Invisible magnetic monopole and spacetime geometry
aa r X i v : . [ g r- q c ] A p r Emergent monopoles and magnetic charge
Suvikranth Gera ∗ Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
Sandipan Sengupta † Department of Physics and Centre for Theoretical Studies,Indian Institute of Technology Kharagpur, Kharagpur-721302, India
Abstract
We introduce the concept of emergent monopoles, weaved out of space-time geometry in vacuum.These are constructed to be nonsingular classical solutions of gravitational field equations, wherethe metric necessarily exhibits a noninvertible phase at the core. If this is the only possiblerealization of monopoles in nature, they can never be observed in principle. This geometry isshown to have a unique continuation to an Einsteinian phase, namely, the Reissner-Nordstromspacetime. In addition, we provide a new interpretation to the ‘magnetic charge’ in terms of ageometric invariant in gravity theory. A topological quantization law is also presented. ∗ [email protected] † [email protected] . INTRODUCTION Long ago, Dirac had developed the theory of singular magnetic monopoles, thus providinga remarkably elegant yet unconfirmed explanation of the quantization of electric charge [1, 2].Later on, t’hooft and Polyakov showed that monopole fields could emerge naturally in non-abelian gauge theories as well, but now as nonsingular solitonic solutions of finite energy, sizeand a definite magnetic charge [3–6]. Their presence also appears to be fundamental in theformulation of grand unified theories. However, the lack of any experimental encouragementtill now has led to a logical shift towards the ever-growing belief that monopoles might notreally exist in nature, at least in the form we have theorized them till now.In this work, we suggest that monopoles could be nothing but a manifestation of aspecial phase of vacuum gravity, rather than of matter (gauge) fields. This phase necessarilycorresponds to a metric of vanishing determinant [7–9], and has nontrivial geometric andtopological properties. While this degenerate phase defines the regular monopole core, theregion outside is equivalent to the extremal Reissner-Nordstrom spacetime upto a (local)coordinate transformation. The full spacetime is spherically symmetric and smooth. Itis the invertible phase where an observer lives, the charge being manifested through thecurvature of this outer spacetime. The core within the phase boundary cannot be reachedalong any timelike radial trajectory (geodesic). Hence, it can never be detected directly.The magnetic charge of the emergent monopole solution is shown to inherit a topologicalinterpretation in terms of fundamental geometric invariants in gravity theory. We alsopresent a quantization law for this charge, implications of which could be radical.The features discussed above are new, compared to the earlier formulations of monopolesin gauge theory sourced by genuine matter fields [1–4]. Furthermore, the continuation of theReissner-Nordstrom exterior here is not associated with a black hole interior that exhibitsa singularity at the point charge. In a sense, this indicates that curved geometries with anoninvertible tetrad phase in general could emerge as possible regulators of such infinitiescharacteristic of the theory of electrodynamics and gravity.2
I. CONSTRUCTION OF A STATIC GEOMETRY
Let us first define a ‘static’ solution in gravity theory, in the sense of a three-geometry em-bedded in four dimensions. An appropriate setting is to consider a spacetime configurationwhere the metric has a degenerate eigenvalue along the timelike direction, which naturallyprecludes any temporal dynamics. In general, such three-geometries could emerge as solu-tions to the field equations of first-order gravity [7–14], where the invertibility of tetrads isnot assumed or required apriori.Let us introduce a radial coordinate u and the smooth functions F ( u ) , R ( u ) in order todefine the 4-metric: ds = 0 + σF ( u ) du + R ( u )( dθ + sin θdφ ) , (1)where σ = ±
1. This defines the spacetime at u ≤ u ; θ, φ being the angular coordinates on atwo-sphere. Clearly, the inverse tetrad (or metric) fields do not exist. The metric functionsand their n -th derivatives with respect to u satisfy the following boundary conditions: F ( u ) → , F ( n ) ( u ) → as u → u ,R ( u ) → Q, R ( n ) ( u ) → as u → u . (2)The function F ( u ) is to be solved through the field equations associated with the zerodeterminant phase of first order gravity.While the metric above implies the following nonvanishing components of tetrad fields :ˆ e u = √ σF ( u ) , ˆ e θ = R ( u ) , ˆ e φ = R ( u ) sin θ, (3)we define the connection fields to be ( i ≡ (1 , , ω i = λ ˆ e i , ˆ ω = − R ′ ( u ) √ σF ( u ) dθ, ˆ ω = √ σF ( u ) α ( u ) du − cos θdφ, ˆ ω = R ′ ( u ) √ σF ( u ) sin θdφ (4)where λ = const. and the prime denotes a derivative with respect to the radial coordinate u .The connection involves a new field α = α ( u ), reflecting the presence of torsion ˆ T I ≡ D ˆ e I :ˆ T = 0 = ˆ T , ˆ T = −√ σF Rα sin θdφ ∧ du, ˆ T = −√ σF Rαdu ∧ dθ. (5) The internal metric is [ − , , , SO (3 ,
1) connection, the field strength are evaluated to be:ˆ R i = λD ˆ e i ˆ R = (cid:20) − (cid:18) R ′ √ σF (cid:19) ′ + √ σλ F R (cid:21) du ∧ dθ + αR ′ sin θdφ ∧ du ˆ R = " λ R − (cid:18) R ′ √ σF (cid:19) sin θdθ ∧ dφ ˆ R = − αR ′ du ∧ dθ + (cid:20) − (cid:18) R ′ √ σF (cid:19) ′ + √ σλ F R (cid:21) sin θdφ ∧ du (6)This field configuration (ˆ e Iµ , ˆ ω IJµ ) must satisfy the equations of motion of first order gravity[7], given by: ˆ e [ I ∧ D ˆ e J ] = 0 , ˆ e [ I ∧ ˆ R JK ] = 0 (7)where the covariant derivative D is defined with respect to the connection ˆ ω µIJ . Whereasthe first set of equations are satisfied identically, the second implies:3 λ R + " − (cid:18) R ′ √ σF (cid:19) − R √ σF (cid:18) R ′ √ σF (cid:19) ′ = 0 (8)This equation has the following solution for the metric function F ( u ) in terms of the radiusof the two-sphere: √ σF = R ′ q − QR + λ R (9)The denominator above has a single real root, which we denote by R = R ∗ ( Q, λ ) (for some u = u ∗ >
0) and is positive. This defines the inner boundary of the metric. This allows usto write the solution for the three-geometry at u ∗ ≤ u < u as: ds = 0 + R ′ du − QR + λ R + R (cid:2) dθ + sin θdφ (cid:3) (10)The interpretation of the two parameters ( Q, λ ) would become clear from the subsequentanalysis.
III. A MONOPOLE GEOMETRY
Since the null timelike direction allows no physical evolution, it is possible to treat the in-vertible part of the four-metric (10) effectively as a spatial three-geometry. This corresponds4o the following set of (emergent) triad fields E ia ≡ ˆ e ia ( a ≡ ( u, θ, φ ) , i ≡ (1 , , E = √ σF ( u ) du, E = R ( u ) , E = R ( u ) sin θ. As it is, this set is invertible ( E ai E ib = δ ab , E ai E ja = δ ji ) and completely determines the torsion-less connection components ¯ ω ija ( E ). However, the connection fields that characterize thisthree geometry are just the spatial part of the fields given in eq.(4), and have nonvanishingtorsion: W ija ≡ ˆ ω ija = ¯ ω ija ( E ) + K ija The contortion fields are explicitly displayed below: K = 0 , K = √ σF αdu, K = 0 . (11)Note that this lower dimensional contortion tensor may in general be rewritten in terms ofa symmetric 3 × N ij as K ija = ǫ ijk e la N kl , where: N ij = α ( u ) 0 00 0 00 0 0 (12)This tensor is manifestly invariant under the group of rotations about the first axis in theinternal space. This is the residual symmetry, represented by the U (1) subgroup of theoriginal symmetry group ( SO (3)).The fields ( E, W ) given above completely characterize the effective three-dimensionalgeometry. It is tedious but straightforward to verify that the associated three-curvaturescalars, which could be essentially built out of the tensors R ijab ( W ), are all regular at u ∗ ≤ u ≤ u . Hence this three-geometry is free of curvature singularity. Emergent magnetic field:
Einsteinian gravity in vacuum, associated with invertible tetrads, corresponds to vanish-ing contortion. From this perspective, the nontrivial contortion K ija has a natural inter-pretation as emergent matter in the effective three dimensional theory. To this end, let usintroduce the following definition of the emergent SO (3) gauge field-strength: F ibc ≡ e ai K abc = −F icb (13)5here, K abc = K ija ˆ e bi ˆ e cj . The U (1) field-strength is obtained from its projection alongthe axis of symmetry, given by the unit normal to the two-sphere: u a = ( √ σF , , F bc = αR δ θ [ b δ φc ] sin θ (14)The necessary condition for this to satisfy the Bianchi identity is given by α ( u ) R ( u ) = const .However, the associated contortion field K uθφ = αR R ′ √ − QR + λ R sin θ then happens to divergeat u = u ∗ . Hence, the demand that the physical fields be regular everywhere necessarilyimplies that the emergent field-strength must violate the Bianchi identity.The spatial dependence of the contortion field α ( u ), which has no temporal dynamics, isdetermined solely by its boundary behaviour. This in turn is dictated by the requirement ofregularity at the inner boundary ( u = u ∗ ) and continuity at the outer one ( u = u ), leadingto: α ( u ) → u → u ∗ , α ( u ) → QR ( u ) as u → u Following this, the most general ansatz for a nonsingular solution is given by: α ( u ) R ( u ) = Qχ ( u ) , (15)where Q is a constant and χ ( u ) is any regular function satisfying the properties below: χ ( u ) → u → u ∗ , χ ( u ) → u → u . The only nonvanishing component of the field strength now becomes: F θφ = Qχ ( u ) sin θ. (16)Remarkably, it approaches a monopole field ( F θφ → Q sin θ ) at the outer boundary, andis nonsingular everywhere all the way upto the inner boundary. The associated magneticcharge of this solution is obtained from the integral of the current j t ≡ ǫ tabc ∂ a F bc = ∂ u F θφ :14 π Z d x j t = 14 π Z S dθdφ F θφ = Q, (17)where we have used the fact the 2-integral in the second equality gets a contribution only fromthe outer S boundary ( u → u ). As a consequence, this configuration has a straightforwardinterpretation as an emergent monopole. Its core has a finite size ∼ d ( λ, Q ) = R ( u ) − R ( u ∗ ),determined by the scale parameter λ and the charge Q .6his completes the explicit demonstration as to how static nonsingular monopoles couldbe seen as a reflection of novel effects produced by a noninvertible vacuum phase of gravitytheory in general. Next, we explore what could be the manifestation of the associated(‘magnetic’) charge to an observer necessarily living in the invertible metric phase in fourdimensions. IV. CONTINUATION TO AN EINSTEINIAN PHASE
To be physically relevant, the regular emergent monopole solution must be connectedsmoothly to a different solution corresponding to the invertible metric phase. From aninspection of the field-strength components (6), we find the appropriate spacetime to be the(magnetic) Reissner-Nordstrom solution of Einstein-Maxwell theory upto a local coordinatetransformation, as demonstrated below.Let us introduce an invertible metric ( g µν = 0), corresponding to the spacetime geometryoutside the degenerate core ( u > u ): ds = − (cid:20) − ¯ QR ( u ) (cid:21) dt + R ′ ( u ) du h − ¯ QR ( u ) i + R ( u )[ dθ + sin θdφ ] . (18)Here R ( u ) is the same function introduced earlier in (2) and it satisfies the following conditionat the asymptotic boundary where the metric becomes flat: R ( u ) → ∞ , R ′ ( u ) = 1 as u → ∞ . Under a local reparametrization of the radial coordinate u → R ( u ), this becomes equivalentto the extremal Reissner-Nordstrom metric at u > u . The parameter ‘ ¯ Q ’ is the (magnetic)charge of this solution.The tetrad fields associated with this metric (assuming the internal metric to be η IJ =[ − , , , ω IJµ , which in turn7ead to the following field-strength components: R = QR (cid:18) − QR (cid:19) R ′ dt ∧ du, R = − ¯ QR (cid:18) − ¯ QR (cid:19) dt ∧ dθ,R = − QR (cid:18) − ¯ QR (cid:19) sin θdt ∧ dφ ; R = − ¯ QR R ′ du ∧ dθ, R = ¯ QR (cid:18) − ¯ QR (cid:19) sin θdθ ∧ dφ,R = − ¯ QR R ′ sin θdφ ∧ du. (19)The gauge-covariant field components (tetrad, torsion and field-strength) must be smoothacross the phase boundary u = u . This implies that the charge of the Reissner-Nordstromspacetime is the same as that of the emergent monopole: ¯ Q = Q . As u → u at the invertiblephase ( R ( u ) > Q ), the limits which the basic fields approach to are displayed below: e t → , e u → , e θ → Q, e φ → Q sin θ ; T Iµν → R IJµν → δ θ [ µ δ φν ] δ [ I δ J ]3 sin θ. (20)Note that these are exactly the same limiting values that are exhibited by their counterpartsat the noninvertible phase (ˆ e Iµ , ˆ T Iµν and ˆ R IJµν , respectively), as is necessary.In the matter sector, the only non vanishing component of the Maxwell field-strengthreads: F θφ = Q sin θ (21)Comparing with eq.(16), this is precisely the value of the emergent field-strength F θφ at thephase boundary. To conclude, these fields are also smooth everywhere.The exterior geometry here, along with the monopole core, completely defines the fullspacetime spanning [ u ∗ , ∞ ). Since the timelike direction is frozen within the region u ≤ u ,there is no way this core could be detected by an observer outside. This is so because thereexists no (analytic) radial geodesic through the phase boundary. However, what the observercan still perceive is the curvature of the outer spacetime, which reflects the presence of a(emergent) monopole core somewhere else. 8 . A TOPOLOGICAL ORIGIN OF EMERGENT MONOPOLE CHARGE Here, we provide a novel topological interpretation of the magnetic monopole charge interms of a geometric invariant that could be constructed at the zero-determinant phase.Let us consider the Chern-Simons current associated with the SO (3 ,
1) connection ˆ ω IJa ≡ (ˆ ω ija , ˆ ω ia = λ ˆ e ia ):12 ǫ tabc ˆ ω IJa (cid:20) ∂ b ˆ ω cIJ + 23 ˆ ω IKb ˆ ω JcK (cid:21) = 12 ǫ tabc ˆ ω ija (cid:20) ∂ b ˆ ω cij + 23 ˆ ω ikb ˆ ω jck (cid:21) − λ ǫ tabc ˆ e ia D b (ˆ ω )ˆ e ci (22)Thus, we can construct a geometric invariant by taking the difference of the Chern-Simonscurrents corresponding to the SO (3 ,
1) and SO (3) connections ˆ ω IJa and ˆ ω ija respectively,and integrating it over the compact boundary hypersurface(s). These boundaries here aregiven by the two spheres ( S B ), as u → u ∗ and u → u , respectively. Explicitly, this geometricinvariant may be viewed as the lower dimensional counterpart of the Nieh-Yan index [15]: N = − λ π Z S B ds a n a ǫ bcd ˆ e ib D c (ˆ ω )ˆ e di , (23)where n a ≡ ( √ ˆ g uu , ,
0) is the unit normal on the two-sphere.Next, let us evaluate the Nieh-Yan charge for the emergent monopole geometry con-structed above. Using the expression for contortion and eqs (15) and (16), we obtain: N = − λ π Z S B ds a n a ǫ bcd ˆ e ib D c (ˆ ω )ˆ e di = λ π Z S B dθdφ √ σF K uθφ = 2 Qλ (24)As remarkable as it seems, the magnetic charge of the emergent monopole has its origin ina fundamental geometric invariant of gravity theory. The factor of 2 reflects the fact thatthere are two independent fields on S which contribute to the integral . VI. WINDING NUMBER AND A CLASSICAL QUANTIZATION LAW
Let us introduce a unit vector n i ( i = 2 , , n i ( θ, φ ) ≡ (sin θ cos φ, sin θ sin φ, cos θ ) (25) Strictly, the integral at the outer boundary has a regularization implicit in it. Since the differential du failsto be analytic at the phase boundary u = u , the integral should be evaluated an infinitesimal distance ǫ away from (inside) the boundary, before taking the regulator to zero. n i n i = 1, this defines a map from the S in the coordinate space to the S in theinternal space. It is possible to define a different set of vierbein fields, which correspond tothe emergent three-metric given by (11):¯ E = √ σF du, ¯ E i = H∂ a n i dx a [ a ≡ θ, φ ] . (26)The winding number of the map S → S , as encoded by these fields, is given by: N w = 18 π Z S B d x ǫ ijk ǫ ab n i ∂ a n j ∂ b n k = 1 , (27)where the integral is evaluated at the phase boundary.We shall now rewrite the topological invariant (24) in terms of the dimensionless numbers n = Nλ and q = λQ : nq = 2 N w . (28)In other words, the torsional flux through the boundary two sphere is quantized in units ofthe magnetic charge. This reflects an important feature of the emergent monopole geometryconstructed here. VII. CONCLUSIONS
The seminal works of Dirac and t’hooft-Polyakov on gauge theory monopoles have lefta long trail. However, from a modern perspective, the subsequent unobservability of thesematter field configurations opens up other intriguing possibilities. A critical question seemsto be, whether there could be a radically different manifestation of monopoles on one hand,and a natural resolution to the issue of their apparent elusiveness on the other. We haveworked towards a potential answer here, by presenting evidence for the possibility thatmagnetic monopoles could exist purely as emergent objects in nature. These are regularsolutions of the zero-determinant phase of first order gravity in vacuum. Its core has a finitesize, defined by a scale λ and the charge Q.We have also demonstrated what could be the manifestation of such a monopole field toan observer living in the invertible metric phase of the spacetime. The continuity conditionsat the phase boundary select a unique geometry which this core could be smoothly connectedto. This turns out to be the magnetic Reissner-Nordstrom solution of Einstein theory. Since10here are no analytic radial geodesic from this outer region to the noninvertible phase,the monopole core as it is remains unaccessible to any observer. However, its long rangeCoulombic field is still perceptible through the curvature of the outer spacetime.Note that the point charge singularity typical of a Reissner-Nordstrom black hole interioris absent here. The fact that the noninvertible metric phase leads to a regular charged coreis an important insight. This could have generic applicability in regularizing point singular-ities (poles) and also in providing a natural cut-off through the length-scale of connection(analogue of λ here) in a quantum theory of gravity.A topological interpretation of the emergent monopole charge is provided, as we showthat it is equal to a lower dimensional counterpart of the Nieh-Yan topological number at thenoninvertible phase. In addition, we obtain a quantization law involving the dimensionlessNieh-Yan index, magnetic charge and the winding number of the monopole geometry. Itshould be emphasized that this is not equivalent to Dirac’s quantization law as applicable togauge theoretic monopole, since there is no presence of the electric charge anywhere. Thisnovel connection between the magnetic charge and a purely geometrical invariant couldhave deep implications. One could explore if this leads to a quantization of geometry ingravity theory from a purely classical reasoning. It is also worth noting that our formulationprovides a natural connect between the emergent electromagnetism in curved spacetime andviscoelastic electromagnetism in the condensed matter context [16]. Systems such as thesecould be interesting testbeds for some of the ideas presented here. ACKNOWLEDGMENTS
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