Exact solutions for shells collapsing towards a pre-existing black hole

The gravitational collapse of a star is an important issue both for general relativity and astrophysics, which is related to the well-known “frozen star” paradox. This paradox has been discussed intensively and seems to have been solved in the comoving-like coordinates. However, to a real astrophysical observer within a finite time, this problem should be discussed in the point of view of the distant rest-observer, which is the main purpose of this Letter. Following the seminal work of Oppenheimer and Snyder (1939), we present the exact solution for one or two dust shells collapsing towards a pre-existing black hole. We find that the metric of the inner region of the shell is time-dependent and the clock inside the shell becomes slower as the shell collapses towards the pre-existing black hole. This means the inner region of the shell is influenced by the property of the shell, which is contrary to the result in Newtonian theory. It does not contradict the Birkhoff's theorem, since in our case we cannot arbitrarily select the clock inside the shell in order to ensure the continuity of the metric. This result in principle may be tested experimentally if a beam of light travels across the shell, which will take a longer time than without the shell. It can be considered as the generalized Shapiro effect, because this effect is due to the mass outside, but not inside as the case of the standard Shapiro effect. We also found that in real astrophysical settings matter can indeed cross a black hole's horizon according to the clock of an external observer and will not accumulate around the event horizon of a black hole, i.e., no “frozen star” is formed for an external observer as matter falls towards a black hole. Therefore, we predict that only gravitational wave radiation can be produced in the final stage of the merging process of two coalescing black holes. Our results also indicate that for the clock of an external observer, matter, after crossing the event horizon, will never arrive at the “singularity” (i.e. the exact center of the black hole), i.e., for all black holes with finite lifetimes their masses are distributed within their event horizons, rather than concentrated at their centers. We also present a worked-out example of the Hawking's area theorem.     

Exact self-gravitating N-body motion in the CGHS model

In the asymptotically flat two-dimensional dilaton gravity, we present an N-body particle action which has a dilaton coupled mass term for the exact solubility. This gives nonperturbative exact solutions for the N-body self-gravitating system, so the infalling particles form a black hole and their trajectories are exactly described. In our two-dimensional case, the critical mass for the formation of black holes does not exist, so even a single particle forms a black hole. The infalling particles give additional time-like singularities in addition to the space-like black hole singularity. However, the latter singularities can be properly cloaked by the future horizons within some conditions.     

A four-level “Ξ-system”: Exact solvability and effective evolution operators via multiple scales

Properties of a four-level atomic system interacting with one and two modes of the electromagneticfield in a “Ξ”-configuration are investigated. By linearization of the Hamiltonians we show that the corresponding mathematical models are exactly solvable. To obtain simpler effective Hamiltonians the perturbative method of multiple scales is applied. The lowest-order corrections to the resulting effective evolution operators are also calculated.     

Exact self-gravitating N-body motion in the CGHS model

In the asymptotically flat two-dimensional dilaton gravity, we present an N-body particle action which has a dilaton coupled mass term for the exact solubility. This gives nonperturbative exact solutions for the N-body self-gravitating system, so the infalling particles form a black hole and their trajectories are exactly described. In our two-dimensional case, the critical mass for the formation of black holes does not exist, so even a single particle forms a black hole. The infalling particles give additional time-like singularities in addition to the space-like black hole singularity. However, the latter singularities can be properly cloaked by the future horizons within some conditions.     

Conformally symmetric vacuum solutions of the gravitational field equations in the brane-world models

A class of exact solutions of the gravitational field equations in the vacuum on the brane are obtained by assuming the existence of a conformal Killing vector field, with non-static and non-central symmetry. In this case, the general solution of the field equations can be obtained in a parametric form in terms of the Bessel functions. The behavior of the basic physical parameters describing the non-local effects generated by the gravitational field of the bulk (dark radiation and dark pressure) is also considered in detail, and the equation of state satisfied at infinity by these quantities is derived. As a physical application of the obtained solutions we consider the behavior of the angular velocity of a test particle moving in a stable circular orbit. The tangential velocity of the particle is a monotonically increasing function of the radial distance and, in the limit of large values of the radial coordinate, tends to a constant value, which is independent on the parameters describing the model. Therefore, a brane geometry admitting a one-parameter group of conformal motions may provide an explanation for the dynamics of the neutral hydrogen clouds at large distances from the galactic center, which is usually explained by postulating the existence of the dark matter.     

Exact soliton solution of spin chain with an external magnetic field in linear wave background

Employing a simple, straightforward Darboux transformation we construct exact N-soliton solution for anisotropic spin chain driven by an external magnetic field in linear wave background. As a special case the explicit one- and two-soliton solution dressed by the linear wave corresponding to magnon in quantum theory is obtained analytically and its property is discussed in detail. The dispersion law, effective soliton mass, and the energy of each soliton are investigated as well. Our result show that the stability criterion of soliton is related with anisotropic parameter and the amplitude of the linear wave.     

What is the fate of a black hole embedded in an expanding universe?

Within a large class of exact solutions of the Einstein equations describing a black hole embedded in a Friedmann universe it is shown that, under certain assumptions, only those with comoving Hawking–Hayward quasi-local mass are generic, in the sense that they are late-time attractors.     

Exact solutions of embedding the four-dimensional perfect fluid in a five- or higher-dimensional Einstein spacetime and the cosmological interpretations

We investigate an exact solution that describes the embedding of the four-dimensional (4D) perfect fluid in a five-dimensional (5D) Einstein spacetime. The effective metric of the 4D perfect fluid as a hypersurface with induced matter is equivalent to the Robertson–Walker metric of cosmology. This general solution shows interconnections among many 5D solutions, such as the solution in the braneworld scenario and the topological black hole with cosmological constant. If the 5D cosmological constant is positive, the metric periodically depends on the extra dimension. Thus we can compactify the extra dimension on S¹${\mathrm{S}}^1$ and study the phenomenological issues. We also generalize the metric ansatz to the higher-dimensional case, in which the 4D part of the Einstein equations can be reduced to a linear equation.     

Stability of Motion of a Nonholonomic System

Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev＇s type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.     

The evolution of chaotic dynamics for fractional unified system

Based on reliable numerical approach, this Letter studies the chaotic behavior of the fractional unified system. The lowest orders for this system to have a complete chaotic attractor (the attractor covers the three equilibrium points of the classical unified system) at different parameter values are obtained. A striking finding is that with the increase of the parameter α of the fractional unified system from 0 to 1, the lowest order for this system to have a complete chaotic attractor monotonically decreases from 2.97 to 2.07. Because of the inherent attribute (memory effects) of fractional derivatives, this finding reveals that the chaotic behavior of fractional (classical) unified system becomes stronger and stronger when α increases from 0 to 1. Furthermore, this Letter introduces a novel measure to characterize the chaos intensity of fractional (classical) differential system.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order

Starting from a specific matrix iso-spectral problem, an associated hierarchy of multi-component Hamiltonian equations is constructed, based on zero curvature equations. The key point is to choose appropriate time parts of Lax pairs which can yield evolution equations, and the existence of a Hamiltonian structure for the obtained hierarchy is established by means of the trace identity. An example with five components is computed, along with its Hamiltonian structure.     

Weyl metrics and the generating conjecture

By means of the Ernst complex potential formalism it is shown that previously studied static axially symmetric Einstein-Maxwell fields obtained though the application of the Horsky-Mitskievitch generating conjecture represent a combination of Kinnersley’s transformations [W. Kinnersley: J. Math. Phys.14 (1973) 651]. New theoretical background for the conjecture is suggested and commented.     

An effective method for finding special solutions of nonlinear differential equations with variable coefficients

There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method.     

A Hierarchy of Differential-Difference Equations and Their Integrable Couplings

Starting from a discrete spectral problem, the corresponding hierarchy of nonlinear differential-difference equation is proposed. It is shown that the hierarchy possesses the bi-Hamiltionian structures. Further, two integrable coupling systems for the hierarchy are constructed through enlarged Lax pair method.     

Solitons and periodic solutions for the fifth-order KdV equation with the Exp-function method

In this Letter the Exp-function method is applied to obtain new generalized solitonary solutions and periodic solutions of the fifth-order KdV equation. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.     

New Symmetries for a Model of Fast Diffusion

The new symmetries for a mathematical model of fast diffusion are determined. A new system method is given to search for new symmetries of differential equations written in a conserved form, several new symmetry generators and exact solutions are presented.     

A lattice hierarchy and its continuous limits

By introducing a discrete spectral problem, we derive a lattice hierarchy which is integrable in Liouville's sense and possesses a multi-Hamiltonian structure. It is show that the discrete spectral problem converges to the well-known AKNS spectral problem under a certain continuous limit. In particular, we construct a sequence of equations in the lattice hierarchy which approximates the AKNS hierarchy as a continuous limit.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

New Taub–NUT–Reissner–Nordström spaces in higher dimensions

We construct new charged solutions of the Einstein–Maxwell field equations with cosmological constant. These solutions describe the nut-charged generalisation of the higher-dimensional Reissner–Nordström spacetimes. For a negative cosmological constant these solutions are the charged generalizations of the topological nut-charged black hole solutions in higher dimensions. Finally, we discuss the global structure of such solutions and possible applications.