Tortoise Coordinate Research Articles

Solution of Dirac equation in Reissner–Nordström de Sitter space

The radial parts of the Dirac equation between the outer black hole horizon and the cosmological horizon are solved in Reissner–Nordström de Sitter (RNdS) space numerically. An accurate approximation, the polynomial approximation, is used to approximate the modified tortoise coordinate , which leads to the inverse function and the potential . The potential is replaced by a collection of step functions in sequence. Then the solution of the wave equation as well as the reflection and transmission coefficients is computed by a quantum mechanical method.

Numerical Solution of Scalar Field for Nariai Case in 5D Ricci-Flat SdS Black String Space with Polynomial Approximation

As one exact candidate of the higher dimensional black hole, the 5D Ricci–Qat Schwarzschild-de Sitter black string space presents something interesting. In this paper, we give a numerical solution to the real scalar field around the Nariai black hole by the polynomial approximation. Unlike the previous tangent approximation, this fitting function makes a perfect match in the leading intermediate region and gives a good description near both the event and the cosmological horizons. We can read from our results that the wave is close to a harmonic one with the tortoise coordinate. Furthermore, with the actual radial coordinate the waves pile up almost equally near the both horizons.

Gravitational anomaly and Hawking radiation near a weakly isolated horizon

Based on the idea of the work by Wilczek and his collaborators, we consider the gravitational anomaly near a weakly isolated horizon. We find that there exists a universal choice of tortoise coordinate for any weakly isolated horizon. Under this coordinate, the leading behavior of a quite arbitrary scalar field near a horizon is a 2-dimensional chiral scalar field. This means we can extend the idea of Wilczek and his collaborators to more general cases and show the relation between gravitational anomaly and Hawking radiation is a universal property of a black hole horizon.

Real scalar field scattering with polynomial approximation around Schwarzschild—de Sitter black-hole

As one of the fitting methods, the polynomial approximation is effective to process sophisticated problem. In this paper, we employ this approach to handle the scattering of scalar field around the Schwarzschild—de Sitter black-hole. The complicated relationship between tortoise coordinate and radial coordinate is replaced by the approximate polynomial. The Schrödinger-like equation, the real boundary conditions and the polynomial approximation construct a full Sturm–Liouville type problem. Then this boundary value problem can be solved numerically for two limiting cases: the first one is the Nariai black-hole whose horizons are close to each other, the second one is the black-hole with the horizons widely separated. Compared with previous results (Brevik and Tian), the field near the event horizon and cosmological horizon can have a better description.

Hawking radiation of Dirac particles in a rectilinearly accelerating Kinnersley black hole

In a uniformly accelerating space time, the coupling Dirac equation can be simplified to the second-order equation. Using new tortoise coordinates, we further simplify the equation to the standard wave-equation near the horizon and get the temperature function of radiation and the Hawking radiation spectrum.

Scalar field probes of power-law space-time singularities

We analyse the effective potential of the scalar wave equation near generic space-time singularities of power-law type (Szekeres-Iyer metrics) and show that the effective potential exhibits a universal and scale invariant leading x^{-2} inverse square behaviour in the ``tortoise coordinate'' x provided that the metrics satisfy the strict Dominant Energy Condition (DEC). This result parallels that obtained in hep-th/0403252 for probes consisting of families of massless particles (null geodesic deviation, a.k.a. the Penrose Limit). The detailed properties of the scalar wave operator depend sensitively on the numerical coefficient of the x^{-2}-term, and as one application we show that timelike singularities satisfying the DEC are quantum mechanically singular in the sense of the Horowitz-Marolf (essential self-adjointness) criterion. We also comment on some related issues like the near-singularity behaviour of the scalar fields permitted by the Friedrichs extension.

Highly damped quasinormal modes of generic single-horizon black holes

We calculate analytically the highly damped quasinormal mode spectra of generic single-horizon black holes using the rigorous WKB techniques of Andersson and Howls (2004 Class. Quantum Grav. 21 1623). We thereby provide a firm foundation for previous analysis, and point out some of their possible limitations. The numerical coefficient in the real part of the highly damped frequency is generically determined by the behaviour of coupling of the perturbation to the gravitational field near the origin, as expressed in tortoise coordinates. This fact makes it difficult to understand how the famous ln(3) could be related to the quantum gravitational microstates near the horizon.

Quantum entropy of the non-static black hole in（2+1）dimensions

In the tortoise coordinates，the quantum entropy of the non-static black hole in (2+1)dimensions due to scalar fields has been investigated by using the brick-wall model.It is shown that the entropy of scalar fields within the thin region near the horizon has the same structure as that of the black ho1e.In particular,the quantum entropy satisfies the perimeter law of the entropy for the static case.

THE SCALAR FIELD IN EXTREME REISSNER–NORDSTRÖM–DE SITTER SPACE

By generalizing the method of I. Brevik et al. the scalar field equation between the outer black hole horizon and the cosmological horizon in the extreme Reissner–Nordström–de Sitter (RNdS) geometry is solved. The field amplitude, as well as the potential, is shown graphically by introducing the "tangent" approximation, which is more exact than that used by I. Brevik et al., of the tortoise coordinate. There are two limiting cases of our special interest. The first one is when the cosmological horizon is very close to the outer horizon of the "black hole." The second one is when they are far apart. And the reflection and transmission coefficients are worked out in the two cases respectively.

Hawking Radiation of Rarita—Schwinger Fields of a Stationary Charged Black Hole

Hawking radiation of the Rarita—Schwinger fields in a stationary charged black hole is studied exactly in region near the event horizon by using the Newman—Penrose formalism and the tortoise coordinate. The result shows that the temperature of the thermal radiation spectrum of Rarita—Schwinger fields is exactly the same as that of the scalar, Dirac, and electromagnetic fields.

Brick Walls for Nonstationary Black Holes

The validity condition of the brick-wall model is analysed innonstationary space-time. It is shown that the model holds only in thinregions, only by using tortoise coordinates, near the event horizon of aslowly varying (quasi-stationary) black hole. The reason for the use oftortoise coordinates is that the stationary state solutions of quantumfield equations in other coordinates do not exist for any region innonstationary space-time. Meanwhile, thequantum corrections to the entropy of the Vaidya-Bonner black hole dueto the spin fields are calculated in terms of the brick-wall model.

Hawking Radiation Arising from the Electromagnetic Fields in the Kerr-Newman Black Hole

Hawking radiation arising from the electromagnetic fields in the Kerr–Newman black hole is studied exactly by using the Newman–Penrose formalism and the tortoise coordinate. It is shown that the thermal radiation spectrum due to the photons in the Kerr–Newman black hole does not depend on the spins of the particles, and the effect is exactly same as that of the Klein–Gordon scalar particles.

Quantum mechanical path integrals and thermal radiation in static curved spacetimes

The propagator of a spinless particle is calculated from the quantum mechanical path integral formalism in static curved spacetimes endowed with event-horizons. A toy model, the Gui spacetime, and the 2D and 4D Schwarzschild black holes are considered. The role of the topology of the coordinates configuration space is emphasised in this framework. To cover entirely the above spacetimes with a single set of coordinates, tortoise coordinates are extended to complex values. It is shown that the homotopic properties of the complex tortoise configuration space imply the thermal behaviour of the propagator in these spacetimes. The propagator is calculated when end points are located in identical or distinct spacetime regions separated by one or several event-horizons. Quantum evolution through the event-horizons is shown to be unitary in the fifth variable.

DIRAC PARTICLE RADIATION FROM EINSTEIN-MAXWELL-DILATON BLACK PLANE

Dirac equation near Einstein-Maxwell-Dilaton black plane's event horizon is analysed and discussed.Under tortoise coordinate translation,it can be verified that there exists thermal radiation of dirac particles from the event horizon.

Quantum thermal effects of a radiating rotating charged black hole

Asymptotic solutions of the Klein-Gordon equation in a region near the event horizon of a radiating rotating charged black hole are obtained by using generalized tortoise coordinates. The location of the event horizon and the Hawking temperature of the black hole are given. Both the horizon and the temperature depend on the angle and time, due to radiation. However, they are independent of the angle if either rotation or radiation vanishes. The treatment encompasses as special cases the results on a number of well-known black holes.

Hawking effect in Vaidya-Bonner space-time

A new method determining the location and the temperature of event horizons of evaporating black holes is suggested. Both the Klein-Gordon equation and the Dirac equation are studied with the method in a Vaidya-Bonner space-time. These equations are reduced near the event horizon when the generalized tortoise coordinates are adopted. The location and the temperature of the event horizon are shown automatically. The first approximation of our result is the same as that obtained by using the calculation of the vacuum expectation value of the renormalized energy-momentum tensor when the evaporation of the black hole is very slow.

Hawking effect in Vaidya space-time

The Hawking effect of the Vaidya black hole is given easily without calculating the vacuum expectation value of the renormalized energy-momentum tensor. By means of the generalized tortoise coordinates, both the Klein-Gordon equation and the Dirac equation are reduced near the event horizon. The location and the temperature of the event horizon are obtained automatically.

Quantum thermal effect of a Kerr black hole moving in a straight line with a non-uniform acceleration

Using a new method, we give the radiative temperature and the event horizon equation of a Kerr black hole moving rectilinearly with a non-uniform acceleration without calculating the vacuum expectation values of the renormalised energy-momentum tensor. The temperature depends not only on the time but also on the polar angle. The location of the event horizon depends on the time and its shape is not spherically symmetric. With the generalized tortoise coordinates, the Klein-Gordon equation can be reduced to the standard wave equation around the event horizon. The outgoing wave is not analytical, but it can be extended from the outside of the black hole to the inside by analytical continuation through the lower half complex plane. The relative scattering probability of the outgoing wave at the horizon is calculated and the spectrum of the Hawking radiation is given. It is just the Planck spectrum at each moment and space point of the event horizon for a temperature that varies with the time and the polar angle.

Thermal Radiation of Dirac Particles from Vaidya Black Hole

With a new method calculating the location and temperature of the event horizon of the non-static black hole, the Dirac equation is reduced and solved near the horizon in Vaidya space-time when the tortoise coordinate is adopted. The location of Vaidya event horizon, the Hawking temperature and the formula of thermal spectrum of the Dirac particles are derived.

Hawking Radiation from a Non-static Black Hole

A method exactly determining an event horizon and its temperature in a non-static space-time is proposed. Using the generalized Tortoise coordinate, we give exact location of event horizon and exact Hawking temperature for a general spherically symmetric evaporating black hole.