does pressure accentuate general relativistic

3rd Reading March 12, 2013 17:25 WSPC/S0218-2718

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International Journal of Modern Physics D Vol. 22, No. 5 (2013) 1350021 (13 pages) c World Scientific Publishing Company DOI: 10.1142/S0218271813500211

Int. J. Mod. Phys. D Downloaded from www.worldscientific.com by BHABHA ATOMIC RESEARCH CENTRE TROMBAY on 03/18/13. For personal use only.

DOES PRESSURE ACCENTUATE GENERAL RELATIVISTIC GRAVITATIONAL COLLAPSE AND FORMATION OF TRAPPED SURFACES?

ABHAS MITRA Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India [email protected] Received 8 January 2013 Revised 24 January 2013 Accepted 28 January 2013 Published 15 March 2013 It is widely believed that though pressure resists gravitational collapse in Newtonian gravity, it aids the same in general relativity (GR) so that GR collapse should eventually be similar to the monotonous free fall case. But we show that, even in the context of radiationless adiabatic collapse of a perfect fluid, pressure tends to resist GR collapse in a manner which is more pronounced than the corresponding Newtonian case and formation of trapped surfaces is inhibited. In fact there are many works which show such collapse to rebound or become oscillatory implying a tug of war between attractive gravity and repulsive pressure gradient. Furthermore, for an imperfect fluid, the resistive effect of pressure could be significant due to likely dramatic increase of tangential pressure beyond the “photon sphere.” Indeed, with inclusion of tangential pressure, in principle, there can be static objects with surface gravitational redshift z → ∞. Therefore, pressure can certainly oppose gravitational contraction in GR in a significant manner in contradiction to the idea of Roger Penrose that GR continued collapse must be unstoppable. Keywords: Gravitational collapse; Newtonian gravity; general relativity; trapped surfaces. 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 topics in physics and astrophysics. Unfortunately, there cannot be any general solution of GR collapse equations both due to the complexity of the ten coupled nonlinear partial differential equations, unknown evolution of equation of state (EOS) of the collapsing fluid and associated complex radiation transport properties. And, at best, there could be particular solutions depending on the various simplifications and 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 1350021-1


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the enormously simplified Newtonian gravity. Given such difficulties, to make the problem tractable, it is natural to assume the collapsing fluid to be a “dust” having no pressure at all: p = 0. Assuming the dust to be homogeneous, ρ(t) > 0, the Oppenheimer and Snyder (OS)1 solution would suggest the formation of a black −1/2 hole in a finite comoving proper time τ ∝ ρ0 , where ρ0 is the initial density. Since a dust is pressureless and undergoes geodesic motion, it can be easily shown that the 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 resistive agent, a given dust shell follows the equation of motion ¨ t) = − GM (r, t) , R(r, R2

(1)

where R is the area coordinate, an overdot denotes partial differentiation by t and a prime denotes differentiation by r. Here we have used the fact that for a dust, one can set comoving g00 = 1 and comoving proper time τ = t. This equation happens to be exactly same as the Newtonian equation of free fall.2,3 Once R˙ < 0, as per this equation, R˙ < 0 for all latter times in an unbounded manner! For this dust case, 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) > 1. Rc2

(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 However since pressure free monotonous collapse equations are just similar to their Newtonian counterparts and yield exactly same solutions,4,5 one cannot a priori demand that the inequality (2) is realized for physical gravitational collapse. Indeed later we shall cite references contradicting Eq. (2). Additionally, it is also mentioned that, in GR, the Active Gravitational Mass Density (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, we shall show that, the negative self-gravitational energy actually decreases the AGMD in GR. The objective of this paper is to see whether pressure must necessarily accentuate gravitational collapse in GR vis-a-vis Newtonian gravity. In order to figure out the real role of pressure, here we would consider only adiabatic gravitational collapse with no dissipation, no heat flow and no radiation. And it would be pointed out that there are many examples of adiabatic collapse which show “bouncing” or “oscillatory” behavior indicating a tug of war between attractive gravity and resistive pressure gradient. On the other hand, had, pressure only accentuated the gravity, there would have only been monotonic collapse. Then it will be shown that, the resistive action of pressure increases even more if local anisotropy will be developed so that tangential pressure pt = p. 1350021-2


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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 the following 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) T11 = −p;

T22 = T33 = −pt ;

T00 = ρ,

(5)

where p is the radial pressure and pt is the transverse pressure. The condition for no heat/mass energy flow in the comoving frame leads to7 ˙ − µν 2µ˙ + µµ ˙ − λµ ˙ = 0.

(6)

It is interesting to see that, by using Eq. (4), the foregoing equation can be written in a simpler form R˙ ν λ˙ R˙ (7) = + . R 2 R 2 Then by using other relevant Einstein equations, one can define a new variable M (r, t) through the equation 8M = µ˙ 2 e3µ/2−ν − µ2 e3µ/2−λ + 4eµ/2 ,

(8)

˙ M˙ = −2π µe ˙ 3µ/2 p = −4πR2 Rp

(9)

M = 2πµ e3µ/2 ρ = 4πR2 R ρ.

(10)

Also, the integration of the last equation leads to7 r M (r, t) = 4πρR2 R dr

(11)

which satisfies

and

0

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

1 ν (ρ + p) + µ (pt − p) 2

(12)

and ˙ ρ˙ = (µ˙ + λ/2)(p + ρ) + µ(p ˙ t − p). 1350021-3

(13)


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By using (4), these two equations can be reorganized as 4R pt − p 2p + ρ+p R ρ+p

(14)

˙ + pt ) −2ρ˙ 4R(ρ λ˙ = − . ρ+p R(ρ + p)

(15)

ν = −


and

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/2 R ,

(17)

Eq. (8) can be rearranged to assume the form 2M . Γ2 = 1 + e−ν R˙ 2 − R

(18)

Note that by differentiating Eq. (17), one may also obtain Γ˙ λ˙ R˙ = − + . Γ 2 R

(19)

Now using Eqs. (7) and (19), one finds that9 R˙ ν Γ˙ = . Γ R 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 ν 2 ¨ = −eν R(4πp + M/R3 ) + e Γ ν + R 2R For further analysis, let us write the three terms equation as A, B, C, respectively:

¨ = A(r, t) + B(r, t) + C(r, t). R

1 ˙ ν˙ R. (21) 2 on the RHS of the above (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 recovers Eq. (1): ¨ = Aff = − M . R R2 1350021-4

(23)


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For further insight into this term, let us momentarily consider the fluid to be of uniform density so that 4π M ρ(t). (24) = 3 R 3 On the other hand by including pressure, for an uniform density case, the first term on the right-hand side (RHS) of Eq. (21) becomes 4πR ν e (ρ + 3p). (25) 3 The notion that inclusion of pressure may make the monotony of collapse in GR even 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. This inference however need not be correct because when pressure is present, g00 = eν < 1.


Apressure = −

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 or ν(0, 0) = 0. Then we must have ν > 0 as we would move outward. Also, for realistic cases, 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, let us use (16) to write B=

eν Γ2 |p | eν Γ2 −p =+ . R ρ+p R ρ+p

(26)

Here we have used the fact that the pressure gradient p is negative. Note, the addition of the p/c2 term in the denominator is usually considered as the enhancement of “inertial mass density” in GR case. The corresponding Newtonian term would obviously be B = +|p |/ρ. 3.3. Analysis of the third term Suppose we define a “compactness factor” −1/2

z = g00

− 1 = e−ν/2 − 1

(27)

so that 1 ν˙ = −ze ˙ ν/2 . (28) 2 At the beginning of the collapse, one may have g00 = 1 and if a spacetime singularity would indeed form, one may have g00 = 0, so that z˙ ≤ 0 and 1 ˙ ≥ 0. C = ν˙ R˙ = eν/2 |z| ˙ R| (29) 2 On the other hand, for the unrealistic dust collapse case ν˙ = 0. Note, in the Newtonian case, there is no such term: CNewtonian = Cff = 0. 1350021-5

(30)


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4. Comparison with Newtonian Case As we found, for the corresponding Newtonian case, one would have


¨ = − M + |p | . R R2 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 idea that addition of pressure accentuates GR collapse is a misconception. Further, for ˙ (Eq. (29)) which the GR case, there is an additional compactness term ∝ |z|| ˙ R| ˙ Thus directly acts like a stabilizing factor because it is proportional to R. even for a perfect fluid having no tangential pressure, no heat flow, GR collapse has more 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 that we must always have ν > 0, we expect pt ≥ p. Physically, shear and heat flow associated with an imperfect fluid increases pt .10,11 This apart, in the presence of a strong radially directed inward gravitational field, motion in the radial direction will be less random, and thus, it is expected that ∆ ≡ pt − p ≥ 0. In such a case, Eq. (26) gets modified into   2R |p | + |∆|  e ν Γ2  R . (32) B=+    R ρ+p Thus, with the inclusion of tangential pressure, collapse is likely to be resisted in a way stronger than the perfect fluid case. And, in case, the collapse would proceed to very deep gravitational well, this effect may increase dramatically because of the following reason: It may be first recalled that given a spherically symmetric body having a gravitational mass Mb and a boundary at R = Rb , the surface gravitational redshift is given by (G = c = 1): −1/2 2Mb zb = 1 − − 1. (33) Rb Suppose the body is shrinking and its gravitational field is increasing. Then the formation 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 to hinder the outflow of heat/radiation or anything due to its stronger gravitational pull.√In particular, a photon sphere is defined by region interior to R = 3Mb or z = 3 − 1 where gravitational pull is already so strong that photons and neutrinos would start moving in closed circular orbits. 1350021-6


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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 tend to move in counter rotating circular orbits.12,13 In such a case, even the matter particles may tend to move in similar way so that the configuration would tend to be an “Einstein Cluster.”14 Formally an Einstein cluster is a spherical cloud comprising point particles moving in closed circular orbits having various radii; and in order to have zero angular momentum for the entire cluster, for every rotating particle, there must be a counter-rotating particle. Clearly, for this dust cloud, radial pressure, p = 0 while the entire pressure is of transverse nature. Such a configuration was first conceived by Einstein to conclude that continued gravitational collapse would be halted by such transverse pressure.14 Although Einstein’s intuition was correct, he had no idea of a likely “photon sphere” and he did not answer why continually contracting matter must shed radial pressure and on the other hand develop entirely transverse pressure. Consequently, his exercise was largely ignored by most of the relativists. But, here we offered a qualitative answer as to why matter may approximately behave like an “Einstein cluster.” Therefore once the fluid would dip into its photon sphere, there should be dramatic increase in the value 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 the gravitational intensity. In GR, if one would consider a perfect fluid, in the local free falling 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 mass energy density and therefore, pressure can only assist collapse (here we have reintroduced c = c). But it is easy to see that the above interpretation is completely incorrect. Of course, mathematically, one can always conceive of a locally free falling frame and a fundamental consequence of Einstein’s Equivalence Principle is that g00 = 1 in such a frame. But in the presence of the pressure, the fluid is subject to pressure gradient force in its own rest frame and therefore, the above equation is not relevant in the comoving frame of the fluid. On the other, for the comoving frame, 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 energy 15 : ρg = ρ − 3p/c2 . 1350021-7

(37)


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Accordingly, we might rewrite Eq. (25) as Apressure = −

4πR (ρ − 3p/c2 ), 3

(38)

so that


|Apressure | < |Aff |.

(39)

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

4πR (ρ − 3p). 3

(40)

Since this term essentially contains the trace of the energy–momentum tensor, for the imperfect fluid, one may have Apressure = −

4πR (ρ − p − 2pt ). 3

(41)

7. Does Trapped Surface and Event Horizon Form? The concept of a “trapped surface” may be an important concept in differential geometry.2,3 But does real physical situation must oblige all novelties of differential geometry? In other words, while by Einstein’s equation, given a certain distribution of matter energy–momentum tensor there will be certain nontrivial spacetime geometry, does all hypothetical spacetime geometries must correspond to finite matter energy–momentum distribution in a real physical situation. Historically, progress in the research of GR gravitational collapse has taken place by assuming an affirmative answer/faith for such a question. It is clear that the idea of trapped surface formation is compatible with the pressureless case with p = pt = ∆ = 0 (provided one can assume ρdust > 0). And this assumption of formation of trapped surfaces 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 the supposed adiabatic collapse of perfect fluid may experience bounce or even stop with the formation of a static object. Since with radial pressure alone, for a static object, zb < 2,16 such a bounce must happen by honoring this constraint. On the other hand, since for pure tangential pressure, there is no upper limit on zb ,17,18 the bounce can happen even from the deepest potential well; or else, the contraction can 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 must correspond to R˙ = 0. If a trapped surface would be formed, of course, under very reasonable conditions, the collapse must be monotonic and “unstoppable.”2,3 However, in order to ensure that trapped surface is formed, one needs to assume that, the GR collapse becomes unstoppable much before the formation of any trapped surface! This seems like a tautology and hence Kriele commented that19 : 1350021-8


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“The presence of a closed trapped surface which is assumed in those singularity theorems that (hopefully) predict the collapse of a star is not entirely global. But under normal circumstances such a trapped surface can only occur inside the black hole region. Thus the singularity theorems seem to be inapplicable for predicting the 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 are fulfilled for realistic cases. The assumptions behind formation of trapped surfaces and singularity theorems were probably first questioned by Donald and he concluded that20 “the assumptions required by the singularity theorems are examined critically. These assumptions are found to be questionable.” Later, by using differential geometry, Kriele showed that a spherically symmetric star of uniform density “cannot contain a trapped surface.”21 Such questions were raised later both from detail physical and mathematical perspectives and it was concluded that neither trapped surfaces nor (finite mass) black holes can form in realistic gravitational collapse.22,23 And a finally an elegant correct proof to this 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 spherically evolving body must remain timelike to all observers, one must have 2M (r, t) ≤ 1. R(r, t)

(42)

This above result does not depend on any exterior boundary condition or any specific 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 assumption of spherical symmetry/isotropy. It shows that (i) if continued collapse would indeed continue upto R = 0, the final state would be a M = 0 black hole. This however does not mean collapse must continue all the way to R = 0; on the other hand, it means that there may be physical mechanisms by which continued collapse may be halted to result in either truly static or at least quasi-static compact objects. In fact if one would accept a class of nonlinear generalizations of the electromagnetic theory, occurrences of trapped surfaces and singularities may be avoided both in the context of gravitational collapse and cosmology in accordance with Eq. (42).25−27 As to the exact mathematical solution for black holes, let us remind that, in general most of the exact solutions in GR could be physically meaningless or not realizable. This is so because GR has complex mathematical structure and finding an exact solution could be a sort of miracle. This is even more true for the problem of gravitational collapse where one must feed those equations with exact (evolving) EOS of matter, exact radiation and heat transportation calculations. And note here, while the integration constant appearing in the vacuum Schwarzschild solution α = 2GM/c2 must be finite for an object with finite radius (Rb > 0) (like the Sun, Galaxy etc.), there no a priori guarantee that it must be so in the limit of 1350021-9


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Rb → 0 or a “point particle.” Since in contrast to the exterior spacetime of say Sun, the Schwarzschild black hole solution corresponds to a “point mass,” it has been 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 upheld by claiming that the Kruskal coordinates offer self-consistent description for the entire associated spacetime. But recently this claim has been critically re-examined and 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˙ = −∞! at R = 0 if indeed ρ > 0. R

(43)

Further, since in GR, pressure increases AGDM in a locally free falling frame, it was 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 lines of thinking which led to the idea that in GR, collapse must be monotonic even if pressure forces will be active. But we found that this idea is incorrect because of the 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 opposes ¨ has positive contribution from pressure gradient. the collapse process because R • In GR, there is an additional term C = (1/2)ν˙ R˙ > 0 which directly opposes ˙ Therefore, collapse by acting as a positive feedback which is proportional to R. unlike the case of a dust, the mathematical collapse of a perfect fluid can reverse or oscillate. In fact, there are innumerable claims of bounce or oscillation in violation of the idea that GR collapse must be monotonic.31–42 In particular, the paper by Bondi was titled as Gravitational Bounce in General Relativity.37 Despite this, one may argue that, for a collapsing perfect fluid having only radial pressure, reversal may be difficult in view of the fact that the surface gravitational redshift of static compact objects must be zb < 2.0.16 • The pressure related resistive effect gets enhanced for an imperfect fluid with transverse pressure. By considering such effects, the eventual evolution equation, for a constant density case, may be written as:

3eν Γ2 |p | + (2R/R)|∆| 3ν˙ R˙ 4πR ¨ ρ − (p + 2pt ) − − . (44) R=− 3 4πRR ρ+p 8πR For the corresponding Newtonian case, one has

¨ = − 4πR ρ − 3|p | . R 3 4πRρ 1350021-10

(45)


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And for the free fall dust case, whether it is Newtonian or Einstein gravity, one ¨ = −(4π/3)ρR. We noted that, for the GR, case, there could be dramatic has R increase of the effect of tangential pressure term ∆/R if the body would plunge √ into 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. 17 and 18); therefore, in principle, the effect of tangential pressure might give rise to ultra compact objects having zb 1. The fact that tangential pressure can stabilize the contracting tendency of self-gravity is known as Lemaitre Vault formation by tangential pressure.18 9. Conclusion For a spherically evolving fluid, GR does not allow formation of trapped surfaces in order that a timelike worldline must always remain so. Then the inequality (42) tells that 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 from a much less general and rather questionable consideration, it has been opined that OS collapse should not lead to black hole formation.44 To ensure the absence of formation of trapped surfaces there must be appropriate physical mechanisms. As already mentioned, one such mechanism could be adoption of appropriate nonlinear electrodynamics.25–27 But we found here that even in the absence of such departures from standard physics, usual effects like pressure gradient and reduction of AGMD must play an important role in ensuring the sanctity of (42). In general, physical gravitational collapse is radiative45,46 and there are innumerable examples that for such radiative collapse, there can be bounce, oscillation or 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 the inward pull of gravity, it has been shown that continued radiative collapse should indeed give rise to ever contracting hot quasi-static objects12,13 as the trapped radiation luminosity would become equal to the corresponding Eddington luminosity. And here we found that, this phenomenon might be also understood from the view point of unabated growth of tangential stresses and formation of radiation supported “Lemaitre Vault.” For practical cases, this scenario will yield quasi-static objects with zb 1 and which would act as “quasi black holes” or “black hole mimickers.” And since they asymptotically evolve towards the true black hole state having zb = ∞ and Mb = 0 (Refs. 28–30) they have been termed as “Eternally Collapsing Objects.”

Acknowledgments The author thanks the anonymous referee for making several suggestions which led to an improved version of this manuscript. 1350021-11


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does pressure accentuate general relativistic

does pressure accentuate general relativistic

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General Relativistic MHD Jets

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General Relativistic Calculations in Mathematica

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General Relativistic Calculations in Mathematica

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