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Development of the Incompatibility Between Einstein's General Noname manuscript No. (will be inserted by the editor) Development of the Incompatibility between Einstein's General Relativity and Heisenberg's Uncertainty Principle Alexandre Georges Received: date / Accepted: date Abstract Are General Relativity and Quantum Mechanics incompatible? Each in their world, that of the infinitely great and that of the infinitely small, they did not seem to interfere as long as they avoided each other. How- ever, it is their fundamental oppositions that prevent the scientific community from achieving a unification of Physics. The proposal of this paper is to pro- vide a mathematical proof of incompatibility, beyond the fact that they have fundamentally different principles, between the foundations of General Rela- tivity and Quantum Mechanics, namely the deformation of the geometry of space-time and the Uncertainty Principle, to provide an absolute limitation in establishing a unifying theory of physics, if any. Keywords Time · General Relativity · Uncertainty Principle · Quantum Mechanics · Unification of Physics · Photon · Particule 1 Introduction It will be demonstrated here that, on the one hand, the conservation of energy is relative and that, on the other hand, by using it, we can precisely and simul- taneously determine the momentum and the position of a subatomic particle, bypassing the problems related to the non-relativistic aspect of said parti- cle. We will thus demonstrate, by an extremely simple mathematical proof, the incompatibility between the time dilatation principle and the Heisenberg's Uncertainty Principle, before interpreting the results on the basis of observa- tions made. Bureau du Professeur Georges Projet Energium (private company laboratory registered: 822 143 707), France E-mail: [email protected] Present address: Professor Alexandre Georges 51 Redcliffe Street, Keighley, United Kingdom, BD21 2PX E-mail: [email protected] 2 Alexandre Georges 2 The relative conservation of energy Let Ω be an object of mass M and Φ a photon of frequency ν and energy E. In the first situation, Φ is at a distance r1 of Ω. In the second situation, Φ is at a distance r2 of Ω. In both situations, Φ is observed from an almost flat space-time. In first situation, the proper time dτ1, relative to the point in which Φ is located in the first situation, is expressed by: s 2GM c2 dτ1 = dt 1 − (1) r1 In second situation, the proper time dτ2, relative to the point in which Φ is located in the second situation, is expressed by: s 2GM c2 dτ2 = dt 1 − (2) r2 If: r1 < r2 (3) Then: dτ1 < dτ2 (4) Thus, the more Φ moves away from Ω, the more the time dτ proper to the point where the photon Φ is, from the point of view of an observer situated in an almost flat space-time, increases with respect to dt. But: E = h × ν (5) We will note E1 the energy of Φ in first situation and E2 the energy of Φ in second situation. So, if we apply the time dilatation to the frequency of the photon wave, we have in first situation: h E1 = (6) r 2GM dt 1 − c2 r1 Then: h E1 = (7) dτ1 And in the second situation: Time Dilatation and Uncertainty Principle 3 h E2 = (8) r 2GM dt 1 − c2 r2 Then: h E1 = (9) dτ1 But: dτ1 < dτ2 (10) So: E1 > E2 (11) The proper times dτ1 and dτ2, derived from the Schwarzschild metric [1], are to be differentiated from the proper time of the photon Φ, these respectively corresponding to the times proper to the positions of the photon Φ in the first and second situations, while dτ corresponds to the time proper to the photon itself, as a particle. Finally, from the point of view of an observer situated in an almost flat space-time, the more Φ moves away from Ω, the lower its energy level decreases, without any energy being transferred from Φ to its environment. Also, the closer Φ gets to Ω, the more its energy increases, again from the point of view of an observer situated in an almost flat space-time, without any energy being brought to the system that Φ constitutes. In conclusion, from the point of view of an observer situated in an almost flat space-time, the energy level of a photon depends on its frequency, depend- ing on the deformation of the geometry of the space-time in which it is located. The conservation of the energy is then relative. 3 Application of the relative conservation of energy to the Heisenberg's Uncertainty Principle Let's start by considering that: ¯h σ .σ ≥ (12) x p 2m Where: h ¯h = (13) 2π It is, according to the Uncertainty Principle, as expressed through its fun- damental equation, essentially impossible to determine with precision both the position x and the momentum p = mv (and therefore by extension the velocity) 4 Alexandre Georges of a particle, regardless of observational qualities [2{4]. We apply this theorem to non-relativistic particles. So, we could not apply relativistic principles to the particle, but we will consider the deformation of the space-time geometry to the point where the particle is located. As seen in section 2, the energy of a photon can vary according to its position. It is then possible, for an observer in an almost flat space-time, to determine the position of the particle moving on an axis, knowing its level of energy. Imagine that we are this observer. Our particle Φ, of known mass m, moves towards another object Ω of known mass M. M is much higher than m. The object Ω is immobile, without rotation, and homogeneous. The axis corre- sponding to the trajectory of the particle Φ is known and rectilinear. Our object Ω is very far, in our simple example, from any other mass object than our particle Φ. As our Φ moves towards our isolated Ω of mass M, it never has the same level of energy from one position to another. If we are able to determine with precision the momentum and the position of the particle, precisely and simultaneously, by applying relativistic principle on the point in which our particle is located, we will prove incompatibility between General Relativity and Quantum Mechanics. Moreover, the fact that the impossibility of determination, namely uncer- tainty, is fundamental and independent of the observational qualities does not make the improbability of the previously proposed observational method re- sponsible for the simultaneous and precise determination of the momentum and the position of the particle. In view of De Broglie's thesis, generalizing the Einstein-Planck relation by applying it to mass particles, we can express: h λ = (14) p h λ = (15) mv As seen by the principles of pair production process and annihilation pro- cess [5], the mass of our particle corresponds to a potential amount of energy that can be produced. We will then study the particle as a quantity (or more generally a set of quantities) of energy. Thus, if we cannot use the dilatation of the time undergone by a non-relativistic particle, we can determine, however, as seen in section 2, the relative energy level corresponding to the mass of the particle, considering the photon that it could be (or more generally the set of quantities of masses that could be removed in the form of extractable energy). Thus, from the quantity (here set of quantities) of mass, we would consider the quantity (here set of quantities) of relative energy producible from it. All this could make it possible to circumvent the problem of the non-relativistic aspect of the studied particle, the relative conservation of the energy being applied to the quanta of energy. Time Dilatation and Uncertainty Principle 5 Considering that Φ is a particle with a mass from which quantity of energy, quantum by quantum, can be withdrawn up to a finite energy quantity e, corresponding to mc2, we will express m by its set of masses: i!n X m = mi = m1 + m2 + m3 + ::: + mn (16) i=1 Where n is a finite positive integer and where each mi is a part of the particle mass m. Then: i!n i!n X X e mi = (17) c2 i i=1 i=1 Also: i!n e X ei e1 e2 e3 en = = + + + ::: + (18) c2 c2 c2 c2 c2 c2 i=1 Where each ei is a quantum of withdrawable energy. So: i!n X e = ei (19) i=1 Considering the proof set out in section 2 and from the point of view of an observer situated in an almost flat space-time, we will express the energy taking into accout the dilatation of time: h E = (20) dτ We can express: i!n i!n X X h E = (21) i dτ i=1 i=1 i And: i!n X h h = (22) dτ dτ i=1 i To finish by: E h µ = = (23) c2 c2dτ The masse µ is here considered as the mass corresponding to the potentially extractable energy, from the point of view of an observer located in an almost 6 Alexandre Georges flat space-time. The proof provided in section 2 covers the case of a quantum of energy, the photon. The mass of the particle does not vary, from the point of view of an observer situated in an almost flat space-time, according to the curvature of the space-time geometry, considering the non-relativistic aspect of the particle requiered by the Uncertainty Principle and the fact that the proof provided in section 2 concerns quanta of energy. But the energy level, from the point of view of an observer situated in an almost flat space-time, varies according to the curvature of the space-time geometry, given the fact that its energy E is relative to the frequency ν of its wave.
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