Tortoise Coordinate Research Articles

Black holes, Achilles and the original tortoise coordinate

Abstract The “frozen star” picture of black holes, based on the fact that an outside observer will never see a collapsing sphere shrink sufficiently to form a black hole horizon, was a historical obstacle to the understanding of black holes, and continues to be a stumbling point for students trying to understand horizons. Promoting the discrete steps in Zeno’s original story of Achilles and the tortoise to a “Zeno time coordinate” provides a quantitative toy model that allows students to understand why the “frozen star” phenomenon does not mean that objects cannot fall into a black hole. The toy model can be used for teaching about this particular feature of black holes in an introductory setting that does not introduce the Schwarzschild metric and its tortoise coordinates.

Greybody factor of uncharged black hole in symmetric teleparallel gravity

In this study, we investigate the greybody factor of a (3+1)-dimensional black hole solution in the context of alternative f(Q) gravity with minimum scalar field coupling. We obtain the radial equation by applying the Klein–Gordon equation and transforming it into a Schrodinger wave equation using the tortoise coordinate. This transformation enables us to determine the effective potential and its graphical behavior for different values of the coupling constant and relevant physical parameters. We obtain two solutions for the radial equation corresponding to the event and cosmological horizons. To determine the greybody factor and its behavior, we combine these two solutions in the intermediate regime. The greybody factor increases with an increase in radius and coupling constant. This indicates that in alternative gravity, the larger black hole will die sooner as compared to cosmic gravity.

Greybody factors, quasi-normal modes and thermal fluctuations of quantum-corrected Schwarzschild black hole surrounded by quintessence

In this study, we examine the effects of a quintessence field-enclosed Schwarzschild black hole that has undergone quantum corrections on a number of different physical phenomena. The effective potential and its graphical features across different values of the quantum correction and quintessence field as well as relevant physical parameters are analyzed by deriving the radial equation from the Klein–Gordon equation and converting it into a Schrodinger wave equation using the tortoise coordinate. Both the event and cosmic horizons are found in the solutions to the radial equation. We provide a detailed analysis of the greybody component and its behavior by merging these solutions in the intermediate regime. By analyzing the black hole metric function, effective potential, and absorption probabilities, the study uncovers the complex relationship between black hole characteristics, quantum corrections, and the surrounding quintessence field. Investigating the BH’s quasinormal modes and thermal fluctuations provides insights into its stability as well as thermodynamic properties. The findings offer a comprehensive understanding of how the black hole interacts with its environment and responds to environmental changes, shedding light on the rich dynamics of black hole systems in the presence of quantum corrections and quintessence fields.

Generalized wormhole models within galactic halo region in torsion and matter coupling gravity formalism

In the current analysis, we investigate the possible existence of generalized Wormhole models within the galactic halo region by analyzing specific dark matter models. The observational data is checked in the framework of torsion within minimal matter coupling gravity. By using the variational approach modified versions of the field equations under the anisotropic source of matter are calculated. To discuss the specific necessary characteristics of the embedded wormhole geometry, we have analyzed the energy conditions under the effect of dark matter halos. We consider two embedded wormhole-specific shape functions in terms of the tortoise coordinate for the current study. Further, we compare different wormhole solutions for the different regional ranges depending on the involved parametric values. To check the effect of the dark matter halos, we use the observational data within the signature of the M87 galaxy and the Milky Way galaxy.

Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics.

This work offers a didactical introduction to the calculations and geometrical properties of a static, spherically symmetric spacetime foliated by hyperboloidal time surfaces. We discuss the various degrees of freedom involved, namely the height function, responsible for introducing the hyperboloidal time coordinate, and a radial compactification function. A central outcome is the expression of the Trautman-Bondi mass in terms of the hyperboloidal metric functions. Moreover, we apply this formalism to a class of wave equations commonly used in black-hole perturbation theory. Additionally, we provide a comprehensive derivation of the hyperboloidal minimal gauge, introducing two alternative approaches within this conceptual framework: the in-out and out-in strategies. Specifically, we demonstrate that the height function in the in-out strategy follows from the well-known tortoise coordinate by changing the sign of the terms that become singular at future null infinity. Similarly, for the out-in strategy, a sign change also occurs in the tortoise coordinate's regular terms. We apply the methodology to the following spacetimes: Singularity-approaching slices in Schwarzschild, higher-dimensional black holes, black hole with matter halo, and Reissner-Nordström-de Sitter. From this heuristic study, we conjecture that the out-in strategy is best adapted for black hole geometries that account for environmental or effective quantum effects. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Echoes of axial gravitational perturbations in stars of uniform density* *Supported by the National Natural Science Foundation of China (42230207) and the Brazilian Agencies Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal

This study investigates the echoes in axial gravitational perturbations in compact objects. Accordingly, we propose an alternative scheme of the finite difference method implemented in two coordinate systems, where the initial conditions are placed on the axis of the tortoise coordinate with appropriate boundary conditions that fully respect the causality. The scheme is then employed to study the temporal profiles of the quasinormal oscillations in the Schwarzschild black hole and uniform-density stars. When presented as a two-dimensional evolution profile, the resulting ringdown waveforms in the black hole metric are split into reflected and transmitted waves as the initial perturbations evolve and collide with the peak of the effective potential. Meanwhile, for compact stars, quasinormal oscillations might be characterized by echoes. Consistent with the causality arguments, the phenomenon is produced by the gravitational waves bouncing between the divergent potential at the star's center and the peak of the effective potential. The implications of the present study are also discussed herein.

Greybody factors emitted by a regular black hole in a non-minimally coupled Einstein–Yang–Mills theory

In this paper, we study the greybody factors (GFs) for fermions with different spins and bosons in the regular black hole (BH) predicted by a non-minimal Einstein–Yang–Mills (EYM) theory. We investigate the effect of magnetic charge on effective potentials and GFs. For this purpose, we consider the Dirac and Rarita–Schwinger, as well as Klein–Gordon equations. First, we study the Dirac equation in curved spacetime for massive and massless spin-1/2 fermions. We then separate the Dirac equation into sets of radial and angular equations. Using the analytical solution of the angular equation, the Schrödinger-like wave equations with potentials are derived by decoupling the radial wave equations via the tortoise coordinate. We also consider the Rarita–Schwinger equation for massless spin-3/2 fermions and derive the one-dimensional Schrödinger wave equation with gauge-invariant effective potential. For bosons, we study the Klein–Gordon equation in the regular non-minimal EYM BH. Afterward, semi-analytic methods were used to calculate the fermionic and bosonic GFs. Finally, we discuss the graphical behavior of the obtained effective potentials and bounds on the GFs. According to graphs, the GF is highly influenced by the potential’s shape, which is determined by the parameterization of the model. This is in line with quantum theory.

Fermionic and bosonic greybody factors as well as quasinormal modes for charged Taub NUT black holes

The paper studies the spinorial wave equations, namely the Dirac and the Klein–Gordon equations, as well as the greybody radiations and quasinormal modes (QNMs) of the charged Taub NUT black hole (CTNBH). To obtain fermionic greybody factors (GFs) and QNMs, we study the charged fermions by employing the Dirac equation. To this end, we use a null tetrad in the Newman–Penrose formalism. Then, we separate the Dirac equation into radial and angular sets. Using the obtained radial equations, we convert them into the typical one dimensional Schrödinger-like wave equations with the aid of tortoise coordinate and derive the effective potentials. For bosonic GFs and QNMs, we study the Klein–Gordon equation in the CTNBH geometry and obtain the radial equation. We then derive the effective potential and investigate the effect of NUT parameter on it. We show that while the fermionic QNMs and GFs individually increase with the increasing NUT parameter, the increase of bosonic GFs with increasing NUT parameter is overwhelmingly greater than that of the bosonic QNMs.

Global conformal diagram of a black hole with angular momentum

We present the first explicit global conformal diagram of Kerr spacetime and discuss some implications on the causal structure. For this construction we use a new double null coordinate system for Kerr spacetime, which we have recently presented. These null coordinates are smooth everywhere and are naturally adapted to the horizons and to the null infinities. In this setting there naturally appears a family of spheres that are parameterized by r s , which are the intersections of both null coordinates, and r s can be thought of as the extension of the tortoise coordinate for the Kerr spacetime.

On the Possibility of Generalized Wormhole Formation in the Galactic Halo Due to Dark Matter Using the Observational Data within the Matter Coupling Gravity Formalism

In this article, we find the possibility of generalized wormhole formation in the galactic halo due to dark matter using observational data within the matter coupling gravity formalism. Keeping this as a target, we specifically employ f(R, T) gravity with (i) a variational approach concerning the metric, and (ii) the anisotropic source of matter. To understand the features of the wormholes, we thoroughly calculated and analyzed the energy conditions under f(R, T) gravity. We discuss the second embedded wormhole solution, known as the generalized Ellis–Bronnikov spacetime (ultrastatic wormhole model), in terms of the tortoise coordinate. Thereafter we generate and compare different wormhole solutions depending on the parametric values. In the second part of our investigation, we presented dark matter halos and provided interesting features by considering a couple of profiles. For the dark matter halos models, we particularly use the observational data of the M87 galaxy and the Milky Way galaxy.

Scalar Particles around a Rindler–Schwarzschild Wormhole

In this paper, we study quantum relativistic features of a scalar field around the Rindler–Schwarzschild wormhole. First, we introduce this new class of spacetime, investigating some energy conditions and verifying their violation in a region nearby the wormhole throat, which means that the object must have an exotic energy in order to prevent its collapse. Then, we study the behavior of the massless scalar field in this spacetime and compute the effective potential by means of tortoise coordinates. We show that such a potential is attractive close to the throat and that it is traversable via quantum tunneling by massive particles with sufficiently low energies. The solution of the Klein–Gordon equation is obtained subsequently, showing that the energy spectrum of the field is subject to a constraint, which induces a decreasing oscillatory behavior. By imposing Dirichlet boundary conditions on a spherical shell in the neighborhood of the throat we can determine the particle energy levels, and we use this spectrum to calculate the quantum revival of the eigenstates. Finally, we compute the Casimir energy associated with the massless scalar field at zero temperature. We perform this calculation by means of the sum of the modes method. The zero-point energy is regularized using the Epstein–Hurwitz zeta-function. We also obtain an analytical expression for the Casimir force acting on the shell.

Quantum time dilation in the near-horizon region of a black hole

In this work, we obtain a relation for average quantum time dilation between two clocks A and B in the near-horizon region of a black hole supported by the Rindler metric and conformal tortoise coordinate. It is indicated that this relation is identified with time dilation in classical and flat background limits.

Revisiting superradiant stability of Kerr–Newman black holes under a charged massive scalar

We revisit the superradiant stability of Kerr–Newman black holes under a charged massive scalar perturbation. We obtain a newly suitable potential which is not singular at the outer horizon when a radial equation is expressed the Schrödinger-type equation in terms of the tortoise coordinate. From the potential analysis, we find a condition for the superradiant stability of Kerr–Newman black holes.

2-2-holes simplified

Quadratic gravity illustrates how a replacement for black holes can emerge from a UV completion of gravity. 2-2-holes are extremely compact horizonless objects with an entropy S22 due to trapped normal matter, and in this way they are conceptually easy to understand. But the field equations are cumbersome and the numerical analysis has so far been restricted to relatively small size solutions. Here we show how the properties of arbitrarily large 2-2-holes can be found, including the time delay for gravitational wave echoes and the result T∞S22=M/2. The starting point is to formulate the metric in terms of the tortoise coordinate, and to have one of the two metric functions be a conformal factor. A large conformally-related volume becomes associated with the interior of a 2-2-hole. We also discuss implications for the weak gravity conjecture.

New Chandrasekhar transformation in Kerr spacetime

We construct a new type of Chandrasekhar transformation in Kerr spacetime using the different tortoise coordinate, which is useful for exact analysis to study Teukolsky equation with arbitrary frequency. We also give the interpretation of our transformation using the formalism of the quantum Seiberg-Witten geometry.

Greybody factor and perturbation of a Schwarzschild black hole with string clouds and quintessence

In this paper, we investigated the Dirac and Klein-Gordan equations, as well as the greybody factor for a Schwarzschild black hole (SBH) immersed in quintessence and associated with a cloud of strings. Primarily, we study the Dirac equation using a null tetrad in the Newman- Penrose (NP) formalism. Next, we separate the Dirac equation into radial and angular sets. Using the radial equations, we study the profile of effective potential by transforming the radial equation of motion into standard Schr% \"{o}dinger wave equations form through tortoise coordinate. Similarly, we study the Klein-Gordan equation in this spacetime. The Miller-Good transformation method is employed to compute the greybody factor of bosons. To compute the Gerybody factor for fermions, we use the general method of semi-analytical bounds. We also investigate the effect of string clouds and quintessence parameters on Hawking radiation. According to the results, the greybody factor is strongly influenced by the shape of the potential, which is determined by the model parameters. This is consistent with the ideas of quantum mechanics; as the potential rises, it becomes harder for the wave to penetrate.

Long-range potential scattering: Converting long-range potential to short-range potential by tortoise coordinate

Inspired by general relativity, we suggest an approach for long-range potential scattering. In scattering theory, there is a general theory for short-range potential scattering, but there is no general theory for long-range potential scattering. This is because the scattering boundary conditions for all short-range potentials are the same, but for different long-range potentials, they are different. In this paper, by introducing tortoise coordinates, we convert long-range potential scattering to short-range potential scattering. This allows us to deal with long-range potential scattering as short-range potential scattering. An explicit expression of the scattering wave function for long-range potential scattering is presented, in which the scattering wave function is represented by the tortoise coordinate and the scattering phase shift. We show that the long-range potential scattering wave function is just the short-range potential scattering wave function with a replacement of a common coordinate by a tortoise coordinate. The approach applies not only to scattering but also applies to bound states. Furthermore, in terms of tortoise coordinates, we suggest a classification scheme for potentials. We also discuss the duality between tortoise coordinates.

On the photon motion near a five-dimensional Schwarzschild black hole

In this report, we discuss the dynamics of photons in a five-dimensional Schwarzschild black hole. We find that the corresponding Eddington–Finkelstein-like coordinate admits two time-coordinate singularities for a particular condition. This condition is due to the existence of a non-zero angular momentum related to the extra spatial dimension. By introducing a new Regge–Wheeler tortoise coordinate, both singularities can be removed from the Eddington–Finkelstein-like coordinate. It is identified that the additional singularity is related to the photon sphere in a hyperplane, and we analyze it using a dynamical system approach. Simulation results of photon motion based on numerical calculation are presented. It is suggested that the related bending motion characteristic can probably be considered as a signature for probing the existence of the extra spatial dimension.

Scalar field in Reissner–Nordström spacetime: Bound state and scattering state (with appendix on eliminating oscillation in partial sum approximation of periodic function)

In this paper, we solve the massive scalar field in the Reissner–Nordström spacetime. The scalar field in the Reissner–Nordström spacetime has both bound states and scattering states. For bound states, we solve the bound-state wave function and the eigenvalue spectrum. For scattering states, we solve the scattering wave function and give an explicit expression for scattering phase shift by the integral equation method. Especially, we introduce the tortoise coordinate for the Reissner–Nordström spacetime. Moreover, in the calculation of scattering cross sections, we encounter a difficulty in partial sum approximation. If the sum of partial waves cannot be performed exactly, one has to turn to the partial sum approximation which approximates a function by the first several terms of the series. However, in the partial sum approximation there exists an incorrect oscillation which cannot be eliminated by keeping more terms. In the appendix we suggest an approach for eliminating such oscillations in the partial sum.

Stability considerations of a Schwarzschild black hole with an dependent mass‐function

AbstractThe axial and polar modes for the ring down of a Schwarzschild black hole are calculated using an ‐dependent mass‐function and finally applying the asymptotic iteration method (AIM). We were able to reach up to 500 iterations, obtaining for the first time convergence for a wide range of large damping modes. The general relativity (GR) and an extended model with an ‐dependent mass‐function are compared. This mass‐function allows an analytical solution for the Tortoise coordinate. The example of the mass‐function corresponds to the leading correction for possible extended theories and serves as a starting point to treat other ‐dependent parameter mass‐functions.