Signatures of the sources in the gravitational waves of a perturbed Schwarzschild black hole

The explicit form of perturbation equation for the Ψ₄ Weyl scalar, containing the matter source terms, is derived for general type D spacetimes. It is described in detail the particular case of the Schwarzschild spacetime using in-going penetrating coordinates. As a practical application, we focused on the emission of gravitational waves when a black hole is perturbed by a surrounding dust-like fluid matter. The symmetries of the spacetime and the simplicity of the matter source allow, by means of a spherical harmonic decomposition, to study the problem by means of a one dimensional numerical code.     

Geometric analysis of particular compactly constructed time machine spacetimes

We formulate the concept of time machine structure for spacetimes exhibiting a compactly constructed region with closed timelike curves. After reviewing essential properties of the pseudo Schwarzschild spacetime introduced by Ori, we present an analysis of its geodesics analogous to the one conducted in the case of the Schwarzschild spacetime. We conclude that the pseudo Schwarzschild spacetime is geodesically incomplete and not extendible to a complete spacetime. We then introduce a rotating generalization of the pseudo Schwarzschild metric, which we call the pseudo Kerr spacetime. We establish its time machine structure and analyze its global properties.     

Evolution of massive fields around a black hole in Hořava gravity

We study the evolution of massive scalar field in the spacetime geometry of Kehagias-Sfetsos black hole in deformed Hořava-Lifshitz (HL) gravity by numerical analysis. We find that the signature of HL theory is encoded in the quasinormal mode (QNM) phase of the evolution of field. The QNM phase in the evolution process lasts for a longer time in HL theory. QNMs involved in the evolution of massive field are calculated and find that they have a higher oscillation frequency and a lower damping rate than the Schwarzschild spacetime case. We also study the relaxation of field in the intermediate and asymptotic range and verified that behaviors of field in these phases are independent of the HL parameter and is identical to the Schwarzschild case.     

Stability problem in Rindler spacetime

The stability problem of the Rindler spacetime is carefully studies by using the scalar wave perturbation. Using two different coordinate systems, the scalar wave equation is investigated. The results are different in the two cases. They are analysed and compared with each other in detail. The following conclusions are obtained: (a) the Rindler spacetime as a whole is not stable; (b) the Rindler spacetime can exist stably only as part of the Minkowski spacetime, and the Minkowski spacetime can be a real entity independently; (c) there are some defects for the scalar wave equation written by the Rindler coordinates, and it is unsuitable for the investigation of the stability properties of the Rindler spacetime. All these results may shed some light on the stability properties of the Schwarzschild black hole. It is natural and reasonable for one to infer that: (a) perhaps the Regge--Wheeler equation is not sufficient to determine the stable properties; (b) the Schwarzschild black hole as a whole might be really unstable; (c) the Kruskal spacetime is stable and can exist as a real physical entity; whereas the Schwarzschild black hole can occur only as part of the Kruskal spacetime.     

On limit behavior of quasi-local mass for ellipsoids at spatial infinity

We discuss the spatial limit of the quasi-local mass for certain ellipsoids in an asymptotically flat static spherically symmetric spacetime. These ellipsoids are not nearly round but they are of interest as an admissible parametrized foliation defining the Arnowitt–Deser–Misner mass. The Hawking mass of this family of ellipsoids tends to-∞. In contrast, we show that the Hayward mass converges to a finite value. Moreover, a positive mass type theorem is established. The limit of the mass has a uniform positive lower bound no matter how oblate these ellipsoids are. This result could be extended for asymptotically Schwarzschild manifolds. And numerical simulation in the Schwarzschild spacetime illustrates that the Hayward mass is monotonically increasing near infinity.     

The ringing of quantum corrected Schwarzschild black hole with GUP

Schwarzschild black holes with quantum corrections are studied under scalar field perturbations and electromagnetic field perturbations to analyze the effect of the correction term on the potential function and quasinormal mode (QNM). In classical general relativity, spacetime is continuous and there is no existence of the so-called minimal length. The introduction of the correction items of the generalized uncertainty principle, the parameter β, can change the singularity structure of the black hole gauge and may lead to discretization in time and space. We apply the sixth-order WKB method to approximate the QNM of Schwarzschild black holes with quantum corrections and perform numerical analysis to derive the results of the method. Also, we find that the effective potential and QNM in scalar fields are larger than those in electromagnetic fields.     

Reducing the calculation workload of the Green function for electromagnetic scattering in a Schwarzschild gravitational field

When the finite difference time domain(FDTD) method is used to solve electromagnetic scattering problems in Schwarzschild space-time, the Green functions linking source/observer to every surface element on connection/output boundary must be calculated.When the scatterer is electrically extended, a huge amount of calculation is required due to a large number of surface elements on the connection/output boundary.In this paper, a method for reducing the calculation workload of Green function is proposed.The Taylor approximation is applied for the calculation of Green function.New transport equations are deduced.The numerical results verify the effectiveness of this method.     

Fermi coordinates in Schwarzschild spacetime: closed form expressions

Fermi coordinates are constructed as exact functions of the Schwarzschild coordinates around the world line of a static observer in the equatorial plane of the Schwarzschild spacetime modulo a single impact parameter determined implicitly as a function of the latter coordinates. This illustrates the difficulty of constructing explicit exact Fermi coordinates even along simple world lines in highly symmetric spacetimes.     

The energy-momentum tensor near a black hole

The energy-momentum tensor for a conformally invariant scalar field near a Schwarzschild black hole in thermal equilibrium with radiation is found by a combination of analytical and numerical techniques. Calculations are performed in the Euclidean section of the spacetime, and divergences isolated using the heat kernel expansion. It is found that the results agree well with those of Candelas [1], but that there are significant differences from the Gaussian approximation of Page [2].     

HORIZONS AND GENERATING FUNCTIONALS FOR THE TEMPERATURE GREEN FUNCTIONS

The generating functional approach to Green functions in the thermal equilibrium is used to explore the geometrical origin of the temperatures of the quantum fields in the Rindler space-time and black hole spacetimes. It is shown that under the transformation from Minkowski space to the Rindler space the path integral representation for the Euclidean generating functionals of Green functions at zero temperature would transform into the corresponding ones of the quantum fields at a certain finite temperature, and the Minkowski vacuum state would have the same properties as that of the quantum mixed state at the same temperatfire. All thermal Green functions for the mixed state are given. Similar results would be obtained for the Schwarzschild, the Reissner-NordstrOm and the Kerr black holes and whereupon the Hawking temperature for the black holes would have geometrical origin as well as that in the Rindler spacetime. The various density operators of the mixed states at the Hawking temperature for the black hole sacetimes are specified.     

APPROXIMATE PARTIAL NOETHER OPERATORS OF THE SCHWARZSCHILD SPACETIME

The objective of this paper is twofold: (a) to find a natural example of a perturbed Lagrangian that has different partial Noether operators with symmetries different from those of the underlying Lagrangian. First we regard the Schwarzschild spacetime as a perturbation of the Minkowski spacetime and investigate the approximate partial Noether operators for this perturbed spacetime. It is shown that the Minkowski spacetime has 12 partial Noether operators, 10 of which are different from the 17 Noether symmetries for this spacetime. It is found that for the perturbed Schwarzschild spacetime we recover the exact partial Noether operators as trivial first-order approximate partial Noether operators and there is no non-trivial approximate partial Noether operator as for the Noether case. As a consequence we state a conjecture. (b) Then we prove a conjecture that the approximate symmetries of a perturbed Lagrangian form a subalgebra of the approximate symmetries of the corresponding perturbed Euler–Lagrange equations and illustrate it by our examples. This is in contrast to approximate partial Noether operators.     

Bondian frames to couple matter with radiation

A study is presented for the non linear evolution of a self gravitating distribution of matter coupled to a massless scalar field. The characteristic formulation for numerical relativity is used to follow the evolution by a sequence of light cones open to the future. Bondian frames are used to endow physical meaning to the matter variables and to the massless scalar field. Asymptotic approaches to the origin and to infinity are achieved; at the boundary surface interior and exterior solutions are matched guaranteeing the Darmois–Lichnerowicz conditions. To show how the scheme works some numerical models are discussed. We exemplify evolving scalar waves on the following fixed backgrounds: (a) an atmosphere between the boundary surface of an incompressible mixtured fluid and infinity; (b) a polytropic distribution matched to a Schwarzschild exterior; (c) a Schwarzschild–Schwarzschild spacetime. The conservation of energy, the Newman–Penrose constant preservation and other expected features are observed.     

Spinning Strings,Black Holes and Stable Closed Timelike Geodesics

The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics.     

MALBEC: a new CUDA-C ray-tracer in general relativity

A new CUDA-C code for tracing orbits around non-charged black holes is presented. This code, named MALBEC, take advantage of the graphic processing units and the CUDA platform for tracking null and timelike test particles in Schwarzschild and Kerr. Also, a new general set of equations that describe the closed circular orbits of any timelike test particle in the equatorial plane is derived. These equations are extremely important in order to compare the analytical behavior of the orbits with the numerical results and verify the correct implementation of the Runge–Kutta algorithm in MALBEC. Finally, other numerical tests are performed, demonstrating that MALBEC is able to reproduce some well-known results in these metrics in a faster and more efficient way than a conventional CPU implementation.     

Cosmological black holes: A black hole in the Einstein-de Sitter universe

As an example of a dynamical cosmological black hole, a spacetime that describes an expanding black hole in the asymptotic background of the Einstein-de Sitter universe is constructed. The black hole is primordial in the sense that it forms ab initio with the big bang singularity and its expanding event horizon is represented by a conformal Killing horizon. The metric representing the black hole spacetime is obtained by applying a time dependent conformal transformation on the Schwarzschild metric, such that the result is an exact solution with a matter content described by a two-fluid source. Physical quantities such as the surface gravity and other effects like perihelion precession, light bending and circular orbits are studied in this spacetime and compared to their counterparts in the gravitational field of the isolated Schwarzschild black hole. No changes in the structure of null geodesics are recorded, but significant differences are obtained for timelike geodesics, particularly an increase in the perihelion precession and the non-existence of circular timelike orbits. The solution is expressed in the Newman-Penrose formalism.     

Cutoff anti-de Sitter space/conformal-field-theory duality and the quest for braneworld black holes

Significant evidence is presented in favor of the holographic conjecture that "4D black holes localized on the brane found by solving the classical bulk equations in AdS5 are quantum corrected black holes and not classical ones." The quantum correction to the Newtonian potential is computed using a numerical computation of in Schwarzschild spacetime for matter fields in the zero-temperature Boulware vacuum state. For the conformally invariant scalar field the leading order term is equivalent to that previously obtained in the weak-field approximation using Feynman diagrams and which has been shown to be equivalent, via the anti-de Sitter space/conformal-field-theory (AdS/CFT) duality, to the analogous calculation in Randall-Sundrum braneworlds. The 4D backreaction equations are used to make a prediction about the existence and the possible spacetime structure of macroscopic static braneworld black holes.     

Structure of spacetime tangent bundle

The universal upper limit on attainable proper acceleration relative to the vacuum imposes restrictions on possible structures in the spacetime tangent bundle. Various features of the differential geometry of the spacetime tangent bundle are presented here. Also, a modified Schwarzschild solution is obtained, and the associated gravitational red shift is calculated.     

Behavior of a spin-1/2 massive charged particle in Schwarzschild immersed in an electromagnetic universe

The Dirac equation is considered in a spacetime that represents a Schwarzschild metric coupled to a uniform external electromagnetic field. Due to the presence of electromagnetic field from the surroundings, the interaction with the spin-1/2 massive charged particle is considered. The equations of the spin-1/2 massive charged particle are separated into radial and angular equations by adopting the Newman–Penrose formalism. The angular equations obtained are similar to the Schwarzschild geometry. For the radial equations we manage to obtain the one dimensional Schrödinger-type wave equations with effective potentials. Finally, we study the behavior of the potentials by plotting them as a function of radial distance and expose the effect of the external parameter, charge and the frequency of the particle on them.     

Dimension in a Radiative Stellar Atmosphere

Dimensional scales are examined in an extended 3 + 1 Vaidya atmosphere surrounding a Schwarzschild source. At one scale, the Vaidya null fluid vanishes and the spacetime contains only a single spherical 2-surface. Both of these behaviors can be addressed by including higher dimensions in the spacetime metric.     

A Complete Foliation of Schwarzschild Spacetime by Free Falling Hypersurfaces

Free falling hypersurfaces in the Schwarzschild geometry have been studied to provide a complete foliation of spacetime. The hypersurfaces do not cross into the maximally extended spacetime and are well behaved everywhere except at the singularity r =0 the mean extrinsic curvature becomes infinity.