David Kubiznak

Reentrant phase transitions in rotating anti–de Sitter black holes

Introduction. In view of the AdS/CFT correspondence, phase transitions in asymptotically AdS black holes allow for a dual interpretation in the thermal conformal field theory (CFT) living on the AdS boundary—the principal example being the well known radiation/Schwarzschild-AdS black hole Hawking–Page transition [1] which can be interpreted as a confinement/deconfinement phase transition in the dual quark gluon plasma [2]. Charged [3–6] and rotating [7, 8] asymptotically AdS back holes possess an interesting feature—they allow for a first order small-blackhole/large-black-holephase (SBH/LBH) transition which is in many ways reminiscent of the liquid/gas transition of the Van der Waals fluid. This superficial analogy was recently found more intriguing [9] by considering a thermodynamic analysis in an extended phase space where the cosmological constant is identified with thermodynamic pressure and its variations are included in the first law of black hole thermodynamics. This notion emerges from geometric derivations of the Smarr formula [10] that i) imply the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime and ii) allow for a computation of the conjugate thermodynamic volume. Intensive and extensive quantities are now properly identified [9] and the SBH/LBH transition can be understood as a liquid/gas phase transition by employing Maxwell’s equal area law to the P V diagram. Coexistence lines and critical exponents are then seen to match those of a Van der Waals fluid. In this paper we report the finding of an interesting phenomena, observed previously in multicomponent fluids, e.g., [11], of black hole reentrant phase transitions (RPTs). A system undergoes an RPT if a monotonic variation of any thermodynamic quantity results in two (or more) phase transitions such that the final state is macroscopically similar to the initial state. We find for a certain range of pressures (and a given angular momentum) that a monotonic lowering of the temperature yields a large-small-large black hole transition, where we refer to the latter ‘large’ state as an intermediate black hole (IBH). This situation is accompanied by a discontinuity in the global minimum of the Gibbs free energy, referred to as a zeroth-order phase transition, a phenomenon seen in superfluidity and superconductivity [12], and recently for Born–Infeld black holes [13]. We find the RPT to be generic for all rotating AdS black holes in d � 6 dimen

Complete Integrability of Geodesic Motion in General Higher-Dimensional Rotating Black-Hole Spacetimes

We explicitly exhibit n-1 constants of motion for geodesics in the general D-dimensional Kerr-NUT-AdS rotating black hole spacetime, arising from contractions of even powers of the 2-form obtained by contracting the geodesic velocity with the dual of the contraction of the velocity with the (D-2)-dimensional Killing-Yano tensor. These constants of motion are functionally independent of each other and of the D-n+1 constants of motion that arise from the metric and the D-n = [(D+1)/2] Killing vectors, making a total of D independent constants of motion in all dimensions D. The Poisson brackets of all pairs of these D constants are zero, so geodesic motion in these spacetimes is completely integrable.

Hidden symmetries of higher-dimensional rotating black holes.

We demonstrate that the rotating black holes in an arbitrary number of dimensions and without any restrictions on their rotation parameters possess the same hidden symmetry as the four-dimensional Kerr metric. Namely, besides the spacetime symmetries generated by the Killing vectors they also admit the (antisymmetric) Killing-Yano and symmetric Killing tensors.

Killing-Yano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions

From the metric and one Killing-Yano tensor of rank D−2 in any D-dimensional spacetime with such a principal Killing-Yano tensor, we show how to generate k = [(D+1)/2] Killing-Yano tensors, of rank D−2j for all 0 ≤ j ≤ k−1, and k rank-2 Killing tensors, giving k constants of geodesic motion that are in involution. For the example of the Kerr-NUT-AdS spacetime (hep-th/0604125) with its principal Killing-Yano tensor (gr-qc/0610144), these constants and the constants from the k Killing vectors give D independent constants in involution, making the geodesic motion completely integrable (hep-th/0611083). The constants of motion are also related to the constants recently obtained in the separation of the Hamilton-Jacobi and Klein-Gordon equations (hep-th/0611245).

Separability of Hamilton-Jacobi and Klein-Gordon equations in general Kerr-NUT-AdS spacetimes

We demonstrate the separability of the Hamilton-Jacobi and scalar field equations in general higher dimensional Kerr-NUT-AdS spacetimes. No restriction on the parameters characterizing these metrics is imposed.

Stationary strings and branes in the higher-dimensional Kerr-NUT-(A)dS spacetimes

We demonstrate complete integrability of the Nambu-Goto equations for a stationary string in the general Kerr-NUT-(A)dS spacetime describing the higher-dimensional rotating black hole. The stationary string in D dimensions is generated by a 1-parameter family of Killing trajectories and the problem of finding a string configuration reduces to a problem of finding a geodesic line in an effective (D?1)-dimensional space. Resulting integrability of this geodesic problem is connected with the existence of hidden symmetries which are inherited from the black hole background. In a spacetime with p mutually commuting Killing vectors it is possible to introduce a concept of a ?-brane, that is a p-brane with the worldvolume generated by these fields and a 1-dimensional curve. We discuss integrability of such ?-branes in the Kerr-NUT-(A)dS spacetime.

Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets

In this paper we derive the most general first-order symmetry operator commuting with the Dirac operator in all dimensions and signatures. Such an operator splits into Clifford even and Clifford odd parts which are given in terms of odd Killing-Yano and even closed conformal Killing-Yano inhomogeneous forms, respectively. We study commutators of these symmetry operators and give necessary and sufficient conditions under which they remain of the first-order. In this specific setting we can introduce a KillingYano bracket, a bilinear operation acting on odd Killing-Yano and even closed conformal Killing-Yano forms, and demonstrate that it is closely related to the Schouten-Nijenhuis bracket. An important nontrivial example of vanishing Killing-Yano brackets is given by Dirac symmetry operators generated from the principal conformal Killing-Yano tensor [hep-th/0612029]. We show that among these operators one can find a complete subset of mutually commuting operators. These operators underlie separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes in all dimensions [arXiv:0711.0078].

Dirac equation in Kerr-NUT-(A)dS spacetimes: Intrinsic characterization of separability in all dimensions

DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK(Dated: April 19, 2011)We intrinsically characterize separability of the Dirac equation in Kerr-NUT-(A)dS spacetimes inall dimensions. Namely, we explicitly demonstrate that in such spacetimes there exists a completeset of first-order mutually commuting operators, one of which is the Dirac operator, that allowsfor common eigenfunctions which can be found in a separated form and correspond precisely tothe general solution of the Dirac equation found by Oota and Yasui [arXiv:0711.0078]. Since allthe operators in the set can be generated from the principal conformal Killing–Yano tensor, thisestablishes the (up to now) missing link among the existence of hidden symmetry, presence of acomplete set of commuting operators, and separability of the Dirac equation in these spacetimes.

Solving parallel transport equations in the higher-dimensional Kerr-NUT-(A)dS spacetimes

We obtain and study the equations describing the parallel transport of orthonormal frames along geodesics in a spacetime admitting a nondegenerate, principal, conformal Killing-Yano tensor h. We demonstrate that the operator F, obtained by a projection of h to a subspace orthogonal to the velocity, has in a generic case eigenspaces of dimension not greater than 2. Each of these eigenspaces is independently parallel propagated. This allows one to reduce the parallel transport equations to a set of first order, ordinary, differential equations for the angles of rotation in the 2D eigenspaces. General analysis is illustrated by studying the equations of the parallel transport in the Kerr-NUT-(A)dS metrics. Examples of three-, four-, and five-dimensional Kerr-NUT-(A)dS are considered, and it is shown that the obtained first order equations can be solved by a separation of variables.

Parallel-propagated frame along null geodesics in higher-dimensional black hole spacetimes

In [arXiv:0803.3259] the equations describing the parallel transport of orthonormal frames along timelike (spacelike) geodesics in a spacetime admitting a nondegenerate principal conformal Killing-Yano 2-form h were solved. The construction employed is based on studying the Darboux subspaces of the 2-form F obtained as a projection of h along the geodesic trajectory. In this paper we demonstrate that, although slightly modified, a similar construction is possible also in the case of null geodesics. In particular, we explicitly construct the parallel-transported frames along null geodesics in D=4, 5, 6 Kerr-NUT-(A)dS spacetimes. We further discuss the parallel transport along principal null directions in these spacetimes. Such directions coincide with the eigenvectors of the principal conformal Killing-Yano tensor. Finally, we show how to obtain a parallel-transported frame along null geodesics in the background of the 4D Plebanski-Demianski metric which admits only a conformal generalization of the Killing-Yano tensor.