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Chaotic systems near black holes satisfy a universal bound, $\lambda \leq \kappa_H$ linking the Lyapunov coefficient $\lambda$ associated with unstable orbits to surface gravity $\kappa_H$ of the event horizon. A natural question is whether this bound is satisfied by unstable circular null geodesics in the vicinity of black holes. However, there are known cases where this bound is violated. It is intriguing to ask whether there exists an alternative universal bound that is valid in such situations. We show that for any spherically symmetric, static black hole that satisfies Einstein's equations and the dominant energy condition, there exist other universal bounds relating the Lyapunov coefficient to a generalized notion of surface gravity at the photon sphere, as well as to the Unruh temperature locally measured by static observers. As applications, we show how these bounds also constrain the imaginary part of quasinormal modes in the eikonal regime and how the Lyapunov coefficient relates to the shadow size and the entropy of the horizon., Comment: 8 pages. References added. Clarifications in the discussion of quasinormal modes and addition of a new bound derived from the relationship between the Lyapunov exponent and the Gaussian curvature of an associated optical metric

Choptuik's critical phenomena in general relativity is revisited in the affine-null metric formulation of Einstein's equations for a massless scalar field in spherical symmetry. Numerical solutions are obtained by evolution of initial data using pseudo-spectral methods. The underlying system consists of differential equations along the outgoing null rays which can be solved in sequential form. A new two-parameter family of initial data is presented for which these equations can be integrated analytically. Specific choices of the initial data parameters correspond to either an asymptotically flat null cone, a black hole event horizon or the singular interior of a black hole. Our main focus is on the interior features of a black hole, for which the affine-null system is especially well adapted. We present both analytic and numerical results describing the geometric properties of the apparent horizon and final singularity. Using a re-gridding technique for the affine parameter, numerical evolution of initially asymptotically flat supercritical data can be continued inside the event horizon and track the apparent horizon up to the formation of the final singularity., Comment: 14 pages, 13 figures

Using a quasi-spherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axi-symmetric vacuum spacetime. The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations. It is further verified its equivalence with the Kerr metric in the slowly rotation approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates.

The shadows of black holes encode significant information about the properties of black holes and the spacetime surrounding them. So far, the effects of dispersive media, such as plasma, and relativistic aberration on the propagation of light around compact objects have been treated separately in the literature. In this paper, we will employ the Konoplya, Stuchlik, and Zhidenko family of stationary, axially symmetric, and asymptotically flat metrics to describe the spacetime around rotating black holes. We will study how the parameters of the black hole, the chromatic effects resulting from the presence of a non-magnetized, pressureless plasma environment, and the effects of relativistic aberration of a moving observer modify the morphology of the shadow., Comment: Several improvements, new graphics, and discussions. Accepted in PRD

The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g. $r=2M$ in the Schwarzschild black hole) or to an implicit dependence of the chosen coordinate to physical relevant coordinates (e.g. the dependence of the null coordinates in the Kruskal metric). Here we discuss two approaches for coordinate choices in spherical symmetry allowing us to discuss explicitly "solitary" and spherically symmetric black holes from a regular horizon to null infinity. The first approach relies on a construction of a regular null coordinate (where regular is meant as being defined from the horizon to null infinity) given an explicit solution of the Einstein-matter equations. The second approach is based on an affine-null formulation of the Einstein equations and the respective characteristic initial value problem. In particular, we present a derivation of the Reissner-Nordstr\"om black holes expressed in terms of these regular coordinates., Comment: 17 pages

We give an overview of literature related to J\"urgen Ehlers' pioneering 1981 paper on Frame theory--a theoretical framework for the unification of General Relativity and the equations of classical Newtonian gravitation. This unification encompasses the convergence of one-parametric families of four-dimensional solutions of Einstein's equations of General Relativity to a solution of equations of a Newtonian theory if the inverse of a causality constant goes to zero. As such the corresponding light cones open up and become space-like hypersurfaces of constant absolute time on which Newtonian solutions are found as a limit of the Einsteinian ones. It is explained what it means to not consider the `standard-textbook' Newtonian theory of gravitation as a complete theory unlike Einstein's theory of gravitation. In fact, Ehlers' Frame theory brings to light a modern viewpoint in which the `standard' equations of a self-gravitating Newtonian fluid are Maxwell-type equations. The consequences of Frame theory are presented for Newtonian cosmological dust matter expressed via the spatially projected electric part of the Weyl tensor, and for the formulation of characteristic quasi-Newtonian initial data on the light cone of a Bondi-Sachs metric., Comment: 22 pages, 1 figure. Matches published version in the 'Golden Oldie' section of the Journal of General Relativity and Gravitation, Vol. 51

We discuss a recently presented boosted Kerr black hole solution which had already been used by other authors. This boosted metric is based on wrong assumptions regarding asymptotic inertial observers and moreover the performed boost is not a proper Lorentz transformation. This note aims to clarify some of the issues when boosting black holes and the necessary care in order to interpret them. As it is wrongly claimed that the presented boosted Kerr metric is of Bondi-Sachs type, we recall out some of the necessary requirements and difficulties, when the casting the Kerr metric into a metric with a surface forming null coordinate.

The Minkowski background intrinsic to the Kerr-Schild version of the Kerr metric provides a definition of a boosted spinning black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principal null directions. The two corresponding Minkowski backgrounds and their associated boosts differ drastically. This has an important implication for the gravitational memory effect. A prior analysis of the transition of a non-spinning Schwarzschild black hole to a boosted state showed that the memory effect in the nonlinear regime agrees with the linearised result based upon the retarded Green function only if the final velocity corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity. We show that this results extends to the transition of a Kerr black hole to a boosted state and apply it to set upper and lower bounds for the boost memory effect resulting from the collision of two spinning black holes., Comment: 17 pages, revised discussion section

We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equations is solved hierarchically on the null hypersurface serving as a boundary. Second, with the obtained boundary values, a set of differential equations, similar to the equations of the Bondi-Sachs formalism, comprising of hypersurface and evolution equations is solved hierarchically to find the entire space-time metric. An example is shown how the double null Israel black hole solution arises after specification to spherical symmetry and vacuum. This black hole solution is then discussed to with respect to Penrose conformal compactification of spacetime., Comment: 28 pages, matches (up to reference listing) published version

The Kerr-Schild version of the Schwarzschild metric contains a Minkowski background which provides a definition of a boosted black hole. There are two Kerr-Schild versions corresponding to ingoing or outgoing principle null directions. We show that the two corresponding Minkowski backgrounds and their associated boosts have an unexpected difference. We analyze this difference and discuss the implications in the nonlinear regime for the gravitational memory effect resulting from the ejection of massive particles from an isolated system. We show that the nonlinear effect agrees with the linearized result based upon the retarded Green function only if the velocity of the ejected particle corresponds to a boost symmetry of the ingoing Minkowski background. A boost with respect to the outgoing Minkowski background is inconsistent with the absence of ingoing radiation from past null infinity., Comment: 13 pages, matches published version

We investigate gravitational radiation memory and its corresponding effect on the asymptotic symmetries of a body whose exterior is a boosted Schwarzschild spacetime. First, in the context of linearized theory, we consider such a Schwarzschild body which is initially at rest, then goes through a radiative stage and finally emerges as a boosted Schwarzschild body. We show that the proper retarded solution of the exterior Schwarzschild spacetime for this process can be described in terms of the ingoing Kerr-Schild form of the Schwarzschild metric for both the initial and final states. An outgoing Kerr-Schild or time symmetric metric does not give the proper solution. The special property of Kerr-Schild metrics that their linearized and nonlinear forms are identical allows us to extend this result to processes in the nonlinear regime. We then discuss how the nonlinear memory effect, and its associated supertranslation, affect angular momentum conservation. Our approach provides a new framework for studying nonlinear aspects of the memory effect., Comment: matches published version

The Bondi-Sachs formalism of General Relativity is a metric-based treatment of the Einstein equations in which the coordinates are adapted to the null geodesics of the spacetime. It provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by a mass loss from the system. The asymptotic behaviour of the Bondi-Sachs metric revealed the existence of the symmetry group at null infinity, the Bondi-Metzner-Sachs group, which turned out to be larger than the Poincare group., Comment: Scholarpedia review article, an up-to-date version can be found under http://scholarpedia.org/article/Bondi-Sachs_Formalism, includes general normalisation of unit sphere dyad, corrected typos, additional references

Since the release of the Hipparcos catalog in 1997, the distance to the Pleiades open cluster has been heavily debated. The distance obtained from Hipparcos and those by alternative methods differ by 10 to 15 %. As accurate stellar distances are key to understanding stellar structure and evolution, this dilemma puts the validity of stellar evolution models into question. Using our model-independent method to determine parallaxes based on twin stars, we report individual parallaxes of 15 FGK type stars in the Pleiades in anticipation of the astrometric mission Gaia. These parallaxes give a mean cluster parallax of $7.42\pm0.09~\mathrm{mas}$ corresponding to a mean cluster distance of ${ 134.8\pm 1.7~\mathrm{pc}}$. This value agrees with the current results obtained from stellar evolution models., Comment: refereed version, accepted for publication in A&A

The gravitational memory effect leads to a net displacement in the relative positions of test particles. This memory is related to the change in the strain of the gravitational radiation field between infinite past and infinite future retarded times. There are three known sources of the memory effect: (i) the loss of energy to future null infinity by massless fields or particles, (ii) the ejection of massive particles to infinity from a bound system and (iii) homogeneous, source-free gravitational waves. In the context of linearized theory, we show that asymptotic conditions controlling these known sources of the gravitational memory effect rule out any other possible sources with physically reasonable stress-energy tensors. Except for the source-free gravitational waves, the two other known sources produce gravitational memory with E-mode radiation strain, characterized by a certain curl-free sky pattern of their polarization. Thus our results show that the only known source of B-mode gravitational memory is of primordial origin, corresponding in the linearized theory to a homogeneous wave entering from past null infinity., Comment: typos corrected, matches published version

We report the discovery of BD+20594b, a high density sub-Neptune exoplanet, made using photometry from Campaign 4 of the two-wheeled Kepler (K2) mission, ground-based radial velocity follow-up from HARPS and high resolution lucky and adaptive optics imaging obtained using AstraLux and MagAO, respectively. The host star is a bright ($V=11.04$, $K_s = 9.37$), slightly metal poor ([Fe/H]$=-0.15\pm 0.05$ dex) solar analogue located at $152.1^{+9.7}_{-7.4}$ pc from Earth, for which we find a radius of $R_*=0.928^{+0.055}_{-0.040}R_\odot$ and a mass of $M_* = 0.961^{+0.032}_{-0.029}M_\odot$. A joint analysis of the K2 photometry and HARPS radial velocities reveal that the planet is in a $\approx 42$ day orbit around its host star, has a radius of $2.23^{+0.14}_{-0.11}R_\oplus$, and a mass of $16.3^{+6.0}_{-6.1}M_\oplus$. Although the data at hand puts the planet in the region of the mass-radius diagram where we could expect planets with a pure rock (i.e. magnesium silicate) composition using two-layer models (i.e., between rock/iron and rock/ice compositions), we discuss more realistic three-layer composition models which can explain the high density of the discovered exoplanet. The fact that the planet lies in the boundary between "possibly rocky" and "non-rocky" exoplanets, makes it an interesting planet for future RV follow-up., Comment: 12 pages, 11 figures. Accepted for publication in ApJ

Perturbations of the linearized vacuum Einstein equations on a null cone in the Bondi-Sachs formulation of General Relativity can be derived from a single master function with spin weight two, which is related to the Weyl scalar \Psi_0, and which is determined by a simple wave equation. Utilizing a standard spin representation of the tensors on a sphere and two different approaches to solve the master equation, we are able to determine two simple and explicitly time-dependent solutions. Both solutions, of which one is asymptotically flat, comply with the regularity conditions at the vertex of the null cone. For the asymptotically flat solution we calculate the corresponding linearized perturbations, describing all multipoles of spin-2 waves that propagate on a Minkowskian background spacetime. We also analyze the asymptotic behavior of this solution at null infinity using a Penrose compactification, and calculate the Weyl scalar, \Psi_4. Because of its simplicity, the asymptotically flat solution presented here is ideally suited for testbed calculations in the Bondi-Sachs formulation of numerical relativity. It may be considered as a sibling of the well-known Teukolsky-Rinne solutions, on spacelike hypersurfaces, for a metric adapted to null hypersurfaces., Comment: accepted for publication in Phys. Rev. D

In the Bondi-Sachs formulation of General Relativity space-time is foliated via a family of null cones. If these null cones are defined such that their vertices are traced by a regular world-line then the metric tensor has to obey regularity conditions at the vertices. We explore these regularity conditions when the world line is a time-like geodesic. In particular, we solve the Einstein equations for the Bondi-Sachs metric near the vertices for axially symmetric vacuum space- times. The metric is calculated up to third order corrections with respect to a flat metric along the time-like geodesic, as this is the lowest order where non- linear coupling of the metric coefficients occurs. We also determine the boundary conditions of the metric to arbitrary order of these corrections when a linearized and axially symmetric vacuum space-time is assumed. In both cases we find that (i) the initial data on the null cone must have a very rigid angular structure for the vertex to be a regular point, and (ii) the initial data are determined by functions depending only on the time of a geodesic observer tracing the vertex. The latter functions can be prescribed freely, but if the vertex is assumed to be regular they must be finite and have finite derivatives along the geodesic., Comment: 31 pages, accepted for publication in CQG