Can we understand the concepts of the General Relativity and of the Quantum Mechanics based on the principles of the strength of materials? Reflections and proposals David Izabel
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David Izabel. Can we understand the concepts of the General Relativity and of the Quantum Me- chanics based on the principles of the strength of materials? Reflections and proposals. 2017. hal- 01562660
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Can we understand the concepts of the General Relativity and of the Quantum Mechanics based on the principles of the strength of materials? Reflections and proposals
David Izabel
Engineer and professor
July 2017
Summary
The quantum mechanics, in the infinitesimal world, is the reign of the quantification of energies, and of the probability of presence of particles with the famous wave function ψ associated at the Schrodinger equation. The General relativity, is for it, reserved at the infinitely large world where the mater, the energy density 푇휇휈influence the curvature 퐺휇휈 of the space time at 4 dimensions and reciprocally. Finally, the strength of materials is the mechanic of objects at the human scale (beams, columns, plates, shells) used for the design of structures, without common measures with this two pillars of the of the physics cited earlier. We are going to show in first part of this paper that the results of the quantum mechanics (quantification of energy, shape of the wave function) are similar at the eigen frequencies and eigen modes of a beam. In second part, we will show on simples cases, that the concepts of curvature linked with energy density present in general relativity are equally at the bases of the fundamental equations of the elasticity and of the strength of material. We will demonstrate finally that the stress energy tensor written for small speeds non relativistic is an extension in 4 dimensions of stress tensor of the elasticity theory.
2
1st part) – Quantum mechanics and beam vibration
1.1) Free vibration of a beam on two supports – natural frequencies and mode of vibration
We consider a beam on two supports, with a span L (m), carried out with a material of elastic modulus E (MPa), that have a section with an inertia I (m4) and an area S (m²) and that have a mass by unit of length (m= S) in kg/m (cf. figure 1). [4]
The beam is in free vibration and as a deflection along the time 푦(푥,푡) :
푀, 퐸, 퐼
푥
푦(푥,푡)
퐿
푦
Figure 1 – Beam on 2 supports under its self-weight –
We can write:
휕²푦(푥,푡) 퐹 = 푚푎 = 푚 = 푚푦̈ (1) 휕푡² (푥,푡)
In addition, we have by connecting the radius of curvature R (m) at the bending moment 푀(푥,푡) :
2 1 휕 푦(푥,푡) 푀(푥,푡) = = (2) 푅 휕푥2 퐸퐼 Knowing that:
휕푀(푥,푡) = 푉 (3) 휕푥 (푥,푡)
휕푉(푥,푡) = −푞 (4) 휕푥 (푥,푡) We obtain by deriving twice respect to 푥 the equation (2):
2 4 2 1 푦(푥,푡) 푀(푥,푡) −푞(푥,푡) ( ) = = ( ) = (5) 푥² 푅 휕푥4 푥² 퐸퐼 퐸퐼 So:
4 푦(푥,푡) 퐸퐼 = −푞 (6) 휕푥4 (푥,푡) By equating equations (1) and (6) in the case of auto-vibration, we obtain the well-known differential equation which controls the self-vibration of a beam on 2 supports: 3
4 휕²푦(푥,푡) 푦(푥,푡) 푚 = −퐸퐼 (7) 휕푡² 휕푥4 So :
4 푦(푥,푡) 푚 휕²푦(푥,푡) + ( ) = 0 (8) 휕푥4 퐸퐼 휕푡²
If we put 푦(푥,푡) = 푞(푡)(푥) (9)
By replacing in equation (8) above, we obtains:
푖푉 퐸퐼푞(푡)(푥) + 푆푞̈(푡)휙(푥) = 0 (10)
We can therefore separate the variables:
퐼푉 푆 푞̈ = − × (11) 퐸퐼 푞
As this equation has to be verified for all 푥 or t, the above ration have to be constant.
We therefore define: 퐼푉 푆 푞̈ 훼4 = − × = (12) 퐸퐼 푞 퐿4
And we define the pulsation (circular natural frequency) ω thus:
훼4 퐸퐼 2 = × (13) 퐿4 푆
We can cut in two parts the differential equation (12) in two equations (separation of variables):
훼4 퐼푉 − = 0 (14) (푥) 퐿4 (푥)
2 푞̈(푡) + 푞(푡) = 0 (15)
The study of the first equation gives the following general equation: 푥 푥 푥 푥 = 퐴푠𝑖푛 (훼 ) + 퐵푐표푠 (훼 ) + 퐶푠𝑖푛ℎ (훼 ) + 퐷푐표푠ℎ (훼 ) (16) (푥) 퐿 퐿 퐿 퐿 And the successive derivative gives: 훼 푥 훼 푥 훼 푥 훼 푥 ′ = 퐴 푐표푠 (훼 ) − 퐵 푠𝑖푛 (훼 ) + 퐶 푐표푠ℎ (훼 ) + 퐷 푠𝑖푛ℎ (훼 ) (17) (푥) 퐿 퐿 퐿 퐿 퐿 퐿 퐿 퐿 훼2 푥 훼2 푥 훼2 푥 훼2 푥 ′′ = −퐴 푠𝑖푛 (훼 ) − 퐵 푐표푠 (훼 ) + 퐶 푠𝑖푛ℎ (훼 ) + 퐷 푐표푠ℎ (훼 ) (18) (푥) 퐿2 퐿 퐿2 퐿 퐿2 퐿 퐿2 퐿 훼3 푥 훼3 푥 훼3 푥 훼3 푥 ′′′ = −퐴 푐표푠 (훼 ) + 퐵 푠𝑖푛 (훼 ) + 퐶 푐표푠ℎ (훼 ) + 퐷 푠𝑖푛ℎ (훼 ) (19) (푥) 퐿3 퐿 퐿3 퐿 퐿3 퐿 퐿3 퐿 훼4 푥 훼4 푥 훼4 푥 훼4 푥 퐼푉 = 퐴 푠𝑖푛 (훼 ) + 퐵 푐표푠 (훼 ) + 퐶 푠𝑖푛ℎ (훼 ) + 퐷 푐표푠ℎ (훼 ) (20) (푥) 퐿4 퐿 퐿4 퐿 퐿4 퐿 퐿4 퐿 The 4 boundary conditions allows to write: 4
0 (x0) 0 (xL)
II 0 (x0)
II 0 (xL) 0 0 0 0 = 퐴푠𝑖푛 (훼 ) + 퐵푐표푠 (훼 ) + 퐶푠𝑖푛ℎ (훼 ) + 퐷푐표푠ℎ (훼 ) (21) (푥=0) 퐿 퐿 퐿 퐿
(푥=0) = 0퐴 + 퐵 + 0퐶 + 퐷 = 0 (22) 퐿 퐿 퐿 퐿 = 퐴푠𝑖푛 (훼 ) + 퐵푐표푠 (훼 ) + 퐶푠𝑖푛ℎ (훼 ) + 퐷푐표푠ℎ (훼 ) (23) (푥=퐿) 퐿 퐿 퐿 퐿
(푥=퐿) = 퐴푠𝑖푛(훼) + 퐵푐표푠(훼) + 퐶푠𝑖푛ℎ(훼) + 퐷푐표푠ℎ(훼) = 0 (24) 훼2 0 훼2 0 훼2 0 훼2 0 ′′ = −퐴 푠𝑖푛 (훼 ) − 퐵 푐표푠 (훼 ) + 퐶 푠𝑖푛ℎ (훼 ) + 퐷 푐표푠ℎ (훼 ) (25) (푥=0) 퐿2 퐿 퐿2 퐿 퐿2 퐿 퐿2 퐿
훼2 훼2 ′′ = 0퐴 − 퐵 + 0퐶 + 퐷 = 0 (26) (푥=0) 퐿2 퐿2 훼2 퐿 훼2 퐿 훼2 퐿 훼2 퐿 ′′ = −퐴 푠𝑖푛 (훼 ) − 퐵 푐표푠 (훼 ) + 퐶 푠𝑖푛ℎ (훼 ) + 퐷 푐표푠ℎ (훼 ) (27) (푥=퐿) 퐿2 퐿 퐿2 퐿 퐿2 퐿 퐿2 퐿
훼2 훼2 훼2 훼2 ′′ = −퐴 푠𝑖푛(훼) − 퐵 푐표푠(훼) + 퐶 푠𝑖푛ℎ(훼) + 퐷 푐표푠ℎ(훼) = 0 (28) (푥=퐿) 퐿2 퐿2 퐿2 퐿2 We can put these equations on the form of a matrix: 0 퐵 0 퐷 퐴푠𝑖푛훼 퐵푐표푠훼 퐶푠𝑖푛ℎ훼 퐷푐표푠ℎ훼
훼² 훼² 0 −퐵 0 퐷 (29) 퐿² 퐿² 훼²푠𝑖푛훼 훼²푐표푠훼 훼²푠𝑖푛ℎ훼 훼²푐표푠ℎ훼 −퐴 −퐵 퐶 퐷 [ 퐿² 퐿² 퐿² 퐿² ] The determinant of this matrix have to be null.
We notice that from the equations (1) and (3) that B = D = 0 푥 푥 푥 푥 = 퐴푠𝑖푛 (훼 ) + 퐵푐표푠 (훼 ) + 퐶푠𝑖푛ℎ (훼 ) + 퐷푐표푠ℎ (훼 ) (30) (푥) 퐿 퐿 퐿 퐿 푥 푥 = 퐴푠𝑖푛 (훼 ) + 퐶푠𝑖푛ℎ (훼 ) (31) (푥) 퐿 퐿
푆 푞̈ 훼4 And we have: − × = (32) 퐸퐼 푞 퐿4 The two equations (2) and (4) gives:
(푥=퐿) = 퐴푠𝑖푛(훼) + 퐶푠𝑖푛ℎ(훼) = 0 (33)
훼2 훼2 ′′ = −퐴 푠𝑖푛(훼) + 퐶 푠𝑖푛ℎ(훼) = 0 (34) (푥=퐿) 퐿2 퐿2 5
So:
(푥=퐿) = 퐴푠𝑖푛(훼) + 퐶푠𝑖푛ℎ(훼) = 0 (35)
′′(푥=퐿) = −퐴푠𝑖푛(훼) + 퐶푠𝑖푛ℎ(훼) = 0 (36)
The sum of these two equations gives:
2퐶푠𝑖푛ℎ(훼) = 0 => 훼 = 0 표푢 퐶 = 0 (37)
The subtraction of these two equations gives:
2퐴푠𝑖푛(훼) = 0 => 훼 = 푛 And so C = 0 (38)
Is the Eigen value of the system.
We obtain the following equation of the Eigen mode: 푛푥 = 퐴푠𝑖푛 ( ) (39) (푥) 퐿 That we write in function of n (quantification):
The Eigen mode is so written: 푛푥 = 퐴 푠𝑖푛 ( ) (40) 푛(푥) 푛 퐿 The norm of the Eigen mode is so written:
퐿 퐿 푛푥 2( ) 2 퐿 ∫ 푛(푥) 푛(푥) = ∫ 퐴푛푠𝑖푛 푑푥 (41) 0 0
1 We replace du sin²(ax) by (1 − cos (2푎푥)) (42) 2 퐿 퐿 2 1 2푛푥 ∫ 푛(푥) 푛(푥) = 퐴푛 ∫ (1 − 푐표푠 ( )) 푑푥 (43) 0 0 2 퐿 2푛푥 We define 푢 = 퐿 2푛 푑푢 = 푑푥 퐿 퐿 푑푥 = 푑푢 2푛 퐿 퐿 퐿 2 1 2 1 퐿 ∫ 푛(푥) 푛(푥) = 퐴푛 ∫ 푑푥 + 퐴푛 ∫ 푐표푠푢 푑푢 (44) 0 0 2 2 0 2푛 The antiderivative of cosu is sinu :
퐿 퐿 2 푥 퐿 2푛푥 퐿 퐴푛퐿 ∫ (푥) (푥) = 퐴2 [( − 푠𝑖푛 ( ))] = 퐴2 ( − 0 − (0 − 0)) = (45) 푛 푛 푛 2 4푛 퐿 푛 2 2 0 0
2 We note that if we chose퐴 = √ (46)) the modes (푥) would form an orthonormal basis for the 푛 퐿 푛 canonical scalar product. In the case of normed Eigen vector we have so: 6
2 푛푥 = √ 푠𝑖푛 ( ) (47) 푛(푥) 퐿 퐿
We have to study the second equation now :
2 푞̈(푡) + 푞(푡) = 0
We fix 푞(푡)as following:
푞(푡) = 푎푐표푠푡 + 푏푠𝑖푛푡
푞̇(푡) = −푎푠𝑖푛푡 + 푏푐표푠푡
2 푞̈(푡) = −푎²푐표푠푡 − 푏 푠𝑖푛푡
At the time t = 0, the system is in static, and so the speed is null:
푞̇(푡=0) = −푎푠𝑖푛0 + 푏푐표푠0
That imply b = 0:
And so the final deflection of the beam is:
푦(푥,푡) = 푞(푡)(푥) (9)
2 푛푥 푦 = 푎√ 푠𝑖푛 ( ) cos(휔푡) (9푏𝑖푠) 푛(푥,푡) 퐿 퐿
Note
We find this result by writing that the determinant of the matrix of the components must be zero.
Determination of circular natural frequency:
훼4 퐸퐼 2 = × (48) 퐿4 푆
With α = n (49)
푛44 퐸퐼 푛²² 퐸퐼 2 = × (50) => = √ (51) 퐿4 푆 퐿² 푆
With these equations, we can define the different Eigen mode of vibration of the beam (cf. figure 2).
1st mode :
² 퐸퐼 1 = √ 퐿² 푆
2 푥 = √ 푠𝑖푛 ( ) 1(푥) 퐿 퐿 7
2nd mode : 4² 퐸퐼 = √ 2 퐿² 푆
2 2푥 = √ 푠𝑖푛 ( ) 2(푥) 퐿 퐿 3rd mode :
9² 퐸퐼 = √ 2 퐿² 푆
2 3푥 = √ 푠𝑖푛 ( ) 3(푥) 퐿 퐿 Figure 2 – Eigen mode and circular natural frequencies of a beam on two supports –
1.2) Particle in a potential well with an infinite dimension
We consider a particle in a potential well as defined below (cf. figure 3).
m v
v m
L
푥
0
Figure 3 – Particle in movement in a potential well –
The Schrodinger equation independent of the time can be written as following: 2 2 ħ 휕 (푥) ( ) + 퐸 = 0 (52) 2푚 휕푥2 푚 (푥) 8
This equation can be also written on the form curvature = k energy 2 푑 (푥) 2푚 = − ( ) 퐸 (52 푏𝑖푠) 푑푥2 ħ2 (푥) 푚
The dimensional equation is so the following: 1 푘𝑔 푘𝑔푚² = 2 × 푚² 푘𝑔푚²푠 푠² ( ) 푠² 1 푠² 푘𝑔푚² = × 푚² 푘𝑔푚4 푠² 1 푠² 푈 = × 푚² 푘𝑔푚 푚3
The boundary conditions are therefore the following: The particle is present in the potential well: The particle is not present in the potential well: 0 ≤ 푥 ≤ 퐿 푥 < 0 ou 푥 > 퐿 2 2 ħ 휕 (푥) ( ) + 퐸 = 0 = 0 2푚 휕푥2 푚 (푥) (푥) The particle follows the Schrodinger equation there is no particle
Otherwise, (푥) is a continuous function. The consequence is so: (0) = (퐿) = 0
So: 2 푑 (푥) + 푘2 = 0 (53) 푑푥2 (푥) With: 2푚퐸푚 푘2 = (54) ħ2
The solution of the wave function for this differential equation is of the following shape:
(푥) = 퐴푠𝑖푛(푘푥) + 퐵푐표푠(푘푥) (55)
We suppose that the energy Em is positive.
By use of the boundary condition we can find the constant A and B:
In 푥 = 0, (0) = 0 : That this imply that 퐵 = 0
(푥) = 퐴푠𝑖푛(푘푥)
In 푥 = 퐿, (퐿) = 0 :
(퐿) = 퐴푠𝑖푛(푘퐿) = 0
It is so necessary that with A and k different from 0 that 푘퐿 = 푛 That imply that: 푛 푘 = (56) 퐿 With n an integer =1,2,3……
As we have posed: 9
2푚퐸푚 푘2 = ħ2 The result is : 2 2 2푚퐸푚 푛 푘2 = = (57) ħ2 퐿2
We finally obtain the quantified values of the energy: 푛22ħ2 퐸 = (58) 푚,푛 2푚퐿2 With n = 1,2,3,…
So, the Eigen values of the energy are quantified.
We can finally write the expression of the wave function: 푛푥 = 퐴푠𝑖푛 ( 푥) (푥) 퐿 We are looking for now the value of the constant A:
As the particle has to be somewhere in the quantum well 2 The function ‖(푥)‖ = 1 to satisfy the density of probability (the particle must be somewhere in the box between 0 and L). 퐿 2 ∫ ‖(푥)‖ 푑푥 = 1 (59) 0 퐿 푛푥 퐴2 퐿 2푛푥 퐴2 퐿 2푛푥 퐿 ∫ 퐴2푠𝑖푛² ( ) 푑푥 = ∫ {1 − 푐표푠 ( )} 푑푥 = [푥 − 푠𝑖푛 ( )] 0 퐿 2 0 퐿 2 2푛 퐿 0 퐴2 퐿 2푛퐿 퐴2퐿 = (퐿 − 푠𝑖푛 ( )) = = 1 2 2푛 퐿 2 2 So 퐴 = √ (60) 퐿
And the solution of the problem in the quantum well, the wave function, is so:
2 푛푥 = √ 푠𝑖푛 ( 푥) (61) (푥) 퐿 퐿 Wave function probability to be present
2 푛푥 2 2 푛푥 = √ 푠𝑖푛 ( 푥) ‖ ‖ = 푠𝑖푛² ( 푥) (62) (푥) 퐿 퐿 (푥) 퐿 퐿
Not about the Energy: ℎ By the following classic notation:퐸 = ℎ푣 = 푇 2 And with 푇 = ℎ 퐸 = = ħ 2 푛²² ħ2 ħ = ( ) 퐿² 2m
푛²² ħ = ( ) (63) 퐿² 2m 10
1.3 Conclusion of this first part
Table 1 below shows the perfect analogy between the vibrations of a beam on two supports (resulting from the strength of the materials) and the quantification of the energy of a particle in a quantum well (derived from quantum mechanics).
Case Eigen value Eigen Modes studied Natural circular Frequencies Quantified Eigen mode of Expression of the Wave energy oscillation of a beam function Beam on Without Without interests 푛²² 퐸퐼 2 푛푥 two interests √ = (√ ) 푛(푥) = 푠𝑖푛 ( ) supports 퐿² 푆 퐿 퐿 퐼푉 4 퐼푉 푆 푞̈ 훼4 푆 푞̈ 훼 = − × = = − × = 4 퐸퐼 푞 퐿4 퐸퐼 푞 퐿 2 4 4 푑 푞(푡) 훼 퐸퐼 퐼푉 훼 − = 0 + ( ) 푞(푡) (푥) 퐿4 (푥) 푑푡² 퐿4푆 푦(푥,푡) = 푞(푡)(푥) = 0 푦(푥,푡) = 푞(푡)(푥) ℎ 2 Particle With the notation :퐸 = ℎ푣 = 푛²² ħ Without interests 푇 E = ( ) 2 푛푥 in the 퐿² 2m = √ 푠𝑖푛 ( ) 푛²² ħ 푛(푥) 퐿 퐿 potential = ( ) 퐿² 2m 2 well 푑 (푥) 2푚퐸 2 + ( 푚) = 0 푑 (푥) 2푚퐸푚 2 2 (푥) + ( ) = 0 푑푥 ħ 푑푥2 ħ2 (푥) Tableau 1 – Analogy between the natural circular frequencies of vibration of a beam on 2 supports and the quantified energy of a particle in a quantum well
The circular natural frequencies of a beam in strength of materials have an analogy with the jump 퐸퐼 ħ of quantified energy Em of a particle in a quantum well. (√ ) => ( ) 푆 2m 푅퐷푀 푞푢푎푛푡푖푞푢푒 The Eigen modes s of vibration of a beam in strength of materials have an analogy with the Eigen mode of the wave function in the quantum well.
2nd part) Relation between curvature and energy in 1 and 2 dimensions (beam and slab in strength of materials) and in 4 dimensions (General relativity)
2.1) Case of the beam on two simple supports Considering the same beam as that defined in figure 1.
The fundamental relation connecting the curvature (1/R) at the bending moment 푀(푥)and at the second derivative of the deflection 푦(푥) writes:
2 푑 푦 푀(푥) 1 = − = (64) 푑푥2 퐸퐼 푅
In this expression, 푦(푥) is the deflection of the beam (m), 푀(푥) the bending moment (N.m), E the young modulus of the material that constitute the beam (N/m²), I the inertia in(m4) and R the curvature radius in (m). 11
The exact expression of the curvature is given in the expression (65). The part in cube root is negligible.
푑²푦 1 = 푑푥² (65) 푅 3 푑푦 2 (√1 + ( ) ) 푑푥
In addition the elastic bending energy of a beam can be written as:
퐿 2 1 푀(푥) 푈 = ∫ 푑푥 (66) 2 0 퐸퐼 Considering to simplify a beam under a constant bending moment M at each extremity (cf. figure 4).
R 푀, 퐸, 퐼 푥 M 푦(푥,푡) M
퐿
푦 Figure 4 – Beam on two supports under a bending moment M at each extremity –
The bending moment equation (pure bending) is the following:
푀(푥) = 푀 (67)
From the equation (64) w obtain:
푑2푦 푀 1 = − = (68) 푑푥2 퐸퐼 푅 퐸퐼 푀 = 푀 = − (69) (푥) 푅 By introducing the expression (69) in the expression of bending energy (66):
1 퐿 (퐸퐼)2 1 퐿 퐸퐼 푈 = ∫ 2 푑푥 = ∫ 2 푑푥 2 0 푅 퐸퐼 2 0 푅 With the curvature that is constant, we obtain: 1 퐸퐼퐿 푈 = 2 푅2 So in pure bending: 1 2 푈 = ( ) (70) 푅2 퐸퐼 퐿 12
We obtain thus a relation between the curvature and the strain energy of a beam
Note: By integral two times from 푥 the expression (68) and considering that the deflection have to be null on each support, we obtain the expression of the deflection.
푀 푀퐿 푦 = − 푥2 + 푥 (71) (푥) 2퐸퐼 2퐸퐼 And we find the well-known result of the strength of materials: 푀퐿2 푀퐿2 푀퐿2 푦 = − + = (72) (퐿/2) 8퐸퐼 4퐸퐼 8퐸퐼 2 푑 푦(푥) 푀 1 − = 푑푥2 퐸퐼 푅
2.2) Case of a thin plate of thickness h Considering a thin plane h, of sides 푥 et 푦 sunder a bending moments Mx (cf. figure 5) :
푀 푥 푦
ℎ
푀 푥 푥
Figure 5 – Plate under a bending moment on its sides –
푀푥 Represent a bending moment by unit of length:
Under the bending moment, the plate is in bending and takes a curvature of radius R as indicated at the figure 6 below:
:
θ R
푑푠 ≈ 푥 Figure 6 – Relation between the radius of curvature R of the plate and each mean fiber –
We have following the figure 6: 푅푥휃 = 푑푠 ≈ 푥 (73) And we have a relation connecting the curvature 1/R and the defection w of the plate:
2 1 휕 푤(푥,푦) = 2 (74) 푅푥 휕푥 13
By introducing the expression (74) in the expression (73) we obtain:
2 푥 휕 푤(푥,푦) 휃 = = − 2 푥 (75) 푅푥 휕푥 The elastic bending energy of the plate can be written so: 1 푈 = 푀 푦휃 (76) 2 푥 2 1 휕 푤(푥,푦) 푈 = − 푀 푥푦 (77) 푥 2 푥 휕푥2 The contribution of energy following 푦 can be written following the same approach if a bending 푀푦 is now applied on the side 푥 :
2 1 휕 푤(푥,푦) 푈 = − 푀 푥푦 (78) 푦 2 푦 휕푦2
The contribution due at the torsion can be also written:
2 1 휕 푤(푥,푦) 푈 = 2푀 푥푦 (79) 푥푦 2 푥푦 휕푦휕푥
Finally the strain energy of the plate can be defined:
푈 = 푈푥 + 푈푦 + 푈푥푦 (80)
2 2 2 1 휕 푤(푥,푦) 1 휕 푤(푥,푦) 1 휕 푤(푥,푦) 푈 = − 푀 푥푦 − 푀 푥푦 + 2푀 푥푦 (81) 2 푥 휕푥2 2 푦 휕푦2 2 푥푦 휕푦휕푥 So :
2 2 2 1 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푈 = [−푀 + 2푀 − 푀 ] 푥푦 (82) 2 푥 휕푥2 푥푦 휕푦휕푥 푦 휕푦2
By using the well-known relation connecting the bending moment at the second derivative of the deflection:
2 2 2 2 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푀 = −퐷 ( + 푣 ) (83) , 푀 = −퐷 ( + 푣 ) (84) 푥 휕푥2 휕푦2 푦 휕푦2 휕푥2
1 1 1 1 푀푥 = 퐷 ( + 푣 ) (85), 푀푦 = 퐷 ( + 푣 ) (86) 푅푥 푅푦 푅푦 푅푥
2 휕 푤(푥,푦) 푀 = 퐷(1 − 푣) (87) 푥푦 휕푥휕푦
And by reporting these expressions into (82) we obtain:
2 2 2 2 2 1 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푈 = [퐷 ( + 푣 ) + 2퐷(1 − 푣) 2 휕푥2 휕푦2 휕푥2 휕푥휕푦 휕푦휕푥 2 2 2 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) + 퐷 ( + 푣 ) ] 푥푦 (88) 휕푦2 휕푥2 휕푦2
By developing the expression (88): 14
2 2 2 2 2 2 2 2 1 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푈 = [퐷 ( ) 푣퐷 + 2퐷 − 2푣퐷 2 휕푥2 휕푥2 휕푦2 휕푥휕푦 휕푦휕푥 휕푥휕푦 휕푦휕푥 2 2 2 2 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) + 퐷 ( ) + 푣퐷 ] 푥푦 휕푦2 휕푥2 휕푦2
2 2 2 2 2 2 2 2 퐷 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푈 = [( ) + 2푣 + 2(1 − 푣) ( ) + ( ) ] 푥푦 2 휕푥2 휕푥2 휕푦2 휕푥휕푦 휕푦2
2 2 2 2 2 2 2 2 퐷 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 푈 = [( ) + 2푣 + 2(1 − 푣) ( ) + ( ) ] 푥푦 (89) 2 휕푥2 휕푥2 휕푦2 휕푥휕푦 휕푦2
We obtain a relation between the curvature and the density of strain energy of the plate:
1 2 1 2 1 2 1 1 24(1 − 휈2) 푈 [( ) + ( ) + 2(1 − 휈) {( ) } + 2휈 { }] = 3 × (90) 푅푥 푅푦 푅푥푦 푅푥 푅푦 퐸ℎ 푥푦
Note:
The expression below allow to reformulate and to simplify the expression (89):
2 2 2 휕2푤 휕2푤 휕2푤 휕2푤 휕2푤 휕2푤 휕2푤 휕2푤 휕2푤 −2(1 − 푣) [ − ( ) ] = −2 + 2 ( ) + 2푣 − 2푣 ( ) (91) 휕푥2 휕푦2 휕푥휕푦 휕푥2 휕푦2 휕푥휕푦 휕푥2 휕푦2 휕푥휕푦
2 2 2 2 2 2 2 2 2 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤 휕 푤 ( + ) = ( ) + ( ) + 2 (92) 휕푥2 휕푦2 휕푥2 휕푦2 휕푥2 휕푦2
The sum of the two expressions (91) et (92) above gives again the expression (89) :
2 2 2 2 2 2 2 2 2 2 2 2 2 2 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤 휕 푤 휕 푤 휕 푤 휕 푤 휕 푤 휕 푤 휕 푤 ( ) + ( ) + 2 − 2 + 2 ( ) + 2푣 − 2푣 ( ) 휕푥2 휕푦2 휕푥2 휕푦2 휕푥2 휕푦2 휕푥휕푦 휕푥2 휕푦2 휕푥휕푦
Therefore the expression (89) can be written from the two expression (91) and (92):
2 2 2 2 2 2 2 퐷 휕 푤(푥,푦) 휕 푤(푥,푦) 휕 푤 휕 푤 휕 푤 푈 = [( + ) − 2(1 − 푣) [ − ( ) ]] 푥푦 (93) 2 휕푥2 휕푦2 휕푥2 휕푦2 휕푥휕푦
With for the bending rigidity D:
퐸ℎ3 퐷 = (94) 12(1 − 푣²)
퐷 1 1 2 1 1 1 2 푈 = [(− − ) − 2(1 − 푣) [ − ( ) ]] 푥푦 (95) 2 푅푥 푅푦 푅푥 푅푦 푅푥푦
If we present the expression to put in first plane the curvature in one side and the energy of the other side:
1 1 2 1 1 1 2 2 푈 [(− − ) − 2(1 − 푣) [ − ( ) ]] = ( ) (96) 푅푥 푅푦 푅푥 푅푦 푅푥푦 퐷 푥푦 15
We are developing now this expression in the case of the pure flexion:
The radius of curvature are all equal. The radius of curvature due to the torsion is null.
휕2푤 1 1 1 = 0, = = (97) 휕푥휕푦 푅푥 푅푦 푅 By introducing these expressions in the equation (96) we obtain:
퐷 2 2 1 푈 = [(− ) − 2(1 − 푣) [ ]] 푥푦 2 푅 푅2
퐷 4 2 1 푈 = [ − + 2푣 ] 푥푦 2 푅2 푅2 푅2 퐷 2 1 푈 = [ + 2푣 ] 푥푦 2 푅2 푅2 퐷 푈 = [1 + 푣]푥푦 푅2 So:
1 1 푈 = (98) 푅2 퐷(1 + 푣) 푥푦
If we take into account of the bending rigidity (94) and if we define fix 푥. 푦 = 퐴
1 12(1 − 푣²) 푈 = 푅2 퐸ℎ3(1 + 푣) 퐴 1 12(1 − 푣²) 푈 = × 푅2 퐸ℎ2(1 + 푣) 퐴ℎ 1 12(1 + 푣)(1 − 푣) 푈 = × 푅2 퐸ℎ2(1 + 푣) 퐴ℎ
We obtain in the case of the pure bending: Energy density Curvature 1 12(1 − 푣) 푈 = × (99) 푅2 퐸ℎ2 퐴ℎ
Proportionality factor And the dimensional equation is: proportionnalité 1 1 푈 = × 푘𝑔푚 푚3 푚² × 푚² 푠²푚² 1 푠² 푈 = × 푚² 푘𝑔푚 푉 16
2.3) Case of the General relativity
2.3.1) Expression Einstein Equation
The Einstein equation has to be written as following:
퐺휇푣 = 휅푇휇푣 (100)
In this equation, the curvature and the energy of the space time are linked (cf. figure 7).
Figure 7 – Symbolic view of the curvature of the space time by projection in a plane space –
With:
퐺휇푣 is the Einstein tensor. 8휋퐺 휅 = (101) 푐4 In addition the dimensional equation of κ is the following:
퐿3푇4 푇² 푠² 휅 = = = (102) 푀푇²퐿4 푀퐿 푘𝑔푚 The developed equation of Einstein becomes: 1 8휋퐺 푅 − 𝑔 푅 + 훬𝑔 = 푇 (103) 휇휈 2 휇휈 휇휈 푐4 휇휈 1 푠² 푈 = × 푚² 푘𝑔푚 푉 2.3.2) Details on the curvature tensor
The Ricci tensor is obtained by contraction of the Riemann tensor on the indices λ:
푅휇푣 = 푅휇푣
R is a contraction of the tensor 푅휇푣 . The curvature tensor or Riemann tensor is written as following:
푅 휇푣훼 = 훤휇훼,푣 − 훤휇푣,훼 + 훤푣훤휇훼 − 훤훼훤휇푣 (104) 17
Or in another way:
휕훤휇훼 휕훤휇푣 푅 = − + 훤 훤 − 훤 훤 (104 푏𝑖푠) 휇푣훼 휕푥푣 휕푥훼 푣 휇훼 훼 휇푣 By definition the definition of the Christoffel symbols is the following:
1 1 휕𝑔휇 휕𝑔푣 휕𝑔휇푣 훤 = 𝑔(𝑔 + 𝑔 − 𝑔 ) = 𝑔 ( + − ) (105) 휇푣 2 휇,푣 푣,휇 휇푣, 2 휕푥푣 휕푥휇 휕푥 gμν is the metric,
The coefficient of the metric are issued of the differential distance (special relativity):
푑푠2 = 푐2푑푡2 − 푑푥2 − 푑푦2 − 푑푧2 (106)
So, considering the Einstein convention of summation:
2 휇 푣 푑푠 = 𝑔휇푣푑푥 푑푥 (107)
휕휉훼 휕휉훽 𝑔 = (108) 휇푣 휇푣 휕푥휇 휕푥푣
휉0 = 푐푡; 휉1 = 푥; 휉2 = 푦; 휉3 = 푧 (109)
1 −0 −0 −0 0 −1 − = [ 0 −0] (110) 휇푣 0 −0 −1 −0 0 −0 −0 −1
Λ is the cosmologic constant (possible candidate to explain the dark energy and dark matter)
The are the first derivative of the metric gμν :
For example in coordinates (t,r,θ,φ) :
푡 푟 휃 휑 휕훤 휕훤푟푟 휕훤 휕훤푟휑 훤 = 훤푡 + 훤푟 + 훤휃 + 훤휑 = 푟푡 + + 푟휃 + (111) 푟,푟 푟푡,푟 푟푟,푟 푟휃,푟 푟휑,푟 휕푟 휕푟 휕푟 휕푟 So the 훤푟,푟 are the second derivatives of the metrics 𝑔휇푣 that we can by analogy compare in one and two dimensions with the expressions of curvatures given in the expression (70) , (90), (96) and (99).
So 퐺휇푣 has the dimensional value of a curvature in 1/m²
For example the first terms 푅 휇푣훼 (see 104 bis)
1 휕𝑔휇 휕𝑔푣 휕𝑔휇푣 휕 { 𝑔 ( + − )} 휕훤휇훼 2 휕푥푣 휕푥휇 휕푥 = 휕푥푣 휕푥푣 18
2.3.3) Detail on the stress energy tensor
We have showing below that the component of the stress energy tensor 푇휇휈 have the dimension of an energy density G = 6.6726x10-11 m3kg-1s-2 1J = 1W.s = 1 N.m = 1kg.m².s-2 kg
8휋퐺 퐸푛푒푟𝑔푦 8휋퐺 푀푎푠푠 퐶푢푟푣푎푡푢푟푒 = × = (112) 푐4 푉표푙푢푚푒 푐2 푉표푙푢푚푒
1/L² (L/T)4 (L3)
The stress energy tensor is a matrix where the component are given below (113):
2 푚훾 푐² 2 2 2 휌훾 푐푣푥 휌훾 푐푣푦 휌훾 푐푣푧 푉 2 2 휌훾2푣 푣 휌훾2푣 푣 푇휇푣 = 휌훾 푐푣푥 휌훾 푣푥푣푥 푥 푦 푥 푧 (113) 2 2 휌훾2푣 푣 2 휌훾 푐푣푦 휌훾 푣푦푣푥 푦 푦 휌훾 푣푦푣푧 2 2 2 2 [휌훾 푐푣푧 휌훾 푣푧푣푥 휌훾 푣푧푣푦 휌훾 푣푧푣푧] 푚 Where γ is le factor of Lorentz (114) (cf. special relativity), the energy density 휌 = with m the mass and V 푉 unitary volume, c the speed of light, vi a speed in the direction i.
The factor of Lorentz becomes from the Lorentz transformation that imply that the speed of light stay constant in all the referential. 1 1 훾 = = (114) 푣2 √1 − 훽2 √1 − 푐2 We show that at low velocity (in a non-relativistic situation) and in 3 dimensions, this tensor with 16 components in four-dimensional space-time actually includes the stress tensor (9 components) of the continuum mechanics from which derive the strength of materials [11].
The stress tensor in two dimensions can be written as following:
휎푥푥 푥푦 휎 = [ ] (115) 푖푗 푥푦 휎푦푦
And in 3 dimensions:
휎푥푥 휏푥푦 휏푥푧 휏 휎 휏 휎푖푗 = [ 푦푥 푦푦 푦푧] (116) 휏푧푥 휏푧푦 휎푧푧
In theory of elasticity, that come from the continuum mechanics, the relation between the stress tensor 휎푖푗 and the applied force Qi on a surface of normal nj can be written as following:
푄푖 = 휎푖푗푛푗 (117)
In a field of a variational approach, the stress tensor can be written as follow: 19
훥푄푖 휎푖푗 = 푤𝑖푡ℎ 훥푛푗 → 0 (118) 훥푛푗
So, with m the mass, ρ the density of mass energy, V the volume and ai the acceleration, we have:
훥푄푖 훥(푚 × 푎푖) 훥(휌. 푉. 푎푖) 휎푖푗 = = = (119) 훥푛푗 훥푛푗 훥푛푗
If we make the hypothesis that the variation of the force is due only to the variation of the Volume V in function of the time t we obtain with: 푣 푎 = 푖 (120) 푖 훥푡
훥(휌. 푉. 푎푖) 1 훥푉 휎푖푗 = = 휌 ( ) 푣푖 (121) 훥푛푗 훥푛푗 훥푡 We can of course define the volume V:
푉 = 훥푥푖. 훥푥푗. 훥푥푘 (122)
Thus we obtain:
1 훥푥푖. 훥푥푗. 훥푥푘 휎푖푗 = 휌 ( ) 푣푖 (123) 훥푛푗 훥푡
We can replace the nj by its value:
푛푗 = 훥푥푖. 훥푥푘 (124)
푣푖 훥푥푖. 훥푥푗. 훥푥푘 휎푖푗 = 휌 ( ) (125) 훥푡 훥푥푖. 훥푥푘 After simplification we obtain:
훥푥푗 휎 = 휌푣 ( ) (126) 푖푗 푖 훥푡 By definition of a speed, we have:
훥푥푗 푣 = ( ) (127) 푗 훥푡 We obtain finally the expression of the stress tensor at low speed in function of the energy density ρ and based on the multiplication of the velocity vi and vj:
휎푖푗 = 휌푣푖푣푗 (128)
The stress energy tensor becomes from the product of the density of energy and the multiplication of the four velocity (4 dimension of the space time) issued from the general relativity.
푇휇푣 = 휌푢휇푢푣 (129)
With for the four velocity: 20
훾푐 훾푣 푢 = | 푥 (130) 휇 훾푣푦 훾푣푧 Under low speed γ=1 the stress energy tensor becomes:
푚푐² 휌푐푣푥 휌푐푣푦 휌푐푣푧 푉 휌푣 푣 휌푣 푣 푇휇푣 = 휌푐푣푥 휌푣푥푣푥 푥 푦 푥 푧 (131) 휌푐푣푦 휌푣푦푣푥 휌푣푦푣푦 휌푣푦푣푧 [휌푐푣푧 휌푣푧푣푥 휌푣푧푣푦 휌푣푧푣푧] Based on the definition of the stress tensor, (cf. equation 128), the stress energy tensor at low speed can be written as following:
푚푐² 휌푐푣 휌푐푣 휌푐푣푥 푦 푧 푉 휏 휏 푇휇푣 = 휌푐푣푥 휎푥푥 푥푦 푥푧 (132) 휌푐푣푦 휏푦푥 휎푦푦 휏푦푧 [휌푐푣푧 휏푧푥 휏푧푦 휎푧푧 ] The Einstein equation build a link in 4 dimensions (space time) with the curvature tensor 3 퐺휇푣(dimension 1/m²) and the stress energy tensor 푇휇푣 (dimension energy/m ) that is itself a generalization in 4 dimensions of the stress tensor of the continuum mechanics.
2.4) Conclusion of this second part
The table 2 below makes a synthesis of the results obtained. We can see that the general relativity is a generalization of the elastic theory in 4 dimensions of the space time.
Theory Number of Formulation between energy and curvature Example of application (pure bending) or expression considered dimensions at low speed Beam on 1 푑²푦 1 2 푈 1 2 = ( ) two = 푑푥² 푅 퐸퐼 퐿 simple 푅 3 푑푦 2 supports (√1 + ( ) ) in 푑푥 elasticity 퐿 2 2 1 푀(푥) 푑 푦 푀(푥) 1 푈 = ∫ 푑푥 2 = − = 2 0 퐸퐼 푑푥 퐸퐼 푅 Thin plate 2 2 2 1 12(1 − 푣) 푈 1 1 1 1 1 2 푈 = × [(− − ) − 2(1 − 푣) [ − ( ) ]] = ( ) 푅2 퐸ℎ3 퐴 푅푥 푅푦 푅푥 푅푦 푅푥푦 퐷 푥푦 General 4 1 8휋퐺 퐺 = 휅푇 푅 − 𝑔 푅 + 훬𝑔 = 푇 휇푣 휇푣 relativity 휇휈 2 휇휈 휇휈 푐4 휇휈 푚푐² 휌푐푣 휌푐푣 휌푐푣푥 푦 푧 푉 휏 휏 푇휇푣 = 휌푐푣푥 휎푥푥 푥푦 푥푧 휌푐푣푦 휏푦푥 휎푦푦 휏푦푧 [휌푐푣푧 휏푧푥 휏푧푦 휎푧푧 ] Tableau 2 – Comparison of the different relations between energy and curvature in function of the number of dimensions considered –
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Conclusions
We have showed on several examples that the strength of materials allows to understand at our scale certain fundamental principles of the quantum mechanics:
- The eigen mode of a beam on two simple supports correspond at the different shape of the wave function connected with the jump energy of a particle in a quantum well, - The natural circular frequencies of a beam on two simple supports correspond at the quantification of the energy of a particle in a potential well.
We have showed that the general relativity is a generalization in 4 dimensions (space time) of the relation curvature/energy equally present in strength of materials for the beam and for the thin plates
Finally, we have showed the stress energy tensor written at low speed give the stress tensor of the elasticity theory.
Extension of this article, next steps:
We have equally shown that in strength of material 2 main equations are necessary. One in static (curvature = K energy density) and one in dynamic (natural frequencies and eigen modes). So if the analogy between the beam (or plate) in strength of material and the space time is total; two equations should be also necessary at large scale. The first is the Einstein Equation connecting the curvature and the energy. The second remain to find.
If the analogy with the strength of the materials is exact we have some clues about this second equation.
This second equation would allow to quantify the space time (analogy with the natural frequency and eigen mode of the beam that represent also the energy quantification and shape of the wave function ψ in quantum mechanics).see table 3 in annex A.
Notice that this equation should be in 4th derivative of the space metric and 2th derivative of the time metric, and of course tensorial written to be valid in all the referential (covariant derivative).
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Annex A - Synthesis of the results – Principle Quantum mechanic Strength of material General relativity Space (1-3) dimensions Space (1-3 dimensions) or time Space-time (4 dimensions) What is Wave function 휓(푥) Deflection of the beam 푦(푥,푡) Metric 𝑔휇휈 with휇휈 = distorted 0(푡) 푡표 3(푥, 푦, 푧) Simplified /symbolic drawing of the case studied
Curvature = Curvature of the wave Curvature and energy density Curvature and energy density K energy function Example in 1 dimension Example in 1 dimension Example in 4 dimensions 2 푑 푦 푀(푥) 1 = − = 푑푥2 퐸퐼 푅 퐿 2 1 푀(푥) 푈 = ∫ 푑푥 2 0 퐸퐼 푳 ퟏ ퟖ흅푮 ퟏ ퟐ 푹 − 품 푹 + 휦품 = 푻 ퟐ ∫ 풅풙 = 푼 흁흂 흁흂 흁흂 ퟒ 흁흂 풅 (풙) ퟐ풎 ퟐ ퟐ 풄 = − ( ) 푬 ퟎ 푹 푬푰 ퟐ ퟐ (풙) 풎 1 2 푈 풅풙 ħ If R constant => = ( ) 푅2 퐸퐼 퐿 Eigen value Example of a particle in a Example of a beam simply supported in To be developed… (natural well potential in 1 dimension 1 dimension (natural frequencies) In 4 dimensions ퟒ th frequency) 풚(풙,풕) 풎 흏²풚(풙,풕) In 4 derivative of the space metric + ( ) = ퟎ th and eigen 흏풙ퟒ 푬푰 흏풕² and 2 derivative of the time metric, mode and of course tensorial written to be 푦(푥,푡) = 푞(푡)(푥) valid in all the referential (covariant ퟐ 풏풙 derivative)? 풚풏(풙,풕) = 풂√ 풔풊풏 ( ) 퐜퐨퐬(흎풕) 푳 푳 푛²² ħ 푛²² 퐸퐼 = ( ) = (√ ) 퐿² 2m 퐿² 푆
퐼푉 4 2 푆 푞̈ 훼 푛²² ħ = − × = E = ( ) 퐸퐼 푞 퐿4 퐿² 2m 훼4 퐼푉 (푥) − 4 (푥) = 0 퐿 In space : 4 4 푑 (푥) 훼 − = 0 푑푥4 퐿4 (푥) 2 푛푥 √ 2 푛푥 푛(푥) = 푠𝑖푛 ( ) = √ 푠𝑖푛 ( ) 퐿 퐿 푛(푥) 퐿 퐿
In time : 2 푑 2 4 (푥) 2푚퐸푚 푑 푞(푡) 훼 퐸퐼 + ( ) = 0 + ( ) 푞 = 0 푑푥2 ħ2 (푥) 푑푡² 퐿4푆 (푡)
Table 3 – Basic evidence of one equation is missing at the space time scale by analogy with the 2 main equations in strength of material –
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Bibliographies References
For the strength of materials: [1] S. Timoshenko,(1949) « Résistance des matériaux Traduit de l'anglais sur la 2e édition par Ch. Laffitte » [2] D. Izabel (2007) « Formulaire de résistance des matériaux tome 1 à 5 » Sebtp [3] S. Timoshenko, (1951) « Théorie des plaques et des coques, Traduit de l'anglais par L. Vial Reliure inconnue » (1951) p 46 chap 12 and p37 chap 10 [4] Ménad CHENAF Jean-Vivien HECKCSTB Guide PPRT (2007) Compléments technique relatif a l’effet de surpression Recommandations et précautions en vue de réduire les risques – rapport d’étude
For the general relativity : [5] T.G. Tenev M.F. Horstemeyer (2016) « The Mechanics of Spacetime { A Solid Mechanics Perspective on the Theory of General» Foundations of Physics manuscript arXiv:1603.07655v3 » p 24 [6] M. R. Beau (2014) « Théorie des champs des contraintes et des déformations en relativité générale et expansion cosmologique ». - Foundations of Physics manuscript arXiv:1209.0611v2 p4 and Annales de la Fondation Louis de Broglie, Volume 40, 2015 [7] T. Damour (2006) – « La Relativité Générale aujourd'hui » Institut des Hautes Etudes Scientifiques Bures-sur-Yvette, France - Séminaire Poincaré 1-40 p [8] T. Damour (2005) « Einstein et la physique du vingtième siècle » Membre de l'Académie des sciences p 4 [9] M. R. Beau (2014) « Sur l'hypothèse concernant l'existence d'un champ de gravité de type vectoriel couplant avec l'accélération des masses » Hal archive ouvertes 00683021v2 point c p13 [10] Collective (2016) Observation of Gravitational Waves from a Binary Black Hole Merger physical review letter [11] Cours de physique « tenseur energie impulsion » http://physique.coursgratuits.net/relativite-generale/tenseur-d-energie-impulsion.php [12] Cours en ligne de richard Taillet univ grenoble– initiation à la relativité générale et restreinte
For the quantum mechanics : [13] C. De Beule (2016) “Graphical interpretation of the Schrodinger equation” university antwerpen [14] Michael Fowler (2007) “Schrödinger’s Equation in 1-D: Some Examples Curvature of Wave Functions” 24
[15] Steven Errede (2016) UIUC Physics 406 Acoustical Physics of Music, Department of Physics, University of Illinois at Urbana-Champaign, IL, [16] Xinzhong Wu (2015) Probability and Curvature in Physics School of History and Culture of Science, Shanghai Jiaotong University, Shanghai, China Journal of Modern Physics, 2015, 6, 2191-2197 [17] Dalibard : Cour en ligne de mécanique quantique X polytechnique [18] Cours en ligne : calcul variational et relativité restreinte X polytechnique
Pour l’ensemble des sujets [19] Feynman (1963) The Feynman Lectures on Physics, Addison-Wesley 1963, vol.II, p.38-10. [20] C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H. Freeman 1973; x17.5, pp.426-428 [21] Paty Michel (2005) HAL L’espace-temps de la théorie de la relativité [22] Bernard Guy (2016) Relier la mécanique et la relativité générale ? Réflexions et propositions