Open Phys. 2018; 16:848–853

Research Article

XiuLin Huang*, Yan Xu, and ChengZhi Liu An attempt to geometrize https://doi.org/10.1515/phys-2018-0106 of electromagnetism without consideration of 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 . 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 . 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 {xa^ } regarding the differ- ent metrics of the curved worldline xa(τ). The coordinates *Corresponding Author: XiuLin Huang: Changchun Observatory, { a^ } 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] {xa}. 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 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 ′ 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 √ ds2 x τ ′ 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 √ = 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 ⎛ ⎞ magnitude, which is the third limiting condition. 1 0 0 0 a In our model, the linear transformation h a is defined ⎜ ⎟ ^ ⎜0 −1 0 0 ⎟ as: ηab = ⎜ ⎟ . (3) ⎝0 0 −1 0 ⎠ e h0 δ0 (︀ Q′)︀ 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 Q′ Q e A Q B i where = /(1 − m 0 + ) and a^ is expressed as of the charged particle is Ua = ( √ 1 , √ 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 √ = 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 √ e ′ 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- ⎛ D D e D B e D B e D B ⎞ 0^ 0^ m 0^ 1^ m 0^ 2^ m 0^ 3^ ha e (︀ e )︀2 (︀ e )︀2 (︀ e )︀2 formation a^ in equation (4). Thus, according to the co- ⎜ D B B B B B B ⎟ a ⎜ m 0^ 1^ m 1^ − 1 m 1^ 2^ m 1^ 3^ ⎟ ordinates x , equation (17) is rewritten as [19] ⎜ e (︀ e )︀2 (︀ e )︀2 (︀ e )︀2 ⎟ , ⎝ m D^ B^ m B^ B^ m B^ − 1 m B^ B^ ⎠ 0 2 1 2 2 2 3 2 a b c e D B (︀ e )︀2 B B (︀ e )︀2 B B (︀ e B )︀2 d x a dx dx m 0^ 3^ m 1^ 3^ m 2^ 3^ m 3^ − 1 + ω bc = 0, (18) dτ2 dτ dτ 0 e ′ Aa D h B Q B e Q where 0^ = 0^ = 1 + m 0^ − , a^ = 1− A +Q , is a a m 0 where ω bc represents the Ricci rotation coefficients of the function of time and Aa is the electromagnetic potential. a a tetrad h a. Besides, the tetrad h a is also obtained using It can be seen that the length element ds2 of the ^ ^ G a the metric a^b^ under “resting constraint conditions” [20]. curved worldline x (τ) has two different expressions with a 2 a b 2 In the torsion-free case, ω bc is given as reported in litera- different characteristics: ds = ηab dx dx and ds = ture [21] G dxa^ dxb^ ds2 η dxa dxb η a^b^ . In the first, = ab , flat metric ab a 1 (︀ a a a )︀ is , which does not reflect the curvature of the ω bc = C bc + Cbc − Cc b , (19) 2 worldline. The change of length element ds2 at each point along the worldline depends on the differential coordinate where a dx , which can describe the motion of the particle. In the a a a b^ a^ C ∂ h ∂ h h hc a b^ bc = ( b^ a^ − a^ b^ ) b (20) second expression, ds2 = G dx^ dx , the “artificial” dif- a^b^ a (︁ a^ a^ )︁ a = h ∂c hb − ∂b hc , ferential coordinate dx^ is regarded as invariant and it a^ does not describe the motion of the particle. Besides, the ∂ ha h b^ ∂ δb^ ∂ h b^ ∂ ds2 G where equation c( a^ a ) = c( a^ ) = 0 and a = a b^ change of length element depends on the metric a^b^ at each point of the worldline. are used. Using the tetrad (9) and equaton (14), equaton (18) becomes

2 a b d x e a dx − F b = 0, (21) 3 Differential geometry for dτ2 m dτ electromagnetism where Fab = ∂a Ab −∂b Aa and Aa is electromagnetic poten- tial. Therefore, the Lorentz equation of motion is obtained using 4-dimensional Riemannian manifold M defined by In this section, we use the length element expression ds2 = the metric Gab^ . G dxa^ dxb^ ^ a^b^ to derive equations for the motion of a charged particle in an electromagnetic field based on differential geometry. Let M be a 4-dimensional Riemannian manifold G 4 Covariance of the Lorentz with metric a^b^ , which is defined by equation (15) at the G Q worldline of the charged particle and a^b^ ( = 0) at other a equation of motion space-time points. Now, the transformation h a^ is called tetrad or vierbein. We can derive the equation of a world- Motion equation (18) is form invariant with respect to local line from an action given in the following equation, which Lorentz transformations, under which [22], it is expressed is proportional to the length of the worldline. as ′ ′ ′ √︃ 2 a ′ b c ∫︁ a^ b^ d x a dx dx a dx dx + ω b′ c′ = 0, (22) S(x^ ) = G dτ, (16) dτ2 dτ dτ a^b^ dτ dτ a′ where ω b′ c′ transform as [21] where proper time τ is a parameter associated with a point xa^ a′ a′ b c a a′ d on the worldline and the coordinate is variable. Varia- ω b′ c′ = Λ a Λb′ Λc′ ω bc + Λ d ∂c′ Λb′ , (23) tion of the action (16) with respect to xa^ provides the tra- a′ ditional geodesic equation where Λ a are local Lorentz transformations, and the re- a′ a^ a′ dx dx Λ a (︃ a^ )︃ b^ ^c lationship between and is given by the , d dx a dx dx ^ ′ + Γ b^c = 0, (17) a′ a a dτ dτ ^ dτ dτ dx = Λ a dx . (24) An attempt to geometrize electromagnetism Ë 851

Equation (22) is considered to describe the equation of mo- charged particle is subjected to a and a ′ tion in the different reference frame with coordinates xa . electromagnetic force. These two forces cancel each other Thus, because equation (18) is equivalent to the Lorentz out which result in the motion equation of the particle as ′ d2 xa equation of motion, the Lorentz equation of motion in the dτ2 = 0 in the proper reference frame. In our method, other reference frames can be studied using equation (22) the motion equation of the particle is expressed as motion and two cases are as follows. equation (22) in the proper reference frame. After being For the case that the local Lorentz transformations are calculated, motion equation (22) becomes motion equa- ′ a 2 a 2 constant, equation (22) becomes tion (28), in which the term −Λ a d x /dτ is fictitious ac- e a′ b′ ′ ′ F ′ dx dτ 2 a b celeration and the term m b / is the electromag- d x e a′ dx − F b′ = 0, (25) netic acceleration, and these two forces cancel each other dτ2 m dτ out. The result is reasonable in the proper reference frame. a′ a′ b a where F b′ = Λ a Λb′ F b, equation (25) is the Lorenz equation of motion in another inertial frame. When the lo- cal Lorentz transformations are time-dependent, the refer- ′ 5 Conclusions ence frame with coordinates xa is a non-inertial frame. It is of significant importance that the motion equation inthe To conclude, this work proposes the new metric (15) in flat non-inertial frame is studied using equation (22) depend- a spacetime to use for the curved worldline x (τ) of charged ing on the corresponding local Lorentz transformations. particles in an electromagnetic field. Then, we show that We consider a special non-inertial frame—the proper equation (18) is the motion equation of the charged par- reference frame (comoving frame) that is attached to the ticles and the equation is equivalent to the Lorentz equa- charged particle in an electromagnetic field. Besides, the tion of motion using the differential geometry of the metric charged particle in the proper reference frame is always a′ (15). As the Lorentz equation of motion can describe the stationary. Thus, dx is the differential coordinates of the a ′ ′ change of worldline x (τ), the motion equation (18) ob- proper reference frame, and dx0 = dτ and dxi = 0 in the a a′ a tained shows that our model is self-consistent. Addition- proper reference frame. From the relation Λa′ dx = dx , ′ a 0 a ally, we have also investigated the local Lorentz covariance we have Λ ′ dx = dx , which becomes 0 of the Lorentz equation of motion based on the equiva- a a dx lent equation (18), which is form invariant under the lo- Λ ′ = . (26) 0 dτ cal Lorentz transformations. The important conclusion is

′ ′ a′ a dxb dxc that the Lorentz equation of motion in different inertial Λ ∂ ′ Λ ′ The term a c b dτ dτ in equation (22) is calcu- or non-inertial reference frames is related using equation lated, (22). One example is that the motion equation (22) in the ′ ′ ′ ′ b c 0 0 proper frame has been discussed and a rational prediction a′ a dx dx a′ a dx dx Λ ∂ ′ Λ ′ Λ ∂ ′ Λ ′ a c b dτ dτ = a 0 0 dτ dτ (27) result has been obtained. In a sense, our method extends 2 a a′ d x the Lorentz covariance of the Lorentz equation of motion = Λ a . dτ2 to the local Lorentz covariance. Admittedly, local Lorentz covariance closely relates to general covariance in general In the proper frame, equation (22) becomes relativity. Despite that general covariance has been pro- ′ ′ 2 a 2 a b d x a′ d x e a′ dx posed for numerous years, physicists do not have a unified + Λ a − F b′ = 0. (28) dτ2 dτ2 m dτ understanding and the discussion continues for general covariance [23-28]. This research may provide a new point Using the Lorentz equation of motion, the latter two items of view for understanding geometrizing electromagnetism in equation (28) satisfy equation and the relationship between electromagnetic interaction ′ 2 a b a′ d x e a′ dx and general covariance. L a − F b′ = 0. (29) dτ2 m dτ The innovation of this paper is that the curve char- acteristic of the worldline of charged particles in an elec- Thus, motion equation (22) becomes tromagnetic field is described by the metric (15), andthe ′ d2xa Lorentz equation of motion is achieved by using the Rie- = 0. (30) dτ2 mannian geometry defined by the metric. Our method has an interesting relationship with general relativity. The We know that the proper reference frame (comoving important similarity between general relativity and our frame) is the non-inertial reference frame, in which the 852 Ë X. Huang et al. method is that a 4-dimensional Riemannian manifold is References used for studying interactions. 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Appendix A √ 1 − v2 is associated with the electromagnetic potential of a charged particle accelerated by an electromagnetic field. According to relativistic mechanics [29],

i dU i e (︁ )︁ = k = √ ⃗E + ⃗v × ⃗B , (A.1) dτ m 1 − v2 and dU0 ki vi dτ = , (A.2) √ √ where U0 = 1/ 1 − v2 and Ui = vi / 1 − v2. Combining equation (A.1) and (A.2), it can see that U0 satisfies the re- lationship (︂ )︂ e ∂A ∂Ai i dU0 = − 0 + dx . (A.3) m ∂xi ∂x0

U0 can be divided into the contributions of scalar field 0 0 0 0 A0 and vector field Ai, that is U = U e + U m, dU e = e ∂A0 i 0 e ∂Ai i − m ∂xi dx and dU m = m ∂x0 dx . For the interaction of a charged particle with a scalar field A0, the following rela- tionship is obtained, e U0 C A e = − m 0, (A.4) where C is a constant determined by the initial conditions. 0 However, U m is not easy to obtain. It is considered that the effect of vector field Ai in equation (A.3) can always be replaced by a a function of time at some fixed spacetime 0 points. Therefore, U m is rewritten as a function of time. We set 0 1 e U = √ = 1 − A0 + Q(t), (A.5) 1 − v2 m Q t where ( ) is the time function and determined√ by the vec- tor field Ai. Based on the above argument, 1 − v2 be- comes a function of the electric potential A0 and time t.