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An Attempt to Geometrize Electromagnetism

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An Attempt to Geometrize Electromagnetism Open Phys. 2018; 16:848–853 Research Article XiuLin Huang*, Yan Xu, and ChengZhi Liu An attempt to geometrize electromagnetism https://doi.org/10.1515/phys-2018-0106 of electromagnetism without consideration of gravity has Received May 16, 2018; accepted Sep 06, 2018 begun to increase [9–11]. Such studies are not complicated and the physical meaning is more straightforward. More- Abstract: This study investigates the curved worldline over, the research on only geometrization of electromag- of a charged particle accelerated by an electromagnetic netism constantly contributes to establishing a final uni- field in flat spacetime. A new metric, which dependes on fied field theory. the charge-to-mass ratio and electromagnetic potential, is Gravity is identified as a geometric phenomenon of proposed to describe the curve characteristic of the world- a curved spacetime and the geometric phenomenon is line. The main result of this paper is that an equivalent the subject investigated in general relativity. Electromag- equation of the Lorentz equation of motion is put forward netism is essentially different from gravity, because it does based on a 4-dimensional Riemannian manifold defined not relate to the geometric phenomenon of the curved by the metric. Using the Ricci rotation coefficients, the spacetime. Finding a suitable geometric phenomenon for equivalent equation is self-consistently constructed. Addi- electromagnetism is the first problem for geometrizing tionally, the Lorentz equation of motion in the non-inertial electromagnetism. In the previous references [12–15], the reference frames is studied with the local Lorentz covari- geometrical description of the worldlines of charged par- ance of the equivalent equation. The model attempts to ge- ticles in homogeneous electromagnetic fields have been ometrize classical electromagnetism in the absence of the studied using Frenet-Serret formulae. The time-like world- other interactions, and it is conducive to the establishment line of a charged particle accelerated by an electromag- of the unified theory between electromagnetism and grav- netic field is possibly a geometric phenomenon ofge- itation. ometrizing electromagnetism. When a particle is acceler- Keywords: Electromagnetism, Worldline, Riemannian ated by an electromagnetic force in flat spacetime, the Metric, Ricci rotation coefficients, Local Lorentz trans- worldline of the particle becomes a curved line xa(τ), and formations it is closely relate to the Lorentz equation of motion PACS: 2 a b 03.50.De; 04.20.Cv; 02.40.Ky d x e a dx = F b . (1) dτ2 m dτ The geometrical characteristic of xa(τ) may be the key to 1 Introduction geometrize electromagneticsm. In this paper, we would like to provide an investigation on the geometrical char- Since Einstein’s general relativity succeeded in geometriz- acteristic of the worldline xa(τ) from a different point of ing gravity, numerous people have made great efforts to view. geometrize electromagnetism in lots of unified-field the- The essential difference between the curved spacetime ories [1–8]. However, these unified-field theories are not and the curved worldline indicates that there could be widely accepted. In recent years, work on geometrization many curved worldlines at one spacetime point. To show the feature of the curved worldline, we structure “arti- ficial” curvilinear coordinates fxa^ g regarding the differ- ent metrics of the curved worldline xa(τ). The coordinates *Corresponding Author: XiuLin Huang: Changchun Observatory, f a^ g National Astronomical Observatories, Chinese Academy of Sciences, x are not real curved spacetime, but are hypothetical Changchun, China; Guizhou Provincial Key Laboratory of Radio curved coordinates associated with the flat coordinates Astronomy and Data Processing; Email: [email protected] fxag. Therefore, a metric could be obtained using the lin- Yan Xu: Changchun Observatory, National Astronomical Observato- ear relationship between the flat differential coordinates ries, Chinese Academy of Sciences, Changchun, China; a a dx and the “artificial" differential coordinates dx^ , which Email: [email protected] ChengZhi Liu: Changchun Observatory, National Astronomical differs from the model that a metric is derived by theco- Observatories, Chinese Academy of Sciences, Changchun, China; ordinates transformation [16–18]. In this current work, we Email: [email protected] attempt to introduce differential geometry by defining the Open Access. © 2018 X. Huang et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License An attempt to geometrize electromagnetism Ë 849 a new metric of the curved worldlines of a charge particle We consider that the linear transformation h a^ needs in an electromagnetic field, and then obtain an equivalent to satisfy three limiting conditions. In the first limiting con- equation of the Lorentz equation of motion using differen- dition, in the absence of the case of electromagnetic in- ha δa tial geometry. teraction, the linear transformation a^ should be a^ . The This paper is organized as follows. In Section 2, a new second limiting condition is provided by the study of the metric is given depending on the worldline of a charged Lorentz equation of motion (1), form which the worldline particle in an electromagnetic field. In Section 3, an equiv- xa(τ) depends on the charge-to-mass ratio e/m and elec- alent equation of the Lorentz equation of motion is pro- tromagnetic potential Aa. As a result, the linear transfor- a posed using the new metric. Section 4 discusses the local mation h a^ should be dependent on e/m and Aa. The third Lorentz covariance of the Lorentz equation of motion. Sec- limiting condition derives from the length element’s mag- tion 5 provides a conclusion. nitude change of the worldline, measured by the multiple static-observers positioned at each point of the worldline. The length element of the observers is 0 2 The new metric for classical ds = dx0, (7) electromagnetism which is constant at each point of spacetime. comparing equations (2) and (7), one can find that: a p ds2 x τ 0 The length element squared of a worldline ( ) in 4- ds = 1 − v2ds , (8) dimension flat spacetime is which indicates that the length element’s magnitude is ds2 η dxa dxb p = ab , (2) proportional to 1 − v2. Thus, the linear transformation ha where metric ηab is a^ should show the characteristic of the length element’s 0 1 magnitude, which is the third limiting condition. 1 0 0 0 a In our model, the linear transformation h a is defined B C ^ B0 −1 0 0 C as: ηab = B C . (3) @0 0 −1 0 A e h0 δ0 (︀ Q0)︀ B 0 0 0 −1 a^ = a^ 1 − + m a^ , (9) i i a h δ The differential coordinate dx describes the motion of a a^ = a^ , charged particle in an electromagnetic field and 4-velocity Q0 Q e A Q B i where = /(1 − m 0 + ) and a^ is expressed as of the charged particle is Ua = ( p 1 , p v ), where vi = 1−v2 1−v2 a i Aa(x ) dx Ba^ = e , (10) 0 , i = 1, 2, 3 and the speed of light c = 1. Because any a dx 1 − m A0(x ) + Q(t) curved worldline obeys the relation (2) in flat spacetime, a where Q(t) is a function of time, Aa(x ) is electromagnetic the flat metric ηab does not reflect the curved properties potential and it is used to structure the equation (see Ap- of a worldline. In this model, using “artificial” curvilinear a^ pendix) coordinates x we give a new metric for the worldline of the 1 e p = 1 − A0 + Q. (11) charged particle in an electromagnetic field and the effects 1 − v2 m of radiation are ignored. a^ Then, the inverse transformation ha of the transforma- Assuming that the transformation between differen- a a a tion h a yields tial coordinates dx^ and dx is linear, the linear relation- ^ 0^ 0^ e ship at each point of the worldline is formulated as ha δa Q Aa = (1 + ) − m , (12) a a a^ ^i ^i dx = h a^ dx . (4) ha = δa . ds2 a Equations (2) and (4) show that the length element is The transformation h a^ and the inverse transformation a^ a^ expressed with the differential coordinate dx , ha satisfy the orthogonal relationship 2 a^ b^ a a^ a a b^ b^ ds G dx dx h h δ and h ha δ = a^b^ , (5) a^ b = b a^ = a^ . (13) a where It is easily demonstrated that the transformation h a^ G η ha hb meets the first and second limiting conditions. Besides, us- a^b^ = ab a^ b^ . (6) G ing equation (11), we have The key issue of obtaining metric a^b^ is the relationship p e 0 dxa^ dxa h0 = 1 − v2 = 1 + B − Q , (14) between differential coordinates and . 0^ m 0^ 850 Ë X. Huang et al. which reflects the change of the length element’s magni- Γa^ 1 Ga^d^ ∂ G ∂ G ∂ G where b^^c = 2 ( ^c d^b^ + b^ ^cd^ − d^ b^^c) is Christoffel tude, thus the third limiting condition is satisfied. The new symbol on M. Equation (17) cannot describe the motion of G a^ metric a^b^ of a worldline’s length element is obtained by the charged particle, because dx /dτ is not the 4-velocity equation (6) and (9): of the charged particle. We know that dxa /dτ is consid- G ered to describe the motion of a charged particle, and the a^b^ = (15) a relationship between dxa and dx^ is given by linear trans- 0 D D e D B e D B e D B 1 0^ 0^ m 0^ 1^ m 0^ 2^ m 0^ 3^ ha e (︀ e )︀2 (︀ e )︀2 (︀ e )︀2 formation a^ in equation (4).
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