Back-reaction of the Hawking radiation flux on a gravitationally collapsing star II:Fireworks instead of firewalls

Laura Mersini-Houghton1, 2 and Harald P. Pfeiffer3, 4

1DAMTP,University of Cambridge, Wilberforce Rd., Cambridge, CB3 0WA, England2Department of Physics and Astronomy, UNC-Chapel Hill, NC 27599, USA

3Canadian Institute for Theoretical Astrophysics, 60 St. George Street, University of Toronto, Toronto, ON M5S 3H8, Canada4Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto, ON M5G 1Z8, Canada

(Dated: September 8, 2014)

A star collapsing gravitationally into a black hole emits a flux of radiation, knowns as Hawkingradiation. When the initial state of a quantum field on the background of the star, is placedin the Unruh vacuum in the far past, then Hawking radiation corresponds to a flux of positiveenergy radiation travelling outwards to future infinity. The evaporation of the collapsing star canbe equivalently described as a negative energy flux of radiation travelling radially inwards towardsthe center of the star. Here, we are interested in the evolution of the star during its collapse. Thuswe include the backreaction of the negative energy Hawking flux in the interior geometry of thecollapsing star and solve the full 4-dimensional Einstein and hydrodynamical equations numerically.We find that Hawking radiation emitted just before the star passes through its Schwarzschild radiusslows down the collapse of the star and substantially reduces its mass thus the star bounces beforereaching the horizon. The area radius starts increasing after the bounce. Beyond this point ourprogram breaks down due to shell crossing. We find that the star stops collapsing at a finite radiuslarger than its horizon, turns around and its core explodes. This study provides a more realisticinvestigation of the backreaction of Hawking radiation on the collapsing star, that was first presentedin [1].

PACS numbers: 04.25.D-, 04.25.dg, 04.30.-w, 04.30.Db

I. INTRODUCTION

The backreaction of Hawking radiation on the evolu-tion of the collapsing star is the most important problemin the quantum physics of black holes. This problemprovides an arena for the interplay of quantum and grav-itational effects on black holes and their respective im-plications for the singularity theorem. A key feature ofHawking radiation, which was well established in seminalworks by [3, 4, 9, 11, 17–20], is that the radiation is pro-duced during the collapse stage of the star prior to blackhole formation. The very last photon making it to futureinfinity and thus contributing to Hawking radiation, isproduced just before an horizon forms. However its effecton the collapse evolution of the star was considered forthe first time only recently [1]. As was shown in [1] oncethe backreaction of Hawking radiation is included in theinterior dynamics of the star, then the collapse stops andthe star bounces. Solving analytically for the combinedsystem of a collapsing star with the Hawking radiationincluded, is quite a challenge. The system studied in [1]was idealized in order to obtain an approximate analyti-cal solution: there the star was taken to be homogeneous;the star’s fluid considered was dust; the star was placedin a thermal bath of Hawking radiation which arises fromthe time-symmetric Hartle-Hawking initial conditions onthe quantum field in the far past. Within these approx-imations, the main finding of [1] was that a singularityand an horizon do not form after the star’s collapse be-cause the star reverses its collapse and bounces at a finiteradius due to the balancing pressure of the negative en-

ergy Hawking radiation in its interior. Yet, the evolutionof the star could not be followed beyond the bounce withthe approximate analytic methods of [1].

Given the fundamental importance of this problem andthe intriguing results of [1], we here aim to study thebackreaction of Hawking radiation on the collapsing starby considering a more realistic setting, namely: we allowthe star to be inhomogeneous and consider an Hawkingradiation flux of negative energy which propagates in theinterior of the star, with its counterpart of positive en-ergy flux travelling outwards to infinity. Hawking radia-tion flux arises when the initial conditions imposed on thequantum field on the background of the star, are chosento be in the Unruh vacuum state in the far past [9, 18]. Incontrast to the Hartle-Hawking initial state which leadsto an idealized time symmetric thermal bath of radia-tion present before and after the collapse, the choice ofthe Unruh vacuum describes a flux of thermal radiationwhich is zero before the collapse and switches on afterthe collapse. We solve numerically the full 4 dimensionalset of Einstein and of total energy conservation equationsleading to a complete set of hydrodynamic equations forthis model. Numerical solutions allows us to follow theevolution of the collapsing star beyond its bounce.

The paper is organized as follows: Section II describesthe metric in the star’s interior, the stress energy tensorfor the star and for the Hawking radiation flux, whichcomprise the model we wish to study. We then set upthe system of the evolution equations that need to besolve for the combined system. In Sec. III we describethe numerical implementation and the two codes written,

arX

iv:1

409.

1837

v1 [

hep-

th]

5 S

ep 2

014

2

and provide a consistency test for the codes by applyingthem to the well known Oppenheimer Snyder model. InSect. IV we provide the results of the numerical solutionsfor the evolution of the interior geometry of the star andconclude in Sect. V.

II. MODEL

Let us start with a spherically symmetric and inhomo-geneous dust star, described by the following metric

ds2 = −e2Φ(r,t)dt2 + eλdr2 +R2dΩ2 (1)

where R(r, t) is the areal radius and dΩ2 = dθ2 +sin(θ)2dφ2. This form of the metric is convenient fordescribing a radiating star. The set of Einstein and en-ergy conservation equations, were originally derived inMisner [12]. We will use the set of equations [12] and[10], jointly known as the hydrodynamic equations, bymodifying them to reflect our particular system. At theonset of collapse the star gradually starts producing aflux of Hawking radiation. The positive energy flux trav-els outwards to future infinity and an equivalent but neg-ative energy flux travels radially inwards in the interiorof the star towards the center. Most of the outgoingnull radiation is produced on the last stages of collapse[9, 11, 19, 20]. We wish to include the backreaction of thenegative energy Hawking radiation flux on the interior ofthe star and solve for its evolution.

The stress energy tensor of the Hawking radiation fluxis given by

τab = qHkakb (2)

where ka is an outgoing null vector kaka = 0 and qHis the Hawking radiation energy density related to theluminosity of radiation by L = 4πR2qH .

The fluid of the inhomogenenous dust star with a 4-velocity ua, normalized such that uaua = −1, has a stressenergy tensor

T ab = εuaub, (3)

where the energy density ε is expressed in terms of a spe-cific inetrnal energy density per baryon e and the numberdensity of baryons n by ε = n(1 + e) = nh with h theenthalpy. We normalize ka by the condition kau

a = 1, sothat qH denotes the magnitude of Hawking energy fluxdensity of the inward moving radiation in the rest frameof the fluid. Therefore the total stress energy tensor ofthe combined system of the star’s fluid (Tab)and of theradiation (τab) is

T abtotal = T ab + τab (4)

The equations that describe the dynamics of the in-terior of the star with the backreaction of the Hawkingradiation flux included are: the Einstein equations,

Gab = T abtotal, (5)

the total energy conservation equations,

∇bT abtotal = 0, ∇bT ab = −∇bτab, (6)

and the baryon number conservation

∇a(nua) = 0. (7)

The form of the renormalized stress energy tensor forthe Hawking radiation flux in the Unruh vacua, at fu-ture infinity and near the surface of the star were cal-culated in Candelas [18]. Based on energy conservation,the net inward negative energy flux in the star’s inte-rior is equal and opposite to its value at infinity, shownin [3, 18, 20, 23] who also showed that the Unruh vac-uum best approximates the state that follows gravita-tional collapse since it such that it is nearly empty in thefar past before the star collapses, but then produces aflux of radiation once the collapse starts. On these ba-sis the quantum mechanical stress energy tensor of theHawking radiation flux in the interior is as follows

τab =

L

R2

1 1 0 0−1 −1 0 00 0 0 00 0 0 0

(8)

The total energy conservation is best written as a set oftwo equations which contain the radiative heat transferfrom the fluid to the Hawking flux, its t-component andthe r-component that are obtained by contracting T ab

in 6 with ua and the unit vector na orthogonal to the3-hypersurface Σt at t = constant as follows

Qa = ∇bτab = (e−ΦnC, e−λ/2nCr, 0, 0) (9)

From Eqn. 9 and the Hawking flux stress energy tensor inthe interior Eqn. 8, it is immediately clear that Cr = −C.The latter simply reflect the fact that the star is receivingnegative energy transfer travelling inwards in its interior.

We can now write explicitly the full set of hydrody-namic equations, given above. First let us define theproper derivates Dr, Dt by the following relation in termsof the metric

Dr = e−λ/2∂r Dt = e−Φ∂t. (10)

The evolutionary equations for baryon number densityn(r, t) from 7, specific internal energy e from 6, the en-ergy of null-radiation with luminosity L given from thet-component of 9 and 6, the proper velocity of the co-moving fluid U , areal radius of comoving fluid elementsR(r, t), and the ’acceleration’ equation obtained from the

3

r-component of 9 and the Einstein equations 5

1

nDtn+

1

R2

∂

∂R(R2U) = 4πRqHαf

−1/2, (11a)

Dte = −C − pDt

(1

n

), (11b)

DtL = 4πR2nC − e−2ΦDr

(Le2Φ

)− 2L

(∂U

∂R− Lα

Rf1/2

),

(11c)

DtR = U,DrR = f, (11d)

DtU = f∂Φ

∂R− m

R2+L

R. (11e)

Two auxiliary quantities are given by radial ODEs,

∂m

∂R= 4πR2

(ε− qH

(1 + αUf−1/2

)), (11f)

(ε+ p)∂Φ

∂R= − ∂p

∂R+ nCf−1/2, (11g)

with boundary conditions

m = 0, L = 0, at r = 0, for all t, (12)

Φ = 0 at r = rstar, for all t. (13)

Finally, ε, f , α and R′ are short-cuts for

f = 1 + U2 − 2m

R, (14)

ε = n(1 + e), (15)

α =R′

|R′|= signR′, (16)

R′ =∂R

∂r. (17)

A. Choice of energy conversion rate C

The remaining quantities pressure p and the heatingtransfer rate C are free data, that must be specifiedthrough their functional dependence on the evolved vari-ables. For the present work, we assume pressure-lessdust, i.e. p = 0 throughout. In the future we will ex-tend the analysis to the case p = wε with w 6= 0 theequation of state for the fluid.

Having to specify C is somewhat inconvenient, as ourgoal is a solution of Eqs. (11a)–(11e) for the scenariowhere the luminosity at infinity is equal to the expectedHawking radiation:

L∞ = LH,S(t)(US +√fS)2 = LH . (18)

where LH,S(t) is the luminosity of the outgoing null radi-ation at the moving surface of the star and LH the Hawk-ing flux at future infinity. Unfortunately, Eqs. (11a)–(11e) do not allow to specify L freely. Rather, L followsfrom the choice of the energy-conversion rate C. There-fore, our task is to choose a suitable function C(r, t).

We being the motivation of our choice C(r, t) by not-ing that for p = 0, Eq. (11b) simplifies to ∂te = −C eΦ.Therefore, we can achieve a spatially constant e (i.e. spa-tially uniform mass evaporation) with the choice

C(r, t) = e−Φ(r,t)C(t), (19)

where C(t) is a not yet determined function of time.To determine C(t), we consider Eq. (11c), which relates

the luminosity L to the energy-conversion rate C. If wedisregard all gradient terms and any other term indepen-dent of C, then Eq. (11c) becomes ∂tL = 4πR2eΦnC.This represents the radiation creation rate in a sphericalshell at radius R. Integrating over radius with volumeelement eλ/2dr, we find a total radiation creation rate of

Ctotal =

∫ rs

0

dr 4πR2eΦnC eλ/2 (20)

= C(t)

∫ rS

0

dr 4πR2nαR′√f. (21)

If we disregard radiation propagation effects, i.e. assumethat the star is unchanged during a light-crossing time,then we expect that LS ≈ Ctotal. Combining this withEq. (18), we arrive at

LH = C(t)(US +

√fS

)2∫ rS

0

dr 4πR2nαR′√f, (22)

from which it follows that

C(t) =LH(

US +√fS)2 (∫ rS

0

dr 4πR2nαR′√f

)−1

. (23)

III. NUMERICAL IMPLEMENTATION

The numerical implementation will use coordinates tand r. t represents the proper time of the fluid-elementon the surface of the star (this identification is enforcedthrough the boundary condition (13). r is the radialcoordinate comoving with the fluid. Using r as radialcoordinate in the numerical implementation ensures thatthe star always covers the same range r ∈ [0, rstar]. Werewrite Eqs. (11) with partial derivatives according toEq. (10).

A. Discretization

We implement Eqs. (11) in a finite-difference code, us-ing the Crank-Nicholson method. A uniform grid in co-moving radius r is employed. Spatial derivatives are dis-cretized with second order accuracy using centered finite-difference stencils in the interior and one-sided stencils atthe boundary. Denoting the vector of evolved variableswith u = n, e, L,R,U, then discretized equations have

4

the form

u(ri, t+ ∆t)− u(ri, t)

∆t=

1

2(R[u](ri, t) + R[u](ri, t+ ∆t)) .

(24)Here, R represents the right-hand-sides of Eq. (11). Rinvolves second order finite-difference stencils (centeredin the interior, one-sided at the boundary). When eval-uating R at the origin, r = R = 0, the rule of L’Hospitalis used to evaluate terms that would lead to a divisionby zero, e.g L/R→ 0, U/R→ ∂rU/∂rR.

We solve Eq. (24) separately for each of the variablesn, e, L,R,U , fixing all other variables to guesses of theirvalues at the new time. Once all variables have beensolved for, the auxiliary variables f , m, and Φ are up-dated. Now the updated variables are copied into theguesses, and if necessary the procedure is repeated. Weiterate until the maximal norm of the correction is lessthan 10−10. This takes typically 3-6 iterations.

When integrating Eq. (11f), care must be taken to cor-rectly represent the m ∝ r3 behavior close to the origin.A second order finite-difference method cannot resolvethis cubic behavior of m(r), and would therefore lead toa loss of accuracy. Instead, we define the auxiliary quan-tity

η ≡ 3m

4πR3, (25)

which is approximately constant close to the origin. Werewrite Eq. (11f) as

∂η

∂R=

3

R

[ε− q

(1 + αUf−1/2

)− η], (26)

and discretize the partial derivative directly in terms ofthe areal radius R. After solving for η, we recover m =43πR

3η.Figure 1 demonstrates the expected second order con-

vergence of this finite-difference code.

B. Diagnostics

Besides monitoring the evolved variables and the aux-iliary variables, we utilize two additional diagnostics.

First, we compute the expansion θ(k) of r =const sur-faces along the outgoing null-normal kµ = Dt + Dr =e−Φ∂t + e−λ/2∂r. This quantity is computed as

θ(k) =2

R

(α√f + U

). (27)

When θ(k) = 0, an apparent horizon has formed. Specif-ically, we will plot Θ(k)R/2, a quantity which is unity inflat space.

Second, we track during the evolution outgoing nullgeodesics, to gain a deepened understanding of the causalstructure of the space-time. The null-geodesics are rep-resented by the level-sets of an auxiliary variable Λ(t, r),

0

0.1

0.2

0.3

0.4

10-7

10-6

10-5

10-4

0 4 8 12

10-7

10-6

10-5

10-4

|dtk - dt

2|

RS/10

mS

|Nrk - Nr

512|

t

FIG. 1: Convergence test for the numerical code. The toppanel shows the evolution of areal radius RS and total massmS for the run discussed in detail in Sec. IV. This evolutionwas performed at five different time-steps, dt = 2, 4, 8, 16, 32(in units of τcollapse/10000), and the middle panel shows thedifference between runs at the larger for time-steps and thesmallest time-step. The evolution was also performed at fivedifferent radial resolutions (Nr = 32, 64, 128, 256, 512), andthe lower panel shows the difference between the finest res-olution and the coarser ones. The thick black lines in thelower two panels are representative of the errors in the runsdiscussed in this paper.

i.e. the lines Λ(r, t)=const are null. To achieve this, Λmust satisfy

0 = k(Λ) = e−φ∂Λ

∂t+ e−λ/2

∂Λ

∂r. (28)

Solving for ∂Λ/∂t results in an evolution equation for Λ,

∂Λ

∂t= −eφα

√f

R′∂Λ

∂r. (29)

This is an advection equation with positive advectionspeed, i.e. the characteristics of Λ (the null rays) al-ways move toward larger values of r. This is an expectedresult for a comoving coordinate r. Because Λ is alwaysadvected outward, we must supply a boundary conditionat r = 0. This boundary condition sets the value of Λ forthe null ray starting at (t, 0), and we choose

Λ(t, 0) = t. (30)

We also need initial conditions at t = 0, where we set

Λ(0, r) = −r. (31)

These initial- and boundary conditions ensure that Λ ≤ 0represents null rays that intersect the t = 0 surface,whereas Λ ≥ 0 represent null rays that emenate from

5

-0.6

0

0.6

1.2

0 5 10 15

-0.4

-0.2

0

0.2

0.4

RS/R

0

θS R

S/2

mS

nS

US

1000 LS 1000 L

∞

1000 LH

eS

t

FIG. 2: Evolution of pressure-free collapse with Hawkingradiation (R0 = 4, M0 = 0.4). The thick lines indicate theevolution of quantities evaluated at the stellar surface: arealradius RS , radial velocity US , total mass mS , number densitynS , expansion θS . The lower panel shows the internal energyeS , luminosity at the stellar surface LS , luminosity at infinityL∞, and our target luminosity LH. For easy of plotting, theluminosities are scaled by a factor of 1000. The thin dashedlines in the upper panel represent the respective quantitiesin a pure Oppenheimer-Snyder collapse of the same initialconditions.

the origin at coordinate time t = Λ. The boundaryconditions (30) and (31) are consistent with the evolu-tion equation (29), because at the origin φ(0, 0) = 0,f(0, 0) = α(0, 0) = R′(0, 0) = 1. Equation (29) is addedas an extra evolution equation to Eqs. (11).

IV. RESULTS

We will investigate stars with different initial radiusR0 and with different initial masses M0. Figure 2shows a rather typical outcome, a star with initial com-pactness R0/M0 = 10. The evolution proceeds firstthrough a phase which is almost identical to standardOppenheimer-Snyder collapse: The star collapses withincreasing inward velocity. The luminosity is very smallin this early phase, resulting in a negligible change ininternal energy e and mass ms. The upper panel ofFig. 2 also shows the various quantities for a standardOppenheimer-Snyder collapse (dashed lines), which es-sentially overlap the one for the Hawking-radiation col-lapse case.

The similarities between Oppenheimer-Snyder collapseand the Hawking-radiation collapse end, however, as thestar approaches its Schwarzschild-radius, which wouldhappen when θ(k) crosses zero. Just before the starcrosses its Schwarzschild-radius, the luminosity LS di-

0 5 10 150

1

2

3

4

5

13.5 13.8 14.10

0.4

0.8R(ri, t)

2mS(t)

RS

Oppenh.-Sny.

t

FIG. 3: Evolution of pressure-free collapse with Hawkingradiation (R0 = 4, M0 = 0.4). Shown are the area radiiat sixteen equally-spaced comoving coordinate radii ri. Thethick red line indicates 2mS , which is scaled such that RS =2mS would indicate the formation of a black hole. The thindashed blue line indicates the areal radius of a Oppenheimer-Snyder collapse.

0 4 8 12

-0.8

-0.4

0

0.4

t

U(ri, t)

FIG. 4: Radial velocity U for the case R0 = 4,M0 = 0.4.Shown are the area radii at sixteen equally-spaced comovingcoordinate radii ri, with the blue thick lines indicating thestellar surface, r = rS and the radius r = rS/2.

verges, resulting in substantial reduction of internal en-ergy e and the total mass mS . The internal energy dropsto e ∼ −0.4, and the total mass drops to about half itsinitial value.

As can be seen, the velocity of the fluid changes be-haviour from negative to positive at the bounce radius,describing a collapsing phase switching to an expandingphase of the fluid. The inner layers of the star pick up alarger positive velocity than the outer layers, resulting inshell crossing. At that point, due to shell crossing which

6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

13.4

13.5

13.6

13.7

13.8

13.9

14.0

t

Areal radius R

0.0

0.6

1.2

1.8

2.4

3.0

3.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

13.4

13.5

13.6

13.7

13.8

13.9

14.0

t

Potential φ

0.70.60.50.40.30.20.1

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

13.4

13.5

13.6

13.7

13.8

13.9

14.0

t

Potential λ

6.55.74.94.13.32.51.70.90.1

0.7

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

0

2

4

6

8

10

12

14

t

Areal radius R

0.0

0.6

1.2

1.8

2.4

3.0

3.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

0

2

4

6

8

10

12

14

t

Potential φ

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r

0

2

4

6

8

10

12

14

t

Potential λ

6.55.74.94.13.32.51.70.90.1

0.7

FIG. 5: Contour plots of metric quantities for the simulation with R0 = 4,M00 = 0.4. The lower panel show the entiretime-range on the y-axis, the upper panels zoom into the late-time phase with strong Hawking radiation.

is an artifact of p = 0 choice in this model, the densityn(r, t) at the surface of the star diverges ∝ (tcrit− t)−1/2,and the simulation crashes. The divergence in n(r, t)at finite stellar radius and mass indicating shell-crossingof the outer layers of the star, can not be handled byour code. A more sophisticated numerical method, incombination with a fluid-description with pressure wouldbe needed to proceed further and will be done in fu-ture work. Nevertheless, we consider the behavior beforeshell-crossing as generic.

The behavior of this collapse with Hawking radiationis explored further in Fig. 3. Shown are 16 areal radii(at constant comoving coordinates), as well as 2mS(t).The main panel shows clearly the homologous collapseat early times t . 13.5. The inset shows an enlargementof the late-time behavior, showing clearly that the massmS is reduced such that the outer surface of the starremains just outside its horizon.

Figure 4 shows radial velocity U at 16 comoving radii.When Hawking radiation becomes dominant during thelast ∼ 1 of the evolution, the inward motion is signifi-cantly slowed down in the outer layers, and even reversedin the inner layers of the star. The radial gradient in theacceleration (with larger outward velocities toward thecenter of the star) is responsible for the diverging num-ber density in the star. To proceed further will eitherrequire a treatment of shocks, or a modification of ourmodel to avoid shocks.

Figure (5), finally presents the metric quantities forthis evolution. The areal radius R is very similar to thestandard Oppenheimer-Snyder collapse; the small devia-tions during the Hawking radiation phase are not easilyvisible. The lapse-potential Φ clearly shows two distinctphases: During the quasi Oppenheimer-Snyder collapseΦ ≈ 0. When the Hawking radiation becomes significant,

Φ drops quickly to values around −0.5. The potential λshows also two distinct phases: During the Oppenheimer-Snyder phase, λ remains approximately spatially con-stant, but slowly decreases. During the Hawking radi-ation phase, λ quickly decreases close to the surface ofthe star, presumably in accord with the diverging numberdensity n close to the stellar surface.

Figure 6 explores the parameter space of different R0

and M0. Generically, we find the behavior shown inFigs. 2 and 3: The stars first collapse at nearly con-stant mass until they reach a size close to the horizon (inFig. 6, this represents a diagonal line at mS = 2RS). Asthe stars approach their horizons, luminosity increasesand the mass drops significantly, with the star remain-ing just outside RS = 2mS . All trajectories approach thesame limiting behavior, thus we find the expected genericbehavior of Hawking radiation.

We have shown results here for very compact stars asthose are the more likely ones to collapse into a blackhole. We conducted the analysis for stars with low mass,see the M = 0.1 data in Fig. 6. As expected these starsbehave differently, and substantially evaporate and ex-plode before they approach their horizon.

V. CONCLUSIONS

Einstein equations tell us that the final destiny of agravitationally collapsing massive star is a black hole [2].This system satisfies all the conditions of the PenroseHawking singularity theorem [25]. However a collaps-ing star has a spacetime dependent gravitational fieldwhich by the theory of quantum fields on curved space-time should give rise to a flux of particles created [11].Hawking discovered in the early ′70′s that this is in-

7

0.3 1 3 10 30 100

0.04

0.1

0.2

0.4

1

2

1 2

0.4

0.8

1.2

RS(t)

mS(t

)

FIG. 6: Collapsing stars with Hawking radiation for severalchoices of initial mass and initial radius. Shown are trajec-tories in the (RS(t),mS(t))-plane, with the circles indicatingthe initial conditions of each run. The filled blue circle marksthe evolution discussed in detail in Figs. 2–5. The dashedorange line indicates the Schwarzschild horizon, RS = 2mS .The inset shows an enlargement, to emphasize the universalnature of the near-horizon behavior.

deed the case for stars collapsing to a black hole, theyproduce Hawking radiation [3]. The conclusions derivedfrom both theories, the existence of black holes from Ein-stein’s theory of gravity and the existence of Hawkingradiation from the theory of quantum fields in curvedspacetime, were soon found to be in high friction with oneanother, (see [26] for an interesting treatment) . Theyled to a series of paradoxes, most notably the informa-tion loss [5] and firewalls [7]. Being forced to give up oneither unitarity or causality is at the heart of the prob-lem. But the quantum theory is violated if unitarity isbroken. Einstein’s theory is violated if causality is bro-ken. Violations of the quantum theory imply Hawkingradiation may not exist. Violations of Einstein’s theoryof gravity, on which the singularity theorem is based, im-ply black holes may not exist. Thus black hole physicsprovides the best arena for understanding the friction be-tween quantum and gravitational physics. In this light,an investigation of the evolution of the star’s interior asit is approaching its singularity with the backreaction ofthe Hawking radiation produced by quantum effects, isof fundamental importance.

This problem was first investigated with semianalyt-ical methods and a series of approximations, such as ahomogeneous star placed in the Hartle Hawking thermalbath in [1].

In this work we extend our investigation to study amore realistic system of a collapsing star with the quan-tum field in the Unruh vacuum, which gives rise to a

Hawking flux produced only during the collapse. Herewe solved the full set of Einstein and energy conserva-tion equations numerically therefore the exact solutionswe obtain are robust. We find for a range of initialmasses and radii of the collapsing star that the evolutionproceeds through two phases: First, a collapsing phasewhere Hawking radiation is unimportant, and the starfollows very closely standard Oppenheimer-Snyder col-lapse. When the star approaches formation of an horizon,then Hawking radiation sets in. This slows down the col-lapse (e.g. Fig. 4) while significantly reducing the massof the star. Both effects (slowdown and mass-loss) bal-ance such that the evaporating star remains very slightlyoutside its event horizon, cf. Fig. 6.

The key idea that enables this program is the fact thatradiation is produced during the collapse stage of thestar, prior to black hole formation. Once the star be-comes a black hole nothing can escape it, not even light.Since Hawking radiation is produced prior to black holeformation then its backreaction on the star’s evolutioncan be included in the set of hydrodynamic equationsfor the coupled system of the quantum field and thestar. Thus we solve numerically the set of hydrodynamicsequations describing the evolutio of collapse for the inho-mogeneous dust star absorbing negative energy Hawkingflux during its collapse. We discover that the stars ex-plode instead of collapsing to a black hole, as they getclose to their last stage of collapse, just when Hawkingradiation reaches a maximum, but they never cross whatwould have been the apparent horizon. Just like Hawk-ing radiation which is universal, we discover that this be-haviour of the collapsing stars bouncing and explodingbefore the horizon and singularity would have formed,is universal, i.e independent of their characteristics suchas mass and size. Physically the backreaction of ingoingnegative energy Hawking radiation reduces the gravita-tional binding energy in the star with the maximum lossnear the last stages of collapse, while taking momentumaway from the star. This is the reason for the universalfeature of the explosion in the star instead of its collapseto a black hole singularity. Independent of what size andmass the star starts from, most if its radiation will be pro-duced as the star nears its future horizon. At that stagethe drop in mass and internal energy is maximum andthe star goes through a bounce from collapse to explo-sion. The negative energy Hawking radiation absorbedby the star, violates the energy condition of the singular-ity theorem [25] thus it is not surprising that a singularityand an horizon do not form, features traditionally associ-ated with the definition of black holes. Stated simply ourfindings indicate that singularities and horizon do not ex-ist due to quantum effect and that universally stars blowup on their last stage of collapse.

More specifically, we find that collapsing stars slowdown their collapse right outside their horizon, whilesubstantially reducing their mass through Hawking ra-diation. In the cases studied, the mass of the black holeis reduced by roughly a factor of 2, with the radius of the

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star shrinking in proportion, to preserve RS/mS > 2,cf. Figs. 3 and 6. The velocity of the collapsing mat-ter then reverses toward an expanding phase, cf. Fig. 4with the inner layers expanding faster than the outerlayers. Unfortunately the latter behaviour, i.e. the ’un-folding’ of the star from inside out leads numerically toshell-crossing which our code can not handle. Once shellcrossing occurs we can not follow the evolution furtherto determine whether in the expanding phase, the stellarremnant explode, or simply evaporate away, despite thatfrom the low mass cases and from the velocity results theexplosion seems to be the case.

The star never crosses its horizon, so neither unitaritynor causality are violated, thereby solving the longstand-ing information loss paradox. This investigation shows

that universally collapsing stars bounce into an expand-ing phase and probably blow up, instead of collapsing toa black hole. Thus ’fireworks’ should replace ’firewalls’.

Acknowledgments

LMH is grateful to DAMTP Cambridge Universityfor their hospitality when this work was done. LMHwould like to thank M.J. Perry, L. Parker, J. Bekenstein,P. Spindel, and G. Ellis for useful discussions. LMH ac-knowledges support from the Bahnson trust fund. HPPgratefully acknowledges support from NSERC of Canadaand the Canada Research Chairs Program.

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