伊形 尚久

We consider the periapsis shifts of bound orbits of stars on static clouds around a black hole. The background spacetime is constructed from a Schwarzschild black hole surrounded by a static and spherically symmetric self-gravitating system of massive particles, which satisfies all the standard energy conditions and physically models the gravitational effect of dark matter distribution around a nonrotating black hole. Using nearly circular bound orbits of stars, we obtain a simple formula for the precession rate. This formula explicitly shows that the precession rate is determined by a positive contribution (i.e. a prograde shift) from the conventional general-relativistic effect and a negative contribution (i.e. a retrograde shift) from the local matter density. The four quantities for such an orbit (i.e. the orbital shift angle, the radial oscillation period, the redshift and the star position mapped onto the celestial sphere) determine the local values of the background model functions. Furthermore, we not only evaluate the precession rate of nearly circular bound orbits in several specific models but also simulate several bound orbits with large eccentricity and their periapsis shifts. The present exact model demonstrates that the retrograde precession does not mean any exotic central objects such as naked singularities or wormholes but simply the existence of significant energy density of matters on the star orbit around the black hole.

We completely classify the Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature K = 0, ±1 for perfect fluids with linear equation of state p = wρ, where ρ and p are the energy density and pressure, without assuming any energy conditions. We extend our previous work to include all geodesics and parallelly propagated (p.p.) curvature singularities, showing that no non-null geodesic emanates from or terminates at the null portion of conformal infinity and that the initial singularity for K = 0, −1 and −5/3 < w < −1 is a null non-scalar polynomial curvature singularity. We thus obtain the Penrose diagrams for all possible cases and identify w = −5/3 as a critical value for both the future big-rip singularity and the past null conformal boundary.

We consider necessary and sufficient conditions for photons emitted from an arbitrary spacetime position of the extremal Kerr black hole to escape to infinity. The radial equation of motion determines necessary conditions for photons emitted from r=r∗ to escape to infinity, and the polar angle equation of motion further restricts the allowed region of photon motion. From these two conditions, we provide a method to visualize a two-dimensional photon impact parameter space that allows photons to escape to infinity, i.e., the escapable region. Finally, we completely identify the escapable region for the extremal Kerr black hole spacetime. This study has generalized our previous result [K.~Ogasawara and T.~Igata, Phys. Rev. D \textbf{103}, 044029 (2021)], which focused only on light sources near the horizon, to the classification covering light sources in the entire region.

We consider a situation where a light source orbiting the innermost stable circular orbit (ISCO) of the Kerr black hole is gently falling from the marginally stable orbit due to an infinitesimal perturbation. Assuming that the light source emits photons isotropically, we show that the last radius at which more than 50% of emitted photons can escape to infinity is approximately halfway between the ISCO radius and the event horizon radius. To evaluate them, we determine emitter orbits from the vicinity of the ISCO, which are uniquely specified for each black hole spin, and identify the conditions for a photon to escape from any point on the equatorial plane of the Kerr spacetime to infinity by specifying regions in the two-dimensional photon impact parameter space completely. We further show that the proper motion of the emitter affects the photon escape probability and blueshifts the energy of emitted photons.

We consider the motion of massive and massless particles in a five-dimensional spacetime with a compactified extra-dimensional space where a black hole is localized, i.e., a caged black hole spacetime. We show the existence of circular orbits and reveal their sequences and stability. In the asymptotic region, stable circular orbits always exist, which implies that four-dimensional gravity is more dominant because of the small extra-dimensional space. In the vicinity of a black hole, they do not exist because the effect of compactification is no longer effective. We also clarify the dependence of the sequences of circular orbits on the size of the extra-dimensional space by determining the appearance of the innermost stable circular orbit and the last circular orbit (i.e., the unstable photon circular orbit).

Reducing motion of particles to a two-dimensional potential problem, we show that there are stable circular orbits around a squashed Kaluza-Klein black hole with a spherical horizon and multi–Kaluza-Klein black holes with two spherical horizons in five dimensions. For a single horizon, we show analytically that the radius of an innermost stable circular orbit monotonically depends on the size of an extra dimension. For two horizons, the radius of an innermost stable circular orbit depends on the separation between two black holes besides the size of an extra dimension. More precisely, the set of the stationary points of the potential is composed of two branches. For a large separation, stable circular orbits exist on the two branches regardless of the size of an extra dimension, and in particular, on one branch, the set of stable circular orbits is connected for the small extra dimension but has two disconnected parts for the large extra dimension. For a small separation, only on one branch it exists, and the radius of an innermost stable circular orbit monotonically increases with an extra-dimension size.

We consider necessary and sufficient conditions for photons emitted from the vicinity of a Kerr black hole horizon to escape to infinity. The radial equation of motion determines necessary conditions for photons to reach infinity, and the polar angle equation of motion further restricts the allowed region of photon motion. Unlike emission from the equatorial plane, the latter restrictions are crucial for photon escape when the initial polar angle of the emission point is arbitrary. We provide a visualization tool to analyze these two conditions and demonstrate a procedure for revealing photon escape. Finally, we completely identify the two-dimensional impact parameter space in which photons can escape.

We investigate photon surfaces and their stability in a less symmetric spacetime, a general static warped product with a warping function acting on a Riemannian submanifold of codimension two. We find a one-dimensional pseudopotential that gives photon surfaces as its extrema regardless of the spatial symmetry of the submanifold. The maxima and minima correspond to unstable and stable photon surfaces, respectively. It is analogous to the potential giving null circular orbits in a spherically symmetric spacetime. We also see that photon surfaces indeed exist for the spacetimes which are solutions to the Einstein equation. The parameter values for which the photon surfaces exist are specified. As we show finally, the pseudopotential arises due to the separability of the null geodesic equation, and the separability comes from the existence of a Killing tensor in the spacetime. The result leads to the conclusion that photon surfaces may exist even in a less symmetric spacetime if the spacetime admits a Killing tensor.

In contrast to five-dimensional Schwarzschild-Tangherlini and Myers-Perry backgrounds, we show that there are stable bound orbits of massive and massless particles in five-dimensional black lens backgrounds, in particular, the supersymmetric black lens with L(2,1) and L(3,1) topologies. We also show that in the zero-energy limit of massless particles, there exist stable circular orbits on the evanescent ergosurfaces.

We consider the dynamics of particles, particularly focusing on circular orbits in the higher-dimensional Majumdar-Papapetrou (MP) spacetimes with two equal mass black holes. It is widely known that in the 5D Schwarzschild-Tangherlini and Myers-Perry backgrounds, there are no stable circular orbits. In contrast, we show that in the 5D MP background, stable circular orbits can always exist when the separation of two black holes is large enough. More precisely, for a large separation, stable circular orbits exist from the vicinity of horizons to infinity; for a medium one, they appear only in a certain finite region bounded by the innermost stable circular orbit and the outermost stable circular orbit outside the horizons; for a small one, they do not appear at all. Moreover, we show that in MP spacetimes in more than 5D, they do not exist for any separations.

We consider test particle motion in a gravitational field generated by a homogeneous circular ring placed in n-dimensional Euclidean space. We observe that there exist no stable stationary orbits in n=6,7,…,10 but exist in n=3, 4, 5 and clarify the regions in which they appear. In n=3, we show that the separation of variables of the Hamilton-Jacobi equation does not occur though we find no signs of chaos for stable bound orbits. Since the system is integrable in n=4, no chaos appears. In n=5, we find some chaotic stable bound orbits. Therefore, this system is nonintegrable at least in n=5 and suggests that the timelike geodesic system in the corresponding black ring spacetimes is nonintegrable.

Newtonian gravitational potential sourced by a homogeneous circular ring in arbitrary dimensional Euclidean space takes a simple form if the spatial dimension is even. In contrast, if the spatial dimension is odd, it is given in a form that includes complete elliptic integrals. In this paper, we analyze the dynamics of a freely falling massive particle in its Newtonian potential. Focusing on circular orbits on the symmetric plane where the ring is placed, we observe that they are unstable in 4D space and above, while they are stable in 3D space. The sequence of stable circular orbits disappears at 1.6095⋯ times the radius of the ring, which corresponds to the innermost stable circular orbit (ISCO). On the axis of symmetry of the ring, there are no circular orbits in 3D space but more than in 4D space. In particular, the circular orbits are stable between the ISCO and infinity in 4D space and between the ISCO and the outermost stable circular orbit in 5D space. There exist no stable circular orbits in 6D space and above.

We investigate the escape of photons from the vicinity of the horizon to infinity in the Kerr-Newman black hole spacetime. We assume that a light source is at rest in a locally nonrotating frame on the equatorial plane and photons are emitted isotropically. Then, we evaluate the escape probability of the emitted photons. The main result of this paper is the following. If the black hole is extremal with the nondimensional spin parameter a*>1/2, however close to the horizon the light source would be, the escape probability remains nonzero. The near-horizon limit value of the escape probability is a monotonically increasing function of a* and takes a maximum ∼29.1% at a*=1, i.e., for the extremal Kerr case. On the other hand, if the black hole is extremal with 0≤a*≤1/2 or if the black hole is subextremal, the near-horizon limit value is zero.

In higher-dimensional Schwarzschild black hole spacetimes, there are no stable bound orbits of particles. In contrast to this, it is shown that there are stable bound orbits in a five-dimensional black lens spacetime.

We investigate how stable circular orbits around a main compact object appear depending on the presence of a second one by using the Majumdar-Papapetrou dihole spacetime, which consists of the two extremal Reissner-Nordström black holes with different masses. While the parameter range of the separation of the two objects is divided due to the appearance of stable circular orbits, this division depends on its mass ratio. We show that the mass ratio range separates into four parts, and we find three critical values as the boundaries.

We consider the escape probability of a photon emitted from the innermost stable circular orbit (ISCO) of a rapidly rotating black hole. As an isotropically emitting light source on a circular orbit reduces its orbital radius, the escape probability of a photon emitted from it decreases monotonically. The escape probability evaluated at the ISCO also decreases monotonically as the black hole spin increases. When the dimensionless Kerr parameter a is at the Thorne limit a=0.998, the escape probability from the ISCO is 58.8%. In the extremal case a=1, even if the orbital radius of the light source is arbitrarily close to the ISCO radius, which coincides with the horizon radius, the escape probability remains at 54.6%. We also show that such photons that have escaped from the vicinity of the horizon reach infinity with sufficient energy to be potentially observed because Doppler blueshift due to relativistic beaming can overcome the gravitational redshift. Our findings indicate that signs of the near-horizon physics of a rapidly rotating black hole will be detectable on the edge of its shadow.

We investigate the spherical photon orbits in near-extremal Kerr spacetimes. We show that the spherical photon orbits with impact parameters in a finite range converge on the event horizon. Furthermore, we demonstrate that the Weyl curvature near the horizon does not generate the shear of a congruence of such light rays. Because of this property, a series of images produced by the light orbiting around a near-extremal Kerr black hole several times can be observable.

We develop a new perturbation method to study the dynamics of massive tensor fields on extremal and near-extremal static black hole spacetimes in arbitrary dimensions. On such backgrounds, one can classify the components of massive tensor fields into the tensor, vector, and scalar-type components. For the tensor-type components, which arise only in higher dimensions, the massive tensor field equation reduces to a single master equation, whereas the vector and scalar-type components remain coupled. We consider the near-horizon expansion of both the geometry and the field variables with respect to the near-horizon scaling parameter. By doing so, we reduce, at each order of the expansion, the equations of motion for the vector and scalar-type components to a set of five mutually decoupled wave equations with source terms consisting only of the lower-order variables. Thus, together with the tensor-type master equation, we obtain the set of mutually decoupled equations at each order of the expansion that govern all dynamical degrees of freedom of the massive tensor field on the extremal and near-extremal static black hole background.

We investigate the positions of stable circular massive particle orbits in the Majumdar-Papapetrou dihole spacetime with equal mass. In terms of qualitative differences of their sequences, we classify the dihole separation into five ranges and find four critical values as the boundaries. When the separation is relatively large, the sequence on the symmetric plane bifurcates, and furthermore, they extend to each innermost stable circular orbit in the vicinity of each black hole. In a certain separation range, the sequence on the symmetric plane separates into two parts. On the basis of this phenomenon, we discuss the formation of double accretion disks with a common center. Finally, we clarify the dependence of the radii of marginally stable circular orbits and innermost stable circular orbits on the separation parameter. We find a discontinuous transition of the innermost stable circular orbit radius. We also find the separation range at which the radius of the innermost stable circular orbit can be smaller than that of the stable circular photon orbit.

We analyze rigidly rotating Nambu-Goto strings in the Kerr spacetime, particularly focusing on the strings sticking in the horizon. From the regularity on the horizon, we find the condition for sticking in the horizon, which is consistent with the second law of black hole thermodynamics. Energy extraction through the sticking string from a Kerr black hole occurs. We obtain the maximum value of the luminosity of the energy extraction.

Scale invariance in the theory of classical mechanics can be induced from the scale invariance of background fields. In this paper we consider the relation between the scale invariance and the constants of particle motion in a self-similar spacetime, only in which the symmetry is well defined and is generated by a homothetic vector. Relaxing the usual conservation condition by the Hamiltonian constraint in a particle system, we obtain a conservation law holding only on the constraint surface in the phase space. By the conservation law, we characterize constants of motion associated with the scale invariance not only for massless particles but for massive particles and classify the condition for the existence of the constants of motion. Furthermore, we find the explicit form of the constants of motion by solving the conservation equations.

We completely classify Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature and equation of state , according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow , thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for w  >  −1/3 and , while there is no big-bang singularity for w  <  −1 and . For K  =  0 or  −1, −1  <  w  <  −1/3 and , there is an initial null big-bang singularity. For each spatial curvature, there is a final spacelike future big-rip singularity for w  <  −1 and , with null geodesics being future complete for but incomplete for w  <  −5/3. For w  =  −1/3, the expansion speed is constant. For  −1  <  w  <  −1/3 and K  =  1, the universe contracts from infinity, then bounces and expands back to infinity. For K  =  0, the past boundary consists of timelike infinity and a regular null hypersurface for  −5/3  <  w  <  −1, while it consists of past timelike and past null infinities for . For w  <  −1 and K  =  1, the spacetime contracts from an initial spacelike past big-rip singularity, then bounces and blows up at a final spacelike future big-rip singularity. For w  <  −1 and K  =  −1, the past boundary consists of a regular null hypersurface. The trapping horizons are timelike, null and spacelike for , and , respectively. A negative energy density () is possible only for K  =  −1. In this case, for w  >  −1/3, the universe contracts from infinity, then bounces and expands to infinity; for  −1  <  w  <  −1/3, it starts from a big-bang singularity and contracts to a big-crunch singularity; for w  <  −1, it expands from a regular null hypersurface and contracts to another regular null hypersurface.

Geometrical symmetry in a spacetime can generate test solutions to the Maxwell equation. We demonstrate that the source-free Maxwell equation is satisfied by any generator of spacetime self-similarity---a proper homothetic vector---identified with a vector potential of the Maxwell theory. The test fields obtained in this way share the scale symmetry of the background.

From a spacetime perspective, the dynamics of magnetic field lines of force-free electromagnetic fields can be rewritten into a quite similar form for the dynamics of strings, i.e., dynamics of “field sheets”. Using this formalism, we explicitly show that the field sheets of stationary and axisymmetric force-free electromagnetic fields have identical intrinsic properties to the world sheets of rigidly rotating Nambu–Goto strings. Thus, we conclude that the Blandford–Znajek process is kinematically identical to an energy-extraction mechanism by the Nambu–Goto string with an effective magnetic tension.

We consider a head-on collision of two massive particles that move in the equatorial plane of an extremal Kerr black hole, which results in the production of two massless particles. Focusing on a typical case, where both of the colliding particles have zero angular momenta, we show that a massless particle produced in such a collision can escape to infinity with arbitrarily large energy in the near-horizon limit of the collision point. Furthermore, if we assume that the emission of the produced massless particles is isotropic in the center-of-mass frame but confined to the equatorial plane, the escape probability of the produced massless particle approaches 5/12, and almost all escaping massless particles have arbitrarily large energy at infinity and an impact parameter approaching 2GM/c2, where M is the mass of the black hole.

In this paper, we show that a rigidly rotating string can extract the rotational energy from a rotating black hole. We consider Nambu-Goto strings stationary with respect to a corotating Killing vector with an uniform angular velocity ω in the Kerr spacetime. We show that a necessary condition of the energy-extraction process is that an effective horizon on the string world sheet, which corresponds to the inner light surface, is inside the ergosphere of the Kerr black hole and the angular velocity ω is less than that of the black hole Ωh. Furthermore, we discuss global configurations of such strings in both of a slow-rotation limit and the extremal Kerr case.

Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equations and using the Levi-Cività solution as a seed, we construct a two-soliton solution with cylindrical symmetry. In our previous work, we constructed the one-soliton solution with a real pole and showed that the singularities that the Levi-Cività background has on an axis can be removed by the choice of certain special parameters, but it still has unavoidable null singularities, as usual one-solitons do. In this work, we show that for the two-soliton solutions, any singularities can be removed by suitable parameter-setting and such solutions describe the propagation of gravitational wave packets. Moreover, in terms of the two-soliton solutions, we mention a time shift phenomenon, the coalescence and the split of solitons as the nonlinear effect of gravitational waves.

We study the self-similar motion of a string in a self-similar spacetime by introducing the concept of a self-similar string, which is defined as the world sheet to which a homothetic vector field is tangent. It is shown that in Nambu-Goto theory, the equations of motion for a self-similar string reduce to those for a particle. Moreover, under certain conditions such as the hypersurface orthogonality of the homothetic vector field, the equations of motion for a self-similar string simplify to the geodesic equations on a (pseudo)Riemannian space. As a concrete example, we investigate a self-similar Nambu-Goto string in a spatially flat Friedmann-Lemaître-Robertson-Walker expanding universe with self-similarity and obtain solutions of open and closed strings, which have various nontrivial configurations depending on the rate of the cosmic expansion. For instance, we obtain a circular solution that evolves linearly in the cosmic time while keeping its configuration by the balance between the effects of the cosmic expansion and string tension. We also show the instability for linear radial perturbation of the circular solutions.

The existence of stable bound orbits of test particles is one of the most characteristic properties in black hole spacetimes. In higher-dimensional black holes, due to the dependence of gravity on the spacetime dimensions, there is no stable bound orbit balanced by Newtonian gravitational monopole force and centrifugal force, although this balance occurs in the four-dimensional Kerr black hole. In this paper, however, the existence of stable bound orbits of massive and massless particles is shown in six-dimensional singly spinning Myers-Perry black holes with a value of the spin parameter larger than a critical value. The innermost stable bound orbits and the outermost stable bound orbits are found on the rotational axis.

Applying the Pomeransky inverse scattering method to the four-dimensional vacuum Einstein equation and using the Levi-Cività solution for a seed, we construct a cylindrically symmetric single-soliton solution. Although the Levi-Cività spacetime generally includes singularities on its axis of symmetry, it is shown that for the obtained single-soliton solution, such singularities can be removed by choice of certain special parameters. This single-soliton solution describes propagation of nonlinear cylindrical gravitational shock wave pulses rather than solitonic waves. By analyzing wave amplitudes and time dependence of polarization angles, we provide a physical description of the single-soliton solution.

The geodesic equation in the five-dimensional singly rotating black ring is nonintegrable, unlike the case of the Myers-Perry black hole. In the Newtonian limit of the black ring, its geodesic equation agrees with the equation of motion of a particle in the Newtonian potential due to a homogeneous ring gravitational source. In this paper, we show that the Newtonian equation of motion allows the separation of variables in the spheroidal coordinates, providing a nontrivial constant of motion quadratic in momenta. This shows that the Newtonian limit of a black ring recovers the symmetry of its geodesic system, and the geodesic chaos is caused by relativistic effects.

We study the geodesic motion of massless particles in singly rotating black ring spacetimes. We find stable stationary orbits of massless particles in a toroidal spiral shape in the case that the thickness parameter of a black ring is less than a critical value. Furthermore, there exist nonstationary massless particles bounded in a finite region outside the horizon. This is the first example of stable bound orbits of massless particles around a black object.

We investigate stationary rotating closed Nambu-Goto strings in five-dimensional flat spacetime. The stationary string is defined as a world sheet that is tangent to a timelike Killing vector. The Nambu-Goto equation of motion for the stationary string is reduced to the geodesic equation on the orbit space of the isometry group action generated by the Killing vector. We take a linear combination of a time-translation vector and space-rotation vectors as the Killing vector, and explicitly construct general solutions of stationary rotating closed strings in five-dimensional flat spacetime. We show a variety of their configurations and properties.

We study high energy charged particle collisions near the horizon in an electromagnetic field around a rotating black hole and reveal the condition of the fine-tuning to obtain arbitrarily large center-of-mass (CM) energy. We demonstrate that the CM energy can be arbitrarily large as the uniformly magnetized rotating black hole arbitrarily approaches maximal rotation under the situation that a charged particle plunges from the innermost stable circular orbit (ISCO) and collides with another particle near the horizon. Recently, Frolov [Phys. Rev. D 85, 024020 (2012)] proposed that the CM energy can be arbitrarily high if the magnetic field is arbitrarily strong, when a particle collides with a charged particle orbiting the ISCO with finite energy near the horizon of a uniformly magnetized Schwarzschild black hole. We show that the charged particle orbiting the ISCO around a spinning black hole needs arbitrarily high energy in the strong field limit. This suggests that Frolov's process is unstable against the black hole spin. Nevertheless, we see that magnetic fields may substantially promote the capability of rotating black holes as particle accelerators in astrophysical situations.

We generalize Killing equations to a test particle system which is subjected to external force. We relax the conservation condition by virtue of reparametrization invariance of a particle orbit. As a result, we obtain generalized Killing equations which have hierarchical structure on the top of which a conformal Killing equation exists. Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd.

We discuss constants of motion of a particle under an external field in a curved spacetime, taking into account the Hamiltonian constraint, which arises from the reparametrization invariance of the particle orbit. As the necessary and sufficient condition for the existence of a constant of motion, we obtain a set of equations with a hierarchical structure, which is understood as a generalization of the Killing tensor equation. It is also a generalization of the conventional argument in that it includes the case when the conservation condition holds only on the constraint surface in the phase space. In that case, it is shown that the constant of motion is associated with a conformal Killing tensor. We apply the hierarchical equations and find constants of motion in the case of a charged particle in an electromagnetic field in black hole spacetimes. We also demonstrate that gravitational and electromagnetic fields exist in which a charged particle has a constant of motion associated with a conformal Killing tensor.

We study bound orbits of a free particle around a singly rotating black ring. We find there exists chaotic motion of a particle which is gravitationally bound to the black ring by using the Poincaré map.

We examine bound orbits of particles around singly rotating black rings. We show that there exist stable bound orbits in toroidal spiral shape near the "axis" of the ring, and also stable circular orbits on the axis as special cases. The stable bound orbits can have arbitrary large size if the thickness of the ring is less than a critical value. © 2010 The American Physical Society.

We present solutions of the Nambu-Goto equation for test strings in a shape of toroidal spiral in five-dimensional spacetimes. In particular, we show that stationary toroidal spirals exist around the five-dimensional Myers-Perry black holes. We also show the existence of innermost stationary toroidal spirals around the five-dimensional black holes like geodesic particles orbiting around four-dimensional black holes.

We report on our progress in research of separability of the Nambu-Goto equation for test strings with a symmetric configuration in a shape of toroidal spiral in a five-dimensional Kerr-AdS black hole. In particular, for a Hopf loop string which is a special class of the toroidal spirals, we show the complete separation of variables occurs in two cases, Kerr background and Kerr-AdS background with equal angular momenta. We also obtain the dynamical solution for the Hopf loop around a black hole and for the general toroidal spiral in Minkowski background.

We examine the separability of the Nambu-Goto equation for test strings in a shape of toroidal spiral in a five-dimensional Kerr-AdS black hole. In particular, for a “Hopf loop” string which is a special class of the toroidal spiral strings, we show the complete separation of variables occurs in two cases, Kerr background and Kerr-AdS background with equal angular momenta. We also obtain the dynamical solution for the Hopf loop around a black hole and for the general toroidal spiral in Minkowski background.