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.

We derive a variety of exact black hole solutions in a subclass of Horndeski's scalartensor theory possessing shift symmetry, φ → φ + c, and reflection symmetry, φ → −φ. The theory admits two arbitrary functions of X := −(∂φ) 2 /2, and our solutions are constructed without specifying the concrete form of the two functions, implying that black hole solutions in specific scalar-tensor theories found in the literature can be extended to a more general class of theories with shift symmetry. Our solutions include a black hole in the presence of an effective cosmological constant, the Nariai spacetime, the Lifshitz black hole, and other nontrivial solutions, all of which exhibit nonconstant scalar-field profile.

We study cosmological perturbations in bimetric theory with two fluids each of which is coupled to one of the two metrics. Focusing on a healthy branch of background solutions, we clarify the stability of the cosmological perturbations. For this purpose, we extend the condition for the absence of the so-called Higuchi ghost, and show that the condition is guaranteed to be satisfied on the healthy branch. We also calculate the squared propagation speeds of perturbations and derive the conditions for the absence of the gradient instability. To avoid the gradient instability, we find that the model parameters are weakly constrained. I. INTRODUCTIONOur universe consists of various particles with different spins and masses. The graviton, the spin-2 particle mediating the gravitational force, is of special interest since gravity is the least understood among fundamental forces in nature. Assuming Lorentz invariance, Weinberg's theorem in 1964 [1] and its extensions [2, 3] exclude more than one interacting massless gravitons in four-dimensional Minkowski spacetime. Those theorems, however, do not exclude massive graviton(s) interacting with a massless graviton.In the present paper we consider a bimetric theory of gravity, i.e. a physical setup involving two dynamical metrics interacting with each other. In this setup, after diagonalizing the mass matrix for metric perturbations around Minkowski background, we end up with a massive graviton and a massless graviton, in accord with the above mentioned general theorems. Bimetric theory thus propagates seven physical degrees of freedom in Minkowski background: five from the massive graviton and two from the massless graviton. Until recently, however, it was thought that nonlinear extension of massive gravity inevitably would have involved a sixth degree of freedom (eighth degree of freedom in nonlinear bimetric theory, e.g. [4]), which would have been a ghost [5]. This ghost degree of freedom, called Boulware-Deser (BD) ghost, was recently excised in the construction of a massive gravity theory by de Rham, Gabadadze and Tolley (dRGT) [6,7] at fully nonlinear level [8,9]. A simple extension of the dRGT massive gravity allows the construction of a fully nonlinear bimetric theory of gravity without the (would-be) BD ghost [10]. It is thus this formulation that the studies of bimetric theory in the present paper are based on.Having a promising candidate for theoretically consistent bimetric theory, it is important to investigate whether it can accommodate viable cosmology. Before starting the study of cosmology in bimetric theory, however, let us briefly review the current status of cosmology in dRGT massive gravity and its extensions.In the covariant formulation of dRGT massive gravity, the basic quantities in the gravity sector are a metric field and four scalar fields called Stückelberg fields. The original dRGT theory respects the Poincaré symmetry in the space of Stückelberg fields so that the Stückelberg fields enter the action only through the so-called fiducial metric, w...

In Lovelock theories, gravity can travel faster or slower than light. The causal structure is determined by the characteristic hypersurfaces. We generalise a recent result of Izumi to prove that any Killing horizon is a characteristic hypersurface for all gravitational degrees of freedom of a Lovelock theory. Hence gravitational signals cannot escape from the region inside such a horizon. We investigate the hyperbolicity of Lovelock theories by determining the characteristic hypersurfaces for various backgrounds. First we consider Ricci flat type N spacetimes. We show that characteristic hypersurfaces are generically all non-null and that Lovelock theories are hyperbolic in any such spacetime. Next we consider static, maximally symmetric black hole solutions of Lovelock theories. Again, characteristic surfaces are generically non-null. For some small black holes, hyperbolicity is violated near the horizon. This implies that the stability of such black holes is not a well-posed problem.

A massless scalar field exhibits an instability at the event horizon of an extreme black hole. We study numerically the nonlinear evolution of this instability for spherically symmetric perturbations of an extreme Reissner-Nordstrom (RN) black hole. We find that generically the endpoint of the instability is a non-extreme RN solution. However, there exist fine-tuned initial perturbations for which the instability never decays. In this case, the perturbed spacetime describes a time-dependent extreme black hole. Such solutions settle down to extreme RN outside, but not on, the event horizon. The event horizon remains smooth but certain observers who cross it at late time experience large gradients there. Our results indicate that these dynamical extreme black holes admit a C 1 extension across an inner (Cauchy) horizon.

We study the non-equilibrium condensation process in a holographic superconductor. When the temperature T is smaller than a critical temperature T c , there are two black hole solutions, the Reissner-Nordström-AdS black hole and a black hole with a scalar hair. In the boundary theory, they can be regarded as the supercooled normal phase and the superconducting phase, respectively. We consider perturbations on supercooled Reissner-Nordström-AdS black holes and study their non-linear time evolution to know about physical phenomena associated with rapidly-cooled superconductors. We find that, for T < T c , the initial perturbations grow exponentially and, eventually, spacetimes approach the hairy black holes. We also clarify how the relaxation process from a far-from-equilibrium state proceeds in the boundary theory by observing the time dependence of the superconducting order parameter. Finally, we study the time evolution of event and apparent horizons and discuss their correspondence with the entropy of the boundary theory. Our result gives a first step toward the holographic understanding of the non-equilibrium process in superconductors.

Motion of a particle near a horizon of a spherically symmetric black hole is shown to possess a universal Lyapunov exponent of a chaos provided by its surface gravity. To probe the horizon, we introduce electromagnetic or scalar force to the particle so that it does not fall into the horizon. There appears an unstable maximum of the total potential where the evaluated maximal Lyapunov exponent is found to be independent of the external forces and the particle mass. The Lyapunov exponent is universally given by the surface gravity of the black hole. Unless there are other sources of a chaos, the Lyapunov exponent is subject to an inequality λ ≤ 2πTBH/ , which is identical to the bound recently discovered by Maldacena, Shenker and Stanford.

In Randall-Sundrum II (RS-II) braneworld model, it has been conjectured according to the AdS/CFT correspondence that brane-localized black hole (BH) larger than the bulk AdS curvature scale ℓ cannot be static, and it is dual to a four dimensional BH emitting the Hawking radiation through some quantum fields. In this scenario, the number of the quantum field species is so large that this radiation changes the orbital evolution of a BH binary. We derived the correction to the gravitational waveform phase due to this effect and estimated the upper bounds on ℓ by performing Fisher analyses. We found that DECIGO/BBO can put a stronger constraint than the current table-top result by detecting gravitational waves from small mass BH/BH and BH/neutron star (NS) binaries. Furthermore, DECIGO/BBO is expected to detect 10 5 BH/NS binaries per year. Taking this advantage, we found that DECIGO/BBO can actually measure ℓ down to ℓ = 0.33µm for 5 year observation if we know that binaries are circular a priori. This is about 40 times smaller than the upper bound obtained from the table-top experiment. On the other hand, when we take eccentricities into binary parameters, the detection limit weakens to ℓ = 1.5µm due to strong degeneracies between ℓ and eccentricities. We also derived the upper bound on ℓ from the expected detection number of extreme mass ratio inspirals (EMRIs) with LISA and BH/NS binaries with DECIGO/BBO, extending the discussion made recently by McWilliams [1]. We found that these less robust constraints are weaker than the ones from phase differences.