Geometrical interpretation of electromagnetism in a 5-dimensional manifold

Transcription

1 Classical and Quantum Gavity PAPER OPEN ACCESS Geometical intepetation of electomagnetism in a 5-dimensional manifold To cite this aticle: TaeHun Kim and Hyunbyuk Kim 017 Class. Quantum Gav View the aticle online fo updates and enhancements. Related content - ft telepaallel gavity and cosmology Yi-Fu Cai, Salvatoe Capozziello, Maiafelicia De Lauentis et al. - Pespective on gavitational self-foce analyses Steven Detweile - Rotating Einstein--Maxwell fields: smoothly matched exteio and inteio spacetimes with chaged dust and suface laye A Geogiou This content was downloaded fom IP addess on 16/08/018 at 10:3

2 Classical and Quantum Gavity Class. Quantum Gav pp Geometical intepetation of electomagnetism in a 5-dimensional manifold TaeHun Kim 1 and Hyunbyuk Kim,3 1 Depatment of Physics & Astonomy, Seoul National Univesity 1 Gwanak-o, Gwanak-gu, Seoul, 0886, Republic of Koea Depatment of Physics & Eath Science, Koea Science Academy of KAIST, , Baegyanggwanmun-o, Busanjin-gu, Busan, 4716, Republic of Koea gimthcha@snu.ac.k and time@kaist.ac.k Received 6 Septembe 016, evised 1 Apil 017 Accepted fo publication 16 May 017 Published 4 July 017 Abstact In this pape, Kaluza Klein theoy is evisited and its implications ae elaboated. We show that electomagnetic 4-potential can be consideed as a sheaing-like defomation of a 5-dimensional 5D manifold along the fifth 5th axis. The chage-to-mass atio has a physical meaning of the atio between the movement along the diection of the 5th axis and the movement in the 4D spacetime. In ode to have a 5D matte which is consistent with the constuction of the 5D manifold, a notion of paticle-thead is suggested. Examinations on the compatibility of efeence fames eveal a covaiance beaking of the 5th dimension. The field equations which extend Einstein s field equations give the total enegy-momentum tenso as a sum of that of matte, electomagnetic field, and the inteaction between electic cuent and electomagnetic potential. Finally, the expeimental implications ae calculated fo the weak potential case. Keywods: Kaluza Klein theoy, Einstein Maxwell equations, geneal elativity, manifold defomation 1. Intoduction It has been nealy a hunded yeas since the attempt fo a unified field theoy was fist made. Although nowadays mainsteam concen about the unification is to obtain a quantum theoy 3 Autho to whom any coespondence should be addessed. Oiginal content fom this wok may be used unde the tems of the Ceative Commons Attibution 3.0 licence. Any futhe distibution of this wok must maintain attibution to the authos and the title of the wok, jounal citation and DOI /17/ $ IOP Publishing Ltd Pinted in the UK 1

3 Class. Quantum Gav of gavity, the fist appoaches wee to make a theoy of electomagnetism in the manne of diffeential geomety of geneal elativity. The most epesentative wok of this kind was done by Theodo Kaluza [1] and Oska Klein []. The following bief intoduction of Kaluza Klein theoy is based on [3, 4, 5]. In 191, Kaluza tied to unify the electomagnetism and the gavitation in 5-dimensions. Kaluza constucted a 5D metic including a new scala field and the whole metic was independent of the 5th dimension. He also assumed the souce-fee condition fo 5-dimensions. The geodesic equation in 5-dimensions poduced a tem that could be thought as the Loentz foce. Howeve, thee was anothe tem containing the scala field and this tem could exceed the Loentz foce tem. This additional tem was a citical defect of Kaluza s model. In 196, Klein poposed a modified vesion of Kaluza s 5D metic, which was g αβ = gab + φ A a A b φ A a φ A b φ, 1 whee φ was a scala field and the epeating bounday condition was imposed along the 5th dimension. This is what we now call as the Kaluza Klein metic. Then, fom Einstein s field equations fo 5D vacuum, 4D Einstein s field equations with enegy-momentum of electomagnetic field could be obtained when φ was fixed to be a constant. The fixation of φ could avoid the citical defect of Kaluza s model. Klein then compactified the 5th dimension in ode to connect his model to quantum mechanics. Howeve, it was discoveed late that the fixation of φ is consistent with the souce-fee condition only if F ab F ab = 0 [6]. In this pape, we conside a model to explain the electomagnetism as a geometical stuctue in 5-dimensions. The pesent model does not compactify the 5th dimension, so it is close to Kaluza s oiginal idea. The suggested model obtains a metic which has an identical fom of Klein s metic o Kaluza Klein metic with a fixed φ though a sheaing-like defomation of a 5D manifold. Even though the metic has the same fom, the constuction of the metic fo the 5D manifold and the 4D space-time thoughout the pesent pape, space-time will mean 4D spacetime is diffeent fom both Kaluza s oiginal idea and Klein s vesion. The model focuses not on the induced matte fom highe dimensional geomety but on the pe-existing matte. Because of this, the model does not inheit the souce-fee condition and gives a new fom of 5D matte. The stuctue of the pesent pape is as follows. Motivations fo intoducing the 5th dimension and the method to get the 4D physics fom the 5D one ae discussed in section. Then, the 5D manifold used thoughout this pape is constucted and the intepetation of electomagnetic 4-potential is given at section 3. Hee, we also suggest that 5D matte is composed of many 1D theads extending paallel to the 5th axis. Section 4 is about physical quantities in the defomed manifold and decides the metic of the 5D manifold. Discussions on the splitting of 5D field equations ae also given hee. Section 5 shows that, with a help fom the intepetation of the chage-to-mass atio, the 4-acceleation due to the Loentz foce oiginates fom the pojection of a 5D geodesic onto the space-time. Section 6 examines the compatibility of efeence fames, and consides that the covaiance of the 5th dimension is boken. Based on all the pevious discussions about the 5th dimension, the 4D field equations which extend Einstein s field equations ae finally suggested in section 7, and this completes the discussions on the field equations. Hee we show that the total enegy-momentum tenso is a sum of enegy-momenta of matte, electomagnetic field, and the inteaction between electic cuent and electomagnetic potential. Section 8 deals with the unit system deived fom the constuction of the field equations. Section 9 discusses the expeimental implications. Lastly, section 10 gives the conclusion and suggests futhe eseaches. Thoughout the pape, indices fo 4D vaiables ae given in Roman alphabets a, b, c,..., wheeas indices fo 5D vaiables ae given in Geek alphabets α, β, γ,... Tilde is used fo

4 Class. Quantum Gav physical quantities in the defomed 5D manifold and the defomed space-time. An uppe ba is used when the indices alone cannot guaantee that the quantity is calculated in the 5D manifold. s is the 5D inteval and τ is the pope time in the space-time. Notations ae summaized in appendix A. We adopt the geometized unit G = c = 1. Deivations of equations ae also given in the appendices.. Design of the model The model constucted in this pape is inspied fom the chaacteistic of a static gavitational field. In a static gavitational field, the metic of space-time is time independent and 3D space at each moment becomes a 3D hypesuface of a constant time. While evey object follows its geodesic in the space-time, its tajectoy in a 3D space is cuved and is not a 3D geodesic in space. The 3D tajectoy is a pojection of the 4D geodesic onto the 3D space and the cuvatue of 3D tajectoy in space depends on the object s velocity in the Newtonian sense. Although the diection of velocity can be known in the 3D space by dealing with the 3D tajectoy, the magnitude of velocity cannot be detemined only fom the 3D tajectoy but should be found in the space-time. Inteestingly, the magnitude of velocity indicates how much the 4D geodesic is paallel to the time axis such that the moe paallel it is, the smalle is the speed. So, the cuvatue of 3D tajectoy becomes smalle as the object s speed becomes highe, o as the 4D geodesic becomes less paallel to the time axis and moe paallel to the 3D space. Analogous aguments can be applied to a model which descibes electomagnetic field in a 5D manifold. Now electomagnetic field is a geometical stuctue of the 5D manifold and evey object follows 5D geodesic motion. Space-time is a 4D hypesuface with a constant 5th coodinate and the 5D metic is independent of the 5th coodinate. An object s tajectoy on the space-time is a pojection of its 5D geodesic onto the space-time and geneally is not a 4D geodesic in space-time. The cuvatue of 4D tajectoy is the 4-acceleation and this should be egaded as the effect of the Loentz foce. Meanwhile, it is known that the 4- acceleation due to the Loentz foce depends on the 4-velocity and the chage-to-mass atio of an object. Fo a given electomagnetic field, it is expeimentally veified that the diection of the 4-acceleation is decided by the 4-velocity and the magnitude of the 4-acceleation is diectly popotional to the chage-to-mass atio. Theefoe, the influence of 4-velocity on the cuvatue of 4D tajectoy in a 5D manifold coesponds to the influence of the diection of 3-velocity on the cuvatue of 3D tajectoy in a static gavitational field. Moe impotantly, the influence of the chage-to-mass atio on the cuvatue of 4D tajectoy in a 5D manifold coesponds to the influence of the magnitude of 3-velocity on the cuvatue of 3D tajectoy in a static gavitational field note that it is meaningless to conside the magnitude of 4-velocity. This compaison encouages the intepetation that an object s chage-to-mass atio can be thought as a quantity indicating how much its 5D geodesic is paallel to the 5th axis o paallel to the space-time. Since highe chage-tomass atio gives highe 4-acceleation, objects with a highe chage-to-mass atio should have 5D geodesics moe paallel to the 5th axis, and neutal objects may have 5D geodesics paallel to the space-time. We popose moe quantitatively that the chage-to-mass atio is a atio of the movement along the diection of the 5th axis to the movement on the space-time. This initial idea about chage-to-mass atio in tems of a dimensionless paticle will be supplemented and moe completely intepeted using a 1D object in sections 3 and 4. Once the physics in the 5D manifold is fully descibed it should be educed to that of the 4-dimensions in ode to get the equations in tems of the obseved quantities in the space-time without consideing the 5th dimension. This eduction will allow us to check the 5D physics 3

5 Class. Quantum Gav with the expeimentally veified 4D physics. The pescibed setup, especially the pat that the space-time is the hypesuface with a constant 5th coodinate, makes the eduction pocess a simple pojection. The coodinate system in the 5D manifold is capable of being constucted in such a way that its 4D pat coincides with the coodinate of the space-time. Fom now on, that coodinate system is assumed. This coodinate constuction of the 5D manifold gives the induced metic of the space-time to be identical to the 4D pat of the 5D metic. Also, a tajectoy of an object in the space-time can be simply obtained by taking the 4D pat of the 5D geodesic due to this coodinate constuction. Although it is common to use the 5th dimension to explain electomagnetism in a geometical manne, the setup of the pesent model and the esultant eduction pocess along with the educed 4D physics ae distinct fom the oiginal Kaluza Klein theoy. Reduction in Kaluza Klein theoy can be found in [3, 7, 8]. In Kaluza Klein theoy, the way of embedding of space-time into 5D manifold is not specified but only the metic elation between 5D manifold and the space-time is given. Instead, in the pesent model, it is established that the space-time is a hypesuface with a constant 5th coodinate. This esults in a diffeent spacetime metic fom the Kaluza Klein case. By showing that the Ricci scalas of the space-time fo the two cases ae diffeent, it is clea that the space-time in the pesent model and the space-time in Kaluza Klein theoy ae actually diffeent, so the ways of embedding ae also. Calculations of the Ricci scala of the space-time is given in appendix E, afte obtaining the final 5D metic via discussions in sections 3 and 4. The diffeence in embedding aises diffeences on the space-time metic see section 4, on the coection tems in the 4-acceleation of the space-time tajectoy see section 5, and even on the signatue of the 5th dimension see section 7. The fist two ae physical and not mee mathematical diffeences as they can be tested by expeiments see section Constuction of the model Ou model inheits the qualitative featues of geneal elativity but in 5-dimensions, so that the 5D manifold is descibed by 5D metic and the metic is affected by 5D enegy-momentum. We conside a 5D manifold which is constucted by dagging the space-time to the diection of the 5th axis so that the 4D pat of 5D metic is independent of the 5th coodinate. Howeve, to validate the independence of 5D metic on the 5th dimension, the independence of matte on the 5th dimension is needed. Theefoe, we conside that paticles in space-time ae also dagged togethe so that they become theads paallel to the 5th axis. Given the notion of the paticle-thead, a paticle in the space-time becomes the coss section of the coesponding paticle-thead along the space-time. 5D matte composed of paticle-theads ensues that the matte distibution is independent of the 5th coodinate, and hence the independence of whole 5D metic on the 5th coodinate due to the tanslational symmety of matte. Actually, the notion of paticle-thead is not additionally intoduced but follows fom the dagging constuction. Consideing that the 5D manifold is constucted by continuously dagging the space-time, a 5D point paticle distibution is unnatual since placing discete 5D mattes will invalidate the constuction pocess of dagging. Discete placing means that thee should be some unknown extenal factos affecting the pocess of dagging fo deciding on which 4D hypesuface will the 5D paticle be placed. Yet, no such extenal factos ae allowable fo ou constuction of the 5D manifold. In addition, the notion of paticle-thead helps to define the distance between paticles. If a point paticle does not extend along the 5th dimension, the poposed intepetation of the chage-to-mass atio implies that the 5D distance between two paticles with diffeent 4

6 Class. Quantum Gav chage-to-mass atios will divege as time goes on, even if they ae stationay in the 3D space. On the othe hand, the distance between two paticle-theads can be defined natually as the distance between thei coss sections on the space-time and then it is educed to the wellestablished distance between the coesponding paticles in the 3D space. Besides these helping featues, it is possible to imagine a mechanical wave popagating along a paticle-thead. A wave will ceate a petubation on the enegy-momentum of the paticle-thead but the change of the enegy-momentum along the 5th dimension should be negligible in lage scale. Although the possibility of a mechanical wave does not seem to aise any immediate issues, it should be examined thooughly to see whethe thee is any contadiction with the obsevations. Also, the physical meanings of quantities associated with the wave, such as its wavelength, amplitude, and popagation speed, need to be investigated. The pesent pape will not pusue the issue elated to the wave along a paticle-thead futhe and deals only with the zeo mode of the wave. The design of the 5D model in section is now to be evised to a paticle-thead vesion. Fotunately, both paticle-thead and 5D metic ae independent of the 5th dimension, so any pai of geodesics which have the same tangent vectos at each of thei stating points with common space-time coodinates but diffeent 5th coodinate ae exactly identical. In othe wods, evey potion of paticle-thead moves identically and shows exactly the same behavio as a geodesic motion of a test paticle in the 5D manifold. This guaantees that a paticlethead which is following its 5D geodesic does not entangle with anothe paticle-theads along its extension unless thee exists some pe-given intenal motion of the paticle-thead. The pojection of 5D geodesic on the space-time will be the tajectoy of the coss section of the paticle-thead along the space-time. Howeve, the chage-to-mass atio should no longe be thought of as the atio of a paticle s movement along the 5th axis to its movement on the space-time, but as the flow speed of a paticle-thead along its extension. Then a paticlethead coesponding to a chaged paticle flows along its extension and a paticle-thead coesponding to a neutal paticle does not flow but only moves paallel to the space-time. Now we stat to constuct the 5D manifold by consideing fist a situation whee thee exist only neutal objects. Stating fom a space-time with its metic tenso g ab, a 5D manifold is constucted by dagging the space-time into the 5th diection. This dagging constuction makes evey hypesuface of constant x 4 to be identical to the space-time, so the 5D manifold will have a metic tenso gab g αβ =, 1 with a pope coodinate along the 5th dimension and evey component of the metic is independent of the 5th dimension. We will take the signatue of the 5th dimension to be positive. The 4a-pat of the metic decides the othogonality of the space-time and the 5th diection. Howeve, fo the case that evey object is neutal, all the paticle-theads ae not flowing so the 5D matte distibution will have a 5th axis-evesal symmety. Theefoe, the whole 5D manifold also should have the 5th axis-evesal symmety and this gives gab 0 g αβ = Let us take this to be an electomagnetic field-fee manifold. Now we conside a geneal situation whee thee exist chaged objects too. In this case, the 5th axis-evesal symmety is boken so it is possible to have non-vanishing 4a-pat. In ou model, a flowing paticle-thead tails its neighboing pat of the 5D manifold so that sheaing 5

7 Class. Quantum Gav occus on the manifold in the diection of the flow of paticle-thead. To epesent the sheaing quantitatively, we intoduce an opeation on a manifold which will be called defomation. A defomation is an opeation that keeps the coodinate values of evey point of the manifold while changing the manifold. This means that d x α = dx α fo any infinitesimal displacement since the coodinate does not change, but its nom can diffe since the manifold and its metic changes. Even though the effect of a defomation is completely diffeent fom that of a coodinate tansfomation, inteestingly, they can give a simila-looking change of metic. Keeping this in mind, a defomation acting on a manifold gives a defomed metic by g αβ = D γ αd δ βg γδ, 4 whee D γ α pefoms the defomation. It is impotant to note that the Riemann tenso, the Ricci tenso, and the Ricci scala of g αβ cannot be computed by teating D γ α as a coodinate tansfomation on g γδ and applying the tenso tansfomation ule on those tensos since, again, the defomation is an opeation on the manifold itself. Moe discussions about the defomation ae given in appendix B. Fo ou case, to find a sheaing defomation, we fist look at a coodinate tansfomation given by x a = x a, x 4 = Ca x a + x 4, whee C a = C a x b. Then, the equations fo the coodinate diffeentials become dx a = dx a, 5a 5b 6a dx 4 = Ca + C b,a x b dx a + dx 4, and we ename the whole paenthesis pat in equation 6b to be C a fo the notational convenience. Then, the equations fo the coodinate diffeentials ae e-witten as dx a = dx a, dx 4 = Ca dx a + dx 4. 7b This tansfomation epesents a sheaing along the 5th dimension since the 5th axis emains the same but the othe fou axes become inclined while keeping the values of the fist fou coodinates. Afte the above tansfomation, the metic of equation 3 is now expessed in the pimed coodinate, g αβ = gab + C a C b C a. 8 C b 1 This is the same fom of Kaluza Klein metic with φ = 1. Howeve, it is not tue that all Kaluza Klein metics with φ = 1 can be thought as a poduct of a coodinate tansfomation fom the metic of the electomagnetic field-fee manifold since the commutativity of the second patial deivatives equies C a,b = C b,a. Because of this, we conside a defomation which esembles the esulting equation of a coodinate tansfomation, equation 8, but now the commutativity of the second patial deivatives is no longe of concen. To emphasize this, instead of C a, we intoduce B a which does the ole of defomation. Due to the intoduction of paticle-thead, the defomation is independent of the 5th coodinate and hence B a is a function of the space-time coodinates alone. Then the defomation given by 6 6b 7a

8 Class. Quantum Gav , if α = β D β α = B a if α = a and β = 4 0, othewise yields, using equations 3 and 4, the defomed 5D manifold with metic gab + B a B b B a g αβ =, 10 B b 1 whee evey component is independent of the 5th coodinate and B a,b is no longe equied to be the same as B b,a. A compaison between g αβ and g αβ pesents that this defomation affects the 5D manifold in the sense of sheaing. It should be noted that since the defomation is not a mee mathematical handling but a physical phenomenon, the B a in equation 10 is the ode paamete deciding how much the distance stuctue of the manifold changes. We will intepet B a, the quantity indicating the sheaing-like defomation, as the electomagnetic 4-potential on each point of the defomed 5D manifold. This is justified by the fact that the 4-acceleation of a pojection of 5D geodesic of that metic on the space-time gives the Loentz foce tem as the leading tem the calculation will be given in section 5. Then the metic g αβ of the defomed 5D manifold becomes Kaluza Klein metic with φ = 1. Befoe deiving moe equations using this metic of defomed manifold, the electomagnetic 4-potential should be defined clealy. This discussion will be continued in section 4. Is the 5th dimension space-like o time-like? It depends on the choice of the signatue of the space-time metic. In section 7, the field equations eveal that the 5th dimension is timelike by showing that the 4D signatue should be +. But then the ultahypebolicity poblem aises. As Tegmak pointed out [9], the patial diffeential equations in two o moe time dimensions become ultahypebolic and the wold becomes unpedictable. Remakably, the paticle-thead of the pesent model hindes this poblem. The paticle-thead guaantees both metic and enegy-momentum to be independent of the 5th dimension, and thanks to this, evey patial deivative with espect to the 5th coodinate vanishes. Then, the 5D patial diffeential equations can be ecast into the hypebolic 4D patial diffeential equations of the space-time. In this espect, one can say that the paticle-thead does the ole of keeping the hypebolicity of the 5D manifold Physical quantities in the defomed manifold Since a defomation acting on a manifold affects the manifold itself, all the obsevable quantities in the pesence of non-zeo electomagnetic potential should be defined in the defomed manifold. In a sense, an undefomed manifold pefoms only the intemediate ole. The fact, which will be poven in section 5, that the Loentz foce does appea in the defomed manifold suppots this. Theefoe, all the obsevable quantities in the space-time including the Loentz foce and the 4-velocities will be defined in the defomed spacetime. In othe wods, the defomed space-time is the ight stage whee the 4D physics is descibed. We also define the chage-to-mass atio of an object in the defomed manifold. It is intepeted as the flow speed of the coesponding paticle-thead along its extension, and its evey potion follows an identical 5D geodesic. Fo this geodesic, the chage-to-mass atio has a meaning of the atio of movement along the diection of the 5th axis to the movement on the space-time. Theefoe, an object s chage-to-mass atio can be expessed by 7

9 Class. Quantum Gav q m = dx4, 11 fo the geodesic that evey potion of the coesponding paticle-thead follows. This allows us to pictue the chage-to-mass atio as a slope of the geodesic in 5-dimensions. Fo a paticle-thead, its coss section along the space-time is a paticle in the space-time, and mass is a chaacte of that coss section. Theefoe, the electic chage q = m dx 4 / becomes a quantity descibing the mass flow ate of a paticle-thead. Fom this, the pictue of the model can be dawn such that the mass flow of a paticle-thead along the 5th diection in the 5D wold is conceived as an electic chage in a 4D hypesuface. So, in the pesent model, thee is no such thing as a 3D bane in which the mattes ae confined, no is thee any division between the bane and the bulk. As discussed in section 3, 5D matte is composed of paticle-theads. This suggests that two physical quantities ae equied to descibe 5D matte. One is a 4D enegy-momentum tenso which coesponds to the distibution and movement of coss sections of paticletheads on the space-time. The othe is a 4-cuent density which descibes the flux of the paticle-theads. These two ae enough to descibe 5D matte distibution consideing that the only appopiate fom of 5D matte is a paticle-thead. Actually, using 4D quantities only is bette than newly intoducing a 5D enegy-momentum tenso because the 5D volume density of a paticle-thead blows up. Note that the 4D suface density of the 5D matte along the space-time is the mass distibution which is finite and non-zeo. Fo a given matte distibution, the two quantities, 4D enegy-momentum tenso and 4-cuent density, can be obtained by conventional means but now in the defomed space-time. This disjunction of 5D matte quantities and the covaiance beaking of the 5th dimension, which is dealt with in section 6, causes the field equations elating the 5D metic and the 5D matte to be descibed by two split equations, one using only the 4-cuent density and the othe using both. Befoe discussing moe about the field equations, the metic of the 5D manifold should be claified fist. As aleady mentioned in section 3, the electomagnetic 4-potential does the ole of the defomation facto B a of the defomed 5D manifold with espect to the undefomed 5D manifold. This defomation is caused by flowing paticle-theads tailing the suounding 5D manifold, and the flux of the paticle-theads is the electic 4-cuent in the defomed space-time, which is an obsevable quantity. Theefoe, the electomagnetic 4-potential will be defined in the defomed space-time as well, unlike Kaluza Klein theoy. Then, by eplacing B a in equation 10 to Ãa, emphasizing that it is the electomagnetic 4-potential defined in the defomed space-time, the metic tenso of the defomed 5D manifold can be witten as g αβ = g ab + à a à b à a à b 1 = g ab à a à b 1, 1 whee the second equality comes fom the discussion on the induced metic tenso of the defomed space-time in section. Hee, evey component is independent of the 5th coodinate. Again, as noted in section 3, Ãa does the ole of an ode paamete indicating the change of 5D manifold. Now we esume the discussion on the field equation that employs only the 4-cuent density. It will be shown in section 5 that when the sheaing-like defomation facto is intepeted as the electomagnetic 4-potential, the leading tem of 4-acceleation fom the pojection of 5D geodesic onto the space-time becomes the Loentz foce. Fom the well established facts that the Loentz foce depends on the electomagnetic fields and these fields follow 8

10 Class. Quantum Gav Maxwell s equations, the equations detemining Ãa will be Maxwell s equations. Howeve, unlike the conventional Maxwell s theoy, the issue of gauge condition aises fo Ãa in this 5-dimensional model because now the physical quantity is not the electomagnetic field but the electomagnetic 4-potential itself. Thus, the gauge of the electomagnetic 4-potential must be fixed. The most natual way of gauge fixing is to take it to be the etaded potential in the defomed space-time to satisfy the causality in the space-time. In shot, the field equations fo à a, o g 4a, will be Maxwell s equations fo electomagnetic 4-potential in Loenz gauge in the defomed space-time and the physically valid solution of these equations is a etaded potential due to the 4-cuent density J b in the defomed space-time. The geneal fom of this solution is à a x = G et ab x, x J b x gx d 4 x, 13 G et ab whee the integal is taken ove the defomed space-time and x, x is the Geen s function fo etaded potential. G et ab x, x and gx ae functions of the metic of the defomed space-time and, as a consequence, functions of the 4-potential itself. To obtain a complete equation fo the 4-potential, the Geen s function should be expessed in tems of the quantities associated with the defomed space-time such as the metic, the Riemann tenso, the Ricci tenso, and the Ricci scala. This is discussed in [10]. Instead of pusuing the exact Geen s function of the defomed space-time in this pape, we will conside a weak potential appoximation in section 9. Until now, we have dealt with the field equations fo g 4a = à a, so the only pat emaining to be detemined in 5D metic is g ab = g ab + à a à b. The field equations fo this pat will be suggested in section 7 afte discussing the covaiance beaking of the 5th dimension in section 6. At this moment, the featues of metic and the metic constuction sepaate the pesent model fom Kaluza Klein theoy. In Kaluza Klein theoy, 5D metic is given as an ansatz, but in the pesent model, 5D metic is obtained by dagging the space-time and applying the sheaing-like defomation to the 5D manifold. Also, the souce of the defomation is said to be paticle-theads of chaged objects, giving us a pictue that flowing paticle-theads tail the suounding 5D manifold. Futhemoe, although the two expessions of the 5D metic in equation 1 look simila to Kaluza Klein metic and Kaluza s oiginal metic, espectively, the decomposed pats of the 5D metic ae diffeent. The fome expession in equation 1 has the electomagnetic 4-potential defined on the defomed space-time unlike Kaluza Klein metic that employs electomagnetic 4-potential defined on the undefomed space-time. Also, the space-time metic fo the latte expession in equation 1 is g ab instead of g ab used in Kaluza s oiginal metic. All these diffeences come fom the embedding of the space-time. Since all the physical quantities ae defined in the defomed manifold, we will use the latte expession of equation 1 except the cases whee it is needed to distinguish the effect of electomagnetic field and the effect of matte without electomagnetic field sepaately. By taking invese of g αβ, the invese metic tenso of the defomed 5D manifold becomes g αβ = g ab + kã a à b kã b kã a k, whee k = 1 1 à c à c. 14 Equations 1 and 14 include the elation between the metic and the invese metic of the defomed space-time. Fo the 4D physics, index aising and loweing of 4D tensos should be done in the space-time without efeencing the 5D manifold. 9

11 Class. Quantum Gav This way of defining the electomagnetic 4-potential pevents loopholes in the model building by making all the obseved quantities defined on the defomed space-time. But, at the same time, this makes the calculation of electomagnetic 4-potential with a given 4-cuent density difficult. To calculate Ãa, the etadation must be decided by g ab, which is the function of Ãa itself. Theefoe, to know Ãa fo a given 4-cuent density distibution, g ab and Ãa should be detemined simultaneously. Fotunately, thee is a moe convenient way if the 4-cuent density is weak enough to make ÃaÃb sufficiently small compaed to g ab. In this case, it will be acceptable to calculate Ãa in the undefomed space-time as the zeoth ode of appoximation. 5. Pojection of 5D geodesic Now we show that the suggested geometical intepetation of the chage-to-mass atio and the electomagnetic 4-potential gives the ight 4-acceleation which can be seen as the Loentz foce fo the pojection of 5D geodesic onto the space-time. In the defomed 5D manifold we shall give the 5D geodesic equation in tems of the 4D pope time τ since the Loentz foce will be expessed with a 4-velocity. The 4D pope time is not an affine paamete, so d x α A lengthy calculation appendix C gives + Γ α dx β dx γ βγ = f τdxα. 15 dx β dx γ dx a f τ = g 4a Γ4 βγ. 16 The 4D pat of the geodesic equation in the defomed 5D manifold becomes d x a + Γa dx b dx c bc = Γ a dx 4 dx b 4b Γ a dx 4 dx 4 44 g dx β dx γ dx d dx a 4d Γ4βγ. 17 This is the equation of the pojected 5D geodesic onto the space-time. The Chistoffel symbols ae given in appendix D. Then, eaangement of tems finalizes the equation of the 4-acceleation in the defomed space-time appendix F whee ã a = d x a ã a = L a + + Γ a dx b dx c bc, 1 L c à c D a à e à e 1 à e à b à c dxb dx c e D a, 18 L a = dx4 F a dx b b, and D a = à a dx d dx a à d. 19 The tem L a is the Loentz foce tem. Equation 18 shows the effect of defomation clealy. It can be easily checked that the 4-acceleation and hence the Loentz foce vanishes when thee is no defomation because the electomagnetic 4-potential becomes zeo in that case. The equation also shows that, in addition to the Loentz foce tem, thee ae coections fo the 4-acceleation in the pesent model. Fo Kaluza Klein theoy, thee ae no such additional coection tems because the pope time is defined on g ab instead of g ab [11]. These coection tems ae expeimentally veifiable and can be used to distinguish the pesent model fom Kaluza Klein theoy. We will calculate the explicit components of those tems in section 9. 10

12 Class. Quantum Gav Covaiance beaking of the 5th dimension The pesented intepetation of the chage-to-mass atio implies that the neutal paticletheads move only along the space-time while chaged paticle-theads flow along the diection of the 5th axis also. But then the motion will be elative between the paticle-theads, so should a chaged one obseve itself as a neutal one and an oiginally neutal one as a chaged one? An examination on the compatibility of efeence fames gives the answe to this question. All of the electomagnetic phenomena have been descibed and successfully explained by an obseve who obseves nealy all of the macoscopic objects in the wold to be neutal. We will call this obseve a natual obseve and its fame as a natual fame. The suggested model in this pape poduces a veified motion of an object at least to the leading tem, the Loentz foce tem, when the metic of the 5D manifold is constucted by a natual obseve. We can examine whethe a given fame is compatible with a natual fame by checking that the metic of the 5D manifold constucted by the coesponding obseve gives the same 5D manifold constucted by a natual obseve and also gives a coect motion fo an object in a natual fame. We denote a given fame with an acute accent and a natual fame without any special mak. Then a geneic coodinate tansfomation between these two can be expessed as d x a =Λ a bdx b + M a dx 4, 0a d x 4 = N a dx a + k dx 4. 0b Hee, only the coefficients M a and N a ae impotant fo the pesent pupose, so we will take Λ a b = δa b and k = 1. Then the equations become d x a = dx a + M a dx 4, 1a d x 4 = N a dx a + dx 4. 1b Now we show that M a should vanish in ode to have a pope metic of the 5D manifold. The metic of the 5D manifold constucted by the given obseve does not change with espect to x 4 so that ǵ αβ,4 = 0. Then the deivative of metic in a natual fame with espect to x 4 is given by g αβ,4 = M a x γ x δ x a x α x β ǵ γδ + M a xγ x δ x α x β ǵγδ,a + x γ x δ x 4 x α x β ǵ γδ. Since we do not have any geneal elation between g αβ and g αβ,a o between ǵ γδ and ǵ γδ,a, the second tem should vanish by itself in ode to have g αβ,4 = 0. Then, to make M a ǵ γδ,a zeo fo abitay cases, M a should be zeo. Theefoe, we conclude M a must vanish fo a compatible fame. Next we show that N a also should vanish in ode to have a pope inteaction between paticles in the space-time obseved by a natual obseve. Fo a given N a, we can find an object fo which the motion of evey potion of the coesponding paticle-thead satisfies dx 4 = N a dx a 0 in a natual fame. Then let us assume that thee ae two identical objects satisfying this equation. We can put all othe objects infinitely fa away fom these two objects in the natual fame and this allows us to deal with only the two objects in the given fame too. While these two objects ae chaged identically in the natual fame they ae neutal in the given fame. If the given obseve constucts the metic of the 5D manifold, thee is no 11

13 Class. Quantum Gav chaged object in its fame within finite distance so Áa = 0 in any finite egion including the two objects. So the two objects will not have 4-acceleation in the given fame. Now if thei motion is obseved by the natual obseve, thee might be a 4-acceleation which aises fom the coodinate tansfomation including N a. Howeve, this cannot give a coect 4-acceleation in the natual fame since this 4-acceleation now depends only on the details of N a which has nothing to do with the physical condition such as the distance between the two objects. Thus we conclude N a also should vanish fo a compatible fame. To summaize, we cannot constuct a compatible fame though a coodinate tansfomation fom a natual fame if thee exists any non-zeo coss tem between the space-time and the 5th dimension. That is, the covaiance of the 5th dimension is boken. This can be undestood in two aspects. One is M a = 0 which means that evey obseve should agee on the 5th axis and the othe is N a = 0 which indicates that evey obseve shaes a common neutal plane, the 4D hypesuface in the 5D manifold, coinciding with the space-time. The covaiance beaking might be associated with the dagging constuction but the exact mechanism emains to be discussed elsewhee. The covaiance is boken only fo the 5th dimension and it esults in the common 5th axis and common neutal plane. Since any motion along othe than the 5th dimension is paallel to the neutal plane, the covaiance of the othe 4-dimensions is unaffected. Because of this, fo a geneal coodinate tansfomation between the compatible obseves, the 4D pat of the 5D tenso behaves as a 4D tenso: X a b1...bj 1...a i = xγ1 x b1 x a1... xγi x ai x δ1 xbj... Xδ1...δj γ δj x 1...γ i = xc1 x a1 xci x b xbj X d1...dj x ai x d1 x dj c 1...c i Field equations and the enegy-momentum tenso In this section we suggest the field equations fo g ab which ae needed to detemine the defomed manifold completely in conjunction with Ãa discussed in section 4. The 4D pat of the Einstein tenso of the defomed 5D manifold becomes appendix G G ab = G ab g ab F cd F cd F a c F bc 1 à a c F c b + à b c F c a, 4 up to the second powe of the electomagnetic 4-potential, the Riemann tenso, the Ricci tenso, and the Ricci scala of the undefomed space-time the latte thee will be called the cuvatue quantities of the undefomed space-time. Fo the 4D signatue of +, the above equation can be witten as G ab = G ab + µ 0 EM T ab µ 0 à a J b + J a à b, 5 whee the supescipt EM stands fo electomagnetic field. Now we choose the pemeability of vacuum to be µ 0 = 16π. Also, by taking Ĝ ab = 8πˆT ab fo the field equations of any manifold i.e. defomed and undefomed manifold, this is emphasized with the hat mak, the field equations of the undefomed manifold become G ab = Ḡ ab = 8πT M ab whee the supescipt M stands fo matte. Finally, the field equations fo the defomed manifold cystallize into [ ] G ab = 8π T M ab + T EM ab à a J b + J a à b. 6 1

14 Class. Quantum Gav Theefoe, the total enegy-momentum tenso in the defomed manifold up to the second powe of the electomagnetic 4-potential and the cuvatue quantities of the undefomed space-time is T ab = T M ab + T EM ab à a J b + J a à b. 7 This enegy-momentum tenso includes the enegy-momenta of matte, the electomagnetic field, and additional tems involving the electic cuent. The latte show that thee is an inteaction between the electomagnetic potential and the electic cuent. Befoe seeing the implications of the field equations, we biefly discuss the signatue of the model. If the 4D signatue was taken to be ++ +, the sign of the electomagnetic enegymomentum tenso and inhomogeneous Maxwell s equations should be changed. Then, the esultant total enegy-momentum tenso would not make sense since then it involves a diffeence between the enegy-momentum of matte and that of the electomagnetic field. Theefoe, + signatue is coect fo the 4D metic. This diffeence between two signatues oiginates fom the defomation of the 5D manifold caused by the electomagnetic 4-potential. To see the effect of the signatue in the 5D metic tenso, assume that the undefomed space-time is flat and the defomation is small enough so that index aising and loweing of the electomagnetic 4-potential can be done with a flat space-time metic. Ãa = Φ, A is common fo both signatue, so Ãa = Φ, A fo + and Ãa = Φ, A fo ++ + the tilde fo Φ and A is omitted hee. Then, the 5D defomed metic in each signatue would be 1 +Φ Φ A Φ 1 +Φ Φ A Φ g αβ = Φ A I + A A A and g αβ = Φ A I + A A A, Φ A 1 Φ A 1 8 espectively. They diffe not only in the signs but also in the absolute values of the 4D diagonal components. This gives the opposite sign fo the electomagnetic enegy-momentum in the total enegy-momentum tenso. At a glance, time-like 5th dimension looks contay to the discussion in [8] that the 5th dimension should be space-like. Howeve, this is not a contadiction but a esult of diffeent embedding. In Kaluza Klein theoy, the space-time metic is g ab while the space-time metic in the pesent model is g ab, thus, the cuvatue quantities ae calculated with diffeent metics. Because of this, while Kaluza Klein theoy obseves that G ab yields exact Einstein Maxwell equations, the pesent model makes G ab to yield field equations containing Einstein Maxwell equations. An inteesting point in equation 5 is that the electomagnetic enegy-momentum tenso should be tansposed to the opposite side if one wants to get Einstein Maxwell equations of G ab by imposing the vacuum condition fo G αβ. This implies diffeent appopiate signs of electomagnetic enegy-momentum tenso in field equations fo two cases, and hence equies diffeent signatues of the 5th dimension. Now we go back to equation 6. Thee ae fou points woth mentioning in the suggested field equations. Fist, the field equations elate a 4D pat of a 5D tenso with a 4D tenso. This does not hold fo geneal coodinate tansfomations in 5-dimensions but the field equations ae justified by the covaiance beaking discussed in section 6. Second, the field equations ae 4D equations and its matte side ight-hand-side of equation 6 is composed of the 4D enegy-momentum tenso, not a 5D one. This is a consequence of the notion of paticle-thead as discussed in section 3. Thid, the field equations have new inteaction tems compaed to Einstein Maxwell equations. Fouth, the suggested field equations become identical to Einstein s field equations in the electomagnetic field-fee case. 13

15 Class. Quantum Gav Unit system Befoe seeing the expeimental implications of the model, we check ou unit. In ou system we take G = c = µ 0 /16π =16πɛ 0 = 1. So, time, mass, and chage ae all expessed in the length dimension. The convesion factos fo mass and chage become 1 kg = Ǧ č m m, ˇµ 0 Ǧ 1C= 16πč m m, 9a 9b whee the checked symbols efe to the numeical pats of the constants in the MKSC units. Fo electic potential, 16πǦ 1V= ˇµ 0 č , 30 which is dimensionless. The Planck mass and the Planck chage become Ǧȟ m p = πč 3 m m, Ǧȟ q p = 8πč 3 m m. The Planck mass is twice the Planck chage. 31a 31b 9. Expeimental implications The effect of defomed space-time metic and the coection tems of the 4-acceleation ae expeimental obsevables fo the test of the pesent model. Fo the explicit calculation of these two, the 4-potential should be known. As discussed in section 4, the etaded electomagnetic 4-potential of the given electomagnetic souce distibution in the undefomed space-time can be used as the zeoth ode of the petubation method unde the assumption of that the 4-potential is small. This assumption is easonable fo the most cases since the convesion facto of the electic potential is ode of 10 7 as shown in section 8. The following discussion will use paticle pictue in the space-time fo the intuitive appeal. Fo the simplicity, we take the electomagnetic souce to be a spheically symmetic chage Q with mass M such that g ab can be consideed as a Schwazschild metic. We assume that Q/ and M/ ae both small paametes in the egion of inteest, so that the 4-potential becomes à a = 4Q, 0, 0, 0 = à a, 3 to the fist ode of those paametes. We poceed with this 4-potential fo the following discussion. Fom now on, the tilde fo the defomed space-time will be omitted in this section. 14

16 Class. Quantum Gav Metic of the defomed space-time We compae the metic of the defomed space-time against the Reissne Nodstöm metic which is the metic of the space-time accoding to Einstein Maxwell equations. Adopting the unit system of the pesent pape, the Reissne Nodstöm metic is given by ds = 1 M + 4Q dt 1 M 1 + 4Q d dθ sin θdφ, 33 while the metic of the defomed space-time is ds = 1 M + 16Q dt 1 M 1 d dθ sin θdφ. 34 The effect of the chage becomes fou times bigge in g 00 and disappeas in g 11. This diffeence will give a chance to check which metic descibes the phenomena bette by measuing the bending of light ay passing nea a chaged object. The bending of light ay in the Reissne Nodstöm metic is given at [1] by using the method in [13]. The deflection angle α is α = 4M 0 + 4M 0 15π πQ 35 0 up to the second powe of 1/ 0, whee 0 denotes the closest appoach distance of the light ay to the object 4. Hee, we adopt the unit system of the pesent pape. By following the same pocedue in [1, 13], the deflection angle fo the metic of the defomed space-time becomes α = 4M 0 + 4M 0 15π πQ The effect of chage Q in the deflection angle is diffeent in two cases by facto 8/3. The deivation of deflection angle is given in appendix H. The equation between the impact paamete b and the closest appoach distance 0 is given at [13] fo a geneal metic. In the case of the Reissne Nodstöm metic, it is b = 0 1 M 1/ + 4Q, and in the case of the defomed space-time metic, it is b = 0 1 M 1/ + 16Q If the measuable paamete is not 0 but b, the above two equations can be used to obtain 0 fo equations 35 and Coection tems in the 4-acceleation To calculate the explicit components of equation 18, we conside a adial acceleation duing a adial motion of a test paticle with chage q and mass m. The 4-velocity of the paticle is 4 The autho of [1] coected the wong tem in the equation 1 in [1] though a pivate communication. 15

17 Class. Quantum Gav U a = γ, γv, 0, 0, whee γ = 1 M + 16Q 1 1 M v 1/. 39 Now, fo the electomagnetic field tenso, the only non-zeo components of F a b ae F 0 1 = 4Q = F so that the Loentz foce tem of the 4-acceleation becomes L a = q 4Q γv, γ, 0, m Meanwhile, to the leading ode, which is the lowest ode in Q, D a = A a A b dx b b A c dxb dτ dτ dx a dτ = 4Q γ v, γ v, 0, 0, 4 dx c dτ = 1A 0 dx0 dx 1 dτ dτ = 4Q γ v. 43 Combining all these tems accoding to equation 18, the 4-acceleation becomes a a = q 4Q q 64Q 3 γv, γ, 0, 0 m m 4 γ v γv, γ, 0, Q 3 γ 3 v γv, γ, 0, The second and the thid tems in the ight-hand-side might give a chance to test the model expeimentally. 10. Conclusion and futhe eseach In the pesent pape, the 5th dimension is intoduced to explain the electomagnetism in tems of the geometic stuctue. The electomagnetic field-fee 5D manifold is constucted by dagging the space-time. Then, electomagnetic 4-potential is intepeted as a sheaing-like defomation facto of the 5D manifold and we obtain the esultant metic of the defomed 5D manifold which has the simila fom of Kaluza Klein metic with φ = 1. But the embedment of the space-time is diffeent and the metic employs a diffeent electomagnetic potential. The notion of the paticle-thead natually aises fom the constuction of the 5D manifold. A paticle existing in the space-time will be dagged along the diection of the 5th axis as the 5D manifold is constucted fom the dagging of the space-time. The paticle-thead automatically guaantees the independence of the enegy-momentum and the metic with espect to the 5th dimension. In addition to that, the paticle-theads make it natual to use the 4D enegymomentum tenso along with the 4-cuent density fo the desciption of 5D matte. By adopting the paticle-thead, the electic chage of a paticle is no longe a pe-given popety but a quantity descibing the motion of a paticle-thead, that is to say the mass flow ate of a paticle-thead along its extension. Futhemoe, the mass flow to the 5th dimension esults in the sheaing of the 5D manifold to the 5th dimension. It is assumed that evey potion of a paticle-thead follows its 5D geodesic. Then a paticle in the space-time is the coss section of the coesponding paticle-thead along the spacetime. A 4D tajectoy of a paticle in the space-time is the tace of the coss section of the paticle-thead on the space-time which makes the 4D tajectoy become the pojection of a 5D 16

18 Class. Quantum Gav geodesic onto the space-time. Afte the pojection, its deviation fom the space-time geodesic depends on the closeness between the diection of the 5D geodesic and the space-time. With this deviation, the pojection in the defomed manifold esults in the tajectoy of a paticle with the Loentz foce exeting on it. In addition to the Loentz foce tem, thee ae additional small coection tems in the pesent model. The coection tems would be much smalle than the Loentz foce tem unless the potential becomes extemely high. By examining the compatibility of efeence fames it is concluded that the covaiance of the 5th dimension is boken and thee exists a common neutal plane. Fom the covaiance beaking, thee should be estictions on the coodinate tansfomations in the 5-dimensions. These estictions make the 4D pat of a 5D tenso behaves as a 4D tenso. Since any elative motions in the space-time ae paallel to the neutal plane, the covaiance of the space-time still holds. The covaiance beaking of the 5th dimension and the existence of the common neutal plane with the common 5th axis seem to oiginate fom the dagging of the space-time. The field equations between the 5D metic and the 5D matte ae split into two equations, as a consequence of the paticle-thead and the covaiance beaking. One is Maxwell s equations in the defomed space-time, and the physically valid solution fo these is the etaded potential in the defomed space-time. The othe field equations that extend Einstein s field equations ae suggested as G ab = 8π T ab. The total enegy-momentum tenso is the sum of enegy-momenta of the matte, the electomagnetic field, and the inteaction between the electomagnetic potential and the electic cuent. The last one is a newly intoduced tem. The 4D metic signatue is fixed to be + to have the appopiate total enegy-momentum tenso. This signatue fixing makes the 5th dimension time-like. Finally, the physical quantities fo expeimental veifications ae suggested. One is the bending angle of a light ay and the othe is the coection tems in the 4-acceleation. The calculations can be easily done by using the weak potential appoximation. The constuction of this model leaves a few topics that equie futhe eseach. Fist, the physical mechanism of dagging and a moe fundamental cause of the covaiance beaking should be discussed. Second, the eason why a specific diectional degee of feedom is selected to be devoted fo the dagging should be studied. One possibility is that the 5th dimension may have been evolved diffeently fom the othe fou dimensions in the ealy univese. Thid, thee might exist a pe-given 5D enegy-momentum even if it is impope unde the constuction of the 5D manifold and the paticle-thead. Quantization of the 5th dimension may allow this by intoducing a finite 5-volume density since then the thickness of the space-time along the 5th dimension becomes non-zeo. Fouth, the possibility of a wave along the paticle-thead is aised. If it exists, its popety should be investigated. Lastly, numeical estimations of the coection tems ae needed to compae with the expeimental values. Acknowledgments TaeHun Kim gives special thanks to Hyunbyuk Kim, the coesponding autho, fo numeous communications and convesations which helped the eseach geatly. HbK also gave a numbe of detailed elaboations fo the pesent pape. THK gives thanks to JinWoo Sung who wee inteested in the pesent topic and willing to have a discussion about it. JWS too gave detailed elaboations even in his militay sevice. Finally, THK thanks fo anonymous acquaintances who gave encouagements. 17