Abstract:
The final fate of the spherically symmetric collapse of a perfect fluid which follows the $\gamma$-law equation of state and adiabatic condition is investigated. Full general relativistic hydrodynamics is solved numerically using a retarded time coordinate, the so-called observer time coordinate. Thanks to this coordinate, the causal structure of the resultant space-time is automatically constructed. Then, it is found that a globally naked, shell-focusing singularity can occur at the center from relativistically high-density, isentropic and time symmetric initial data if $\gamma \alt 1.01$ within the numerical accuracy. The result is free from the assumption of self-similarity. The upper limit of $\gamma$ with which a naked singularity can occur from generic initial data is consistent with the result of Ori and Piran based on the assumption of self-similarity.

Abstract:
Einstein's field equations in general relativity admit a variety of solutions with spacetime singularities. Numerical relativity has recently revealed the properties of somewhat generic spacetime singularities. It has been found that in a variety of systems self-similar solutions can describe asymptotic or intermediate behaviour of more general solutions in an approach to singularities. The typical example is the convergence to an attractor self-similar solution in gravitational collapse. This is closely related to the cosmic censorship violation in the spherically symmetric collapse of a perfect fluid. The self-similar solution also plays an important role in critical phenomena in gravitational collapse. The critical phenomena are understood as the intermediate behaviour around a critical self-similar solution. We see that the convergence and critical phenomena are understood in a unified manner in terms of attractors of codimension zero and one, respectively, in renormalisation group flow.

Abstract:
General relativity as well as Newtonian gravity admits self-similar solutions due to its scale-invariance. This is a review on these self-similar solutions and their relevance to gravitational collapse. In particular, our attention is mainly paid on the crucial role of self-similar solutions in the critical behavior and attraction in gravitational collapse.

Abstract:
A direct method is developed to reconstruct the equation of state for high-density nuclear matter from the relationship between any two properties of neutron stars, such as masses, radii, moments of inertia, baryonic masses, binding energies, gravitational redshifts, and their combinations.

Abstract:
A stability analysis of a spherically symmetric star in scalar-tensor theories of gravity is given in terms of the frequencies of quasi-normal modes. The scalar-tensor theories have a scalar field which is related to gravitation. There is an arbitrary function, the so-called coupling function, which determines the strength of the coupling between the gravitational scalar field and matter. Instability is induced by the scalar field for some ranges of the value of the first derivative of the coupling function. This instability leads to significant discrepancies with the results of binary-pulsar-timing experiments and hence, by the stability analysis, we can exclude the ranges of the first derivative of the coupling function in which the instability sets in. In this article, the constraint on the first derivative of the coupling function from the stability of relativistic stars is found. Analysis in terms of the quasi-normal mode frequencies accounts for the parameter dependence of the wave form of the scalar gravitational waves emitted from the Oppenheimer-Snyder collapse. The spontaneous scalarization is also discussed.

Abstract:
Applying the first and generalised second laws of thermodynamics for a realistic process of near critical black hole formation, we derive an entropy bound, which is identical to Bekenstein's one for radiation. Relying upon this bound, we derive an absolute minimum mass $\sim0.04 \sqrt{g_{*}}m_{\rm Pl}$, where $g_{*}$ and $m_{\rm Pl}$ is the effective degrees of freedom for the initial temparature and the Planck mass, respectively. Since this minimum mass coincides with the lower bound on masses of which black holes can be regarded as classical against the Hawking evaporation, the thermodynamical argument will not prohibit the formation of the smallest classical black hole. For more general situations, we derive a minimum mass, which may depend on the initial value for entropy per particle. For primordial black holes, however, we show that this minimum mass can not be much greater than the Planck mass at any formation epoch of the Universe, as long as $g_{*}$ is within a reasonable range. We also derive a size-independent upper bound on the entropy density of a stiff fluid in terms of the energy density.

Abstract:
Gravitational collapse is one of the most striking phenomena in gravitational physics. The cosmic censorship conjecture has provided strong motivation for researches in this field. In the absence of general proof for the censorship, many examples have been proposed, in which naked singularity is the outcome of gravitational collapse. Recent development has revealed that there are examples of naked singularity formation in the collapse of physically reasonable matter fields, although the stability of these examples is still uncertain. We propose the concept of ``effective naked singularities'', which will be quite helpful because general relativity has the limitation of its application for high-energy end. The appearance of naked singularities is not detestable but can open a window for new physics of strongly curved spacetimes.

Abstract:
We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

Abstract:
A stability criterion is derived for self-similar solutions with perfect fluids which obey the equation of state $P=k\rho$ in general relativity. A wide class of self-similar solutions turn out to be unstable against the so-called kink mode. The criterion is directly related to the classification of sonic points. The criterion gives a sufficient condition for instability of the solution. For a transonic point in collapse, all primary-direction nodal-point solutions are unstable, while all secondary-direction nodal-point solutions and saddle-point ones are stable against the kink mode. The situation is reversed in expansion. Applications are the following: the expanding flat Friedmann solution for $1/3 \le k < 1$ and the collapsing one for $0< k \le 1/3$ are unstable; the static self-similar solution is unstable; nonanalytic self-similar collapse solutions are unstable; the Larson-Penston (attractor) solution is stable for this mode for $0

Abstract:
Ho？ava–Lifshitz gravity has covariance only under the foliation-preserving diffeomorphism. This implies that the quantities on the constant-time hypersurfaces should be regular. In the original theory, the projectability condition, which strongly restricts the lapse function, is proposed. We assume that a star is filled with a perfect fluid with no-radial motion and that it has reflection symmetry about the equatorial plane. As a result, we find a no-go theorem for stationary and axisymmetric star solutions in projectable？Ho？ava–Lifshitz？ gravity under the physically reasonable assumptions in the matter sector. Since we do not use the gravitational action to prove it, our result also works out in other projectable theories and applies to not only strong gravitational fields, but also weak gravitational ones.