Black Holes in Spherical Gravitational Collapse

Non-occurrence of Trapped Surfaces and Black Holes in Spherical Gravitational Collapse

Abhas Mitra

Theoretical Physics Division, Bhabha Atomic Research Center,Mumbai-400085,

 India E-mail: amitra@apsara.barc.ernet.in 

We carefully analyze the apparently commonplace yet subtle concepts associated with the no-tion of existence of Black Holes. We point out that although the pioneering work of Oppen-heimer and Snyder (OS), technically, indicated the formation of an event horizon for a collaps-ing homogeneous dust ball of massM bin that the circumference radius the outermost surface,Rb→Rgb= 2GM b/Rbc2, in a proper time τ gb ∝ R−1/2gb, it never explicitly showed the formation of a “trapped surface” where Rb< R gb On the other hand, the Eq. (36) of their paper (T ∼ln yb+1yb−1)categorically demands thaty b=Rb/Rgb≥1, or 2GM b/Rbc2≤1, so that, if one has to pursue thecentral singularity,Rb→0, it is necessary that Rgb→0 at a faster rate. Consequently, actually,τ gb→ ∞, and more importantly, the (fixed) gravitational mass of the dust M b = 0. Further, byanalyzing the general inhomogeneous dust solutions of Tolman in a proper physical perspective, weshow that, all dust solutions obey the same general constraint and are characterized by M b= 0.Next, for a collapsing fluid endowed with radiation pressure, in a most general fashion , we discoverthat the collapse equations obey the same Global Constraint 2 GM/Rc2 ≤1 and which specificallyshows that, contrary to the traditional intuitive Newtonian idea, which equates the gravitationalmass (M b) with the fixed baryonic mass (M 0), the trapped surfaces are not allowed in general theoryof relativity (GTR). Now by invoking, the “positive mass theorems”, it follows that for continuedcollapse, the final gravitational mss M f →0 asR→0. Thus we confirm Einstein’s and Rosen’s ideathat Event Horizons and Schwarzschild Singularities are unphysical and can not occur in Nature.This, in turn, implies that, if there would be any continued collapse , the initial gravitational massenergy of the fluid must be radiated away Q→M ic2.1

Irrespective of the gravitational collapse problem , by analyzing the properies of the Lemaitre andKruskal transformations, in a straightforward manner, we show that finite mass Schwarzschild BlackHoles can not exist at all.

I. INTRODUCTION

One of the oldest and most fundamental problem of physics and astrophysics is that of gravitational collapse, and, specifically, that of the ultimate fate of a sufficiently massive collapsing body. Most of the astrophysical objects that we know of, viz. galaxies, stars, White Dwarfs (WD), Neutron Stars (NS), in a broad sense, result from gravitational collapse. And in the context of classical General Theory of Relativity (GTR), it is believed that the ultimate fate of sufficiently massive bodies is collapse to a Black Hole (BH). A spherical charge less BH of (gravitational) mass M b is supposed to occupy a region of space time which is separated by a hypothetical one-way membrane of “radius” R gb= 2GM b/c2, where G is the Newtonian gravitational constant and c is the speed of light. This membrane,called, an event horizon, is supposed to contain a central singularity at R= 0, where most of the physically relevantquantities like (local) energy density, (local) acceleration due to gravity, (local) tidal acceleration, and components of the Rimmenian curvature tensor diverge. However, although such ideas are, now, commonly believed to be elementsof ultimate truth, the fact remains that, so far, it has not been possible to obtain any analytical solution of GTRcollapse equations for a physical fluid endowed with pressure ( p ), temperature (

T ) and an equation of state (EOS).And the only situation when these equations have been solved (almost) exactly, is by setting p≡ 0, and further byneglecting any density gradient, i.e., by considering ρ = constant [26]. It is believed that these (exact) asymptoticsolutions actually showed the formation of BH in a finite comoving proper time τ gb. However, this, assumption of perfect homogeneity is a very special case, and, now many authors believe that for a more realistic inhomogeneousdust, the results of collapse may be qualitatively different. These authors, on the strength of their semi-analyticaland numerical computations, claim that the resultant singularity could be a “naked” one i.e., one for which thereis no “event horizon” [7, 12, 13, 22 36]. Therefore light may emanate from a naked singularity and reach a distantobserver. A naked singularity may also spew out matter apart from light much like the White Holes. In other words,unlike BHs, the naked singularities are visible to a distant observer and, if they exit, are of potential astrophysicalimportance. However, according to a celebrated postulate by Penrose [42], called “Cosmic Censorship Conjecture”,for all realistic gravitational collapse, the resultant singularity must be covered by an event horizon, i.e, it must be aBH. And many authors believe that the instances of occurrences of “naked singularities” are due to fine tuned artificialchoice of initial conditions or because of inappropriate handling and interpretation of the semi-analytical treatments.In this paper, we are not interested in such issues and would avoid presenting and details about the variants of nakedsingularities (strong, weak, local, global, etc) or the variants of the censorship conjecture. Also in this paper, we areinterested only in the case of collapse of physical matter consisting of baryons and leptons, and would completelyavoid any discussion on collapse of hypothetical fields like various “scalar fields”.When we say that BHs result from collapse of “sufficiently” massive bodies, it is in order to qualify the term“sufficient”. Very crudely, the cores of moderately massive stars end up as WDs, a configuration supported bypressure of degenerate electrons. And it was shown by Chandrasekhar [46] and independently by Landau [29] thatthere is an upper limit on the mass of the WDs, called, “Chandrasekhar Mass”

1

whereµeis the number of electrons per nucleon, so that, for a He- WD, µe= 2 When the main sequence mass of a star is such that, the final mass of its coreM c> M ch, the core continues to collapse without ever resting in astate of hydrostatic equilibrium supported by degenerate electron pressure. At a (baryon)density of ∼1011g/cm3,neutronization of matter starts, and a new state of hydrostatic equilibrium may be reached where the pressure is dueto degenerate neutrons. In other words, the collapse process ends with the formation of a NS. But again, there isan upper limit on the mass of stable NSs, called, Oppenheimer and Volkoff limit, M OV . The original value of thislimiting mass was first obtained by Oppenheimer and Volkoff (1938) by treating the NS as a self-gravitating gas of freeneutrons, M OV ≈0.7M ⊙. However, in the past few decades, with the progress of nuclear physics, there havebeen enormous amount of work to find the value of M OV using actual EOS of nuclear matter. We would only mentionhere a particular value obtained by using an EOS which incorporates the fact the sound speed in nuclear matter islimited by the speed of light, dp/dρ ≤c 2, [8, 53]

2

Thus, more massive stellar cores are supposed to undergo “continued collapse” without reaching a new state of hydrostatic equilibrium, and are believed to end up as BHs (or naked singularities). However, to fully appreciate this

2

conviction it is necessary to understand another point. All along this chain of previous discussion it was implicitlyassumed that, during the collapse process, the role of GTR is negligible except for the stage beyond the NS stage,and the instantaneous gravitational mass of the core

 M f ≈M i≈M b=M 0=Nm (3)

where the constant the baryonic mass of the core (by ignoring the mass of the leptons and assuming no anti baryonsto be present)M 0=mN , with m as the mean nucleon mass and N to be the number of nucleons. Under thisassumption, an event horizon is formed at a density

where

freedom and the value of γ t→4/3 or even, momentarily, be<4/3 even when all the constituent particles are not in astate of extreme relativistic degeneracy. After the neutronization process, such a thing happens during the supernovacollapse prior to the attainment of nuclear density of the collapsing matter; and the collapse during this stage is nearadiabatic [47]. But in the limit of a monoatomic gas, when new degrees of freedom are not suddenly liberated, thevalue of γ t→4/3 only if all the fluid particles become relativistically degenerate with individual momenta→ ∞ Thus, this can happen only asymptotically, and, very strictly, it can not be exactly realized except at a physicalsingularity. It may appear that, if the fluid is buried under an event horizon and yet Ris finite, the emission of radiation will stop because then the fluid can not communicate withS ∞ However, by the Principle of Equivalence,to be elaborated latter, the local laws of thermodynamics remains unchanged, and although, it is not possible todefineE gmeaningfully in such cases, it is possible that the fluid will still require to radiate to honor thermodynamics.This difficulty can be alleviated if we assumeγ t= 4/3 inside the event horizon. But, the pressure, internal energyand all other physical quantities remain finite for a finiteR, and, as discussed above,γ tshould actually be>4/3.The virial theorem (VT) used in the present discussion is actually due to the Newtonian inverse square law of gravitational acceleration. There is no counterpart of a GTR VT, in an easily usable form, yet. It is interesting torecall that, for spherical symmetry and in the absence of any angular momentum of the “test particle”, the effectivepotential felt by a test particle still has this Newtonian R−1form. In general, all the peculiarities associated withgravity get accentuated by GTR, and it is possible that a VT almost similar to the one used here might be applicablein the GTR case too. However, when GTR becomes important, this Newtonian expression for gravitational energyhas to be modified. In GTR, like all global energies,E g too is defined only with respect to S ∞ [9, 48] :

depending on the finite grid sizes used in the analysis and limitation of the computing machine, one may concludethat the formalism adopted is really satisfying, and then find thatQ≪M ic2 [14, 51, 52, 54]. Meanwhile, one has toextend the presently known (cold) nuclear EOS at much higher densities and maintain the assumption that the risein temperature is moderate. Because if T is indeed high, in the diffusion limit, the emitted energy Q∼T 4 would bevery high, and the value of M f could drop to an alarmingly low value. Thus, for the external spacetime, one needsto consider the Vaidya metric [34]. Actually, even when, T is low, it is extremely difficult to self consistently handlethe coupled energy transport problem.It may appear that, the practical difficulties associated with the study of collapse involving densities much higherthan the nuclear density can be avoided if one starts with a very high value of  M i, say, 1010 M ⊙. Then if one retainsthe assumption that M i=M f , one would conclude that an “event horizon” is formed at a density of ∼10−4 gcm−3, where the EOS of matter is perfectly well known. What is overlooked in this traditional argument is that,once we are assuming that an event horizon is about to form, we are endorsing the fact that we are in the regime of extremely strong gravity, and, therefore for all the quantities involved in the problem, a real GTR estimate has tobe made without making any prior Newtonian approximation. To further appreciate this important but convenientlyoverlooked point by the numerical relativists note that the strength of the gravity may be approximately indicatedby the “surface redshift”, zs, of the collapsing object, and while a Supermassive Star may have an initial value of zs as small as 10−10, a canonical NS has zs∼0.

1, while the Event Horizon, irrespective of the initial conditions of thecollapse, has gotzs=∞! Therefore all Newtonian or Post Newtonian estimates or the conclusions based on suchestimates have

little relevance for actual gravitational collapse problem .We would see in the latter part of this paper that, for a “test particle” in the External Schwarzschild metric[28], the maximum value of the local free fall speed of the fluid appears to b evex= (Rg/R)1/2c. Andvex→casR→Rg= 2GM/c2. But then for a finite value of R g= 2GM/c2, it is believed that this anomalous behavior results from a “coordinate singularity”. When, this “coordinate singularity” is removed, it is expected thatthe actual valueof v

would→conly when the fluid collapses to the central singularity R→0. Therefore, the actual value of 

v mustbe considerably less than light speed at R=Rg, if Rg= 0. For a fluid endowed with pressure, the collapse processis bound to be slower, and therefore, one may legitimately expectvto be appreciably lower than c for R ≥Rg, andconsequently, the bulk flow kinetic energy to be considerably smaller than

Mc2(if Rg>0). Then one may crudelyuse the Eqs. (1.9) and (1.21) to find that Γs∼0, so that|E g|increases drastically. As a result, the integratedvalue of Qwould tend to increase drastically, and this would pull down the running value of M f =M 0−Q/c 2 and Rgbto an alarming level! In fact Eq. (1.6)indeed shows that the value of M should drop significantly as Γsis excepted to decrease substantially near the horizon irrespective of the value of M 0andM . At the same time, of course, the value of R is decreasing. But how would the value of Rb / Rgb would evolve in this limit? Unfortunately, nobody has ever, atleast in the published literature, tried to look at the problem in the way it has been unfolded above. On the other hand, in Newtonian notion, the value of  M f is permanently pegged at M 

0 because energy has no mass-equivalence(although in the corpuscular theory of light this is not so, but then nobody dragged the physics to the R→Rg limitseriously then). So, in Newtonian physics, or in the intuitive thinking process of even the GTR experts, the value of

cnt...