SSpacetime: Arena or Reality?
H. I. Arcos and J. G. Pereira Universidad Tecnol´ogica de Pereira, A.A. 97, La Julita, Pereira, Colombia [email protected] Instituto de F´ısica Te´orica, Universidade Estadual Paulista, Rua Pamplona 145,01405-900, S˜ao Paulo SP, Brazil [email protected]
The concept of fundamental particle has been quite elusive along the his-tory of physics. The term fundamental is commonly used as a synonymous ofstructureless particles. However, this assumption is clearly contradictory. Forexample, it is impossible to explain spin without assuming a structure for theparticle. In fact, a point particle is by definition spherically symmetric, a sym-metry violated by the presence of spin. This problem is usually circumventedby saying that spin is a purely quantum property, which cannot be explainedby classical physics. This means to keep it as a mysterious property of nature.If one assumes that a fundamental particle is a point-like object, severalarguments against this idea show up immediately. First, as discussed above, apoint-like object seems to be inconsistent with the existence of spin. Second,if we try to reconcile general relativity with point-particles, which are singularpoints in a pseudo-Riemannian spacetime, unwanted features, like for exampleultraviolet divergences, will appear. A natural alternative would be to assumethat a fundamental particle is a string-like object, a point of view adoptedby string theory [1]. Similarly, one can introduce membranes as fundamen-tal objects, or even extended objects with certain geometries. These models,however, are also plagued by problems. The membrane model has failed togenerate a theory free of negatively-normed states, or tachyons, and theo-ries with extended objects have failed to explain the existence of supportinginternal forces that avoid the collapsing of the model.With the evolution of particle physics and gravitation, the idea that afundamental particle should somehow be connected to spacetime began toemerge. This is the case, for example, of Wheeler’s approach, which was basedon the concept of spacetime foam. At the Planck scale, uncertainty in energyallows for large curvature values. At this energy, spacetime can undergo deeptransformations, which modify the small scale topology of the continuum. Thisis where the “foam” notion becomes important. Small regions of spacetimecan join and/or separate giving rise to non-trivial topological structures. The a r X i v : . [ g r- q c ] O c t H. I. Arcos and J. G. Pereira simplest of these structures is the so called wormhole, a quite peculiar solu-tion to Einstein’s equation. It represents a topological structure that connectsspacetime points separated by an arbitrary spatial distance. An interestingproperty of the wormhole solution is that it can trap an electric field. Since,for an asymptotic observer, a trapped electric field is undistinguishable froma charge distribution, Wheeler introduced the concept of “charge withoutcharge” [2]. However, as Wheeler himself stated, these Planckian wormholescould not be related to any particle model for several reasons: charge is notquantized, they are not stable, their mass/charge ratio is very different fromthat found in known particles, and half-integral spin cannot be defined fora simple wormhole solution. There was the option to interpret a particle asformed by a collective motion of wormholes, in the same way phonons behaveas particles in a crystal lattice. None of these ideas were developed further.The discovery of the Kerr-Newman (KN) solution [3, 4, 5] in the early six-ties opened the door for new attempts to explore spacetime-rooted models forfundamental particles [6, 7, 8, 9]. In particular, using the Hawking and Ellisextended interpretation of the KN solution [10], as well as the Wheeler’s con-cept of “charge without charge”, a new model has been put forward recently[11]. The purpose of this chapter is to present a glimpse on the characteris-tics of this model, as well as to analyze the consequences for the concept ofspacetime. We begin by reviewing, in the next section, the main propertiesand the topological structure of the KN solution.
The stationary axially-symmetric Kerr-Newman (KN) solution of Einstein’sequations was found by performing a complex transformation on the tetradfield for the charged Schwarzschild (Reissner-Nordstr¨om) solution [3, 4, 5].For m ≥ a + q , it represents a black hole with mass m , angular momentumper unit mass a , and charge q (we use units in which ¯ h = c = 1). In the socalled Boyer-Lindquist coordinates r, θ, φ , the KN solution is written as ds = dt − ρ ∆ dr − ( r + a ) sin θ dφ − ρ dθ − Rrρ ( dt − a sin θ dφ ) , (1)where ρ = r + a cos θ, ∆ = r − Rr + a , R = 2 m − q /r. This metric is invariant under the change ( t, a ) → ( − t, − a ). It is also invariantunder ( m, r ) → ( − m, − r ) and q → − q . This black hole is believed to be thefinal stage of a very general stellar collapse, where the star is rotating and itsnet charge is different from zero. pacetime: Arena or Reality? 3 The structure of the KN solution changes deeply for m < a + q . Dueto the absence of a horizon, it does not represent a black hole, but a circularnaked singularity in spacetime. In fact, it represents a singular disk of radius a , whose border is a true singularity in the sense that it cannot be removed byany coordinate transformation. This means that there is a true singularity atthe border. However, the metric singularity at the interior points of the diskcan be removed by introducing a specific interpretation of the KN solution,as described by Hawking and Ellis [10]. In what follows we give a detaileddescription of the topological structure behind such interpretation. The lack of smoothness of the metric components across the enclosed disk canbe remedied by considering the extended spacetime interpretation of Hawkingand Ellis [10]. The basic idea of this extension is to consider that our spacetimeis connected to another one through the interior points of the disk. This ex-tended solution does not necessarily implies that the dimensionality of space-time is greater than four, but rather that the manifold volume is greater thanexpected. In other words, the disk surface (with the upper points considereddifferent from the lower ones) is interpreted as a shared border between ourspacetime, denoted by M , and another similar one, denoted by M’ . According A CB D
M M’
Fig. 1.
To better visualize the intrinsic geometry of the KN manifold, the KNdisk is drawn as if it presented a finite thickness, and consequently there is a spaceseparation between the upper and lower surfaces of the disk. The left-hand siderepresents the upper and lower surfaces of the disk in M , whereas the right-handside represents the upper and lower surfaces of the disk in M’ . to this construction, the KN metric components are no longer singular acrossthe disk, making it possible to smoothly join the two spacetimes, giving rise toa single 4-dimensional spacetime, denoted M . This link can be seen in Fig. 1as solid cylinders going from M to M’ . In this figure, to clearly distinguishthe upper from the lower side, the disk was drawn as if it presented a finite H. I. Arcos and J. G. Pereira thickness. In order to cross the disk, therefore, an electric field line that hitsthe surface A will forcibly emerge from surface D, in M’ . Then, it must gothrough surface C to finally emerge from surface B, in M . This picture givesa clear idea of the topological structure underlying the KN solution.Now, the singular disk is located at θ = π/ r = 0. Therefore, if r isassumed to be positive in M , it will be negative in M’ . Since the KN metricmust be the same on both sides of the solution, the mass m will be negativein M’ . Furthermore, the magnitude of the electric charge q on both sides ofthe solution is, of course, the same. Taking into account that the source of theKN solution is represented by the electromagnetic potential A = − qrρ ( dt − a sin θdφ ) , (2)which is clearly singular along the ring, and since r has different sign ondifferent sides of the solution, we see from this expression that, if the chargeis positive in one side, it must be negative in the other side. As already remarked, the above extended interpretation does not eliminatethe singularity at the rim of the disk. However, there are some arguments thatcan be used to circumvent this problem. First, it is important to observe thatthere is a torus-like region around the singular ring, in which the coordinate φ becomes timelike. Inside this region, defined by r + a + (cid:18) rRρ (cid:19) a sin θ < , (3)there will exist closed timelike curves [12]. In fact, when crossing the sur-face of this region, the signature of the metric changes from ( − , − , − , +) to( − , − , + , +). This reduction in the number of spatial dimensions is a drawbackof the solution.Now, when the values of a , q and m are chosen to be those of the electron,the surface of the torus-like region is separated from the singular ring by adistance of the order of 10 − cm, which coincides roughly with the Plancklength. At this scale, as is well known, topology changes are expected toexist, and consequently changes in the connectedness of spacetime topologyare likely to occur. A solution to this problem is to excise the infinitesimalregion around the singular ring on both the positive and negative r sides, andthen glue back the manifold. A simple drawing of the region to be excisedcan be seen in Fig. 2, where the direction of the gradient of r has been drawnat several points. As an example, note that the point A on the positive r sidemust be glued to the point A on the negative r side. If we glue all points of the This kind of singularity removal has already been explored by Punsly for the caseof the Kerr solution [13].pacetime: Arena or Reality? 5
Singular ring r > 0 A r ! r ! A r < 0 r ! r ! Fig. 2.
Tubular-like regions around the singular ring, which is to be excised. Several ∇ r directions are also depicted, which show how the borders in the positive andnegative r sides can be continuously glued. torus border, we obtain a continuous path for the electric field lines that flowthrough the disk, even for those lines that would hit the disk at the singularring. Furthermore, since the extrinsic curvature does not change sign whencrossing the hypersurface g φφ = 0, the above gluing process does not generatestress-energy [8].An important point of the above structure is that, after removing thetubular region around the singular ring, the surface delimiting both spacetimesturns out to be defined by a reversed topological product between two 2-torus. As is well known, this is nothing, but the Klein bottle [14]. This is acrucial property because, as we are going to see, in order to present a spinorialbehavior, any spacetime topological structure must somehow involve the Kleinbottle. And of course, in order to be used as a model for any fundamentalparticle, a topological structure must necessarily be a spacetime spinorialstructure. We are going now to explore the possibility of using the KN solution as amodel for the electron. To begin with, let us observe that the total internalangular momentum L of the KN solution, on either side of M , can be writtenas L = m a. (4)If we take for a , m and q the experimentally known electron values, andconsidering that, for a spin 1 / L = 1 /
2, it is easy to see that thedisk has a diameter equal to the Compton wavelength λ/ π = 1 /m of theelectron. Consequently, the angular velocity ω of a point in the singular ringturns out to be ω = 2 m, (5)which corresponds to the so called Zitterbewegung frequency [15, 16] for apoint-like electron orbiting a ring of diameter equal to λ e . This means that H. I. Arcos and J. G. Pereira the KN solution has a gyromagnetic ratio g = 2 [4, 12]. Due to this property,several attempts to model the electron by using the KN solution have beenmade. In most of these models, however, the circular singularity was alwayssurrounded by a massive ellipsoidal shell (bubble), so that it was actually un-reachable. In other words, the singularity was considered to be non-physicalin the sense that the presence of the massive bubble would preclude its for-mation.Using the extended interpretation of Hawking and Ellis, a different modelhas been proposed recently [11]. Its main property is that, differently fromolder models, it is represented by an empty KN solution, that is, no surround-ing massive bubble is supposed to exist around the singular ring. Instead,we make use of the excision procedure to circumvent the problems related tothe naked singularity and the non-causal regions. The fundamental propertyof this model is that Wheeler’s idea of “charge without charge” and “masswithout mass” can be extended to spin. As a consequence, it is able to providea topological explanation for the concepts of charge, mass, and spin.Charge can be interpreted as arising from the multi-connectedness of thespatial section of the KN solution. In other words, we can associate the elec-tric charge of the KN solution with the net flux of a topologically trappedelectric field. In fact, remember that, from the point of view of an asymptoticobserver, a trapped electric field is indistinguishable from the presence of acharge distribution. Then, in analogy with the geometry of the wormhole solu-tion, there must exist a continuous path for each electric field line going fromone space to the other. Furthermore, the equality of magnetic moment onboth sides of M implies that the magnetic field lines must also be continuouswhen passing through the disk enclosed by the singularity.Mass can be associated with the degree of non-flatness of the KN solution.It is given by Komar’s integral [18], m = (cid:90) ∂Σ (cid:63)dξ, (6)which holds for any stationary, asymptotically flat spacetime. In this expres-sion, (cid:63) denotes the Hodge dual operator, ξ is the stationary Killing one-form ofthe background metric, and ∂Σ is a spacelike surface of the background met-ric. It should be noticed that the mass m is the total mass of the system, thatis, the mass-energy contributed by the gravitational and the electromagneticfields [19].Finally, spin can be consistently interpreted as an internal rotational mo-tion of the singular ring. Of course, after the excision process, it turns out tobe interpreted as an internal rotation of the infinitesimally-sized Klein bottle.It is important to remark that the KN solution is a singular ring in spacetime,not in the three-dimensional space. In fact, if the singularity were, let us say,in the xy plane, the angular momentum would be just a component of the A similar approach has been used by Burinskii; see [17], and references therein.pacetime: Arena or Reality? 7 orbital angular momentum, for which the gyromagnetic factor is well knownto be g = 1. Since the gyromagnetic factor of the KN solution is g = 2, therotation plane must necessarily involve the time axis. In fact, we know fromNoether’s theorem that conservation of spin angular momentum is related tothe invariance of the system under a rotation in a plane involving the timeaxis. If one tries to compute the size of the KN particle, a remarkable result isobtained. To see it, we write down the spatial metric of the KN solution,which is given by [20] dl = ρ (cid:20) ∆ dr + dθ + ∆ sin θ∆ − a sin θ dφ (cid:21) . (7)If we use this metric to compute the spatial length L of the singular ring, wefind it to be zero: L ≡ (cid:90) π dl = 0 . (8)This result is consistent with previous analysis made by some authors [8, 12],who pointed out that an external observer is unable to “see” the KN solutionas an extended object, but only as a point-like object. We can then say that the“particle” concept is validated in the sense that the non-trivial KN structureis seen, by all observers, as a point-like object. Although the spatial dimensionof the disk is zero, its spacetime dimension is of the order of the Comptonwavelength for the particle, which for the electron is λ = 10 − cm.It is well known that a fundamental particle fulfills the de Broglie rela-tionship λ = 1 p = 1 mv , (9)where p is its total linear momentum. This relationship can be given a the-oretical fundamentation in our model. The wave-lenght λ is associated withthe diameter of the singular ring, and at first glance it seems to be unrelatedto mass. But since the radius a is also the angular momentum per unit mass,which is a particular property of the solution, the mass m and a are linked by ma = 1 / A simple analysis of the structure of the extended KN metric shows thatit is possible to isolate four physically non-equivalent states on each side of M , that is, on M and on M’ . These states can be labeled by the senseof rotation ( a can be positive or negative), and by the sign of the electric H. I. Arcos and J. G. Pereira charge (positive or negative). Each one of these non-equivalent states in M must be joined continuously through the KN disk to another one in M’ ,but with opposite charge. Since we want a continuous joining of the metriccomponents, this matching must take into account the sense of rotation of therings. In Fig. 3, just as in Fig. 1, the tubular joining between M and M’ aredrawn for one specific value of the electric charge, but taking into accountthe different spin directions in each disk, which are drawn as small arrows.The differences among the configurations are the orientation of the spin vectorand the geometry of the tubes. M M' M M'M M' M M'
Fig. 3.
The four possible geometric configurations of KN states for a specific valueof the electric charge. The arrows indicate the sense of the spin vector.
It is important to remark that the model considers both sides of the so-lution, that is, M and M’ , as part of a single spacetime. The use of twospacetimes is just a mathematical necessity to describe the topological struc-ture behind the KN solution. The question then arises on how to interpret thefact that the mass, and consequently the energy, acquires a negative value in M’ , if they are assumed to be positive in M . The same happens with the senseof rotation, or equivalently, with the arrow of time. At this point it is possibleto see the close analogy that exists between the topological structure of theKN solution and the structure of a Dirac spinor. In fact, the same questionson the interpretation of M and M’ could be made on the interpretation of thetwo upper and the two lower components of the Dirac spinor. The answer tothe latter question, as is well known, requires both special relativity and quan- Two signs for the electric charge q in M or M’ are allowed since the KN metricdepends quadratically on q . This is similar to the wormhole solution, which connects two points of the samespacetime.pacetime: Arena or Reality? 9 tum mechanics, and consequently the notion of anti-particles to comply withnegative energies [21]. We can then say that the necessity of two spacetimesto describe a spinorial structure in spacetime is quite similar to the necessityof a four-component spinor to describe a spin-half particle.
The excision process used to eliminate the non-causal region gives rise tohighly non-trivial topological structure. Now, it is a well known result that,in order to exhibit gravitational states with half-integral angular momentum,a 3-manifold must fulfill certain topological conditions. These conditions werestated by Friedman and Sorkin [22], whose results were obtained from a pre-vious work by Hendricks [23] on the obstruction theory in three dimensions.Interesting enough, the KN solution can be shown to satisfy these conditions,which means that it is actually a spacetime spinorial structure [11].An alternative way to verify this result is to analyze the behavior of theKN topological structure under rotations. In general, when rotated by 2 π ,a classical object returns to its initial orientation. However, the topologicalstructure of the KN solution presents a different behavior: it returns to itsinitial position only after a 4 π rotation. This result can be understood fromthe topology of the 2-dimensional surface that is formed in the excision andgluing procedure. This surface, as we have already seen, is just a Klein bottle.A 2 π rotation of the positive r side is equivalent to moving a point on theKlein bottle surface halfway from its initial position. Only after a 4 π rotationit returns to its departure point. This is a well known property of M¨obiusstrip, and consequently of the Klein bottle since the latter is obtained by atopological product of two M¨obius strips. As we have seen, the extended KN solution represents a spacetime spinorialstructure. It can, therefore, be naturally represented in terms of spinor vari-ables of the Lorentz group SL(2, C ). A crucial point towards this possibility isthe fact that the KN solution presents four non-equivalent states, defined bythe sense of rotation and by the sign of the electric charge. Since a Dirac spinoralso has four independent components, it is not difficult to find an algebraicrepresentation for the KN solution. Considering then an asymptotic observerin a Lorentz frame moving with a constant velocity, the evolution of the KNstate vector is found to be governed by the Dirac equation [11]. Taking intoaccount that the KN solution represents a spacetime spinorial structures, wecan say this is a natural and expected result. By using the extended spacetime interpretation of Hawking and Ellis, togetherwith Wheeler’s idea of “charge without charge”, the KN solution was shownto exhibit properties that are quite similar to those presented by an electron.Apart from the eventual importance of this result for particle physics, thereis also deep consequence for the concept of spacetime. At the early times ofgravitation theory, space was considered simply an arena where all phenomenawould take place. In other words, space was just a relation between the existingobjects; without objects, there would be no space. Later on, the existence of anaether was considered, which in a sense would give some reality to the space.Since all experiments to detect such aether gave null results, space continuedfor some time to be this mysterious nothing in which we live in.The advent of special relativity introduced the first important changes inour concept of space. Time lost its absolute character, and became just onemore coordinate. Instead of living in a three-dimensional space, we discoveredthat we actually live in a four-dimensional spacetime. The advent of generalrelativity introduced further and deeper conceptual changes in our notion ofspacetime. We discovered, for example, that spacetime can storage energy.This means essentially that it could not anymore be interpreted as a simplearena because, if it can storage energy, it must have a concrete existence.In addition to simple configurations, like a curved spacetime, general rel-ativity allows the existence of much more complex spacetime structures. Oneexample is the KN solution of Einstein’s equation, which presents a very pe-culiar topological structure. Its main property is to be a spinorial spacetimestructure, which is revealed by the fact that only after a 4 π rotation it returnsto its initial position. The presence of the Klein bottle in the topologicalstructure makes it easier to understand this property.Now, if we consider that the topological structure is able to trap an elec-tric field, an asymptotic observer would see it as if the structure presentedan electric charge. Furthermore, because the curved spacetime associated tothe topological structure has a non-vanishing energy, the same asymptotic ob-server would see it as if the structure presented a mass. When the experimentalvalues for the electron charge and mass are used, the internal angular momen-tum of the KN solution is found to present a gyromagnetic factor g = 2. Inaddition to storage energy, therefore, spacetime can also carry electric chargeand spin angular momentum.Due to the fact that it represents a spacetime spinorial structure, theKN solution can be represented in terms of the spinor variables of the Lorentzgroup SL(2, C ). Its spacetime evolution is then naturally found to be governedby the Dirac equation. The KN structure, therefore, can be interpreted as aspacetime-rooted electron model. Of course, it is not a finished model, andmany points remain to be understood and clarified. For example, it is anopen question whether it is applicable or not to other particles of nature. If,however, it shows to be a viable model, spacetime will acquire a new and more pacetime: Arena or Reality? 11 important status. In fact, it will be not only the arena, but will also provide— through its highly non-trivial Planck-scale topological structures — thebuilding blocks of all existing matter in the universe, including ourselves. Acknowledgments
The authors would like to thank A. Burinskii and T. Nieuwenhuizen for usefulcomments. They would like to thank also FAPESP, CNPq and CAPES forfinancial support.
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