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

We present a family of new exact solutions for relativistic anisotropic stellar objects by considering a four-dimensional spacetime embedded in a five-dimensional pseudo Euclidean space, known as Class I solutions. These solutions are well behaved in all respects, satisfy all energy conditions, and the resulting compactness parameter is also within Buchdahl limit. The well-behaved nature of the solutions for a particular star solely depends on the index n. We have discussed the solutions in detail for the neutron star XTE J1739-285 (
 $$M=1.51M_\odot , ~R=10.9$$
 km). For this particular star, the solution is well behaved in all respects for $$8 \le n \le 20$$
 . However, the solutions with $$n<8$$
 possess an increasing trend of the sound speed and the solutions belonging to $$n>20$$
 disobey the causality condition. Further, the well-behaved nature of the solutions for PSR J0348+0432 (
 $$2.01M_\odot , ~11$$
 km), EXO 1785-248 (1.3
 $$M_\odot $$
 , 8.85 km), and Her X-1 (0.85
 $$M_\odot $$
 , 8.1 km) are specified by the index n with limits $$24 \le n \le 54$$
 , $$1.5 \le n \le 4$$
 , and $$0.8 \le n \le 2.7$$
 , respectively.

/pdf/a-family-of-well-behaved-karmarkar-spacetimes-describing-vaww7ea7nj.pdf

A family of well-behaved Karmarkar spacetimes describing interior of relativistic stars

In this paper we have studied the behavior of static spherically symmetric relativistic objects with locally anisotropic matter distribution considering the Tolman VII form for the gravitational potential $g_{rr}$
 in curvature coordinates together with the linear relation between the energy density and the radial pressure. The interior spacetime has been matched continuously to the exterior Schwarzschild geometry. We have investigated and analyzed different physical properties of the stellar model and presented graphically.

Relativistic anisotropic stellar models with Tolman VII spacetime

We obtain a new static model of the TOV equation for an anisotropic fluid distribution by imposing the Karmarkar condition. In order to close the system of equations we postulate an interesting form for the $$g_{rr}$$
 gravitational potential, which allows us to solve for $$g_{tt}$$
 metric component via the Karmarkar condition. We demonstrate that the new interior solution has well-behaved physical attributes and can be utilized to model relativistic static fluid spheres. By using observational data sets for the radii and masses for compact stars such as 4U 1538-52, LMC X-4, and PSR J1614-2230 we show that our solution describes these objects to a very good degree of accuracy. The physical plausibility of the solution depends on a parameter c for a particular star. For 4U 1538-52, LMC X-4, and PSR J1614-2230 the solutions are well behaved for $$0.1574 \le c \le 0.46$$
 , $$0.1235 \le c \le 0.35$$
 and $$0.05 \le c \le 0.13$$
 , respectively. The behavior of the thermodynamical and physical variables of these compact objects leads us to conclude that the parameter c plays an important role in determining the equation of state of the stellar material and observed that smaller values of c lead to stiffer equation of states.

https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-017-4612-4.pdf

Physical viability of fluid spheres satisfying the Karmarkar condition

We present a new class of solutions to the Einstein field equations for an anisotropic matter distribution in which the interior space-time obeys the Karmarkar condition. The necessary and sufficient condition required for a spherically symmetric space-time to be of Class One reduces the gravitational behavior of the model to a single metric function. By assuming a physically viable form for the g(rr) metric potential we obtain an exact solution of the Einstein field equations which is free from any singularities and satisfies all the physical criteria. We use this solution to predict the masses and radii of well-known compact objects such as Cen X-3, PSR J0348+0432, PSR B0943+10 and XTE J1739-285.

Anisotropic compact stars in Karmarkar spacetime

We are presenting a family of new exact solutions for relativistic anisotropic stellar objects by considering four dimensional spacetime embedded in five dimensional Pseudo Euclidean space known as Class I solutions. These solutions are well-behaved in all respects, satisfy all energy conditions and the resulting compactness parameter is also within Buchdahl limit. The well-behaved nature of the solutions for a particular star solely depends on index n. We have discussed the solutions in detail for the neutron star XTE J1739-285 (M = 1.51M$\odot$, R = 10.9 km). For this particular star, the solution is well behaved in all respects for $8 \le n \le 20$. However, the solutions with n 20 disobey causality condition. Further, the well-behaved nature of the solutions for PSR J0348+0432 (2.01M$\odot$, 11 km), EXO 1785-248 (1.3M$\odot$, 8.85 km) and Her X-1 (0.85M$\odot$, 8.1 km) are specified by the index n with limits $24 \le n \le 54$, $1.5 \le n \le 4$ and $0.8 \le n \le 2.7$ respectively.

/pdf/a-family-of-well-behaved-karmarkar-spacetime-describing-52u20ewv8q.pdf

Neeraj Pant

Papers

A family of well-behaved Karmarkar spacetimes describing interior of relativistic stars

Relativistic anisotropic stellar models with Tolman VII spacetime

Physical viability of fluid spheres satisfying the Karmarkar condition

Anisotropic compact stars in Karmarkar spacetime

A family of well-behaved Karmarkar spacetime describing interior of relativistic stars