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Journal ArticleDOI

First-order flow equations for extremal and non-extremal black holes

TL;DR: In this paper, a general form of first-order flow equations for extremal and non-extremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity was derived.
Abstract: We derive a general form of first-order flow equations for extremal and non-extremal, static, spherically symmetric black holes in theories with massless scalars and vectors coupled to gravity. By rewriting the action as a sum of squares a la Bogomol'nyi, we identify the function governing the first-order gradient flow, the `generalised superpotential', which reduces to the `fake superpotential' for non-supersymmetric extremal black holes and to the central charge for supersymmetric black holes. For theories whose scalar manifold is a symmetric space after a timelike dimensional reduction, we present the condition for the existence of a generalised superpotential. We provide examples to illustrate the formalism in four and five spacetime dimensions.
Citations
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Journal ArticleDOI
Kyosuke Hotta1
TL;DR: In this article, the authors extend the discussion of the Kerr/CFT correspondence to the more general gauge/gravity correspondence in the full extremal black hole space-time of the bulk by using a technique of the holographic renormalization group (RG) flow.
Abstract: We extend the discussion of the 'Kerr/CFT correspondence' and its recent developments to the more general gauge/gravity correspondence in the full extremal black hole space-time of the bulk by using a technique of the holographic renormalization group (RG) flow. It is conjectured that the extremal black hole space-time is holographically dual to the chiral two-dimensional field theory. Our example is a typical four-dimensional Reissner-Nordstrom black hole, a system in which the M5-brane is wrapped on four cycles of Calabi-Yau threefold. In the five-dimensional supergravity viewpoint, this near horizon geometry is AdS{sub 3}xS{sup 2}, and three-dimensional gravity coupled to moduli fields is effectively obtained after a dimensional reduction on S{sup 2}. Constructing the Hamilton-Jacobi equation, we define the holographic RG flow from the three-dimensional gravity. The central charge of the Virasoro algebra is calculable from the conformal anomaly at the point where the beta function defined from the gravity side becomes zero. In general, we can also identify the c function of the dual two-dimensional field theory. We show that these flow equations are completely equivalent to not only BPS but also non-BPS attractor flow equations of the moduli fields. The attractor mechanism by which the values of the moduli fieldsmore » are fixed at the event horizon of the extremal black hole can be understood equivalently to the fact that the RG flows are fixed at the critical points in the dual field theory.« less

76 citations

Journal ArticleDOI
TL;DR: In this article, the authors define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time.
Abstract: We define and study projective special para-Kahler manifolds and show that they appear as target manifolds when reducing five-dimensional vector multiplets coupled to supergravity with respect to time. The dimensional reductions with respect to time and space are carried out in a uniform way using an epsilon-complex notation. We explain the relation of our formalism to other formalisms of special geometry used in the literature. In the second part of the paper we investigate instanton solutions and their dimensional lifting to black holes. We show that the instanton action, which can be defined after dualising axions into tensor fields, agrees with the ADM mass of the corresponding black hole. The relation between actions via Wick rotation, Hodge dualisation and analytic continuation of axions is discussed.

51 citations

Journal ArticleDOI
TL;DR: In this paper, a systematic study of the state space of non-extremal, stationary black hole bound states in four-dimensional N = 2 supergravity was initiated, and it was shown that an exponential multitude of classically stable "halo" bound states can be formed between large finite temperature D4-D0 black hole cores and much smaller, arbitrarily charged black holes at the same temperature.
Abstract: We initiate a systematic study of the state space of non-extremal, stationary black hole bound states in four-dimensional N = 2 supergravity. Specifically, we show that an exponential multitude of classically stable "halo" bound states can be formed between large finite temperature D4-D0 black hole cores and much smaller, arbitrarily charged black holes at the same temperature. We map out in full the regions of existence for thermodynamically stable and metastable bound states in terms of the core's charges and temperature, as well as the region of stability of the core itself. Several features of these systems, such as a macroscopic configurational entropy and exponential relaxation timescales, are similar to those of the extended family of glasses. We draw parallels between the two with a view toward understanding complex systems in fundamental physics.

44 citations

Journal ArticleDOI
TL;DR: The authors generalize Denef's method of deriving and solving first-order equations describing multicenter extremal black holes in four-dimensional N = 2 supergravity to allow non-supersymmetric solutions.
Abstract: Using the superpotential approach we generalize Denef's method of deriving and solving first-order equations describing multicenter extremal black holes in four-dimensional N = 2 supergravity to allow non-supersymmetric solutions. We illustrate the general results with an explicit example of the stu model.

34 citations

Journal ArticleDOI
TL;DR: In this article, a class of five-dimensional Einstein-Maxwell type Lagrangians, which contains the bosonic Lagrangian of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations, was constructed.
Abstract: We find a class of five-dimensional Einstein-Maxwell type Lagrangians which contains the bosonic Lagrangians of vector multiplets as a subclass, and preserves some features of supersymmetry, namely the existence of multi-centered black hole solutions and of attractor equations. Solutions can be expressed in terms of harmonic functions through a set of algebraic equations. The geometry underlying these Lagrangians is characterized by the existence of a Hesse potential and generalizes the very special real geometry of vector multiplets. Our construction proceeds by first obtaining instanton solutions for a class of four-dimensional Euclidean sigma models, which includes those occuring for four-dimensional Euclidean N=2 vector multiplets as a subclass.

31 citations

References
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Journal ArticleDOI
TL;DR: It is shown that extremal magnetic black hole solutions of N=2 supergravity coupled to vector multiplets with a generic holomorphic prepotential F can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon.
Abstract: It is shown that extremal magnetic black hole solutions of N=2 supergravity coupled to vector multiplets ${\mathit{X}}^{\mathrm{\ensuremath{\Lambda}}}$ with a generic holomorphic prepotential F(${\mathit{X}}^{\mathrm{\ensuremath{\Lambda}}}$) can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the ${\mathit{X}}^{\mathrm{\ensuremath{\Lambda}}}$ are real, and it is seen that the logarithm of the conformal factor of the spatial metric equals the K\"ahler potential on the vector multiplet moduli space. Several examples are discussed in detail.

1,248 citations

Journal ArticleDOI
TL;DR: The results provide an explicit model-independent expression for the macroscopic Bekenstein-Hawking entropy of {ital N}=2 black holes which is manifestly duality invariant.
Abstract: We find a general principle which allows one to compute the area of the horizon of $N=2$ extremal black holes as an extremum of the central charge. One considers the ADM mass equal to the central charge as a function of electric and magnetic charges and moduli and extremizes this function in the moduli space (a minimum corresponds to a fixed point of attraction). The extremal value of the square of the central charge provides the area of the horizon, which depends only on electric and magnetic charges. The doubling of unbroken supersymmetry at the fixed point of attraction for $N=2$ black holes near the horizon is derived via conformal flatness of the Bertotti-Robinson-type geometry. These results provide an explicit model-independent expression for the macroscopic Bekenstein-Hawking entropy of $N=2$ black holes which is manifestly duality invariant. The presence of hypermultiplets in the solution does not affect the area formula. Various examples of the general formula are displayed. We outline the attractor mechanism in $N=4, 8$ super-symmetries and the relation to the $N=2$ case. The entropy-area formula in five dimensions, recently discussed in the literature, is also seen to be obtained by extremizing the $5d$ central charge.

955 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric, and they showed that extremal value of the central charge provides the minimal value of BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point.

744 citations

Journal ArticleDOI
TL;DR: It is suggested that the universal duality symmetric formula for the energy of the ground state in supersymmetric gravity is given by the modulus of the maximal central charge at the attractor point in any supers asymmetric theory in any dimension.
Abstract: The macroscopic entropy-area formula for supersymmetric black holes in {ital N}=2,4,8 theories is found to be universal: in {ital d}=4 it is always given by the square of the largest of the central charges extremized in moduli space. The proof of universality is based on the fact that the doubling of unbroken supersymmetry near the black hole horizon requires that all central charges other than {ital Z}={ital M} vanish at the attractor point for {ital N}=4,8. The ADM mass at the extremum can be computed in terms of duality symmetric quartic invariants which are moduli independent. The extension of these results for {ital d}=5, {ital N}=1,2,4 is also reported. A duality symmetric expression for the energy of the ground state with spontaneous breaking of supersymmetry is provided by the power 1/2 (2/3) of the black hole area of the horizon in {ital d}=4 ({ital d}=5). It is suggested that the universal duality symmetric formula for the energy of the ground state in supersymmetric gravity is given by the modulus of the maximal central charge at the attractor point in any supersymmetric theory in any dimension. {copyright} {ital 1996 The American Physical Society.}

629 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider generalizations in 4 dimensions of the Einstein-Maxwell equations which typically arise from Kaluza-Klein theories and specify conditions such that stationary solutions lead to non-linearσ-models for symmetric spaces.
Abstract: In this paper we consider generalizations in 4 dimensions of the Einstein-Maxwell equations which typically arise from Kaluza-Klein theories. We specify conditions such that stationary solutions lead to non-linearσ-models for symmetric spaces. Using both this group theoretic structure and some properties of harmonic maps we are able to generalize many of the known existence and uniqueness theorems for black holes in Einstein-Maxwell theory to this more general setting.

536 citations