General Relativistic Collapse Research Articles

Theory of frozars and its observable effects. 1. Structure of stars frozen during general relativistic collapse

Theory of frozars and its observable effects. 1. Structure of stars frozen during general relativistic collapse

Driven neutron star collapse: Type I critical phenomena and the initial black hole mass distribution

We study the general relativistic collapse of neutron star (NS) models in spherical symmetry. Our initially stable models are driven to collapse by the addition of one of two things: an initially in-going velocity profile, or a shell of minimally coupled, massless scalar field that falls onto the star. Tolman-Oppenheimer-Volkoff (TOV) solutions with an initially isentropic, gamma-law equation of state serve as our NS models. The initial values of the velocity profile's amplitude and the star's central density span a parameter space which we have surveyed extensively and which we find provides a rich picture of the possible end states of NS collapse. This parameter space survey elucidates the boundary between Type I and Type II critical behavior in perfect fluids which coincides, on the subcritical side, with the boundary between dispersed and bound end states. For our particular model, initial velocity amplitudes greater than 0.3c are needed to probe the regime where arbitrarily small black holes can form. In addition, we investigate Type I behavior in our system by varying the initial amplitude of the initially imploding scalar field. In this case we find that the Type I critical solutions resemble TOV solutions on the 1-mode unstable branch of equilibrium solutions, and that the critical solutions' frequencies agree well with the fundamental mode frequencies of the unstable equilibria. Additionally, the critical solution's scaling exponent is shown to be well approximated by a linear function of the initial star's central density.

Modeling quasar central engine as a relativistic radiating star

Long ago Hoyle & Fowler attempted to model the central engine of quasars as hot super-massive stars supported by radiation pressure. Whereas the model of Hoyle & Fowler was Newtonian, here we make a toy model of quasar central engines as ultra relativistic ultrahot plasma or as a ball of radiation. Accordingly, we consider general relativistic gravitational collapse including emission of radiation. More specifically, we discuss a new class of radiating fluid ball exact solution in conformally-flat metric which is quasi-static and contracting at negligible rate. The problem is solved by assuming that the metric potential is separable in to radial and time dependent parts. It is found the gravitational mass of the radiating ball M→0 as comoving time t→∞ in conformity of the idea of an “Eternally Collapsing Object” (ECO) which has been claimed to be the true nature of the so-called “Black Holes”. In particular, we consider here a quasi-static radiation ball having M≈9.507×107M⊙, a radius of ≈2×1014 km, and a luminosity L∞≈9.1×1046 erg/s. Prima-facie, such an ECO solution is compatible with the central compact object of a quasar having comoving lifetime of ≈107 yr and a distantly observed lifetime (u) which could be higher by many orders of magnitude.

Revisiting the old problem of general-relativistic adiabatic collapse of a uniform-density self-gravitating sphere

The problem of general-relativistic adiabatic collapse of a uniform-density perfect sphere has been studied since Wyman (Phys. Rev. 70, 396, (1946)) [1]. Apparently, there could be bouncing and oscillating solutions in such a case, as claimed by numerous authors since then. Consequently, various authors invoked such models for explaining pulsations of compact objects. However, here, for this age-old problem, we prove that for an assumed nonstatic adiabatically evolving sphere, density homogeneity implies (isotropic) pressure homogeneity too. This proof is based on the simple fact that in general relativity (GR), given one time coordinate t, one can employ another time coordinate t → t* = f(t) without any loss of generality. Since this proof does not use any exterior boundary condition, it is valid in a cosmological scenario too. However, here we focus on the evolution of an isolated sphere having a boundary. And the proof obtained here shows that a uniform-density perfect fluid collapse can occur only if the (isotropic) pressure is p = 0, i.e., only when the problem is reduced to the one treated by Oppenheimer and Snyder. For such an isolated sphere, we offer a supporting proof. This result is important and nontrivial because in the past 65 years innumerable authors working on this problem failed to see that the collapse of a supposed homogeneous sphere is (actually) synonymous to the old Oppenheimer-Snyder problem.

The fallacy of Oppenheimer Snyder collapse: no general relativistic collapse at all, no black hole, no physical singularity

By applying Birkhoff’s theorem to the problem of the general relativistic collapse of a uniform density dust, we directly show that the density of the dust ρ=0 even when its proper number density n would be assumed to be finite! The physical reason behind this exact result can be traced back to the observation of Arnowitt et al. (Phys. Rev. Lett. 4: 375, 1960) that the gravitational mass of a neutral point particle is zero: m=0. And since, a dust is a mere collection of neutral point particles, unlike a continuous hydrodynamic fluid, its density ρ=mn=0. It is nonetheless found that for k=−1, a homogeneous dust can collapse and expand special relativistically in the fashion of a Milne universe. Thus, in reality, general relativistic homogeneous dust collapse does not lead to the formation of any black hole in conformity of many previous studies (Logunov et al., Phys. Part. Nucl. 37: 317, 2006; Kiselev et al., Theor. Math. Phys. 164: 972, 2010; Mitra, J. Math. Phys. 50: 042502, 2009a; Suggett, J. Phys. A 12: 375 1979b). Interestingly, this result is in agreement with the intuition of Oppenheimer and Snyder (Phys. Rev. 56: 456, 1939) too:“Physically such a singularity would mean that the expressions used for the energy-momentum tensor does not take into account some essential physical fact which would really smooth the singularity out. Further, a star in its early stages of development would not possess a singular density or pressure, it is impossible for a singularity to develop in a finite time.”

New open-source approaches to the modeling of stellar collapse and the formation of black holes

We present new approaches to the simulation of stellar collapse, the formation of black holes, and explosive core-collapse supernova nucleosynthesis that build upon open-source codes and microphysics. We discuss the new spherically-symmetric general-relativistic (GR) collapse code GR1D that is endowed with an approximate 1.5D treatment of rotation, comes with multiple nuclear equations of state, and handles neutrinos with a multi-species leakage scheme. Results from a first set of spinning black hole formation simulations are presented. We go on to discuss the derivative code GR1D+N which is tuned for calculations of explosive nucleosynthesis and includes a NSE/non-NSE equation of state treatment, and a nuclear reaction network. We present sample results showing GR1D+N’s performance in reproducing previous results with thermal-bomb-driven explosions. Finally, we introduce the 3 + 1 GR Zelmani core collapse simulation package and present first results obtained in its application to the 3D modeling of failing core-collapse supernovae.

Likely formation of general relativistic radiation pressure supported stars or ‘eternally collapsing objects’

Abstract Hoyle & Fowler showed that there could be radiation pressure supported stars (RPSSs) even in Newtonian gravity. Much later, Mitra found that one could also conceive of their general relativistic (GR) version, ‘relativistic radiation pressure supported stars’ (RRPSSs). While RPSSs have z≪ 1, RRPSSs have z≫ 1, where z is the surface gravitational redshift. Here, we elaborate on the formation of RRPSSs during continued gravitational collapse by recalling that a contracting massive star must start trapping radiation as it enters its photon sphere. It is found that, irrespective of the details of the contraction process, the trapped radiation flux should attain the corresponding Eddington value at sufficiently large z≫ 1. This means that continued GR collapse may generate an intermediate RRPSS with z≫ 1 before a true black hole state with z=∞ is formed asymptotically. An exciting consequence of this is that the stellar mass black hole candidates, at the present epoch, should be hot balls of quark–gluon plasma, as has been discussed by Royzen in a recent article entitled ‘QCD against black holes?’.

Ohm’s Law in the Fast Lane: General Relativistic Charge Dynamics

Fully relativistic and causal equations for the flow of charge in curved spacetime are derived. It is believed that this is the first set of equations to be published that correctly describes the flow of charge, as well as the evolution of the electromagnetic field, in highly dynamical relativistic environments on timescales much shorter than the collapse time (GM/c3). The equations will therefore be important for correctly investigating problems such as the dynamical collapse of magnetized stellar cores to black holes and the production of jets. Both are potentially important problems in the study of gamma-ray burst engine models and in predicting the dynamical morphology of the collapse and the character of the gravitational waves generated. This system of equations, given the name of "charge dynamics," is analogous to those of hydrodynamics (which describe the flow of mass in spacetime rather than the flow of charge). The most important equation in the system is the relativistic generalized Ohm's law, which is used to compute time-dependent four-current. Unlike other equations for the current now in use, this one ensures that charge drift velocities remain less than the speed of light, takes into account the finite current rise time, is expressed in a covariant form, and is suitable for general relativistic computations in an arbitrary metric. It includes the standard known effects (Lorentz force, Hall effect, pressure effect, and resistivity) and reduces to known forms of Ohm's law in the appropriate limits. In addition, the plasma particles are allowed to have highly relativistic drift velocities, resulting in an implicit equation for the "current beaming factor" γq. It is proposed that, short of solving the multifluid plasma equations or the relativistic Boltzmann equation itself, these are the most general expressions for relativistic current flow in the one-fluid approximation, and they should be made part of the general set of equations that are solved in extreme black hole accretion and fully general numerical relativistic collapse simulations.

Critical phenomena in the Einstein-massless-Dirac system

We investigate the general relativistic collapse of spherically symmetric, massless spin-$\frac{1}{2}$ fields at the threshold of black hole formation. A spherically symmetric system is constructed from two spin-$\frac{1}{2}$ fields by forming a spin singlet with no net spin-angular momentum. We study the system numerically and find strong evidence for a type II critical solution at the threshold between dispersal and black hole formation, with an associated mass scaling exponent $\ensuremath{\gamma}\ensuremath{\sim}0.26.$ Although the critical solution is characterized by a continuously self-similar (CSS) geometry, the matter fields exhibit discrete self-similarity with an echoing exponent $\ensuremath{\Delta}\ensuremath{\sim}1.34.$ We then adopt a CSS ansatz and reduce the equations of motion to a set of ODEs. We find a solution of the ODEs that is analytic throughout the solution domain, and show that it corresponds to the critical solution found via dynamical evolutions.

Quantum general relativity and Hawking radiation

In a previous paper we have set up the Wheeler-DeWitt equation which describes the quantum general relativistic collapse of a spherical dust cloud. In the present paper we specialize this equation to the case of matter perturbations around a black hole, and show that in the WKB approximation, the wave-functional describes an eternal black hole in equilibrium with a thermal bath at Hawking temperature.

Does the Principle of Equivalence Prohibit Trapped Surfaces from Forming in the General Relativistic Collapse Process

It has recently been shown that time-like spherical collapse of a physical fluid in General Relativity does not permit formation of “trapped surfaces.” This result followed from the fact that the formation of a trapped surface in a physical fluid would cause the time-like world lines of the collapsing fluid to become null at the would-be trapped surface, thus violating the Principle of Equivalence in General Theory of Relativity (GTR). For the case of the spherical collapse of a physical fluid, the “no trapped surface condition” 2GM(r, t)/R(r, t) c2 0, where zs is the surface red shift seen by a zero-angular momentum observer. When this condition is applied to the first integral of the time-time component of the Einstein equation, it leads to the “no trapped surface condition” 2GM(rs, t)/R(rs, t) c2<1 consistent with the condition obtained above for the interior co-moving metric. The Principle of Equivalence enforces the “no trapped surface condition” by constraining the physics of the general relativistic radiation transfer process in a manner that requires it to establish and maintain an Eddington limited secular equilibrium on the dynamics of the collapsing radiating surface so as to always keep the physical surface of the collapsing object outside of its Schwarzschild radius. The important physical implication of the “no trapped surface condition” is that galactic black hole candidates GBHC do not possess event horizons and hence do possess intrinsic magnetic fields. In this context the spectral characteristics of galactic black hole candidates offer strong evidence that their central nuclei are highly red-shifted Magnetospheric Eternally Collapsing Objects (MECO) within the framework of General Relativity.

Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry

We perform a numerical study of the critical regime at the threshold of black hole formation in the spherically symmetric, general relativistic collapse of collisionless matter. The coupled Einstein-Vlasov equations are solved using a particle-mesh method in which the evolution of the phase-space distribution function is approximated by a set of particles (or, more precisely, infinitesimally thin shells) moving along geodesics of the spacetime. Individual particles may have non-zero angular momenta, but spherical symmetry dictates that the total angular momentum of the matter distribution vanish. In accord with previous work by Rein et al, our results indicate that the critical behavior in this model is Type I; that is, the smallest black hole in each parametrized family has a finite mass. We present evidence that the critical solutions are characterized by unstable, static spacetimes, with non-trivial distributions of radial momenta for the particles. As expected for Type I solutions, we also find power-law scaling relations for the lifetimes of near-critical configurations as a function of parameter-space distance from criticality.

NON-OCCURRENCE OF TRAPPED SURFACES AND BLACK HOLES IN SPHERICAL GRAVITATIONAL COLLAPSE

By using the most general form of Einstein equations for General Relativistic (GTR) spherical collapse of an isolated fluid having arbitrary equation of state and radiation transport properties, we show that they obey a Global Constraint, 2GM(r, t)/R(r, t)c2≤1, where R is the “invariant circumference radius”, t is the comoving time, and M(r, t) is the gravitational mass enclosed within a comoving shell r. This inequality specifically shows that, contrary to the traditional intuitive Newtonian idea, which equates the total gravitational mass (Mb) with the fixed baryonic mass (M0), the trapped surfaces are not allowed in general theory of relativity (GTR), and therefore, for continued collapse, the final gravitational mass Mf→0 as R→0. This result should be valid for all spherical collapse scenarios including that of collapse of a spherical homogeneous dust as enunciated by Oppenheimer and Snyder (OS). Since the argument of a logarithmic function cannot be negative, the Eq. (36) of the O–S paper (T∼In\(T \sim \ln \tfrac{{y_b + 1}}{{y_b - 1}}\)) categorically demands that yb=Rb/Rgb≥1, or 2GMb/Rbc2≤1, where Rb referes to the invariant radius at the outer boundary. Unfortunately, OS worked with an approximate form of Eq. (36) [Eq. 37], where this fundamentalconstraint got obfuscated. And although OS noted that for a finite value of M(r, t) the spatial metric coefficient for an internal point fails to blow up even when the collapse is complete\(e^{\lambda (r < r_b )} \) ≠ ∞ for R→0, they, nevertheless, ignored it, and, failed to realize that such a problem was occurring because they were assuming a finite value ofMf, where Mf is the value of the finite gravitational mass, in violation of their Eq. (36).

Towards a Probabilistic Paradigm for Gravitational Collapse, Black Hole Creation and Singularity Smearing

A prototype probability interpretation ispresented for the Oppenheimer-Snyder model ofspherically symmetric, gravitational collapse of apressureless ensemble of n point particles. A transitionprobability P(R(t), t; R1, t1) isderived for an initial sphere or fluid star of radius Rat comoving time t, collapsing smoothly andhomogeneously to any finite radii R(t, r) t1 and R(t) = 0 at t = tf. The transitionprobability is evaluated in two cases. In the firstcase, Planck's constant is assumed zero and smoothdifferential limits exist for space and matter on alllength scales down to zero. The probability for singularityformation converges smoothly to unity as R → 0 ort → tf: the collapse is deterministic atall scales. There is also a finite, nonzero probabilityof event horizon formation at R = Rh = 2GM, but the starcontinues to collapse through this radius since there isalways a higher probability of reaching any smallerradius R < Rh. An event horizon forms sothe collapsed state is still a black hole. In the classical limit(as ℏ → 0) the singularity returns with unitprobability. Finally, we briefly discuss how the final,fuzzy, collapsed state may be related to aspects ofstring theory. The emphasis of the paper is on theconceptual ideas and general possibilities which couldarise when incorporating stochastic mechanics andanalysis into general relativistic collapse.

An Implicit Lagrangian Code for Spherically Symmetric General Relativistic Hydrodynamics with an Approximate Riemann Solver

An implicit Lagrangian hydrodynamics code for general relativistic spherical collapse is presented. This scheme is based on an approximate linearized Riemann solver (Roe-type scheme) and needs no artificial viscosity. This code is aimed especially at the calculation of the late phase of collapse-driven supernovae and the nascent neutron star, where there is a remarkable contrast between the dynamical timescale of the proto-neutron star and the diffusion timescale of neutrinos, without such severe limitation of the Courant condition at the center of the neutron star. Several standard test calculations have been done, and their results show the following: (1) This code captures the shock wave accurately, although some erroneous jumps of specific internal energy are found at the contact discontinuity in the shock tube problems. (2) The scheme shows no instability even if we choose the Courant number larger than 1. (3) However, the Courant number should be kept below ~0.2 at the shock position, so that the shock can be resolved with a few meshes. (4) The scheme reproduces the well-known analytic solutions to the point blast explosion, the gravitational collapse of the uniform gas with γ = 4/3, and the general relativistic collapse of uniform dust. In addition, what is more important, calculations of hydrostatic configurations and the onset of radial instability, a preliminary neutrino transport in a proto-neutron star, and adiabatic collapses of a stellar core have also been done in order to show the performance of the code in the context of the collapse-driven supernovae. It is found that the time step can be extended far beyond the Courant limitation at the center of the neutron star, which fact is crucially significant for the purpose of this project. The details of the scheme and the results of these test calculations are discussed.

General relativistic collapse of homothetic ideal gas spheres and planes.

This paper presents in a succinct but self-contained style of our underatanding of the gravitational collapse of homothetic, ideal gas spheres and planes. The physical problem is reduced to a study of a nonlinear autonomous system of differential equations. It is first shown that this system is a Cauchy system everywhere in the projective space t/r ≡ ξ ∈ R. The concept of sonic Cauchy and apparent horizons is introduced, and it is shown that the set of globally analytic naked solutions is discrete as mentioned by Ori and Piran but is finite and even empty for very strong equations of state. Even when singularities may be «seen,» we are able to show that they cannot be «heard»

General relativistic textures and their interactions with matter and radiation

Time-dependent solutions of the coupled Einstein-σ-model equations are studied analytically and numerically. We analyse in detail a spherically symmetric self-similar solution which describes the general relativistic collapse of a global texture. In view of the cosmological implications of the texture scenario for large scale structure formation, we derive exact analytical formulae for the light deflection, photon red shift and density fluctuations in the gravitational field of this texture solution. We also discuss the behavior of collisionless particles. Numerical results of a parameter study of other texture solutions are presented and are compared with the special self-similar solution.

General relativistic collapse of textures

We present an exact self-similar solution of the coupled Einstein-σ model equations which describes the general relativistic collapse of global textures. In one coordinate system the texture geometry has a simple interpretation in terms of a deficit solid angle. We also briefly discuss the behavior of matter and light in this geometry. In particular we show that the weak field approximation for the metric perturbations of flat space texture solutions is quantitatively quite reliable.

General Relativistic Collapse to Black Holes and Gravitational Waves from Black Holes

General Relativistic Collapse to Black Holes and Gravitational Waves from Black Holes

General Relativistic Collapse of Accreting Neutron Stars with Rotation

General Relativistic Collapse of Accreting Neutron Stars with Rotation