Mechanical Conversion of the Gravitational Einstein's Constant

Pramana – J. Phys. (2020) 94:119 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-01954-5

Mechanical conversion of the gravitational Einstein’s constant κ

IZABEL DAVID

Institut National des Sciences Appliquées de Rennes 20, Avenue des Buttes de Coësmes, CS 70839 35708, Rennes Cedex 7, France E-mail: [email protected]

MS received 3 May 2019; revised 20 December 2019; accepted 18 February 2020

Abstract. This study attempts to answer the question of what space is made of and explores in this objective the analogy between the Einstein’s gravitational geometrical theory in one- and two-dimensional linear deformations and a possible space material based on strain measures done on the Ligo or Virgo interferometers. It draws an analogy between the Einstein’s gravitational constant κ and the Young’s modulus and Poisson’s ratio of an elastic material that can constitute the space fabric, in the context of propagation of weak gravitational waves. In this paper, the space is proposed to have an elastic microstructure of 1.566 × 10−35 mgrainsize as proposed in string theory, with an associated characteristic frequency f . The gravitational constant G is the macroscopic manifestation of the said frequency via the formula G = π f 2/ρ, where ρ is the density of the space material.

Keywords. Space–time fabric; general relativity; quantum mechanics; Young’s modulus; strength of the materials; gravitational waves; gravity Probe B; Hubble’s law; space–time curvature; Einstein’s constant; dark matter; string theory; graviton.

PACS Nos 04.50.Kd; 46.90.+s

1. Introduction is inversely proportional to the square of the radius r which separates these two masses (1). Quantum mechanics and general relativity are the twin M × m F = G × . pillars of modern physics, but while they have coexisted 2 (1) they have remained broadly irreconcilable. r In order to solve this dilemma, we must go back to Considering that this Newton’s mathematical expres- the foundations of these two theories to see if something sion of the gravitation is only a weak field simplification must not be changed at a fundamental level so as to bring of general gravitation, that an illusion of force, should them closer. we not also consider that the constant G is also an To date, the general relativity [1–3] clearly dethroned illusion disappearing with the Newton’s formula who the gravitation according to Newton. It is clear that the created it? concept of Newton’s gravitational force is in fact an illu- Just as it is necessary to abandon Newton’s formu- sion. General relativity shows indeed that two masses lation in strong gravitational field, should we not also fall against each other not because they attract each abandon G as an indivisible universal constant because other but because they follow the curvature, the defor- it is at the basis of this Newtonian formula (1)? mation of the space–time. But have we really drawn all But in this case, is it possible to reconstruct the Ein- the consequences of this conceptual error in Newton’s stein’s constant κ without going through G but by going gravitation? through a different theory? Can we separate G from the In fact, Newton’s formula is the basis of the defini- more fundamental parameters? tion of the gravitational constant since the Newtonian To answer these questions it is interesting to compare gravitational force F is proportional to the gravitational the strong and weak points of the gravitation according constant G, to the product of the masses M and m,and to Newton and Einstein.

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1 The strong points of Newton’s gravitation are G(ρ, f 2) = cte × f 2. ρ (2b) (a) It allowed the discovery, mathematically, of the planet Neptune by Urban Le Verrier. Therefore it In general relativity [1], G is introduced because it works obviously well. comes from the use of Poisson’s equation (3a) to cal- κ (b) It explains all the effects of gravity on Earth and ibrate the constant (eqs (4a)and(4b)) (analysis at the in the solar system except the Mercury perihelion component 00, time t of the different tensors of the Ein- delay (effect of strong gravitational ﬁeld near the stein’s gravitation formula): see (3b). φ Sun). Indeed, the Laplacian of the gravitational ﬁeld, , follows the Poisson’s equation The weak points of Newton’s gravitation are: φ = 4πGρ (3a)

(a) It only works if the objects have masses. It is not, and the 00 component of the metric gμν is then therefore, possible to predict the curvature of a ray of light tangent at the Sun by the effect of 2φ g ≈ 1 + . (3b) gravitation, though predicted by general relativ- 00 c2 ity and verified by Arthur Eddington on May 29, It is also surprising, in hindsight, to see that Einstein pre- 1919 [4]. cisely calibrates his equations on the time component of (b) The forces are instantaneous (therefore applied his tensors (3b), while precisely the formulae of New- faster than the speed of light) and their mode of ton (1) and Poisson (3a) that follow are independent of transmission is as yet unexplained while the spe- time! cial relativity shows that no phenomenon can be κ (4a) connects the Ricci tensor Rμν (issued of the ten- faster than the speed of light. sor contraction of the curvature tensor, a function of the (c) Its results are imprecise for the action of gravity metric gμν and of its second partial derivatives), with the in strong field. Indeed, the delay in the perihe- stress energy tensor Tμν (external mass/energy applied lion of the Mercury planet is defined exactly by at the space–time fabric) in the Einstein’s gravitational the general relativity at 43 arcsec [5] while it is field equation (5a). much weaker using the Newton’s gravitational approach. 8πG κ = (4a) (d) The formulation depends on the parameter r,and c4 if the objects of mass M and m rotate with respect 3 m 2 kg s2 s 1 − to each other at speeds tending towards the speed κ = = = = N 1. (4b) of light, the effects of the special relativity change m4 kg m Newton 4 the notion of distance for each observer. Which s value of r should be used in calculations in this The Einstein’s gravitational field equation is case? 1 (e) In this formulation, space–time is a rigid non- Gμν = Rμν − gμν R deformable object, whereas in general relativity 2 8πG μν μν it is precisely the deformation of space–time that =− T =−κT (5a) generates gravitation giving us the illusion that c4 forces act and attract objects between them. 1 1 Nm = × . (5b) m2 N m3 In view of all these points, it is therefore clear that New- ton’s concept of force is meaningless. It is an illusion. We do not show here the cosmological constant as This formulation is only a simplification of a larger the- the possible source of dark energy [7]. In addition R is ory, the general relativity. This formulation works in low a tensorial contraction of Rμν. gravity fields but is false in strong fields. Additionally, as the Newton’s expression (1) is false We can also ask ourselve, why the Universe would for the concept of force (it is the deformation of space– depends on a constant G with strange dimensions: time which gives the illusion of an attractive force inverse of a density by a frequency squared (see (2a) between two objects of mass M and m andsothere and (2b)). Some authors propose, based on this dimen- is no attractive force), and as the constant of gravitation sional equation, that G depends effectively on density is directly related to this concept of force (see Newton’s and frequency [6]. gravitation formulation), is it not necessary to abandon the proportionality factor G associated with this force G = 1/(kg/m3) × 1/s2 (2a) in general relativity? Pramana – J. Phys. (2020) 94:119 Page 3 of 37 119

Therefore, the question which needs to be asked is the R following: If Newton’s gravitation did not exist, could m,E,I κ Einstein have been able to calibrate without going M through the Newton’s limit, without using the Poisson’s M L equation and the time component (00) of its tensor, but by using directly the spatial components (1,2,3) of its tensors indicated in bold in (5c)? ⎡ ⎤ G00 G01 G02 G03 ⎢ ⎥ ⎢ G10 G11 G12 G13 ⎥ ⎣ G20 G21 G22 G23 ⎦ Figure 1. Timoshenko beam, with radius of curvature R, G30 G31 G32 G33 deflection y, loaded by two equal bending moments M. ⎡ ⎤ T 00 T01 T02 T03 ⎢ 10 11 12 13 ⎥ = κ⎢ T T T T ⎥. of 2 of the graviton in quantum field theory on the other ⎣ 20 21 22 23 ⎦ (5c) T T T T hand. Indeed, when the facet carrying the stresses makes 30 31 32 33 T T T T a complete rotation of one turn on the elastic model Or is it possible to find the Einstein constant κ by going in reality, the facet carrying the stresses on the Mohr’s through a different theory using the spatial components circle makes two turns (see Feynman lectures on grav- of the gravitational field tensors? itation – lecture 3, paragraph 3.4, figures 3.3 and 3.4 To define which theory to use, we must now explore [10]). the strong and weak points of general relativity. The So, this beam (see figure 1)hasaspanL (unit strong points of the general relativity are: m), is made of an elastic material of Young’s modulus Y = E (unit MPa = MN/m2), has a section (a) It addresses all the gaps of Newton’s formulation S = bh (unit m2), an inertia I = bh3/12 (unit m4), with an extraordinary precision. a mass m (unit kg/m) and a radius of curvature R (unit (b) It predicts the curvature of a ray of light passing m). near the Sun during an eclipse. The equivalent of the curvature equation of this beam (c) It introduces time in tensorial writing in four analogy, is the equation which defines in elasticity dimensions. The phenomena are no longer instan- (strength of the material is a simplification of this the- taneous and respect the special relativity. ory) the deflection of the beam y(x) and the rotation θ(x) (d) It predicts exactly the delay of the perihelion of of the beam sections under the influence of two external the Mercury planet, in a strong gravitational field. bending moments M applied at each end which act as (f) It no longer considers space–time as a rigid phys- the outer mass curving the beam. ical object but as a deformable object, compatible The relation to find the equivalent geodesic y(x), with observations made more than a century ago. depending on the curvature is The weak points of the general relativity are: dy(x) (a) The Einstein’s field equation is a mathematical d2 y( ) d dx dθ( ) 1 x = = x = . (6) description of the space–time deformation due to energy dx2 dx dx R density present in this space. Gravitation is therefore mathematically described by the metric of the space– As we are in pure bending, we have (M(x) = cte): time gμν. This metric is a mathematical description in four dimensions of the space–time deformation. EI M( ) = M =− . (7) But this mathematical (geometrical) description of the x R space said nothing about the physical nature and the possible mechanical properties of this deformed elastic The strain energy U (= work done by internal forces) medium constituting the space fabric [8,9]. of the beam in pure bending is If we take the analogy of a simple beam in pure bending in Timoshenko’s strength of the material, a 1 L (M)2 M2 L U = dx = . (8a) simplification of the elasticity theory, it is possible to 2 0 EI 2EI understand the fundamental concept used in Einstein’s general relativity (space–time curvature = κ energy den- In expression (8a), we see the link between the external sity) on the one hand and to assimilate the Mohr’s work, function of M on the right and the strain energy stresses circle in elasticity (see figure 12)atthespin U due to the inner work on the left. By substituting M in 119 Page 4 of 37 Pramana – J. Phys. (2020) 94:119 μν 8πG μν μν (8a) for its value given in (7), we obtain for the internal G =− T =−κ T . (10c) forces or internal strain energy of the beam: c4 1 EIL For memory, the scalar curvature R of a sphere of radius U = . (8b) / 2 2 R2 r,is2 r . And this parallelism also teaches us that it must exist as an internal work of the fabric of space– We can reformulate eq. (8b) to extract the curvature term time taking into account expression (10d); tensor not μν 1 2 U M2 developed by Einstein. We shall call M this tensor = = . (8c) (see (10e)and(10h)): R2 EI L (EI)2 1 2 W ( ) W ( ) Or to have a similar presentation to Einstein’s formal- = ext total = K ext total 2 ism, we replace 2/EI by K and we obtain R EI L L 2 U 1 U M2 = (10d) = K = . (8d) EI L R2 L (EI)2 μν 8πG μν μν μν = G =− T =−κT =−κ M . (10e) When K 2/EI, the coupling constant (8d) is between c4 the curvature, ( 1 ), on the left and the linear strain R2 Indeed, the masses/energies applied within the structure U energy density, L , on the right. The constant K is of the space bend the fabric which constitutes it. As a therefore identical to flexibility (8c). In addition, we result, its fibres compress, stretch, shear and bend. These can write the expression of the external work created actions generate an internal work, a strain energy. by the two moments on the beam. The expression of the The equation to the dimensions of the fundamental external work of moment M is principle linking curvature and strain energy density is 1 (10f) Wext(1moment) = Mθ. (9a) 2 kgm2 2 θ 1 s s2 The rotation on each support of our beam under two = × . (10f) constant moments is: m2 kg m3 m ML What we can introduce again θ = . (9b) 2EI 1 s2 U 1 1 U = × = = × . (10g) By referring (9b)to(9a) we obtain for the moment, a new m2 kg m m3 m2 N m3 expression of the external work applied on the beam: But the deflection line y(x) of the beam is not the beam M2 L itself. There is a physical object in the form of the beam W ( ) = . (9c) ext 1moment 4EI at the start! So if there is a parallelism between the concepts of strength of the material (and by extension We can express M2 according to the total external work of the beam under two constant moments M at each end: elasticity theory) and general relativity (see table 1), it must exist as an elastic substance in the space, a fabric, 2 M L a sort of frame in correspondence with the beam. W ( ) = . (9d) ext total 2EI In addition, the beam has a characteristic frequency f So we have (eigenvalue) and a material of density ρ, and so if space is like a beam in each of its three dimensions, it must also 2 2EI M = Wext(total) (9e) intrinsically have these two fundamental characteristics. L In this parallelism, there are only two major differ- That we can substitute in the expression of the inner ences: work U (8c): 1. The constant K depends on the rigidity of the 2 1 2 U M 2 Wext(total) medium and is expressed from the bending inertia = = = . (10a) R2 EI L (EI)2 EI L of the beam I and the Young’s modulus E characterising the material of the beam which is not the Thus, the equation of curvature of a beam in pure case for κ (in appearance) which is defined defini- bending (10a) can be considered as a one-dimensional tively according to G and c (non-variable constant analogy of the equation of the Einstein field in four value as a function of the space–time fabric char- dimensions (5a) (see table 1): acteristics). 1 2 W ( ) W ( ) 2. The strain energy U depends on the beam itself. = ext total = K ext total (10b) R2 EI L L In general relativity, the stress energy tensor is Pramana – J. Phys. (2020) 94:119 Page 5 of 37 119

Table 1. Parallelism between strength of the material in bending (1D) and general relativity (4D). Parameters General relativity (four dimensions) Unit (s) Strength of the material (one dimension) Unit (s) Principle Gμν = κT μν 1 1 = 2 U = 2 Wext(total) 1 m2 R2 EI L EI L m2 Curvature Gμν 1 1 1 m2 R2 m2 Strain energy density of Have to be built Mμν Nm U Nm m3 L m the space fabric External work T μν Nm Wext(total) Nm m3 L m π Proportionality factor κ = 8 G 1 K = 2 1 c4 N EI Nm2

associated with the charge (mass) applied to the do not overlap. Gravitation has not yet been quantified space, not with the space itself. According to the (independent of h) and is deterministic (not probabilis- mechanical principle, external work = internal tic as in quantum mechanics)! the graviton, boson vector work (see 10d), we can reconstruct the bridge of the force of gravity and keystone of quantum gravi- between the two approaches. Indeed, if the anal- tation in connection with the quantum field theory has ogy with the beam in 1D (or plate theory in 2D) is not yet been measured. exact, it must exist as a tensor of stresses/(normal Nevertheless, we can quote string theory with strings efforts, bending and torsion moments and shear open or closed of 1.0 × 10−35 m length and quantum loads) associated with the curvature of the space. gravity that try to solve this problem. In this case, expression (5a) will become, with a Based on the analogy of the beam in pure bending in mechanical stress tensor acting within the space– one dimension, it seems natural to consider an elastic time fabric (interior work) (10h)andκ a coupling substance to associate with the space fabric. Conse- ⎡ constant: ⎤ quently, a special elastic material should constitute the G00 G01 G02 G03 space described in the Einstein’s field equation [8,9]. ⎢ ⎥ But, do we have solid proofs of this elastic material ⎢ G10 G11 G12 G13 ⎥ and of this space behaviour in weak field which would ⎣ G20 G21 G22 G23 ⎦ allow us to study the space following the elasticity the- G30 G31 G32 G33 ⎡ ⎤ ory? The answer is yes and it will be studied in more σ tt τ tx τ ty τ tz detail in §3. ⎢ τ xt σ xx τ xy τ xz ⎥ =−κ⎣ ⎦, (10h) Our aim is to explore these issues in five different τ yt τ yx σ yy τ yz ways simultaneously by first considering the time and τ zt τ zx τ zy σ zz space separately and then reassembling them at the end where σ μν are the normal stresses in the space–time of the article as a second step: fabric and τ μν are the shear stresses in the space–time fabric both created by the curvature of the space fabric – Consider that space is made of an elastic substance/ under the external masses. material, The constant of proportionality κ is the same as we – calibrate κ using the spatial components rather than consider the work of the internal forces (Wint = U)or the temporal components of the Einstein’s tensor. the work of the external forces (Wext), that is to say (K , Thus, the temporal part is studied separately from the κ). Later in our reasoning, we shall progress by con- spacial part of the gravitation field tensors (see §10): sidering U and therefor the reader must keep in mind use the elasticity theory, the strains measured on the expression (10d) connecting the external work to the interferometers Ligo and Virgo, the concept curva- inner work. We shall therefore make the shortcut asso- ture = K energy density, the gravitational waves in ciating T with U. weak field, (b) There is a fracture in the quantum field theory – replace G by mechanical and physical parameters, that allows to describe the standard model, the vacuum – build a link with the quantum field theory by using energy, the Casimir force and general relativity. The gen- the vacuum energy property to physically calibrate eral relativity wonderfully explains the mechanics of the G and κ, infinitely large and the quantum mechanics that of the – conclude on the possible new characteristics of the infinitely small. The problem is that these two theories space material. 119 Page 6 of 37 Pramana – J. Phys. (2020) 94:119

Moving on, we shall propose a new mechanical and of the Einstein constant κ based on longitudi- physical interpretation of the coupling constant κ in nal, torsional and shear waves in the medium order to provide a different perspective on the under- [12,13]. standing of the vacuum and of the potential material (6) Proposition of numerical values of different constituting the space elastic medium. By handling the parameters of these new expressions of the grav- nature of time separately from the nature of the space itational constant G and κ from the calculated material it is hoped to contribute to the building of a vacuum energy. bridge between the general relativity, the quantum field (7) Checking the orders of magnitude of different theory and the string theory. parameters of these new expressions from the constants of physics. (8) Proposition of the Young’s modulus value E = Y and Poisson’s ratio ν for the space medium. 2. Methods (9) Proposition of a method for measuring the Young’s modulus of the space material from the Casimir The following methodology has been used focussing on effect and the possible shear effect in the plane of the space part of the gravitational field equation with the interferometers. time being treated separately (see [11]): (10) Possible reformulation of the Einstein’s gravity field equation from this new approach to the con- (l) Analysis of the behaviour of space from the stant κ. measurements made for more than 100 years in (11) Consequences of the nature and characteristics of general relativity and quantum mechanics (vac- the space material constituting the microscopic uum energy) deducing an elastic behaviour. fabric of the Universe (quantification of space and (2) Research of the elastic theory which capture this time). behaviour: Hooke’s law, Timoshenko’s bending theory, beam, plates. Use analogy between the spi- ral form of the galaxies, the whirlwinds forming 3. Analysis of the space behaviour from the during cyclones or whirlpools on the sea as seen measurements made for more than 100 years in during the tsunami of 11 march 2011 in Japan and general relativity and quantum mechanics the potential presence of a space-material fluid (vacuum energy) at low speed, behaving for objects (gravitational waves) moving at the speed of light like an equiv- The Einstein’s gravitational field equation (5a) has been alent solid elastic material (see [8,9]). well verified experimentally. Indeed, all the gravitation (3) Demonstration, in weak gravity field, of the par- tests and measurements carried out for more than 100 allelism between the expression of the Einstein’s years constitute proofs which confirm that space is an gravity field (curvature = κ× energy density) in elastic deformable physical object, which is not made four dimensions and the expression obtained in of ‘nothing’ but that it is filled with ‘something’, a field, elasticity/strength of the materials in two dimen- a material that is elastic, [8,9]. Indeed sions (curvature = K × the elastic strain energy density of two linear elements as the interferome- (a) The acceleration of the expansion of the Universe ter arms): use elasticity theory and assimilate each has been observed [14]: the galaxies move away void volume of space inside the interferometer arm from us faster when they are away from us (Hubble’s as a harmonic elastic oscillator made of vacuum law (11)). In other words, the galaxies are ‘motion- full of an equivalent virtual elastic material. less’ and it is the space between them that dilates like (4) Application of the analogy obtained to the space dots on a balloon, which is inflated. So, the space material included in the two orthogonal tubes of fabric is an elastic body that stretches. / the Ligo Virgo interferometers: demonstration of v = H d, (11) the parallelism between the Einstein’s field equa- 0 tion (tensorial equation in 4D) and the strain field where v is the recessional velocity of the galaxy, H0 of these two tubes like that of a strain gauge (ten- is the Hubble constant and d is the distance of the sorial equation in 2D) by using the work principle, galaxy from the observator. Wexternal = Winternal. (b) The rays of light tangent at the Sun are curved by (5) Deduction of this parallelism, via the elastic the deformation of the space around it. The light rays behaviour of the space, of a new mechanical coming from stars follow the curvatures of an elastic expression of the gravitational constant G and body deformed by the present mass ([4] Eddington Pramana – J. Phys. (2020) 94:119 Page 7 of 37 119

in 1919 measured the deflection angle of the stars From the aforementioned observations, it seems that: around the Sun during an eclipse of 1.75). (a) The space fabric appears to be physically made up (c) The space is elastic. As the Sun moves, the space of ‘Something’ since the galaxies are trained by this curvature in its wake disappears and becomes flat. ‘Something’ that is expanding accelerated. The rays of light become straight again. (b) The space bends (analogy with the notion of cur- (d) The rotation of the Earth twists the space fabric vature in elasticity theory see (6)) in the presence around it, indeed gyroscopes placed in orbit at 400 of energy density, mass. The sum of the angles of a km around the Earth are deflected by this space ◦ triangle in the presence of mass is no longer 180 rotation/torsion/curvature (see Probe B experiment (see [8,9]). [15]. Thus, a horizontal angle of 0.000010833◦ has (c) The space fabric is deformed and transmits gravita- been measured from the gyroscopes placed in vac- tional waves by elongations and shortenings. So, uum in accordance with the prediction of general small strains (interferometers) or angles (Gravity relativity). Probe B) are measured when we place it sufficiently (e) The space deformations produced by the coales- far from the said mass/energy (weak field princi- cence, for example, of two black holes in the form ple). of gravitational waves [16–18] were measured for (d) The space is elastic, the curvature disappears when the first time by the Ligo interferometers in 2015 the mass that created it disappears. (official announcement of GW150914 detected by (e) A black hole is a type of yielding of space loaded to Ligo on February 11, 2016). Other observations the extreme [12,13]. followed, including a fusion of neutron stars (pul- (f) In the application of elasticity theory, space is sars) GW170817 also observed in electromagnetic proposed to have a quantified microstructure with radiation. So the interferometers have effectively an associated characteristic frequency f . G is the measured signals which are moving at the speed of macroscopic manifestation of the said frequency light c inside the frame of the vacuum space medium. (see eqs (2a)and(2b)) So, if there are strains, then there is an elastic physical body that is deformed and we measure these deformations. Consequently, following the elastic- 4. Research of the elastic theory which approaches ity theory, the material of the physical body can the space behaviour: Hooke’s law, Timoshenko’s be characterised by the Young’s modulus E and theory Poisson’s ratio ν. A strain tensor εij can thus be established from the space strains δL/L measured 4.1 Hooke’s law in the arms of the interferometers. The magnitude − of these strains is 10 21. Mechanically speaking, All these elements suggest that these strains measured are in the principal direction. (a) An elastic material constitutes the space (E, ν). They correspond to the transverse elastic waves of (b) Space is proposed to have a quantified microstruc- the medium with a propagation direction perpen- ture of radius r with an associated characteristic dicular to the plane build by the two arms of the frequency f . interferometers. The formulas below describe the (c) The Hooke’s law applies to space medium. space metric gij built from hij (12b) a disturbance of the flat metric ηij (12a) in weak gravitational field See T Damour’s book “if Einstein was told to me, and the link with the strain tensor εij: chapter 3, the elastic space–time”, where Einstein’s equation is simplified by D(g) = K ·T where D is a gij = ηij + hij = ηij + 2εij (12a) deformation tensor, T is a tensile tensor and K is a fac- hij = 2εij (12b) tor of proportionality δ δ Consequently, it seems logical to apply the elasticity L i −21 1 = = 10 = εij = hij. (12c) parameters to the deformable and elastic medium that L L 2 constitutes it. This is reflected mathematically by the two (f) And finally, we can consider that the space rings expressions below relating to the normal stresses σ (13) like any elastic material according to the study con- and tangential stresses τ (14) connected respectively to ducted by Ringermacher and Mead [19]. the strain ε by the elasticity modulus E (the so-called Young’s modulus) and to the shear strain (angle γ )bythe Based on all these points, it seems logical to consider shear modulus μ (some times noted as G) (see [22,23]). that space is made of a strange elastic material, a new δl σ = εE = E (13) type of Ether as explained by Einstein himself [20,21]. L 119 Page 8 of 37 Pramana – J. Phys. (2020) 94:119

E τ = γμ= γ . (14) or 2(1 + ν) σij = 2μεij + λεkkδij. (16d) It is important to see that With the Lame coefﬁcients: δL , (a) at strain, L is associated with Young’s modulus E σ μ = E and normal stress , ( + ν) (17) (b) at the shear strain, (angle γ) is associated with a 2 1 μ ν Eν shear modulus , Poisson’s ratio and shear stress λ = . (18) τ. (1 + ν)(1 − 2ν) ε This approach of general relativity by the elasticity the- In this expression, kk is the trace of the strain tensor, δ ν ory is not new and has been studied by many researchers ij is the Kronecker symbol and is the Poisson’s ratio. (see [11,24–48]). We review now the main points concerning, in particular, the relationship between the stress 4.3 Relationship between the elastic strain tensor and tensor and the stress–energy tensor (§4.2) on the one the metric tensor in weak gravitational ﬁelds hand and between the metric tensor and the strain tensor (§4.3) on the other hand. 4.3.1 Case of elasticity in four dimensions. Following [11], Chapter 2.6, formula 2.9, we have in four dimen- 4.2 Similarity between the stress tensor in elasticity sions: and the stress–energy tensor in general relativity gμν = (ημν + hμν) = (ημν + 2εμν). (19)

The Einstein’s gravitational field formula (5a) can also The metric tensor gμν in a weak field is therefore equiv- be expressed in the following form: alent to the metric in flat field to which is added a perturbation which is only twice the strain tensor εμν. 4 μν c μν 1 μν ε T =− × R − g R . (15a) The general structure of the strain tensor ij in the the- 8πG 2 ory of three-dimensional elasticity is given in (20a)and (20b). The equation to dimensions is ⎡ ⎤ 2 ε ε ε kg m m 4 xx xy xz 2 Nm kg 1 kg m ⎣ ⎦ s = = = s × = εij = εyx εyy εyz . (20a) 3 3 2 3 2 2 m m ms m m s εzx εzy εzz kg s2 1 With the displacements ui and u j we have × . 2 (15b) m 1 ∂ui ∂u j εij = + . (20b) Or in a compact form, it becomes 2 ∂x j ∂xi Nm N 1 = = N × . (15c) 4.3.2 Case of gravitational waves m3 m2 m2 We then notice that the energy density has the same 4.3.2.1 Theoretical aspects. The linearised form of the dimension as the stress in Newton’s classical mechan- Einstein’s equation in weak gravitational fields is, see ics. Moreover, it is possible to demonstrate that the (21a)and[50]: stress–energy tensor (16a) is like stress tensor (16b) λ 16πG by replacing in it velocities vi with four velocities ∂ ∂λhμν = h =− Tμν. μν 4 (21a) uμ and moving from a three-dimensional space to a c four-dimensional space–time of density ρ (see [49]and In vacuum (case of the gravitational waves), we have Appendix A). λ ∂ ∂λhμν = hμν = 0. (21b) Tμν = ρuμuν (16a) The d’Alembertian wave operator, σij = ρvi v j . (16b) = + 1η The stress tensor σij in three-dimensional elasticity is hμν hμν μνh (22a) 2 written depending on the strain tensor εij and two con- σ hμν = Aμν cos kσ x (22b) stants according to the Young’s modulus of the medium Y = E and the Poisson’s ratio ν: σ ω k = ; k quadri vector wave of the plane wave, E ν c σij = εij + εkkδij (16c) σ (1+ν) (1 − 2ν) x = (ct, x, y, z) (23) Pramana – J. Phys. (2020) 94:119 Page 9 of 37 119 2 ω2 k = (24) c2 ω is the circular frequency of the wave. ⎡ ⎤ 0000 ⎢ 0 +100⎥ Aμν = A+⎣ ⎦ 00−10 0000 ⎡ ⎤ 0000 Figure 2. Example of particle coordinates subjected to a gravitational wave polarised A+ particles propagating per- ⎢ 00+10⎥ +A×⎣ ⎦ (25) pendicular to the plane xy. 0 +100 0000 ⎡ ⎤ 1000 The particle position measured from the centre of the ⎢ 0 −100⎥ circle is ημν = ⎣ ⎦ (26) 00−10 ω 2 2 2 000−1 l = R − R A+(cos(2θ)) cos (ct − z) , (29) c h is the trace of hμν and where l is the final length after deformation, z is the θ h =−h. (27) direction of the wave propagation, t is the time, is the angle between the abcissa x and R, the radius of the cir- Using (22a), to have a new expression of (21a) function cle where the particles are positioned, ω is the circular of εij, we obtain a new expression of Einstein’s gravi- frequency, c is the speed of light, A+ is the first wave tational field (28a) polarisation and A× is the second wave polarisation. With the metric gμν (19) and dimensionless perturba- λ 1 1 ∂ ∂λ hμν + ημνh = hμν + ημνh tion hμν: 2 2 16πG For a polarised wave A+: =− Tμν. (28a) c4 ⎡ ⎤ 0000 By replacing hμν by 2εμν (see (12c)) in (28a) we obtain: ω ⎢ 0 +100⎥ hμν = A+ cos (ct − z) ⎣ ⎦. c 0 0 −10 λ 1 1 ∂ ∂λ 2εμν + ημν2ε = 2εμν + ημν2ε 0 000 2 2 (30a) 16πG =− Tμν. (28b) c4 For a polarised wave A×: After simplification by 2, we obtain the same constant ⎡ ⎤ κ 0000 in weak field: ω ⎢ 0 0 +10⎥ hμν = A× cos (ct − z) ⎣ ⎦. λ 1 1 c 0 +100 ∂ ∂λ εμν + ημνε = εμν + ημνε 2 2 0 000 8πG (30b) =− Tμν. (28c) c4 The spatial part of hμν is indicated in bold. l2 is the final The formulation of particle position variations arranged length squared and R2 is the initial length squared. along a circle during the passage of a gravitational wave We can calculate the variation in length due to the perpendicular to the plane xy, allows to find expression displacement of the particle. (29) by considering, for example, polarisation A+ (see figure 2). 2 − 2 ( )2 − ( )2 l R = final length initial length The movements of these particles correspond always R2 (initial length)2 to pure compression or pure traction of the space ω medium inside the interferometric tube. The deforma- =−A+(cos(2θ)) cos (ct − z) , (31a) c tion of the circle containing the particles is identical but ◦ rotates by 45 according to the type of polarisation (A+ where L F = Li + δi is the final length, Li is the initial or A×). length and δi is the length variation. 119 Page 10 of 37 Pramana – J. Phys. (2020) 94:119

Figure 3. Torsional waves created by a binary system in rotation.

With δi Li At the end of this chapter, with the links between the σ , 2 − 2 2 − 2 2+ δ + δ2 − 2 tensors Tij and ij on the one hand and between gij hij l R L F Li Li 2Li i i Li = = . and εij on the other hand, and taking into account the 2 2 2 (31b) R Li Li common principle curvature = K × the energy density, With δ2 δ we have all the bridges necessary to interpret general i i relativity according to the elasticity theory. l2 − R2 2δ =∼ i 2 (31c) R Li 4.3.2.2 Interpretation of the results of the calculation of and with the strain definition general relativity on gravitational waves in weak field − + δ − δ from the angle of elasticity theory. We consider that ε = L F Li = Li i Li = i . (31d) the rotation of a binary system (like two black holes, for Li Li Li example) creates a sort of ‘torsional waves’, in the space So finally we obtain fabric (see figure 3 and [3,12,58]). 2 − 2 2 − 2 δ Indeed, this point was presented and demonstrated l R L F Li ∼ 2 i = = = 2ε. (31e) by Professor Kip Thorne (Nobel Prize 2017) during his R2 L2 L i i conference on March 6, 2018 at UCI USA [12] “explor- To correlate with the Hooke‘s law (13) ing the universe with gravitational waves from big bang 2 2 to black holes”. At this conference accessible on the l − R ∼ ω ∼ = −A+( ( θ)) (ct − z) = ε. 2 cos 2 cos 2 net, he expains with numerical simulations based on R c the general relativity, the behaviour of two black holes (31f) during their coalescence. In fact there is the creation So, the perturbation hμν and consequently the metric gμν of two types of vortices (turning to right, turning to are close to 2εμν. So, in weak gravity field we demon- left) emerging from rotating black holes, dragging the strate eq. (19) and the relationship between the metric space medium around them, which merge to create a tensor and the strain tensor well. Thus, the metric tensor ring which makes these swirl mixtures with compres- in the general relativity approach in weak field is assim- sion and tensile tendencies measured by Ligo and Virgo. ilated into elasticity to the flat metric tensor to which the The consequences are that gravitational waves are not strain tensor is added twice (12a). a classical shear wave but a mixture of vortices whose Pramana – J. Phys. (2020) 94:119 Page 11 of 37 119

Figure 4. Movements of the laser mirror inside the interferometric tubes formed by the gravitational transversal wave. results are compression/tensile tendencies measured in the interferometers. If we compare the results of the general relativity in weak field (hμν) with the elastic strain tensor (εij) (see eq. (19)and[10]) we can conclude on the deformation states of the elastic medium in the xyplane of the arms of the interferometer during the passage of a gravitational wave coming from the z direction. Kip Thorne explains also in [13] the geometrodynamics of the space–time made of warped space. First point: Therefore, based on expressions (30a)and(30b), it is known that there are two clearly separated types of polarisation of the gravitational wave produced by the coalescence of two massive objects rotating relative to one another: A+ and A× (see [10]). δL γ Second point: Figure 5. Strains L measured and shear strain (angle ) not measured on the interferometers Ligo and Virgo to this day. By the relation between hμν and εμν (see (12c)and(31f) and [11]), there exists for each polarisation of hμν the equivalent of an associated strain tensor (see [10]): deformation state of the space layers (multisandwich) (a) Based on (30a), (31f)and(12c) associated with perpendicular to the direction z of the gravitational wave the polarisation A+ we have (see figure 6). It also proves the elastic behaviour of ⎡ ⎤ space in perfect correlation with the elastic theory and εxx 00 consequently the existence of an equivalent elastic mate- ε = ⎣ −ε ⎦. xy(A+) 0 yy 0 (32a) rial in the space vacuum. This fundamental research also 000 proves that it is possible to unify the theory of elastic- This state of deformations is obtained in the tubes of the ity with general relativity and thus that it is possible to interferometer by analysing the forward and backward define an elastic material constituting space. motions (δL) or strains (δL/L) of the laser mirrors in Third point: the two tube sections (see figure 4). (b) Based on (30b), (31f)and(12c) associated with By the theory of elasticity, therefore, there exists, for the polarisation A× we have: each deformation tensor, a stress tensor (see [10]): ⎡ ⎤ (a) On the basis of (32a)and(30a), associated with ε 0 xy 0 the polarisation A+, the corresponding stress tensor of ε = ⎣ ε ⎦. xy(A×) yx 00 (32b) pure compression/traction of the space medium in the 000 xy plane is ⎡ ⎤ This state of deformations should be obtained in the σxx 00 tubes of the interferometer by analysing the lateral ⎣ σ ⎦ σxy(A+) = 0 yy 0 . (33a) movements of the laser mirrors in the two tube sections 000 (see figures 4 and 5). These two deformation states (32a)and(32b) cer- (b) On the basis of (32b)and(30b), associated with tainly prove that hμν corresponds to a pure shear the polarisation A×, the corresponding stress tensor of 119 Page 12 of 37 Pramana – J. Phys. (2020) 94:119

Figure 8. Normal stresses and shear stresses measured on interferometers as a function of their orientations on a space cylinder in torsion.

According to their orientations, the interferometers do δL not measure anything (Case a) or strains L (Case b) as seen in ﬁgure 8 and [54]. Fourth point: Each of its stress states corresponds to a wave velocity characteristic of the elastic medium measured in the interferometric plane by the longitudinal oscillation of the laser mirror.

(a) Associated with the polarisation A+, a pure longitudinal tensile compression wave velocity in each x or y direction is measured by the laser mirror of Figure 6. Plane deformations formed by the transverse grav- the Ligo and Virgo interferometers connected to a itational wave. tensor according to the expression (33a) (see [53]). ∂2u( , ) 1 ∂2u( , ) x t − x t = 0. (34a) ∂x2 c2 ∂t2 The Alembert equation is ∂2u(x, t) ρ ∂2u(x, t) − × = 0. (34b) ∂x2 E ∂t2 Comparing (34a)and(34b), we have the well- known equation ρ 1 = Figure 7. Creation of normal stresses by the combination of 2 (35a) shear stresses. c E E c = . pure shear of the space medium in the xy plane is ρ (35b) ⎡ ⎤ τ = σ 0 xy xx 0 (b) Associated with the polarisation A×, a pure shear ⎢ ⎥ ⎢ ⎥ wave velocity, not yet measured by the laser mirror σ ( ) = τ = σ 00. (33b) xy A× ⎣ yx yy ⎦ of interferometers Ligo and Virgo (lateral move- 000 ments of the mirrors (see figures 4 and 5), connected to a tensor according to expression (33b) and a tor- The two stress tensors above, according to the ori- sion torque [54]. entation of the facet considered (see figure 8), are characteristic of a pure shear associated with a pure Indeed, if we consider that the fabric which constitutes torsion of space following the rotation of two massive space has a dynamic behaviour similar to the one we objects which merge. Figure 6 shows the plane defor- have on Earth in the event of an earthquake but with mation of the space created perpendicular at the wave only transverse waves, we only need to consider shear direction with the gravitational wave as a transverse wave S instead of S and P (pressure) waves. Indeed, in wave. line with what we measure, only strains in the plane xy Figure 7 shows how a combination of shear stresses perpendicular to the direction of the gravitational wave τ create normal stresses σ. are seen (see figures 4–6). Pramana – J. Phys. (2020) 94:119 Page 13 of 37 119

So, the following expression (36) of elasticity theory direction of propagation (z). A shear with a transversal applies, with ui as the component i of the displacement wave of velocity cshear: vector u, t as the time, σij as the stress tensor and ρ as the mass density (see formula (3.10), and (3.11), chapter μ E c = = . (41) 3.3 of [11]): shear ρ 2(1 + ν)ρ 2 ∂ u ∂σij This hypothesis of shear waves is also strictly speak- ρ i = . (36) ∂t2 ∂ j ing not acceptable, because the strains measured on the interferometer are not shear strains (angles γ ) linked at δ With for the strain tensor εij: L shear waves but, L , strains that are always linked at 1 eventual compression waves; but we shall consider it εij = ∂i u j + ∂ j ui (37) because it is possible that shear strain exists and is not 2 already measured. The classical elastic wave motion is so with (16c)in(36): So strictly speaking, the gravitational waves can- not be assimilated to classical elastic waves in an ∂2u −→ −→ ρ = (λ + 2μ)∇ (div(u)) − μrot rot(u) elastic medium. They have the particularity to create ∂t2 strains perpendicular to the wave propagation direc- + fexternal (38a) tion (transversal shear wave characteristics) but with 2 / ∂ u elongation and shortening (compression traction wave ρ = (λ + 2μ)∇(div(u)) − μ∇(div(u)) characteristics); see [12,13]. The proposition of a space ∂t2 medium made up of a multisandwich of thin sheets +μ () + . u fexternal (38b) sheared perpendicularly to the direction of propagation After calculation of the gravitational wave therefore seems a reasonable hypothesis (see figure 11). ∂2 u ρ = (λ + μ)∇(div(u)) + μ(u) + fexternal ∂t2 4.4 Determination of the Poisson’s ratio intensity (38c) We can conclude from these wave speed equations, on or again a potential value of the Poisson’s ratio ν. ∂2 u 2 First approach: Analysis of the particle movements ρ = (λ + μ)∇ ∇•u + μ∇ u + fexternal, ∂t2 on a circle under a gravitational wave (see figure 2) (38d) The analysis of figure 2 from the calculation of general xy where relativity shows that an object on the plane positioned perpendicular to the z direction of the propagation → ∇ u = grad(u) (38e) of gravitational transverse waves, is simultaneously compressed in one direction and stretched in the perpen- ∇ 2u = (u). (38f) dicular direction. The strains are equal but of opposite sign: εxx =−νεyy. When fexternal = 0, the solution of the equation follows With the definition of Poisson’s ratio we have the Helmholtz’s decomposition that gives two waves relative transverse shrinkage that propagate in the elastic medium: v = = 1. relative longitudinal elongation u =upressure,longitudinal +ushear,transversal. (39) Second approach: In the z direction, the gravitational A pressure with a longitudinal wave of velocity cpressure: wave is a transverse wave and not a compression wave λ + 2μ c = . pressure ρ (40) There is no compression wave perpendicular to the plane of the interferometer. So eq. (40) must be equal to This hypothesis of compression waves in the direction 0. With eqs (17)and(18) we obtain ν = 1again. z is not acceptable because in this case the strains are Third approach: Based on current data (see [11]) in the same direction as the propagation of the waves. Of course this is not the case for gravitational waves Based on the results of ref. [11] and following eq. (31) where the strains are in a plane perpendicular to the wave we have, the Young’s modulus E = Y = 4.4×10113 Pa 119 Page 14 of 37 Pramana – J. Phys. (2020) 94:119

(see §3.4, formula 3.13 [11]) and density ρ = 1.30 × between the strength of the material and the general rela- 1096 kg/m3 (see §3.4, formula 3.14 [11]). tivity about the principle: curvature = K energy density on the one hand (see Introduction) and the transversality E ν = − 1. (42) of the physical terms (curvature, metric, strain energy, 2ρ 2c external work, stresses, strains, coefficient of propor- With eq. (42) and the results of [11]weget tionality K and κ) between the general relativity and the elasticity theory on the other hand (see §4.2 and §4.3). ν = . . z 0 8829 The parallelism between the general relativity and the strength of material formulas on the principle curvature This last value of the Poisson’s ratio remains accept- = able taking into account the uncertainty on the intensity K energy density is as follows (see “Introduction”): of the vacuum energy (very important according to the quantum field theory and very low according to the value measured in the vacuum space) which is the object of many discussions within the international scientific community. The transversality between general relativity and the strength of the material on the key parameters is as fol- ν = Conclusion: We retain for the Poisson’s ratio: 1. lows: (link between εij, hij and gij (see (31f), (12c)and (19)) and between Tμν and σij (see (16a)and(16b)and 5. Highlighting the parallelism and differences Appendix A): between the strain energy density in elasticity and the mass energy density in general relativity

5.1 The strain energy density in elasticity

5.1.1 Strain energy in general. Thestrainenergyden- From this analysis, it is clear that the constant κ in sity in elasticity is [22] 8πG this case, c4 would be closer to mechanical constant of the space medium function of the Young’s modulus = 1σ εij. Uij ij (43a) E and the Poisson’s ratio ν. In other terms κ should be 2 (1+ν) proportional to E . By introducing in (43a) the expression of the stress ten- To finalise the construction of parallelism between sor (16c), we obtain the following expression of strain general relativity and the theory of elasticity (or the energy density of an elastic body: strength of materials which results from it) and to find ν a mechanical transposition of the Einstein’s constant E ij Uij = εij + εkkδij ε (43b) κ κ 2(1+ν) 1 − 2ν , we must now try to find the expression of from the curvature = K energy density principle expressed or from simple equations of strength of the materials. The ν ( +ν) ij 2 1 comparison of the two formulas (5a)and(21a) with εij + εkkδij ε = Uij 1 − 2ν E the formula (43c), terms to terms, allows to compare K κ ( +ν) and and to identify the mechanical correspondences 1 ij ν = σijε . (43c) between (1 + )/E and the parameters G and c). E The examination of (43c) shows that these equations In plate theory there are relations between the strain ten- will be of the form sor and the curvature tensor (see eqs (78)–(81)) which f(ε ) = K +ν U. (43d) ij2 1 brings us closer to Einstein’s formalism (see 5a)ifwe E consider that the external work produced by the external masses applied on the space fabric is equal to the inter- 5.2 Proposition of a tensor equation between nal work of the space fabric curved by these masses. A curvature and space strain energy based on the strains tensorial space approach is also developed in [42]. measured on the interferometers Ligo and Virgo

5.1.2 Consequences of the parallelism between the 5.2.1 Study of a horizontal space cylinder in one direc- elasticity theory and the general relativity. Thus, to tion solicited by a gravitational wave – without the effect build a bridge between the elasticity theory and general of Poisson’s ratio – use of the longitudinal velocity relativity formalism, we have studied the parallelism of the correlated compression/traction wave with the Pramana – J. Phys. (2020) 94:119 Page 15 of 37 119 compression/traction stress tensor in the interferometric tube. We assume in this section that the coalescence of two black holes, for example, creates by their rotations, a torsion of the space as shown in figure 3 and ref. [12,13]. This twisting of the space sheets creates, in the succes- Figure 9. Tube loaded by a normal force N. sive xy planes, tensiles and compressions of the space material sheets (multisandwich) that finally arrive on the Earth in each arm xy of the interferometers (see fig- The Hooke’s law (13) can be written as a function of ures 6 and 11). In weak field, for space only, the metric the displacement u(x): is given by (12a). Einstein’s gravitational equation in a N u(x+dx) − u(x) δL weak field for space is (21b). The result is for a polarised σxx = εE = = E = E, wave A+, the tensor given in (30a). S dx L (44a) Note: Wechoose this polarisation because it corresponds to the displacements measured in the interferometric where σ is the normal stress in N/m2, ε is the strain in arms (compression /traction of the volume which create ε = N %, ( ES) is measured by Ligo and Virgo, N is the the advances and retreats of the laser mirrors). normal force in Newton, S is the tube section in m2, L With formula (31f) demonstrated in §4.3.2.1, we have is the tube length in m, V = S × L is the tube volume a link between the general relativity (disturbance of the in m3, E = Y is the Young’s modulus of the material ) spatial part of the metric hij and the theory of elastic- constituting the tube in N/m2, δL is the length variation ity (the strain ε of the elastic medium) (see eq. (12c)). ij in m under the normal force N, u(x) is the longitudinal So, based on (31f)and(12c), at the spatial perturba- displacement in m along the longitudinal axis of x. ( ) tion of the metric, hμν(A+) corresponds to the strain The stress–displacement relation as a function of ε tensor ij (32a). Thanks to the Hooke’s and elasticity for- rigidity K = ES/L, is written as follows: mula, it corresponds to the strain tensor εij (32a) a stress tensor σ (33a). This type of stress tensor (33a)rep- ES ij N = δL = K δL. (44b) resents a longitudinal pure compression/traction wave L (35a) associated with the normal effort N (see figure 9 The strain energy U (N m) of the tube in static, when N and [10]). is constant and taking into account (44b), is This section considers therefore a tube of a Ligo/Virgo type interferometer of length L and section S loaded 1 L N 2 1 N 2 with the normal force N as defined in figure 9.Inside U = dx = . (45a) 2 ES 2 K the tube, it is considered that the vacuum consists of a 0 space elastic substance made from very small particles The strain energy of the stiffness spring K (N/m) can be to constitute a granular substance, fluid whose granular- written by substituting N by expression (44b)in(45a): ity of quantum dimension r. 1 U = K (δL)2. (45b) Note: In this simplified approach we deliberately sep- 2 arate the correlation between the x and y directions of the tubes seen in figure 2. This was done in order to Moreover, from the strain definition, the displacement see already if the basic principle curvature = K × the variation δL is energy density is respected on the one hand and with ε × L = δL. (46) the eventual stresses on the perpendicular direction of the transverse wave to be in correlation with the stress By introducing (46) in expression (45b)ofthestrain plane state on the other hand. The consequence is that energy, we obtain in this section we do not take into account the Poisson’s ratio to be in agreement with the information of the ten- 2 U (ε)2 = . (47a) sor considered here (see eq. (33a)). K L2 Since there are displacements of the laser mir- Starting from the rigidity K depending on length L,the ror in the direction of the tube corresponding to the Young’s modulus E, the tube section S and substituting compression/traction of the space medium (see figure 4 the expression of L by K in (47a), we obtain and [53]), there are strains and stresses in this frame- work and therefore a dynamic normal force N and a K U / (ε)2 = 2 . (47b) pure compression traction wave inside the tube. E2 S2 119 Page 16 of 37 Pramana – J. Phys. (2020) 94:119

Considering the tube volume V we extract the section By substituting (49b)in(53e) we obtain S: 1 (ε)2 2 V L = T = S. (48) K 1 (54a) L 2 V E2 By substituting the expression of S (48)in(47b), we or in an equivalent form: obtain the strain energy density U/V : 1 K 1 1 K 1 U (ε)2 = 2 T. (54b) (ε)2 = 2 × (49a) L2 V E2 L2 V E2 V or By substituting V (53c) in expression (54b)weget 1 (ε)2 ω2 U 2 1 2 = L . (49b) (ε) = 2ρ × T. (54c) V K 1 L2 E2 2 V E2 By substituting the circular frequency ω (53b) by fre- We now consider the tube under a dynamic behaviour quency f in (54c), we obtain and with a pure longitudinal wave compression/traction following x in the arm as a consequence of polarised 1 f 2 ρ (ε)2 = 8π 2ρ × T. (54d) gravitational waves A+. We note the density as L2 E2 m ρ = . (50) In the plane of the interferometer, the movements of V the mirror are perpendicular to the tube section (see The fundamental dynamic equation allows us to find figure 3). Thus, in the plane of the interferometer, the the eigencircular frequency (harmonic oscillator) of the mirror follows the compression/traction movements tube made of the space material: of the spatial material inside the volume of the tube δLx (, , in connection with , u(x),). These oscillations 1 2 1 2 Lx Uc + U = mx˙( ) + Kx( ) = E = 0, (51a) 2 t 2 t 0 in the plane of the interferometer are therefore pure compression/traction longitudinal waves correlated to where U is the kinetic energy. c the transversal waves perpendicular to the plane of the By a derivative with respect to t we have interferometers (see [12]and[53]). So in this section, K following the type of tensor considered (33a), the veloc- x¨( ) + x( ) = 0 (51b) t m t ity of the wave is given in formula (35a) and the Young’s = = ρ 2 and of course following the Newton’s force definition modulus E Y c . The speed is limited because it propagates in an elas- Force = mx¨ =−Kx (52) tic medium of density ρ. A similar experiment would be which allows us to express the circular frequency ω to pull more or less quickly a metal ball in the middle of according to the tube rigidity K and its mass m: a pile of sand. Depending on the density and intensity of sand, the ball will move more or less quickly. In our K ω2 = = (2π f )2 (53a) case, the bullets should be the photons and the medium m would consist of a material of extremely fine granulom- ω = 2π f. (53b) etry (1×10−35 m). We come back now at the tube under a gravitational wave. We multiply and divide by ρ the By substituting (53a)in(50) we obtain a new expression expression (54d): for the volume V as

ρ 2 m K 1 2 2 2 1 V = = , (53c) (ε) = 8π f × T. (55) ρ ω2ρ L2 E ρ

The total energy density of the system mass-spring UT Finally, we have the relation between the dynamic cur- is a function of the kinetic energy density Uc and strain vature and the strain in the tube: energy density U: 1 π f 2 ρ 2 U (ε)2 = 8π × . (56) Uc U UT 2 ρ + = . (53d) L E V V V V Taking into account eq. (35a) we obtain The strain energy density is π 2 U U U 1 2 f 1 U = T − c = T. (53e) (ε) = 8π × . (57) V V V L2 ρ c4 V Pramana – J. Phys. (2020) 94:119 Page 17 of 37 119

Figure 10. Double perpendicular tube loaded by a normal force.

π f 2 We notice that the term ρ has the dimension of the Figure 11. Sandwich structure of the space medium under a G ( 3/( · 2)) 1 (ε)2 gravitational wave. gravitational constant m kg s ,theterm L2 has the dimension of a curvature (1/m2) (see eqs (78) ( / ) and (81)), the term U V has the dimension of the the spatial material is considered as multisandwiched ( / 3) ( π f 2 ) 1 energy density Nm m ,theterm ρ c4 has the thin sheets of thickness 2r (see [10–13]) successively dimension of the inverse of the load (N−1). twisted during the passage of the shear wave (see fig- The formula (57) thus satisfies the principle curvature ure 11). = K energy density. Consequently, we learn that the With the actual measurements made at Ligo and π 2 term ( f ) can be identified with the Einstein’s constant Virgo, we know that when one of the arm is in com- ρ pression the other is in tension simultaneousely. κ if G π f 2 First step: Determination of the strains and stresses G = , ρ (58) in the sheet space in torsion ; ε =−νε . where f is the natural frequency of the spatial material Following the axis x y,wehave xx yy These inside the tube and ρ is the density of the spatial material. two deformations are correlated via the general relativity data by hμν = 2εμν [8,9,11]. The strain tensor accord- In the next section, we take into account the Poisson’s ; ratio. ing to the axes system x y is given in (32a). So, the relations between the strains and the stresses are 5.2.2 Study of two perpendicular horizontal space 1 ε = σ − νσ (59) cylinders solicited by a gravitational wave – effect of xx E xx yy the Poisson’s ratio – use of the longitudinal velocity 1 / εyy = σyy − νσxx . (60) of the correlated compression traction wave with the E compression/traction stress tensor in the interferometric tubes. We assume the same hypothesis on the strain Using the stress tensor definition (16c), we get and stress tensors as in §5.2.1. We consider now the two ν perpendicular tubes undergoing (z direction) a gravita- σ = E ε + ε + ε xx ( +ν) xx ( − ν) xx yy (61a) tional wave perpendicular to their plane compressing 1 1 2 and dilating them simultaneously as shown in figure 10. E ν σyy = εyy + εxx + εyy . (61b) Therefore, a Poisson’s ratio has to be taken into account (1+ν) (1 − 2ν) in this section. In this case, the two tubes behave like a gigantic Taking into account that, εxx =−νεyy (with ν = 1, see stress/strain gauge. Each arm constitutes a principal 4.4), expressions (61a)and(61b) become direction in the xyplane. We can therefore write a tensor expression in two dimensions and consider a strain ten- E σxx = {εxx} (62a) sor and a strain energy tensor. We also assume that the (1+ν) xy plane is deconnected from the z direction (elasticity E σyy =− εyy . (62b) theory: plane stress problem). The consequences is that (1+ν) 119 Page 18 of 37 Pramana – J. Phys. (2020) 94:119

=90° ―

Mohr circle in stresses

Figure 12. Mohr’s circle of the stress state [54]. Figure 13. Morh’s circle of the strain state [54].

On the Morh’s circle (figure 13) we see that Note: If one of the tube section is in compression (x γXY = εXX. (68) direction) the other one in y is in traction (origin of the 2 minus sign). According to the system of axes x;y, the stress tensor According to this equation, it seems that depending on (, ) is as shown below: the orientation of the interferometer in the plane x y ⎡ ⎤ ( , ) σ and X Y with respect to the direction of propaga- xx 00 () σ = ⎣ −σ ⎦. tion of the gravitational wave z , lateral movements xy 0 yy 0 (63) of the laser mirrors are possible with the same shape 000 and same wave intensity as the conventional one in Using Mohr’s circle (see figure 12) we can confirm the compression/traction (see figure 4). It should be inter- global shear behaviour along a 45◦ facet (axes system esting to measure these movements especially when the / X; Y) (see figure 11)and[10]. traditional compression traction motions are not mea- When we turn 90◦ on the Mohr’s circle, we turn 45◦ sured because of the position of the interferometer with on the real facet (image of the spin 2 of the graviton, see respect to the direction of propagation of the gravita- [10]). So on a 45◦ facet we are in pure shear as shown in tional wave. From (67a)and(68) we obtain figure 7. The strain tensor according to the axes system εXX = εXY. (69) X; Y is (see figure 13)and[10]: ⎡ ⎤ So the deformation of the circle containing the particles 0 εXY 0 ε = ⎣ ε ⎦. is the same (traction in one direction and compression XY XY 00 (64) in the other) but the circle rotates by an angle of 45◦, 00 0 see [10]. The stress tensor according to the axes system X; Y Second step: Determination of the strain energy of is ⎡ ⎤ the two connected tubes in the main system of axes 0 τXY = σ XX = σ 0 x y σ = ⎣ τ = σ = σ 00⎦. XY YX YY We now consider the strain energy of the two tubes con- 000 nected in traction/compression according to the axes (65) system xy: The shear stress is defined in eq. (11): Note: The results will be the same if we consider the E axes system X; Y because of the equivalence between τXY = γXY = μγXY. (66) 2(1 + ν) shear stresses and normal stresses on the one hand (see (65)) and the equivalence between the strains and angles In addition, we have in elasticity: on the other hand (see (64), (68)and(69)and[11]). 1 The energy density or energy per unit volume is εXY = γXY. (67a) 2 U 1 ij Uij = = σijε . (70) So from (66)and(67a)weget V 2 E In the axes system x;y, we are in pure compression/ τXY = εXY. (67b) (1 + ν) traction in the tube, and the total strain energy per unit Pramana – J. Phys. (2020) 94:119 Page 19 of 37 119 of volume is with i = x or y. Introducing (75f)into(75e), we get U 1 1 = σ ε + σ ε . 1 2 2 xx xx yy yy (71) U = K (εii) L (75g) V 2 2 (1+ν) As the section S of the tube is constant and as the stresses or and strains are constant on S, for a fixed section S of ( ) ( ) 2 1 U abcissa x or y ,wehave (εii) = (1+ν) . (76a) K L2 U 1 L 1 L = σxxεxxdx+ σyyεyydx. (72) Starting from the rigidity K depending on the length L, S 2 0 2 0 the Young’s modulus E, the tube section S and substi- When one arm is in compression the other is in traction tuting the expression of L resulting from K in (76a), we and so obtain U N (ε )2 = ( +ν)K . ε =−ε = (73) ii 1 2 2 (76b) xx yy ES E S and from the Hooke’s law: Considering the tube volume V , the section S is extracted (see (48)). Substitute the expression of S (48) ES N = N = N = δL = K δL. (74) in (76b)givestheU/V strain energy density: x y L K U 1 (ε )2 = ( + ν) 1 × The expressions of the stresses are given in (62a) 2 ii 1 2 (76c) and (62b). We can introduce the expressions of theses L V E V stresses in eq. (72): or 1 (ε )2 L U 2 ii U = E {ε }2 = L . (76d) xx dx ( +ν)K 1 S 2(1+ν) V 1 2 0 V E L E 2 We now consider the tubes under dynamic behaviour + εyy dx. (75a) 2(1 + ν) 0 and with a compression/traction wave following x and By replacing the strains by their expressions (73), we y in both arms simultaneousely due to gravitational obtain transversal waves. By proceeding as in §5.2.1 in one dimension we obtain: L 2 E N ω2 U = Sdx 1 2 2(1+ν) ES (εii) = (1+ ν)ρ × T. (76e) 0 L2 E2 E L N 2 + S x. By substituting the circular frequency ω (53b) by fre- ( +ν) d (75b) 2 1 0 ES quency f in (76e), we obtain We obtain the generalisation of expression (45a)intwo 2 1 2 2 f dimensions: (εii) = 4π (1+ ν)ρ × T. (76f) L2 E2 1 L N 2 1 L N 2 U = ( + ν) x+ x We have a pure compression/traction in the inter- 1 d ( +ν) d (75c) 2 0 ES 2 1 0 ES ferometer (not in 3D with shear wave), and so the or after simplification compression/traction wave equation is (see §4.3.2.1 and figure 4)thesameasin§5.2.1 (see (35a)and(35b)). We 1 N 2 L obtain U = . ( + ν) (75d) 1 ES 1 f 2 (ε )2 = 4π 2(1 + ν)ρ × T. (76g) The strain energy of a stiffness spring K (N/m) can be L2 ii c4ρ2 written by substituting N by expression (44b)in(75d): On the basis of (75f), we can write this equation accord- 1 ing to each strain in each arm of the interferometer: U = K (δL)2. (75e) (1+ν) 1 π f 2 1 U (ε )2 = 4(1+ν) × π × × × (76h) Moreover, according to the strain definition, the dis- L2 xx ρ c4 V δ ν = placement variation L is with 1: 2 1 2 π f 1 U ε =−ε =±ε × =±ε × =±δ εyy = 4(1+ν) × π × × × (76i) xxL yyL L ii L L (75f) L2 ρ c4 V 119 Page 20 of 37 Pramana – J. Phys. (2020) 94:119 and it is interesting to see that with ν = 1 (see 4.4)and 24 1−ν2 dU 2 = × . π f 3 (77a) G = ρ we obtain Eh dxdy With the plate bending rigidity 1 2 G U (εii) = 8π × (76j) L2 c4 V Eh3 D = (77b) or with (75f): 12(1 − ν2) π 1 2 8 G U or (εxx) = × (76k) ⎛ ⎞⎛ ⎞ L2 c4 V 1 ν 0 εxx 1 8πG U 1 εxy ε 2 = × . ε ;ε ; ⎝ ν 10⎠⎝ ε ⎠ yy (76l) 2 xx yy εyy L2 c4 V z 2 ( − ν) xy 0021 2 So we can construct the tensorial expression (76m)on 24 1−ν2 dU the basis of (76h)and(76i) and following the principle = × (77c) curvature = K strain energy density which is also equal Eh2 dxdyh to K times the work of external forces. 1 1 Nm = × . (77d) 1 (ε )2 m2 N m3 L2 0 xx 0 / 3 1 ε 2 We note that the right term in N m m is like an energy 0 L2 0 yy −2 density and the left term in m is like a curvature. π 2 = 8 G Txx 0 . 24 1−ν (76m) We note that 2 has the same dimension as c4 0 Tyy Eh κ = 8πG −1) κ c4 (N . We can concider as an equivalent Note: If complementary measurements are carried out flexibility of the space fabric and 1/κ as an equivalent on an interferometer and if we can confirm that there rigidity of this frame. With the relationships between the γ are also shear strains (angles ), then the hypothesis curvatures and the second derivatives of z displacements of plate behaviour is confirmed, eq. (76m) will be built (w) we have with 3×3 matrix completed by Poisson’s ratio ν,strains ε ε ∂2w ∂2w xy and yx and strain energies Txy and Tyx. 1 = (x,y) , 1 = (x,y) , 2 2 By opposition, if measurements are carried out and if Rx ∂x Ry ∂y there are no shear strains (angles γ ), it is necessary to 1 ∂2w( , ) consider the hypothesis of torsional wave cited above. = x y ∂ ∂ (78) In this case, the mechanical model could not be a plate Rxy x y but a space cylinder in torsion due to the rotation of and the relations between strains and curvatures are black holes’ binary system (see §5.2.3). But all these z z elements do not change the κ coefficient subject of this εxx =− ,εyy =− , article. We continue so on the basis of the actual data, Rx Ry δL the measured in each direction of the interferometer ∂2w( , ) L γ = ε =− x y =− 1 . arms. xy 2z 2z (79) ∂x∂y Rxy It must be remembered that in weak field there is a ε relationship between the metric gij and the strain ten- The strain tensor ij can then be expressed in terms of sor εij (see (12a)) and a relationship between the stress curvature tensor Rij of the thin plate: energy tensor T ij and the mechanical stress tensor σ ij ⎡ ⎤ ε ε 1 2 1 (see (16a), (16b) and Appendix A). The parallelism xx xy Rx Rxy ε = =−z⎣ ⎦ =−zR ij 1 1 ij between expression (76m) and the plate bending expres- εyx εyy 2 sion is noted for the diagonal terms of (77c) but not the Rxy Ry (80) terms concerning the Poisson’s ratio (seeε above). What is important here, is that the term , ij , can be inter- L2 or pretated as a curvature by comparison with the terms, ⎛ ⎞ εij ⎛ ⎞ 1 , that we have in a plate case ((77a), (78)and(79)). z2 1 ν 0 ⎜ Rx ⎟ Indeed, in the case of the plate hypothesis [51,52], the 1 1 1 ⎝ ν ⎠⎜ 1 ⎟ ; ; 10⎝ R ⎠ strain energy is with z, the thickness of the plate perpen- R R R y x y xy 002(1 − ν) 1 dicular to the plane xy: R xy −ν2 1 2 2 1 2 24 1 dU (εxx) + εyy +2(1− ν) εxy +2 ν εxxεyy = × . (81) z2 4 Eh2 dxdyh Pramana – J. Phys. (2020) 94:119 Page 21 of 37 119

Therefore, expression (82) from (76m), can be consid- tensor [54]. We assume in this section that the coales- ered as equivalent at a curvature tensor whose radii of cence of two black holes for example, creates by their curvature in each direction are infinite (R → L). rotations, a torsion of the space as shown in figure 3 and refs [12,13]. In weak field, for space only, the metric 1 0 (ε )2 L2 xx 0 is given in (12a). Einstein’s gravitational equation in a 2 . (82) 1 0 εyy weak field for space is 0 L2 So, we define λ 16πG ∂ ∂λh = h =− T . ij ij 4 ij (88) 1 (ε )2 c L2 0 xx 0 Rij = (83a) 1 ε 2 The result for a polarised wave A× is eq. (30b). With 0 L2 0 yy formula (31f) demonstrated in §4.3.2.1, we have a link and we have tensor of strain energy density as between the general relativity (disturbance of the spatial ) T 0 part of the metric hij and the theory of elasticity (the T = xx . (83b) ε ij 0 T strain ij of the elastic medium) (see 12c). So, based on yy (30b)and(12c), the spatial perturbation of the metric Expression (76k) can therefore be written as hμν(A×) is in correspondence with the following strain 8πG tensor [10,54]: R = T . (84) ⎡ ⎤ ij c4 ij 0 εxy = ε 0 ⎣ ⎦ We consider into this expression (76m) an infinite radius εxy = εyx = ε 00. (89) of curvature which therefore implies a zero scalar cur- 000 vature. Indeed, we assume εxx =−εyy and after rising the second index of the Ricci tensor we get the diagonal Thanks to the Hooke’s and elasticity formula, there is a part εxx;−εyy , for which the trace is zero. correspondence with the strain tensor εxy (89)andthe stress tensor σ (90): 1 1 Nm xy = × . (85) ⎡ ⎤ ⎡ ⎤ m2 N m3 0 τyx 0 0 τ 0 ⎣ ⎦ ⎣ ⎦ In this expression (eq. (84)) Rij is the equivalent cur- σxy = τxy 00= τ 00. (90) vature tensor of the space and Tij is the strain energy 000 000 tensor of the deformed space. To reach the parallelism with the Einstein’s field equa- This type of stress tensor is representative of a cylinder tion where Gμν cover the curvature tensor of the space in pure torsion, twisted by a torque Mt (see figure 14). and Tμν cover the energy density out of the space (e.g., For this section, we consider the cube of normal to the the Sun curved the space but is not the space itself), facet z at a point Q (τ stresses distribution). we have to consider in our analogy that the work of the external forces is equal to the work of the internal forces Note: The attentive reader will, however, not have failed created by the strain energy (see (8d), (5a)and(10e)). to notice that the facets subjected to traction and compression are not in the shear plane perpendicular to the = . Texternal Tinternal (86) direction of the wave (figures 6, 8 and 11), as is the case ◦ In this case, our analogy in 2D (analogy (76m)) is close with gravitational waves, but inclined to 45 (figure 14). to the Einstein’s field equation in four dimensions (5a). This approach is therefore only a simplified model to We thus managed to find κ (see §6), by passing check the possible value of κ in accordance with the through the mechanical components of the stress ten- stress tensors (33b) connected to the pure torsion stud- sor (and not via the temporal component of the tensor ied according to the elasticity theory [54]. as Einstein did it to be correlated with the Newton’s The torsional strain energy is approach in weak gravitational field ) corresponding to 1 L M2 1 M2 L the internal work of the space fabric which is equal to U = t dx = t . (91a) the external work of the applied masses. 2 0 μIt 2 μIt

U = Wint = Wext. (87) With θ the angular displacement of the point Q located on the outer surface of the cylinder, we have: 5.2.3 Study of a vertical cylinder in pure torsional space (see ﬁgures 5 and 14) – use of the shear veloc- dθ M = t . (91b) ity of the correlated shear wave with the shear stress dx μIt 119 Page 22 of 37 Pramana – J. Phys. (2020) 94:119

r θ = σ 45°

=―σ

L Figure 15. Deﬁnition of the shear strain γ .

Introducing (95)into(91a) yields 1 M2 L 1 (M )2 U = t = t . (96a) 2 μIt 2 k The introduction of (94a)in(96a)gives 1 U = kθ 2. (96b) 2 Figure 14. Cylinder clamped in pure torsion [54]. By the Hooke’s law (14) we have a relationship between the stress τ and the shear stain γ . Using ﬁgure 15,we We assume that the torsion torque is constant. Introduc- determine the relationship between the shear strain γ ing (91b)in(91a)weget and the angular displacement θ. We have geometrically: θ 2 1 d θ U = μIt L. (92) r 2 dx tangγ ≈ γ = (97a) L From expression (92), the equivalent torsional curvature and the relationship between the stress and the strain is extracted: (14) is as follows: dθ 2 2 U rθ = . (93) τ = Gγ = G . (97b) dx μIt L L By introducing θ from expression (97a)into(96b)we The relationship between the torque Mt and the rotation θ gives the torsional stiffness k according to expression obtain a new formula of the torsional strain energy of (94a): the cylinder: = θ. 1 γ 2 L2 Mt k (94a) U = k . (98) 2 r 2 By integrating (91b) with respect to x, we can extract a That we can rewrite as new expression for θ: γ 2 2U Mt L = . (99) θ = (94b) r 2 kL2 μIt We can now replace the torsional stiffness k by its value or (95). We extract the length of the cylinder and we obtain μIt M = θ. (94c) μIt t L L = . (100) k Comparing (94c) with (94a), the expression of the tor- The introduction of (100)into(99) yields sional stiffness is therefore: μ γ 2 2kU = It . = . (101) k (95) 2 μ2 2 L r It Pramana – J. Phys. (2020) 94:119 Page 23 of 37 119

The torsional inertia It of a cylinder expressed as a func- Introducing the expression of It (102)in(110c)gives tion of S, the surface of the cylinder, is written as π 4 d2θ 2μ r π 4 π 4 2 + 2 θ = 0 (110d) d r r 2 ρπ 4 It = = = S . (102) dt L r L 32 2 2 and after simplification we get By introducing It in (101) we obtain d2θ μ 2 2 8kU + θ = 0. (110e) γ r = . (103) 2 ρ 2 μ2 S2 dt L Introducing (104)into(103), we obtain We verify the dimensional equation of the term before θ: V π 4 Sc = (104) μ μ r μ L ω2 = k = It = 2 = 1 k U J JL 1 ρπr 4 L ρL2 γ 2 = 8 × . (105) 2 L2 r 2μ2V V kg m 2 2 1 = s m = . (111a) We now consider the dynamic behaviour of the cylinder kg m2 s2 consisting of an elastic substance of density ρ, Young’s m3 modulus E, Poisson’s ratio ν obtained from of the vac- By replacing the shear modulus μ as a function of k (95) uum energy. The fundamental equation of dynamics is in expression (111a) we obtain derived from the sum of the kinetic energy and torsional k strain energy: ω2 = . (111b) ρLIt 1 2 1 2 UC + U = Jω + kθ . (106) 2 2 In expression (111b), L is extracted and multiplied each side by the surface S of the tube to obtain a new expres- For the mechanical characteristics of the cylinder made sion of the volume V of the cylinder: of a space equivalent material, the following definitions k are given: Lπr 2 = V = πr 2. 2 (112a) The moment of inertia of a rotating cylinder is ω ρ It 1 By introducing V (112a)in(105)weget J = ρπr 4 L. (107) 2 1 k U γ 2 = 8 × (112b) The circular frequency is 2 L r 2μ2 k πr 2 V ω2ρ I dθ t ω = = 2π f (108) dt or after simplification or 1 ω2ρ I U γ 2 = 8 t × . (112c) μIt L2 πr 4μ2 V ω2 = . (109) JL Introducing the expression of It (102)into(112c)we By introducing (108)into(106)weget get 2 θ( ) ω 2 ω 2 1 d t 1 2 1 2 U UC + U = J + kθ( ). (110a) γ = 4ρ × = 4ρ × T. (112d) 2 dt 2 t L2 μ V μ By derivative with respect to t (110a) we obtain, looking By replacing ω and the shear modulus μ by their values, for the natural frequency of the cylinder: we obtain 2 2 d2θ( ) 1 4π f ρ U t + θ = . γ 2 = × J k (t) 0 (110b) 2 4 2 (112e) dt2 L E V 2(1+ν) By introducing the formula of k (95)andJ (107)in (110b) we obtain or after simplification 1 d2θ μI 1 16π 2 f 2ρ U ρπr 4 L + t θ = 0. (110c) γ 2 = 4(1+ν)2 × . (112f) 2 dt2 L L2 E2 V 119 Page 24 of 37 Pramana – J. Phys. (2020) 94:119

By multiplying and dividing expression (112f)byρ we 6. Deduction of the parallelism of a new get mechanical expression of the gravitational constant G and Einstein’s constant κ based on the Ligo and 1 16π 2 f 2ρ2 U Virgo measurements γ 2 = 4(1+ν)2 × (112g) L2 E2ρ V 6.1 General or The parallelism between the Timoshenko approaches 1 π f 2 ρ 2 U (see (57), (76h), (76i)or(114)) and the Einstein’s γ 2 = 64π(1+ν)2 × . (112h) L2 ρ E V approach (5a) is therefore demonstrated three times (see §5.2). We note that this parallelism is obtained This time we use the definition of the speed of a trans- by using the wave propagation in the medium as the verse wave (torsion wave) acting by shear (see 41)and transverse gravitational waves in eq. (5a) is a tensor we obtain expression (tensor of the curvatures Rij connected to the strain tensor εij and strain energy density tensor ρ 2 1 ) = . (112i) Tij (see (84)) and is based on the Hooke’s law (13), E 4c4(1+ν)2 the spring elastic strain energy and the eigen circular frequency of the spring/mass system assimilating the We introduce (112i)in(112h) and we obtain tube to a stiffness spring k and mass m. In all the calculations carried out, we thus obtain that the gravitational 1 π f 2 1 U constant G can be expressed as a function of the density γ 2 = 64π(1+ν)2 × (112j) L2 ρ 4c4(1+ν)2 V ρ and of the natural characteristic frequency f of an elastic microstructure constituting the vacuum space. or after simplification we get the final result π f 2 G = . γ 2 π f 2 1 U ρ = 16π × . (112k) L2 ρ c4 V On the basis of this equation of G, we can re-express now the Einstein’s constant κ (4a) in mechanical terms. Assuming that the external work T is equal to the inter- As the displacements of the laser mirrors in the internal work U, and comparing with (88), we confirm that ferometers are limited to forward or backward motions, we only consider the associated traction/compression π f 2 waves (§5.2.1 and §5.2.2) as representatives of that G = (113) ρ actual data. and 6.2 In the case of §5.2.1: One arm of the interferometer 1 G γ 2 = π × T. 2 16 4 (114) L c Taking into account the following points: That we have to compare (see 28b) with (a) With compression, tensile wave velocities (35a) correlated with the stress tensor (33a)andthe 1 16πG εμν + ημν ε =− Tμν. polarised wave A+ [53], 2 2 4 (115) 2 c (b) with the definition of the circular frequency ω of the elastic medium, we obtain based on (54c), We therefore have a factor 2 on the left term of (115) 2 (dimension 1/m ) which allows us to find the usual we obtain value of κ with a factor 8. 8πG ω 2 κ = = ρ . 4 2 (117a) 1 2 1 c E γ => 2εμν + ημν2ε , (116) L2 2 We verify the dimensional equation (117b)ofκ, indeed: " # 2 with the d’Alembertian. ω 2 kg 1 s2 1 So, via (112d)and(116), it will be necessary to divide ρ = > = = . (117b) E m3 s kg kgm N by 2 the factor before U/V in (112d) to obtain κ. ms2 Pramana – J. Phys. (2020) 94:119 Page 25 of 37 119

6.3 In the case of §5.2.2 – two arms of the 1. Particles and virtual antiparticles are created and interferometer correlated via the Poisson’s ratio ν annihilated spontaneously, generating a force that brings together two parallel plates placed in the vac- Taking into account the following points: uum. There is energy in the vacuum. 2. There is a fundamental state different from 0 from (a) With compression, tensile wave velocities (35a) the quantified vacuum (QED). are correlated with the stress tensor (33a)andthe 3. There is a scalar field (Brout, Englert, Higgs field) polarised wave A+ [53], in vacuum space. (b) with (76f)or(76h)and(76i), 4. Emptiness is not a void. (c) with the definition of the circular frequency ω of the elastic medium, In addition, the vacuum energy has been measured and calculated: we obtain

8πG ω 2 – According to the cosmological constant measure- κ = = (1 + ν)ρ . (118) − − c4 E ments: ρ = 1 × 10 29 g/cm3 (1 × 10 26 kg/m3), – According to quantum field theory: ρ = 1.11 × Note: We confirm well the parallelism previously 93 /cm3 demonstrated in §5.1.2. Indeed κ is well proportional 10 g , – According to the cosmological constant measure- to (1 + ν) in the expression, see (43c). In addition, with ments: T = 8.987551787 × 10−10 kg m2/ ν = 1 we obtain (117a). vacuum s2/m3, – According to the quantum field theory: E = 6.4 In the case of §5.2.3 – Pure shear and torsion vacuum 1×10113 kg m2/s2/m3. approach Even if the values are so different, the vacuum energy Taking into account the following points: is not null. That is the fundamental point here. (a) With shear wave velocities (41) correlated with the The best way to know which value is the right one is to measure the Young’s modulus via the Casimir test stress tensor (33b) and the polarised wave A× [54]. (b) With eqs (112d)and(28b) and taking into account (see §9.4) and to determine the good value of the energy that we have a coefficient 2 on the left term which by the formula given in tables 2 and 3. So, following the must be taken into account to reach κ. quantum field theory, the vacuum energy Ev is quantified at the fundamental state: Following these hypotheses, we thus obtain from (112d), 1 1 ω 1 Ev = hf = h = h¯ ω, (115), (116) π (120) 2 2 2 2 π ω 2 8 G Ev h κ = = 2ρ . (119) where is the vacuum energy, is the Planck’s con- c4 μ stant and ω is the circular frequency. In addition, the special relativity laws apply to the We obtain a logical result for this case of pure torsion. vacuum. The energy of emptiness is The formula is the same as that obtained in eqs (117a) 2 and (117b), but this time the shear modulus μ plays the Ev = mc . (121) role of Young’s modulus E. Using (120)and(121), we deduce the equivalent mass expression m present in vacuum: 7. Physical approach or simple mathematical 1 h¯ ω artefact? = m (122) 2 c2 Does all this have such a physical reality, or is it just and the expression of the vacuum density ρ is a mathematical coincidence? Let us take a look at this 1 h¯ ω m about the vacuum. The Casimir force [55] in connection = = ρ. 2 (123) with quantum field theory shows that the vacuum has a 2 c V V non-zero ground state, a non-zero energy and therefore, It seems therefore coherent that formula (123) is related according to the special relativity, a non-zero equivalent to the density ρ and to the specific circular frequency ω mass. The space is therefore a physical object and so is (via f ) that can be, in this case, those of the vacuum (via not empty. The measurement of Casimir’s force in the vacuum energy). We note also that eqs (113), (117a), vacuum [55] confirms that (118), (119) are related to ω and ρ. 119 Page 26 of 37 Pramana – J. Phys. (2020) 94:119

8. Results and expressions (130)and(53b) give the circular frequency: 8.1 Numerical application at the vacuum energy – Gh longitudinal wave in the interferometric tubes ω = . (131) Vc2 8.1.1 Theoretical development. We note T as a vac- We must therefore check the small volume hypothesis uum energy density measured in J/m3 with V a volume (quantum approach). According to the quantum field 93 3 of space elastic material that have to be determined. theory, vacuum density ρ = 1.11 × 10 g/cm and the vacuum energy density T = 1.00 × 10113 J/m3.In 2 2 T × V = Ev = mc = ρ×V × c . (124a) addition, the fundamental physic constants are: − So we can extract the density ρ of (124a): G = 6.67408 × 10 11 m3/(kg s2) −34 2 Ev h = 6.62607004 × 10 m kg/s ρ = . (124b) Vc2 c = 299 792 458 m/s − − Considering eq. (120) we extract the frequency f : κ = 2.0766 × 10 43N 1. 2Ev As all the formulae are functions of volume V ,nowwe f = . (124c) h are looking for the value of this volume V that allowes to satisfy all the physic constants. From this volume V , From (120), the energy density of the vacuum Ev/V is therefore written as a function of the circular frequency it will then be possible to determine the r dimension of ω and the Planck’s constant h: the space fabric fibres and the Young’s modulus E of the space material. Ev hω = T = . (125) V 4πV 8.1.2 Magnitude obtained for the new parameters of G We extract from (125) the circular frequency: based on vacuum energy. So, the question is therefore what volume V does to simultaneously satisfy 4π Ev ω = . (126) expressions (4a), (35a)and(113). Once all iterative cal- h culations have been carried out, we obtain the results By substituting expressions (124b)and(126) in expres- given in table 2. sion (117a), we obtain the Young’s modulus E: 8.2 Numerical application at the vacuum energy – π 3/2 c global approach by a shear torsion wave E = Y = 2Ev . (127a) h GV 8.2.1 Theoretical development. The approach is the By substituting expressions (124b)and(127a)in(35a), same as in §8.1 but the Young’s modulus E is replaced we obtain the vacuum energy Ev which depends on vol- throughout the equation by the shear modulus μ.The ume V : energy density T follows eqs (124a)and(125). The den- 2 ρ 2 Gh sity follows eq. (124b). The frequency f is defined in Ev = ρVc = (127b) 4πVc2 eq. (124c). The circular frequency is equal to (126). By substituting expressions (124b)and(126) in expression and from the expression (127b) we determine the vac- (119), we obtain the Young’s modulus E. uum density ρ: π 2 3/2 c Gh μ = 2Ev . (132) ρ = . (128) h GV 4πV 2c4 With the definition of the shear modulus μ we obtain By expressions (35a)and(128) we deduce a new expres- the Young’s modulus E: sion of the Young’s modulus E: 3 c π G h 2 E = Y = 4(1 + ν)Ev2 . (133) E = ρc2 = . (129) h GV 4π cV By substituting expressions (124b)and(127a)in(41), Expressions (124c)and(127b) draw the expression of eq. (127b) is again obtained on the vacuum energy Ev f : as a function of the volume V considered: 2 Gh 2 Gh f = (130) Ev = ρVc = μV = (134) 2πVc2 4πVc2 Pramana – J. Phys. (2020) 94:119 Page 27 of 37 119

Table 2. Numerical application (case of Young’s modulus approach)

Parameters Physical objects Units Values with Evacuum With v = 1 = 1×10113 J/m3 Case of longitudinal waves (pure compression/traction)

Volume V m3 1.61 × 10−104 = 4 π 3 . × −35 Radius of V (link with r if V 3 r m1566 10 the string theory and Planck’s length) Gh2 kg m2 Vacuum energy Ev = ρVc2 = = J1.61 × 109 4πVc2 s2 Vacuum energy density T = Ev kg = J 1.00 × 10113 V s2 m m3 (link with the quantum ﬁeld heory) 2 Density ρ = Gh kg 1.11 × 1096 4πV 2c4 m3 2 Young’s modulus (link E = Y = ρc2 = G h kg = Pa 1.00 × 10113 4π cV s2 m with the elasticity $ / theory) = = 3 2 c π E Y 2Ev h GV $ = E m Speed of light (link c ρ s 299792458 with the special relativity) Frequency f = Gh 1/s4.861 × 1042 2πVc2 = 1 . × −43 Period (link with the T f s205684 10 Planck’s time) 3 Gravitation constant G=π f 2 1 m 6.67408 × 10−11 ρ kg s2 (link with the Newton’s gravitation) π Circular frequency ω = Gh = 2π f = 4 Ev 1/s3.05 × 1043 Vc 2 h κ= ρ ω 2 1 . × −43 Einstein’s constant 2 E Newton 2 07658 10 (link with the general relativity)

and from expression (127b) the vacuum density ρ (128) or with the new deﬁnition of G: is again obtained. By expressions (41)and(128)anew 2 1/6 9 f / expression of the shear modulus μ is deduced: r = × h1 3 (138) 64π 2 ρμc2 2 μ = G h and the diameter d is π (135) 4 cV 1/6 9 f 2 d = × h1/3 and of the Young’s modulus: π 2 ρμc2 / / (1 + ν)G h 2 9 1 6 fh 1 3 E = . (136) = . (139) 2π cV π 2ρμ c

Expressions (124c)and(134) restore again expression 8.2.2 Magnitude obtained for the new parameters of G of f (130) and the expressions (130)and(53b)givethe based on the vacuum energy circular frequency (131). The space is thus quantiﬁed via radius r of the space ﬁbres: So, the question is therefore what volume V does to simultaneously satisfy expressions (4a), (41)and(113). 1/6 9 G / Once all iterative calculations have been carried out, we r = × h1 3 (137) 64π 3 μc2 obtain the results given in table 3. 119 Page 28 of 37 Pramana – J. Phys. (2020) 94:119

Table 3. Numerical application (case of shear modulus approach).

Parameters Physical objects Units Values with Evacuum With v = 1 = 1×10113 J/m3 Case of shear waves (torsion)

Volume V m3 1.61 × 10−104 = 4 π 3 . × −35 Radius of V (link with r if V 3 r m1566 10 the string theory and Planck’s length)

Gh2 kg m2 Vacuum energy Ev = ρVc2 = μV = = J1.61 × 109 4πVc2 s2 Vacuum energy density T = Ev kg = J 1.00 × 10113 V s2m m3 (link with the quantum ﬁeld theory) 2 Density ρ = Gh kg 1.11 × 1096 4πV 2c4 m3 Young’s modulus/shear E = Y = 2ρc2(1 + ν) kg = Pa 4.00 × 10113 s2 m modulus (link with the elasticity theory) = (1+ν)G h 2 2π cV μ = G h 2 4π cV $ = = ( + ν) 3/2 c π E Y 4 1 Ev h GV $ $ = μ = E m Speed of light (link c ρ 2(1+ν)ρ s 299792458 with the special $ relativity) = 1 GE c f 2π(1+ν) Frequency f = Gh 1/s4.861 × 1042 2πVc2 = 1 . × −43 Period (link with the T f s205684 10 Planck’s time) 3 Gravitation constant G = π f 2 1 m 6.67408 × 10−11 ρ kg s2 (link with the Newton’s gravitation) π ω = Gh = π f = 4 Ev / . × 43 Circular frequency Vc2 2 h 1 s305 10 κ = ρ(ω )2 1 . × −43 Einstein’s constant 2 μ Newton 2 07658 10 (link with the general relativity)

The results are identical to table 2 except for the Weobtain ρ = 1.11×1096 kg/m3 compatible with the Young’s modulus that is multiplied by 4. results of ref. [11](1.30×1096 kg/m3, see §3.4, formula 3.14). We find a period equal to 2.05684 × 10−43 s near the Planck’s time that is logical because at this time all the interactions have to merge. So the magnitude 9. Discussion obtained is correct. In addition, with this volume V we satisfy all the constant values of physics: The Planck’s 9.1 About the numerical values of the results obtained time where all the forces merge, G, c, κ. With expression (58)ofG and taking into account eqs (129)and(130) Are our numerical results physically acceptable? We we obtain obtain radius r of volume V in correlation with the string − dimension defined in the string theory: 1.566×10 35 m. 64 c4ρv G = π 3 r 6 (140) We obtain Young’s modulus as 4.00 × 10113 Pa, which 9 h2 is compatible with the results of ref. [11](4.4 × 10113 Pa, see §3.4, formula 3.13). and the numerical application gives Pramana – J. Phys. (2020) 94:119 Page 29 of 37 119 64 G = π 3 (d) We can see the shear strain of space created by the 9 rotation of the Earth measured by the satellite probe 6 (299792458)4 × 1.11 × 1096 × 1.566 × 10−35 B, as the expression of the torsion of space is akin × . to [15] a certain Ether characterised by gμν.This . × −34 2 6 62607004 10 experience can be considered as the right one to G = 6.67408000 × 10−11 m3/kg s2. measure the twist of the space and the dynamic state Thus this formulation unites all the theories of the of this new relativistic Ether. current physics: The light follows the curvature of the space but does not (a) quantum mechanics with the Planck constant h, need a luminescence medium to spread. (b) special relativity with the speed of light c, If the Ether of luminescence is dead [56], the rela- (c) quantum field theory with the vacuum energy (vac- tivistic Ether [20,21] appears to be alive.... In Einstein’s uum density) ρv, letter to Lorentz of June 17, 1916 [20], we can read: (d) string theory with the size of the medium grain size “I agree with you that the general theory of relativity r (string size), is closer to the Ether hypothesis than the special theory. (e) The gravitation with G. This new theory of Ether, however, would not violate the = principle of relativity, because the state of this gmn 9.2 About the two approaches: Longitudinal and The Ether would not be that of the rigid body in an shear waves independent state of motion, but each state of motion would depend on the position determined by the material Irrespective of whether we adopt the approach follow- processes”. ing the longitudinal waves of pure compression/traction or pure shear waves (pure torsion) we obtain identical 9.4 Experimental checking of the Young’s modulus of numerical values for all the parameters except Young’s thespacemedium modulus recalibrated by the effect of the Poisson’s ratio (see tables 2 and 3). An experimental way to establish the Young’s modulus E of vacuum is to perform the Casimir test. This 9.3 Quid de Michelson–Morley and the existence of a test consists of taking two metal plates spaced at a very certain Ether short distance L0, and to position them in the vacuum by simultaneously measuring force F and the horizon- Considering an elastic body for the space medium tal displacements (see figure 16). Generally, the force immediately raises the question of Ether. We recall first alone is measured, and the displacement calculated in the conclusions of Michelson and Morley’s [56]which the Casimir experiment. The result of the test is to are: draw the stress/strain curve. Of course, the slope of the “From all of the above, it seems reasonably certain stress/strain curve corresponds to the Young’s modulus that if there is a relative movement between the Earth E of the space medium according to eq. (142). and the luminous Ether, it must be small, small enough to refute the explanation of Fresnel’s aberration. ...” 9.5 Experimental checking of the shear of the space Few comments: (a) It is important not to confuse the vibration of a / medium supposed to be a luminescent Ether that It will be interesting in the interferometer Ligo Virgo does not exist (light spreads alone without need of to measure the eventual lateral movements of the laser beam (or the variation of angles between the two laser medium) with a fabric of curved space whose light γ follows curvatures (e.g. around the Sun). beams) to determine the eventual shear strain created (b) Contrary to what is often said, Michelson and Mor- on the space by the passage of gravitational waves (see ley did not write that there was no Ether but that it figure 5). had to be small enough not to be detected, (c) It is well known that the Higgs field does not inter- fere with the photon or with the light. It is then 10. Consequence on the time of this research obvious that it is impossible to detect this field with light. It is therefore possible that the experience of 10.1 Behaviour of time as an elastic material Michelson and Morley with the light is not the right experience to detect the material that constitutes the In this article, we focussed on the spatial part of Ein- elastic space fabric. stein’s tensor. Based on the new definition of G, we can 119 Page 30 of 37 Pramana – J. Phys. (2020) 94:119 now return to the temporal part of the Einstein’s tensor (see also [11], chapters 2.4.3 and 2.6). Einstein has demonstrated that the stress energy tensor Tμν curves space but also the time at the location of this mass via a proportionality factor κ. This space–time deformation, mathematically characterised by the metric variation gμν is the gravitation. Near a black hole the curvature of time is such that time expands endlessly giving the impression that it stops. The question is therefore the following: can we consider time as elastic? For memory, some measured facts are: (a) At high speed v, time expands: A clock placed in a rapidly moving airplane (t)sloweddowncompared to a clock stayed on Earth (t) (special relativity principle): Figure 16. Test to measure the equivalent Young’s modulus 1 E = Y of the space medium. t = $ t. (141) − v 2 1 c (b) At high speed the distance contracts: The gravitational potential of a sphere of radius R and mass M, is written as v 2 3 z = 1− z. (142) m kg c GM kg s2 φ = = . (146) Therefore, time has an elastic behaviour, it lengthens R m or shortens and time has a behaviour opposite to that By introducing (144)into(143b) we obtain of space (dilation = negative contraction) (see (149)). 2 GM The metric is connected to the interval, ds .Soina ε00 ≈ . (147a) non-inertial frame of reference for example (if the coor- Rc2 , , dinates according to x y z do not vary), we have: Taking into account the new definition of the function G 2 2 2 (143) of the density ρ and the frequency f of the space ds = g00c dt (143a) material: or in the general case, the unknowns of the Einstein’s π f 2 M equation are the 10 components of the metric tensor gμν: ε ≈ . (147b) 00 ρ 2 2 2 2 Rc ds = g00c dt + 2g10dxcdt We square the expression (145b): +2g20dycdt + 2g30dzcdt +g dx2 + 2g dxdy + 2g dxdz π 2 4 2 11 12 13 ε 2 ≈ f M . + 2 + 00 (148a) g22dy 2g23dydz ρ2 R2c4 +g dz2. (143b) 33 Taking into account eqs (76a)and(43c), the strain For this we have to focus on the time component of squared is proportional at (1 + ν): the metric (3b): π 2 4 2 2 f M 2φ ε00 ≈ (1 + ν) . (148b) g = η + h = 1 + 2ε ≈ 1 + . (144) ρ2 2 4 00 00 00 00 c2 R c Thus, the expression of the perturbation hμν for the Given the new definition of G we obtain time component (00 index) is: π f 2 1 f 2 M2 ε 2 ≈ (1 + ν)π . (148c) 2φ 00 ρ 4 2ρ h = 2ε ≈ . (145a) c R 00 00 c2 We multiply each side of eq. (148c)by1/R2: The equivalent strain of the time is therefore π 2 2 2 φ 1 2 f 1 f M ε ≈ . (145b) ε00 ≈ (1 + ν)π . (148d) 00 c2 R2 ρ c4 R3ρ R Pramana – J. Phys. (2020) 94:119 Page 31 of 37 119

Figure 17. An instantaneous photo (taken at the speed of light) of the xy plane space deformed by the gravitational wave propagating along z direction.

For memory the volume V of a sphere is We obtain, for the temporal component of the dis- 4 turbance of the metric, an expression similar to that V = π R3. (148e) which one obtains by considering a beam in pure 3 compression/traction (see (57)). So, time behaves like We introduce this volume into eq. (147a)viaR3: an elastic material. 1 π f 2 1 f 2 M2 ε 2 ≈ (1 + ν)π (148f) 10.2 Relates time lapses to the thickness of the space R2 00 ρ c4 3V ρ 4π R fibres and we get after some calculations: For memory, some measured facts are 1 4 π f 2 1 f 2 M2 ε 2 ≈ π 2(1 + ν) . (148g) R2 00 3 ρ c4 Vρ R (a) Time slows down when it is immersed in a gravitational field (see general relativity). Therefore, Noting that ρV = M and the last term of the equation as dimension of energy density U/V (see (148b)): (1) Gravitation curves space–time. Time is no longer absolute and become an illusion. 2 2 kg m (2) The more there is gravitation, the more there f M = kg = s2 = U 2 3 (148h) is curvature and the more there is tension in R ms m V the material and the more the slow down time and we obtain expands. π 2 1 2 4 f 1 U ε00 ≈ ππ(1 + ν) . (148i) (b) The time acts with a different sign of the space, R2 3 ρ c4 V 2 2 2 2 2 Taking into account that 3.1 ≈ 3, we obtain ds = dτ −dx −dy −dz . (149) π 2 1 2 f 1 U For a gravitational wave, the passage of time is a suc- ε00 ≈ 4π(1 + ν) . (148j) R2 ρ c4 V cession of instantaneous pictures following z direction. With (ν = 1) (see §4.4): Figure 17 shows an instantaneous photo of space–time taken during the passage of a gravitational wave. 1 π f 2 1 U In figure 17, R is the radius of the xy plane, n is the ε 2 ≈ 8π . (148k) xy R2 00 ρ c4 V whole number of small quantum distances c × t,2r is 119 Page 32 of 37 Pramana – J. Phys. (2020) 94:119

Table 4. Formula obtained in modified general relativity, new expression of κ. Parameters Formula Formula ε = δL ε = γ Strains ii L ij 2 ω ( − ) ω ( − ) Polarised wave hμν A+⎡cos c ct z ⎤ A×⎡cos c ct z ⎤ 00 00 00 00 ⎢ + ⎥ ⎢ + ⎥ ×⎣ 0 100⎦ ×⎣ 0 0 10⎦ 0 0 −10 0 +100 00 00 0 000 σ 00 0 τ 0 Associated stress tensor σxy = 0 − σ 0 σxy = τ 00 000 000 Associated wave in the interferometer tube Longitudinal Shear$ / μ compression$ traction c = ρ E c = ρ 2 π 2 = k 1 (ε) = π( + ν) 1 (γ )2 = π f 1 × U Curvature energy density L2 4 1 L2 16 ρ c4 V × π f 2 1 × U ρ c4 V μ ν 8πG μν λ 16πG G =− T ∂ ∂λh = h =− T Einstein’s gravitational field c4 ij ij c4 ij ω ω 2 Proposed Izabel’s field equation Gμ ν=−(1+ν)ρ 2T μν Gμν =−2ρ T μν E μ ω Gμ ν =−2ρ 2 T μν E = π f 2 2 = π f 2 2 GGE c G μ c = π 2 1 = ω2 = π 2 1 = ω2 G f ρ 4πρ G f ρ 4πρ Poisson’s ratio ν = 1 ν = 1 Young’s modulus E = Y = ρc2 E = Y = 2ρc2(1 + ν) E=Y E = Y = 4ρc2 $ $ = 1 GE = 1 GE Speed ccf π c f 2π(1+ν) = Gh = Gh Frequency ffπ 2 f π 2 2 Vc 2 Vc κκ=−( +ν)ρ ω 2 κ =− ( + ν)2ρ ω 2 Einstein’s constant 1 E 8 1 E 2 κ =− ρ ω 2 κ =− ρ ω 2 E 2 μ

the thickness of the space sheet (r is the radius of the to make up space (following z in the case of gravitational fibre of space = 1 × 10−35 m). wave). As the gravitational wave passes from one plane The passage of time is the succession of its instanta- of fibres to another (see figure 17), the different time neous photos following z. We notice that lapses can be counted by these successive passages from one spatial diameter of the fibre to another (quantum of (1) The perturbation hμν of the metric depends on the time). Time and space therefore appear to be somewhat − − variable , ct z,wherect and z are at the same quantified: level but with opposite sign. (2) In the hμν matrix, we see that all the time terms and z terms are correlated at zero (see (30a)and(30b)). 2r tq = . (150) In adddition, as the z-axis is confused with the time c axis, we therefore have thin sheets of xy plane space, of thickness 2r (2 × 10−35 m) which follow each other Time has a minimum duration corresponding to the time in time along the z-axis.Therefore, we propose to relate necessary to transmit information at the speed c from rate of time lapse to the thickness of the fibre that are said one fibre plane to another (multisandwich sheets). Based Pramana – J. Phys. (2020) 94:119 Page 33 of 37 119 on (134)and(17), taking into account the definition V (5) The intrinsic quantum characteristics of this elastic (148e) function of r, we obtain the quantified time lapse space medium with a ground state different from 0 tq : (ρ/vacuum energy and circular frequency ω). (6) The quantified space microstructure with a fibre 1/6 2r 18 G(1 + ν) / t = = h1 3. (151a) dimension of the Planck scale, and quantified lapse q c π 3 Ec8 time. (7) The new definition of G (macroscopic manifesta- With the new definition of G: tion of the said frequency f of space medium (see 2 1/6 2 2 1/6 58)). 9 f 1/3 f h tq = h ≈ . (151b) π 2 ρμc8 ρμc8 We obtain a new set of formulas defined in table 4 as With the values of table 3: a function of the tensors (polarisations) considered: Numerically we obtain the following values: −43 tq = 1.0451 × 10 s (1) The fibre length constituting the substance texture Value compatible with the Planck time: of the space fabric is 1.566 × 10−35 m compati- ≈ . × −43 . ble with the string dimension defined in the string tp 1 0000 10 s theory. = = . × 113 If we define Et = hf the time motor (energy), we get (2) The Young’s modulus E Y 1 00 10 Pa (from the longitudinal wave) and 4.00 × 10113 Pa 1 1 6 1 (from the shear wave) compatible with the results t ≈ E 3 . (152) . × 113 q ρμc8 t of ref. [11](44 10 Pa, see §3.4, formula 3.13). This value is compatible with the extreme stiffness We confirm that if the frequency f is zero, the spatial of space. material disappears and time too disappears. Gravitation (3) The density ρ = 1.11×1096 kg/m3 is also compat- can bend time because it bends the fibres of the space ible with the results of ref. [11](1.30×1096 kg/m3, fabric which therefore expands. The time passes step see §3.4, formula 3.14). by step through each of its fibre, one after the other to transmit information, and so it expands like these fibres. Asaresult: This elasticity of time is a characteristic of its elastic (1) The gravitational constant G does not seem to be a ε behaviour defined by the strain 00. universal stable constant as it would depend on the characteristics of the material constituting the space elastic fabric (ρ, f ) calculated from the vacuum 11. Conclusions properties. (2) G could thus have varied in time as the density or In the light of the above, it seems logical to propose a the vacuum natural frequency. new mechanical and physics expression of the Einstein’s (3) The speed limit of light could find an explanation constant κ based on: via the limited ratio ρ/E characterising the penetra- tion degree of light inside the elastic space medium (1) The elastic behaviour of space and time. (analogy of water that becomes a concrete wall at (2) The postulate that space is a substance, an elastic high speed). material characterised by its characteristic frequency f , its elastic properties (E = Y , ν) and This vision of the medium of space is a new vision its density ρ (or energy via E = mc2). of a relativistic Ether [20,21] without any correlation (3) The parallelism between the elasticity theory and with the luminescent Ether which does not exist. This general relativity (see §5.1.2 and [10]). medium can be made up of infinitesimal beams of small (4) The perfect correlation between the two gravita- sizes (quantum) in perpetual oscillation forming a three- tional waves polarisations (hμν(A+), hμν(A×)) with dimensional frame of stiffness density rather than strings the two compression/traction and pure shear ten- without a bending stiffness of 1 × 10−35 m. This three- sors of the space medium as a function of the facet dimensional fabric can be characterised by an elastic considered in elasticity (case of a space medium material itself, characterised by the Young’s modulus cylinder twisted by rotation of two massive objects E, Poisson’s ratio ν and density ρ. In this case, we char- that merge) [10–13] and the link between the gravi- acterise the space by a sort of elastic substance, at the ton of spin 2 with the image of the Mohr’s circle in fundamental state (minimum vacuum energy Ev) based terms of rotation (see [10]). on the quantum field theory. 119 Page 34 of 37 Pramana – J. Phys. (2020) 94:119

In addition, we have shown that we could consider a massive black hole in the centre), made of elastic mate- new type of wave, space torsion waves, which can gen- rial (elastic fabric of the space) rotating at high speed, erate longitudinal and shear waves. It would therefore be we see by an elastic calculation [57] that the shape taken extremely interesting to use current interferometers to by it redistributes the material and changes the apparent measure possible shear strains, that is to say, use the density ρ of the disc along the radius (thinning near the potential lateral movements of the laser in the inter- centre, thickening in the periphery). With the new defini- ferometers to determine the angular shear strain γ and tion of G as a function of this density ρ (153), it becomes thus the shear modulus μ of space. This will also con- possible to consider a variation of G along the radius of firm whether, in addition to transverse shear wave, we the galaxy (rather to increase the mass M andtolookfor should also consider longitudinal waves in the gravita- dark matter) and thus to recalculate different velocities tional waves. of the stars with a variable value of G. Indeed,we see Wehaveshownintables2 and 3 that it is possible to that the spinning disc gets thinner and stretches. There- recalculate the Einstein’s constant κ based on the theory fore, the density ρ decreases which, taking into account of elasticity and volume wave theory. A research con- the new definition of G, implies that 1/ρ increases. This ducted by Ringermacher and Mead [19] seems to show can compensate the decrease of 1/R when R approaches that the Universe can sound like a crystal. The analysis the periphery of the disc and can thus make it possible of these ‘special frequencies’ is also an open door to to maintain an almost constant speed of rotation of stars explore and to obtain information about the space fabric in the disc (see also the MOND’s approach). In addi- structure. tion, the tensions inside the space fabric disc in rotation In this study, we focussed on the elastic approach of allow to maintain the stars in their place, ensuring a space in weak fields. The presence of black hole being global motion of the rotating disc, and a constant speed confirmed, it seems logical that the gravitation in strong of the stars in the centre and in the periphery of the disc. field approaches the plasticity of the space material, in which case, it would be interesting to study what M M π f 2 v = × G = × , (153) becomes of the Einstein’s constant in strong fields using R R ρ the plastic theory of the strength of the materials [12,13]. In this paper we found an expression of the Ein- where M is the rotating mass and v is the speed of rota- stein’s constant κ based on the spatial part only of the tion of the stars around the centre of the galaxy. gravitational field tensors. For this, we used the two- In addition, if the vacuum is full of fluctuating sub- dimensional stress tensor and the Poisson’s ratio ν.We stance as this article seems to prove via the quantum recalled that Einstein established its coupling constant κ field theory of the vacuum energy at the fundamen- by using the temporal part of its gravitational field tensor tal state, it implies that in Young’s experience of in weak fields (Poisson’s formula and Newton’s gravi- the double slit experiment, this substance has to be tational field equation). It naturally comes from the idea taken into account on the interpretation of the duality of extending the stress tensor obtained from the elastic wave/particle. Indeed, the fluctuations of the vacuum theory at four dimensions and to deduce informations present everywhere in the medium during the experi- about the potential elastic characteristics of time. Such ment disrupts the trajectory of the particles between the approaches are adopted in [11]andin§10 we showed moment it is drawn and the moment it strikes the screen that it is possible to relate time lapse to the thickness of having passed through the two slits. the space fibres with an elastic time. It seems therefore that in order to make progress in Dark energy and the associated cosmological con- physics today, to renounce the Newton’s gravitation constant are characterised by an accelerated expansion cept, as proposed by the Eisntein’s general relativity, is of space. Based on the new definition of G = π f 2/ρ, not sufficient. As this gravitation force is only an illu- we might consider that G varied over time as a function sion, it seems logical that the universal and constant of density ρ. This variation of density can be explained character of this constant G created with this force also by the intrinsic nature of the space material related to the have to be given up. We show in this article that if dark energy (correlation with the expansion coefficient Newton and his gravitational force had not existed, Ein- of the elastic material). stein could have found his coupling constant κ without Dark matter was proposed to explain why the stars on going through the temporal component of his theory (μ, the periphery of a galaxy were spinning at high speed ν = 0), but going through the mechanical and spatial without being ejected from galaxies (normally speeds parts of its theory (μ, ν = 1, 2, 3) via the elasticity decrease when moving from the centre of the disc to its theory, the quantum field theory and the string theory periphery (153)). Considering the elastic deformation of data. If this had been the case, we would not have had a disc with a hole reamed (analogy of a galaxy with a the idea of constraining, to a universal constant, the Pramana – J. Phys. (2020) 94:119 Page 35 of 37 119 parameters πf2/ρ related to the substance constituting Appendix A. Demonstration of the equivalence the space that without knowing it Newton had called between the stress tensor σμν and the stress energy G in his equation. We also show that by proceeding in tensor Tμν [49] this way, and by calculating the parameter G asafunc- tion of the vacuum data via quantum field theory, we In the theory of elasticity, resulting from the continuum find elements on the infinitesimal substance constituting mechanics, the relation between the stress tensor σij, × −35 the space: the dimension of its granulometry 1 10 (with Ti = σijn j where T is a stress vector attached to is compatible with string theory; a high Young modu- the facet of normal vector n), and the applied force Qi lus is necessary considering the very small curvatures on a surface S j can be written as follows: generated at the space by extremely massive bodies; = σ . Y = E = 1 × 10113; a Poisson’s ratio equal to 1. The Qi ijS j (A1) × 101 ‘metal’ constituting the space is 5 10 times harder In the field of variational approach, the stress tensor can than steel at the speed of light (and rather fluid at very be written as follows: low speed)! To advance further in the elastic approach of space, it seems necessary to conduct measurements Qi σij = with S j → 0, (A2a) of Young’s modulus (see Casimir test described in this S j article) and of shear modulus (measurement of the lat- where Si is an area. eral motion of the laser in the large interferometer). This So, with m as the mass, ρ as the density of mass will also make it possible to settle the paradox of the energy, V as the volume and a as acceleration, we have: energies of the vacuum resulting from the cosmological i constant on the one hand and the quantum field the- Qi (m × ai ) (ρ × V × ai ) σij = = = . (A2b) ory on the other hand by quantifying once and for all S j S j S j the energy of the vacuum that have to be considered in the calculation. To summarise, let G vary, and probably Assuming that the variation of the force is due only to the variation of volume V as a function of time t we many of today’s physics problems will in turn disappear. v = i So we shall conclude by quoting one of the first great obtain with, ai t , scientist Alhazen Ibn al-Haytham (965–1040) who said (ρ × V × ai ) 1 V that: σ = = ρ v . (A3a) ij S S t i “The search for the truth is difficult, the road that leads j j to it is full of pitfalls, to find the truth, it is advisable to Thus, we get with V = xi × x j × xk leave aside its opinions and not to trust the writings of × × the ancients. You must question them and submit each 1 xi x j xk σij = ρ vi . (A3b) of their assertions to your critical mind. Trust only logic S j t and experimentation, never the affirmation of one or the We can replace S by its value: other, because every human being is subject to all sorts j of imperfections; in our quest for truth, we must also S j = xi × xk. (A4) question our own theories, each of our research to avoid So the new expression of the stress tensor is succumbing to prejudices and intellectual laziness. Do this and the truth will be revealed to you. ” vi xi × x j × xk σij = ρ . (A5) t xi × xk After simplification we obtain x j Acknowledgements σ = ρv . (A6) ij i t

The author would like to thank Christian Liegeois, Doc- By deﬁnition of speed v j ,wehave tor in Physics, for the useful comments and discussions, Daniel Spagni for the English and the two Pramana x j v j = . (A7) reviewers for their very effective and thoughtful reading t that allowed the author to further develop and improve We ﬁnally obtain the expression of the stress tensor at this article. The author would also like to thank the late R low speed as a function of energy density ρ and based Gregoire, the great mechanician, who through his teach- on the multiplication of the velocities v and v : ing based on the research of “how it works” guided the i j author. σij = ρvi v j . (A8) 119 Page 36 of 37 Pramana – J. Phys. (2020) 94:119

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