arXiv subjects

Hideo Kodama

Publications and source records attributed to Hideo Kodama.

At least 55 records · Page 3Linked to original sources

arXiv2003

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.

hep-th↗

arXiv2003

An Investigation of the Tomimatsu-Sato Spacetime

We investigate the structure of the $δ=2$ Tomimatsu-Sato spacetime. We show that this spacetime has degenerate horizons with two components, in contrast to the general belief that the Tomimatsu-Sato solutions with even $δ$ do not have horizons.

gr-qc↗

arXiv2002

Vacuum branes in D-dimensional static spacetimes with spatial symmetry IO(D-2), O(D-1) or O+(D-2,1)

In this paper, we give a complete classification of vacuum branes, i.e., everywhere umbilical time-like hypersurfaces whose extrinsic curvature is a constant multiple of the induced metric, K_mn=k g_mn, in D-dimensional static spacetimes with spatial symmetry G(D-2,K), where G(n,K) is the isometry group of an n-dimensional space with constant sectional curvature K. D>=4 is assumed. It is shown that all possible configurations of a brane are invariant under an isometry subgroup G(D-3,K') for some K'>= K. In particular, configurations of a brane with non-zero k are always G(D-2,K) invariant, except for those in five special one-parameter families of spacetimes. Further, such G(D-2,K)-invariant configurations are allowed only in spacetimes whose Ricci tensors are isotropic in the two planes orthogonal to each G(D-2,K)-orbit, or for special values of k, which do not exist in generic cases. On the basis of these results, we prove the non-existence of a vacuum brane with black hole geometry in static bulk spacetimes with spatial symmetry G(D-2,K). We also discuss mathematical implications of these results.

gr-qc↗

arXiv2002

Rigidity theorems in the braneworld model

In the present paper, we give some theorems representing ridigity of a vacuum brane in static bulk spacetimes. As an application, we show that a static bulk spacetime with dimension D>3 and spatial symmetry IO(D-2), O(D-1) or O_+(D-2,1) does not allow a vacuum brane with a black hole on it. We also show that if a static bulk spacetime with dimension D>4 satsifying the vacuum Einstein equations can be foliated by a continuous family of vacuum branes with asymptotically constant curvature, it is a black string solution.

gr-qc↗

arXiv2001

Phase Space of Compact Bianchi Models with Fluid

The structure of phase space is determined for spatially compact and locally homogeneous universe models with fluid. Analysis covers models with all possible space topologies except for those covered by S^3, H^3 or S^2xR which have no moduli freedom. We show that space topology significantly affects the number of dynamical degrees of freedom of the system. In particular, we give a detailed proof of the result that for the systems modeled on the Thurston types H^2xR and SL_2, which have locally the Bianchi type III or VIII symmetry, the number of dynamical degrees of freedom increases without bound when the space topology becomes more and more complicated, which was first pointed out by Koike, Tanimoto and Hosoya in an incomplete form.

gr-qc↗

arXiv2001

All Universes Great and Small

If the topology of the universe is compact we show how it significantly changes our assessment of the naturalness of the observed structure of the universe and the likelihood of its present state of high isotropy and near flatness arising from generic initial conditions. We also identify the most general cosmological models with compact space.

gr-qc↗

arXiv2001

The Isotropy of Compact Universes

We discuss the problem of the stability of the isotropy of the universe in the space of ever-expanding spatially homogeneous universes with a compact spatial topology. The anisotropic modes which prevent isotropy being asymptotically stable in Bianchi-type $VII_h$ universes with non-compact topologies are excluded by topological compactness. Bianchi type $V$ and type $VII_h$ universes with compact topologies must be exactly isotropic. In the flat case we calculate the dynamical degrees of freedom of Bianchi-type $I$ and $VII_0$ universes with compact 3-spaces and show that type $VII_0$ solutions are more general than type $I$ solutions for systems with perfect fluid, although the type $I$ models are more general than type $VII_0$ in the vacuum case. For particular topologies the 4-velocity of any perfect fluid is required to be non-tilted. Various consequences for the problems of the isotropy, homogeneity, and flatness of the universe are discussed.

gr-qc↗

arXiv2000

Behavior of Cosmological Perturbations in the Brane-World Mode

In this paper we present a gauge-invariant formalism for perturbations of the brane-world model developed by the author, A. Ishibashi and O. Seto recently, and analyze the behavior of cosmological perturbations in a spatially flat expanding universe realized as a boundary 3-brane in AdS$^5$ in terms of this formalism. For simplicity we restrict arguments to scalar perturbations. We show that the behavior of cosmological perturbations on superhorizon scales in the brane-world model is the same as that in the standard no-extradimension model, irrespective of the initial condition for bulk perturbations, in the late stage when the cosmic expansion rate $H$ is smaller than the inverse of the bulk curvature scale $\ell$. Further, we give rough estimates which indicate that in the early universe when $H$ is much larger than $1/\ell$, perturbations in these two models behave quite differently, and the conservation of the Bardeen parameter does not hold for superhorizon perturbations in the brane-world model.

hep-th↗

arXiv2000

Gravitational waves in cosmological models of Hořava-Witten theory

We study the behavior of gravitational waves and their backreaction on the background in cosmological solutions of the five-dimensional Hořava-Witten theory. As a dynamical background, we consider two cosmological solutions with spatially flat expanding FRW branes, called $(\uparrow)$- and $(\downarrow)$-solutions, in which the orbifold size increases and decreases in time, respectively. % For these background solutions, the wave equation for the tensor perturbation can be solved by the method of separation of variables, and the mode functions are classified by a separation constant which can be regarded as a graviton mass. We show that the spatial behavior of the mode functions are the same for both background solutions, but the temporal behavior is significantly different. We further show that for the $(\uparrow)$ solution, the background bulk geometry is unstable against the backreaction of the perturbation, while for the $(\downarrow)$ solution, the backreaction on the bulk geometry can be neglected. We also show that, in contrast to the effect to the bulk geometry, the backreaction of the perturbation significantly alters the intrinsic geometry of the brane for the $(\downarrow)$ solution.

hep-th↗

arXiv2000

Brane World Cosmology - Gauge-Invariant Formalism for Perturbation

In the present paper the gauge-invariant formalism is developed for perturbations of the brane-world model in which our universe is realized as a boundary of a higher-dimensional spacetime. For the background model in which the bulk spacetime is $(n+m)$-dimensional and has the spatial symmetry corresponding to the isometry group of a $n$-dimensional maximally symmetric space, gauge-invariant equations are derived for perturbations of the bulk spacetime. Further for the case corresponding to the brane-world model in which $m=2$ and the brane is a boundary invariant under the spatial symmetry in the unperturbed background, relations between the gauge-invariant variables describing the bulk perturbations and those for brane perturbations are derived from Israel's junction condition under the assumption of $\ZR_2$ symmetry. In particular, for the case in which the bulk spacetime is a constant-curvature spacetime, it is shown that the bulk perturbation equations reduce to a single hyperbolic master equation for a master variable, and that the physical condition on the gauge-invariant variable describing the intrinsic stress perturbation of the brane yield a boundary condition for the master equation through the junction condition. On the basis of this formalism it is pointed out that it seems to be difficult to suppress brane perturbations corresponding to massive excitations for a brane motion giving a realistic expanding universe model.

hep-th↗

arXiv1999

What happens when the inflaton stops during inflation

The spectrum of adiabatic density perturbation generated during inflation is studied in the case the time derivative of an inflation-driving scalar field (inflaton) vanishes at some time during inflation. It is shown that the nondecaying mode of perturbation has a finite value even in this case and that its amplitude is given by the standard formula with the time derivation of the scalar field replaced by the potential gradient using the slow-roll equation.

astro-ph↗

arXiv1998

Thermodynamics of entanglement in Schwarzschild spacetime

Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on the Schwarzschild spacetime. Contrary to the flat case, the entanglement energy $E_{ent}$ turns out to be proportional to area radius of the boundary if it is near the horizon. This peculiar behavior of $E_{ent}$ can be understood by the red-shift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as the black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of the black hole thermodynamics.

gr-qc↗

arXiv1998

Summary report of the workshop D2: Conceptual Issues, Foundational Questions and Quantum Cosmology

This is a summary report of the workshop D2 in GR15. In this workshop rather diverging problems related to quantum gravity and quantum cosmology were discussed. Since it is almost impossible to summarize all these discussions in coherent and integrated expressions, in this report, I give a brief description of the content of each paper in order, with some additional comments on their backgrounds.

gr-qc↗

arXiv1998

Entanglement thermodynamics

Entanglement entropy is a statistical entropy measuring information loss due to coarse-graining corresponding to a spatial division of a system. In this paper we construct a thermodynamics (entanglement thermodynamics) which includes the entanglement entropy as the entropy variable, for a massless scalar field in Minkowski, Schwarzschild and Reissner-Nordstr{ö}m spacetimes to understand the statistical origin of black-hole thermodynamics. It is shown that the entanglement thermodynamics in Minkowski spacetime differs significantly from black-hole thermodynamics. On the contrary, the entanglement thermodynamics in Schwarzschild and Reissner-Nordstr{ö}m spacetimes has close relevance to black-hole thermodynamics.

gr-qc↗

arXiv1997

Canonical Structure of Locally Homogeneous Systems on Compact Closed 3-Manifolds of Types $E^3$, Nil and Sol

In this paper we investigate the canonical structure of diffeomorphism invariant phase spaces for spatially locally homogeneous spacetimes with 3-dimensional compact closed spaces. After giving a general algorithm to express the diffeomorphism-invariant phase space and the canonical structure of a locally homogeneous system in terms of those of a homogeneous system on a covering space and a moduli space, we completely determine the canonical structures and the Hamiltonians of locally homogeneous pure gravity systems on orientable compact closed 3-spaces of the Thurston-type $E^3$, $\Nil$ and $\Sol$ for all possible space topologies and invariance groups. We point out that in many cases the canonical structure becomes degenerate in the moduli sectors, which implies that the locally homogeneous systems are not canonically closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces.

gr-qc↗

arXiv1997

Can the entanglement entropy be the origin of black-hole entropy ?

Entanglement entropy is often speculated as a strong candidate for the origin of the black-hole entropy. To judge whether this speculation is true or not, it is effective to investigate the whole structure of thermodynamics obtained from the entanglement entropy, rather than just to examine the apparent structure of the entropy alone or to compare it with that of the black hole entropy. It is because entropy acquires a physical significance only when it is related to the energy and the temperature of a system. From this point of view, we construct a `thermodynamics of entanglement' by introducing an entanglement energy and compare it with the black-hole thermodynamics. We consider two possible definitions of entanglement energy. Then we construct two different kinds of thermodynamics by combining each of these different definitions of entanglement energy with the entanglement entropy. We find that both of these two kinds of thermodynamics show significant differences from the black-hole thermodynamics if no gravitational effects are taken into account. These differences are in particular highlighted in the context of the third law of thermodynamics. Finally we see how inclusion of gravity alter the thermodynamics of the entanglement. We give a suggestive argument that the thermodynamics of the entanglement behaves like the black-hole thermodynamics if the gravitational effects are included properly. Thus the entanglement entropy passes a non-trivial check to be the origin of the black-hole entropy.

gr-qc↗

arXiv1996

Evolution of cosmological perturbations in a stage dominated by an oscillatory scalar field

In the investigation of the evolution of cosmological perturbations in inflationary universe models the behavior of perturbations during the reheating stage is the most unclear point. In particular in the early reheating phase in which a rapidly oscillating scalar field dominates the energy density, the behavior of perturbations are not known well because their evolution equation expressed in terms of the curvature perturbation becomes singular. In this paper it is shown that in spite of this singular behavior of the evolution equation the Bardeen parameter stays constant in a good accuracy during this stage for superhorizon-scale perturbations except for a sequence of negligibly short intervals around the zero points of the time derivative of the scalar field. This justifies the conventional formula relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the Friedmann stage, except for possible corrections produced by the energy transfer from the scalar field to radiation in the late stage of reheating. It is further shown that outside the above sequence of time intervals the behavior of the perturbations coincides in a good accuracy with that for a perfect fluid system obtained from the original scalar field system by the WKB approximation and a spacetime averaging over a Hubble horizon scale.

gr-qc↗

arXiv1996

Evolution of Cosmological Perturbations during Reheating

The behavior of scalar perturbations on superhorizon scales during the reheating stage is investigated by replacing the rapidly oscillating inflaton field by a perfect fluid obtained by spacetime averaging and the WKB approximation. The influence of the energy transfer from the inflaton to radiation on the evolution of the Bardeen parameter is examined for realistic reheating processes. It is shown that the entropy perturbation generated by the energy transfer is negligibly small, and therefore the Bardeen parameter is conserved in a good accuracy during reheating. This justifies the conventional prescription relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the post-Friedmann stage.

gr-qc↗