QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

When all thermonuclear sources of energy are exhausted a sufficiently heavy star will collapse. Unless fission due to rotation, the radiation of mass, or the blowing off of mass by radiation, reduce the star's mass to the order of that of the sun, this contraction will continue indefinitely. In the present paper we study the solutions of the gravitational field equations which describe this process. In I, general and qualitative arguments are given on the behavior of the metrical tensor as the contraction progresses: the radius of the star approaches asymptotically its gravitational radius; light from the surface of the star is progressively reddened, and can escape over a progressively narrower range of angles. In II, an analytic solution of the field equations confirming these general arguments is obtained for the case that the pressure within the star can be neglected. The total time of collapse for an observer comoving with the stellar matter is finite, and for this idealized case and typical stellar masses, of the order of a day; an external observer sees the star asymptotically shrinking to its gravitational radius.

On Continued Gravitational Contraction

Physical acceptability of isolated, static, spherically symmetric, perfect fluid solutions of Einstein's equations

The Einstein–Maxwell equations with anisotropic pressures and electromagnetic field are studied with a polytropic equation of state. New exact solutions to the field equations are generated in terms of elementary functions. Special cases of the uncharged solutions of Feroze and Siddiqui (Gen Relativ Gravit 43:1025, 2011) and Maharaj and Mafa Takisa (Gen Relativ Gravit 44:1419, 2012) are recovered. We also obtain exact solutions for a neutral anisotropic gravitating body for a polytrope from our general treatment. Graphical plots indicate that the energy density, tangential pressure and anisotropy profiles are consistent with earlier treatments which suggest relevance in describing relativistic compact stars.

Some charged polytropic models

We consider the general situation of a compact relativistic body with anisotropic pressures in the presence of the electromagnetic field. The equation of state for the matter distribution is linear and may be applied to strange stars with quark matter. Three classes of new exact solutions are found to the Einstein–Maxwell system. This is achieved by specifying a particular form for one of the gravitational potentials and the electric field intensity. We can regain anisotropic and isotropic models from our general class of solutions. A physical analysis indicates that the charged solutions describe realistic compact spheres with anisotropic matter distribution. The equation of state is consistent with dark energy stars and charged quark matter distributions. The masses and central densities correspond to realistic stellar objects in the general case when anisotropy and charge are present.

Charged anisotropic matter with a linear equation of state, Classical and Quantum Gravity

In this paper, we are proposing a three-layered hybrid compact star model with a distinct equation of states (EOSs) in the realm of general relativity. The core is assumed to be quark matter described by the MIT-bag model, an intermediate layer filled with neutron liquid and a thin envelope of matter satisfying a quadratic EoS. Three pairs of interfaces are matched by using Darmois–Israel conditions. For better and easier tuning, we have chosen $$\alpha $$
 as a free parameter for core, k for intermediate layer and g and t for envelope, while the rest of the constant parameters are linked with mass and radius. Most of the physical parameters such as density, pressures and EoS parameters are continuous in all the three regions; however, $$v_t^2$$
 and stability factor are discontinuous. This is because of the non-differentiability of $$p_t$$
 ’s at the interfaces. Hence, stability is not defined at the interfaces. Further, matching of $$p_t$$
 ’s can be performed by tuning suitable values of the free parameters $$\alpha , ~k, ~g$$
 and t. Further, the most prevailing aspect of this method is that we can arbitrarily choose the radii of each region. As per Buchler and Barkat (PRL 27: 48, 1971) and Baym et al.(PRL 175: 225, 1971) , there exists a smooth transition density between quark core and intermediate neutron-liquid layer at about $$\rho > 10^{14}~{\mathrm{g/cc}}$$
 . Our calculation shows that the smooth transition density is at about $$\rho _I=4.16 \times 10^{14}~ {\mathrm{g/cc}}$$
 which is in good agreement with the above prediction.

Three-layered relativistic hybrid star with distinct equation of states

A relativistic model of stellar objects with core-crust-envelope division

We present a spherically symmetric solution of the general relativistic field equations in isotropic coordinates for perfect fluid, compatible with a super dense star modeling. The solution is well behaved for all the values of u lying in the range 0<u≤0.12. Further, we have constructed a super-dense star model with all degree of suitability and by assuming the surface density ρ
 b
 =2×1014 g/cm3. Corresponding to u=0.12, the resulting well behaved model has maximum mass M=0.912M
 Θ
 with radius R
 b
 ≈11.27 km and Moment of inertia 0.97×1045 gm cm2. The good matching of our results for Vela pulsars show the stoutness of our model.

An Exact Solution of Perfect Fluid in Isotropic Coordinates, Compatible with Relativistic Modeling of 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≈9.507×107
M
⊙, 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.

Modeling quasar central engine as a relativistic radiating star

We present a well behaved class of charged analogue of M.C. Durgapal (J. Phys. A, Math. Gen. 15:2637, 1982) solution. This solution describes charged fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one and causality condition is obeyed at the centre. The outmarch of pressure, density, pressure-density ratio and the adiabatic speed of sound is monotonically decreasing, however, the electric intensity is monotonically increasing in nature. This solution gives us wide range of parameter for every positive value of n for which the solution is well behaved hence, suitable for modeling of super dense stars. Keeping in view of well behaved nature of this solution, one new class of solution is being studied extensively. Moreover, this class of solution gives us wide range of constant K (0≤K≤2.2) for which the solution is well behaved hence, suitable for modeling of super dense stars like strange quark stars, neutron stars and pulsars. For this class of solution the mass of a star is maximized with all degree of suitability, compatible with quark stars, neutron stars and pulsars. By assuming the surface density ρb=2×1014 g/cm3 (like, Brecher and Capocaso, Nature 259:377, 1976), corresponding to K=0 with X=0..235, the resulting well behaved model has the mass M=4.03MΘ, radius rb=19.53 km and moment of inertia I=1.213×1046 g cm2; for K=1.5 with X=0.235, the resulting well behaved model has the mass M=4.43MΘ, radius rb=18.04 km and moment of inertia I=1.136×1046 g cm2; for K=2.2 with X=0.235, the resulting well behaved model has the mass M=4.56MΘ, radius rb=17.30 km and moment of inertia I=1.076×1046 g cm2. These values of masses and moment of inertia are found to be consistent with the crab pulsars.

Neeraj Pant

Papers

Three-layered relativistic hybrid star with distinct equation of states

A relativistic model of stellar objects with core-crust-envelope division

An Exact Solution of Perfect Fluid in Isotropic Coordinates, Compatible with Relativistic Modeling of Star

Modeling quasar central engine as a relativistic radiating star

A well-behaved class of charged analogue of Durgapal solution