Generalization of Klein-Gordon Equation in Curved Space (PDF)

NeuroQuantology| | August 2023 | Volume 21 | Issue 6 |Page 1795-1800| doi: 10.48047/nq.2023.21.6.nq23178 Abeer Mohammed Khairy Ahmed et al/ Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution Generalization of Klein Gordon Equation in a Curved Spac...

NeuroQuantology| | August 2023 | Volume 21 | Issue 6 |Page 1795-1800| doi: 10.48047/nq.2023.21.6.nq23178 Abeer Mohammed Khairy Ahmed et al/ Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution Abeer Mohammed Khairy Ahmed 1, Mohamed Yahia Shirgawi 2, ZoalnoonAhmed Abeid Allah Saad3 1 Dep. of physics, College of science and Arts at Al- Muthnib, Qassim University, Kingdom of Saudi Arabia. 2 Dep. of physics, College of Science and Arts at Al Muthnib, Qassim University, Kingdom of Saudi Arabia 3 Dep. of Physics, Faculty of Arts & Sciences, Dhahran Aljanoub, King Khalid University, Kingdom of Saudi Arabia https://orcid.org/0000-0003-0895-6683,[email protected], [email protected],[email protected] Abstract In this work the equation of motion of a particle in a curved space time which proved that all fields deform the space time as derived by many researchers was used to derive Klein Gordon equation in a curved space time. This expression indicates that the rest mass is potential dependent. This confirms the recent observations of the dependence of the rest mass of elementary particles by the potential. It also confirms the effective mass relationships in material science, which was proposed to explain the observed change of the electron mass by the crystal and bulk matter internal potential. This expression confirms the calculations of the loops contributions to the mass term, which also include the effect of the potential on the rest mass. The time dependent rest mass was 1795 incorporated into the Klein Gordon equation. The results obtained for harmonic oscillator solution gives frequency and rest mass dependent terms. The appearance of the additional rest mass term beside the frequency term confirms the particle wave duality of the atomic entities. Where the rest mass term exhibits the particle nature while the frequency dependent term manifests the wave nature. The association of the curved space with the potential which is responsible for the appearance of the frequency term indicates that the space curvature represents a new ether that can generate gravitational waves produced by oscillating masses Keywords: curved space, Klein Gordon equation, quantum, harmonic oscillator, wave particle duality, rest mass, potential, ether DOI Number: 10.48047/nq.2023.21.6.nq23178 NeuroQuantology2023;21(6): 1795-1800 dual nature. This dual nature lead to the formation Introduction: of quantum mechanics[4,5].The oldest quantum Newton’s laws are the oldest physical laws equation was formulated by Schrodinger.His which successfully describe the behavior of the quantum equation is mainly based on the wave gravitational field. On the other hand Huygens packet wave function beside the classical Newton formulate his well-known principle to describe the energy momentum relation. Later on Klein and nature of the light. According to his principle light Gordon beside Dirac formulated quantum behave as waves which was confirmed by equations based on the special relativity energy Maxwell's equations.The wave nature of light momentum relation.One of the most known explains the laws of reflectanction, refraction and issues that faced quantum laws are the unification interference as well as diffraction phenomenon of gravity with the quantum laws.These. But unfortunately the black body radiation, processes can be described either by describing cannot be described by the wave model. This the quantum equations in a curved space time or in encourages Max Plank to suggest that the light and a potential field or by quantizing gravity laws. electromagnetic fields behave as a discrete quanta These models looks quite different but they are or a particle [2,3,4].This means that they have a eISSN1303-5150 www.neuroquantology.com 1 NeuroQuantology| | August 2023 | Volume 21 | Issue 6 |Page 1795-1800| doi: 10.48047/nq.2023.21.6.nq23178 Abeer Mohammed Khairy Ahmed et al/ Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution effective mass change with velocity and helpful for describing the behavior of realistic momentum in graphine and Cu can be explained particles in different fields beside the quantum by perturbation approach for electron density gravity phenomenon [8,9].The emergence of Nano functional theory. The results obtained by Sami science and technology gives quantum laws utilized the invariance of the space - time remarkable push. This comes from the fact that distance expression in a curved space - time to find Nano materials are described by quantum laws potential dependent energy and mass. In the work [10,11]. done by E. Goulart etal the momentum and Different attempts were made by many researchers energy beside mass are affected by the space time to modify quantum equations. One of them is the metric. The paper of Axel etal was used to work done by Najwa and others , where derive Schrödinger time dependent equation for electrons were treated as strings moving in a electrons having effective mass in crystals. Using frictional medium. Then new energy conservation particular point canonical transformation which equation was found and used to describe new preserves the L^2 normalizability. The solution of Schrodinger equation.. The frequency dependent this equation is possible when transforming this energy term has been found. Since the red shift Schrodinger equation with variable effective mass phenomenon indicates that the frequency is to the ordinary one. In the paper of Frank etal affected by the gravity field this means that they derived the effective mass expression for treating electrons as strings which have the same electrons and holes in the anisotropic multi and frequency as the photon causes the energy to be sem metals like AR, Sb and Bi. The plasma sensitive to the potential. The paper of HassabAlla, treatment was averaged to get rid of the effective etal used also frictional dependent mass anisotropy. The work of Viktor Ariel etal Schrodinger equation successfully describe the generalized the conventional effective mass scattering of x-rays and phonons by Nano particles relation based on the parabolic energy-momentum like proteins. The energy expression is again 1796 relation to include non parabolic case. They used frequency dependent which means that it can be wave-particle duality to make this generalization. affected by the field potential. Similarly another The paper published by J. Laflamme etal used paper done by HassabAlla etal indicates that density functional perturbation theory to obtain a the frictional and frequency dependent new useful expression for the effective mass. The energy relation leads to a new modified solution was made by adopting the concent of Schrodinger equation can be used to fabricate transport equivalent effective mass to the K. P Nano capacitors and inductors, depending on the framework. When applied to silicon, graphine, and frictional coefficient of the medium. In the work arsenic the work was found to conform with done by Lamia Kasmi etal verified previous studies. In the model proposed by experimentally the theoretically the change of the Savickas he generalized the energy and momentum electron mass which was observed even outside concepts for the gravitational field to be metric and the crystal at distances in the range of 5-7 A. This potential dependent. Unfortunately his model is means that the change of the electron mass results only restricted to gravity without recognizing other from potential field not from atoms and matter. fields The paper of Oleg Rubel etal show that the Klein Gordon equation in a curved space-time: One can start by reproving that all fields can cause the space as derived also by Dirar and others. According to their version, the affine connection in a weak field takes the form 1 𝜕ℎ 𝛤00 𝜆 = − 2 𝜕𝑥00 (1) Thus, the acceleration is given by 𝑑2 𝑥 𝑐 2 𝑑𝑡 2 = - 2 𝛻ℎ00 (2) According to the Newtonian mechanics the force is defined in terms of the potential to be in the form 𝑚𝑑 2 𝑥 𝛻𝑉 = 𝛻𝑚𝜙 = −𝑚𝛻𝜙 = (3) 𝑑𝑡 2 Defining the potential per unit mass the acceleration is given to be eISSN1303-5150 www.neuroquantology.com 2 NeuroQuantology| | August 2023 | Volume 21 | Issue 6 |Page 1795-1800| doi: 10.48047/nq.2023.21.6.nq23178 Abeer Mohammed Khairy Ahmed et al/ Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution 𝑑2 𝑥 𝜕𝜙 𝛻𝜙 = = (4) 𝑑𝑡 2 𝜕𝑥 Comparing equations (2)and (4) gives 𝑐2 ℎ =𝜙 2 00 2𝜙 ℎ00 = 𝑐 2 (5) Therefore, the time metric is given by 2𝜙 𝑔00 = 1 + 2 (6) 𝑐 On the other hand, the space-time interval takes the form 𝑐 2 𝑑𝑡 2 = 𝑔00 𝑑𝑥 𝑢 𝑑𝑥 𝑣 (7) To write the Klein Gordon equation in a curved space-time, one has to recognize the conventional one in the Euclidean space which is given by 𝜕2 𝜓 −ℏ2 𝜕𝑡 2 = −𝑐 2 ℏ2 𝛻 2 𝜓 + 𝑚0 2 𝑐 4 𝜓 (8) using the separation of variables method, the wave function can be written as time dependent and time independent functions to get 𝜓(𝑟, 𝑡) = 𝑓(𝑡)𝑢(𝑟) (9) Hence equation (8)becomes ℏ2 𝜕2 𝑓 𝑐 2 ℏ2 2 − 𝑓 𝜕𝑡 2 = − 𝑢 𝛻 𝑢 + 𝑚0 2 𝑐 4 = 𝐸 2 (10) where the time part is given by ℏ2 𝜕2 𝑓 − 𝑓 𝜕𝑡 2 = 𝐸2 (11) with the solution f=A𝑒 −𝑖𝜔𝑡 (12) where the energy E is given to be 1797 2 2 2 𝐸 =ℏ 𝜔 𝐸 = ℏ𝜔 (13) Thus 𝜕2 𝑓 −ℏ2 = 𝐸2𝑓 (14) 𝜕𝑡𝑐 2 Consider the case when the particle is at rest. Thus the momentum P and the energy E are given by P = m𝑣 = 0 E = 𝑚0 𝑐 2 (15) Using equation (7) for the space time in a curved space time the time in a curved space time takes the form 𝑡𝑐 = √𝑔00 𝑡 (16) Therefore, the time dependent part in a curved space-time is thus given by ℏ2 𝑑 2 𝑓 − = 𝐸2𝑓 (17) 𝑔00 𝑑𝑡 2 On the other hand, the effect of curvature can be incorporated in the energy term instead of the space-time to get 𝑑2 𝑓 −ℏ2 𝑑𝑡 2 = 𝐸𝑐 2 𝑓 (18) Comparing equations (17) and (18) gives 𝐸𝑐 = √𝑔00 𝐸 (19) This means that the rest mass is modified to get the rest energy and rest mass energy to satisfy 𝐸𝑐 = √𝑔00 𝑚0 𝑐 2 = 𝑚0𝑐 𝑐 2= 𝑚0∗ 𝑐 2 (20) ∗ 𝑚0𝑐 = 𝑚0 = √𝑔00 𝑚0 (21) eISSN1303-5150 www.neuroquantology.com 2 NeuroQuantology| | August 2023 | Volume 21 | Issue 6 |Page 1795-1800| doi: 10.48047/nq.2023.21.6.nq23178 Abeer Mohammed Khairy Ahmed et al/ Generalization of Klein Gordon Equation in a Curved Space and Harmonic Oscillator's Solution The expression of the rest mass in a curved space time in the presence of a field can be simplified by taking into consideration that, E>V Therefore 𝑚𝑐 2 >𝑚𝜑 𝜑 𝑐2