It is argued that the general four-dimensional extremal Kerr-Newman-AdS-dS black hole is holographically dual to a (chiral half of a) two-dimensional CFT, generalizing an argument given recently for the special case of extremal Kerr. Specifically, the asymptotic symmetries of the near-horizon region of the general extremal black hole are shown to be generated by a Virasoro algebra. Semiclassical formulae are derived for the central charge and temperature of the dual CFT as functions of the cosmological constant, Newton's constant and the black hole charges and spin. We then show, assuming the Cardy formula, that the microscopic entropy of the dual CFT precisely reproduces the macroscopic Bekenstein-Hawking area law. This CFT description becomes singular in the extreme Reissner-Nordstrom limit where the black hole has no spin. At this point a second dual CFT description is proposed in which the global part of the U (1) gauge symmetry is promoted to a Virasoro algebra. This second description is also found to reproduce the area law. Various further generalizations including higher dimensions are discussed. 1

Aretakis has proved that a massless scalar field has an instability at the horizon of an extreme Reissner-Nordström black hole. We show that a similar instability occurs also for a massive scalar field and for coupled linearized gravitational and electromagnetic perturbations. We present numerical results for the late time behaviour of massless and massive scalar fields in the extreme RN background and show that instabilities are present for initial perturbations supported outside the horizon, e.g. an ingoing wavepacket. For a massless scalar we show that the numerical results for the late time behaviour are reproduced by an analytic calculation in the near-horizon geometry. We relate Aretakis' conserved quantities at the future horizon to the Newman-Penrose conserved quantities at future null infinity.

The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

Numerical studies of black hole greybody factors indicate that Hawking emission from a highly rotating black hole is strongly spin dependent, with particles of highest spin (gravitons) dominating the energy spectrum. So far, there has been no analytic explanation or description of this effect. Using "gravitomagnetism", or the formal analogy between the Maxwell's field equations for electro-magnetism and Einstein's equations for gravity, we were able to establish a link between the spin of the rotating black hole and spin of an emitted particle. Namely, the intrinsic spin of the particle creates a "mass dipole moment" which interacts with external gravitomagnetic field whose source is the rotation of the black hole. We showed that a rotating black hole prefers to shed its spin, i.e. tends to emit particles with the spin parallel to its own. We also showed that the probability for emission grows with the increasing spin of the emitted particles. The amplification factors can be huge if a black hole is highly rotating, i.e. close to extremal. When applied to central galactic black holes, the same physical mechanism indicate that particles orbiting around these black holes should have spins strongly correlated with the spin of the black hole, which may have implications for cosmic rays believed to be coming from these regions of space.