The Bell-Szekeres Solution and Related Solutions of the Einstein-Maxwell Equations

A novel technique for solving some head-on collisions of plane homogeneous light-like signals in Einstein-Maxwell theory is described. The technique is a by-product of a re-examination of the fundamental Bell-Szekeres solution in this field of study. Extensions of the Bell-Szekeres collision problem to include light-like shells and gravitational waves are described and a family of solutions having geometrical and topological properties in common with the Bell-Szekeres solution is derived.Comment: 18 pages, Latex fil

The Effect of Sources on the Inner Horizon of Black Holes

Single pulse of null dust and colliding null dusts both transform a regular horizon into a space-like singularity in the space of colliding waves. The local isometry between such space-times and black holes extrapolates these results to the realm of black holes. However, inclusion of particular scalar fields instead of null dusts creates null singularities rather than space-like ones on the inner horizons of black holes.Comment: Final version to appear in PR

Second and higher-order perturbations of a spherical spacetime

The Gerlach and Sengupta (GS) formalism of coordinate-invariant, first-order, spherical and nonspherical perturbations around an arbitrary spherical spacetime is generalized to higher orders, focusing on second-order perturbation theory. The GS harmonics are generalized to an arbitrary number of indices on the unit sphere and a formula is given for their products. The formalism is optimized for its implementation in a computer algebra system, something that becomes essential in practice given the size and complexity of the equations. All evolution equations for the second-order perturbations, as well as the conservation equations for the energy-momentum tensor at this perturbation order, are given in covariant form, in Regge-Wheeler gauge.Comment: Accepted for publication in Physical Review

Stellar Pulsations excited by a scattered mass

We compute the energy spectra of the gravitational signals emitted when a mass m is scattered by the gravitational field of a star of mass M >> m. We show that, unlike black holes in similar processes, the quasi-normal modes of the star are excited, and that the amount of energy emitted in these modes depends on how close the exciting mass can get to the star.Comment: 23 pages, 6 figures, RevTe

The spatial correlations in the velocities arising from a random distribution of point vortices

This paper is devoted to a statistical analysis of the velocity fluctuations arising from a random distribution of point vortices in two-dimensional turbulence. Exact results are derived for the correlations in the velocities occurring at two points separated by an arbitrary distance. We find that the spatial correlation function decays extremely slowly with the distance. We discuss the analogy with the statistics of the gravitational field in stellar systems.Comment: 37 pages in RevTeX format (no figure); submitted to Physics of Fluid

On the stability of naked singularities

We study the linearised stability of the nakedly singular negative mass Schwarzschild solution against gravitational perturbations. There is a one parameter family of possible boundary conditions at the singularity. We give a precise criterion for stability depending on the boundary condition. We show that one particular boundary condition is physically preferred and show that the spacetime is stable with this boundary condition.Comment: 20 pages. 5 figure

Quasinormal modes of plane-symmetric anti-de Sitter black holes: a complete analysis of the gravitational perturbations

We study in detail the quasinormal modes of linear gravitational perturbations of plane-symmetric anti-de Sitter black holes. The wave equations are obtained by means of the Newman-Penrose formalism and the Chandrasekhar transformation theory. We show that oscillatory modes decay exponentially with time such that these black holes are stable against gravitational perturbations. Our numerical results show that in the large (small) black hole regime the frequencies of the ordinary quasinormal modes are proportional to the horizon radius $r_{+}$ (wave number $k$). The frequency of the purely damped mode is very close to the algebraically special frequency in the small horizon limit, and goes as $ik^{2}/3r_{+}$ in the opposite limit. This result is confirmed by an analytical method based on the power series expansion of the frequency in terms of the horizon radius. The same procedure applied to the Schwarzschild anti-de Sitter spacetime proves that the purely damped frequency goes as $i(l-1)(l+2)/3r_{+}$, where $l$ is the quantum number characterizing the angular distribution. Finally, we study the limit of high overtones and find that the frequencies become evenly spaced in this regime. The spacing of the frequency per unit horizon radius seems to be a universal quantity, in the sense that it is independent of the wave number, perturbation parity and black hole size.Comment: Added new material on the asymptotic behavior of QNM

On the stable configuration of ultra-relativistic material spheres. The solution for the extremely hot gas

During the last stage of collapse of a compact object into the horizon of events, the potential energy of its surface layer decreases to a negative value below all limits. The energy-conservation law requires an appearance of a positive-valued energy to balance the decrease. We derive the internal-state properties of the ideal gas situated in an extremely strong, ultra-relativistic gravitational field and suggest to apply our result to a compact object with the radius which is slightly larger than or equal to the Schwarzschild's gravitational radius. On the surface of the object, we find that the extreme attractivity of the gravity is accompanied with an extremely high internal, heat energy. This internal energy implies a correspondingly high pressure, the gradient of which has such a behavior that it can compete with the gravity. In a more detail, we find the equation of state in the case when the magnitude of the potential-type energy of constituting gas particles is much larger than their rest energy. This equation appears to be identical with the general-relativity condition of the equilibrium between the gravity and pressure gradient. The consequences of the identity are discussed.Comment: 12 pages (no figure, no table) Changes in 3-rd version: added an estimate of neutrino cooling and relative time-scale of the final stage of URMS collaps

Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes

It is shown that the equations governing linearized gravitational (or electromagnetic) perturbations of the near-horizon geometry of any known extreme vacuum black hole (allowing for a cosmological constant) can be Kaluza-Klein reduced to give the equation of motion of a charged scalar field in AdS_2 with an electric field. One can define an effective Breitenlohner-Freedman bound for such a field. We conjecture that if a perturbation preserves certain symmetries then a violation of this bound should imply an instability of the full black hole solution. Evidence in favour of this conjecture is provided by the extreme Kerr solution and extreme cohomogeneity-1 Myers-Perry solution. In the latter case, we predict an instability in seven or more dimensions and, in 5d, we present results for operator conformal weights assuming the existence of a CFT dual. We sketch a proof of our conjecture for scalar field perturbations.Comment: 24 pages (+ 16 pages appendices), 2 figures. v2: Corrected error in CFT operator dimensions (they are all integers). v3: Various improvements and corrections, in particular for electromagnetic perturbations. Accepted by Physical Review