[PDF] Electromagnetism from 5D gravity: Beyond the Maxwell equations

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hQ +Bi2 i?Bb p2`bBQM, Electromagnetism from 5D gravity: Beyond the Maxwell equations

Romulus Breban

Institut Pasteur, Paris, France

Abstract

Ever with the work of Kaluza, it has been known that 4D Einstein- and Maxwell-type equations can emerge from the eld equations of 5D gravity in Ricci- at space-times, having a space-like Killing vector. We revisit these equations and compare them with the Maxwell equations and the Ohm's law.

Although 5D gravity and traditional electromagnetic theory are mathematically related, a paradigm shift

in traditional electromagnetic concepts is required for unication. First, both the Maxwell equations and

the Ohm's law are found among the eld equations for 5D gravity, which gives Ohm's law the status of a eld equation. Second, the concept of 4-current density, bringing together electrical charge and

current density, is not fundamental for the electromagnetic theory resulting from 5D gravity. Third, the

5D gravity eld equations suggest a generalization for the model of the perfect conductor, to describe a

broader range of electromagnetic phenomena and bypass the concept of 4-current density. To see these ideas at work in a simple context, we propose an amendment to the Maxwell equations, based on the eld equations of 5D gravity, and discuss a few applications. Namely, we discuss the propagation of

electromagnetic waves through an imperfect conductor and topics in electrostatics and magnetostatics,

touching ground with the Hall eect and the phenomenon of persistent current.

1 Introduction

The history of the electromagnetic theory is marked by philosophical choices, standardizations and leaps in

understanding [1]. These may be revisited with progress in physics, to shed new light on the exceptional

body of work in electromagnetism. The core equations for explaining the phenomenology are the Maxwell

equations, written in space and time. The electromagnetic eld is dened using three spatial coordinates,

x j(j;k;l;:::= 1;2;3), in 3D Euclidian geometry, while the timetstands only as a parameter. However, the very structure of the Maxwell equations denes a 4D Lorentzian geometry, where time appears as a

coordinate,x0. In CGS units, the modern formulation of the Maxwell equations is (e.g., Refs. [2, Ch. 1]

and [3, Ch. 6]) rD= 4;(1.1) rB= 0;(1.2) r E=1c @B@t ;(1.3) r H=1c @D@t +4c j;(1.4) supplemented by constitutive/material equations

D=E+ 4P;(1.5)

B=H+ 4M;(1.6)

(n.b., in vacuum,P=M= 0) and Ohm's law j=&E:(1.7) For mathematical convenience, the potentialsA0andAare introduced for the electromagnetic eld

E=rA01c

@A@t ;(1.8)

B=r A:(1.9)

1

A recent critique of the Maxwell equations can be found in Ref. [4]. We make two additional notes. First, the

Maxwell equations require the concepts of both eld and source. Brie y, source generates eld; the observer

can actdirectlyon the source, but not on the eld. It is conjectured that electrical charge in motion is the

source of the electromagnetic eld. Second, as evidenced by Eqs. (1.5)-(1.7), the Maxwell equations make

explicit assumptions about the structure of matter. In fact, many concepts such as conductor, dielectric,

electron, were developed in parallel with the equations for the electromagnetic eld, itself. Electromagnetic

phenomena are attributed to particles, crystalline/amorphous structures and domain dynamics of matter.

The relationship between electromagnetic phenomenology and the Maxwell equations remains complex;

e.g., Ref. [5]. Here, our discussion is limited to elds, sources and particle electromagnetism in vacuum.

Features of the Maxwell equations can be recovered from a 5D geometry without compact dimensions. In 1921, Kaluza [6] proposed to unify the description of electromagnetic eld in vacuum, with that of the 4D gravitational eld, using a 5D space-time geometry where the metric is independent of the fth

dimension,x5. Later, Wesson [7] elaborated the work of Kaluza, inspired by progress in 4D gravitation

theory, particularly resisting Klein's idea [8] that the fth dimension be compact. Specically, Wesson [7]

proposed an interpretation for the covariance breaking in the gravitational eld variables, from ve to four

dimensions. We adopt the Kaluza-Wesson interpretation ofinduced matter, whereby sourceless 5D gravity

can be regarded, in four dimensions, as electromagnetic eld in vacuum, onto a space-time hosting 4D gravity. Thus, an empty (i.e., Ricci- at) 5D space-time appears to containactive matter(i.e., sources of

elds), when it is interpreted in four dimensions. Several authors published eld equations resulting for

5D Ricci-

at space-times, which are compatible with the induced-matter interpretation [6,9{12]. However,

covariance breaking of the 5D gravitational eld, to obtain eld equations of type Einstein and Maxwell, is

not unique. There are many ways to parameterize the 5D metric and still obtain 4D eld equations with desired properties. Here, we impose two requirements, which make the 5D eld parameterization unique.

First, the parameterization submits to the geometric picture of the foliation of the 5D metric alongx5and,

second, it provides an anomaly-free theory for single-particle propagation in 4D space-time, according to the

principles of general relativity [13{16]. Geometrically speaking, there is nothing peculiar about the fth dimension. Rather than using anx5- independent 5D metric, we can use anx0-independent 5D metric, foliate alongx0, and obtain Einstein- and Maxwell-type equations for 4D elds from the condition of 5D Ricci atness;

1It is claimed that time-

independent particle propagation can be interpreted within a statistical physics formalism [13{15]. However,

the Maxwell-type equations resulting in this case are associated with a 4D Euclidian metric rather than a 4D

Lorentzian metric and it remains unclear how they link to the current phenomenology of electromagnetism.

An alternative is to maintain the parameterization induced by the foliation alongx5, when discussingx0-

independent physics, as it guarantees an interpretation for the eld variables [13{15]. These topics remain

for further work. The paper is structured as follows. In Sec. 2, we brie y review the theory for 5D null-propagation of

passiveparticles; i.e., material particles which are not sources of elds.2In Sec. 3, we explain how, in the case

where the 5D metric isx5-independent, 5D physics can undergo covariance breaking, to be interpreted as 4D

mechanics. Indeed, the resulting 4D theory of passive particle propagation is anomaly free, in full agreement

to the traditional equations of 4D, classical and quantum, relativistic mechanics. The reader familiar with

these developments may proceed directly to Sec. 4, where we discuss the eld equations resulting in Ricci-

at

5D gravity, from the perspective of the Maxwell equations and Ohm's law. As with previous work [6,9{11],

we make the case that 5D gravity is a successful physical theory, in touch with experiment. However, the

traditional electromagnetic theory can gain, as well, from the parallel with eld equations for 5D gravity. In

particular, it can gain sharper concepts and, perhaps, a richer basis, as phenomenology nds reason to be

included with the fundamentals. These aspects and others are discussed in Secs. 4.1 and 5. In Sec. 6, we

conclude our work.1

This type of foliation is performed in the Hamiltonian formulation of 4D general relativity, for which the Arnowitt-Deser-

Misner foliation [17,18] was originally intended.

2We do not consider the propagation ofactivemassive (charged) particles propagating within the gravitational (electromag-

netic) eld that themselves produce. 2

2 5D physics for 4D formalisms

We postulate that reality is 5D, yet an observer perceives/acknowledges only 4 dimensions out of 5. For a

complete physical picture or formalism, the observer relies on 4D geometry and an array of non-geometrical

concepts. The physical pictures employed by a 4D observer result from dierent foliations of a 5D geometry

[15]. It is postulated that, if the 5D geometry is independent of the fth coordinate then the 5D physics

appears as 4D mechanics, while if the 5D geometry is independent of time then the 5D physics appears

as 4D statistical physics. Throughout the manuscript, we use bold face to indicate that some indices of

the corresponding mathematical objects are suppressed, particularly for 3D vectors. We start with a 5D

Lorentzian space-time,M(5), with metrichMN(M;N;= 0;1;2;3;5), having the signature (;+;+;+;+),

that is suciently smooth and subject to the principle of general covariance. We further consider thatM(5)

is Ricci at; i.e.,R(5) MN= 0 and the 5D gravity elds have no sources. This is just a minimalist choice, not required by the induced matter principle.

We assume that the observed particle is passive, not a source of eld interacting with the space-time met-

ric. Particle propagation can be described using suitable equations dened on (M(5);h). We are particularly

interested in null propagation, therefore 5D optics, dened in two avors. First, we can deneclassicalor

geometrical optics, based on 5D null geodesics. Second, we can dene quantum optics. A spinless formulation

is provided by the 5D wave equation, where the wavefunction (;) may be regarded as the number of null

paths between any two given points ofM(5). A spin-1/2 quantum optics requires a wavefunction, which

is a pair of bispinors or abiquaternion[13], a vector with two quaternionic components, at every point of

M

(5). It can be argued thatrepresents two tangent null vectors, one incoming and one outgoing, at every

point ofM(5)[13]. The wavefunction equation is of Dirac-type, based on a funfbein eld3and a special

Cliord algebra; see Eq. (14) in Ref. [13]. The focus is, thus, solving null-propagation problems for passive

particles, given the 5D gravity eld.

4The propagation problems are invariant to conformal transformations

ofhMN, which cannot be fully recovered by observing only particles in null propagation. We add tilde to

the metric symbol,~hMN, to denote a conformally transformed metrichMN. Among space-times suitable for applications are 5D black holes, where the null structure has been a

topic of major interest [19{22]. It is interesting to note that, while the event horizon of a 4D black hole

is always of spherical topology, the event horizon of a 5D black hole can have spherical, toroidal, or other

topologies [22]. Here, we propose 4D interpretations of 5D space-times and passive-particle null-propagation

in 5D space-times [13{15].

3 4D classical and quantum mechanics from 5D physics

Interpretation of the 5D geometry and dimensional reduction are greatly facilitated by symmetry. The

isometries of (M(5);h) form a Lie group, denoted Iso(M(5);h). The generators of this group are called

Killing vectors,, satisng the Killing equation@MN+@NM= 0. We select a nontrivial subgroup of Iso(M(5);h), which we call theformalism isometry group, FIso(M(5);h), and its elements,formalism isometries. Here, we consider that (M(5);h) has only one formalism isometry: the translation along a

space-like dimension,x5.5We foliate (M(5);h) along the formalism Killing eld, into 4D space-times, using

the Arnowitt-Deser-Misner procedure [17,18], and indicate the lapse and shift elds of the foliation in the

5D metric. In turn, this yields a geometrical parameterization ofhMN, which we further write in adapted

coordinates, so it does not depend onx5. We thus obtain h MN=gN N NN+N2 ,hMN=g+NN=N2N=N2

N=N21=N2

;3 The funfbein eld species a choice for a local frame at each point of the 5D manifold.

4The 5D observer has no notion of particle charge or mass and does not distinguish between electric and gravitational elds.

5Mathematics allows for much more general theories. For example, we may consider a space-time with a time-like Killing

vector or several Killing vectors as formalism isometries, possibly requiring foliation over several dimensions of the 5D manifold.

The diculty resides in the physical interpretation and link to experiment. 3 where (M(4);g) is the foliated 4D manifold and,NandN(;;:::= 0;1;2;3), respectively, are the lapse and shift of the 4D foliation. We introducephysical notation N qc

2A; N1;

whereArepresents the electromagnetic potential in CGS units;qis specic charge andcis the speed of

light in vacuum. As it turns out, the 4D metric, which yields the well-known laws of 4D particle propagation,

is notg, but ~gg=2[15]. The 5D metric observed through particle propagation in 4D mechanics is ~hMN=hMN=2; this gives the second requirement for the parameterization of the 5D metric. We write the metrichMNusing the elds that are observed in particle propagation; see Sec. 4. The 4D observer may perform anycylindricaltransformation of coordinates, consisting of dieomorphisms of (M(4);~g) leavingx5unchanged; i.e., x ! F(x); x5!x5:(3.10)

Other coordinate transformations, not changing 4D physical observables, are gauge transformations of the

shift of the foliation,N, or the electromagnetic potential,A x!x; x

5!x5qc

2A(x);,8

:~g!~g; A !A+@A(x); !;(3.11) whereA(x) is the gauge function forA. It is unclear how the 4D observer would use other coordinate transformations and what is their physical interpretation within 4D mechanics.

The symmetry of (M(5);h) to translation alongx5is key for describing 5D passive particles of null mass

(i.e., 5D photons), in four dimensions, as 4D passive particles carrying mass and electrical charge. The 4D

observer considers himself/herself uncharged, which leads to important choices in the process of dimensional

reduction. For classical propagation, 5D null geodesics in (M(5);h) are given 4D interpretation [15]. If the

uncharged 4D observer adopts the path element of (M(4);~g) for the innitesimal proper time, then 5D null

geodesics are represented as 4D time-like Randers geodesics [23,24] in (M(4);~g), where the eccentricity of

the path element is determined byA. Furthermore, since thex5-translational symmetry yields a conserved

quantity for 5D geodesic motion, we say that the 4D particle propagates with constant passive mass [15]. For

quantum propagation, we perform Fourier transforms with respect withx5of the 5D wave equation and the

5D Dirac-type equation, which, in addition, requires a foliated funfbein eld [13].

6It is further postulated

that the coordinatex5is conjugated with the inverse Compton wavelength1=mc=~[13,14]. In this fashion, 5D photons are interpreted as 4Don-shellmassive quantum particles.

A priori, there is no reason for the 5D space-time be symmetric, in which case, we do not oer a physical

picture for the 5D geometry. However, if the 5D space-time isnearlysymmetric, then a interpretation can be

constructed based on the alleged symmetry and a top-down hierarchy of physical concepts. The rst order

concepts are those of the physical picture assuming that the symmetry is exact. They provide a satisfactory

description for particle propagation in manifold elds, for a small enough proper-time interval and space-time

neighborhood. Next to the rst order physical concepts, second order concepts can be added for improving

the description of particle propagation, so it holds for longer proper-time intervals and larger space-time

neighborhoods. In general, the physical concepts of second order rely on those of rst order. Therefore, the

hierarchical physical interpretation is not covariant. Reference [16] demonstrates how particle propagation

in a 5D space-time with approximatex5-translation symmetry can be given a physical picture. In the rst

order, we use 4D mechanics [15]. For the second order, we invoke additional phenomena such as Stokes friction and non-hermitian quantum decoherence [16].

4 Field equations in 4D mechanics

Several authors [9{12] published the eld equations forx5-independent 5D gravity,R(5)

MN= 0, in the

parametrization given byg,Aand . We performed an additional conformal transformation and rewrote6 The choice for the foliated funfbein rests with the uncharged 4D observer [14]. 4 the equations for 5D gravity in the elds ~g,Aand , which have more physical content because ~gis the 4D metric observed for 4D passive particle propagation [15]

R(4)12

~g~R(4)=q22c4~TEM+~T;(4.12)

5~F=3~F~5ln;(4.13)

1 ~5~5 =q24c4~F~F+16 ~T~g;(4.14) where ~R(4)and~5are, respectively, the 4D Ricci tensor and the Levi-Civita connection of the metric ~g.

Furthermore,

~FF=@A@Aand

TEM= ~g~F~F+14

~g~F~F;(4.15) T=1 ~5~52 (~5)(~5) 1 ~g~g ~5~52 (~5)(~5) ;(4.16)

are the stress-energy tensors for the electromagnetic and elds, respectively. These equations are in-

terpreted according to the induced matter principle. An empty 5D space-time, equipped with a space-like

Killing eld, is regarded as a 4D space-time containing electromagnetic and elds, as sources of 4D gravity.

Equation (4.12) represents the 4D Einstein equations where the dichotomy between elds and sources occurs

only with the 4D induced-matter interpretation of the 5D geometry. The RHS of Eq. (4.12) is separated into

electromagnetic and sources. It is important to note that the 4D gravity sources are dierent in nature:~TEMcan be inferred from passive particle propagation, while~Tcannot. Therefore, from this perspective,

appears like adark eld[25,26].

4.1 The Maxwell equations and Ohm's law from the equations for 5D gravity

Equation (4.13) is of type Maxwell and helps recover Eqs. (1.1) and (1.4), considered in vacuum.

7For this,

we break Eq. (4.13) to introduce the 4-current density,j 5~F4c j;(4.17) 4c j 3~F~5ln:(4.18) Eq. (4.17) corresponds to the Maxwell equations (1.1) and (1.4); the RHS of Eq. (4.17) expresses the sources for the electromagnetic eld in the LHS. Equation (4.18) further explicatesjin terms of and

the electromagnetic eld, and is interpreted in the same way: the RHS designates the sources for the LHS.

The alternative interpretation, where the LHS designates the source for the electromagnetic and elds

in the RHS, yields a contender to the Maxwell equations, and a line of argument far from the traditional

electromagnetic theory. We will not pursue it here. Applying~5on both sides of Eq. (4.18) and using Eq. (4.13) yields the 4D continuity equation forjas an identity8 4c ~5j=3(~5~F)(~5ln)3~F~5~5ln = 6~F~5~5ln = 0:(4.19)

It is signicant thatjdepends on , which does not appear in the LHS of Eq. (4.17). Still, as a divergence-

free vector eld,jretains more freedom than . From the perspective of a 4D eld theory,jrequires no7

The concept of vacuum is vague in terms of Lorentzian geometry. According to the induced matter principle,Fis an

active matter eld dened on 4D space-time. However, the reference, empty 4D space-time is Ricci at, with eld equations R (4)

MN= 0, whereFand do not not exist. Therefore, the reference 4D space-time is not sucient for describing particle

propagation in electromagnetic eld, in vacuum. We are lead to postulate that vacuum represents 5D emptiness rather than

4D emptiness; it is unclear how this relates to experiment. Another possibility is to associate vacuum to the limit of weak elds

evolving in a 4D space-time. The termelectrovacuumis often used for solutions of both Einstein and Maxwell equations.

8This is also obtained by applying~5on both sides of Eq. (4.17).

5

further analysis. It is very important to note that, in the theory of 5D gravity,jresults from a dispensable

ansatz (i.e., Eqs. (4.17) and (4.18)), of no particular benet for solving the 5D eld equations. This is in

stark contrast with the fundamental role thatjplays in the theories of electromagnetism and condensed

matter, where a fundamental SI unit is dedicated to measurements of electrical current. These theories

further describej, postulating conduction mechanisms which rely on the structure of matter. For example,

in solid state physics (e.g., Refs. [27, Chs. 1, 2] and [28, Ch. 6]), the current density is described using classical

or quantum formalisms for passive particles/quasiparticles in motion, trapped within matter, subject to the

Lorentz force. In addition, the particles/quasiparticles are subject to collisions or phenomenological friction

forces, due to additional electromagnetic interactions associated to the atomic structure of matter. 9In

plasma physics (e.g., Refs. [29, Ch. 1] and [30, Chs. 5, 6]), conduction mechanisms have similar building

principles, although, sometimes, they simply postulate that the Lorentz force is zero for particles within

plasma. A 3D velocity eld for charged carriers is often dened using 4D active matterv=cj=j0.10

There are two main uses for the microscopic pictures ofj. First, they explain the Laplace force [31], the

force acting on a conductor placed in electromagnetic eld, in terms of the Lorentz force;

11see e.g., [3, Ch. 5.2]

for an argument based on Ampere's work [32]. However, the Laplace force does not require an explanation

via the Lorentz force, because the Laplace force can be obtained from the divergence of the electromagnetic

stress-energy tensor,~5~TEM[3, Ch. 11.11]. Second, they provide microscopic explanations for the Ohm's

law and other laws in condensed matter physics. However, as we explain below, a conduction mechanism is

not required to explain Ohm's law, which here appears as a eld equation. To see this, we provide a physical

interpretation for Eq. (4.18), breaking 4D covariance in the limit of weak elds.

5 A minimal amendment to the Maxwell equations

The rigorous approach to Eqs. (4.12)-(4.14) is through exact solutions, analyzed for their relevance to ex-

periment. Many exact solutions for empty 5D space-times without compact dimensions have already been

found [20,33{39], but they primarily describe time-independent geometries, which are not invariant to trans-

lation alongx5, and do not satisfy Eqs. (4.12)-(4.14). Perhaps, the only exception is a class of time-dependent

solutions, obtained through the Wick rotation of the 5D Myers-Perry black-hole metric [12,40,41]. 12 An alternate approach is using the 5D eld equations for an amendment to the Maxwell equations, which

are already close to experiment. Evidently, the physics of the amended Maxwell equations is superseded

by that of corresponding exact 5D solutions. We propose a strong approximation of the 5D gravity equa-

tions (4.12)-(4.14) in the limit of weak elds. We consider that the 4D metric, ~g, is approximately the

Minkowski metric,; n.b., (q=c2)2~TEM2+~T0. We further break covariance to distinguish 3D spacequotesdbs_dbs47.pdfusesText_47

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