We present the key elements of the proof of an upper bound for
angular-momentum and charge in terms of the mass for electro-vacuum
asymptotically flat axisymmetric initial data sets with simply connected orbit
space

Costa J L

Dain S

Gibbons G W

Hennig J Cederbaum C Ansorg M

João Lopes Costa

Piotr T Chruściel

English

arXiv

Crossref

Mass, angular-momentum and charge inequalities for axisymmetric initial data

WOS:000271759400013 (Nº de Acesso Web of Science)We present the key elements of the proof of an upper bound for angular momentum and charge in terms of the mass for electro-vacuum asymptotically flat axisymmetric initial data sets with simply connected orbit space

Chruściel, Piotr T.

Costa, João Lopes

Repositório Institucional do ISCTE-IUL

                       This is an Author's Accepted Manuscript of an article published as: CHRUSCIEL, P. and COSTA, J.L. (2009) Mass, angular-momentum and charge inequalities for axisymmetric initial data. Classical and Quantum Gravity. 26(23):235013 available online at: http://dx.doi.org/10.1088/0264-9381/26/23/235013    arXiv:0909.5625v1  [gr-qc]  30 Sep 2009Mass, angular-momentum, and chargeinequalities for axisymmetric initial dataPiotr T. ChruścielLMPT, Fédération Denis Poisson, ToursMathematical Institute and Hertford College, OxfordJoão Lopes CostaLisbon University Institute – ISCTEMathematical Institute and Magdalen College, OxfordSeptember 30, 2009AbstractWe present the key elements of the proof of an upper bound forangular-momentum and charge in terms of the mass for electro-vacuumasymptotically flat axisymmetric initial data sets with simply con-nected orbit space.1 IntroductionIn important recent work, Dain [6] has proved an upper bound for angular-momentum in terms of the mass for a class of maximal, vacuum, axisymmet-ric initial data sets.1 The aim of this work is to generalize Dain’s inequalityto include electric and magnetic charges. In fact, the heuristic argumentbehind Dain’s original inequality generalizes to electro-vacuum as follows:the “standard picture of gravitational collapse” [7] is that the formation of1The analysis of [6] has been extended in [3] to include all axisymmetric vacuum initialdata, with simply connected orbit space, and manifolds which are asymptotically flat inthe standard sense, allowing moreover several asymptotic ends.1event horizons should occur generically in large families of space-times, withthe exterior region approaching a Kerr-Newman metric asymptotically withtime. Now, mass and Maxwell charges are conserved quantities, and thesame is true for angular momentum if one further assumes axisymmetry, sothe inequality (1.1) follows for all initial data for such a collapse. Besidesits intrinsic interest, our inequality provides some support for this “standardpicture”, and in particular for weak cosmic censorship.More precisely, consider a three dimensional electro-vacuum smooth ini-tial data set (M, g,K,E,B), where M is the union of a compact set and oftwo asymptotically flat regions M1 and M2. Here g is a Riemannian metricon M , K is the extrinsic curvature tensor, E is the electric field and B themagnetic one, both divergence-free in electro-vacuum. We suppose that theinitial data set is axisymmetric, by which we mean that it is invariant un-der an action of U(1), and maximal: trgK = 0. It is further assumed thatM/U(1) is simply connected, so that the results of [2] can be used. The no-tion of asymptotic flatness is made precise in (2.1) and (2.3), where moreoverk ≥ 6 needs to be assumed when invoking [2]. We then have the following:Theorem 1.1 Under the conditions just described, let m, ~J , QE and QBdenote respectively the ADM mass, the ADM angular momentum, the totalelectric charge and the total magnetic charge of M1:QE = −14π∫S∞∗F , QB =14π∫S∞F .Thenm ≥√| ~J |2m2+Q2E +Q2B . (1.1)A slightly more general version of Theorem 1.1 can be found in Theo-rem 2.1 below.The reader should note an inequality relating area, angular momentum,and charge, proved for stationary Einstein-Maxwell black holes in [9], as wellas the discussion of the Penrose inequality in electrovacuum of [11].Remark 1.2 We expect the equality to be attained only for the magneti-cally and electrically charged extreme Kerr-Newman space-times, which areunlikely to satisfy the hypotheses of Theorem 1.1.2Remark 1.3 If M contains only one asymptotic flat end and ∂M = ∅ wehave ~J = QE = QB = 0 (see (2.9) below), whence our interest in initial datasets containing two ends. One expects the result to generalize to several endsalong the lines of [3], but a proof of this lies beyond the scope of this work.Remark 1.4 The proof applies to Einstein–Abelian Yang-Mills fields con-figurations, giving in this casem ≥√√√√| ~J |2m2+ (∑iQEi)2 + (∑iQBi)2 , (1.2)where the QEi ’s and the QBi ’s are the electric and magnetic charges associ-ated with the i’th Maxwell field.2 Angular momentum and charge inequali-tiesRecall that an asymptotically flat end is a region Mext ⊂M diffeomorphic toR3 \ B(R), where B(R) is a coordinate ball of radius R, such that in localcoordinates on Mext obtained from R3 \B(R) we have, for some k ≥ 1,2gij = δij + ok(r−1/2) , ∂kgij ∈ L2(Mext) , Kij = Ok−1(r−λ) , λ >52. (2.1)(The asymptotic conditions on g arise from the requirement of well de-fined ADM mass, with the integrability condition satisfied if, e.g., ∂kgij =O(r−α−1), for some α > 1/2. The restriction on the decay rate of K abovearose already in the vacuum case, and can be traced back to the unnumberedequation after (2.37) in [3].)The electric and magnetic fields E and B are defined to be the orthogonalprojections to TM of their space-time analoguesEµ = F µνnν , Bµ = ∗F µνnν , (2.2)2We write f = ok(r−α) if the limits limr→∞ rα+ℓ∂k1 . . . ∂kℓf vanish for all 0 ≤ ℓ ≤ k,and f = Ok(r−α) if there exists a constant C such that |rα+ℓ∂k1 . . . ∂kℓf | ≤ C for all0 ≤ ℓ ≤ k.3where F is the Maxwell two-form, and where n is a unit normal to M , whenembedded in a space-time. We assume that in the manifestly asymptoticallyflat coordinates we haveEi = Ok−1(r−γ−1) , Bi = Ok−1(r−γ−1) , γ > 3/4 . (2.3)We will use the Einstein-Maxwell scalar constraint equation, which formaximal initial data reads(3)R = 16πµ+ |K|2g + 2(|E|2g + |B|2g), (2.4)where the function µ ≥ 0 represents the non-electromagnetic energy densityand | · |g denotes the norm of a vector with respect to the metric g.To obtain our inequality we start by bounding (2.4) from below, as follows.By [2] there exists a coordinate system, with controlled asymptotic behaviour,in which the metric takes the formg = e−2U+2α(dρ2 + dz2)+ ρ2e−2U (dϕ+ ρ Wρdρ+Wzdz)2 . (2.5)Consider an orthonormal frame ei such that e3 is proportional to the rota-tional Killing vector fieldη := ∂ϕ .Let θi denote the dual co-frame; for definiteness we takeθ1 = e−U+αdρ , θ2 = e−U+αdz , θ3 = ρe−U (dϕ+ ρWρdρ+Wzdz) .We assume that the initial data are invariant under the flow of η; this impliesthe space-time equations LηF = 0 and Lη ∗ F = 0, where a star stands forthe Hodge dual, and L denotes a Lie-derivative. A standard calculation,together with the hypothesis of simple-connectedness of M/U(1), impliesthen the existence of functions χ and ψ such that∂αχ = Fµαηµ , ∂αψ = ∗Fµαηµ . (2.6)We then have|E|2g + |B|2g =e2Uρ2((∂nχ)2 + |Dχ|2g + (∂nψ)2 + |Dψ|2g)≥e2Uρ2(|Dχ|2g + |Dψ|2g)=e4U−2αρ2((∂ρχ)2 + (∂zχ)2 + (∂ρψ)2 + (∂zψ)2), (2.7)4where ∂n denotes the derivative in the direction normal to the initial datahypersurface.Writing4πTµνηµ = FµαFναηµ −14FαβFαβην = ∂αψFνα −14FαβFαβην ,we are now able to justify Remark 1.3, under the (reasonable) condition thatFµνψ = o(r−2) . (2.8)First, the vanishing of the electric and magnetic charges is immediate. Next,the symmetry of the problem implies that ~J is aligned along the axis ofrotation. Letting Jz denote the component of angular-momentum along therotation axis, by the Komar identity we obtain16πJz =∫S∞∇µηνdSµν =12∫M∇µ∇µηνdSν=12∫MRµνηµdSν = 4π∫MTµνηµdSν= −∫S∞ψ∇αFνα︸ ︷︷ ︸0dSν +∫M∇α(ψFνα)dSν −14∫MFαβFαβηνdSν= 2∫S∞ψF ναdSνα −14∫MFαβFαβ ηµnµ︸︷︷︸0d3µg = 0 , (2.9)where we have used the fact that η is tangent to M , and where the firstintegral in the last line vanishes by (2.8).As discussed in [6], in vacuum the one-form3λ := 2ǫijkKjℓηkηℓdxi= 2ǫ(∂A, ∂B, ∂ϕ)K(dxB, ∂ϕ)dxA= 2g(η, η)ǫ(ea, eb, e3)K(θb, e3)θa (2.10)is closed. Here, as before, the upper case indices A,B = 1, 2 correspondto the coordinates (ρ, z), while the lower case indices a, b = 1, 2 are frameindices. In electro-vacuum we have instead (see, e.g., [10])d(λ− 2(χdψ − ψdχ))= 0 , (2.11)3We take this opportunity to point out a factor of 2 missing in the left-hand-side ofEq. (2.6) in [3], which affects numerical factors in some subsequent equations, but has noother consequences.5and, since we have assumed that M/U(1) is simply connected, there exists afunction v such thatλ = 2(dv + χdψ − ψdχ) . (2.12)Then, writing Kb3 for K(eb, e3), and using ǫab := ǫ(ea, eb, e3) ∈ {0,±1}, wehave2ρ2e−2U(K23θ1 −K13θ2) = λρdρ+ λzdz ;equivalentlyK13 = −e3U−α2ρ2λz , K23 =e3U−α2ρ2λρ , (2.13)so thate2(α−U) |K|2g ≥ 2e2(α−U)(K213 +K223) =e4U2ρ4|λ|2δ . (2.14)In [2] (compare [1, 6, 8]) it has been shown thatm =116π∫ [(3)R +12ρ2e−4α+2U (ρWρ,z −Wz,ρ)2]e2(α−U)d3x+18π∫(DU)2 d3x . (2.15)Inserting (2.7) and (2.14) into (2.15) we obtainm ≥116π∫ [(3)Re2(α−U) + 2 (DU)2]d3x≥18π∫ [(DU)2 +e4Uρ4(Dv + χDψ − ψDχ)2 +e2Uρ2((Dχ)2 + (Dψ)2) ]d3x ,(2.16)where, from now on, we use the symbol Df to denote the gradient of afunction f with respect to the flat metric δ, and where (Df)2 ≡ |Df |2δ.It follows from (2.6) that ψ and χ are constant on each connected com-ponent Aj of the “axis”A := {ρ = 0} \ {z = 0} ;(2.12)-(2.13) then show that so is v. We setvj := v|Aj , ψj := ψ|Aj , χj := χ|Aj . (2.17)We have the following, from which Theorem 1.1 immediately follows:6Theorem 2.1 Let (M, g,K, v, χ, ψ) be a three dimensional smooth data setinvariant under an action of U(1), where M is the union of a compact set andof two asymptotically flat regions M1 and M2, in the sense of (2.1) togetherwith (2.3), and where v, ψ and χ are global potentials as in (2.12)-(2.13) and(2.6). Assume that (2.4) holds with µ ≥ 0. Let m and ~J denote the ADMmass and angular momentum of M1, and let QE and QB be the global electricand magnetic charges of M1. If M/U(1) is simply connected, thenm ≥√| ~J |2m2+Q2E +Q2B .The proof of Theorem 2.1 proceeds as follows. If the mass is infinite thereis nothing to prove; under the current hypotheses this will be the case iff(3)R 6∈ L1(M). Otherwise, in view of (2.16), one considers the actionI :=∫ [(DU)2 +e4Uρ4(Dv + χDψ − ψDχ)2 +e2Uρ2((Dχ)2 + (Dψ)2) ]d3x .(2.18)One wishes to show that I is bounded from below by the right-hand-side ofour bound, which is the value of the corresponding action for the extremeKerr-Newman solution with the same global Poincaré and Maxwell charges.The proof can be obtained by following the argument in [3], except for sup-plementary difficulties in several places related to the new structure of theterm e4Uρ4(Dv + χDψ − ψDχ)2. For example, when a cut-off function ϕη isused in [3], the estimates do not appear to go through using the argumentsthere. This can be circumvented as follows: Let the quantities decoratedwith tildes refer to the extreme Kerr-Newman solution. Let θ = U, v, χ, ψand writeθη := ϕηθ + (1 − ϕη) θ̃ = ϕη(θ − θ̃) + θ̃ .Setλ η := Dvη + χηDψη − ψηDχη . (2.19)While a direct estimate of λη in the most singular integrals in [3] does notappear to be immediate, it turns out that each summand in the identityλη = ϕηλ+ (1 − ϕη)λ̃+Dϕη(v − ṽ) +Dϕη(χ̃ψ − ψ̃χ)+ϕη(1 − ϕη){(ψ − ψ̃)D(χ− χ̃) − (χ− χ̃)D(ψ − ψ̃)}(2.20)7can be handled in a way similar to the original integrals in [3]. For example,one of the steps of the proof is to establish that∫η/2<r<ηe4Uηρ4|λη|2d3x→η→0 0 , (2.21)where r2 =√ρ2 + z2. Recall that near r = 0 the coordinates (ρ, z) canbe obtained from the usual cylindrical coordinates in the other asymptoti-cally flat region, which we denote by (ρ̂, ẑ), by an inversion (ρ̂, ẑ) = ( ρr2, zr2),compare [2, Theorem 2.9, p. 2580]. This leads to estimates for small r,equivalently for large r̂, such as|Dχ|δ =1r2|D̂χ|δ̂ .1r2ρ̂r̂−γ−1 = ρrγ−3 , r → 0 . (2.22)From this and the known asymptotic behaviour of extreme Kerr-Newman oneobtains, when the decay exponent λ of K (see (2.1)) satisfies λ ≥ 2γ + 1,4v − ṽ = O(r2γ−2) . (2.23)This allows us to estimate the contribution of the term Dϕη(v− ṽ) of (2.19)in the subregion ρ ≥ z of the integral (2.21) as∫ ηη/2r4ρ4r2(r2γ−2)2r2dr = O(η4γ−3) →η→0 0 provided that γ > 3/4 .This explains our ranges of λ and γ in (2.1) and (2.3).A detailed presentation, with some simplifications of the argument of [3],will be given elsewhere [4, 5].Acknowledgements: We are grateful to S. Dain and E. Delay formany useful discussions, and for sharing their calculations of the potentialsfor the extreme Kerr-Newman solution.References[1] D. Brill, On the positive definite mass of the Bondi-Weber-Wheeler time-symmetric gravitational waves, Ann. Phys. 7 (1959), 466–483.4For λ ≤ 2γ + 1 the dominating behaviour in (2.12) is governed by λ, which leads tov − ṽ = O(rλ−3) and the necessity to impose λ > 5/2, as in vacuum [3, p. 2602].8[2] P.T. Chruściel, Mass and angular-momentum inequalities for axi-symmetric initial data sets. I. Positivity of mass, Annals Phys. 323(2008), 2566–2590, doi:10.1016/j.aop.2007.12.010, arXiv:0710.3680 [gr-qc].[3] P.T. Chruściel, Y. Li, and G. Weinstein, Mass and angular-momentuminequalities for axi-symmetric initial data sets. II. Angular momentum,Annals Phys. 323 (2008), 2591–2613, doi:10.1016/j.aop.2007.12.011,arXiv:0712.4064v2 [gr-qc].[4] J. Lopes Costa, A Dain inequality with charge, in preparation.[5] , On black hole uniqueness theorems, Ph.D. thesis, Oxford, inpreparation.[6] S. Dain, Proof of the angular momentum-mass inequality for axisymmet-ric black holes, Jour. Diff. Geom. 79 (2006), 33–67, arXiv:gr-qc/0606105.MR MR2414749[7] , The inequality between mass and angular momentum for axiallysymmetric black holes, Int. Jour. Modern Phys. D 17 (2008), 519–523.MR MR2441278 (2009e:83085)[8] G.W. Gibbons and G. Holzegel, The positive mass and isoperimetricinequalities for axisymmetric black holes in four and five dimensions,Class. Quantum Grav. 23 (2006), 6459–6478, arXiv:gr-qc/0606116. MRMR2272015[9] J. Hennig, C. Cederbaum, and M. Ansorg, A universal inequality foraxisymmetric and stationary black holes with surrounding matter in theEinstein-Maxwell theory, (2008), arXiv:0812.2811 [gr-qc].[10] G. Weinstein, N-black hole stationary and axially symmetric solutionsof the Einstein/Maxwell equations, Commun. Part. Diff. Eqs. 21 (1996),1389–1430.[11] G. Weinstein and S. Yamada, On a Penrose inequality with charge,arXiv:math.dg/0405602.9

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Mass, angular-momentum, and charge inequalities for axisymmetric initial data

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Mass, angular-momentum, and charge inequalities for axisymmetric initial data

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