Lorentzian-Euclidean singularity-free solutions to gravitational collapse

There is a possibility that the event horizon of a Kerr-like black hole with perfect fluid dark matter (DM) can be destroyed, providing a potential opportunity for understanding the weak cosmic censorship conjecture of black holes. In this study, we analyze the influence of the strength parameter of perfect fluid DM on the destruction of the event horizon of a Kerr-like black hole with spinning after injecting a test particle and a scalar field. We find that, when a test particle is incident on the black hole, the event horizon is destroyed by perfect fluid dark matter for extremal black holes. For nearly extremal black holes, when the dark matter parameter satisfies α∈-rh,0∪rh,k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in \\left( -r_{h} , 0\\right) \\cup \\left( r_{h} ,k_2\\right) $$\\end{document} i.e. (A<0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(A<0),$$\\end{document} the event horizon of the black hole will not be destroyed; when the dark matter parameter satisfies α∈k1,-rh∪0,rh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in \\left( k_1 ,-r_{h} \\right] \\cup \\left[ 0,r_{h}\\right] $$\\end{document} i.e. (A≥0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(A\\ge 0),$$\\end{document} the event horizon of the black hole will be destroyed. When a classical scalar field is incident into the black hole in the extremal black hole case, we find that the range of mode patterns of the scalar field that can disrupt the black hole event horizon is different for different values of the perfect fluid dark matter strength parameter. In the nearly extremal black hole case, through our analysis, we have found when α≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \ e 0 $$\\end{document} and α≠±rh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \ e \\pm \\ r_h$$\\end{document} i.e. A≠0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\ e 0,$$\\end{document} the event horizon of the black hole can be disrupted. Our research results indicate that dark matter might be capable of breaking the black hole horizon, thus potentially violating the weak cosmic censorship conjecture.

This paper is a technical review for a more deliberate paper (Bhattacharya &amp; Lahiri, 2007) where it has been shown that on a positive cosmological scale with Λ&gt;0 having a cosmic horizon scale ~1/√Λ, there exists the soft electric hairs for the solution having the T_00 components of the stress-energy tensor T_μν i.e., ρ=0 on black hole horizon B_H having the maximum density at black hole singularity B_S where cosmic horizon C_H and black hole horizon B_H has only been considered. KEYWORDS: Black Hole – Cosmic Horizon – TOV Limit – Stress-Energy Tensor – Positive Cosmological Constant

This research is an extension of our earlier published (2+1) dimensional cosmological models in f(R,T) gravity with Λ(R,T) (IOP Conf. Ser. J. Phys. Conf. Ser. 2019, 1258, 012026). A different class of cosmological space model is studied under modified theories of f(R,T) gravity, where the cosmological constant Λ is expressed as a function of the Ricci scalar R and the trace of the stress-energy momentum tensor T. We call such a model as “Λ(R,T) gravity”. Such a specific form of Λ(R,T) has been defined in the dust as well as in the perfect fluid case. We intend to search for a combined model that satisfies the equation of state for dark energy matter or quintessence matter or perfect matter fluid. Some geometric and intrinsic physical properties of the model are also described. The energy conditions, pressure and density are discussed both when Λ=Λ(r) is a function of the radial parameter r, as well as when Λ is zero. We study the effective mass function and also the gravitational redshift function, both of which are found to be positive as per the latest observations. The cosmological model is studied in f(R,T) modified theory of gravity, where the gravitational Lagrangian is expressed both in terms of the Ricci scalar R and the trace of the stress-energy tensor T. The equation of state parameter is discussed in terms of ω corresponding to the three cases mentioned above. The behaviour of the cosmological constant is separately examined in the presence of quintessence matter, dark energy matter and perfect fluid matter.

We consider a way to avoid black hole singularities by gluing a black hole exterior to an interior with a tube-like geometry consisting of a direct product of two-dimensional AdS, dS, or Rindler spacetime with a two-sphere of constant radius. As a result we obtain a spacetime with either “cosmological” or “acceleration” (event) horizons but without an apparent horizon. The inner region is everywhere regular and supported by matter with the vacuum-like equation of state pr+ρ=0 where pr=Trr is the longitudinal pressure, ρ=−T00 is the energy density, Tμν is the stress–energy tensor. When the surface of gluing approaches the horizon, surface stresses vanish, while pr may acquire a finite jump on the boundary. Such composite spacetimes accumulate an infinitely large amount of matter inside the horizon but reveal themselves for an external observer as a sphere of a finite ADM mass and size. If the throat of the inner region is glued to two black hole exteriors, one obtains a wormhole of an arbitrarily large length. Wormholes under discussion are static but not traversable, so the null energy condition is not violated. In particular, they include the case with an infinite proper distance to the throat. We construct also gravastars with an infinite tube as a core and traversable wormholes connected by a finite tube-like region.

In this chapter, solutions of the conservation equations in partial-differential equation form are sought for a simple case—namely, steady, incompressible, inviscid two-dimensional flow. Each of these crucial assumptions is discussed in detail and their applicability as models of real flow-field situations are justified. Body forces such as gravity effects are neglected because they are negligible in most aerodynamics problems. Simple geometries are considered first. The analysis is then extended so that finally it is possible to represent the complex flow field around realistic airfoil shapes, such as those needed to efficiently produce lift forces for flight vehicles. Chapter 5 is a detailed treatment of two-dimensional airfoil flows.

Within the general theory of relativity the Bianchi type VIII cosmological models with rotationand expansion have been built. The first case includes a field of radiation, the second one -a perfect fluid with dust-like equation of state. Perfect anisotropic fluid imitates the rotatingdark energy. Static and dynamic cosmological modes have been observed, at the same timethe equations of state are partly postulated in the first case for the anisotropic fluid and inthe second case - for the perfect isotropic fluid, that imitates baryon matter. The analysisof absence of closed time-like curves has been done, so the models have been proved to becasual when the metric parameters satisfy the found conditions. Also the conditions, when theanisotropic fluid’s equation of state becomes vacuum-like, the energy of the fluid dominatesand it becomes asymptotically isotropic, have been cleared out. Specialities of the oscillatingmode have been observed. The order of present angular velocity value, calculated within thecosmological models, has been found to be quite satisfactory when expanding from the Plank scale to the present size of observed part of the Universe. The found solutions may be used foreffects taking place nowadays and also during the inflationary stage.

In this work, a four-dimensional cylindrical symmetric and non-static or static space-times in the backgrounds of anti-de Sitter (AdS) spaces with perfect stiff fluid, anisotropic fluid and electromagnetic field as the stress–energy tensor, is presented. For suitable parameter conditions in the metric function, the solution represents non-static or static non-rotating black hole solution. In addition, we show for various parameter conditions, the solution represents static and/or non-static models with a naked singularity without an event horizon.

The initial state of the spherical gravitational collapse in general relativity has been studied with different methods, especially by using a priori given equations of state that describe the matter as a perfect fluid. We propose an alternative approach, in which the energy density of the perfect fluid is given as a polynomial function of the radial coordinate that is well-behaved everywhere inside the fluid. We then solve the corresponding differential equations, including the Tolman–Oppenheimer–Volkoff equilibrium condition, using a fourth-order Runge–Kutta method and obtain a consistent model with a central perfect-fluid core surrounded by dust. We analyze the Hamiltonian constraint, the mass-to-radius relation, the boundary and physical conditions, and the stability and convergence properties of the numerical solutions. The energy density and pressure of the resulting matter distribution satisfy the standard physical conditions. The model is also consistent with the Buchdahl limit and the speed of sound conditions, even by using realistic values of compact astrophysical objects such as neutron stars.

We show that the stress-energy tensor has additional terms with respect to the ideal form in states of global thermodynamic equilibrium in flat spacetime with nonvanishing acceleration and vorticity. These corrections are of quantum origin and their leading terms are second order in the gradients of the thermodynamic fields. Their relevant coefficients can be expressed in terms of correlators of the stress-energy tensor operator and the generators of the Lorentz group. With respect to previous assessments, we find that there are more second-order coefficients and that all thermodynamic functions including energy density receive acceleration and vorticity dependent corrections. Notably, also the relation between $\ensuremath{\rho}$ and $p$, that is, the equation of state, is affected by acceleration and vorticity. We have calculated the corrections for a free real scalar field---both massive and massless---and we have found that they increase, particularly for a massive field, at very high acceleration and vorticity and very low temperature. Finally, these nonideal terms depend on the explicit form of the stress-energy operator, implying that different stress-energy tensors of the scalar field---canonical or improved---are thermodynamically inequivalent.

The first law of black hole mechanics (in the form derived by Wald) is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism-invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter, and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. One would expect that Wald's methods, applied to a Lagrangian Einstein-perfect fluid formulation, would convert these terms to surface integrals. However, because the fields appearing in the Lagrangian of a gravitating perfect fluid are typically nonstationary (even in a stationary black-hole--perfect-fluid spacetime) a direct application of these methods generally yields restricted results. We therefore first approach the problem of incorporating general nonstationary matter fields into Wald's analysis, and derive a first-law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter, and Hawking when the theory is general relativity coupled to a perfect fluid. We then turn to consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis, assuming that both the metric and fluid fields are stationary and axisymmetric in the black hole spacetime. The first law we derive contains only surface integrals at the black hole horizon and spatial infinity, but the assumptions of stationarity and axisymmetry of the fluid fields make this relation much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter, and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.

Accretion of matter onto astronomical objects is an important phenomenon of long-standing interest to astrophysicists and the most likely scenario to explain the high energy output from quasars and active galactic nuclei. Some analytic solutions for simple cases have been obtained in literatures. Here we obtain an analytic solution for accretion of a gaseous medium onto a Kerr-Newman black hole which moves at a constant velocity through the medium. The gaseous medium has an adiabatic equation of state: P=ρ, which implies that the adiabatic index is equal to 2 and that the speed of sound is equal to the speed of light, so the flow velocity must be subsonic everywhere and therefore no shock waves arise. We consider the flow in the Kerr-Newman black hole rest frame. The flow is into, and not out from, a Kerr-Newman black hole. We seek a stationary solution, assuming a homogeneous fluid moving at constant velocity at large distances. The flow of matter is approximated as a perfect fluid with zero vorticity, which means the velocity of the perfect fluid can be expressed as the gradient of a potential: huμ=ψ,μ. Assuming that no particles are created or destroyed, then the particle density n is conserved, which results in the equation of continuity for the particle density: (nuα);α=0 and it reduces to (ψ,α);α=0 for the case of an adiabatic equation of state. By assuming the asymptotic boundary condition in spherical coordinates and the general formula of the solution in the Kerr-Newman spacetime, we obtain the concrete form of the equation for the potential (ψ,α);α=0 and derive the solution under two boundary conditions: the expression of the potential ψ at infinity and the particle density n are finite everywhere, including at the event horizon of a black hole. We present the mass accretion rate which depends on Lorentz factor, the mass, the electric charge, and the angular momentum, but it is independent of the orientation of the black hole’s spin with respect to the incident direction of the flow. The flow is fully three dimensional for Kerr-Newman black hole, which is a function of spherical coordinates (r, θ, ϕ) and has no spherical symmetry and axial symmetry. The results obtained here may provide valuable physical insight into the more complicated cases and can be generalized to other types of black hole.

This chapter is concerned with a thermodynamic analysis on a flat FLRW universe admitting both apparent and event horizons. As the physical system within the cosmological apparent horizon forms a Bekenstein system, so we have considered the Bekenstein entropy and the Hawking temperature on the apparent horizon. However, since the system bounded by the event horizon may not be a Bekenstein system, we have assumed the Clausius relation on the event horizon to determine its entropy variation. Moreover, we have assumed the modified Hawking temperature on the event horizon. Three types of dark energy cosmic fluids have been considered—a perfect fluid with a constant equation of state, an interacting holographic dark energy, and a Chaplygin gas.

We analyze Hawking radiation as perceived by a freely-falling observer and try to draw an inference about the region of origin of the Hawking quanta. To do so, first we calculate the energy density from the stress energy tensor, as perceived by a freely-falling observer. Then we compare this with the energy density computed from an effective temperature functional which depends on the state of the observer. The two ways of computing these quantities show a mismatch at the light ring outside the black hole horizon. To better understand this ambiguity, we show that even taking into account the (minor) breakdown of the adiabatic evolution of the temperature functional which has a peak in the same region of the mismatch, is not enough to remove it. We argue that the appearance of this discrepancy can be traced back to the process of particle creation by showing how the Wentzel–Kramers–Brillouin approximation for the field modes breaks down between the light ring at 3M and 4M, with a peak at r=3.3M exactly where the energy density mismatch is maximized. We hence conclude that these facts strongly support a scenario where the Hawking flux does originate from a “quantum atmosphere” located well outside the black hole horizon.

The theory of distributions is applied to determine the source of the Kerr geometry. The stress-energy tensor is obtained, via Einstein’s field equations, by adopting the «branch cut» interpretation of the equatorial disk which spans the ring singularity. The stress-energy tensor is thus assumed to be a distribution having its support throughout the disk and along the ring. A simple model, which accounts for the tensor so obtained, is set up. Upon the disk it consists in a mixture of gas and dust rotating as a whole about the symmetry axis with constant angular velocity ω=a −1. The linear velocity, in turn, increases from zero at the centre of the disk up to the speed of light on its edge. The energy densities of both perfect fluids have the same absolute value, but are of opposite sign. On the other hand, the source on the ring is made up of dustlike particles having infinite positive energy and a finite radial dipole moment. Although these particles circulate along the ring with the speed of light, they cannot escape because they are pulled by an infinite tension (negative pressure) of the gas lying on the disk. It is further verified that the stress-energy tensor correctly reproduces the values of the mass and angular momentum of the Kerr black hole.

In this paper, the model of radial motion of dusty matter for the spherically symmetric case in General Relativity (GR) was developed. The model can be used, in particular, to describe galactic halos of dark matter. The question of the qualitative and quantitative composition of dark matter is of great importance both for understanding the current structure of the Universe and for choosing the most realistic scenario of its evolution. Since dark matter effectively manifests itself only gravitationally, its pressure can be neglected, and the equation of state, despite its physical nature, can be considered dusty. The presence of supermassive black holes in the centers of galaxies and the scale of the phenomena necessitate the use of GR equations. The spherical shapes of galactic halos require the use of spherical symmetry. The aim of this work is to model the radial motion of dark matter for a spherically symmetric GR interval, taking into account a possible central mass. Dark matter is assumed to be dusty and moving both toward and away from the center. It is proved that such a stationary case corresponds to a static interval of spacetime in the coordinates of curvatures. The system of Einstein's equations for this case is significantly simplified and solved numerically. The Lichnerowicz-Darmois conditions of crosslinking the spacetime of the proposed model with the external spherically symmetric Schwarzschild spacetime are chosen as boundary conditions. In the proposed model, there is an event horizon, approaching which the motion of particles seems to “freeze”, similar to the motion near the event horizon in the Schwarzschild field. Thus, the model also takes into account the presence of a black hole in the center, which should have been formed as a result of the considered motion of matter. The proposed model can be applied to spherical galaxy clusters and even to stellar systems in space beyond the star, assuming the existence of a halo of cold dark matter particles around them.