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We study the energy issue in critical collapse of a spherically symmetric scalar field. It is found that in critical collapse, the contribution from the material energy is greater than that from the gravitational energy. The quantity $m/r$ plays an important role in identifying the formation of apparent horizon in gravitational collapse, where $m$ is the Misner-Sharp mass and $r$ the areal radius. We observe that in critical collapse, the maximum value of $m/r$ fluctuates between $2/15$ and $4/15$. This denotes a large gap between critical collapse and black hole formation for which the criterion is $m/r=1/2$., Comment: 8 pages, 7 figures

We study the collisions of two scalar wave packets in the asymptotically flat spacetime and asymptotically anti-de Sitter spacetime in spherical symmetry. An energy transfer formula is obtained, $y=Cm_{i}m_{o}/r$, where $y$ is the transferred energy in the collisions of the two wave packets, $m_i$ and $m_o$ are the Misner-Sharp energies for the ingoing and outgoing wave packets, respectively, $r$ is the areal radius and collision place, and $C=1.873$ and $C=1.875$ for the asymptotically flat spacetime and asymptotically anti-de Sitter spacetime circumstances, respectively. The formula is universal, independent of the initial profiles of the scalar fields., Comment: 8 pages, 9 figures

We study the dynamics of the critical collapse of a spherically symmetric scalar field. Approximate analytic expressions for the metric functions and matter field in the large-radius region are obtained. In the central region, owing to the boundary conditions, the equation of motion for the scalar field is reduced to the flat-spacetime form., Comment: 8 pages, 5 figures

We study the dynamics near the central singularity in spherically symmetric collapse of a massless scalar field toward Schwarzschild black hole formation. The equations of motion take different simplified forms in the early and late stages of the singularity curve. We report some fine structures of the analytic solutions and universal features for the metric functions and matter near the singularity., Comment: 10 pages, 10 figures

The critical collapse of a scalar field is a threshold solution of black hole formation, in which a naked singularity arises. We study here the curvature strength of this singularity using a numerical ansatz. The behavior of the Jacobi volume forms is examined along a non-spacelike geodesic in the limit of approach to the singularity. These are seen to be vanishing, thus showing that all physical objects will be crushed to zero size near the singularity. Consequently, although the critical collapse is considered to be a fine-tuning problem, the naked singularity forming is gravitationally strong. This implies that the spacetime cannot be extended beyond the singularity, thus making the singularity genuine and physically interesting. These results imply that the nature of critical collapse may need to be examined and explored further., Comment: 5 pages, 3 figures

We investigate here the thermal properties of accretion disks in a spacetime for some galactic density profiles in spherical symmetry. The matter distributions have a finite outer radius with a naked central singularity. The luminosities of the accretion disks for some density profile models are found to be higher than those for a Schwarzschild black hole of the same mass. The slopes for the luminosity distributions with respect to frequencies are significantly different, especially at higher frequencies, from that in the Schwarzschild black hole case. Such features may be used to distinguish black holes from naked singularities. The efficiencies for the conversion of the mass energy of the accreting gas into radiation and the strength of naked singularities are analyzed. The novel feature that we find is, the strength of the singularity is different depending on the profiles considered, and the stronger the singularity is, the higher is the efficiency for the accretion disk., Comment: 23 pages, 9 figures, 2 tables

Critical collapse of a massless scalar field in spherical symmetry is systematically studied. We combine numerical simulations and asymptotic analysis, and synthesize critical collapse, spacetime singularities, and complex science. First set of approximate analytic expressions near the center are obtained. We observe that, near the center, the spacetime is nearly conformally flat, the dynamics is not described by the Kasner solution, and the Kreschmann scalar is proportional to r^(-5.30), where r is the areal radius. These features are significantly different from those in black hole singularities. It is speculated that the scalar field in critical collapse may be a special standing wave., Comment: Title changed. 11 pages, 8 figures, 1 table

The superspinar proposed by Gimon and Horava is a rapidly rotating compact entity whose exterior is described by the over-spinning Kerr geometry. The compact entity itself is expected to be governed by superstringy effects, and in astrophysical scenarios it can give rise to interesting observable phenomena. Earlier it was suggested that the superspinar may not be stable but we point out here that this does not necessarily follow from earlier studies. We show, by analytically treating the Teukolsky equations by Detwiler's method, that in fact there are infinitely many boundary conditions that make the superspinar stable, and that the modes will decay in time. It follows that we need to know more on the physical nature of the superspinar in order to decide on its stability in physical reality., Comment: 5 pages

Spherical vacuum and scalar collapse for the Starobinsky R^2 model is simulated. Obtained by considering the quantum-gravitational effects, this model would admit some cases of singularity-free cosmological spacetimes. It is found, however, that in vacuum and scalar collapse, when f' or the physical scalar field is strong enough, a black hole including a central singularity can be formed. In addition, near the central singularity, gravity dominates the repulsion from the potential, so that in some circumstances the Ricci scalar is pushed to infinity by gravity. Therefore, the semiclassical effects as included here do not avoid the singularity problem in general relativity. A strong physical scalar field can prevent the Ricci scalar from growing to infinity. Vacuum collapse for the RlnR model is explored, and it is observed that for this model the Ricci scalar can also go to infinity as the central singularity is approached. Therefore, this feature seems universal in vacuum and scalar collapse in f(R) gravity., Comment: 10 pages, 6 figures. Major change

We study spherical scalar collapse toward a black hole formation and examine the asymptotic dynamics near the central singularity of the formed black hole. It is found that, in the vicinity of the singularity, due to the strong backreaction of a scalar field on the geometry, the mass function inflates and the Kretschmann scalar grows faster than in the Schwarzschild geometry. In collapse, the Misner-Sharp mass is a locally conserved quantity, not providing information on the black hole mass that is measured at asymptotically flat regions., Comment: 7 pages, 3 figures. Minor change. Related paper. arXiv:1507.01806

In this paper, we explore the interior dynamics of neutral and charged black holes in $f(R)$ gravity. We transform $f(R)$ gravity from the Jordan frame into the Einstein frame and simulate scalar collapses in flat, Schwarzschild, and Reissner-Nordstr\"om geometries. In simulating scalar collapses in Schwarzschild and Reissner-Nordstr\"om geometries, Kruskal and Kruskal-like coordinates are used, respectively, with the presence of $f'$ and a physical scalar field being taken into account. The dynamics in the vicinities of the central singularity of a Schwarzschild black hole and of the inner horizon of a Reissner-Nordstr\"om black hole is examined. Approximate analytic solutions for different types of collapses are partially obtained. The scalar degree of freedom $\phi$, transformed from $f'$, plays a similar role as a physical scalar field in general relativity. Regarding the physical scalar field in $f(R)$ case, when $d\phi/dt$ is negative (positive), the physical scalar field is suppressed (magnified) by $\phi$, where $t$ is the coordinate time. For dark energy $f(R)$ gravity, inside black holes, gravity can easily push $f'$ to $1$. Consequently, the Ricci scalar $R$ becomes singular, and the numerical simulation breaks down. This singularity problem can be avoided by adding an $R^2$ term to the original $f(R)$ function, in which case an infinite Ricci scalar is pushed to regions where $f'$ is also infinite. On the other hand, in collapse for this combined model, a black hole, including a central singularity, can be formed. Moreover, under certain initial conditions, $f'$ and $R$ can be pushed to infinity as the central singularity is approached. Therefore, the classical singularity problem, which is present in general relativity, remains in collapse for this combined model., Comment: 35 pages, 22 figures. (Special Issue. Modified Gravity Cosmology: From Inflation to Dark Energy). Minor change. arXiv admin note: substantial text overlap with arXiv:1507.01806

In this paper, we explore the interior dynamics of neutral and charged black holes. Scalar collapses in flat, Schwarzschild, and Reissner-Nordstrom geometries are simulated. We examine the dynamics in the vicinities of the central singularity of a Schwarzschild black hole and of the inner horizon of a Reissner-Nordstrom black hole. In simulating scalar collapses in Schwarzschild and Reissner-Nordstrom geometries, Kruskal and Kruskal-like coordinates are used, respectively, with the presence of a scalar field being taken into account. It is found that, besides near the inner horizons of Reissner-Nordstrom and Kerr black holes, mass inflation also takes place near the central singularity in neutral scalar collapse. Approximate analytic expressions for different types of mass inflation are partially obtained via a close interplay between numerical and analytical approaches and an examination of the connections between Schwarzschild black holes, Reissner-Nordstrom black holes, neutral collapse, and charge scattering. We argue that the mass inflations near the central singularity and the inner horizon are related to the localness of the dynamics in strong gravity regions. This is in accord with the Belinskii, Khalatnikov, and Lifshitz conjecture., Comment: 38 pages, 23 figures. Minor change

Spherical scalar collapse in f(R) gravity is studied numerically in double-null coordinates in the Einstein frame. Dynamics in the vicinity of the singularity of the formed black hole is examined via mesh refinement and asymptotic analysis. Before the collapse, the scalar degree of freedom f' is coupled to a physical scalar field, and general relativity is restored. During the collapse, the major energy of the physical scalar field moves to the center. As a result, f' loses the coupling and becomes light, and gravity transits from general relativity to f(R) gravity. Due to strong gravity from the singularity and the low mass of f', f' will cross the minimum of the potential and approach zero. Therefore, the dynamical solution is significantly different from the static solution of the black hole in f(R) gravity---it is not the de Sitter-Schwarzschild solution as one might have expected. f' tries to suppress the evolution of the physical scalar field, which is a dark energy effect. As the singularity is approached, metric terms are dominant over other terms. The Kasner solution for spherical scalar collapse in f(R) theory is obtained and confirmed by numerical results. These results support the Belinskii-Khalatnikov-Lifshitz conjecture well., Comment: Results in the Jordan frame and for the Starobinsky model are added, see Sections V and VI. 27 pages, 14 figures