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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. 0123456789().: V,-vol 119 Page 2 of 37 Pramana – J. Phys. (2020) 94:119 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 mod- ulus 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.
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