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arXiv:1001.0019v1 [gr-qc] 30 Dec 2009On the instability of Reissner-Nordstr¨ om black holes in de Sitter backgrounds |
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Vitor Cardoso∗ |
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CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico, |
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Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal & |
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Department of Physics and Astronomy, The University of Miss issippi, University, MS 38677-1848, USA |
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Madalena Lemos†and Miguel Marques‡ |
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CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecn ico, |
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Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal |
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(Dated: November 3, 2018) |
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Recent numerical investigations have uncovered a surprisi ng result: Reissner-Nordstr¨ om-de Sitter |
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black holes are unstable for spacetime dimensions larger th an 6. Here we prove the existence of |
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such instability analytically, and we compute the timescal e in the near-extremal limit. We find very |
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good agreement with the previous numerical results. Our res ults may me helpful in shedding some |
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light on the nature of the instability. |
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PACS numbers: 04.50.Gh,04.70.-s |
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I. INTRODUCTION |
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In physics, stability of a given configuration (solution |
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of some set of equations), is a useful criterium for rele- |
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vance of that solution. Unstable configurations are likely |
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not tobe realizablein practice, and representaninterme- |
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diate stage in the evolution of the system. Nevertheless, |
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the instability itself is of great interest, since an under- |
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standing of the mechanism behind it may help one to |
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better grasp the physics involved. In particular, it is of |
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interest to be able to predict which other systems display |
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similar instabilities, or even have a deeper understanding |
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of the physics behind the instability (why is the system |
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unstable? is there some fundamental principle behind |
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the instability?). |
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In General Relativity, the Kerr family exhausts the |
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blackhole solutionsto the electro-vacEinstein equations. |
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Kerr black holes are stable, and can therefore describe |
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astrophysicalobjects. However,there aremanyinstances |
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of instabilities afflicting objects with an event horizon, |
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such as the Gregory-Laflamme [1], the ultra-spinning [2] |
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or superradiant instabilities [3] and other instabilities of |
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higher-dimensional black holes in alternative theories [4, |
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5](for a review see Ref. [6]). |
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Konoplya and Zhidenko (hereafter KZ) recently stud- |
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ied small perturbations in the vicinity of a charged black |
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hole in de Sitter background, a Reissner-Nordstr¨ om de |
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Sitter black hole (RNdS) [7]. Their (numerical) results |
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show that when the spacetime dimensionality D >6, the |
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spacetime is unstable, provided the charge is larger than |
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agiventhreshold, determined byKZforeach D. Because |
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∗Electronic address: [email protected] |
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†Electronic address: [email protected] |
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‡Electronic address: [email protected] results are so surprising (the mechanism behind it is |
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not yet understood), we set out to to investigate this in- |
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stability and hopefully understand it better. Our results |
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can be summarized as follows: (i) we can prove analyti- |
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cally the existence of unstable modes for charge Qhigher |
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thanacertainthreshold. (ii)inthenear-extremalregime, |
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we are able to find an explicit solution for the unstable |
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modes, determining the instability timescale analytically. |
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We hope that our incursion in this topic helps to better |
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understand the physics at work. |
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II. EQUATIONS |
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This work focuses on the higher dimensional RNdS ge- |
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ometry, described by the line element |
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ds2=−f dt2+f−1dr2+r2dΩ2 |
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n, (1) |
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wheredΩ2 |
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nis the line element of the nsphere and |
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f= 1−λr2−2M |
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rn−1+Q2 |
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r2n−2. (2) |
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the background electric field is E0=q/rn, withqthe |
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electric charge. The quantities MandQare related to |
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the physical mass M and charge qof the black hole [8], |
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andλto the cosmological constant. The spacetime di- |
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mensionality is D=n+2. |
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The above geometry possesses three horizons: the |
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black-holeCauchyhorizonat r=ra, the black hole event |
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horizon is at r=rband the cosmological horizon is at |
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r=rc, whererc> rb> ra, the only real, positive zeroes |
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off. For convenience, we set rb= 1, i.e., we measure all |
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quantities in terms of the event horizon rb. We thus get |
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2M= 1+Q2−λ, (3)2 |
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Furthermore, we can also write |
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λ=r−4−n |
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c(rn+2 |
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c−r3 |
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c)(rn+2 |
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c−Q2r3 |
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c) |
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rn+2c−rc.(4) |
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For a fixed rcand spacetime dimension D, the existence |
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ofaregulareventhorizonimposesthatthecharge Qmust |
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be smaller than a certain value Qext. With our units this |
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maximum charge is |
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Q2 |
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ext=rn |
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c/parenleftbig |
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−2rc+(n+1)rn |
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c−(n−1)rn+2 |
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c/parenrightbig |
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−rc/parenleftbig |
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rc(n+1)−2nrnc+(n−1)r2n+1c/parenrightbig.(5) |
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Gravitational perturbations of this spacetime couple to |
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the electromagnetic field, and were completely character- |
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ized by Kodama and Ishibashi [8]. They can be reduced |
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to a set of two second order ordinary differential equa- |
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tions of the form, |
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d2 |
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dr2∗Φ±+/parenleftbig |
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ω2−VS±/parenrightbig |
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Φ±= 0, (6)where the tortoise coordinate r∗and the potentials VS± |
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are defined through |
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r∗≡/integraldisplay |
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f−1dr, V S±=fU± |
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64r2H2 |
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±.(7) |
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We have |
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H+= 1−n(n+1) |
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2δx, (8) |
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H−=m+n(n+1) |
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2(1+mδ)x, (9) |
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and the quantities U±are given by |
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U+=/bracketleftbig |
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−4n3(n+2)(n+1)2δ2x2−48n2(n+1)(n−2)δx |
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−16(n−2)(n−4)]y−δ3n3(3n−2)(n+1)4(1+mδ)x4 |
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+4δ2n2(n+1)2/braceleftbig |
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(n+1)(3n−2)mδ+4n2+n−2/bracerightbig |
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x3 |
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+4δ(n+1)/braceleftbig |
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(n−2)(n−4)(n+1)(m+n2K)δ−7n3+7n2−14n+8/bracerightbig |
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x2 |
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+/braceleftbig |
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16(n+1)/parenleftbig |
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−4m+3n2(n−2)K/parenrightbig |
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δ−16(3n−2)(n−2)/bracerightbig |
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x |
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+64m+16n(n+2)K, (10) |
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U−=/bracketleftbig |
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−4n3(n+2)(n+1)2(1+mδ)2x2+48n2(n+1)(n−2)m(1+mδ)x |
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−16(n−2)(n−4)m2/bracketrightbig |
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y−n3(3n−2)(n+1)4δ(1+mδ)3x4 |
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−4n2(n+1)2(1+mδ)2/braceleftbig |
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(n+1)(3n−2)mδ−n2/bracerightbig |
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x3 |
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+4(n+1)(1+ mδ)/braceleftbig |
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m(n−2)(n−4)(n+1)(m+n2K)δ |
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+4n(2n2−3n+4)m+n2(n−2)(n−4)(n+1)K/bracerightbig |
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x2 |
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−16m/braceleftbig |
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(n+1)m/parenleftbig |
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−4m+3n2(n−2)K/parenrightbig |
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δ |
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+3n(n−4)m+3n2(n+1)(n−2)K/bracerightbig |
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x |
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+64m3+16n(n+2)m2K. (11) |
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The variables x,yand parameters µ,mare defined |
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through |
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x≡2M |
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rn−1, y≡λr2, (12) |
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µ2≡M2+4mQ2 |
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(n+1)2, m≡k2−nK,(13) |
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andthe quantity δis implicitly givenby µ= (1+2mδ)M. |
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Note that the following relations holds Q2= (n+ |
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1)2M2δ(1+mδ). |
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Note also that for the spacetime considered in this pa- |
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perK= 1, whichmeansthatthe eigenvalues k2aregivenbyk2=l(l+n−1), where lis the angular quantum |
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number, that gives the multipolarity of the field. The |
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behavior of the potentials varies considerably over the |
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range of parameters. In Fig. 1 we show V−forD= 8, |
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rc= 1/0.95,l= 2andthreedifferentvaluesofthecharge, |
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Q= 0.2,0.35,0.44. |
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III. A CRITERIUM FOR INSTABILITY |
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A sufficient (but not necessary) condition for the exis- |
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tence of an unstable mode has been proven by Buell and3 |
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/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51 /s48/s44/s48/s52 /s48/s44/s48/s53/s45/s50/s48/s50/s52/s54 |
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/s49/s48/s52 |
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/s32/s86 |
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/s45/s49/s48/s52 |
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/s32/s86 |
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/s45 |
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/s32/s32/s86 |
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/s45 |
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/s114/s45/s49/s32/s81/s61/s48/s46/s50/s48 |
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/s32/s81/s61/s48/s46/s51/s53 |
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/s32/s81/s61/s48/s46/s52/s52/s49/s48/s51 |
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/s32/s86 |
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/s45 |
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FIG. 1: Behavior of V−for different parameters, for D= 8. |
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Here we fix the event horizon at rb= 1, and the cosmological |
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horizon at rc= 1/0.95. We consider l= 2 modes and three |
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different charges, Q= 0.2,0.35,0.44. |
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Shadwick [9] and is the following, |
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/integraldisplayrc |
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rbV |
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fdr <0. (14) |
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The instability region is depicted in figure 2 for several |
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/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s32 |
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/s32/s81/s47/s81 |
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/s101/s120/s116 |
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/s114 |
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/s98/s47/s114 |
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/s99 |
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FIG. 2: The parametric region of instability in Q/Qext−rb/rc |
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coordinates, according to criterim (14), for l= 2. Top to |
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bottom, D= 7,8,9,10,11. |
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spacetime-dimension D, which can be compared with the |
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numerical results by KZ, their figure 4. It is apparent |
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that condition (14) very accurately describes the numer- |
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ical results for rb/rc∼1, a regime we explore below in |
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Section IV. As one moves away from extremality cri- |
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terium (14) is just too restrictive. An improved analysis |
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and refined criterium would be necessary to describe the |
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whole rangeofthe numericalresults. Nevertheless, figure2 is very clear: higher-dimensional ( D >6) RNdS black |
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holes are unstable for a wide range of parameters. |
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IV. AN EXACT SOLUTION IN THE NEAR |
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EXTREMAL RNDS BLACK HOLE |
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Let us now specialize to the near extremal RNdS black |
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hole, which we define as the spacetime for which the cos- |
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mological horizon rcis very close (in the rcoordinate) |
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to the black hole horizon rb, i.e.rc−rb |
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rb≪1. The wave |
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equationin this spacetime can be solvedexactly, in terms |
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of hypergeometric functions [10]. The key point is that |
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the physical region of interest (where the boundary con- |
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ditions are imposed), lies between rbandrc. Thus, |
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f∼2κb(r−rb)(rc−r) |
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rc−rb, (15) |
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where we have introduced the surface gravity κbassoci- |
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ated with the event horizon at r=rb, as defined by the |
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relationκb=1 |
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2df/drr=rb. For near-extremal black holes, |
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it is approximately |
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κb∼(rc−rb)(n−1) |
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2r2 |
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b/parenleftbig |
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1−nQ2/parenrightbig |
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.(16) |
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In this limit, one can invert the relation r∗(r) of (7) to |
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get |
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r=rce2κbr∗+rb |
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1+e2κbr∗. (17) |
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Substituting this on the expression (15) for fwe find |
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f=(rc−rb)κb |
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2cosh(κbr∗)2. (18) |
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As such, and taking into account the functional form of |
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the potentials for wave propagation, we see that for the |
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near extremal RNdS black hole the wave equation (6) is |
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of the form |
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d2Φ(ω,r) |
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dr2∗+/bracketleftBigg |
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ω2−V0 |
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cosh(κbr∗)2/bracketrightBigg |
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Φ(ω,r) = 0,(19) |
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with |
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V0=(rc−rb)κb |
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2VS±(rb) |
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f(20) |
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The potential in (19) is the well known P¨ oshl-Teller po- |
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tential [11]. The solutions to (19) were studied and they |
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are of the hypergeometric type, (for details see Refs. |
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[12, 13]). Itshouldbesolvedunderappropriateboundary |
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conditions: |
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Φ∼e−iωr∗, r∗→ −∞ (21) |
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Φ∼eiωr∗, r∗→ ∞. (22)4 |
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These boundary conditions impose a non-trivial condi- |
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tion onω[12, 13], and those that satisfy both simultane- |
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ously are called quasinormal frequencies. For the P¨ oshl- |
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Teller potential one can show [12, 13] that they are given |
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by |
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ω=κb/bracketleftBigg |
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−/parenleftbigg |
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j+1 |
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2/parenrightbigg |
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i+/radicalBigg |
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V0 |
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κ2 |
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b−1 |
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4/bracketrightBigg |
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, j= 0,1,.... |
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(23) |
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We conclude therefore that an instability is present |
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TABLE I: The threshold of instability for near-extremal |
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RNdS black holes (i.e., black holes for which the cosmologic al |
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and event horizon almost coincide) for l= 2 modes. We show |
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the prediction from the exact, analytic expression obtaine d |
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in the near extremal limit (24), which we label Q/QN |
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extand |
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the one from criterium (14) which we label as Q/QV |
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ext. Both |
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these results are compared to the numerical results by KZ. |
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D |
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7 8 9 10 11 D→ ∞ |
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Q/QN |
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ext0.913 0.774 0.683 0.617 0.567p |
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2/D |
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Q/QV |
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ext0.913 0.775 0.684 0.618 0.568p |
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2/D |
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Q/QNum |
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ext0.94 0.78 0.68 0.61 0.55 — |
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whenever V0is negative. The threshold of stability in |
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the near-extremal regime is therefore given by |
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VS±(rb) |
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f= 0, (24) |
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The expression for VS±(rb)/fis lengthy, and we won’t |
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presentit here. Thevaluesofthe charge Q/Qextthat sat- |
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isfy the condition above are given in Table I (for l= 2), |
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and compared to the prediction from the analysis in Sec- |
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tion III, criterium (14). The agreement is excellent. Fur-thermore, we compare these predictions against the nu- |
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merical results by KZ, extrapolated to the extremal limit |
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(ρ= 1 in KZ notation). The agreement is remarkable. |
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V. CONCLUSIONS |
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We have shown analytically that charged black holes |
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in de Sitter backgrounds are unstable for a wide range of |
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charge and mass of the black hole, confirming previous |
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numerical studies [7]. The stability properties of the ex- |
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tremalD= 6 black hole remain unknown. Our methods |
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and results and inconclusive at this precise point, further |
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dedicated investigations would be necessary. |
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Ouranalyticalresultinthenear-extremalregimecould |
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be used to investigate further the nature of this instabil- |
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ity, something we have not attempted to do here. A |
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possible refinement concerns the large- Dlimit of the in- |
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stability, where it couldbe possible to find an analytical |
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expression throughout all range of parameters. We have |
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inmind resultsandtechniquessimilartothoseofKoland |
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Sorkin [14]. It would also be interesting to investigate |
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the stability properties, using this or other techniques, of |
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near-extremal Kerr-dS black holes, which have recently |
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been conjectured to have an holographic description [15]. |
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Acknowledgements |
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We warmly thank Roman Konoplya and Alexander |
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Zhidenko for useful correspondence and for sharing their |
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numerical results with us. This work was partially |
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funded by Funda¸ c˜ ao para a Ciˆ encia e Tecnologia (FCT)- |
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Portugal through projects PTDC/FIS/64175/2006, |
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PTDC/ FIS/098025/2008,PTDC/FIS/098032/2008and |
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CERN/FP/109290/2009. |
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