Roger Penrose's Assumption Behind the HYPOTHESIS of ``Trapped Surfaces'' Falsified: Int. J. Mod....

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March 12, 2013 17:25 WSPC/S0218-2718 142-IJMPD 1350021

International Journal of Modern Physics DVol. 22, No. 5 (2013) 1350021 (13 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218271813500211

DOES PRESSURE ACCENTUATE GENERALRELATIVISTIC GRAVITATIONAL COLLAPSE AND

FORMATION OF TRAPPED SURFACES?

ABHAS MITRA

Theoretical Astrophysics Section,Bhabha Atomic Research Centre, Mumbai, India

[email protected]

Received 8 January 2013Revised 24 January 2013

Accepted 28 January 2013Published 15 March 2013

It is widely believed that though pressure resists gravitational collapse in Newtoniangravity, it aids the same in general relativity (GR) so that GR collapse should eventuallybe similar to the monotonous free fall case. But we show that, even in the context ofradiationless adiabatic collapse of a perfect fluid, pressure tends to resist GR collapse in amanner which is more pronounced than the corresponding Newtonian case and formationof trapped surfaces is inhibited. In fact there are many works which show such collapseto rebound or become oscillatory implying a tug of war between attractive gravity andrepulsive pressure gradient. Furthermore, for an imperfect fluid, the resistive effect ofpressure could be significant due to likely dramatic increase of tangential pressure beyond

the “photon sphere.” Indeed, with inclusion of tangential pressure, in principle, therecan be static objects with surface gravitational redshift z → ∞. Therefore, pressure cancertainly oppose gravitational contraction in GR in a significant manner in contradictionto the idea of Roger Penrose that GR continued collapse must be unstoppable.

Keywords: Gravitational collapse; Newtonian gravity; general relativity; trappedsurfaces.

PACS Number(s): 04.40.−b, 04.40.Nr, 04.40.Dg, 95.30.Lz

1. Introduction

General Relativistic (GR) gravitational collapse is one of the most important topicsin physics and astrophysics. Unfortunately, there cannot be any general solution ofGR collapse equations both due to the complexity of the ten coupled nonlinearpartial differential equations, unknown evolution of equation of state (EOS) of thecollapsing fluid and associated complex radiation transport properties. And, atbest, there could be particular solutions depending on the various simplificationsand assumptions made for particular cases. In fact, in view of such difficulties,there cannot be any general solution of the gravitational collapse problem even in

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the enormously simplified Newtonian gravity. Given such difficulties, to make theproblem tractable, it is natural to assume the collapsing fluid to be a “dust” havingno pressure at all: p = 0. Assuming the dust to be homogeneous, ρ(t) > 0, theOppenheimer and Snyder (OS)1 solution would suggest the formation of a blackhole in a finite comoving proper time τ ∝ ρ

−1/20 , where ρ0 is the initial density. Since

a dust is pressureless and undergoes geodesic motion, it can be easily shown thatthe mass energy content within a surface having a fixed number of dust particles(proportional to r) is constant: M(r, t) = const. In the absence of any resistiveagent, a given dust shell follows the equation of motion

R(r, t) = −GM(r, t)R2

, (1)

where R is the area coordinate, an overdot denotes partial differentiation by t anda prime denotes differentiation by r. Here we have used the fact that for a dust, onecan set comoving g00 = 1 and comoving proper time τ = t. This equation happensto be exactly same as the Newtonian equation of free fall.2,3 Once R < 0, as perthis equation, R < 0 for all latter times in an unbounded manner! For this dustcase, as R decreases monotonically and M(r, t) remains constant, sooner or later,the dust shell must satisfy (both in Newtonian and Einstein gravity)

2GM(r, t)Rc2

> 1. (2)

In the context of GR, the foregoing inequality would indicate formation of“trapped surfaces” from which no light (or anything else) can escape out.2,3 Howeversince pressure free monotonous collapse equations are just similar to their Newto-nian counterparts and yield exactly same solutions,4,5 one cannot a priori demandthat the inequality (2) is realized for physical gravitational collapse. Indeed laterwe shall cite references contradicting Eq. (2).

Additionally, it is also mentioned that, in GR, the Active Gravitational MassDensity (AGMD) is ρg = ρ + 3p/c2, where p is isotropic pressure.3 And if so,increase of pressure would only accentuate the collapse process. As we progress, weshall show that, the negative self-gravitational energy actually decreases the AGMDin GR.

The objective of this paper is to see whether pressure must necessarily accentu-ate gravitational collapse in GR vis-a-vis Newtonian gravity. In order to figure outthe real role of pressure, here we would consider only adiabatic gravitational col-lapse with no dissipation, no heat flow and no radiation. And it would be pointedout that there are many examples of adiabatic collapse which show “bouncing”or “oscillatory” behavior indicating a tug of war between attractive gravity andresistive pressure gradient. On the other hand, had, pressure only accentuated thegravity, there would have only been monotonic collapse. Then it will be shownthat, the resistive action of pressure increases even more if local anisotropy will bedeveloped so that tangential pressure pt = p.

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GR Gravitational Collapse and Formation of Trapped Surfaces

2. General Formalism for Adiabatic Collapse

The interior spacetime of a spherically symmetric fluid can be described by thefollowing diagonal metric in terms of comoving coordinates r and t6–9:

ds2 = eν(r,t)dt2 − eλ(r,t)dr2 − R2(r, t)dΩ2, (3)

where dΩ2 = (dθ2+sin2 θdφ2) and R(r, t) is the invariant circumference coordinate.If we write R = R(r, t) = eµ/2, we will have

µ =2R

R; µ′ =

2R′

R. (4)

The comoving components of the stress-energy tensor are (G = c = 1)

T 11 = −p; T 2

2 = T 33 = −pt; T 0

0 = ρ, (5)

where p is the radial pressure and pt is the transverse pressure. The condition forno heat/mass energy flow in the comoving frame leads to7

2µ′ + µµ′ − λµ′ − µν′ = 0. (6)

It is interesting to see that, by using Eq. (4), the foregoing equation can be writtenin a simpler form

R′

R′ =λ

2+

R

R′ν′

2. (7)

Then by using other relevant Einstein equations, one can define a new variableM(r, t) through the equation

8M = µ2e3µ/2−ν − µ′2e3µ/2−λ + 4eµ/2, (8)

which satisfies

M = −2πµe3µ/2p = −4πR2Rp (9)

and

M ′ = 2πµ′e3µ/2ρ = 4πR2R′ρ. (10)

Also, the integration of the last equation leads to7

M(r, t) =∫ r

0

4πρR2R′dr (11)

and we indentify the new variable as the mass function. On the other hand, thelocal energy–momentum conservation leads to7

p′ =12ν′(ρ + p) + µ′(pt − p) (12)

and

ρ = (µ + λ/2)(p + ρ) + µ(pt − p). (13)

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By using (4), these two equations can be reorganized as

ν′ = − 2p′

ρ + p+

4R′

R

pt − p

ρ + p(14)

and

λ =−2ρ

ρ + p− 4R(ρ + pt)

R(ρ + p). (15)

For the special case of a perfect fluid with p = pt, Eq. (14) becomes:

ν′ = − 2p′

ρ + p. (16)

Further, if we would use a compact notation6

Γ(r, t) = e−λ/2R′, (17)

Eq. (8) can be rearranged to assume the form

Γ2 = 1 + e−νR2 − 2M

R. (18)

Note that by differentiating Eq. (17), one may also obtain

ΓΓ

= − λ

2+

R′

R′ . (19)

Now using Eqs. (7) and (19), one finds that9

ΓΓ

=R

R′ν′

2. (20)

3. Evolution of a Perfect Fluid

For a perfect fluid, pt = p. Then by differentiating (18) by t and invoking Eqs. (9)and (20), we obtain

R = −eνR(4πp + M/R3) +eνΓ2

2R′ ν′ +12νR. (21)

For further analysis, let us write the three terms on the RHS of the aboveequation as A, B, C, respectively:

R = A(r, t) + B(r, t) + C(r, t). (22)

3.1. Analysis of the first term A

In the free fall case, one can set p = ν = 0 so that, B = C = 0 and one recoversEq. (1):

R = Aff = −M

R2. (23)

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For further insight into this term, let us momentarily consider the fluid to be ofuniform density so that

M

R3=

4π

3ρ(t). (24)

On the other hand by including pressure, for an uniform density case, the first termon the right-hand side (RHS) of Eq. (21) becomes

Apressure = −4πR

3eν(ρ + 3p). (25)

The notion that inclusion of pressure may make the monotony of collapse in GReven worse than the free fall case arose by noting that, apparently |Apressure| > |Aff |because of the addition of the 4πp/c2 term with M/R3 in the former case. Thisinference however need not be correct because when pressure is present, g00 = eν < 1.

3.2. Analysis of the second term B

We expect the spacetime to be regular at r = 0 at least initially; i.e. g00(0, 0) = 1 orν(0, 0) = 0. Then we must have ν′ > 0 as we would move outward. Also, for realisticcases, we expect the increasing two spheres to enclose more and more matter; i.e.R′ > 0. Then we would always have B > 0. For further appreciating this term, letus use (16) to write

B =eνΓ2

R′−p′

ρ + p= +

eνΓ2

R′|p′|

ρ + p. (26)

Here we have used the fact that the pressure gradient p′ is negative. Note, the addi-tion of the p/c2 term in the denominator is usually considered as the enhancementof “inertial mass density” in GR case. The corresponding Newtonian term wouldobviously be B = +|p′|/ρ.

3.3. Analysis of the third term

Suppose we define a “compactness factor”

z = g−1/200 − 1 = e−ν/2 − 1 (27)

so that12ν = −zeν/2. (28)

At the beginning of the collapse, one may have g00 = 1 and if a spacetime singularitywould indeed form, one may have g00 = 0, so that z ≤ 0 and

C =12νR = eν/2|z|R| ≥ 0. (29)

On the other hand, for the unrealistic dust collapse case ν = 0. Note, in theNewtonian case, there is no such term:

CNewtonian = Cff = 0. (30)

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4. Comparison with Newtonian Case

As we found, for the corresponding Newtonian case, one would have

R = −M

R2+

|p′|ρ

. (31)

The positive term B present in both Newtonian and Einstein case, shows that,pressure resists collapse in GR almost as much as it does so in Newtonian gravity.Therefore, even granting for a moment that |Apressure| > |ANewtonian|, the ideathat addition of pressure accentuates GR collapse is a misconception. Further, forthe GR case, there is an additional compactness term ∝ |z||R| (Eq. (29)) whichdirectly acts like a stabilizing factor because it is proportional to R. Thuseven for a perfect fluid having no tangential pressure, no heat flow, GR collapse hasmore chance of stabilization in comparison to the corresponding Newtonian case.

5. Evolution of an Imperfect Fluid

For an adiabatically evolving imperfect fluid, we will have pt = p, and in order thatwe must always have ν′ > 0, we expect pt ≥ p. Physically, shear and heat flowassociated with an imperfect fluid increases pt.10,11 This apart, in the presence ofa strong radially directed inward gravitational field, motion in the radial directionwill be less random, and thus, it is expected that ∆ ≡ pt − p ≥ 0. In such a case,Eq. (26) gets modified into

B = +eνΓ2

R′

|p′| +

(2R′

R

)|∆|

ρ + p

. (32)

Thus, with the inclusion of tangential pressure, collapse is likely to be resisted ina way stronger than the perfect fluid case. And, in case, the collapse would proceedto very deep gravitational well, this effect may increase dramatically because of thefollowing reason:

It may be first recalled that given a spherically symmetric body having a grav-itational mass Mb and a boundary at R = Rb, the surface gravitational redshift isgiven by (G = c = 1):

zb =(

1 − 2Mb

Rb

)−1/2

− 1. (33)

Suppose the body is shrinking and its gravitational field is increasing. Then theformation of its “Event Horizon” would correspond to a situation Rb(t) = 2Mb(t).But much before formation of any such event horizon, the body would tend tohinder the outflow of heat/radiation or anything due to its stronger gravitationalpull. In particular, a photon sphere is defined by region interior to R = 3Mb orz =

√3−1 where gravitational pull is already so strong that photons and neutrinos

would start moving in closed circular orbits.

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If the collapsing object would dip below its photon sphere, i.e. if 1 + zb >√

3,then even radiation/heat quanta generated within the contracting object would tendto move in counter rotating circular orbits.12,13 In such a case, even the matterparticles may tend to move in similar way so that the configuration would tendto be an “Einstein Cluster.”14 Formally an Einstein cluster is a spherical cloudcomprising point particles moving in closed circular orbits having various radii; andin order to have zero angular momentum for the entire cluster, for every rotatingparticle, there must be a counter-rotating particle. Clearly, for this dust cloud, radialpressure, p = 0 while the entire pressure is of transverse nature. Such a configurationwas first conceived by Einstein to conclude that continued gravitational collapsewould be halted by such transverse pressure.14 Although Einstein’s intuition wascorrect, he had no idea of a likely “photon sphere” and he did not answer whycontinually contracting matter must shed radial pressure and on the other handdevelop entirely transverse pressure. Consequently, his exercise was largely ignoredby most of the relativists. But, here we offered a qualitative answer as to whymatter may approximately behave like an “Einstein cluster.” Therefore once thefluid would dip into its photon sphere, there should be dramatic increase in thevalue of ∆/R and it would be much more likely that the collapse could be arrested.

6. Active Gravitational Mass Density

In Newtonian gravitation, the Poisson’s equation is ∇ · E = 4πρ, where E is thegravitational intensity. In GR, if one would consider a perfect fluid, in the local freefalling frame, the corresponding equation would be

∇ · E = 4π(ρ + 3p/c2). (34)

This equation leads to the apparent idea that, in GR, pressure enhances the massenergy density and therefore, pressure can only assist collapse (here we have rein-troduced c = c). But it is easy to see that the above interpretation is completelyincorrect. Of course, mathematically, one can always conceive of a locally free fallingframe and a fundamental consequence of Einstein’s Equivalence Principle is thatg00 = 1 in such a frame. But in the presence of the pressure, the fluid is subjectto pressure gradient force in its own rest frame and therefore, the above equationis not relevant in the comoving frame of the fluid. On the other, for the comovingframe, the correct form of Poisson’s law is15

∇ ·E = 4π√

g00(ρ + 3p/c2). (35)

Therefore the AGMD is

ρg =√

g00(ρ + 3p/c2) < (ρ + 3p/c2). (36)

At least for a spherically symmetric static fluid, it has been found that, actually,ρg < ρ because of the effect of global negative self-gravitational energy15:

ρg = ρ − 3p/c2. (37)

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Accordingly, we might rewrite Eq. (25) as

Apressure = −4πR

3(ρ − 3p/c2), (38)

so that

|Apressure| < |Aff |. (39)

If we would again set c = 1, we may write

Apressure = −4πR

3(ρ − 3p). (40)

Since this term essentially contains the trace of the energy–momentum tensor, forthe imperfect fluid, one may have

Apressure = −4πR

3(ρ − p − 2pt). (41)

7. Does Trapped Surface and Event Horizon Form?

The concept of a “trapped surface” may be an important concept in differentialgeometry.2,3 But does real physical situation must oblige all novelties of differentialgeometry? In other words, while by Einstein’s equation, given a certain distributionof matter energy–momentum tensor there will be certain nontrivial spacetime geom-etry, does all hypothetical spacetime geometries must correspond to finite matterenergy–momentum distribution in a real physical situation. Historically, progressin the research of GR gravitational collapse has taken place by assuming an affir-mative answer/faith for such a question. It is clear that the idea of trapped surfaceformation is compatible with the pressureless case with p = pt = ∆ = 0 (pro-vided one can assume ρdust > 0). And this assumption of formation of trappedsurfaces ignored the fact that in the presence of pressure, for the collapse process,there will be a tug of war between terms of opposite signs. Accordingly, even thesupposed adiabatic collapse of perfect fluid may experience bounce or even stopwith the formation of a static object. Since with radial pressure alone, for a staticobject, zb < 2,16 such a bounce must happen by honoring this constraint. On theother hand, since for pure tangential pressure, there is no upper limit on zb,17,18 thebounce can happen even from the deepest potential well; or else, the contractioncan asymptotically attempt to result in a compact object with zb → ∞, R → 0.Note, for physical consistency, it is necessary that by definition, the final state mustcorrespond to R = 0.

If a trapped surface would be formed, of course, under very reasonable condi-tions, the collapse must be monotonic and “unstoppable.”2,3 However, in order toensure that trapped surface is formed, one needs to assume that, the GR collapsebecomes unstoppable much before the formation of any trapped surface! This seemslike a tautology and hence Kriele commented that19:

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“The presence of a closed trapped surface which is assumed in those singularitytheorems that (hopefully) predict the collapse of a star is not entirely global. Butunder normal circumstances such a trapped surface can only occur inside the blackhole region. Thus the singularity theorems seem to be inapplicable for predictingthe formation of a singularity before a black hole forms.”

Kriele has also pointed out that though one can formulate “necessary condition”for formulation of “trapped surfaces,” nobody has shown that such conditions arefulfilled for realistic cases. The assumptions behind formation of trapped surfacesand singularity theorems were probably first questioned by Donald and he con-cluded that20 “the assumptions required by the singularity theorems are examinedcritically. These assumptions are found to be questionable.”

Later, by using differential geometry, Kriele showed that a spherically symmetricstar of uniform density “cannot contain a trapped surface.”21 Such questions wereraised later both from detail physical and mathematical perspectives and it wasconcluded that neither trapped surfaces nor (finite mass) black holes can formin realistic gravitational collapse.22,23 And a finally an elegant correct proof tothis effect was offered25 by removing some subtle confusion in earlier proofs.22,23

Essentially it was shown that in order that a timelike worldline of a sphericallyevolving body must remain timelike to all observers, one must have

2M(r, t)R(r, t)

≤ 1. (42)

This above result does not depend on any exterior boundary condition or any spe-cific EOS of the fluid or on the question whether pressure is radial or transverse.Thus it is valid both for isolated bodies and the cosmos subject to the assump-tion of spherical symmetry/isotropy. It shows that (i) if continued collapse wouldindeed continue upto R = 0, the final state would be a M = 0 black hole. Thishowever does not mean collapse must continue all the way to R = 0; on the otherhand, it means that there may be physical mechanisms by which continued col-lapse may be halted to result in either truly static or at least quasi-static com-pact objects. In fact if one would accept a class of nonlinear generalizations of theelectromagnetic theory, occurrences of trapped surfaces and singularities may beavoided both in the context of gravitational collapse and cosmology in accordancewith Eq. (42).25−27

As to the exact mathematical solution for black holes, let us remind that, ingeneral most of the exact solutions in GR could be physically meaningless or notrealizable. This is so because GR has complex mathematical structure and findingan exact solution could be a sort of miracle. This is even more true for the problemof gravitational collapse where one must feed those equations with exact (evolving)EOS of matter, exact radiation and heat transportation calculations. And notehere, while the integration constant appearing in the vacuum Schwarzschild solutionα = 2GM/c2 must be finite for an object with finite radius (Rb > 0) (like theSun, Galaxy etc.), there no a priori guarantee that it must be so in the limit of

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Rb → 0 or a “point particle.” Since in contrast to the exterior spacetime of saySun, the Schwarzschild black hole solution corresponds to a “point mass,” it hasbeen found that in such a case, the corresponding integration constant α → α0 =2GM/c2 = 0.24,29 The paradigm of Schwarzschild black holes is stubbornly upheldby claiming that the Kruskal coordinates offer self-consistent description for theentire associated spacetime. But recently this claim has been critically re-examinedand it has been shown that Kruskal black hole too corresponds to α0 = 0.30

8. Summary

For the pressureless uniform density spherical case, Eq. (1) becomes, R =−(4π/3)Rρ. And this leads to runaway process by which

R = R = −∞! at R = 0 if indeed ρ > 0. (43)

Further, since in GR, pressure increases AGDM in a locally free falling frame, itwas postulated that in deep gravitational collapse, where pressure could be large,the collapse process must be monotonic like the dust case. It is these two linesof thinking which led to the idea that in GR, collapse must be monotonic even ifpressure forces will be active. But we found that this idea is incorrect because ofthe following reasons:

• In the comoving frame of the fluid, pressure actually reduces AGMD.15

• For both Newtonian and GR collapse of a perfect fluid, pressure gradient opposesthe collapse process because R has positive contribution from pressure gradient.

• In GR, there is an additional term C = (1/2)νR > 0 which directly opposescollapse by acting as a positive feedback which is proportional to R. Therefore,unlike the case of a dust, the mathematical collapse of a perfect fluid can reverseor oscillate. In fact, there are innumerable claims of bounce or oscillation inviolation of the idea that GR collapse must be monotonic.31–42 In particular, thepaper by Bondi was titled as Gravitational Bounce in General Relativity.37

Despite this, one may argue that, for a collapsing perfect fluid having onlyradial pressure, reversal may be difficult in view of the fact that the surfacegravitational redshift of static compact objects must be zb < 2.0.16

• The pressure related resistive effect gets enhanced for an imperfect fluid withtransverse pressure. By considering such effects, the eventual evolution equation,for a constant density case, may be written as:

R = −4πR

3

[ρ − (p + 2pt) − 3eνΓ2

4πRR′|p′| + (2R′/R)|∆|

ρ + p− 3νR

8πR

]. (44)

For the corresponding Newtonian case, one has

R = −4πR

3

[ρ − 3|p′|

4πRρ

]. (45)

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And for the free fall dust case, whether it is Newtonian or Einstein gravity, onehas R = −(4π/3)ρR. We noted that, for the GR, case, there could be dramaticincrease of the effect of tangential pressure term ∆/R if the body would plungeinto its photon sphere zb >

√3 − 1. Further for an object completely dominated

by tangential pressure, there is no upper limit on the value of zb < ∞ (Refs. 17and 18); therefore, in principle, the effect of tangential pressure might give rise toultra compact objects having zb 1. The fact that tangential pressure can stabilizethe contracting tendency of self-gravity is known as Lemaitre Vault formation bytangential pressure.18

9. Conclusion

For a spherically evolving fluid, GR does not allow formation of trapped surfaces inorder that a timelike worldline must always remain so. Then the inequality (42) tellsthat if any situation would appear to violate it one must intrinsically have ρ = 0.For the case of OS dust collapse this has been explicitly shown.43 Incidentally froma much less general and rather questionable consideration, it has been opined thatOS collapse should not lead to black hole formation.44

To ensure the absence of formation of trapped surfaces there must be appro-priate physical mechanisms. As already mentioned, one such mechanism could beadoption of appropriate nonlinear electrodynamics.25–27 But we found here thateven in the absence of such departures from standard physics, usual effects likepressure gradient and reduction of AGMD must play an important role in ensuringthe sanctity of (42).

In general, physical gravitational collapse is radiative45,46 and there are innu-merable examples that for such radiative collapse, there can be bounce, oscillationor formation of hot quasi-static objects.47–51 Further, in view of the existence of“Eddington Luminosity” at which repulsive effects of radiation pressure balances theinward pull of gravity, it has been shown that continued radiative collapse shouldindeed give rise to ever contracting hot quasi-static objects12,13 as the trapped radi-ation luminosity would become equal to the corresponding Eddington luminosity.And here we found that, this phenomenon might be also understood from the viewpoint of unabated growth of tangential stresses and formation of radiation sup-ported “Lemaitre Vault.” For practical cases, this scenario will yield quasi-staticobjects with zb 1 and which would act as “quasi black holes” or “black holemimickers.” And since they asymptotically evolve towards the true black hole statehaving zb = ∞ and Mb = 0 (Refs. 28–30) they have been termed as “EternallyCollapsing Objects.”

Acknowledgments

The author thanks the anonymous referee for making several suggestions which ledto an improved version of this manuscript.

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Roger Penrose's Assumption Behind the HYPOTHESIS of ``Trapped Surfaces'' Falsified: Int. J. Mod....

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