on the Cauchy stress tensor, the two tensors must have thesame meaning (Fig. E-3). This means that the trace of the Energy–Momentum Tensor must be anisostatic pressure force, not an attractive force (main article and Appendix A).

Figure E-3. Since the Cauchy Stress Tensor (a) is included in the Energy–Momentum Tensor (b), the two tensors musthave the same meaning, i.e. a pressure force, not an attractive force.

Appendix E

E-10 Inconsistency 3: Mass or Volume?In fluid mechanics, the displacement of the fluid is made by the volume of the object, not by its

mass, as shown in Fig. E-2a. Here too, we must keep the original meaning of the fluid mechanicsand replace “Mass” by “Volume”. The main article covers this subject.

E-11 DiscussionInconsistencies 1 and 2 are easily solved because they cancel out each other. For example, a

pressure on one side of a sheet of paper produces the same effects than an attractive force on theother side (Appendix A). The curvature of the sheet is also inverted: concave on one side, convexon the other.

Convex curvature + Pressure force = Concave curvature + Attractive force

If we assign by convention a positive sign to a convex curvature of spacetime and to a pressureforce, the above assertion gives (++) = (– –). In both cases, the results are identical. As we shall see,this does not change the mathematics of GR.

Inconsistency 3 is less obvious. Indeed, experimentations prove that spacetime is curved bymass, but Fig. E-2a shows that spacetime is curved by volume. In reality, spacetime is not curvedby mass as is thought, nor by volume, but by a special kind of volume, as explained in the mainarticle.

(Next pages : full development of the EFEs)

Appendix E (additional information)

Full development of the EFEs

E-12 - General Relativity OriginsIntroductionIn the 1910s, Einstein studied gravity. Following the reasoning of Faraday and Maxwell, he

thought that if two objects are attracted to each other, there would be some medium. The onlymedium he knew in 1910 was spacetime. He then deduced that the gravitational force is an indirecteffect carried by spacetime. He concluded that any mass perturbs spacetime, and that the spacetime,in turn, has an effect on mass, which is gravitation. So, when an object enters in the volume of thecurvature of spacetime made by a mass, i.e. the volume of a gravitational field, it is subject to anattracting force. In other words, Einstein assumed that the carrier of gravitation is the curvature ofspacetime. Thus, he tried to find an equation connecting:

1. The curvature of spacetime. This mathematical object, called the “Einstein tensor”, is the lefthand side of the EFE (Eq. 1).

2. The properties of the object that curves spacetime. This quantity, called the “Energy–Momentum tensor”, is Tµν in the right hand side of the EFE.

Curvature of spacetime ≡ Object producing this curvature

Rµν −1

2gµνR=

8πG

c4Tµν (1)

Einstein Tensor (curvature of spacetime)Thus, Einstein early understood that gravity is a consequence of the curvature of spacetime.

Without knowing the mechanism of this curvature, he posed the question of the special relativityin a curved space. He left aside the flat Minkowski space to move to a Gaussian curved space. Thelatter leads to a more general concept, the “Riemannian space”. On the other hand, he identifiedthe gravitational acceleration to the inertial acceleration (see the Appendix G “New Version of theEquivalent Principle”).

The curvature of a space is not a single number, though. It is described by “tensors”, whichare a kind of matrices. For a 4D space, the curvature is given by the Riemann-Christoffel tensorwhich becomes the Ricci Tensor after reductions. From here, Einstein created another tensor called"Einstein Tensor" (left hand side of equation 1) which combines the Ricci Curvature Tensor Rµν ,the metric tensor gµν and the scalar curvature R (see the explanations below).

Fluid MechanicsIn fluid mechanics, the medium has effects on objects. For example, the air (the medium) makes

pressures on airplanes (objects), and also produces perturbations around them. So, Einstein thoughtthat the fluid mechanics could be adapted to gravity. He found that the Cauchy-Stress Tensor wasclose to what he was looking for. Thus, he identified 1/ the "volume" in fluid mechanics to "mass",and 2/ the "fluid" to "spacetime".


Energy-Momentum TensorThe last thing to do is to include the characteristics of the object that curves spacetime in the

global formulation. To find the physical equation, Einstein started with the elementary volumedx.dy.dz in fluid mechanics. The tensor that describes the forces on the surface of this elementaryvolume is the Cauchy Tensor, often called Stress Tensor. However, this tensor is in 3D. To convertit to 4D, Einstein used the “Four-Vectors” in Special Relativity. More precisely, he used the “Four-Momentum” vector Px, Py, Pz and Pt. The relativistic “Four-Vectors” in 4D (x, y, z and t) are anextension of the well-known non-relativistic 3D (x, y and z) spatial vectors. Thus, the original 3DStress Tensor of the fluid mechanics became the 4D Energy-Momentum Tensor of EFE.

Einstein Field Equations (EFE)Finally, Einstein identified its tensor that describes the curvature of spacetime to the Energy-

Momentum tensor that describes the characteristics of the object which curves spacetime. Headded an empirical coefficient, 8πG/c4, to homogenize the right hand side of the EFE. Today,this coefficient is calculated to get back the Newton’s Law from EFE in the case of a static spherein a weak field. If no matter is present, the energy-momentum tensor vanishes, and we come backto a flat spacetime without gravitational field.

The Proposed TheoryHowever, some unsolved questions exist in the EFE, despite the fact that they work perfectly.

For example, Einstein built the EFE without knowing 1/ what is mass, 2/ the mechanism ofgravity, 3/ the mechanism by which spacetime is curved by mass . . . To date, these enigmas remain.Considering that "mass curves spacetime" does not explain anything. Only a few know, such asGupta & Pradhan and Jerome, by which strange phenomenon a mass can curve spacetime. Itseems obvious that if a process makes a deformation of spacetime, it may reasonably be expectedto provide information about the nature of this phenomenon. Therefore, the main purpose of thepresent paper is to try to solve these enigmas, i.e. to give a rational explanation of mass, gravity andspacetime curvature. The different steps to achieve this goal are:

- Special Relativity (SR). This section gives an overview of SR.- Einstein Tensor. Explains the construction of the Einstein Tensor.- Energy-Momentum Tensor. Covers the calculus of this tensor.- Einstein Constant. Explains the construction of the Einstein Constant.- EFE. This section assembles the three precedent parts to build the EFE.

E-13 - Special Relativity (Background)Lorentz Factor

β =v

c(2)

γ =1√

1− v2/c2=

1√1− β2

(3)

Minkowski Metric with signature (−,+,+,+):

ηµν ≡

−1 0 0 00 1 0 00 0 1 00 0 0 1

(4)

ds2 =−c2dt2 + dx2 + dy2 + dz2 = ηµνdxµdxν (5)


Minkowski Metric with signature (+,−,−,−):

ηµν ≡

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

(6)

ds2 = c2dt2 − dx2 − dy2 − dz2 = ηµνdxµdxν (7)

Time Dilatation“τ” is the proper time.

ds2 = c2dt2 − dx2 − dy2 − dz2 = c2dτ2 (8)

dτ2 =ds2

c2⇒ dτ2 = dt2

(1− dx2 + dy2 + dz2

c2dt2

)(9)

v2 =dx2 + dy2 + dz2

dt2(10)

dτ = dt√

1− β2 (11)

Lenght Contractions“dx′” is the proper lenght.

dx′ =dx√

1− β2(12)

Lorentz Transformation

ct′ = γ(ct− βx) (13)x′ = γ(x− βct)y′ = yz′ = z

ct′

x′

y′

z′

=

γ −βγ 0 0−βγ γ 0 0

0 0 1 00 0 0 1

c txyz

(14)

Inverse transformation on the x-direction:

ct= γ(ct′ + βx′) (15)x= γ(x′ + βct′)y= y′

z = z′

c txyz

=

γ +βγ 0 0+βγ γ 0 0

0 0 1 00 0 0 1

c t

′

x′

y′

z′

(16)


Four-PositionEvent in a Minkowski space:

X = xµ = (x0, x1, x2, x3) = (ct, x, y, z) (17)

Displacement:

∆Xµ = (∆x0,∆x1,∆x2,∆x3) = (c∆t,∆x,∆y,∆z) (18)

dxµ = (dx0, dx1, dx2, dx3) = (cdt, dx, dy, dz) (19)

Four-Velocityvx, vy, vz = Traditional speed in 3D.

U= uµ = (u0, u1, u2, u3) =

(cdt

dτ,dx

dτ,dy

dτ,dz

dτ

)(20)

from (7):

ds2 = c2dt2(

1− dx2 + dy2 + dz2

c2dt2

)(21)

v2 =dx2 + dy2 + dz2

dt2(22)

ds2 = c2dt2(

1− v2

c2

)⇒ ds= cdt

√1− β2 (23)

Condensed form:

uµ =dxµdτ

=dxµdt

dt

dτ(24)

Thus:

u0 =dx0dτ

=cdt

dt√

1− β2=

c√1− β2

= γc (25)

u1 =dx1dτ

=dx

dt√

1− β2=

vx√1− β2

= γvx (26)

u2 =dx2dτ

=dy

dt√

1− β2=

vy√1− β2

= γvy (27)

u3 =dx3dτ

=dz

dt√

1− β2=

vz√1− β2

= γvz (28)

Four-Acceleration

ai =duidτ

=d2xidτ2

(29)


Four-Momentumpx, py, pz = Traditional momentum in 3DU = Four-velocity

P =mU =m(u0, u1, u2, u3) = γ(mc, px, py, pz) (30)

(p = mc = E/c)

‖P‖2 =E2

c2− |~p|2 =m2c2 (31)

pµ =muµ =mdxµdτ

(32)

hencep0 =mc

dt

dτ=mu0 = γmc= γE/c (33)

p1 =mdx

dτ=mu1 = γmvx (34)

p2 =mdy

dτ=mu2 = γmvy (35)

p3 =mdz

dτ=mu3 = γmvz (36)

Force

F= Fµ =dpµdτ

=

(d(mu0)

dτ,d(mu1)

dτ,d(mu2)

dτ,d(mu3)

dτ

)(37)

E-14 - Einstein TensorSince the Einstein Tensor is not affected by the presented theory, one could think that it is not

useful to study it in the framework of this document. However, the knowledge of the construction ofthe Einstein Tensor is necessary to fully understand the four inconsistencies highlighted and solvedin this document. Therefore, this section is only a summary of the Einstein Tensor. A more accuratedevelopment of EFE can be obtained on books or on the Internet.

The Gauss Coordinates

Consider a curvilinear surface with coordinates u and v (Fig. 1A). The distance between twopoints, M(u, v) and M ′(u+ du, v + dv), has been calculated by Gauss. Using the gij coefficients,this distance is:

ds2 = g11du2 + g12dudv + g21dvdu+ g22dv

2 (38)

The Euclidean space is a particular case of the Gauss Coordinates that reproduces thePythagorean theorem (Fig. 1B). In this case, the Gauss coefficients are g11 = 1, g12 = g21 = 0, andg22 = 1.

ds2 = du2 + dv2 (39)


Figure 1: Gauss coordinates in a curvilinear space (A) and in an Euclidean space (B).

Equation (39) may be condensed using the Kronecker Symbol δ, which is 0 for i 6= j and 1 fori= j, and replacing du and dv by du1 and du2. For indexes i, j = 0 and 1, we have:

ds2 = δijduiduj (40)

The Metric Tensor

Generalizing the Gaussian Coordinates to “n” dimensions, equation (38) can be rewritten as:

ds2 = gµνduµduν (41)

or, with indexes µ and ν that run from 1 to 3 (example of x, y and z coordinates):

ds2 = g11du21 + g12du1du2 + g21du2du1 · · ·+ g32du3du2 + g33du

23 (42)

This expression is often called the “Metric” and the associated tensor, gµν , the “Metric Tensor”.In the spacetime manifold of RG, µ and ν are indexes which run from 0 to 3 (t, x, y and z).Each component can be viewed as a multiplication factor which must be placed in front of thedifferential displacements. Therefore, the matrix of coefficients gµν are a tensor 4× 4, i.e. a set of16 real-valued functions defined at all points of the spacetime manifold.

gµν =

g00 g01 g02 g03g10 g11 g12 g13g20 g21 g22 g23g30 g31 g32 g33

(43)

However, in order for the metric to be symmetric, we must have:

gµν = gνµ (44)

...which reduces to 10 independent coefficients, 4 for the diagonal in bold face in equation (45),g00, g11, g22, g33, and 6 for the half part above - or under - the diagonal, i.e. g01 = g10, g02 = g20,g03 = g30, g12 = g21, g13 = g31, g23 = g32. This gives:


gµν =

g00 g01 g02 g03(g10 = g01) g11 g12 g13(g20 = g02) (g21 = g12) g22 g23(g30 = g03) (g31 = g13) (g32 = g23) g33

(45)

To summarize, the metric tensor gµν in equations (43) and (45) is a matrix of functions whichtells how to compute the distance between any two points in a given space. The metric componentsobviously depend on the chosen local coordinate system.

The Riemann Curvature Tensor

The Riemann curvature tensorRαβγδ is a four-index tensor. It is the most standard way to expresscurvature of Riemann manifolds. In spacetime, a 2-index tensor is associated to each point of a2-index Riemannian manifold. For example, the Riemann curvature tensor represents the forceexperienced by a rigid body moving along a geodesic.

The Riemann tensor is the only tensor that can be constructed from the metric tensor and its firstand second derivatives. These derivatives must exist if we are in a Riemann manifold. They are alsonecessary to keep homogeneity with the right hand side of EFE which can have first derivative suchas the velocity dx/dt, or second derivative such as an acceleration d2x/dt2.

Christoffel Symbols

The Christoffel symbols are tensor-like objects derived from a Riemannian metric gµν . They areused to study the geometry of the metric. There are two closely related kinds of Christoffel symbols,the first kind Γijk, and the second kind Γkij , also known as “affine connections” or “connectioncoefficients”.

At each point of the underlying n-dimensional manifold, the Christoffel symbols are numericalarrays of real numbers that describe, in coordinates, the effects of parallel transport in curvedsurfaces and, more generally, manifolds. The Christoffel symbols may be used for performingpractical calculations in differential geometry. In particular, the Christoffel symbols are used inthe construction of the Riemann Curvature Tensor.

In many practical problems, most components of the Christoffel symbols are equal to zero,provided the coordinate system and the metric tensor possesses some common symmetries.

Comma Derivative

The following convention is often used in the writing of Christoffel Symbols. The componentsof the gradient dA are denoted A,k (a comma is placed before the index) and are given by:

A,k =∂A

∂xk(46)

Christoffel Symbols in spherical coordinates

The best way to understand the Christoffel symbols is to start with an example. Let’s considervectorial space E3 associated to a punctual space in spherical coordinates E3. A vector OM in afixed Cartesian coordinate system (0, e0i ) is defined as:

OM= xie0i (47)


orOM= r sinθ cosϕ e01 + r sinθ sinϕ e02 + r cosθ e03 (48)

Calling ek the evolution of OM, we can write:

ek = ∂k(xie0i ) (49)

We can calculate the evolution of each vector ek. For example, the vector e1 (equation 50)is simply the partial derivative regarding r of equation (48). It means that the vector e1 will besupported by a line OM oriented from zero to infinity. We can calculate the partial derivatives forθ and ϕ by the same manner. This gives for the three vectors e1, e2 and e3:

e1 = ∂1M= sinθ cosϕ e01 + sinθ sinϕ e02 + cosθ e03 (50)

e2 = ∂2M= r cosθ cosϕ e01 + r cosθ sinϕ e02 − r sinθ e03 (51)

e3 = ∂3M=−r sinθ sinϕ e01 + r sinθ cosϕ e02 (52)

The vectors e01, e02 and e03 are constant in module and direction. Therefore the differential ofvectors e1, e2 and e3 are:

de1 = (cosθ cosϕ e01 + cosθ sinϕ e02 − sinθ e03)dθ . . .· · ·+ (−sinθ sinϕ e01 + sinθ cosϕ e02)dϕ (53)

de2 = (−r sinθ cosϕ e01 − r sinθ sinϕ e02 − r cosθ e03)dθ . . .· · ·+ (−r cosθ sinϕ e01 + r cosθ cosϕ e02)dϕ . . .· · ·+ (cosθ cosϕ e01 + cosθ sinϕ e02 − sinθ e03)dr (54)

de3 = (−r cosθ sinϕ e01 + r cosθ cosϕ e02)dθ . . .· · ·+ (−r sinθ cosϕ e01 − r sinθ sinϕ e02)dϕ . . .· · ·+ (−sinθ sinϕ e01 + sinθ cosϕ e02)dr (55)

We can remark that the terms in parenthesis are nothing but vectors e1/r, e2/r and e3/r. Thisgives, after simplifications:

de1 = (dθ/r)e2 + (dϕ/r)e3 (56)

de2 = (−r dθ)e1 + (dr/r)e2 + (cotangθ dϕ)e3 (57)

de3 = (−r sin2θ dϕ)e1 + (−sinθ cosθ dϕ)e2 + ((dr/r) + cotangθ dθ)e3 (58)

In a general manner, we can simplify the writing of this set of equation writing ωji the contra-variant components vectors dei. The development of each term is given in the next section. Thegeneral expression, in 3D or more, is:

dei = ωji ej (59)


Christoffel Symbols of the second kind

If we replace the variables r, θ and ϕ by u1, u2, and u3 as follows:

u1 = r; u2 = θ; u3 =ϕ (60)

. . . the differentials of the coordinates are:

du1 = dr; du2 = dθ; du3 = dϕ (61)

. . . and the ωji components become, using the Christoffel symbol Γjki:

ωji = Γjki duk (62)

In the case of our example, quantities Γjki are functions of r, θ and ϕ. These functions can beexplicitly obtained by an identification of each component of ωji with Γjki. The full development ofthe precedent expressions of our example is detailed as follows:

ω11 = 0

ω21 = 1/r dθ

ω31 = 1/r dϕ

ω12 =−r dθ

ω22 = 1/r dr

ω32 = cotang θ dϕ

ω13 =−r sin2θ dθ

ω23 =−sinθ cosθ dϕ

ω33 = 1/r dr + cotangθ dθ

(63)

Replacing dr by du1, dθ by du2, and dϕ by du3 as indicated in equation (61), gives:

ω11 = 0

ω21 = 1/r du2

ω31 = 1/r du3

ω12 =−r du2

ω22 = 1/r du1

ω32 = cotang θ du3

ω13 =−r sin2θ du2

ω23 =−sinθ cosθ du3

ω33 = 1/r du1 + cotang θ du2

(64)

On the other hand, the development of Christoffel symbols are:


ω11 = Γ1

11 du1 + Γ1

21 du2 + Γ1

31 du3

ω21 = Γ2

11 du1 + Γ2

21 du2 + Γ2

31 du3

ω31 = Γ3

11 du1 + Γ3

21 du2 + Γ3

31 du3

ω12 = Γ1

12 du1 + Γ1

22 du2 + Γ1

32 du3

ω22 = Γ2

12 du1 + Γ2

22 du2 + Γ2

32 du3

ω32 = Γ3

12 du1 + Γ3

22 du2 + Γ3

32 du3

ω13 = Γ1

13 du1 + Γ1

23 du2 + Γ1

33 du3

ω23 = Γ2

13 du1 + Γ2

23 du2 + Γ2

33 du3

ω33 = Γ3

13 du1 + Γ3

23 du2 + Γ3

33 du3

(65)

Finally, identifying the two equations array (64) and (65) gives the 27 Christoffel Symbols.

Γ111 = 0 Γ1

21 = 0 Γ131 = 0

Γ211 = 0 Γ2

21 = 1/r Γ231 = 0

Γ311 = 0 Γ3

21 = 0 Γ331 = 1/r

Γ112 = 0 Γ1

22 =−r Γ132 = 0

Γ212 = 0 Γ2

22 = 0 Γ232 = 0

Γ312 = 0 Γ3

22 = 0 Γ332 = cotang θ

Γ113 = 0 Γ1

23 =−r sin2θ Γ133 = 0

Γ213 = 0 Γ2

23 = 0 Γ233 =−sinθ cosθ

Γ313 = 1/r Γ3

23 = cotang θ Γ333 = 0

(66)

These quantities Γjki are the Christoffel Symbols of the second kind. Identifying equations (59)and (62) gives the general expression of the Christoffel Symbols of the second kind:

dei = ωji ej = Γjki duk ej (67)

Christoffel Symbols of the first kind

We have seen in the precedent example that we can directly get the quantities Γjki byidentification. These quantities can also be obtained from the components gij of the metric tensor.This calculus leads to another kind of Christoffel Symbols.

Lets write the covariant components, noted ωji, of the differentials dei:

ωji = ej dei (68)

The covariant components ωji are also linear combinations of differentials dui that can bewritten as follows, using the Christoffel Symbol of the first kind Γkji :

ωji = Γkji duk (69)

On the other hand, we know the basic relation:

ωji = gjl ωli (70)

Porting equation (69) in equation (70) gives:

Γkji duk = gjl ω

li (71)


Let’s change the name of index j to l of equation (62):

ωli = Γlki duk (72)

Porting equation (117) in equation (116) gives the calculus of the Christoffel Symbols of thefirst kind from the Christoffel Symbols of the second kind:

Γkji duk = gjl Γjki du

k (73)

Geodesic Equations

Let’s take a curve M0-C-M1. If the parametric equations of the curvilinear abscissa are ui(s),the length of the curve will be:

l=

∫M1

M0

(gijdui

ds

duj

ds

)1/2

ds (74)

If we pose u′i = dui/ds and u′j = duj/ds we get:

l=

∫M1

M0

(giju

′iu′j)1/2

ds (75)

Here, the u′j are the direction cosines of the unit vector supported by the tangent to the curve.Thus, we can pose:

f(uk, u′j) = giju′iu′j = 1 (76)

The length l of the curve defined by equation (74) has a minimum and a maximum that can becalculated by the Euler-Lagrange Equation which is:

∂L∂fi−

d

dx

(∂L∂f ′i

)= 0 (77)

In the case of equation (74), the Euler-Lagrange equation gives:

d

ds(giju

′j)− 1

2∂igjk u

′j u′k = 0 (78)

or

giju′j + (∂kgij −

1

2∂igjk) u

′j u′k = 0 (79)

After developing derivative and using the Christoffel Symbol of the first kind, we get:

gijdu′j

ds+ Γjik u

′j u′k = 0 (80)


The contracted multiplication of equation (79) by gilgives, with gijgil and gilΓijk = Γljk:

d2ul

ds2+ Γljk

duj

ds

duk

ds= 0 (81)

Parallel TransportFigure 2 shows two points M and M’ infinitely close to each other in polar coordinates.

Figure 2: Parallel transport of a vector.

In polar coordinates, the vector V1 will become V2. To calculate the difference between twovectors V1 and V2, we must before make a parallel transport of the vector V2 from point M’ to pointM. This gives the vector V3. The absolute differential is defined by:

dV = V3 − V1 (82)

Variations along a Geodesic

The variation along a 4D geodesic follows the same principle. For any curvilinear coordinatessystem yi, we have (from equation 81):

d2yi

ds2+ Γikj

dyj

ds

dyk

ds= 0 (83)

where s is the abscissa of any point of the straight line from an origin such as M in figure 2.


Let’s consider now a vector ~v having covariant components vi. We can calculate the scalarproduct of ~v and ~n= dyk/ds as follows:

~v · ~n= vidyi

ds(84)

During a displacement from M to M’ (figure 2), the scalar is subjected to a variation of:

d

(vidyi

ds

)= dvk

dyk

ds+ vi d

(d2yi

ds2

)(85)

or:

d

(vidyi

ds

)= dvk

dyk

ds+ vi

d2yi

ds2ds (86)

On one hand, the differential dvk can be written as:

dvk = ∂jvkdyj

dsds (87)

On the other hand, the second derivative can be extracted from equation (83) as follows:

d2yi

ds2=−Γikj

dyj

ds

dyk

ds(88)

Porting equations (87) and (88) in (86) gives:

d

(vidyi

ds

)= ∂jvk

dyj

ds

dyk

dsds− viΓikj

dyj

ds

dyk

dsds (89)

or:

d

(vidyi

ds

)= (∂jvk − viΓikj)

dyj

ds

dyk

dsds (90)

Since (dyj/ds)ds= dyj , this expression can also be written as follows:

d(~v · ~n) = (∂jvk − viΓikj) dyjdyk

ds(91)

The absolute differentials of the covariant components of vector ~v are defined as:

Dvkdyk

ds= (∂jvk − viΓikj) dyj (92)

Finally, the quantity in parenthesis is called “affine connection” and is defined as follows:

∇jvk = ∂jvk − viΓikj (93)


Some countries in the world use “;” for the covariant derivative and “,” for the partial derivative.Using this convention, equation (93) can be written as:

vk;j = vk,j − viΓikj (94)

To summarize, given a function f , the covariant derivative ∇vf coincides with the normaldifferentiation of a real function in the direction of the vector ~v, usually denoted by ~vf and df(~v).

Second Covariant Derivatives of a VectorRemembering that the derivative of the product of two functions is the sum of partial derivatives,

we have:

∇a(tbrc) = rc .∇atb + tb . ∇arc (95)


∇a(tbrc) = rc (∂atb − tlΓlab) + tb (∂arc − rlΓlac) (96)

or:

∇a(tbrc) = rc∂atb − rctlΓlab + tb∂arc − tbrlΓlac (97)

Hence:

∇a(tbrc) = rc∂atb + tb∂arc − rctlΓlab − tbrlΓlac (98)

Finally:

∇a(tbrc) = ∂a(tbrc)− rctlΓlab − tbrlΓlac (99)

Posing tbrc =∇jvi gives:

∇k(∇jvi) = ∂k(∇jvi)− (∇jvr)Γrik − (∇rvi)Γrjk (100)


∇k(∇jvi) = ∂k(∂jvi − vlΓlji)− (∂jvr − vlΓljr)Γrik − (∂rvi − vlΓlri)Γrjk (101)

Hence:

∇k(∇jvi) = ∂kjvi − (∂kΓlji)vl − Γlji∂kvl − Γrik∂jvr + ΓrikΓ

ljrvl − Γrjk∂rvi + ΓrjkΓ

lrivl (102)

The Riemann-Christoffel Tensor

In expression (102), if we make a swapping between the indexes j and k in order to get adifferential on another way (i.e. a parallel transport) we get:

∇j(∇kvi) = ∂jkvi − (∂jΓlki)vl − Γlik∂jvl − Γrij∂kvr + ΓrijΓ

lkrvl − Γrkj∂rvi + ΓrkjΓ

lrivl (103)


A subtraction between expressions (102) and (103) gives, after a rearrangement of some terms:

∇k(∇jvi)−∇j(∇kvi) = (∂kj − ∂jk)vi + (∂jΓlki − ∂kΓlji)vl + (Γlik∂j − Γlji∂k)vl . . .

· · ·+ (Γrij∂k − Γrik∂j)vr + (ΓrikΓljr − ΓrijΓ

lkr)vl + (Γrkj − Γrjk)∂rvi + (ΓrjkΓ

lri − ΓrkjΓ

lri)vl (104)

On the other hand, since we have:

Γrjk = Γrkj (105)

Some terms of equation (104) are canceled:

∂kj − ∂jk = 0 (106)

Γrkj − Γrjk = 0 (107)

ΓrjkΓlri − ΓrkjΓ

lri = 0 (108)

And consequently:

∇k(∇jvi)−∇j(∇kvi) = (∂jΓlki − ∂kΓlji)vl + (Γlik∂j − Γlji∂k)vl . . .

· · ·+ (Γrij∂k − Γrik∂j)vr + (ΓrikΓljr − ΓrijΓ

lkr)vl (109)

Since the parallel transport is done on small portions of geodesics infinitely close to each other,we can take the limit:

∂jvl, ∂kvl, ∂jvr, ∂kvr → 0 (110)

This means that the velocity field is considered equal in two points of two geodesics infinitelyclose to each other. Then we can write:

∇k(∇jvi)−∇j(∇kvi)∼= (∂jΓlki − ∂kΓlji + ΓrikΓ

ljr − ΓrijΓ

lkr)vl (111)

As a result of the tensorial properties of covariant derivatives and of the components vl, thequantity in parenthesis is a four-order tensor defined as:

Rli,jk = ∂jΓlki − ∂kΓlji + ΓrikΓ

ljr − ΓrijΓ

lkr (112)

In this expression, the comma in Christoffel Symbols means a partial derivative. The tensorRli,jk is called Riemann-Christoffel Tensor or Curvature Tensor which characterizes the curvatureof a Riemann Space.

The Ricci Tensor

The contraction of the Riemann-Christoffel Tensor Rli,jk defined by equation (112) relative toindexes l and j leads to a new tensor:

Rik =Rli,lk = ∂lΓlki − ∂kΓlli + ΓrikΓ

llr − ΓrilΓ

lkr (113)


This tensor Rik is called the “Ricci Tensor”. Its mixed components are given by:

Rjk = gjiRik (114)

The Scalar Curvature

The Scalar Curvature, also called the “Curvature Scalar” or “Ricci Scalar”, is given by:

R=Rii = gijRij (115)

The Bianchi Second Identities

The Riemann-Christoffel tensor verifies a particular differential identity called the “BianchiIdentity”. This identity involves that the Einstein Tensor has a null divergence, which leads to aconstraint. The goal is to reduce the degrees of freedom of the Einstein Equations. To calculatethe second Bianchi Identities, we must derivate the Riemann-Christoffel Tensor defined in equation(112):

∇tRli,rs = ∂rtΓlsi − ∂stΓlri (116)

A circular permutation of indexes r, s and t gives:

∇rRli,st = ∂srΓlti − ∂trΓlsi (117)

∇sRli,tr = ∂tsΓlri − ∂rsΓlti (118)

Since the derivation order is interchangeable, adding equations (116), (117) and (118) gives:

∇tRli,rs +∇rRli,st +∇sRli,tr = 0 (119)

The Einstein Tensor

If we make a contraction of the second Bianchi Identities (equation 119) for t= l, we get:

∇lRli,rs +∇rRli,sl +∇sRli,lr = 0 (120)

Hence, taking into account and the definition of the Ricci Tensor of equation (113) and thatRli,sl =−Rli,ls, we get:

∇lRli,rs +∇sRir −∇rRis = 0 (121)

The variance change with gij gives:

∇sRir =∇s(gikRkr ) (122)

or

∇sRir = gik∇sRkr (123)


Multiplying equation (121) by gik gives:

gik∇lRli,rs + gik∇sRir − gik∇rRis = 0 (124)

Using the property of equation (123), we finally get:

∇lRkl,rs +∇sRkr −∇rRks = 0 (125)

Let’s make a contraction on indexes k and s:

∇kRkl,rk +∇kRkr −∇rRkk = 0 (126)

The first term becomes:

∇kRkr +∇kRkr −∇rRkk = 0 (127)

After a contraction of the third term we get:

2∇kRkr −∇rR= 0 (128)

Dividing this expression by two gives:

∇kRkr −1

2∇rR= 0 (129)

or:

∇k(Rkr −

1

2δkrR

)= 0 (130)

A new tensor may be written as follows:

Gkr =Rkr −1

2δkrR (131)

The covariant components of this tensor are:

Gij = gik Gkj (132)

or

Gij = gik

(Rkj −

1

2δkjR

)(133)

Finally get the Einstein Tensor which is defined by:

Gij =Rij − 12gijR (134)


E-15 The Einstein ConstantThe Einstein Tensor Gij of equation (134) must match the Energy-Momentum Tensor Tuv

defined later. This can be done with a constant κ so that:

Gµν = κTµν (135)

This constant κ is called “Einstein Constant” or “Constant of Proportionality”. To calculate it,the Einstein Equation (134) must be identified to the Poisson’s classical field equation, which is themathematical form of the Newton Law. So, the weak field approximation is used to calculate theEinstein Constant. Three criteria are used to get this "Newtonian Limit":

1 - The speed is low regarding that of the light c.2 - The gravitational field is static.3 - The gravitational field is weak and can be seen as a weak perturbation hµν added to a flat

spacetime ηµν as follows:

gµν = ηµν + hµν (136)

We start with the equation of geodesics (83):

d2xµ

ds2+ Γµαβ

dxα

ds

dxβ

ds= 0 (137)

This equation can be simplified in accordance with the first condition:

d2xµ

ds2+ Γµ00

(dx0

ds

)2

= 0 (138)

The two other conditions lead to a simplification of Christoffel Symbols of the second kind asfollows:

Γµ00 =1

2gµλ(∂0gλ0 + ∂0g0λ − ∂λg00) (139)

Or, considering the second condition:

Γµ00 =−1

2gµλ∂λ g00 (140)

And also considering the third condition:

Γµ00 ≈−1

2(ηµλ + hµλ)(∂λη00 + ∂λh00) (141)

In accordance with the third condition, the term ∂λη00 is canceled since it is a flat space:

Γµ00 ≈−1

2(ηµλ + hµλ)∂λh00 (142)

Another simplification due to the approximation gives:

Γµ00 ≈−1

2ηµλ∂λh00 (143)


The equation of geodesics then becomes:

d2xµ

dt2− 1

2ηµλ∂λh00

(dx0

dt

)2

= 0 (144)

Reduced to the time component (µ= 0), equation (144) becomes:

d2xµ

dt2− 1

2η0λ∂λh00

(dx0

dt

)2

= 0 (145)

The Minkowski Metric shows that η0λ = 0 for λ> 0. On the other hand, a static metric (thirdcondition) gives ∂0h00 = 0 for λ= 0. So, the 3x3 matrix leads to:

d2xi

dt2− 1

2∂ih00

(dx0

dt

)2

= 0 (146)

Replacing dt by the proper time dτ gives:

d2xi

dτ2− 1

2∂ih00

(dx0

dτ

)2

= 0 (147)

Dividing by (dx0/dτ)2 leads to:

d2xi

dτ2

(dτ

dx0

)2

=1

2∂ih00 (148)

d2xi

(dx0)2=

1

2∂ih00 (149)

Replacing x0 by ct gives:

d2xi

d(ct)2=

1

2∂ih00 (150)

or:

d2xi

dt2=c2

2∂ih00 (151)

Let us pose:

h00 =− 2

c2Φ (152)

or

4h00 =− 2

c24Φ (153)


Since the approximation is in an euclidean space, the Laplace operator can be written as:

4h00 =− 2

c2∇2Φ (154)

4h00 =− 2

c2(4πG0ρ) (155)

4h00 =−8πG0

c2ρ (156)

On the other hand, the element T00 defined later is:

T00 = ρc2 (157)

or

ρ=T00c2

(158)


4h00 =−8πG0

c2T00c2

(159)

or:

4h00 =−8πG0

c4T00 (160)

The left hand side of equation (160) is the perturbation part of the Einstein Tensor in the caseof a static and weak field approximation. It directly gives a constant of proportionality which alsoverifies the homogeneity of the EFE (equation 1). This equation will be fully explained later in thisdocument. Thus:

Einstein Constant=8πG0

c4(161)

E-16 Movement Equations in a Newtonian FluidThe figure 3 (next page) shows an elementary parallelepiped of dimensions dx, dy, dz which is a

part of a fluid in static equilibrium. This cube is generally subject to volume forces in all directions,as the Pascal Theorem states. The components of these forces are oriented in the three orthogonalaxis. The six sides of the cube are: A-A’, B-B’ and C-C’.

E-17 Normal ConstraintsOn figure 4 (next page), the normal constraints to each surface are noted “σ”. The tangential

constraints to each surface are noted “τ”. Since we have six sides, we have six sets of equations.In the following equations, “σ” and “τ” are constraints, dF is an elementary force, and dS is anelementary surface:


Figure 3: Elementary parallelepiped dx, dy, dz

d~F+x

dS= ~σx + ~τ ′yx + ~τxz (162)

d~F−xdS

= ~σ′x + ~τ ′xy + ~τzx (163)

d~F+y

dS= ~σy + ~τyz + ~τ ′′yx (164)

d~F−ydS

= ~σ′y + ~τzy + ~τ ′′xy (165)

d~F+z

dS= ~σz + ~τ ′′xz + ~τ ′yz (166)

d~F−zdS

= ~σ′z + ~τ ′′zx + ~τ ′zy (167)

We can simplify these equations as follows:

~σx + ~σ′x = ~σX (168)

~σy + ~σ′y = ~σY (169)

~σz + ~σ′z = ~σZ (170)

So, only three components are used to define the normal constraint forces, i.e. one per axis.


Figure 4: Forces on the elementary parallelepiped sides.

E-18 Tangential Constraints

If we calculate the force’s momentum regarding the gravity center of the parallelepiped, we have12 tangential components (two per side). Since some forces are in opposition to each other, only6 are sufficient to describe the system. Here, we calculate the three momenta for each plan, XOY,XOZ and YOZ, passing through the gravity center of the elementary parallelepiped:

For the XOY plan:

MXOY = (~τzydzdx)dy

2+ (~τzxdydz)

dx

2+ (~τyzdxdz)

dy

2+ (~τxzdzdy)

dx

2(171)

MXOY =1

2dV [(~τzy + ~τyz) + (~τzx + ~τxz)] (172)

MXOY =1

2dV [(~τZY + ~τZX)] (173)

MXOY =1

2dV ~τXOY (174)


For the XOZ plan:

MXOZ = (~τ ′yxdydz)dx

2+ (~τ ′yzdxdy)

dz

2+ (~τ ′xydzdy)

dx

2+ (~τ ′zydydx)

dz

2(175)

MXOZ =1

2dV[(~τ ′yx + ~τ ′xy) + (~τ ′yz + ~τ ′zy)

](176)

MXOZ =1

2dV [(~τY X + ~τY Z)] (177)

MXOZ =1

2dV ~τXOZ (178)

For the ZOY plan:

MZOY = (~τ ′′xzdxdy)dz

2+ (~τ ′′yxdxdz)

dy

2+ (~τ ′′zxdydx)

dz

2+ (~τ ′′xydzdx)

dy

2(179)

MZOY =1

2dV[(~τ ′′xz + ~τ ′′zx) + (~τ ′′yx + ~τ ′′xy)

](180)

MZOY =1

2dV [(~τZX + ~τXY )] (181)

MZOY =1

2dV ~τZOY (182)

So, for each plan, only one component is necessary to define the set of momentum forces. Sincethe elementary volume dV is equal to a*a*a (a = dx, dy or dz), we can come back to the constraintequations dividing each result (174), (178) and (182) by dV/2.

E-19 Constraint Tensor

Finally, the normal and tangential constraints can be reduced to only 6 terms with ~τxy = ~τyx,~τxz = ~τzx, and ~τzy = ~τyz:

~σX = ~σxx (183)

~σY = ~σyy (184)

~σZ = ~σzz (185)

~τXOY = ~τxy (186)

~τXOZ = ~τxz (187)

~τZOY = ~τzy (188)

Using a matrix representation, we get:[FxFyFz

]=

[σxx τxy τxzτyx σyy τyzτzx τzy σzz

][SxSySz

](189)

The constraint tensor at point M becomes:

T(M) =

[σxx τxy τxzτyx σyy τyzτzx τzy σzz

]=

[σ11 τ12 τ13τ21 σ22 τ23τ31 τ32 σ33

](190)


This tensor is symmetric and its meaning is shown in Figure 5.

Figure 5: Meaning of the Constraint Tensor.

Since all the components of the tensor are pressures (more exactly constraints), we can representit by the following equation where Fi are forces and si are surfaces:

Fi = σijsj =∑

j σijsj (191)

E-20 Energy-Momentum Tensor

We can write equation (191) as:

σij =Fisj

=∆Fi∆sj

=∆(mai)

∆sj(192)

Here we suppose that only volumes (x, y and z) and time (t) make the force vary. Therefore, themass (m) can be replaced by volume (V) using a constant density (ρ):

m= ρV (193)

Porting (193) in (192) gives:

σij =∆(m.ai)

∆sj=

∆(ρV.ai)

∆sj=ρV

∆sj∆ai (194)

As shown in figure 3, the volume V and surface Sj (for surface on side j), concern anelementary parallelepiped.

V = (∆Xj)3 and Sj = (∆Xj)

2 (195)


Thus:

V

Sj=

(∆Xj)3

(∆Xj)2= ∆Xj (196)

Hence

σij = ρ ∆Xj∆ai (197)

Since ∆ai is an acceleration, or vi/dt:

σij = ρ∆Xjvi∆t

= ρ∆Xj

∆tvi (198)

Finally

σij = ρvivj (199)

This tensor comes from the Fluid Mechanics and uses traditional variables vx, vy and vz . Wecan extend these 3D variables to 4D in accordance with Special Relativity (see above). The new 4Dtensor created, called the “Energy-Momentum Tensor’, has the same properties as the old one, inparticular symmetry. To avoid confusion, let’s replace σij by Tµν as follows:

Tµν = ρuµuν (200)

or

Tµν =


=

ρu0u0 ρu0u1 ρu0u2 ρu0u3ρu1u0 ρu1u1 ρu1u2 ρu1u3ρu2u0 ρu2u1 ρu2u2 ρu2u3ρu3u0 ρu3u1 ρu3u2 ρu3u3

(201)

...with uµ and uν as defined in equations (25) to (28).

This tensor may be written in a more explicit form, using the “traditional” velocity vx, vy andvz instead of the relativistic velocities, as shown in equations (25) to (28):

Tµν =

ργ2c2 ργ2cvx ργ2cvy ργ2cvz

ργ2cvx ργ2vxvx ργ2vxvy ργ2vxvzργ2cvy ργ2vyvx ργ2vyvy ργ2vyvzργ2cvz ργ2vzvx ργ2vzvy ργ2vzvz

(202)

For low velocities, we have γ = 1:

Tµν =

ρc2 ρcvx ρcvy ρcvz

ρcvx ρvxvx ρvxvy ρvxvzρcvy ρvyvx ρvyvy ρvyvzρcvz ρvzvx ρvzvy ρvzvz

(203)


Replacing the spatial part of this tensor by the old definitions (equation 189) gives:

Tµν =

ρc2 ρcvx ρcvy ρcvz

ρcvx σx τxy τxzρcvy τyx σy τyzρcvz τzx τzy σz

(204)

Replacing ρ by m/V (equation 193) and mc2 by E leads to one of the most commons form ofthe Energy-Momentum Tensor, V being the volume:

Tµν =


=

E/V ρcvx ρcvy ρcvzρcvx σx τxy τxzρcvy τyx σy τyzρcvz τzx τzy σz

(205)

E-21 Einstein Field EquationsFinally, the association of the Einstein Tensor Gµν =Rµν − 1/2gµνR (equation 134), the

Einstein Constant 8πG0/c4 (equation 161), and the Energy-Momentum Tensor Tµν (equations

204/205), gives the full Einstein Field Equations, excluding the cosmological constant Λ whichis not proven:

Rµν −1

2gµνR=

8πG0

c4Tµν (206)

Figure 6: Meaning of the Energy-Momentum Tensor.


To summarize, the Einstein field equations are 16 nonlinear partial differential equations thatdescribe the curvature of spacetime, i.e. the gravitational field, produced by a given mass. As aresult of the symmetry of Gµν and Tµν , the actual number of equations are reduced to 10, althoughthere are an additional four differential identities (the Bianchi identities) satisfied by Gµν , one foreach coordinate.

E-22 Dimensional Analysis

The dimensional analysis verifies the homogeneity of equations. The most common dimensionalquantities used in this chapter are:

Speed V⇒ [L/T ]Energy E⇒ [ML2/T 2]Force F⇒ [ML/T 2]Pressure P⇒ [M/LT 2]Momentum M⇒ [ML/T ]Gravitation constant G⇒ [L3/MT 2]

Here are the dimensional analysis of the Energy-Momentum Tensor:

T00 is the energy density, i.e. the amount of energy stored in a given region of space per unitvolume. The dimensional quantity of E is [ML2/T 2] and V is [L3]. So, the dimensional quantityof E/V is [M/LT 2]. Energy density has the same physical units as pressure which is [M/LT 2].

T01,T02,T03 are the energy flux, i.e. the rate of transfer of energy through a surface. Thequantity is defined in different ways, depending on the context. Here, ρ is the density [M/V ], and cand vi (i = 1 to 3) are velocities [L/T ]. So, the dimensional quantity of T0i is [M/L3][L/T ][L/T ].It is that of a pressure [M/LT 2].

T10,T20,T30 are the momentum density, which is the momentum per unit volume. Thedimensional quantity of Ti0 (i = 1 to 3) is identical to T0i, i.e. that of a pressure [M/LT 2].

T12,T13,T23,T21,T31,T32 are the shear stress, or a pressure [M/LT 2].

T11,T22,T33 are the normal stress or isostatic pressure [M/LT 2].

Note: The Momentum flux is the sum of the shear stresses and the normal stresses.

As we see, all the components of the Energy-Momentum Tensor have a pressure-likedimensional quantity:

Tµν⇒ Pressure [M/LT 2] (207)

Appendix F

Von Laue Geodesics

F-1 Introduction

A set of concentric circles is drawn in (fig. F-1a). These lines represent the geodesics ofspacetime far from any mass, in a Minkowski spacetime. If a static spherical symmetry mattervolume is inserted in the center (fig. F-1b), spacetime will be curved, as explained in the mainarticle. This figure F-1b has been duplicated in fig. F-2.

Figure F-1: Curvature of spacetime

Figure F-2: Von Laue Geodesics

The Von Laue Geodesics [32] has been drawn over these concentric circles. The result is thatthe Von Laue Geodesics match EXACTLY the concentric circles.

In other words, it seems that Von Laue, early as the 1920s, predicted the theory presented here.It is obvious that his diagram shows volumes, not masses, even if the Von Laue Formulas arerelated to the mass of the body.

Appendix G

Explanation of the Increase of theMass of Relativistic Particles

G-1 IntroductionThe increase of the mass of relativistic particles is covered by special relativity. However, this

phenomenon remains somewhat obscure. To date, we are still unable to explain with simple words,i.e. without using mathematics, why the mass of relativistic particles increases. This appendixproposes a simple and rational explanation of this strange phenomenon using the curvature ofspacetime and “Matter Volume” instead of mass.

G-2 Length contractionSpecial relativity states that, at relativistic speed, times expand, lengths contract and angles are

modified [1-11]. The length contraction is defined by the formula

lm = l0

√1 − v2

c2(1)

where• lm = Measured length• l0 = Proper length• v = Speed of object• c = Speed of light

G-3 Mass increaseConsider a particle at rest (Fig. G-1a, next page). Its matter volume produces a curvature of the

spacetime. Geodesics of spacetime are spaced of l0.

If this particle moves at a relativistic speed “v” (Fig. G-1b), spacetime geodesics seems to shrink(length contraction). As a result, the curvature of spacetime, ∆R0 and ∆Rm below, increases. Inother words, the density of the curvature of spacetime is inversely proportional to the space betweentwo geodesics (also see note 1). Therefore, Eq. (1) becomes:

∆Rm =∆R0√

1 − v2/c2(2)

where:• ∆Rm = Infinitesimal element of the curvature of spacetime measured (external observer)• ∆R0 = Infinitesimal element of the proper curvature of spacetime• v = Speed of the particle• c = Speed of light

Appendix G

Figure G-1: Since the spacetime is denser in (b), the mass effect increases.

Using Eq. (13) of Appendix B, ∆R=KM , we can replace the curvatures of spacetime ∆Rm

and ∆R0, by the measured mass effect “m” and the proper mass effect “m0”. See notes 2 and 3.After simplification by K, we get:

m =m0√

1 − v2/c2(3)

In other words, as shown in Fig. G-1, the mass or volume of a relativistic particle does notincrease. In reality, the curvature of spacetime becomes more condensed at relativistic speed andproduces an opposition force which has the same effect as an increase of mass.

Note 1: The spacetime curvature is the difference of displacement ∆R of a geodesic vs. tothe same geodesic in a Minkowski space. As shown in Appendix C, the curvature of spacetime isinfinitesimal. Thus, regardless of the function used, the portion of the curve on which we work islinear. Taking this linearity into account, there is no objection to consider that the curvature ofspacetime is inversely proportional to the space between two geodesics.

Note 2: The nature of expression ∆R=KM is not relevant because this section coversexclusively the calculation of the coefficient to be applied to a proper value to get the measuredvalue. This coefficient is noted γ (or 1/γ) in scientific literature. This means that the relationshipbetween the spacetime curvature and the mass effect is not affected by this study. For example,if we make 5 measurements of the curvature of spacetime at different speeds, we will not have 5different relationships between ∆R and M, but only one applicable in all cases ...but we will have5 different coefficients γ.

Note 3: The principles of special relativity state that the measurement of the mass of a relativisticparticle increases. However, the converse is also true if we swap the reference systems. If wecould pick up a measuring device on a particle in movement, this device would indicate that ourspacetime, that in which we live, is much more dense as we see it. Thus, a section of the LHC forexample, with a mass of 3 tons, measured from a device located on the particle in motion, wouldhave a mass of 3000 tons if γ = 1000. From our point of view, the mass of a relativistic particleincreases, but from the particle’s point of view, it is our world that increases. In all cases, theproper mass of the particle or that of our world remains unchanged. This “relative view” is oftenmisunderstood.

Appendix G

Note 4: It is wrong to think that the mass or the “matter volume” of relativistic particlesreally increases. In reality, it is the mass effect due to the apparent compression of spacetime thatincreases. The volume remains unchanged. On Earth, we consider that “mass” is an intrinsic valueof a particle, such as the volume. It is not true. Since the mass effect comes from the pressure ofspacetime on the particle, it is a virtual effect, such as pressure, speed, force, energy... On the otherhand, the mass effect depends of the surrounding density of spacetime. Thus, for example, if thespacetime density was two times higher, the mass effect would be twice as important as well, but theintrinsic characteristics of the particle, more specifically the volume, would remain unchanged.

Appendix H

Equivalence Principleusing the Curvature of Spacetime

H-1 DemonstrationLet’s consider a static object on Earth (fig. H-1). “Matter volume” cause a curvature of spacetime

that exerts a gravitational force on the surface of the object. Gravity is g= 9.81m.s−2 on the surfaceof Earth.

Figure H-1: Gravitational force using the spacetime curvature

Let’s now consider the same object accelerated out of any gravitational field. We can representthis object in two different views (fig. H-2a and H-2b).

Figure H-2: Inertial acceleration using the spacetime curvature

In both cases, acceleration “a” is supposed to be identical to g:

a= g= 9.81m.s−2

Without any reference, a local observer cannot know if the curvature of spacetime is due to apressure on the object (fig. H-2a) or to its acceleration (fig. H-2b). In fact, these two figures areidentical and depend on where the observer stands, as described in Special Relativity.

Appendix H

Since:

• By definition, g = 9.81 m.s−2 (fig. H-1) is identical to a = 9.81 m.s−2 (fig. H-2a and H-2b).• These examples use the same object. Therefore, the curvature of spacetime produced by the

matter volume of this object is identical.• The two precedent points show that the “mass effect” produced by these curvatures are

identical.

We deduce that the “gravitational mass effect” (fig. H-1) is identical to the “inertial mass effect”(fig. H-2):

Gravitational mass effect=

Inertial mass effect=

Effect from spacetime curvature

As shown, this demonstration of the Equivalence Principle is fully explained using “MatterVolume” instead of “Mass”.

Appendix J

The Bethe-WeizsackerSemi-Empirical Mass Formula (SEMF)

J-1 HistoryThe Bethe-Weizsacker’s formula, sometimes also called semi-empirical mass formula (SEMF),

is used to approximate the binding energy of an atomic nucleus from its number of protons andneutrons. It is based partly on theory and partly on empirical measurements.

The theory is based on the liquid drop model proposed by George Gamow, which can accountfor most of the terms in the formula and gives rough estimates for the values of the coefficients. Itwas first formulated in 1935 by Carl Friedrich von Weizsacker, and although refinements have beenmade to the coefficients over the years, the structure of the formula remains the same today.

The SEMF gives a good approximation but, as explained, this formula is grounded on the “liquiddrop model”, which is not a “pure theory”.

This appendix shows that the SEMF and Eq. (8) of Appendix D have strong similarities. Inparticular, the two main terms of each expressions, the Volume V , and the Surface S, are present inboth equations. It means that the theory here proposed also explains the SEMF.

J-2 The formulaIn the following formula, let “A” be the total number of nucleons, “Z” the number of protons,

and “N” the number of neutrons, so that “A=Z+N”.

The mass of an atomic nucleus is given by:

m=Zmp +Nmn − EB

c2(1)

wheremp andmn are the rest mass of a proton and a neutron, respectively, andEB is the bindingenergy of the nucleus. The SEMF states that the binding energy will take the following form:

EB = aVA− aSA2/3 − aC

Z2

A1/3− aA

(A− 2Z)2

A− δ(A,Z) (2)

The coefficients are determined empirically. Each term is summarized below.

Volume termThe term “aVA” is known as the “volume term”. The volume of the nucleus is proportional to

“A”, so this term is proportional to the volume.

Surface termThe term “aSA2/3” is known as the “surface term”. This term, based on the strong force, is a

correction to the volume term. If the volume of the nucleus is proportional to “A”, then the radiusshould be proportional to “A1/3” and the surface area to “A2/3”. This explains why the surfaceterm is proportional to “A2/3”. It can also be deduced that “aS” should have a similar order ofmagnitude as “aV ”.

Appendix J

Coulomb termThe term “aC(Z2/A1/3)” is known as the “Coulomb” or “electrostatic term”. The basis for this

term is the electromagnetic force electrostatic repulsion between protons.

Asymmetry termThe term “aA((A− 2Z)2/A)” is known as the “asymmetry term”.

Pairing termThe term “δ(A,Z)” is known as the “pairing term”, possibly also known as the pairwise

interaction.

J-3 Explanation of the SEMFThe proposed theory gives the mass formula in Appendix D, Eq. (8):

M =c2

G0εvV

Sρs (3)

with:M = Mass effect (kg)V = Volume of the “Matter Volume” (m3)S = Surface of the “Matter Volume” (m2)... etc... (see Appendix D).

In this equation, we see that the mass is directly proportional to the volume V , and inverselyproportional to the surface S.

The SEMF equation is different because it represents the binding energy, not the mass of a givennucleus. However, these two expressions (fig. J-1 next page) are grounded on a common basis: thevolume V , and the surface S.

Figure J-1: The mass expression of Appendix D, Eq. (8), and SEMF equation are grounded on thesame basis: the volume V , and the surface S.

Appendix J

J-4 ConclusionsAs fig. J-1 shows, the mass expression of Appendix D, Eq. (8), and SEMF equation are grounded

on the same basis: the volume V , and the surface S.

It is not a validation per se but a strong argument that supports the theory proposed here.

Appendix K

Other Classes of Volumes

In reality, it exists five classes of volumes, not three as indicated in the main article. The twoextra classes are described below.

K-1 Hermetic Volumes

These volumes are combinations of matter and empty volumes but their global volume ishermetic regarding spacetime. Therefore, Hermetic Volumes are like Matter Volumes, they getmass. For example, a nucleus is made of nucleons (matter volumes), separated by empty space(empty volume). Whatever the content of this empty space (a vacuum, π0 mesons...), this volumeexists. Since this global volume is hermetic regarding spacetime, the whole volume of the nucleusdeforms spacetime and, therefore, gets mass. This explains why the volume (≡ mass) of the nucleusis greater than those of its nucleons (mass excess). This also explains why the volume of a proton,938 MeV/c2, is much greater than the “matter volumes” of the three quarks, 2.3+2.3+4.8=9.4MeV/c2.

K-2 Special Volumes

These kinds of volumes are called “special” because we do not know their behavior regardingspacetime. This is the case of 6He, 8He, 14Be... For example, 11Li has a core with 3p6n and a haloof 2n. Since we do not know the penetration of spacetime inside these nuclei, these volumes cannotbe classified in any of the preceding categories.

K-3 Can we replace mass by volume?

The following questions may arise about volumes:

1/ Can we replace mass by “volume”?No, because the word “Volume” is undefined.

2/ Can we replace mass by “apparent volumes”?No, because apparent volumes are a combination of matter volumes, with mass, and massless

empty volumes. Only matter volumes (with mass) are significant.

3/ Can we replace mass by “matter volumes” or “hermetic volumes”?Yes, because there is a relation between matter volumes or hermetic volumes and mass (see the

main article and appendix D). It is obvious that if we have V = f(M) and reciprocally M = f(V ),we can use indifferently V or M in our calculus.

Appendix L

Applications and Simulations

The following document is not part of these appendices.Applications are listed without guarantee, for information only.

The subjects listed below are available in a separate PDF file. Experiments have beenimplemented by students in physics (not by PhD physicists) and the PDF has been written fornon-scientific people. It contains simplified explanations and cartoon-like figures. Therefore, thiskind of document has not its place into this scientific paper. It is proposed for information only.

These applications can be downloaded from [80] but without guarantee as they are not ascientific document. On the other hand, these applications are subject to periodical change.

Important note : This PDF [80] covers several explanations of domains of physics whichare not validated by experimentation. Therefore, some explanations are speculative, especially inastrophysics. Interpretation of results must be done carrefully.

Summary

• Variation of the mass of relativistic particle;

• Mass Excess;

• Nuclear Fission;

• Light Deviation;

• E=mc2;

• Nuclear Fusion

• Dilatation of Time - Twin Paradox;

• Galactic Clusters;

• Filaments of Galaxy;

• Neutrons Stars;

• Could Le Sage Push Gravity Explain Dark Matter?

• Black Hole Simulation;

• Additional Validation of Le Sage Push Gravity;

• Suggestions of Additional Validations.

Bibliography

[1] A. Einstein (1905), "Zur Elektrodynamik bewegter Körper" Annalen der Physik 17: 891.[2] A. Einstein (1907), "Uber das Relativitatsprinzip" Jahrbuch der Radioaktivitat 4: 411.[3] A. Einstein (1907), "On the Inertia of Energy Required by SR" Annalen der Physik 23: 371-384.[4] A. Einstein (1915), "Die Feldgleichungen der Gravitation" SPA der Wissenschaften: 844-847.[5] A. Einstein (1916), "Der allgemeinen Relativitatstheorie" Annalen der Physik 49.[6] A. Einstein (1916), "Die Grundlage der allgemeinen Relativitatstheorie" Annalen der Physik 354 (7): 769-822.[7] A. Einstein (1917), "Kosmologische und Relativitatstheorie" SPA der Wissenschaften 142.[8] A. Einstein (1920), "Relativity, The Special and the General Theory" Routledge ISBN 0-415-25384-5.[9] A. Einstein; M. Grossmann (1913), "Outline of a Generalized Theory of Relativity and of a Theory of Gravitation"Zeitschrift für Mathematik und Physik 62: 225–261.[10] A. Einstein; M. Grossmann (1914), "Covariance Properties of the Field Equations of the Theory of GravitationBased on the Generalized Theory of Relativity" Zeitschrift für Mathematik und Physik 63: 215–225.[11] A. Einstein, H.A. Lorentz, H. Minkowski, H. Weyl (1952, Original: 1923), "The Principle of Relativity: acollection of original memoirs on SR and GR" Courier Dover Publications ISBN 0-486-60081-5.[12] G. J. Stoney (1894), "Of the Electron or Atom of Electricity" Philosophical Magazine 5 38 (233): 418–420.[13] J. J. Thomson (1904), "On the Structure of the Atom" Philosophical Magazine 6 7 (39): 237.[14] H. Friedrich (2005), "Is general relativity understood?" Annalen Physik 15 (1-2): 84-108[15] Von Laue (1921) "Die allgemeine relativitatstheorie, vol. 2" Friedr. Vieweg & Sohn Braunschweig, Fig. 23,page 226[16] R. Caldwell (2004), "Dark Energy", Physics World 17 (5): 37-42[17] A. Ashtekar (1987), "New Hamiltonian formulation of GR", Phys. Rev. D 36 (6): 1587-1602[18] C. H. Brans (1961), "Relativistic Theory of Gravitation", Physical Review 124 (3): 925-935[19] K. Schwarzschild (1916-a), "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie"Sitzungsber. Preuss. Akad. D. Wiss.: 189-196.[20] K. Schwarzschild (1916-b), " Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie ",Sitzungsber. Preuss. Akad. D. Wiss.: 424-434.[21] P. Havas (1964), "Newtonian mechanics their relation to GR", Rev. Mod. Phys. 36 (4): 938-965[22] A. Komar (1959), "Covariant Conservation Laws in GR", Phys. Rev. 113 (3): 934-936[23] D. Lovelock (1972), "The 4D Space and the Einstein Tensor", J. Math. Phys. 13 (6): 874-876[24] J. A. Font (2003), "Numerical Hydrodynamics in GR", Living Rev. Relativity 6 doi:10.12942[25] R. Penrose (1969), "Gravitational collapse: the role of GR", Riv. Nuovo Cim. 1: 252-276[26] I. Shapiro (1964), "Fourth test of general relativity", Phys. Rev. Lett. 13 (26): 789-791[27] I. Shapiro (1968), "Fourth test of general relativity", Phys. Rev. Lett. 20 (22): 1265-1269226 https://archive.org/stream/dierelativitts02laueuoft#page/226/mode/2up[28] C. M. Will (2006), "General Relativity and Experiment", Living Review of Relativity 9.[29] N. Fatio de Duillier (1690a), "Oeuvres complètes de Christiaan Huygens", Société Hollandaise des Sciences, LettreN° 2570: 381–389, http://gallica.bnf.fr/ark:/12148/bpt6k778578/f388.table[30] N. Fatio de Duillier (1701), "Die wiederaufgefundene Abhandlung von Fatio de Duillier: De la cause de laPesanteur", Texte repris par Karl Bopp, http://fr.wikisource.org/wiki/De_la_cause_de_la_pesanteur[31] N. Fatio de Duillier, (1743), "De la Cause de la Pesanteur", Notes and Records, Royal Society of London (1949) 6(2): 125–160[32] G. L. Le Sage (1748), "Lucrèce Newtonien", Memoires de l’Academie Royale des Sciences et Belles Lettres deBerlin: 404–432, http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=03-nouv/1782&seite:int=0495[33] G. L. Le Sage (1748), "The Le Sage Theory of Gravitation", Annual Report of the Board of Regents of theSmithsonian Institution: 139–160, http://en.wikisource.org/wiki/The_Le_Sage_Theory_of_Gravitation[34] P. Prevost (1818), Physique Mécanique des Georges-Louis Le Sage, in: Deux Traites de Physique Mécanique,Geneva & Paris: J.J. Paschoud, pp. 1–186,http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN521099943&IDDOC=304083

Books and websites on General Relativity[50] R. Arnowitt (1962), "The Dynamics of General Relativity", Wiley[51] J. Barreau (2011), "Relativité Générale", Dunod[52] F. Belabar (1999), "Einstein, Relativité Générale, Cosmologie et Theories Unitaires", Seuil[53] L. Brillouin (1938) "Les Tenseurs", Valentin[54] Brown, Kevin (2005), "Reflections on Relativity", Cambridge University Press[55] S. Carroll (2004), "An Introduction to General Relativity", Addison-Wesley, ISBN 0-8053-8732-3

[56] J. E. Charon (1963), "15 Leçons sur la Relativité Générale", Imprimerie Firmin Didot[57] T. Damour (2005), "Einstein Aujourd'hui", EDP Sciences[58] Denis-Papin, Kauffman (1953), "Cours de Calcul Tensoriel", Eyrolles-C4pf[59] G. Efstathiou (2009), "Relativité Générale", De Boeck[60] J. Ehlers (1973), "Survey of General Relativity Theory", Reidel[61] A. Einstein (1917-1990), "La Relativité", Payot[62] A. Einstein (1916-2012), "La Théorie de la Relativité", Dunod[63] A. Einstein (1915), "Die Feldgleichungen der Gravitation", Preussische Akademie der Wissenschaften[64] J. Eisenstaedt (2002), "Einstein et la Relativité Générale", CNRS[65] E. Elbaz (1998), "Relativité Générale et Gravitation", Ellipses[66] J. Emile (1987), "La Relativité Complexe", Albin Michel[67] R. Feynman, Richard (1956), "Cours de Physique", Dunod[68] Gron, Oyvind, Hervik, Einstein (2007) "General Relativity", Springer, ISBN 978-0-387-69199-2[69] J. B. Hartle, "An Introduction to Einstein's General Relativity", ISBN 0-8053-8662-9[70] J. Hladik (2000), La Relativité selon Einstein (Ellipses, 2000), ISBN 2-7298-0124-3[71] J. Hladik (1999), Le Calcul Tensoriel en Physique (Ellipses, 1999)[72] J. Hladik (2006), "Introduction à la Relativité Générale Niveau M1", Ellipses[73] Hughston, "Introduction to General Relativity", Cambridge Univ. Press, ISBN 0-1985-2957-0[74] Landau, "The Classical Theory of Fields", ISBN 0-7506-2768-9[75] M. Ludvigsen, "Relativité Générale, une Approche Géométrique", Dunod ISBN 2-1004-9688-3[76] Misner, Wheeler (1973), "Gravitation", Freeman, ISBN 0-7167-0344-0[77] Moor, Rafi (2006), "Understanding General Relativity", http://www.rafimoor.com/english/GRE.htm[78] G. Nordstrom, Gunnar (1918) "On the Energy of the Gravitational Field", Wetenschap.[79] B. F. Schutz (2001), "A First Course in General Relativity", ISBN 9780-521-88705-2[80] http://www.spacetime-model.com/files/applications.pdf

CopyrightJ. Jerome, F. Jerome (2005…2014). The main article and appendices are the theoretical part of patents filed at INPI(National Institute of Industrial Property), under the following references:238268, 238633, 244221, 05 13355, FR 2 895 559, 248427, 258796, 261255, 268327, 297706, 297751, 297811,297928, 298079, 298080, 329638, 332647, 335152, 335153, 339797, 12-01112, FR 2 989 507.

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