An attempt to geometrize electromagnetism

XiuLin Huang; Yan Xu; ChengZhi Liu

doi:10.1515/phys-2018-0106

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An attempt to geometrize electromagnetism

XiuLin Huang , Yan Xu and ChengZhi Liu

Published/Copyright: December 31, 2018

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From the journal Open Physics Volume 16 Issue 1

Abstract

This study investigates the curved worldline of a charged particle accelerated by an electromagnetic field in flat spacetime. A new metric, which dependes on the charge-to-mass ratio and electromagnetic potential, is proposed to describe the curve characteristic of the world-line. The main result of this paper is that an equivalent equation of the Lorentz equation of motion is put forward based on a 4-dimensional Riemannian manifold defined by the metric. Using the Ricci rotation coefficients, the equivalent equation is self-consistently constructed. Additionally, the Lorentz equation of motion in the non-inertial reference frames is studied with the local Lorentz covariance of the equivalent equation. The model attempts to geometrize classical electromagnetism in the absence of the other interactions, and it is conducive to the establishment of the unified theory between electromagnetism and gravitation.

Keywords: Electromagnetism; Worldline; Riemannian Metric; Ricci rotation coefficients; Local Lorentz transformations

PACS: 03.50.De; 04.20.Cv; 02.40.Ky

1 Introduction

Since Einstein’s general relativity succeeded in geometrizing gravity, numerous people have made great efforts to geometrize electromagnetism in lots of unified-field theories [1, 2, 3, 4, 5, 6, 7, 8]. However, these unified-field theories are not widely accepted. In recent years, work on geometrization of electromagnetism without consideration of gravity has begun to increase [9, 10, 11]. Such studies are not complicated and the physical meaning is more straightforward. Moreover, the research on only geometrization of electromagnetism constantly contributes to establishing a final unified field theory.

Gravity is identified as a geometric phenomenon of a curved spacetime and the geometric phenomenon is the subject investigated in general relativity. Electromagnetism is essentially different from gravity, because it does not relate to the geometric phenomenon of the curved spacetime. Finding a suitable geometric phenomenon for electromagnetism is the first problem for geometrizing electromagnetism. In the previous references [12, 13, 14, 15], the geometrical description of the worldlines of charged particles in homogeneous electromagnetic fields have been studied using Frenet-Serret formulae. The time-like world-line of a charged particle accelerated by an electromagnetic field is possibly a geometric phenomenon of geometrizing electromagnetism. When a particle is accelerated by an electromagnetic force in flat spacetime, the worldline of the particle becomes a curved line x^a(τ), and it is closely relate to the Lorentz equation of motion

(1)d2xadτ2=emFabdxbdτ.

The geometrical characteristic of x^a(τ) may be the key to geometrize electromagneticsm. In this paper, we would like to provide an investigation on the geometrical characteristic of the worldline x^a(τ) from a different point of view.

The essential difference between the curved spacetime and the curved worldline indicates that there could be many curved worldlines at one spacetime point. To show the feature of the curved worldline, we structure “artificial” curvilinear coordinates {x^â} regarding the different metrics of the curved worldline x^a(τ). The coordinates {x^â} are not real curved spacetime, but are hypothetical curved coordinates associated with the flat coordinates {x^a}. Therefore, a metric could be obtained using the linear relationship between the flat differential coordinates dx^a and the “artificial" differential coordinates dx^â ,which differs from the model that a metric is derived by the coordinates transformation [16, 17, 18]. In this current work, we attempt to introduce differential geometry by defining the new metric of the curved worldlines of a charge particle in an electromagnetic field, and then obtain an equivalent equation of the Lorentz equation of motion using differential geometry.

This paper is organized as follows. In section 2, a new metric is given depending on the worldline of a charged particle in an electromagnetic field. In section 3, an equivalent equation of the Lorentz equation of motion is proposed using the new metric. section 4 discusses the local Lorentz covariance of the Lorentz equation of motion. section 5 provides a conclusion.

2 The new metric for classical electromagnetism

The length element squared ds² of a worldline x^a(τ) in 4-dimension flat spacetime is

(2)ds2=ηabdxadxb,

where metric η_ab is

(3)ηab=10000−10000−10000−1.

The differential coordinate dx^a describes the motion of a charged particle in an electromagnetic field and 4-velocity of the charged particle is Ua=(11−v2,vi1−v2)where vi=dxidx0,i=1,2,3and the speed of light c = 1. Because any curved worldline obeys the relation (2) in flat spacetime, the flat metric η_ab does not reflect the curved properties of a worldline. In this model, using “artificial” curvilinear coordinates x^â we give a new metric for the worldline of the charged particle in an electromagnetic field and the effects of radiation are ignored.

Assuming that the transformation between differential coordinates dx^â and dx^a is linear, the linear relationship at each point of the worldline is formulated as

(4)dxa=haa^dxa^.

Equations (2) and (4) show that the length element ds² is expressed with the differential coordinate dx^â,

(5)ds2=Ga^b^dxa^dxb^,

where

(6)Ga^b^=ηabhaa^hbb^.

The key issue of obtaining metric G_âb̂ is the relationship between differential coordinates dx^â and dx^a.

We consider that the linear transformation h^a_â needs to satisfy three limiting conditions. In the first limiting condition, in the absence of the case of electromagnetic interaction, the linear transformation h^a_â should be δa^aThe second limiting condition is provided by the study of the Lorentz equation of motion (1), form which the worldline x^a(τ) depends on the charge-to-mass ratio e /m and electromagnetic potential A_a. As a result, the linear transformation h^a_â should be dependent on e /m and A_a. The third limiting condition derives from the length element’s magnitude change of the worldline, measured by the multiple static-observers positioned at each point of the worldline. The length element of the observers is

(7)ds′=dx0,

which is constant at each point of spacetime. comparing equations (2) and (7), one can find that:

(8)ds=1−v2ds′,

which indicates that the length element’s magnitude is proportional to 1−v2. Thus, the linear transformation h^a_â should show the characteristic of the length element’s magnitude, which is the third limiting condition.

In our model, the linear transformation h^a_â is defined as:

(9)h0a^=δa^01−Q′+emBa^,hia^=δa^i,

where Q′=Q/(1−emA0+Q)and B_â is expressed as

(10)Ba^=Aa(xa)1−emA0(xa)+Q(t),

where Q(t) is a function of time, A_a(x^a) is electromagnetic potential and it is used to structure the equation (see Appendix)

(11)11−v2=1−emA0+Q.

Then, the inverse transformation h_a^â of the transformation h^a_â yields

(12)ha0^=δa0^1+Q−emAa,hai^=δai^.

The transformation h^a_â and the inverse transformation h_a^â satisfy the orthogonal relationship

(13)haa^hba^=δbaandhaa^hab^=δa^b^.

It is easily demonstrated that the transformation h^a_â meets the first and second limiting conditions. Besides, using equation (11), we have

(14)h00^=1−v2=1+emB0^−Q′,

which reflects the change of the length element’s magnitude, thus the third limiting condition is satisfied. The new metric G_â_b̂ of a worldline’s length element is obtained by equation (6) and (9):

(15)Ga^b^=D0^D0^emD0^B1^emD0^B2^emD0^B3^emD0^B1^emB1^2−1em2B1^B2^em2B1^B3^emD0^B2^em2B1^B2^emB2^2−1em2B2^B3^emD0^B3^em2B1^B3^em2B2^B3^emB3^2−1,

where D0^=h00^=1+emB0^−Q′,Ba^=Aa1−emA0+Q,Qis a function of time and A_a is the electromagnetic potential.

It can be seen that the length element ds² of the curved worldline x^a(τ) has two different expressions with different characteristics: ds2=ηabdxadxbandds2=Ga^b^dxa^dxb^. In the first, ds² = η_abdx^adx^b, flat metric η_ab is invariant, which does not reflect the curvature of the worldline. The change of length element ds² at each point along the worldline depends on the differential coordinate dx^a, which can describe the motion of the particle. In the second expression, ds2=Ga^b^dxa^dxb^, the “artificial” differential coordinate ds_â is regarded as invariant and it does not describe the motion of the particle. Besides, the change of length element ds² depends on the metric G_â_b̂ at each point of the worldline.

3 Differential geometry for electromagnetism

In this section, we use the length element expression ds2=Ga^b^dxa^dxb^, 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 with metric Gâ_b̂, which is defined by equation (15) at the worldline of the charged particle and Gâb̂ (Q = 0) at other space-time points. Now, the transformation haâ is called tetrad or vierbein. We can derive the equation of a world-line from an action given in the following equation, which is proportional to the length of the worldline.

(16)S(xa^)=∫Ga^b^dxa^dτdxb^dτdτ,

where proper time τ is a parameter associated with a point on the worldline and the coordinate xâ is variable. Variation of the action (16) with respect to xâ provides the traditional geodesic equation

(17)ddτdxa^dτ+Γa^b^c^dxb^dτdxc^dτ=0,

where Γa^b^c^=12Ga^d^(∂c^Gd^b^+∂b^Gc^d^−∂d^Gb^c^)is Christoffel symbol on M. Equation (17) cannot describe the motion of the charged particle, because dxâ/dτ is not the 4-velocity of the charged particle. We know that dx^a/dτ is considered to describe the motion of a charged particle, and the relationship between dx^a and dx^â is given by linear transformation h^a^â in equation (4). Thus, according to the coordinates x^a, equation (17) is rewritten as [19]

(18)d2xadτ2+ωabcdxbdτdxcdτ=0,

where ω^a_bc represents the Ricci rotation coefficients of the tetrad h^a_â. Besides, the tetrad h^a_â is also obtained using the metric G_âb̂ under “resting constraint conditions” [20]. In the torsion-free case, ω^a_bc is given as reported in literature [21]

(19)ωabc=12Cabc+Cbca−Ccab,

where

(20)Cabc=(∂b^haa^−∂a^hab^)hbb^hca^=haa^∂chba^−∂bhca^,

where equation ∂c(haa^hab^)=∂c(δa^b^)=0and∂a=hab^∂b^are used. Using the tetrad (9) and equaton (14), equaton (18) becomes

(21)d2xadτ2−emFabdxbdτ=0,

where F _ab = ∂_aA_b−∂_bA_a and A_a is electromagnetic potential. Therefore, the Lorentz equation of motion is obtained using 4-dimensional Riemannian manifold M defined by the metric G_âb̂.

4 Covariance of the Lorentz equation of motion

Motion equation (18) is form invariant with respect to local Lorentz transformations, under which [22], it is expressed as

(22)d2xa′dτ2+ωa′b′c′dxb′dτdxc′dτ=0,

where ω^a' _b'_c' transform as [21]

(23)ωa′b′c′=Λa′aΛb′bΛc′cωabc+Λa′d∂c′Λb′d,

where Λ^a' _a are local Lorentz transformations, and the relationship between dx^a' and dx^â is given by the Λ^a' _a,

(24)dxa′=Λa′adxa.

Equation (22) is considered to describe the equation of motion in the different reference frame with coordinates x^a. Thus, because equation (18) is equivalent to the Lorentz equation of motion, the Lorentz equation of motion in the other reference frames can be studied using equation (22) and two cases are as follows.

For the case that the local Lorentz transformations are constant, equation (22) becomes

(25)d2xa′dτ2−emFa′b′dxb′dτ=0,

where Fa′b′=Λa′aΛb′bFabequation (25) is the Lorenz equation of motion in another inertial frame. When the local Lorentz transformations are time-dependent, the reference frame with coordinates x^a' is a non-inertial frame. It is of significant importance that the motion equation in the non-inertial frame is studied using equation (22) depending on the corresponding local Lorentz transformations.

We consider a special non-inertial frame—the proper reference frame (comoving frame) that is attached to the charged particle in an electromagnetic field. Besides, the charged particle in the proper reference frame is always stationary. Thus, dx^a' is the differential coordinates of the proper reference frame, and dx⁰' = dτ and dx^i' = 0 in the proper reference frame. From the relation Λa′adxa′=dxawe have which becomes

(26)Λ0′a=dxadτ.

The term Λa′a∂c′Λb′adxb′dτdxc′dτin equation (22) is calculated,

(27)Λa′a∂c′Λb′adxb′dτdxc′dτ=Λa′a∂0′Λ0′adx0′dτdx0′dτ=Λa′ad2xadτ2.

In the proper frame, equation (22) becomes

(28)d2xa′dτ2+Λa′ad2xadτ2−emFa′b′dxb′dτ=0.

Using the Lorentz equation of motion, the latter two items in equation (28) satisfy equation

(29)La′ad2xadτ2−emFa′b′dxb′dτ=0.

Thus, motion equation (22) becomes

(30)d2xa′dτ2=0.

We know that the proper reference frame (comoving frame) is the non-inertial reference frame, in which the charged particle is subjected to a fictitious force and a electromagnetic force. These two forces cancel each other out which result in the motion equation of the particle as d2xa′dτ2=0in the proper reference frame. In our method, the motion equation of the particle is expressed as motion equation (22) in the proper reference frame. After being calculated, motion equation (22) becomes motion equation (28), in which the term −Λa′ad2xa/dτ2is fictitious acceleration and the term emFa′b′dxb′/dτis the electromagnetic acceleration, and these two forces cancel each other out. The result is reasonable in the proper reference frame.

5 Conclusions

To conclude, this work proposes the new metric (15) in flat spacetime to use for the curved worldline x^a(τ) of charged particles in an electromagnetic field. Then, we show that equation (18) is the motion equation of the charged particles and the equation is equivalent to the Lorentz equation of motion using the differential geometry of the metric (15). As the Lorentz equation of motion can describe the change of worldline x^a(τ), the motion equation (18) obtained shows that our model is self-consistent. Additionally,we have also investigated the local Lorentz covariance of the Lorentz equation of motion based on the equivalent equation (18), which is form invariant under the local Lorentz transformations. The important conclusion is that the Lorentz equation of motion in different inertial or non-inertial reference frames is related using equation (22). One example is that the motion equation (22) in the proper frame has been discussed and a rational prediction result has been obtained. In a sense, our method extends the Lorentz covariance of the Lorentz equation of motion to the local Lorentz covariance. Admittedly, local Lorentz covariance closely relates to general covariance in general relativity. Despite that general covariance has been proposed for numerous years, physicists do not have a unified understanding and the discussion continues for general covariance [23, 24, 25, 26, 27, 28]. This research may provide a new point of view for understanding geometrizing electromagnetism and the relationship between electromagnetic interaction and general covariance.

The innovation of this paper is that the curve characteristic of the worldline of charged particles in an electromagnetic field is described by the metric (15), and the Lorentz equation of motion is achieved by using the Riemannian geometry defined by the metric. Our method has an interesting relationship with general relativity. The important similarity between general relativity and our method is that a 4-dimensional Riemannian manifold is used for studying interactions. Our method significantly differs from general relativity in the following ways:

Our method investigates the relative motion of an object in flat spacetime. We introduce differential geometry to describe the curved worldline rather than curved spacetime. In the present method, the coordinates {x^â} represent the characteristics of a “curved” coordinate with respect to flat coordinates {x^a}, but the coordinates {x^â} are hypothetical.
In general relativity, the metric does not depend on the properties of the object that is accelerated by a gravity indicating that objects of different masses in the gravitational effect have the same geodesic equation (equation of motion). However, in our method, the metric depends on the particle’s charge-to-mass ratio e/m properties. These results are significant, as charged particles with different e /m ratios in an electromagnetic field have different worldlines, consequently introducing different metrics and different motion equations.
In our method, the motion equation (18) is constructed based on the Ricci rotation coefficients rather than the Christoffel symbols. The Ricci rotation coefficients are employed to describe the physical laws in flat spacetime.

To some extent, the present research is significant to unify electromagnetism and gravitation with differential geometry.

Further studies of differential geometry are needed to perfect our model. At first, the study of the classical dynamics equation may be extended into quantum electrodynamics using differential geometry. Secondly, the problem that electromagnetic potential satisfies the Maxwell equation need to be studied in the framework of differential geometry. We believe that the methods and proofs described in this paper bring us a step closer to the final unified theory which will tightly associate differential geometry with fundamental electromagnetic interactions.

Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing;

Acknowledgement

The work is supported by Open Foundation of Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Youth Innovation Promotion Association, Chinese Academy of Sciences under Grant No. 2016056, the Development Project of Science and Technology of Jilin Province under Grant No. 20180520077JH and the National Natural Science Foundation of China Nos. 11805022 and 11803057.

Appendix A

1−v2is associated with the electromagnetic potential of a charged particle accelerated by an electromagnetic field. According to relativistic mechanics [29],

(A.1)dUidτ=ki=em1−v2E→+v→×B→,

and

(A.2)dU0dτ=kivi,

where U0=1/1−v2and Ui=vi/1−v2Combining equation (A.1) and (A.2), it can see that U⁰ satisfies the relationship

(A.3)dU0=em−∂A0∂xi+∂Ai∂x0dxi.

U⁰ can be divided into the contributions of scalar field A₀ and vector field A_i, that is U0=U0e+U0m,dU0e=−em∂A0∂xidxiand dU0m=em∂Ai∂x0dxiFor the interaction of a charged particle with a scalar field A₀, the following relationship is obtained,

(A.4)U0e=C−emA0,

where C is a constant determined by the initial conditions. However, U⁰_m is not easy to obtain. It is considered that the effect of vector field A_i in equation (A.3) can always be replaced by a a function of time at some fixed spacetime points. Therefore, U⁰_m is rewritten as a function of time. We set

(A.5)U0=11−v2=1−emA0+Q(t),

where Q(t) is the time function and determined by the vector field A_i. Based on the above argument, 1−v2becomes a function of the electric potential A₀ and time t.

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Received: 2018-05-16

Accepted: 2018-09-06

Published Online: 2018-12-31

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Keywords for this article

Electromagnetism; Worldline; Riemannian Metric; Ricci rotation coefficients; Local Lorentz transformations

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