Emergent Abelian Gauge Fields from Noncommutative Gravity

We construct exact solutions to noncommutative gravity following the formulation of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for generating cosmological electromagnetic fields in an...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2010
Main Author:	Stern, A.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2010
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/146347
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Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Emergent Abelian Gauge Fields from Noncommutative Gravity / A. Stern // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 32 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Stern, A.
author_facet	Stern, A.
citation_txt	Emergent Abelian Gauge Fields from Noncommutative Gravity / A. Stern // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 32 назв. — англ.
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We construct exact solutions to noncommutative gravity following the formulation of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for generating cosmological electromagnetic fields in an expanding space-time background, and also leads to multipole-like fields surrounding black holes. Exact solutions to noncommutative Einstein-Maxwell theory can give rise to first order corrections to the metric tensor, as well as to the electromagnetic fields. This leads to first order shifts in the horizons of charged black holes.
first_indexed	2025-12-07T17:22:34Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 019, 15 pages Emergent Abelian Gauge Fields from Noncommutative Gravity? Allen STERN Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Al 35487, USA E-mail: astern@bama.ua.edu Received December 30, 2009, in final form February 14, 2010; Published online February 18, 2010 doi:10.3842/SIGMA.2010.019 Abstract. We construct exact solutions to noncommutative gravity following the formula- tion of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for genera- ting cosmological electromagnetic fields in an expanding space-time background, and also leads to multipole-like fields surrounding black holes. Exact solutions to noncommutative Einstein–Maxwell theory can give rise to first order corrections to the metric tensor, as well as to the electromagnetic fields. This leads to first order shifts in the horizons of charged black holes. Key words: noncommutative gravity; Groenewold–Moyal star; exact solutions 2010 Mathematics Subject Classification: 83D05; 53D55; 81T75 1 Introduction A number of arguments point to a breakdown of the Riemannian description of space-time at the Planck scale. In particular, it was been argued by Doplicher et al. [1] that a new set of uncertainty relations for distance and time measurements are relevant at the Planck energy scale. These uncertainty relations are naturally realized within the context of noncommutative geometry, where space-time coordinates are replaced by noncommuting operators. This indicates that noncommutative space-time geometry may be the appropriate setting for Planck scale gravity. Among other things, it has the potential of resolving the point singularities which plague general relativity, since the standard notion of ‘points’ is absent in such a setting1. Corrections to the metric tensor due to noncommutative geometry have been computed for a number of solutions to general relativity [4, 5, 6]. However, so far, the prospects of observing these corrections are not very promising. This is due to the smallness of the noncommutative scale2, plus the fact that the previously obtained corrections to the metric tensor are quadratic in the noncommutative scale [7, 8]. Moreover, the procedure used to obtain the above corrections has some ambiguity. It does not involve solving the noncommutative analogue of Einstein equations, which are not unique. Rather, one maps known solutions of general relativity to the noncommutative theory using the Seiberg–Witten map [9]. A noncommutative analogue of the metric tensor can be defined in order to then give a physical interpretation of the results, but the definition is not unique. ?This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html 1This issue was addressed in the simpler context of a noncommutative deformation of electrodynamics where it was shown that noncommutative effects tend to screen the Coulomb singularity at leading order in the defor- mation [2, 3]. 2Conservative estimates from Hydrogen spectra put limits on the noncommutative scale at around a few GeV [2, 3]. mailto:astern@bama.ua.edu http://dx.doi.org/10.3842/SIGMA.2010.019 http://www.emis.de/journals/SIGMA/noncommutative.html 2 A. Stern Here we shall examine exact solutions to noncommutative gravity, and show that they can give rise to effects which are linear in the noncommutative scale when additional gauge fields are present in the theory. Such additional fields are in fact necessary if one follows the approach to noncommutative gravity of Chamseddine [10, 11]. The latter is based on the gauge theory formulation of general relativity. Because the relevant gauge group for gravity is non unitary, one needs to enlarge the gauge group in order to pass to the noncommutative theory. This requires adding at least two Abelian gauge fields to the theory. Their coupling to point particles was shown to be identical to that of electromagnetism (although one of them is associated with a noncompact direction of the gauge group) [12]. We shall examine several standard solutions of general relativity, and show that the additional gauge fields pick up first order contributions when going to the noncommutative gravity theory. Starting with the Robertson–Walker metric, this then provides a mechanism for generating cosmological electromagnetic fields. Conversely, if an electromagnetic field is present in a solution to the Einstein–Maxwell equations, it can induce first order corrections in the metric tensor. Different formulations of noncommutative gravity have been proposed. Among them is the approach of Wess and collaborators [13], where the full diffeomorphism group is preserved. Exact solutions to the latter have been recently discussed in [14, 15] which preserve the isome- tries of the corresponding commutative solution. (See also [16].) Actually, exact solutions can be obtained for a wide class of noncommutative gravity theories. For this one only needs to re- quire a) that the commutative solutions are associated with some isometry group I and b) that the noncommutative field equations are deformations of the commutative equations obtained in a canonical fashion. Here the term ‘canonical’ means that pointwise products appearing in the commutative theory get replaced by star products. As discussed below, we shall utilize the Seiberg–Witten map [9] in order to make contact with a theory of space-time. (This differs from [14, 15].) Because the Seiberg–Witten map can be problematic for some noncommutative manifolds, we shall specialize to embeddings of the Moyal–Weyl plane, with its realization in terms of the Groenewold–Moyal star product [17, 18]. It is defined in terms of dimensionfull space-time independent parameters Θµν , setting the noncommutative scale. One can choose Θµν such that the Groenewold–Moyal star product acts trivially (i.e., as a pointwise product) be- tween any two functions associated with the isometry group I. Noncommutative field equations evaluated for such functions are then equivalent to commutative field equations, and so solutions to commutative gravity with isometry group I must also be solutions to the noncommutative theory. The final task is then to give a space-time interpretation to the noncommutative solutions. Unlike in previous treatments, we do not try to define a noncommutative analogue of the metric tensor for this purpose. We instead use the Seiberg–Witten map [9] to map noncommutative solutions onto the space of commutative field configurations, where the standard metric tensor, as well as other familiar quantities of general relativity, can be employed. The Seiberg–Witten map can then potentially generate first order shifts in both the metric tensor and the Abelian gauge fields of the commutative solution. Because Θµν are space-time independent, these shifts will in general break various space-time symmetries associated with the commutative solution. We carry out the procedure for three examples: the flat expanding universe, the Schwarzschild black hole and the Reissner–Nordström black hole. First order Abelian gauge fields emerge in the first two examples, and first order corrections to the metric tensor result in the last example. The organization of this article is as follows. We review the noncommutative GL(2, C) gauge theory of [11] in Section 2, along with its Seiberg–Witten map from the commutative theory in Section 3. A prescription for finding exact solutions to the noncommutative Einstein equations is described in Section 4. We apply the procedure to the case of the flat expanding universe in Section 5 and the Schwarzschild metric in Section 6. We generalize the procedure in Section 7 in order to find exact solutions to the noncommutative Einstein–Maxwell system and apply it Emergent Abelian Gauge Fields from Noncommutative Gravity 3 to the case of Reissner–Nordström black hole in Section 8. Concluding remarks are made in Section 9. Expressions for the noncommutative gauge variations, the curvature and torsion and the Seiberg–Witten map are given for component one forms in the appendix. 2 Noncommutative GL(2, C) gauge theory of gravity The standard gauge theory formalism for gravity [19, 20, 21, 22] is expressed in terms of vier- beins ea µ and spin connections ωab µ = −ωba µ . a, b, . . . = 0, 1, 2, 3 denote Lorentz indices which are raised and lowered with the flat metric tensor ηab = diag(−1, 1, 1, 1), while µ, ν, . . . are space-time indices which are lowered with the space-time metric gµν = ea µeb νηab. (2.1) For convenience we use spinor notation of [23] and introduce the matrix-valued one forms e = ea µγadxµ, ω = 1 2ωab µ σabdxµ, with {γa, γb} = 2ηab1l (1l denotes the unit matrix) and SL(2, C) generators σab = − i 4 [γa, γb]. [ , ] and { , } denote the matrix commutator and anticommutator, respectively. Infinitesimal variations δλ of ω and e induced by local SL(2, C) transformations are given by δλω = dλ + i[ω, λ], δλe = i[e, λ], (2.2) where λ = 1 2λabσab is infinitesimal. Such transformations leave the metric tensor (2.1) invariant. The spin curvature R and torsion T are constructed from ω and e according to R = 1 2Rab µνσabdxµ ∧ dxν = dω + iω ∧ ω, T = T a µνγadxµ ∧ dxν = de + i(ω ∧ e + e ∧ ω), the latter of which is set to zero in the usual metric description of general relativity. R and T satisfy the Bianchi identities dR = i(R ∧ ω − ω ∧R), dT = i(R ∧ e− e ∧R + T ∧ ω − ω ∧ T ). The canonical procedure to go from a commutative to a noncommutative theory is to re- place the pointwise products of the commutative theory by star products, more specifically Groenewold–Moyal star products [17, 18], given by ? = exp { i 2 Θµν←−∂µ −→ ∂ν } , (2.3) where Θµν = −Θνµ are constant matrix elements denoting the noncommutativity parameters and ←− ∂µ and −→ ∂µ are left and right derivatives, respectively, with respect to some space-time coordinates xµ. If we call Â, B̂, . . . the noncommutative analogues of the matrix-valued forms A,B, . . . of the commutative theory, then the matrix-valued star commutator [Â, B̂]? = Â ? B̂ − B̂ ? Â replaces the matrix commutator [A,B] when we go to the noncommutative theory. Gauge theories based on non unitary groups, such as SL(2, C), do not have a straightforward generalization to noncommutative gauge theories. (See [24, 25].) For this note that the SL(2, C) algebra is not realized with the star commutator. This follows because the star commutator of matrix valued forms can be decomposed in terms of commutators, as well as anticommutators, of the algebra generators. The star commutators of forms proportional to σab and γa then generate 4 A. Stern the entire Clifford algebra. Such star commutators will appear upon trying to lift the gauge variations in (2.2) to noncommutative variations. The commutators [ω, λ] and [e, λ] in (2.2) get replaced by [ω̂, λ̂]? = 1 8{ω̂ ab, λ̂cd}? [σab, σcd] + 1 8 [ω̂ab, λ̂cd]? {σab, σcd}, (2.4) [ê, λ̂]? = 1 4{ê a, λ̂bc}? [γa, σbc] + 1 4 [êa, λ̂bc]? {γa, σbc}, (2.5) where ω̂ = 1 2 ω̂ab µ σabdxµ, ê = êa µγadxµ and λ̂ = 1 2 λ̂abσab are the noncommutative analogues of the connection one form ω, the vierbein one form e and the gauge parameters λ, respectively, and { , }? denotes the star anticommutator, {α̂, β̂}? = α̂ ? β̂ + β̂ ? α̂. Using the Groenewold– Moyal star, {α̂, β̂}? is real for any two real-valued functions α̂ and β̂, while [α̂, β̂]? is imaginary. The anticommutator {σab, σcd} appearing in (2.4) is a linear combination of γ5 and the unit matrix 1l. Closure therefore requires that we enlarge the gauge group to GL(2, C) in order to include these two generators. Following [11] one then introduces GL(2, C) connections Â and gauge parameters Λ̂ which can be decomposed as Â = ω̂ + Â(1)1l + iÂ(2)γ5, Λ̂ = λ̂ + α̂(1)1l + iα̂(2)γ5, where Â(S), S = 1, 2, are Abelian connection one forms, Â(S) = Â (S) µ dxµ, and α̂(S) are functions on space-time. The anticommutator {σab, γc} appearing in (2.5) is a linear combination of γ5γc, and so closure requires that one generalizes the vierbein one forms ê to Ê = ê + f̂ , f̂ = f̂a µγ5γadxµ. A consistent set of noncommutative GL(2, C) gauge variations can then be defined according to δΛ̂Â = dΛ̂ + i[Â, Λ̂]?, δΛ̂Ê = i[Ê , Λ̂]?. (2.6) These variations are decomposed in terms of component fields ω̂ab, Â(S), êa and f̂a in (A.1) in the appendix. The noncommutative GL(2, C) curvature F̂ and torsion T̂ are given by F̂ = dÂ+ iÂ ? ∧ Â = 1 2R̂abσab + F̂ (1)1l + iF̂ (2)γ5, T̂ = dÊ + i(Â ? ∧ Ê + Ê ? ∧ Â) = T̂ aγa + Ûaγ5γa, where ? ∧ denotes an exterior product where the usual pointwise product between components of the forms replaced by the Groenewold–Moyal star product. The components R̂ab, F̂ (S), T̂ a and Ûa of the noncommutative curvature and torsion two forms are given in (A.2) in the ap- pendix. The Bianchi identities now read dF̂ = i(F̂ ? ∧ Â − Â ? ∧ F̂), dT̂ = i(F̂ ? ∧ Ê − Ê ? ∧ F̂ + T̂ ? ∧ Â − Â ? ∧ T̂ ). 3 First order Seiberg–Witten map The Seiberg–Witten map [9] can be used to map the noncommutative gravity theory back to a commutative gravity theory, where quantities like the metric tensor can be straightforwardly defined. However, the commutative gravity theory here is not the standard SL(2, C) gauge theory description of gravity, but rather it is a GL(2, C) gauge theory. The latter is defined in terms of the one forms A = Aµdxµ = ω + A(1)1l + iA(2)γ5 and E = Eµdxµ = eaγa + faγ5γa, Emergent Abelian Gauge Fields from Noncommutative Gravity 5 which are the commutative analogues of Â and Ê , respectively. The commutative GL(2, C) gauge variations are given by δΛA = dΛ + i[A,Λ], δΛE = i[E ,Λ], with infinitesimal parameters Λ = 1 2λabσab + α(1)1l + iα(2)γ5. In terms of component fields ωab, A(S), ea and fa, this means δΛωab = dλab + ωacλ b c − ωbcλ a c , δΛA(S) = dα(S), S = 1, 2, δΛea = ebλ a b + 2faα(2), δΛfa = f bλ a b + 2eaα(2). (3.1) Equation (2.1) is not suitable as a metric tensor for this theory since it is not invariant under the full set of GL(2, C) transformations. The following GL(2, C) invariant metric tensor was given in [12]: gµν = 1 4 tr EµEν = ea µeaν − fa µfaν . (3.2) Because this gravity theory possesses two vierbein fields ea and fa, one can construct two sets of torsion two-forms T a = dea + 1 2 [ωab, eb], Ua = dfa + 1 2 [ωab, fb], (3.3) which are the commutative analogues of T̂ a and Ûa. The Seiberg–Witten map relates the noncommutative degrees of freedom in Âµ and Êµ, along with transformation parameters Λ̂, to their commutative counterparts Aµ, Eµ and Λ, and is defined such that gauge transformations in the commutative theory induce gauge trans- formations in the corresponding noncommutative theory. Up to first order in Θµν it can be given by Âµ = Aµ + 1 4Θρσ{Aρ, ∂σAµ + Fσµ}+O ( Θ2 ) , Êµ = Eµ + 1 2Θρσ { Aρ, ∂σEµ + i 2 [Aσ, Eµ] } +O ( Θ2 ) , Λ̂ = Λ + 1 4Θρσ{Aρ, ∂σΛ}+O ( Θ2 ) . (3.4) The resulting first order expressions for the component one forms ω̂ab, Â(S), êa and f̂a are given in (A.3) of the appendix3. 4 Exact solutions to noncommutative gravity Here we do not specify any particular dynamics for noncommutative gravity, but only demand that the field equations are deformations of the standard Einstein equations, with deformation parameters Θµν , and are obtained in the canonical way, i.e., by replacing pointwise products with Groenewold–Moyal star products. Such a procedure is, of course, not unique. One can nevertheless write down exact solutions for the noncommutative theory, as we describe below. Say a solution to (commutative) gravity is associated with some nontrivial isometry group I, which is generated by n Killing vectors Kα, α = 1, . . . , n. Denote by VA, A = 1, . . . , 4− n, the remaining independent vectors normal to Kα. Now define Θµν such that ΘABVA ⊗ VB = 0. (4.1) 3For a related Seiberg–Witten map, see [26]. 6 A. Stern If all Kα vanishes on a set of functions F , then the bivectors ΘαBKα ⊗ VB and ΘαβKα ⊗Kβ vanish on F ⊗ F . Moreover, from (4.1) it follows that Θµν∂µ ⊗ ∂ν vanishes on F ⊗ F . From the definition of the Groenewold–Moyal star (2.3), one gets that the star product of any pair of functions in F is identical to its corresponding pointwise product. So when evaluated on elements of F , any noncommutative field equation, obtained using the canonical procedure, reduces to the corresponding commutative field equation. Therefore solutions to commutative gravity with isometry group I must also be solutions to the noncommutative theory. The former solutions are specified by SL(2, C) vierbein and spin connection one forms, which we denote respectively by ea (0) = ea (0)µdxµ and ωa (0) = ωa (0)µdxµ. Then the corresponding noncommutative solution is given by êa µ = ea (0)µ, ω̂ab µ = ωab (0)µ, Â(S) µ = f̂a µ = 0. (4.2) The above procedure can be adapted for obtaining exact solutions can be applied to a wide class of noncommutative theories, i.e., those obtained in the canonical way. Exact solutions were found in [14, 15] to the noncommutative gravity theory of [13]. They were also used to find exact solutions to noncommutative electrodynamics [3]. In order to give a physical interpretation to the exact noncommutative solution (4.2), one can use (3.4) to map the solution to the commutative theory. The SL(2, C) vierbeins ea (0)µ and spin connections ωab (0)µ pick up no first order corrections from the Seiberg–Witten map. On the other hand, the map does generate nonvanishing first order results for A (S) µ and fa µ : A(1) µ = − 1 16Θρσ[ω(0)ab]ρ ( ∂σωab (0)µ + Rab (0)σµ ) +O ( Θ2 ) , A(2) µ = 1 32Θρσεabcdω ab (0)ρ ( ∂σωcd (0)µ + Rcd (0)σµ ) +O ( Θ2 ) , fa µ = 1 4Θρσεa bcdω bc (0)ρ ( ∂σed (0)µ + 1 2 ωdg (0)σe(0)gµ ) +O ( Θ2 ) , (4.3) where R(0) is the Lorentz curvature for the solution. Since the GL(2, C) invariant metric ten- sor (3.2) is quadratic in fa µ , it receives no first order corrections. This is consistent with the usual result that noncommutative gravity corrections to the space-time metric tensor are second order in the noncommutativity parameter [7, 8]. This, however, would not be the case for solutions of the Einstein–Maxwell equations which are associated with a nonzero Abelian curvature, since then Â(S) is nonzero. Therefore an Abelian gauge field can generate first order corrections to the metric tensor. We show this in Section 7 and give the example of the Reissner–Nordström black hole solution in Section 8. The induced fa µ in (4.3) can lead to a first order GL(2, C) torsion Ua = dfa +O(Θ2), but its physical meaning is not immediately evident. In the next two sections we carry out the above procedure to find exact solutions of the noncommutative theory for the case of the flat expanding universe and the Schwarzschild black hole and we obtain the leading order induced Abelian gauge fields in these two examples. For the former example, any choice for the constants Θµν satisfies (4.1) when expressed in terms of comoving coordinates, while for static solutions, (4.1) is always satisfied for the case of space- time noncommutativity. In addition to (4.1), there is the requirement that the generators of the star product algebra are self-adjoint operators. This is the case for all the examples which follow. 5 Flat expanding universe The invariant measure for a flat expanding universe is ds2 (0) = −dt2 + a(t)2dxidxi, i = 1, 2, 3, (5.1) Emergent Abelian Gauge Fields from Noncommutative Gravity 7 where xi span R3 and define comoving coordinates. Equation (5.1) gives a solution to the Einstein equations with stress-energy tensor of the form TSE µν = diag ( ρ(t), a(t)2p(t), a(t)2p(t), a(t)2p(t) ) , (5.2) where ρ(t) and p(t) are the proper energy density and pressure, respectively, and are related to a(t). Vierbein and spin connection one forms consistent with the zero torsion condition of general relativity can be given by e0 (0) = dt, ei (0) = a(t)dxi, ω0i (0) = ȧ(t)dxi, ωij (0) = 0. Here all spatial directions correspond to Killing vectors Ki = ∂ ∂xi . Since only one independent vector V0 = ∂ ∂t remains, the condition (4.1) is satisfied for any choice of Θµν . Therefore, (4.2) is an exact solution to the noncommutative gravity equations (obtained using the canonical procedure) with stress-energy tensor (5.2), for any choice of Θµν . We can thus choose space- space noncommutativity [xi, xj ]? = Θij , (5.3) or time-space noncommutativity [t, xi]? = Θ0i, (5.4) or both. (5.3) and (5.4) break the three dimensional rotation symmetry. This symmetry breaking will not appear at first order in the metric tensor, but does appear in the electric field at first order. As was remarked previously [15], these commutation relations apply for the comoving coordinates, while the commutators between ‘physical’ spatial coordinates yi involve the scale factor, yi = a(t)xi. As a result, the ‘physical’ noncommutative scale will become extremely small at earlier scales, and one would expect that this prevents the noncommutative effects from becoming too large at earlier times. This could be desirable, since such effects break rotational invariance. Despite this, as we argue below, the fields might become significant during the inflation era. Next we substitute into (4.3). A nonvanishing first order potential A (1) µ only arises for the case of time-space noncommutativity (5.4) A (1) 0 = O ( Θ2 ) , A (1) i = −1 4Θ0iȧ(t)ä(t) +O ( Θ2 ) , leading to a uniform electric field at first order, directed along Θ0i, E (1) i = −1 4Θ0i ( ä(t)2 + ȧ(t)a(3)(t) ) +O ( Θ2 ) . (5.5) A nonvanishing first order potential A (2) µ only arises for the case of space-space noncommuta- tivity (5.3) A (2) 0 = O ( Θ2 ) , A (2) i = 1 8εijkΘjkȧ(t)3 +O ( Θ2 ) , leading to a uniform electric field at first order, in the εijkΘjk direction, E (2) i = 3 8εijkΘjkȧ(t)2ä(t) +O ( Θ2 ) . (5.6) Since the associated magnetic fields vanish, it follows that the standard Maxwell equations in vacuum are not satisfied. Rather, the dynamics for the Abelian gauge fields is governed by 8 A. Stern a deformed Maxwell action, and as a result, magnetic fields are shielded by the noncommutative vacuum. If one applies the results (5.5) and (5.6) to the universe at the current time one gets incredibly tiny electric fields, since they go like powers of the Hubble parameter (not to mention the small noncommutative scale Θ). The fields grow when we evolve back in time. Yet, they remain insignificant even if we go all the way back to the beginning of the radiation dominated era. For the radiation dominated era we can apply a(t) = arm ( t trm )1/2 , (5.7) where arm ∼ 3 × 10−4 and trm ∼ 5 × 104 yr, respectively, are the scale factor and time of the radiation-matter equality. Substituting into (5.5) and (5.6) gives E (1) i = − 1 16 Θ0i a2 rm trmt3 +O ( Θ2 ) , E (2) i = − 3 128 εijkΘjk a3 rm t 3/2 rm t5/2 +O ( Θ2 ) . If we take known upper limits [2, 3] for the time-space and space-space noncommutativity scale of Θ . GeV−2, and set t = trm, we get \|E(1)\| . (10−67 eV)2 and \|E(2)\| . (10−69 eV)2. These scales are nowhere near those claimed needed for the primordial (magnetic) field (10−15–10−25 eV2) to seed an amplification process which can then produce the currently observed intergalactic magnetic fields [27]4. If we assume that (5.7) is valid all the way back to, say t ∼ 10−30 sec (and we take the previous limit for Θ), we get \|E(1)\| . (10−3 eV)2 and \|E(2)\| . (10−16 eV)2. These energy scales are still far below the radiation energy (∼ 106 GeV) at the time t ∼ 10−30 sec, and so the fields are not expected to play a role. On the other hand, the fields \|E(1)\| and \|E(2)\| can be very large during a prior exponential inflationary era. (Note that the fields depend on derivatives of a(t), which are not continuous across the transition from the inflationary era to the radiation era if one assumes that the exponential inflation was suddenly switched off. The fields therefore undergo a discontinuous jump across the transition.) Upon taking a(t) ∝ eHt, both \|E(1)\| and \|E(2)\| go like H4 times the noncommutative scale. Since here H is associated with a large energy scale, these fields could be significant during inflation. 6 Schwarzschild solution The Schwarzschild invariant measure may be written as ds2 (0) = ( −1 + α(r)2 ) dt2 + ( α(r)2 1− α(r)2 x̂ix̂j + δij ) dxidxj , α(r)2 = rs r , where the spatial coordinates xi again span R3. r = √ xixi is the radial coordinate, x̂i = xi/r are unit vectors and rs = 2GM is the Schwarzschild radius. A consistent set of vierbein one forms e(0) is e0 (0) = dt + α(r)dr 1− α(r)2 , ei (0) = α(r)x̂ie 0 (0) + dxi. (6.1) The torsion vanishes with the following assignment for the spin connections ωi0 (0) = d ( α(r)x̂i ) + α(r)α′(r)x̂ie 0 (0), ωij (0) = 0, (6.2) 4The application of noncommutative physics to the primordial magnetic field problem was suggested earlier in [28], but the field there originates from the matter content of the universe, and is not generated from the Robertson–Walker background, as is the case here. Emergent Abelian Gauge Fields from Noncommutative Gravity 9 where the prime denotes differentiation in r. Equations (6.1) and (6.2) are valid both inside and outside the event horizon and lead to zero torsion and zero curvature (for r > 0). As with all static solutions, the Schwarzschild solution has a time-like Killing vector K0 = ∂ ∂t . Now we identify the spatially directed vectors with Vi = ∂ ∂xi . From (4.1), we get an exact solution of the noncommutative gravity equations when all space-space components of Θµν vanish. So here we only consider time-space noncommutativity (5.4). The Groenewold–Moyal star product when acting between t-independent functions reduces to the pointwise product, and so if the noncommutative gravity equations are obtained in the canonical way, they are equivalent to the commutative gravity equations when evaluated for static fields. Equation (4.2) is then an exact solution of the noncommutative field equations with time-space noncommutativity. Now substitute into (4.3) to obtain the induced commutative Abelian gauge potentials A (1) µ and A (2) µ at leading order. Here we find that the former leads to an electrostatic field and the latter leads to a magnetostatic field originating from the black hole. The vector potential A (1) i is a pure gauge at first order and hence there is no associated first order magnetic field B (1) i , while A (1) 0 = −Θ0ir2 s x̂i 8r5 +O ( Θ2 ) , resulting in the electrostatic field E (1) i = Θ0jr2 s 8 δij − 6x̂ix̂j r6 +O ( Θ2 ) . (6.3) On the other hand, A (2) 0 = O(Θ2) and hence there is no associated first order electric field E (2) i , while A (2) i = Θ0kr2 s 8r5 εijkx̂j +O ( Θ2 ) , resulting in the magnetostatic field B (2) i = Θ0jr2 s 4 2δij − 3x̂ix̂j r6 +O ( Θ2 ) . (6.4) The results (6.3) and (6.4) break the rotational symmetry of the Schwarzschild solution and resemble higher multipole fields, with Θ0ir2 s playing a role analogous to that of a multipole moment. As the fields fall off as 1/r6, they are very weak far from the black hole. At the Schwarzschild radius r = rs, they both go like ∼ Θ0i/r4 s . Since Θ0i is constant in this theory, the moments of multiple black holes are aligned, which may lead to a peculiar signature for microscopic black holes. 7 Exact solutions to noncommutative Einstein–Maxwell theory It is straightforward to generalize the procedure of Section 4 to find exact solutions to the com- bined noncommutative Einstein–Maxwell system. As before, we do not specify any particular dynamics for the noncommutative theory, but only demand that the field equations are defor- mations of the standard Einstein–Maxwell equations, obtained in the canonical way. In the examples in Sections 5 and 6, solutions to the noncommutative Einstein equations generated Abelian gauge fields which were first order in Θ. Conversely, here we show that solutions to noncommutative Maxwell equations can generate first order corrections to the metric tensor. For this we shall require, as before, that (4.1) is satisfied, and that the generators of the star product algebra are self-adjoint operators. 10 A. Stern Solutions to the commutative Einstein–Maxwell equations are specified by SL(2, C) vierbein and spin connection one forms, ea (0) and ωab (0), along with an Abelian potential one form A(0) = A(0)µdxµ. Since GL(2, C) gauge theory contains two Abelian gauge fields, we can construct two different exact solutions to noncommutative Einstein–Maxwell theory. Below we first assume that Â(1) satisfies the noncommutative Maxwell equation and hence identify A(0) with Â(1), and then we consider the case for Â(2). 1. Â(1) satisfies the noncommutative Maxwell equation. If (4.1) holds we have the exact noncommutative solution êa µ = ea (0)µ, ω̂ab µ = ωab (0)µ, Â(1) µ = A(0)µ, A(2) µ = f̂a µ = 0. (7.1) Upon applying the Seiberg–Witten map back to the commutative theory, one now can pick up first order corrections to all of the GL(2, C) fields. For the two vierbein fields ea µ and fa µ , one has ea µ = ea (0)µ −ΘρσA(0)ρ ( ∂σea (0)µ + 1 2ωag (0)σe(0)gµ ) +O ( Θ2 ) , fa µ = 1 4Θρσεa bcdω bc (0)ρ ( ∂σed (0)µ + 1 2ωdg (0)σe(0)gµ ) +O ( Θ2 ) . The former can then give rise to first order corrections to the classical metric tensor g(0)µν . After substituting the expression for ea µ into (3.2), one gets the simple result that gµν(x) = g(0)µν(x)−ΘρσA(0)ρ(x)∂σg(0)µν +O ( Θ2 ) = g(0)µν ( x + ΘA(0) ) +O ( Θ2 ) . (7.2) Since gµν transforms under diffeomorphisms as a rank 2 tensor, the first order correction cannot be removed by the coordinate redefinition x → x − ΘA(0). From (4.1), the bivector Θµν∂µ ⊗ ∂ν gets contributions from ΘαBKα ⊗ VB and ΘαβKα ⊗ Kβ. However, only the for- mer produces first order corrections to g(0)µν , and from (7.2), they are of the form δg(0)µν = −ΘαBA(0)αVBg(0)µν . Such correction generally break the space-time symmetry associated with the metric tensor g(0)µν . As before, the Seiberg–Witten map produces first order terms in the Abelian gauge fields. Here one gets some additional contributions to A (1) µ A(1) µ = A(0)µ − 1 2Θρσ ( A(0)ρ ( ∂σA(0)µ + F(0)σµ ) + 1 8 [ω(0)ab]ρ ( ∂σωab (0)µ + Rab (0)σµ )) +O ( Θ2 ) , (7.3) where F(0) is the Abelian curvature for the solution. A (2) µ has the same form as in (4.3). There are now also first order correction to the spin connections ωab µ = ωab (0)µ − 1 2Θρσ ( ωab (0)ρ ( ∂σA(0)µ + F(0)σµ ) + A(0)ρ ( ∂σωab (0)µ + Rab (0)σµ )) +O ( Θ2 ) . (7.4) This will result in nonvanishing torsion two forms (3.3) at first order. 2. Â(2) satisfies the noncommutative Maxwell equation. If (4.1) holds we have the exact noncommutative solution êa µ = ea (0)µ, ω̂a µ = ωa (0)µ, Â(2) µ = A(0)µ, A(1) µ = f̂a µ = 0. (7.5) Upon applying the Seiberg–Witten map back to the commutative theory, one gets ea µ = ea (0)µ − 1 4Θρσεa bcdω bc (0)ρA(0)σed (0)µ +O ( Θ2 ) , fa µ = 1 4Θρσεa bcdω bc (0)ρ ( ∂σed (0)µ + 1 2ωdg (0)σe(0)gµ ) +O ( Θ2 ) , Emergent Abelian Gauge Fields from Noncommutative Gravity 11 which now produces no first order correction to g(0)µν . The corrections to the Abelian gauge potentials and spin connections in this case are given by A(1) µ = −1 2Θρσ ( −A(0)ρ ( ∂σA(0)µ + F(0)σµ ) + 1 8 [ω(0)ab]ρ ( ∂σωab (0)µ + Rab (0)σµ )) +O ( Θ2 ) , A(2) µ = A(0)µ + 1 32Θρσεabcdω ab (0)ρ ( ∂σωcd (0)µ + Rcd (0)σµ ) +O ( Θ2 ) , ωab µ = ωab (0)µ − 1 4εab cdΘ ρσ ( ωcd (0)ρ ( ∂σA(0)µ + F(0)σµ ) + A(0)ρ ( ∂σωcd (0)µ + Rcd (0)σµ )) +O ( Θ2 ) . Of course, a final possibility is that the noncommutative Maxwell equation involves both Â(1) and Â(2), i.e., a linear combination of Â(1) and Â(2) satisfies the noncommutative Maxwell equa- tions. The first order corrections to the commutative solution will then be a linear combination of those for the two cases, and would in general lead to first order corrections to g(0)µν . 8 Reissner–Nordström solution We now use the results of the previous section to obtain first order corrections to the Reissner– Nordström metric tensor. The invariant interval for the Reissner–Nordström solution is ds2 (0) = −∆(r)dt2 + ∆(r)−1dr2 + r2 ( dθ2 + sin2 θdφ2 ) , (8.1) ∆(r) = 1− 2GM r + GQ2 r2 . (8.2) Along with the Coulomb gauge field F(0) = Q r2 dr ∧ dt, (8.1) and (8.2) describe a black hole with charge Q and mass M . The metric tensor gives rise to two horizons at r = r±, r± = G ( M ± √ M2 − Q2 G ) . (8.3) Introducing coordinates xi, i = 1, 2, 3, spanning R3, we can re-write the invariant measure to have metric components g(0)00 = −∆(r), g(0)ij = ( ∆(r)−1 − 1 ) x̂ix̂j + δij , g(0)0i = 0. As this is a static solution, the condition (4.1) is satisfied for the case of time-space noncom- mutativity (5.4). Following the discussion in Section 7, we can construct two exact solutions, (7.1) and (7.5), of the noncommutative Einstein–Maxwell equations. We take the former, since we want to obtain first order corrections to the metric tensor. For simplicity, choose only Θ03 nonzero and take A(0) = −Q r dt. Upon substituting into (7.2) we obtain the new invariant measure ds2 = ds2 (0) + ds2 (1), where the first order correction is ds2 (1) = −Θ03Q r ( ∆′(r) cos θ ( dt2 + ∆(r)−2dr2 ) + 2 ( ∆(r)−1 − 1 ) sin θ drdθ ) , the prime denoting a derivative in r. Thus ds2 = −Σ(r, θ)dt2 + Σ(r, θ)−1dr2 + r2 ( dθ2 + sin2 θdφ2 ) − 2Θ03Q r ( ∆(r)−1 − 1 ) sin θ drdθ +O ( Θ2 ) , 12 A. Stern where Σ(r, θ) = ∆(r) + Θ03Q r ∆′(r) cos θ. Rotation invariance is therefore broken at first order in Θ03. The two horizons (8.3) are shifted to r = r± + δr±, where δr± is Θ03−dependent δr± = −Θ03Q r± cos θ +O ( Θ2 ) . The first order corrections to the metric tensor vanish in the Q → 0 limit, consistent with the results obtained in Section 6 for the Schwarzschild black hole. Additional roots of Σ(r, θ) may also occur, and they appear to be nontrivial. Noncommutative corrections to black hole horizons and their effect on the Hawking temperature have been previously computed in [29, 30, 31, 32]. Here such effects will be first order Θ. Using (7.3), first order corrections will also result in the Abelian gauge fields5, as well as in the spin connections (7.4), and they generalize the results of Section 6. 9 Concluding remarks We have constructed exact solutions to noncommutative gravity and shown that they can gener- ate Abelian gauge fields which are first order in the noncommutative scale. Conversely, Abelian gauge fields, if present in the solution, give rise to first order corrections to the metric tensor. In both cases, the effects are expected to break the space-time symmetries associated with the solution. For the case of black holes, we found that both electrostatic and magnetostatic type fields are generated when Θ0i 6= 0. They fall off as 1/r6 and resemble higher multipole fields, with Θ0ir2 s playing a role analogous to the multipole moment. Since Θ0i is constant in this theory, the moments of multiple black holes are aligned, which may lead to a peculiar signature for the detection of microscopic black holes. We got extremely tiny electric-type fields at the current time for the case of the flat expanding universe. The fields grow when we evolve back in time. If we go back to the time of the radiation- matter equality, their strength is nowhere near that claimed needed for the primordial field to seed an amplification process which can produce the currently observed intergalactic magnetic fields. The energy scales of the fields remain well below the radiation energy even if we go back to the beginning of the radiation era. On the other hand, the electric fields could be significant during an exponential inflationary era. We plan on addressing their effect on inflationary models in the future. One possibility is that they reach the critical strength for the creation of charged particle pairs from the vacuum. The system of [11] which was studied here contains two Abelian gauge fields. We have entertained the notion that one of them may correspond to ordinary electromagnetism. Although from (3.1), the infinitesimal gauge variations of A(1) and A(2) are identical in the commutative theory, A(1) is the natural candidate for the electromagnetic potential since it is associated with the compact direction in GL(2, C). Here note that finite (commutative) GL(2, C) gauge transformations are of the form A → ΩAΩ−1 + iΩdΩ−1, E → ΩEΩ−1, Ω = eiΛ, where Λ = 1 2λabσab + α(1)1l + iα(2)γ5 now denote finite parameters. Concerning A(2), it will couple to the axial vector current when including fermions. The possibility of generalizing the 5They will include contributions found in [3] in the absence of gravity. Emergent Abelian Gauge Fields from Noncommutative Gravity 13 internal Abelian gauge symmetries of this theory to standard model symmetries is of further interest. Other avenues of research involve examining alternatives to (5.3) and (5.4), which are asso- ciated with constant commutation relations for the comoving coordinates. As was remarked in Section 5 and in [15], one would get larger noncommutative effects at earlier times if one can instead implement constant commutation relations for the ‘physical’, rather than the comoving, spatial coordinates, as this would introduce extra factors of a(t)−1. However, even after includ- ing such factors, the fields may still be too small to play a role as a primordial field which seeds an amplification process that can produce the currently observed intergalactic magnetic fields. The electric fields which emerge from this gravity theory play a passive role and only appear as a result of the Seiberg–Witten map to the commutative theory. It is of interest to promote them to dynamical fields. They are then expected to contribute symmetry breaking terms to the stress-energy tensor ((5.2), for the example of the flat expanding universe) and generate a back reaction on the space-time geometry. Abelian gauge fields which are linear in Θ will lead to second order back reaction terms in the metric tensor. A Appendix Here we give expressions for the noncommutative gauge variations, the curvature and torsion and the Seiberg–Witten map in terms of the component one forms ω̂ab, Â(S), êa and f̂a. From (2.6), the noncommutative GL(2, C) gauge variations can be written as δΛ̂ω̂ab = dλ̂ab + 1 2 ( {ω̂ac, λ̂ b c }? − {ω̂bc, λ̂ a c }? ) + i[ω̂ab, α̂(1)]? + i[Â(1), λ̂ab]? + i 2εabcd([ω̂cd, α̂ (2)]? + [Â(2), λ̂cd]?), δΛ̂Â(1) = dα̂(1) + i[Â(1), α̂(1)]? − i[Â(2), α̂(2)]? + i 8 [ω̂ab, λ̂ab]?, δΛ̂Â(2) = dα̂(2) + i[Â(1), α̂(2)]? − i[Â(2), α̂(1)]? − i 16εabcd[ω̂ab, λ̂cd]?, δΛ̂êa = 1 2{ê b, λ̂ a b }? − i 4εabcd[f̂b, λ̂cd]? + i[êa, α̂(1)]? + {f̂a, α̂(2)}?, δΛ̂f̂a = 1 2{f̂ b, λ̂ a b }? − i 4εabcd[êb, λ̂cd]? + i[f̂a, α̂(1)]? + {êa, α̂(2)}?. (A.1) The components R̂ab, F̂ (S), T̂ a and Ûa of the noncommutative curvature and torsion two forms are given by R̂ab = dω̂ab + 1 2 [ω̂ac, ω̂ b c ]? + i{ω̂ab, Â(1)}? + i 2εabcd{ω̂cd, Â (2)}?, F̂ (1) = dÂ(1) + i 8 ω̂ab ? ∧ ω̂ab + iÂ(1) ? ∧ Â(1) − iÂ(2) ? ∧ Â(2), F̂ (2) = dÂ(2) − i 16εabcdω̂ ab ? ∧ ω̂cd + i{Â(1), Â(2)}?, T̂ a = dêa + 1 2 [ω̂ab, êb]? − i 2εabcd{ω̂bc, f̂c}? + i{Â(1), êa}? − [Â(2), f̂a]?, Ûa = df̂a + 1 2 [ω̂ab, f̂b]? − i 2εabcd{ω̂bc, êc}? + i{Â(1), f̂a}? − [Â(2), êa]?. (A.2) Up to first order, the Seiberg–Witten maps for the component one forms ω̂ab, A(S), êa and f̂a are ω̂ab µ = ωab µ + 1 2Θρσ ( ωab ρ ( ∂σA(1) µ + F (1) σµ ) + A(1) ρ ( ∂σωab µ + Rab σµ ) + 1 2εab cdω cd ρ ( ∂σA(2) µ + F (2) σµ ) + 1 2εab cdA (2) ρ ( ∂σωcd µ + Rcd σµ )) , Â(1) µ = A(1) µ + 1 2Θρσ ( A(1) ρ ( ∂σA(1) µ + F (1) σµ ) −A(2) ρ ( ∂σA(2) µ + F (2) σµ ) + 1 8 [ωab]ρ ( ∂σωab µ + Rab σµ )) , Â(2) µ = A(2) µ + 1 2Θρσ ( A(1) ρ ( ∂σA(2) µ + F (2) σµ ) + A(2) ρ ( ∂σA(1) µ + F (1) σµ ) − 1 16εabcdω ab ρ ( ∂σωcd µ + Rcd σµ )) , 14 A. Stern êa µ = ea µ + ΘρσA(1) ρ ( ∂σea µ + 1 2ωag σ egµ −A(2) σ fa µ ) − 1 4Θρσεa bcdω bc ρ ( ∂σfd µ + 1 2ωdg σ fgµ −A(2) σ ed µ ) , f̂a µ = fa µ + ΘρσA(1) ρ ( ∂σfa µ + 1 2ωag σ fgµ −A(2) σ ea µ ) − 1 4Θρσεa bcdω bc ρ ( ∂σed µ + 1 2ωdg σ egµ −A(2) σ fd µ ) . (A.3) Acknowledgements I am very grateful to P. Aschieri, S. Fabi, B. Harms and N. Okada for valuable discussions. 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[32] Banerjee R., Majhi B.R., Modak S.K., Noncommutative Schwarzschild black hole and area law, Classical Quantum Gravity 26 (2009), 085010, 11 pages, arXiv:0802.2176. http://dx.doi.org/10.1007/BF02798862 http://dx.doi.org/10.1007/BF02798862 http://dx.doi.org/10.1007/s100520000487 http://arxiv.org/abs/hep-th/0006246 http://dx.doi.org/10.1016/S0550-3213(00)00527-7 http://arxiv.org/abs/hep-th/0006091 http://dx.doi.org/10.1103/PhysRevD.79.025004 http://arxiv.org/abs/0808.2066 http://dx.doi.org/10.1103/PhysRevD.60.021301 http://arxiv.org/abs/astro-ph/9904022 http://dx.doi.org/10.1103/PhysRevLett.87.011301 http://dx.doi.org/10.1103/PhysRevLett.87.011301 http://arxiv.org/abs/hep-ph/0012363 http://dx.doi.org/10.1007/s10714-005-0183-z http://arxiv.org/abs/hep-th/0508051 http://dx.doi.org/10.1103/PhysRevD.77.124035 http://arxiv.org/abs/0801.3583 http://dx.doi.org/10.1088/0264-9381/26/8/085010 http://dx.doi.org/10.1088/0264-9381/26/8/085010 http://arxiv.org/abs/0802.2176 1 Introduction 2 Noncommutative GL(2,C) gauge theory of gravity 3 First order Seiberg-Witten map 4 Exact solutions to noncommutative gravity 5 Flat expanding universe 6 Schwarzschild solution 7 Exact solutions to noncommutative Einstein-Maxwell theory 8 Reissner-Nordström solution 9 Concluding remarks A Appendix References
id	nasplib_isofts_kiev_ua-123456789-146347
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-07T17:22:34Z
publishDate	2010
publisher	Інститут математики НАН України
record_format	dspace
spelling	Stern, A. 2019-02-09T09:13:15Z 2019-02-09T09:13:15Z 2010 Emergent Abelian Gauge Fields from Noncommutative Gravity / A. Stern // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 83D05; 53D55; 81T75 DOI:10.3842/SIGMA.2010.019 https://nasplib.isofts.kiev.ua/handle/123456789/146347 We construct exact solutions to noncommutative gravity following the formulation of Chamseddine and show that they are in general accompanied by Abelian gauge fields which are first order in the noncommutative scale. This provides a mechanism for generating cosmological electromagnetic fields in an expanding space-time background, and also leads to multipole-like fields surrounding black holes. Exact solutions to noncommutative Einstein-Maxwell theory can give rise to first order corrections to the metric tensor, as well as to the electromagnetic fields. This leads to first order shifts in the horizons of charged black holes. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html. I am very grateful to P. Aschieri, S. Fabi, B. Harms and N. Okada for valuable discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Emergent Abelian Gauge Fields from Noncommutative Gravity Article published earlier
spellingShingle	Emergent Abelian Gauge Fields from Noncommutative Gravity Stern, A.
title	Emergent Abelian Gauge Fields from Noncommutative Gravity
title_full	Emergent Abelian Gauge Fields from Noncommutative Gravity
title_fullStr	Emergent Abelian Gauge Fields from Noncommutative Gravity
title_full_unstemmed	Emergent Abelian Gauge Fields from Noncommutative Gravity
title_short	Emergent Abelian Gauge Fields from Noncommutative Gravity
title_sort	emergent abelian gauge fields from noncommutative gravity
url	https://nasplib.isofts.kiev.ua/handle/123456789/146347
work_keys_str_mv	AT sterna emergentabeliangaugefieldsfromnoncommutativegravity

Emergent Abelian Gauge Fields from Noncommutative Gravity

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