Perturbations and Stability of Static Black Holes in Higher Dimensions

In this chapter we consider perturbations and stability of higher dimensional black holes focusing on the static background case. We first review a gauge-invariant formalism for linear perturbations in a fairly generic class of (m+n)-dimensional spacetimes with a warped product metric, including black hole geometry. We classify perturbations of such a background into three types, the tensor, vector and scalar-type, according to their tensorial behaviour on the n-dimensional part of the background spacetime, and for each type of perturbations, we introduce a set of manifestly gauge invariant variables. We then introduce harmonic tensors and write down the equations of motion for the expansion coefficients of the gauge invariant perturbation variables in terms of the harmonics. In particular, for the tensor-type perturbations a single master equation is obtained in the (m+n)-dimensional background, which is applicable for perturbation analysis of not only static black holes but also some class of rotating black holes as well as black-branes. For the vector and scalar type, we derive a set of decoupled master equations when the background is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a cosmological constant. As an application of the master equations, we review the stability analysis of higher dimensional charged static black holes with a cosmological constant. We also briefly review the recent results of a generalisation of the perturbation formulae presented here and stability analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1 table, v2: references added, the notations slightly modified to match PTPS published versio

Stability of Higher-Dimensional Schwarzschild Black Holes

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. We show that, for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.Comment: 19 pages, 9 figures, references and comments on the generalised black hole case are added, minor changes in text, version to appear in PT

A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions

We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single 2nd-order wave equation in a two-dimensional static spacetime for a gauge-invariant master variable, irrespective of the mode of perturbations. Our formulation applies to the case of vanishing as well as non-vanishing cosmological constant Lambda. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background, this master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with K=0,1, our equation coincides with those derived by Cardoso and Lemos recently. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Lambda. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimensions. Although we only treat maximally symmetric black holes in the present paper, the final master equations are valid even when the hirozon geometry is described by a generic Einstein manifold.Comment: 22 pages in the PTP TeX style, no figure. The published versio

Master equations for perturbations of generalised static black holes with charge in higher dimensions

We extend the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper (hep-th/0305147) to a charged black hole background whose horizon is described by an Einstein manifold. For charged black holes, perturbations of electromagnetic fields are coupled to the vector and scalar modes of metric perturbations non-trivially. We show that by taking appropriate combinations of gauge-invariant variables for these perturbations, the perturbation equations for the Einstein-Maxwell system are reduced to two decoupled second-order wave equations describing the behaviour of the electromagnetic mode and the gravitational mode, for any value of the cosmological constant. These wave equations are transformed into Schr\"odinger-type ODEs through a Fourier transformation with respect to time. Using these equations, we investigate the stability of generalised black holes with charge. We also give explicit expressions for the source terms of these master equations with application to the emission problem of gravitational waves in mind.Comment: 46 pages in the PTP-TEX style including 7 figures. The published versio

Perturbations and Stability of Static Black Holes in Higher Dimensions

In this chapter we consider perturbations and stability of higher dimensional black holes focusing on the static background case. We first review a gauge-invariant formalism for linear perturbations in a fairly generic class of (m+n)-dimensional spacetimes with a warped product metric, including black hole geometry. We classify perturbations of such a background into three types, the tensor, vector and scalar-type, according to their tensorial behaviour on the n-dimensional part of the background spacetime, and for each type of perturbations, we introduce a set of manifestly gauge invariant variables. We then introduce harmonic tensors and write down the equations of motion for the expansion coefficients of the gauge invariant perturbation variables in terms of the harmonics. In particular, for the tensor-type perturbations a single master equation is obtained in the (m+n)-dimensional background, which is applicable for perturbation analysis of not only static black holes but also some class of rotating black holes as well as black-branes. For the vector and scalar type, we derive a set of decoupled master equations when the background is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a cosmological constant. As an application of the master equations, we review the stability analysis of higher dimensional charged static black holes with a cosmological constant. We also briefly review the recent results of a generalisation of the perturbation formulae presented here and stability analysis to static black holes in generic Lovelock theory.Comment: Invited review for Prog. Theor. Phys. Suppl, 45 pages, 2 figures, 1 table, v2: references added, the notations slightly modified to match PTPS published versio

The 33-colorable subgroup of Thompson's group and tricolorability of links

Starting from the work by Jones on representations of Thompson's group $F$, subgroups of $F$ with interesting properties have been defined and studied. One of these subgroups is called the $3$-colorable subgroup $\mathcal{F}$, which consists of elements whose ``regions'' given by their tree diagrams are $3$-colorable. On the other hand, in his work on representations, Jones also gave a method to construct knots and links from elements of $F$. Therefore it is a natural question to explore a relationship between elements in $\mathcal{F}$ and $3$-colorable links in the sense of knot theory. In this paper, we show that all elements in $\mathcal{F}$ give 3-colorable links.Comment: 9 pages, 11 figure

Alexander's theorem for stabilizer subgroups of Thompson's group

In 2017, Jones studied the unitary representations of Thompson's group $F$ and defined a method to construct knots and links from $F$. One of his results is that any knot or link can be obtained from an element of this group, which is called Alexander's theorem. On the other hand, Thompson's group $F$ has many subgroups and it is known that there exist various subgroups which satisfy or do not satisfy Alexander's theorem. In this paper, we prove that almost all stabilizer subgroups under the natural action on the unit interval satisfy Alexander's theorem.Comment: 14 pages, 13 figure

Equation of motion for a domain wall coupled to gravitational field

The equation of motion for a domain wall coupled to gravitational field is derived from the Nambu-Goto action. The domain wall is treated as a source of gravitational field. The perturbed equation is also obtained with gravitational back reaction on the wall motion taken into account. For general spherically symmetric background case, the equation is expressed in terms of the gauge-invariant variables.Comment: 13 pages, latex, no figures, uses REVTe

Effect of Injection Flow Rate on Product Gas Quality in Underground Coal Gasification (UCG) Based on Laboratory Scale Experiment: Development of Co-Axial UCG System

Underground coal gasification (UCG) is a technique to recover coal energy without mining by converting coal into a valuable gas. Model UCG experiments on a laboratory scale were carried out under a low flow rate (6~12 L/min) and a high flow rate (15~30 L/min) with a constant oxygen concentration. During the experiments, the coal temperature was higher and the fracturing events were more active under the high flow rate. Additionally, the gasification efficiency, which means the conversion efficiency of the gasified coal to the product gas, was 71.22% in the low flow rate and 82.42% in the high flow rate. These results suggest that the energy recovery rate with the UCG process can be improved by the increase of the reaction temperature and the promotion of the gasification area