Seiberg-Witten Instability of Various Topological Black Holes
aa r X i v : . [ h e p - t h ] A p r August 8, 2018 9:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in main SEIBERG-WITTEN INSTABILITY OF VARIOUS TOPOLOGICALBLACK HOLES
YEN CHIN ONG ∗ and PISIN CHEN Graduate Institute of Astrophysics, National Taiwan University, Taipei 10617, Taiwan. Department of Physics, National Taiwan University, Taipei 10617, Taiwan. Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan. Kavli Institute for Particle Astrophysics and Cosmology,SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, U.S.A. ∗ E-mail: [email protected]
We review the Seiberg-Witten instability of topological black holes in Anti-de Sitterspace due to nucleation of brane-anti-brane pairs. We start with black holes in generalrelativity, and then proceed to discuss the peculiar property of topological black holesin Hoˇrava-Lifshitz gravity – they have instabilities that occur at only finite range ofdistance away from the horizon. This behavior is not unique to black holes in Hoˇrava-Lifshitz theory, as it is also found in the relatively simple systems of charged black holewith dilaton hair that arise in low energy limit of string theory.
Keywords : Topological black holes; Black hole instability; Hoˇrava-Lifshitz gravity
1. Introduction: Seiberg-Witten Instability
In view of the various applications in the Anti-de-Sitter/Conformal Field Theory(AdS/CFT) correspondence, black hole solutions in AdS have received much atten-tion in the literature.
In the original formulation by Maldacena, it is conjecturedthat type IIB superstring theory in AdS × S is dual to N = 4 U( N ) super-Yang-Mills theory in (3 + 1)-dimensions. This has been generalized to other dimensions.In principle, the geometry of spacetimes can be affected by the presence of branes inasymptotically locally AdS spacetime. Seiberg and Witten showed quite genericallythat if a certain function (the brane action) defined on the Wick-rotated spacetimebecomes negative, the spacetime becomes unstable. Specifically, given a BPS (Bogomol’ny-Prasad-Sommerfield) brane Σ in Wick-rotated d -dimensional spacetime, the Seiberg-Witten brane action in the appropri-ate unit is a function of radial coordinate r defined by S [Σ( r )] = A (Σ) − ( d − V (Σ) , where A is the area of Σ and V its volume. The instability is caused by nucleationsof brane-anti-brane pairs at the regions where the brane action is negative, a phe-nomenon analogous to Schwinger pair-production in strong electromagnetic fields. The presence of copious amount of branes alters the geometry. Prior to Seiberg andWitten’s work, Maldacena, Michelson, and Strominger already pointed out thatvarious AdS geometries are prone to such drastic changes, which they referred toas “fragmentation”. It is worth emphasizing that Seiberg-Witten instability applies to any d -dimensional spacetime ( d ≥
4) with an asymptotically hyperbolic Euclidean ver-sion (so that it has well-defined conformal boundary), even for string theory on W d × Y − d , where W d is a d -dimensional non-compact asymptotically hyperbolic ugust 8, 2018 9:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in main manifold (generalizing AdS d ) and Y − d a compact space (generalizing S − d ).
2. Topological Black Holes in General Relativity
Topological black holes can have event horizon with positive, zero, or negative scalarcurvature k . The positively curved black holes include the usual Schwarzschild blackhole with S d − topology, where d is the spacetime dimension, and also black holesof S d − / Γ topology, i.e. quotient of S d − by the action of some discrete group Γ.Similarly, the event horizon of k = 0 and k = − R d − / Γ and H d − / Γ, respectively.In the context of general relativity, it was shown that k = 1 black holes have pos-itive brane action, while for k = − always become negativeand stay negative. Thus positively curved black holes are stable (of course beingstable in Seiberg-Witten sense do not preclude the possibility that it is unstable dueto other effects) but negatively curved ones are inherently unstable. Of course theonset of every instability is associated with a time scale , thus even unstable blackholes could be effectively meta-stable.
3. Topological Black Holes in Hoˇrava-Lifshitz Gravity
Noting that AdS/CFT correspondence is likely to occur in any quantum theory ofgravity, as the existence of holographic dualities is not contingent on the validityof string theory, we expect that something similar, if not identical, to Seiberg-Witten instability is likely to be a feature in any quantum gravity theory thatadmits extended objects (e.g. branes) propagating in asymptotically AdS spaces.We therefore investigate the stability of topological black holes in the context ofHoˇrava-Lifshitz Gravity. We find that in certain range of the detailed balanceparameter ǫ (general relativity is recovered with ǫ = 1, while Hoˇrava-Lifshitz gravitywith detailed balance condition corresponds to ǫ = 0), the black holes in Hoˇrava-Lifshitz theory can have brane action that is only negative in some finite range ofradial coordinate. This is markedly different from black holes in general relativityin which once the brane action becomes negative, it always stays negative. Braneaction with this property was previously found in the context of cosmology byMaldacena and Maoz. Such black holes are expected to be stable in the sensethat backreaction is very likely to set in and the systems eventually settle into anew, stable configurations.
4. Flat Black Holes in Einstein-Maxwell-Dilaton Theory
The Maldacena-Maoz type of instability that occurs in Hoˇrava-Lifshitz Gravitynaturally raises the suspicion that it could be due to the non-relativistic and Lorentz-violating nature of Hoˇrava-Lifshitz Gravity. However, we find that such instabilityalso arises in the relatively simpler Einstein-Maxwell-Dilaton theory, which is a lowenergy limit of string theory. ugust 8, 2018 9:19 WSPC - Proceedings Trim Size: 9.75in x 6.5in main In particular, extending previous work in Ref. 14, we showed that for dilatoncoupling α >
1, asymptotically locally AdS charged dilaton Gao-Zhang black holeswith flat horizon have positive brane action and thus is stable in Seiberg-Wittensense. For 0 < α <
1, the stability is of Maldacena-Maoz type. We proved that inboth cases, the asymptotic behavior of the brane action is logarithmically divergentfor finite α .
5. Conclusion
With the examples from Hoˇrava-Lifshitz and Einstein-Maxwell-Dilaton theories, weemphasize that we still lack a quantitative way to investigate the sufficient andnecessary condition for a spacetime with Maldacena-Maoz instability capable ofsettling down to a stable solution due to backreaction from the brane. We knowthat there are manifolds where once brane action becomes negative at the conformalboundary, cannot be deformed to have non-negative brane action. However, thesituation in general, especially in the case of Maldacena-Maoz type of instability, isfar from settled.
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