Indexación: Scopus.In the context of the anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, 4D AdS gravity is suitably renormalized by adding the Gauss Bonnet term to the Einstein-Hilbert action. The subsequent addition of the Pontryagin term, with a specific coupling, allows to write the on-shell action in terms of the Weyl tensor and its dual, such that the action becomes stationary for asymptotic (anti) self-dual solutions in the Weyl tensor. The addition to the action of both topological invariants mentioned above does not modify the bulk dynamics, but it does modify the expression of the Noether current and, therefore, the conserved quantities of the theory. Here, we show that the method of Iyer and Wald leads to a fully-covariant Noether charge, which contains both the electric and magnetic parts of the Weyl tensor. For configurations which are globally (anti) self-dual in the Weyl tensor, both the action and the Noether charge identically vanish. This means that, for such spacetimes, the magnetic mass is equal to the electric mass. © Published under licence by IOP Publishing Ltd.This work was funded in part by FONDECYT Grants No. 1131075, 1140296 and 1151107, CONICYT Grant DPI 2014-0115, UNAB Grants DI-735-15/R and DI-1336-16/RG and VRIEA-PUCV Grant 039.345/2016. R.Araneda received financial support from Facultad de Física, Pontificia Universidad Católica de Chile to participate in SOCHIFI Congress.https://iopscience.iop.org/article/10.1088/1742-6596/1043/1/01201

Araneda, R.

Aros, R.

Miskovic, O.

Olea, R.

Journal of Physics Conference Series

English

Repositorio Institucional Académico (RIA) de la Universidad Andrés Bello

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSPontryagin Term and Magnetic Mass in 4D AdSGravityTo cite this article: René Araneda et al 2018 J. Phys.: Conf. Ser. 1043 012016 View the article online for updates and enhancements.Related contentAdS/CFT Correspondence in CondensedMatter: IntroductionA S T Pires-Ward Identities for W, Conformal FieldTheories on a Torus*Chao-Shang Huang, Liang-Xin Li and De-Hai Zhang-Topological M-theory, self-dual gravity andthe Immirzi parameterJavier Chagoya and Miguel Sabido-This content was downloaded from IP address 190.21.229.141 on 08/11/2019 at 13:201Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890 ‘’“”XX Chilean Physics Symposium IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1043 (2018) 012016  doi :10.1088/1742-6596/1043/1/012016Pontryagin Term and Magnetic Mass in 4D AdSGravityRene´ Araneda1, Rodrigo Aros2, Olivera Miskovic3 and Rodrigo Olea21 Instituto de F´ısica, Pontificia Universidad Cato´lica de Chile, Av. Vicun˜a Mackenna 4860,Santiago, Chile.2 Departamento de Ciencias F´ısicas, Universidad Andres Bello, Sazie´ 2212, Piso 7, Santiago,Chile.3 Instituto de F´ısica, Pontificia Universidad Cato´lica de Valpara´ıso, Casilla 4059, Valpara´ıso,Chile.E-mail: rraraneda@uc.cl, raros@unab.cl, olivera.miskovic@pucv.cl,rodrigo.olea@unab.clAbstract. In the context of the anti-de Sitter/Conformal Field Theory (AdS/CFT)correspondence, 4D AdS gravity is suitably renormalized by adding the Gauss Bonnet termto the Einstein-Hilbert action. The subsequent addition of the Pontryagin term, with a specificcoupling, allows to write the on-shell action in terms of the Weyl tensor and its dual, suchthat the action becomes stationary for asymptotic (anti) self-dual solutions in the Weyl tensor.The addition to the action of both topological invariants mentioned above does not modify thebulk dynamics, but it does modify the expression of the Noether current and, therefore, theconserved quantities of the theory. Here, we show that the method of Iyer and Wald leads to afully-covariant Noether charge, which contains both the electric and magnetic parts of the Weyltensor. For configurations which are globally (anti) self-dual in the Weyl tensor, both the actionand the Noether charge identically vanish. This means that, for such spacetimes, the magneticmass is equal to the electric mass.1. IntroductionIt has been shown that in asymptotically anti-de Sitter (AAdS) gravity in four dimensions theinfrared divergences are cancelled by the Gauss-Bonnet (GB) term with a given coupling [1].The addition of this topological invariant cancels divergent terms in the asymptotic region, suchas those present in the Euclidean gravity action, only for the specific value of coupling constantmentioned above [2],I4 =116piG∫Md4x√g[R+6`2+`24(RµναβRµναβ − 4RµνRµν +R2)]. (1)With the addition of this term, the action has an extremum for AAdS spacetimes. On the otherhand, the renormalized action (1) can be rewritten in terms of the Weyl tensor [3],I4 =`264piG∫Md4x√gWµναβWαβµν , (2)21234567890 ‘’“”XX Chilean Physics Symposium IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1043 (2018) 012016  doi :10.1088/1742-6596/1043/1/012016for any Einstein space. The GB invariant is not the only topological invariant of the Lorentzgroup in four dimensions. There exists yet another topological term, of opposite parity, knownas Pontryagin term,P4 = −14µναβRσλµνRσλαβ , (3)which is locally equivalent to the derivative of the gravitational Chern-Simons term in threedimensions. The addition of P4 to the gravity action does not induce additional divergencesin the asymptotic region. To our best knowledge, the only way to fix the Pontryagin couplingis to consider (in the Euclidean sector) an asymptotic (anti) self-duality condition in the Weyltensor,1Wαβµν = ± ∗Wαβµν , (4)such that the total action is stationary under arbitrary variations of the fields.Then, the renormalized action (1) acquires an additional term which can be written in termsof the Weyl tensor and its dual,I = I4 ± `232piG∫Md4x√gWµναβ∗Wαβµν . (5)This is analogous to the form in which the Pontryagin density for U(1) group may be addedto the Euclidean action in the Maxwell electromagnetism,IE =14∫M(FµνFµν ∓ ∗FµνFµν) d4x , (6)where the Fµν is the Abelian field strength. Due to the Bianchi identity (∂µ∗Fµν = 0), the secondterm in the Lagrangian does not modify the equations of motion. The Pontryagin coupling ineq.(6) is a consequence of an asymptotic (anti) self-dual condition for F ,Fµν = ±∗Fµν at ∂M . (7)The analogy between AdS gravity and self-dual electromagnetism can be pushed forward, asboth theories portray striking similarities at the level of the conserved charges.The Noether theorem allows to obtain an expression of conserved quantities in a gauge theoryand gravity, as well, though the symmetries present in each case are of different nature. As longas the Noether current is written (globally) as a total derivative, the conserved charges are givenby surface integrals in the asymptotic region, without the need of introducing a source at theorigin.Gauge symmetry U(1) in electromagnetism is realized by the transformation in the gauge fieldδλAν = ∂νλ. As this is an internal symmetry (δxµ = 0), the Lagrangian density (6) produces aNoether currentJµ = ∂ν [(Fµν + γ ∗Fµν)λ] . (8)Due to the conservation law, ∂µJµ = 0, and the fact that Jµ can be written as a total derivative,the conserved charge is an integral on a two-dimensional sphereQ[λ] = −∫S2(Fµν ∓ ∗Fµν)λ dΣµν , (9)1 The Hodge star dual is defined by ∗Wαβµν = 12√g αβγδWγδµν .31234567890 ‘’“”XX Chilean Physics Symposium IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1043 (2018) 012016  doi :10.1088/1742-6596/1043/1/012016in terms of the electromagnetic tensor and its dual. The expression (9), when λ is covariantlyconstant, leads to total charge which is the sum of the electric and the magnetic charges.Therefore, the magnetic charge is understood as a topological charge which is the Noether chargeassociated to a topological invariant. The total charge, for a global (anti) self-dual condition inF , will be identically zero, and so the Euclidean action. This defines a subclass of Euclideaninstantons which can be taken as ground states of the theory.In what follows, we apply the method developed by Wald and Iyer [4] to the 4D AdS gravityaction supplemented by the topological invariants discussed above. The resulting charge featuresstriking similarities to the case of self-dual electromagnetism.2. Noether-Wald charges and Topological InvariantsIn Riemannian gravity, the action∫d4xL depends on the metric and its derivatives combinedin the form of the Christoffel symbol. Invariance under coordinate transformations of the type(δxµ = ξµ(x)) leads to a conserved current√g Jµ[ξ] = Θ(δξg)µ + Θ(δξΓ)µ + Lξµ ,where we have assumed that the surface term Θ which appears in the arbitrary variation of theaction is separable in contributions along the Lie derivative of both the metric and Christoffelconnection.When ξµ is a Killing vector, the diffeomorphic variation of the metric, δξgµν =− (∇µξν +∇νξµ), vanishes.In an arbitrary gravity theory with a Lagrangian L given in terms of the metric, the Riemanntensor and its contractions,L = L(gµν , Rµναβ , Rµν , R) , (10)the conserved current turns out to be√g Jµ[ξ] = EµναβgαλδξΓλνβ + Lξµ, (11)where the tensor Eµναβ corresponds to the variation of L with respect to the Riemann tensorRµναβ , and the Lie derivative of the Christoffel symbol takes the formδξΓλνβ = −12(∇ν∇βξλ +∇β∇νξλ)+12(Rλβνσ +Rλνβσ)ξσ .In addition, we can use the following identities for the Riemann tensor[∇β,∇ν ]ξα = Rβνασξσ and Eµναβ (Rαβνσ − 2Rανβσ) = 0 , (12)where the last one is obtained by contracting the first type Bianchi identity with Eµναβ tensor.In doing so, one can isolate the contribution from the field equations, such that the Noethercurrent can be finally rewritten as√gJµ = 2Eµναβ∇ν∇αξβ . (13)This is the key formula for the conserved current in any gravitational theory defined by theLagrangian (10).For the particular case of the AdS gravity action in eq.(5), the Noether current isJµ[ξ] =`264piG∇ν[δ[µνσλ][αβγδ]∇αξβ(W γδσλ ∓ ∗W γδλσ)], (14)41234567890 ‘’“”XX Chilean Physics Symposium IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1043 (2018) 012016  doi :10.1088/1742-6596/1043/1/012016as it was shown in ref.[5]. It is an analog of the current (8) found in electromagnetism. As it isa total derivative, the corresponding Noether charge is a surface integral on a two-dimensionalsurface, given in terms of the Weyl tensor and its dual. The spacetime topology is M ' R×Σ,where Σ is the space section with unitary normal uµ = (−N˜ , 0, 0, 0), thus we can write theconserved charge asQ[ξ] =`264piG∫∂Σd2x√σ nµuνδ[µνλσ][αβγδ] ∇αξβ(W γδλσ ∓ ∗W γδλσ). (15)Here, nµ = (0, N, 0, 0) is the unitary normal at the boundary ∂M ' R × ∂Σ and σ is thedeterminant of the induced metric at the surface ∂Σ, which satisfies the relation√g = N√h =NN˜√σ.As the first term in the Noether charge is a function of the Weyl tensor, it allows to connectthe addition of the topological invariant of GB to the notion of conformal mass in AAdS gravitydefined by Ashtekar, Magnon and Das [6], given in terms of the electric part of the Weyl tensor.In analogous way to the electromagnetic case, the form of eq. (15) makes manifest the factthat configurations globally (anti) self-dual in the Weyl tensor have a vanishing Noether charge.For these solutions, the magnetic mass is equal to the conformal mass, as it is the case forTaub-NUT-AdS solution [7, 8, 9].3. Magnetic MassThe discussion above is unnecessary for Schwarzschild-AdS or Kerr-AdS solutions as they havea vanishing Pontryagin density. Some of simplest examples to consider are Taub-NUT-AdS andTaub-Bolt-AdS solutions. Taub-NUT-AdS has an (anti) self-dual Weyl tensor and Taub-Bolt-AdS meets that condition in the asymptotic region. Indeed, both solutions present topologicaldefects and have a non-vanishing magnetic mass [5].Both Taub-NUT-AdS and Taub-Bolt-AdS solutions can be written, in the coordinates(τ, r, θ, φ), asds2 = f(r)(dτ + 2n cos θdφ)2 +dr2f(r)+ (r2 − n2)(dθ2 + sin2 θdφ2),wheref(r) =r2 − 2Mr + n2 − 3`2(n4 + 2n2r2 − r43)r2 − n2 . (16)Here, n is a parameter related to a topological defect present in the geometry [11], and M is amass parameter [12].For either NUT or Bolt solution the largest root of the equation of f(r) = 0, namely r = r∗,defines a horizon. The difference between these two geometries is that, for NUT solution, therelation r∗ = |n| holds exactly while, for Bolt solution, it is the inequality r∗ = rb > |n|.Moreover, it must be satisfied f ′(r∗) = 12n to avoid a singularity at r∗. The mass for the NUTsolution, computed in ref.[1], isMNUT = ±n(1− 4`−2n2) , (17)while for the Bolt solutionMBolt =r2b + n22rb− 32`2(n4rb+ 2n2rb − r3b3).51234567890 ‘’“”XX Chilean Physics Symposium IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 1043 (2018) 012016  doi :10.1088/1742-6596/1043/1/012016As mentioned above, we are interested in analyzing the conserved charges as defined by eq.(15).For NUT solution, since its Weyl tensor is self-dual, we haveQNUT[ξ] = 0, (18)for any Killing vector ξ = ξµ∂µ. For Bolt solution, the only non-vanishing conserved charge ineq.(15) is the one associated to a timelike Killing vectorQBolt[∂τ ] = MBolt ±MNUT. (19)4. ConclusionsIn this article, we have shown that the addition of topological invariants (GB and Pontryaginterms), with specific couplings, modifies the value of the Noether charges of the theory. Whilethe electric part of the Weyl tensor gives rise to the energy of AAdS black holes, the magneticpart produces a non-vanishing magnetic mass for gravitational instantons in AdS. Furthermore,we have proved that solutions with (anti) self-dual Weyl tensors are suitable candidates forground states. Since the mass Q[∂τ ] obtained from eq.(15) is a thermodynamic variable, wemay expect a change in the thermodynamics of the system when solutions with a non vanishPontryagin term, as Taub-NUT-AdS and Taub-Bolt-AdS, are considered.AcknowledgementsThis work was funded in part by FONDECYT Grants No. 1131075, 1140296 and 1151107,CONICYT Grant DPI 2014-0115, UNAB Grants DI-735-15/R and DI-1336-16/RG and VRIEA-PUCV Grant 039.345/2016. R.Araneda received financial support from Facultad de F´ısica,Pontificia Universidad Cato´lica de Chile to participate in SOCHIFI Congress.References[1] R. Aros, M. Contreras, R. Olea, R. Troncoso, and J. Zanelli 2000, Phys. Rev. Lett. 84, 1647[2] R. Olea 2005, J. High Energy Phys. 06, 023[3] O. Miskovic and R. Olea 2009, Phys. Rev. D79, 124020[4] V. Iyer and R. Wald 1994, Phys. Rev. D 50, 846[5] R. Araneda, R. Aros, O. Miskovic and R. Olea 2016, Phys. Rev. D 93, 084022[6] A. Ashtekar and A. Magnon 1984, Classical Quantum Gravity 1, L39; A. Ashtekar and S. Das 2000, ClassicalQuantum Gravity 17, L17[7] E. Newman, L. Tamburino and T. Unti 1963, J. Math. Phys. 4, 915[8] D. N. Page and C. N. Pope 1987, Classical Quantum Gravity 4, 213[9] D. Kramer, H. Stephani, M. MacCallum and E. Herlt, Exact solutions of Einstein’s field equations, CambridgeUniversity Press, 1980[10] A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers 1999, Phys. Rev. D 59, 064010[11] T. Eguchi, P. B. Gilkey and A. J. Hanson 1980, Phys. Rept. 66, 213[12] S. W. Hawking, C. J. Hunter and D. N. Page 1999, Phys. Rev. D 59, 044033

Pontryagin Term and Magnetic Mass in 4D AdS Gravity

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Pontryagin Term and Magnetic Mass in 4D AdS Gravity

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