Fate of gravitational collapse in semiclassical gravity
Abstract
While the outcome of gravitational collapse in classical general relativity is unquestionably a black hole, up to now no full and complete semiclassical description of black hole formation has been thoroughly investigated. Here we revisit the standard scenario for this process. By analyzing how semiclassical collapse proceeds we show that the very formation of a trapping horizon can be seriously questioned for a large set of, possibly realistic, scenarios. We emphasise that in principle the theoretical framework of semiclassical gravity certainly allows the formation of trapping horizons. What we are questioning here is the more subtle point of whether or not the standard black hole picture is appropriate for describing the end point of realistic collapse. Indeed if semiclassical physics were in some cases to prevent formation of the trapping horizon, then this suggests the possibility of new collapsed objects which can be much less problematic, making it unnecessary to confront the information paradox or the runaway end point problem.
pacs:
04.20.Gz, 04.62.+v, 04.70.s, 04.70.Dy, 04.80.CcI Introduction
Although the existence of astrophysical black holes is now commonly accepted, we still lack a detailed understanding of several aspects of these objects. In particular, when dealing with quantum field theory in a spacetime where a classical event horizon forms, one encounters significant conceptual problems, such as the informationloss paradox linked to black hole thermal evaporation Hawking:science ; hawkingter ; hawkingparadox ; preskill .
The growing evidence that black hole evaporation may be compatible with unitary evolution in stringinspired scenarios (see, e.g., reference AdS )^{1}^{1}1See, however, a recent article by D. Amati Amati:2006fr for an alternative point of view on the significance of these results. has in recent years led to a revival of interest in, and extensive modification of, early RomanBergmann alternative semiclassical scenarios for the late stages of gravitational collapse hayward ; ashtekarbojowald . (See also tipler ; bardeen ; york .) Indeed, while it is by now certain that the outcome of a realistic classical collapse is necessarily a standard black hole delimited by an event horizon (that is, a region of the total spacetime which does not overlap with the causal past of future null infinity: ), it has recently been suggested that only apparent or trapping horizons might actually be allowed in nature, and that somehow semiclassical or quantum gravitational ashtekarbojowald ; hawkinginfo ; mathur effects could prevent the formation of a (strict, absolute) event horizon,^{2}^{2}2“The way the information gets out seems to be that a true event horizon never forms, just an apparent horizon”. (Stephen Hawking in the abstract to his GR17 talk hawkinginfo .) and hence possibly evade the necessity of a singular structure in their interior.
Note that Hawking radiation would still be present, even in the absence of an event horizon hajicek ; essential . Moreover, the present authors have noticed that, kinematically, a collapsing body could still emit a Hawkinglike Planckian flux even if no horizon (of any kind) is ever formed at any finite time quasiparticleprl ;^{3}^{3}3Recently, it was brought to our attention that this possibility was also pointed out in a paper by P. Grove grove . all that is needed being an exponential approach to apparent/trapping horizon formation in infinite time. Since in this case the evaporation would occur in a spacetime where information by construction cannot be lost or trapped, there would be no obstruction in principle to its recovery by suitable measurements of quantum correlations. (The evaporation would be characterized by a Planckian spectrum and not by a truly thermal one.)
Inspired by these investigations we wish here to revisit the basic ideas that led in the past to the standard scenario for semiclassical black hole formation and evaporation. We shall see that, while the formation of the trapping horizon (or indeed most types of horizon) is definitely permitted in semiclassical gravity, nonetheless the actual occurrence or nonoccurrence of a horizon will depend delicately on the specific dynamical features of the collapse.
Indeed, we shall argue that in realistic situations one may have alternative end points of semiclassical collapse which are quite different from black holes, and intrinsically semiclassical in nature. Hence, it may well be that the compact objects that astrophysicists currently identify as black holes correspond to a rather different physics. We shall here suggest such an alternative description by proposing a new class of compact objects (that might be called ‘‘black stars’’) in which no horizons (or ergoregions) are present.^{4}^{4}4These “black stars” are nevertheless distinct from the recently introduced “gravastars” gravastar . The absence of these features would make such objects free from some of the daunting problems that plague black hole physics.
Ii Semiclassical collapse: The standard scenario
Let us begin by revisiting the standard semiclassical scenario for black hole formation. For simplicity, in this paper we shall consider only nonrotating, neutral, Schwarzschild black holes; however, all the discussion can be readily generalized to other black hole solutions.
Consider a star of mass in hydrostatic equilibrium in empty space. For such a configuration the appropriate quantum state is well known to be the Boulware vacuum state boulware , which is defined unambiguously as the state with zero particle content for static observers, and is regular everywhere both inside and outside the star (this state is also known as the static, or Schwarzschild, vacuum birrelldavies ). If the star is sufficiently dilute (so that the radius is very large compared to ), then the spacetime is nearly Minkowskian and such a state will be virtually indistinguishable from the Minkowski vacuum. Hence, the expectation value of the renormalized stressenergymomentum tensor (RSET) will be negligible throughout the entire spacetime. This is the reason why, when calculating the spacetime geometry associated with a dilute star, one only needs to care about the classical contribution to the stressenergymomentum tensor (SET).
Imagine now that, at some moment, the star begins to collapse. The evolution proceeds as in classical general relativity, but with some extra contributions as spacetime dynamics will also affect the behaviour of any quantum fields that are present, giving place to both particle production and additional vacuum polarization effects. Contingent upon the standard scenario being correct, if we work in the Heisenberg picture there is a single globally defined regular quantum state that describes these phenomena.
For simplicity, consider a massless quantum scalar field and restrict the analysis to spherically symmetric solutions. Every mode of the field can (neglecting backscattering) be described as a wave coming in from (i.e., from , ), going inwards through the star till bouncing at its center (), and then moving outwards to finally reach . As in this paper we are going to work in dimensions (i.e., we shall ignore any angular dependence), for later notational convenience instead of considering wave reflections at we will take two mirrorsymmetric copies of the spacetime of the collapsing star glued together at (see Fig. 1). In one copy will run from to , and in the other from to . Then one can concentrate on how the modes change on their way from (i.e., , ) to (i.e., , ). Hereafter, we will always implicitly assume this construction and will not explictly specify “left” and “right” except where it might cause confusion.
Now, one can always write the field operator as
(1) 
where are the modes that near behave asymptotically as^{5}^{5}5We work in natural units.
(2) 
with and . One can then identify the state as the one that is annihilated by the destruction operators associated with these modes: . (One could also expand using a wave packet basis hawkingter , which is a better choice if one wants to deal with behaviour localized in space and time.) Since the spacetime outside the star is isometric with a corresponding portion of Kruskal spacetime, and is static in the far past, the modes have the same asymptotic expression as the Boulware modes boulware near (i.e., for ). Hence , the quantum state corresponding to the physical collapse, is (near ) indistinguishable from the Boulware vacuum . (But this will of course no longer be true as one moves significantly away from .)
Now, the semiclassical collapse problem consists of studying the evolution of the geometry as determined by the semiclassical Einstein equations
(3) 
where is the classical part of the SET. Significant deviations from the classical collapse scenario can appear only if the RSET in equation (3) becomes comparable with the classical SET. In this analysis there are (at least) two important results from the extant literature that have to be taken into account:

If a quantum state is such that the singularity structure of the twopoint function is initially of the Hadamard form, then Cauchy evolution will preserve this feature FSW , at least up to the edge of the spacetime (which might be, for instance, a Cauchy horizon HawBook ). The state certainly satisfies this Hadamard condition at early times fnw , hence it must satisfy it also in the future, even if a trapping/event horizon forms. (A trapping/event horizon is not a Cauchy horizon, and is not an obstruction to maintaining the Hadamard condition.) As a consequence of this fact the RSET cannot become singular anywhere on the collapse geometry, independently of whether or not a trapping/event horizon is formed.^{6}^{6}6It is important to understand exactly what this theorem does and does not say: If we work in a wellbehaved coordinate system (where the matrix of metric coefficients is nonsingular and has finite components), then the coordinate components of the RSET are likewise finite. But note that finite does not necessarily imply small.

For specific semiclassical models of the collapsing star it has been numerically demonstrated (modulo several important technical caveats) that the value of the RSET remains negligibly small throughout the entire collapse process, including the moment of horizon formation pp .^{7}^{7}7Similar results were, after some discussion, found in dimensional models based on dilaton gravity 2d . Subsequently, in this scenario quantum effects manifest themselves via the slow evaporation of the black hole.
Thus in this standard scenario nothing prevents the formation of trapped regions (or trapped/apparent/event horizons). Given that quantuminduced violations of the energy conditions visser ; cosmo99 are taken to be small enough at this stage of the collapse, one can still use Penrose’s singularity theorem to argue that a singularity will then tend to form. Assuming that quantum gravity effects will not conspire to avoid this conclusion, then, in conformity with all extant calculations and the cosmic censorship conjecture, a spacelike singularity and a true event horizon will form. The collapsed star settles down in a quasistatic black hole and then ultimately evaporates.
This last feature can be easily derived by considering an expansion of the field in a basis which contains modes that near (i.e., for , ), behave asymptotically as
(4) 
with and , so defining creation and annihilation operators that differ from those associated with the modes of equation (2). In a static configuration a (spherical) wave coming from is blueshifted on its way towards the center of the star, and is then equally redshifted on its way out to , arriving there undistorted. However, in a dynamically collapsing configuration the redshift exceeds the blueshift, so that an initial wave at is distorted once it reaches . In this sense the dynamical spacetime acts as a “processing machine” for the normal modes of the field. Expanding the distorted wave in terms of the undistorted basis at tells us the amount of particle creation due to the dynamics. In particular one can take a wave packet centered on frequency on and ask what its typical frequency, say , will be when it arrives on . The Bogoliubov coefficients that allow us to express the annihilation operators related to the modes (4) in terms of the creation operators pertaining to the modes (2) are related to the number of particles seen by asymptotic observers on , which is nothing else than the thermal flux of Hawking radiation Hawking:science ; hawkingter ; birrelldavies .
This can be rephrased saying that the physical state corresponding to the collapse behaves like the Unruh vacuum unruh of Kruskal spacetime near the event horizon, , and near (i.e., for ). Indeed, in the Kruskal spacetime the Unruh state is a zeroparticle state for a freelyfalling observer crossing the horizon, and corresponds to a thermal flux of particles at the Hawking temperature for a static observer at infinity birrelldavies ; scd . Given that at late times classical black holes generated via classical gravitational collapse are virtually indistinguishable from eternal black holes (see, for instance, the classical theorem in wald ), the Unruh vacuum is the only quantum state on Kruskal spacetime which appropriately (near and ) simulates the physical vacuum in a spacetime with an event horizon formed via gravitational collapse.
However as previously mentioned, this standard scenario leads to several well known problems (or at the very least, disquieting features):

Modes corresponding to quanta detected at have an arbitrarily high frequency on (this is the socalled trans–Planckian problem unruh ).

The runaway end point of the evaporation process (the Hawking temperature is inversely proportional to the black hole mass) prevents any welldefined semiclassical answer regarding the ultimate fate of a black hole Hawking:science .

If eventually the black hole completely evaporates, leaving just thermal radiation in flat spacetime, then it would seem that nothing would prevent a unitarityviolating evolution of pure states into mixed states, contradicting a basic tenet of (usual) quantum theory (this is one aspect of the socalled informationloss paradox hawkingparadox ; preskill ). Such a difficulty for reconciliating quantum mechanics with general relativity seems to persist even when imagining many alternative scenarios for the end point of the evaporation, so that one can still continue to talk about an informationloss problem hawkingparadox ; preskill .
All in all, it is clear that this semiclassical collapse scenario is evidently plagued by significant difficulties and obscurities that still need to be understood. For this reason we think it is worthwhile to step back to a clean slate, and to revisit the above story uncovering all the hidden assumptions.
Iii Semiclassical collapse: A critique
It is easy to argue that one cannot trust a semiclassical gravity analysis once a collapsing configuration has entered into a highcurvature (Planckscale) regime; this is expected in the immediate neighborhood of the region in which the classical equations predict the appearence of a curvature singularity. Once the formation of a trapped region is assumed, any solution of the problems mentioned above seems (naively) to demand an analysis in a fullfledged theory of quantum gravity. Here, however, we are questioning the very formation of a trapped region in astrophysical collapse. In analyzing this question we will see that semiclassical gravity provides a useful and sensible starting point. Moreover, we will also show that it provides some indications as to how the standard scenario might be modified.
iii.1 The trans–Planckian problem
One potential problem with the semiclassical gravity framework, when used to analyze the onset of horizon formation, is the trans–Planckian problem. While this problem is usually formulated in static spacetimes, for our purposes we wish to look back to its origin in a collapse scenario.
We can, as usual, encode the dynamics of the geometry in the relation between the affine null coordinates and , regular on and , respectively. Neglecting backscattering, a mode of the form (2) near takes, near , the form
(5) 
This can be regarded, approximately, as a mode of the type presented in equation (4), but now with dependent frequency , where a dot denotes differentiation with respect to . (Of course, this formula just expresses the redshift undergone by a signal in travelling from to .)
In general we can expect a mode to be excited if the standard adiabatic condition
(6) 
does not hold. It is not difficult to see that this happens for frequencies smaller than
(7) 
One can then think of as a frequency marking, at each instant of retarded time , the separation between the modes that have been excited () and those that are still unexcited ().
Moreover, Planckscale modes (as defined on ) are excited in a finite amount of time, even before the actual formation of any trapped region. Indeed, they start to be excited when the surface of the star is above the classical location of the horizon by a proper distance of about one Planck length, as measured by Schwarzschild static observers. We can see this by observing that the redshift factor satisfies
(8) 
where is the surface gravity. This then implies , where we have used . Hence
(9) 
Hence, the trans–Planckian problem has its roots at the very onset of the formation of the trapping horizon. Furthermore, any complete description of the semiclassical collapse cannot be achieved without at least some assumptions about trans–Planckian physics.
Of course, one can simply assume that there is a natural Planckscale frequency cutoff for effective field theory in curved spacetimes. Although one cannot completely exclude this possibility, we find that this way of avoiding the trans–Planckian problem is perhaps worse than the problem itself, as it would automatically also imply a shutdown of the Hawking flux in a finite (very small) amount of time. This would eliminate the thermodynamical behaviour of black holes, thus undermining the current explanation for the striking similarity between the laws of black hole mechanics and those of thermodynamics — that they are, in fact, just the same laws davies .
Moreover, such a “hard cutoff” obviously corresponds to a breakdown of Lorentz invariance at the Planck scale. If one is ready to accept such a departure from standard physics, then it seems more plausible (less objectionable?) to conjecture a milder breaking of Lorentz invariance in the form of a modified dispersion relation, a possibility explored in several works on the trans–Planckian problem Jacobson:1999zk . While it is seemingly well understood that the Hawking radiation would survive in this case US , it is however less clear what effect such modified dispersion relations might have on the possibility of forming a (presumably frequencydependent) trapping horizon, and indeed, on the very definition of such a concept Barcelo:2006yi .
In what follows we shall adopt a conservative approach and stick, as is usually done, to the standard framework of quantum field theory in curved spacetime, assuming its validity up to arbitrarily high frequencies. Even in the presence of Lorentz violating effects, this would remain a valid framework if, for example, the scale at which Lorentz violations might appear was much higher than the Planck scale Jacobson .
iii.2 Vacuum polarization
The other difficulties of the standard scenario previously listed have been linked by different authors to the presence of horizons and of trapping regions in general. As we have previously discussed, several departures from semiclassical gravity have often been called for in order to solve these problems. However, the specific question we now want to raise here is rather different: Is the scenario just described guaranteed to be the one actually realized in semiclassical gravity? Or is it possible that semiclassical gravity allows for alternative endpoints of gravitational collapse, in which these problems are not present? In order to answer these questions we look for possible semiclassical effects which could modify the collapse before the very formation of a trapped region.
In any calculation of semiclassical collapse the choice of the propreties of the matter involved (which will be encoded in the characteristics of the classical SET) is, obviously, of crucial importance. Normally the initial conditions at early times are chosen so that one has a static star with any quantum field in their “natural” vacuum state. As we have discussed, this will be virtually indistinguishable from the Boulware vacuum state. In this initial configuration we are sure that the RSET is practically zero throughout spacetime, at least before the collapse is initiated. We now want to inquire into the possibility that such a RSET becomes nonnegligible during the collapse.
In the standard semiclassical scenario, it is crucial that the initial Boulwarelike structure of the field modes at is somehow “excited” by the collapse and converted into a Unruhlike structure at both and — this is necessary for compatibility with the presence of a trapping horizon. In fact, if this excitation and conversion were not to be sufficiently effective so as to to get rid of Boulwarelike modes in the proximity of the wouldbe horizon, then a potential obstruction to the very formation of the horizon may arise. We know in fact that in static geometries there is an intrinsic incompatibility between the Boulware vacuum and the existence of a trapping horizon, as the RSET near the horizon (in a simplified calculation in 1+1 dimensions) is found to be dfu
(10) 
where we work in an orthonormal basis. A similar result remains valid in the more complicated dimensional case scd . The important point is that the denominator vanishes at the horizon. Hence the RSET acquires a divergent (and energy condition violating visser ) contribution. Note that the divergence is present even if the components of the RSET are evaluated in a freelyfalling basis scd . (To see that something intrinsic is going on at the horizon it is sufficient to calculate the scalar invariant , and to note that this scalar diverges at the horizon.)
Of course the above result applies to a static spacetime, while we are interested in investigating an intrinsically dynamical scenario, which we moreover know, due to the Fulling–Sweeny–Wald theorem FSW , should act in such a way as to avoid the above divergence. We are hence interested in seeing the precise way in which this happens, and in exploring whether it might leave a route to possibly obtaining large, albeit finite, contributions to the RSET at the onset of horizon formation.
Iv The RSET
In calculating the RSET in a dynamical collapse several choices must be made. The major assumption is that we shall for the time being restrict attention to dimensions, since then there is a realistic hope of carrying out a complete analytic calculation. Physically, this is not as bad a truncation as it at first seems, since we can always view it as an wave approximation to full dimensional problem, with at most a few actors of being inserted at strategic places. (For instance, this analytic approximation underlies the subsequent numerical calculation of Parentani and Piran pp .) A second significant choice we will make is to specifically work in a regular coordinate system, in particular, in Painlevé–Gullstrand coordinates pg ; analogue . In regular coordinate systems (where the matrix of metric coefficients is both finite and nonsingular), the values of the stressenergymomentum components are direct and useful diagnostics of the “size” of the stressenergymomentum tensor.
iv.1 Preliminaries
With reference to the diamondshaped conformal diagram of Fig. 1, we shall start by considering a set of affine coordinates and , defined on and respectively. These coordinates are globally defined over the spacetime and the metric can be written as
(11) 
Given that we shall be concerned with events which lie outside of the collapsing star on the righthand side of our diagram, we can also choose a second doublenull coordinate patch , where is taken to be affine on , in terms of which the metric is
(12) 
Of course,
(13) 
where describes the coordinate transformation. Then
(14) 
Furthermore, as long as we are outside the collapsing star it is safe to assume that a Birkhofflike result holds, and take as being that of a static spacetime.
Now for any massless quantum field, the RSET (corresponding to a quantum state that is initially Boulware) has components dfu ; birrelldavies
(15) 
(16) 
(17) 
The coefficients arising here are not particularly important, and will in any case depend on the specific type of quantum field under consideration.
The components and will necessarily be well behaved throughout the region of interest; in particular they are the same as in a static spacetime and are known to be regular. On the contrary shows a more complex structure due to the nontrivial relation between and . A brief computation yields
(18) 
The key point here is that we have two terms, one () arising purely from the static spacetime outside the collapsing star, and the other () arising purely from the dynamics of the collapse. If, and only if, the horizon is assumed to form at finite time will the leading contributions of these two terms cancel against each other — this is the standard scenario.
Indeed the first term is exactly what one would compute from using standard Boulware vacuum for a static star. As the surface of the star recedes, more and more of the static spacetime is “uncovered”, and one begins to see regions of the spacetime where the Boulware contribution to the RSET is more and more negative, in fact diverging as the surface of the star crosses the horizon.
iv.2 Regular coordinates
To probe the details of the collapse, it is useful to introduce yet a third coordinate chart — a Painlevé–Gullstrand coordinate chart in terms of which the metric is quasiparticleprl ; pg ; analogue
(19) 
This coordinate chart is particularly useful because it is regular at the horizon, so that the finiteness of the stressenergymomentum components in this chart has a direct physical meaning in terms of regularity of the stressenergymomentum tensor.^{8}^{8}8These coordinates are also useful as they allow to straightforwardly apply our calculations to acoustic analogue spacetimes (provided one is in a regime in which one could neglect the existence of modified dispersion relations) quasiparticleprl ; analogue . By setting the spacetime interval to zero, it is easy to see that the null rays are given by
(20) 
Although inside the collapsing star the metric can depend on and in a complicated way, the geometry outside the surface of the star is taken to be static, so the functions and do not depend on . Under these conditions we can integrate along the history of an outgoing ray from an event just outside the collapsing star to another event at asymptotic future infinity :
(21) 
Assuming asymptotic flatness, and , we find for the null coordinate in the “out” region,
(22) 
Hence, denoting partial derivatives by subscripts:
(23) 
(24) 
In contrast, along an incoming ray leaving asymptotic past infinity at an event and remaining outside the star,
(25) 
so we have, for the null coordinate:
(26) 
Hence
(27) 
In addition, by substituting and comparing coefficients of the line element, it is easy to see that the and coordinates are related by
(28) 
(29) 
and
(30) 
Therefore the components of the RSET can be calculated in any of the equivalent forms:
(31)  
(32)  
(34)  
(35)  
(37)  
(38) 
Some of these formulae are more useful for calculating the static Boulware contribution, others are more useful for calculating the dynamical contribution. Since at a horizon, while is regular, this is enough to guarantee that the and components of the RSET are always better behaved (less divergent) than the component. Note that no divergence can arise from the terms proportional to .
iv.3 Calculation assuming normal horizon formation
Hereafter, we shall for simplicity restrict our attention to the case . Placing the horizon at for convenience, we can write the asymptotic expansion
(40) 
where can be identified with the surface gravity quasiparticleprl ; analogue .
Consider first the static Boulware term in equation (18). We have (placing the horizon at for convenience)
(41) 
The relevant derivative in is then that with respect to , and we can write
(42)  
In fact, keeping the subleading terms one finds
(43) 
By equations (IV.2) and (38), it is clear that because of the constant term , the components and of the RSET contain contributions that diverge as and , respectively, as . (The subleading terms lead to finite contributions of order and respectively.)
In counterpoint, assuming horizon formation, let us now calculate the dynamical contribution to the RSET (). It is well known that any configuration that produces a horizon at a finite time leads to an asymptotic (large ) form
(44) 
where and are suitable constants. Taking into account the asymptotic expression (40) for near , it is very easy to see that the potential divergence at the horizon due to the static term is exactly cancelled by the dynamical term. In this way we have recovered the standard result that the RSET at the horizon of a collapsing star is regular.
However, the previous relation is an asymptotic one, and for what we are most interested in (the value of the RSET close to horizon formation) it is important to take into account extra terms that will be subdominant at late times. Indeed, we can describe the location of the surface of a collapsing star that crosses the horizon at time by
(45) 
where the expansion makes sense for small values of , and represents the velocity with which the surface crosses the gravitational radius. Let be the time at which a rightmoving light ray corresponding to null coordinates and crosses the surface of the star. Then on the one hand
(46) 
which for (implying ) can be approximated by
(47) 
so that
(48) 
On the other hand, since is simply some regular function, we have
(49) 
Inserting (48) into (49) we obtain an asymptotic expansion
(50) 
which it is useful to write as
(51) 
where is a regular function such that . Then
(52)  
The point is that this has a universal contribution coming from the surface gravity, plus messy subdominant terms that depend on the details of the collapse. It is important to note, however, that the corresponding additional contributions to the RSET are finite, in contrast to the one associated with the first term. Indeed, for small values of ,
(53) 
so
(54) 
and so the second term in the righthand side of equation (52) is , and by equation (38) gives an contribution to that does not depend on , but depends on time as . In addition, from a comparison of equations (48)–(50) we see that
(55) 
so the leading subdominant term in the RSET is inversely proportional to the square of the speed with which the surface of the star crosses its gravitational radius. In particular, at horizon crossing, that is at , the value of the RSET can be as large as one wants provided one makes very small. This would correspond to a very slow collapse in the proximity of the trapping horizon formation. Thus, there is a concrete possibility that (energy condition violating) quantum contributions to the stressenergymomentum tensor could lead to significant deviations from classical collapse when a trapping horizon is just about to form.
iv.4 Calculation assuming asymptotic horizon formation
Another interesting case one may want to consider is one in which the horizon is never formed at finite time, but just approached asymptotically as time runs to infinity. In particular, in reference quasiparticleprl it was shown that collapses characterized by an exponential approach to the horizon,
(56) 
lead to a function of the form
(57) 
where is half the harmonic mean between and the rapidity of the exponential approach ,
(58) 
so that one always has . In this case, the calculation of the dynamical part of the RSET leads to exactly the same result that when using expression (44), modulo the substitution of by . However, the nondynamical part of the RSET remains unchanged. This implies that now, at leading order
(59)  
which obviously diverges in the limit . We stress that this result does not contradict the Fulling–Sweeny–Wald theorem FSW , as the calculation applies only outside the surface of the star (i.e., for ), and so the divergence appears only at the boundary of spacetime. Nevertheless, particularizing to , this again indicates that there is a concrete possibility that energy condition violating quantum contributions to the stressenergymomentum tensor could lead to significant deviations from classical collapse when a trapping horizon is on the verge of being formed.
iv.5 Physical insight
The key bits of physical insight we have garnered from this calculation are:

In the standard collapse scenario the regularity of the RSET at horizon formation is due to a subtle cancelation between the dynamical and the static contributions.

Contributions that can be neglected at late times can indeed be very large at the onset of horizon formation. The actual value of these contributions depends on the rapidity with which the configuration approaches its trapping horizon.

Once the horizon forms, the above contributions will be exponentially damped with time. However, the analysis of the configuration that approches horizon formation asymptotically tells us that, while horizon formation is delayed, there are contributions that will keep growing with time.
Hence apparently the RSET can acquire large (and energy condition violating visser ) contributions when a collapsing object approaches its Schwarzschild radius, depending on the details of the dynamics. The final lesson to draw from this part of our investigation is that not all the classical matter configurations compatible with the formation of a trapping horizon in classical general relativity necessarily lead to the same final state when semiclassical effects are taken into account. In particular, for classical collapses that exhibit a slow approach to horizon formation, our calculation indicates that there will be a large (albeit always finite in compliance with FSW ) contribution from the RSET, a contribution which can potentially lead the semiclassical collapse to classically unforeseeable end points. For these reasons we wish next to further explore the alternative situation in which the horizon is only formed asymptotically.
V A quasiblack hole scenario
The history of the confrontation between general relativity and quantum physics has already shown several times that the quantum mechanical effects in matter can prevent the formation of black holes in situations in which classically such formation would seem unavoidable. Without quantum mechanics, objects such as white dwarfs and neutrons stars would have never been predicted in the first place. Similarly, in this paper we have seen that if for any reason the collapse of the matter forces it into some (metastable) state in which horizon formation is approched sufficiently slowly, then large quantum vacuum effects could prevent the very formation of a trapping horizon. The resulting object could then be considered the most compact and quantum mechanical kind of star. These objects, which we shall tentatively call ‘‘black stars’’,^{9}^{9}9Newtonian versions of “black stars”, more often called “dark stars”, have a very long history in astrophysics, dating back to Michell Michell and Laplace Laplace . For recent commentary on the historical connections between Michell, Cavendish, and Laplace, see LyndenBell . would be supported by a form of quantum pressure of universal nature, being characterized only by their closeness to the formation of a trapping horizon.
Lacking an understanding of the physics of matter at densities well beyond that characterizing neutron stars, we cannot reliably assert anything about the stability of black stars. However, the first motivation for our investigation was to see whether semiclassical physics can allow for compact objects closely mimicking black hole features, including Hawking radiation, without incurring in the same problems plaguing the standard scenario. In this sense, static configurations do not seem viable candidates as the absence of a trapping horizon together with the staticity prevents any possibility of emission of a Hawking flux.^{10}^{10}10It is perhaps worthwhile to stress here that such static black stars do not belong to the class of objects known in the literature as gravastars, (at least not without the addition of considerable extra assumptions), given than the former are compact aglomerates of matter while the latter have a de Sitterlike interior gravastar . On the other side, evolving configurations that continue to asymptotically approach their wouldbe horizon^{11}^{11}11This approach could be completely monotonic or have oscillating components. These oscillations can also produce burst of radiation at the Hawking temperature thooft . would produce quantum radiation at late times.
In order for such a scenario to be realized in nature one can speculate that in some cases, once matter has slowed down the collapse so allowing for the piling up of a sizeable RSET, the latter would not be able to completely stop the collapse, but would instead lead to an evolving configuration where every layer of the collapsing star would lie very close to where the classical horizon of the matter inside it would be located, continually and asymptotically approaching it. We can call this object a “quasiblack hole”.
In order to know exactly how the star asymptotically approaches the horizon in this scenario, one should solve Einstein’s semiclassical field equations with backreaction — obviously a very difficult task. Without the result of such an explicit calculation, it is nevertheless reasonable to conjecture that the approach can either follow a power law, or be exponential with a timescale , say. The case of a power law seems, however, uninteresting for our purposes, because it would not lead to a Planckian emission quasiparticleprl . On the contrary, an exponential approach is associated with the emission of radiation at a modified temperature quasiparticleprl . At least for astrophysical black holes, it is also reasonable to think that at the beginning of the evaporation process, so that , indistinguishable from the standard Hawking temperature. During evaporation increases so, in the long run, is determined by and tends to zero. Hence we could in principle have a “graceful exit” from the evaporation process; that is, one could avoid the standard runaway endpoint. Meanwhile, the evaporation could be visualized as a continuous chasing between the surface of the star and its (receding) Schwarzschild radius.
Indeed, possibilities for such a neverending collapse were already envisaged in 1976, soon after the discovery of Hawking radiation boulware2 ; gerlach and have been recently proposed again Vachaspati:2006ki (although via different backreaction mechanisms). It is important, however, to understand that in the quasiblack hole scenario we discuss here the Hawking flux only affects latetime evolution, and is not the agent that prevents horizon formation in the first place. The initial slowdown of the collapse is in this case due to matterrelated high energy physics. This provides the time necessary for the vacuum polarization to grow and finally modify the evolution of the collapse toward an asymptotic regime.
Of course, the state at is reached only after a very long time (for typical estimates of the evaporation timescale, see reference birrelldavies ), so according to this scenario a collapsing star forms an object that, for a long period, is indistinguishable from a standard black hole, further justifying our nomenclature of “quasiblack hole”. This object would still evaporate with a Planckian spectrum quasiparticleprl , but (since there is no event horizon) it would not be truly “thermal” (the quantum state is indeed a squeezed state squeezed ), hence there would be no informationloss problem. The partners of the particles emitted towards infinity, instead of being accumulated inside the trapping horizon as in the standard scenario, would now simply be emitted with a (significant) temporal delay. The radiation received at one instant of time would be correlated with that arriving some time later, so all the information would be recovered in the resulting radiation.
How does backreaction work in this scenario? During the late time asymptotic collapse, two processes unfold at the same time: (1) the energy associated with vacuum polarization becomes more and more negative; (2) radiation is emitted towards infinity. During a time interval as measured on , an arrival of energy is recorded by observers at infinity. Correspondingly, vacuum polarization leads to an extra energy (due to the fact that the star becomes more compact), so the Bondi mass of the object decreases by an amount . By energy conservation, one expects that , so the emission of radiation is balanced by the increase of vacuum polarization nearby the central object. This balance makes the Bondi mass of the object decrease as if it were taken away by radiation, eventually reducing to zero as . Note that the expression (10) for the RSET can be rewritten in such a way as to exhibit the fact that vacuum polarization corresponds to the absence of blackbody radiation at the temperature scd . Although this does not constitute a proof, it is a strong plausibility argument in favour of the energy balance between radiation and vacuum polarization. Also, it strongly suggests that the asymptotic approach to the wouldbe horizon must be of the exponential type, rather than a power law. Indeed, since a power law would not lead to a Planckian emission, it would be hard to reconcile it with the result presented in reference scd .
Thus, provided that trapping horizons do not form, we have described a plausible scenario for the progressive collapse and evaporation of quasiblack holes. However, the end point of this process seems to still share a problem with the standard scenario: The apparent accumulation of baryon number within the collapsing object boulware2 . The least massive baryon one can find is the proton. Baryon number is conserved in all experiments realized up to now, and in particular, the proton has been found to be stable (nevertheless, Grand Unification Theories predict it should eventually disintegrate into leptons). In the standard paradigm for the evaporation of a black hole, the trapping horizon and its surroundings is an empty region of spacetime. Therefore, there is only one physical quantity characterizing the quantum emission: The value of its Hawking temperature. For a standard evaporating black hole to be able to nucleate a protonantiproton pair, it seems necessary that it reaches a temperature larger than K, or equivalently, a tiny mass of less than , where is the mass of one proton. However, for example, a black hole having initially one solarmass would contain a baryon number of around . During the evaporation it would conserve this baryon number till it reaches a Bondi mass of . But then, even emitting all its remaining energy in the form of baryons (with emission in the form of protons being the most efficient way of removing baryon number), it would end up either: (1) leaving an almost massless relic having a baryon number of (a rather peculiar state); or (2) completely evaporating producing an enormous violation of baryonnumber conservation.
The quasiblack hole scenario, however, adds one extra ingredient to the previous discussion: The wouldbe horizon and its surroundings is now not an empty region of spacetime. In the vicinity of the wouldbe horizon there is always matter progressively being compressed. This fact could significantly affect the way the quasiblack hole radiates its energy. For example, an upper bound for the average density of a solar mass quasiblack hole is given by that of the corresponding black hole (a few times bigger than that of a typical neutron star). At these densities and higher, it is quite plausible that new particle physics effects could come into play and deplete the baryon number much more efficiently than the evaporation process.^{12}^{12}12Of course, for very massive quasiblack holes such effects will be negligible for a very long time, but will eventually become important as the Bondi mass is decreased by the combined effect of Hawking radiation and vacuum polarization.
Up to this point we have only considered spherically symmetric configurations. However, current observations tell us that most of the observed black hole candidates have a high rate of rotation, sometimes very close to extremality rotation . Hence, for a quasiblack hole scenario to be a feasible description of these objects, it would be necessary to generalize our proposal to rotating configurations. Given the complexity of the vacuum structure around rotating black holes ottewill it is very difficult to have a precise proposal in this sense. However, we know that any rotating object possessing an ergoregion but not a horizon would be highly unstable cardoso . Hence we expect that any viable model of a rotating quasiblack hole should be characterized by a matter distribution extending up to the outer boundary of the ergoregion.
The fact that most of the progenitors of the observed black hole candidates are characterized by supercritical rotations (, where is the angular momentum of the progenitor) is often used as evidence of the validity of the cosmic censorship conjecture. It is interesting to note that if such conjecture holds for standard general relativity it would also be effective in preventing supercritical quasiblack holes. In order to understand this point it is enough to realize that a generalization of the calculation of this article to more general metrics allowing for extremality (e.g., Reissner–Nordström, Kerr, …) would still imply a pile up of the RSET in proximity of the “wouldbe horizon” if and only if such a horizon can form in the first place. That is, a large quantuminduced RSET can arise only if the collapsing object has already shed the extra charges (e.g., electric charge or angular momentum) so as to be subcritical in proximity of the horizon crossing. So supercritical configurations are likely to be unaffected by the vacuum polarization and behave as in classical general relativity. On the contrary subcritical configurations will develop (or not develop) trapping horizons according to the details of the dynamics.
Vi Conclusions
Quantum physics imposed upon the description of the collapse of astrophysical objects in situations that would classically lead to black hole formation could unexpectedly lead to observable effects at early times, when the trapping horizon is about to form. In particular we have shown that before forming a trapping horizon, trans–Planckian modes are excited. Hence, whether the trapping horizon forms or not depends critically on assumptions concerning the net effect of any trans–Planckian physics that might be at work.
Assuming that quantum field theory holds unmodified up to arbitrarily high energy (as is commonly done in most of the extant literature) we have shown that there can be large deviations from classical collapse scenarios, if the latter do allow in the first place a piling up of vacuum energy. Most of the classical collapse scenarios so far considered do not allow for such a piling up, due to their intrinsic rapidity. In this sense the prediction of horizon formation in many of these models pp ; massar seems completely correct.
We have argued however, that alternative classical collapse scenarios in which horizon formation is approached in a slow manner are not only foreseeable, but possibly natural in more realistic situations. If this is indeed the case one then would have to add a new class of compact, horizonless, objects (possibly the most compact objects apart from black holes themselves) to the astrophysical bestiary: the black stars.
In the final part of this work we have then considered a particular subclass of these objects, the quasiblack holes, which could closely mimic all the most relevant features of black hole physics, while avoiding at the same time most of its intrinsic problems (such as singularities, the information paradox, and the question of the end point of Hawking evaporation).
Summarizing, the quasiblack hole scenario for collapse and evaporation is the following one (see Fig. 2):
As a star of mass implodes we conjecture that its matter will try to adjust in new, possibly unstable, configurations so to reach a new equilibrium against gravity. If there is ever a significant slowing down of the collapse, for any reason whatsoever, then this allows the vacuum polarization to progressively grow, and further slow down the approach to trapping horizon formation. Provided such an approach is asymptotic with an exponential law controlled by a timescale , then the quantum radiation produced during this process is still Planckian, with a temperature , where is inversely proportional to the total Bondi mass of the star quasiparticleprl . For a long time, and , so and (from the point of view of an external observer) the object is essentially indistinguishable from a standard evaporating black hole. The emission of radiation is accompanied by an increase in vacuum polarization, that progressively diminishes the Bondi mass of the star, so the wouldbe horizon shrinks and is never crossed by the matter configuration. When the Bondi mass has become sufficiently small, is negligible and the temperature is approximately equal to . This quantity is also decreasing, because backreaction is in fact slowing down collapse, so the temperature, after reaching a maximum value, decreases and approaches zero.
We do not yet have a definitive proposal as to the endpoint of the evaporation process. This could only be achieved by understanding the physics of baryon nucleation in the presence of highdensity states of matter. The end state of the evaporation could correspond to a zerotemperature relic^{13}^{13}13Note that the nature of such a relic would be quite different from that of a standard black hole remnant, because the relic could be regarded just as a peculiar case of a very compact star. For this reason, the usual issues related to remnants (like the compatibility with CPT invariance or their capacity for storing information) are not present in this scenario. with vanishing Bondi mass (hence would at large distances be gravitationally inert), with an inner structure formed by a core with mass and a nonvanishing baryon number, immersed into a cloud of polarized vacuum with negative energy . Alternatively, it might correspond to plain vacuum.
Acknowledgments
We are grateful to Daniele Amati, Larry Ford, Ted Jacobson, John Miller, José Navarro–Salas, Tom Roman, and Bernard Whiting for critically reading a preliminary version of this paper and for stimulating discussions. We would like to also thank Renaud Parentani and Robert Wald for their comments. CB has been funded by the Spanish MEC under project FIS200505736C0301 with a partial FEDER contribution. CB and SL are also supported by an INFNMEC collaboration. MV was supported by a Marsden grant administered by the Royal Society of New Zealand, and also wishes to thank both SISSA/ISAS (Trieste) and IAA (Granada) for hospitality.
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