Eugenio Bianchi
email: ebianchi@gravity.psu.edu

Marios Christodoulou
email: christod.marios@gmail.com

Fabio D'

Carlo Rovelli
email: rovelli@cpt.univ-mrs.fr

Hal M Haggard
email: haggard@bard.edu

White Holes as Remnants: A Surprising Scenario for the End of a Black Hole

Quantum tunneling of a black hole into a white hole provides a model for the full life cycle of a black hole. The white hole acts as a long-lived remnant, solving the black-hole information paradox. The remnant solution of the paradox has long been viewed with suspicion, mostly because remnants seemed to be such exotic objects. We point out that (i) established physics includes objects with precisely the required properties for remnants: white holes with small masses but large finite interiors; (ii) non-perturbative quantum-gravity indicates that a black hole tunnels precisely into such a white hole, at the end of its evaporation. We address the objections to the existence of white-hole remnants, discuss their stability, and show how the notions of entropy relevant in this context allow them to evade several no-go arguments. A black hole's formation, evaporation, tunneling to a white hole, and final slow decay, form a unitary process that does not violate any known physics.

I. INTRODUCTION

The conventional description of black hole evaporation is based on quantum field theory on curved spacetime, with the back-reaction on the geometry taken into account via a mean-field approximation [START_REF] Hawking | Black hole explosions?[END_REF]. The approximation breaks down before evaporation brings the black hole mass down to the Planck mass (m P l = c/G ∼ the mass of a 1 2 -centimeter hair). To figure out what happens next we need quantum gravity.

A quantum-gravitational process that disrupts a black hole was studied in [START_REF] Rovelli | Planck stars[END_REF][START_REF] Haggard | Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling[END_REF][START_REF] Lorenzo | Improved black hole fireworks: Asymmetric black-hole-to-white-hole tunneling scenario[END_REF][START_REF] Christodoulou | Planck star tunneling time: An astrophysically relevant observable from background-free quantum gravity[END_REF][START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF]. It is a conventional quantum tunneling, where classical equations (here the Einstein equations) are violated for a brief interval. This alters the causal structure predicted by classical general relativity [START_REF] Frolov | Quantum Gravity removes Classical Singularities and Shortens the Life of Black Holes[END_REF][START_REF] Frolov | Spherically symmetric collapse in quantum gravity[END_REF][START_REF] Stephens | Black hole evaporation without information loss[END_REF][START_REF] Modesto | Disappearance of the black hole singularity in loop quantum gravity[END_REF][START_REF] Modesto | Evaporating loop quantum black hole[END_REF][START_REF] Mazur | Gravitational vacuum condensate stars[END_REF][START_REF] Ashtekar | Black hole evaporation: A paradigm[END_REF][START_REF] Balasubramanian | Information Recovery From Black Holes[END_REF][START_REF] Hayward | Formation and Evaporation of Nonsingular Black Holes[END_REF][START_REF] Hossenfelder | A model for non-singular black hole collapse and evaporation[END_REF][START_REF] Hossenfelder | Conservative solutions to the black hole information problem[END_REF][START_REF] Frolov | Information loss problem and a "black hole" model with a closed apparent horizon[END_REF][START_REF] Gambini | A scenario for black hole evaporation on a quantum Geometry[END_REF][START_REF] Gambini | Quantum shells in a quantum space-time[END_REF][START_REF] Bardeen | Black hole evaporation without an event horizon[END_REF][START_REF] Giddings | Quantum emission from two-dimensional black holes[END_REF], by modifying the dynamics of the local apparent horizon. As a result, the apparent horizon fails to evolve into an event horizon.

Crucially, the black hole does not just 'disappear': it tunnels into a white hole [START_REF] Narlikar | High energy radiation from white holes[END_REF][START_REF] Hájíček | Singularity avoidance by collapsing shells in quantum gravity[END_REF][START_REF] Ambrus | Quantum superposition principle and gravitational collapse: Scattering times for spherical shells[END_REF][START_REF] Olmedo | From black holes to white holes: a quantum gravitational, symmetric bounce[END_REF] (from the outside, an object very similar to a black hole), which can then leak out the information trapped inside. The likely end of a black hole is therefore not to suddenly pop out of existence, but to tunnel to a white hole, which can then slowly emit whatever is inside and disappear, possibly only after a long time [START_REF] Aharonov | The unitarity puzzle and Planck mass stable particles[END_REF][START_REF] Giddings | Black holes and massive remnants[END_REF][START_REF] Callan | Evanescent black holes[END_REF][START_REF] Giddings | Constraints on black hole remnants[END_REF][START_REF] Preskill | Do Black Holes Destroy Information?[END_REF][START_REF] Banks | Classical and quantum production of cornucopions at energies below 1018 GeV[END_REF][START_REF] Banks | Lectures on black holes and information loss[END_REF][START_REF] Ashtekar | Information is Not Lost in the Evaporation of 2-dimensional Black Holes[END_REF][START_REF] Ashtekar | Evaporation of 2-Dimensional Black Holes[END_REF][START_REF] Ashtekar | Surprises in the Evaporation of 2-Dimensional Black Holes[END_REF][START_REF] Rama | Remarks on Black Hole Evolution a la Firewalls and Fuzzballs[END_REF][START_REF] Almheiri | An Uneventful Horizon in Two Dimensions[END_REF][START_REF] Chen | Black Hole Remnants and the Information Loss Paradox[END_REF][START_REF] Malafarina | Classical collapse to black holes and quantum bounces: A review[END_REF].

The tunneling probability may be small for a macroscopic black hole, but becomes large toward the end of the evaporation. This is because it increases as the mass decreases. Specifically, it will be suppressed at most by the standard tunneling factor p ∼ e -S E /

(1)

where S E is the Euclidean action for the process. This can be estimated on dimensional grounds for a stationary black hole of mass m to be S E ∼ Gm 2 /c, giving

p ∼ e -(m/m P l ) 2 , (2)
which becomes of order unity towards the end of the evaporation, when m → m P l . A more detailed derivation is in [START_REF] Christodoulou | Planck star tunneling time: An astrophysically relevant observable from background-free quantum gravity[END_REF][START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF]. As the black hole shrinks towards the end of its evaporation, the probability to tunnel into a white hole is no longer suppressed. The transition gives rise to a long-lived white hole with Planck size horizon and very large but finite interior. Remnants in the form of geometries with a small throat and a long tail were called "cornucopions" in [START_REF] Banks | Are horned particles the end point of Hawking evaporation?[END_REF] by Banks et.al. and studied in [START_REF] Banks | Lectures on black holes and information loss[END_REF][START_REF] Giddings | Dynamics of extremal black holes[END_REF][START_REF] Banks | Black hole remnants and the information puzzle[END_REF][START_REF] Giddings | Constraints on black hole remnants[END_REF]. As far as we are aware, the connection to the conventional white holes of general relativity was never made.

This scenario offers a resolution of the information-loss paradox. Since there is an apparent horizon but no event horizon, a black hole can trap information for a long time, releasing it after the transition to white hole. If we have a quantum field evolving on a black hole background metric and we call S its (renormalized) entanglement entropy across the horizon, then consistency requires the metric to satisfy non-trivial conditions:

(a) The remnant has to store information with entropy S ∼ m 2 o / (we adopt units G=c=1, while keeping explicit), where m o is the initial mass of the hole, before evaporation [START_REF] Marolf | The Black Hole information problem: past, present, and future[END_REF]. This is needed to purify Hawking radiation.

(b) Because of its small mass, the remnant can release the inside information only slowly-hence it must be long-lived. Unitarity and energy considerations impose that its lifetime be equal to or larger than τ R ∼ m 4 o / 3/2 [START_REF] Preskill | Do Black Holes Destroy Information?[END_REF][START_REF] Bianchi | Entanglement entropy production in gravitational collapse: covariant regularization and solvable models[END_REF].

(c) The metric has to be stable under perturbations, so as to guarantee that information can be released [START_REF] Lorenzo | Improved black hole fireworks: Asymmetric black-hole-to-white-hole tunneling scenario[END_REF][START_REF] Frolov | Black Hole Physics: Basic Concepts and New Developments[END_REF][START_REF] Barrab\'es | Death of white holes[END_REF][START_REF] Ori | Death of cosmological white holes[END_REF].

In this paper we show that under simple assumptions the effective metric that describes standard black hole evaporation followed by a transition to a Planck-mass white hole satisfies precisely these conditions. This result shows that this scenario is consistent with known physics and does not violate unitarity.

One reason this scenario may not have been recognised earlier is because of some prejudices (including against white holes), which we discuss below. But the scenario presented here turns out to be consistent with general expectations that are both in the AdS/CFT community (see for instance [START_REF] Engelhardt | Holographic consequences of a no transmission principle[END_REF][START_REF] Fitzpatrick | On information loss in AdS3/CFT2[END_REF]) and in the quantum gravity community (see for instance the 'paradigm' [START_REF] Ashtekar | Black hole evaporation: A paradigm[END_REF]).

II. THE INTERNAL GEOMETRY BEFORE QUANTUM GRAVITY BECOMES RELEVANT

We begin by studying the geometry before any quantum gravitational effect becomes relevant. The standard classical conformal diagram of a black hole formed by collapsing matter is depicted in Figure 1, for the case of spherical symmetry.

Classical general relativity becomes insufficient when either (a) curvature becomes sufficiently large, or (b) sufficient time has ellapsed. The two corresponding regions, A and B, where we expect classical general relativity to fail are depicted in the figure .  Consider the geometry before these regions, namely on a Cauchy surface Σ that crosses the horizon at some (advanced) time v after the collapse. See Figure 1. We are interested in particular in the geometry of the portion Σ i of Σ which is inside the horizon. Lack of focus on this interior geometry is, in our opinion, one of the sources of the current confusion. Notice that we are here fully in the expected domain of validity of established physics.

The interior Cauchy surface can be conveniently fixed as follows. First, observe that a (2d, spacelike) sphere S in (4d) Minkowski space determines a preferred (3d) ball Σ i bounded by S: the one sitting on the same linear subspace-simultaneity surface-as S; or, equivalently, the one with maximum volume. (Deformations from linearity in Minkowski space decrease the volume). The first characterisation-linearity-makes no sense on a curved space, but the second-extremized volume-does. Following [START_REF] Christodoulou | How big is a black hole?[END_REF], we use this characterization to fix Σ i , which, incidentally, provides an invariant definition of the "Volume inside S". Large interior volumes and their possible role in the information paradox have also been considered in [START_REF] Stanford | Complexity and shock wave geometries[END_REF][START_REF] Perez | No firewalls in quantum gravity: the role of discreteness of quantum geometry in resolving the information loss paradox[END_REF][START_REF] Ori | Firewall or smooth horizon?[END_REF][START_REF] Ashtekar | The Issue of Information Loss: Current Status[END_REF][START_REF] Susskind | Black Holes and Complexity Classes[END_REF].

The interior is essentially a very long tube. As time passes, the radius of the tube shrinks, while its length increases, see Figure 2.

It is shown in [START_REF] Christodoulou | How big is a black hole?[END_REF][START_REF] Bengtsson | Black holes: Their large interiors[END_REF][START_REF] Ong | Never Judge a Black Hole by Its Area[END_REF][START_REF] Wang | Maximal volume behind horizons without curvature singularity[END_REF], that for large time v the volume of Σ i is proportional to the time from collapse:

V ∼ 3 √ 3 m 2 o v. (3)
Christodoulou and De Lorenzo have shown [START_REF] Christodoulou | Volume inside old black holes[END_REF] that this picture is not changed by Hawking evaporation: toward the end of the evaporation the area of the (apparent) horizon of the black hole has shrunk substantially, but the length of the interior tube keeps growing linearly with time elapsed from the collapse. This can be huge for a black hole that started out as macroscopic (m o m P l ), even if the horizon area and mass have become small. The key point is that (3) still hold, with m o being the initial mass of the hole [START_REF] Christodoulou | Volume inside old black holes[END_REF], see also [START_REF] Ong | The Persistence of the Large Volumes in Black Holes[END_REF].

The essential fact that is often neglected, generating confusion, is that an old black hole that has evaporated down to mass m has the same exterior geometry as a young black hole with the same mass, but not the same interior: an old, largely evaporated hole has an interior vastly bigger than a young black hole with the same mass. This is conventional physics.

To understand the end of a black hole's evaporation, it is important to distinguish the phenomena concerning along collapsing shell, ing their normal derivatives undetermined. Thus there seems to be plenty of room for patching in a nonsingular vacuum solution.

To obtain some feeling for the motion of the collapsing shell we have made the fairly arbitrary assumption that a fi(r)- (2.13) This gives us a single first-order ordinary differential equation for R (r). The solution so obtained behaves like R(r)=Q+e r', as ~~oo. We can then use this solution to check that the other coefficient functions, to leading order, are well behaved for all finite values of ~.

We can continue this procedure perturbatively, to verify that the coefficients in the expansion in powers of n are smooth functions of ~. Of course, this demonstration of a smooth perturbation expansion around the shell, does not guarantee the existence of an everywhere smooth solu-tion. We continue to search for a sensible ansatz that will enable us to demonstrate explicitly the existence of a smooth collapsing solution, but we feel confident that such a solution exists. The collapsing solution that we have described, begins as a dimple on Aat space. At any finite time after its for- mation, it will have the geometry shown in Fig. 2. We will refer to such an object as a finite volume cornu- copion. It is a solution of the field equations that is static over most of space. The time dependence occurs only in the tip of the horn. FIG. side the shell, but this is inconsistent with the field equa- tions. Similarly, an attempt to keep the three-dimensionally conformally Oat form of the metric, with conformal factor tied to the dilaton, is inconsistent. We have not been able to come up with a natural ansatz. Nonetheless, we believe that smooth solutions exist.

There are many smooth solutions of the vacuum field equations restricted to a manifold with the topology of a hemi-three-sphere cross time. Our matching conditions fix only the values of the metric functions and dilaton along the timelike world line of the collapsing shell, leav-ing their normal derivatives undetermined. Thus there seems to be plenty of room for patching in a nonsingular vacuum solution.

To obtain some feeling for the motion of the collapsing shell we have made the fairly arbitrary assumption that a fi(r)- (2.13) This gives us a single first-order ordinary differential equation for R (r). The solution so obtained behaves like R(r)=Q+e r', as ~~oo. We can then use this solution to check that the other coefficient functions, to leading order, are well behaved for all finite values of ~.

We can continue this procedure perturbatively, to verify that the coefficients in the expansion in powers of n are smooth functions of ~. Of course, this demonstration of a smooth perturbation expansion around the shell, does not guarantee the existence of an everywhere smooth solu-tion. We continue to search for a sensible ansatz that will enable us to demonstrate explicitly the existence of a smooth collapsing solution, but we feel confident that such a solution exists. The collapsing solution that we have described, begins as a dimple on Aat space. At any finite time after its for- mation, it will have the geometry shown in Fig. 2. We will refer to such an object as a finite volume cornu- copion. It is a solution of the field equations that is static over most of space. The time dependence occurs only in the tip of the horn.

(2.12) he fields there is m solu-FIG. hemi-three-sphere cross time. Our matching conditions fix only the values of the metric functions and dilaton along the timelike world line of the collapsing shell, leav-ing their normal derivatives undetermined. Thus there seems to be plenty of room for patching in a nonsingular vacuum solution.

To obtain some feeling for the motion of the collapsing shell we have made the fairly arbitrary assumption that a fi(r)- (2.13) This gives us a single first-order ordinary differential equation for R (r). The solution so obtained behaves like R(r)=Q+e r', as ~~oo. We can then use this solution to check that the other coefficient functions, to leading order, are well behaved for all finite values of ~.

We can continue this procedure perturbatively, to verify that the coefficients in the expansion in powers of n are smooth functions of ~. Of course, this demonstration of a smooth perturbation expansion around the shell, does not guarantee the existence of an everywhere smooth solu-tion. We continue to search for a sensible ansatz that will enable us to demonstrate explicitly the existence of a smooth collapsing solution, but we feel confident that such a solution exists. The collapsing solution that we have described, begins as a dimple on Aat space. At any finite time after its for- mation, it will have the geometry shown in Fig. 2. We will refer to such an object as a finite volume cornu- copion. It is a solution of the field equations that is static over most of space. The time dependence occurs only in the tip of the horn. Nonetheless, we believe that smooth solutions exist.

There are many smooth solutions of the vacuum field equations restricted to a manifold with the topology of a hemi-three-sphere cross time. Our matching conditions fix only the values of the metric functions and dilaton along the timelike world line of the collapsing shell, leav-ing their normal derivatives undetermined. Thus there seems to be plenty of room for patching in a nonsingular vacuum solution.

To obtain some feeling for the motion of the collapsing shell we have made the fairly arbitrary assumption that a fi(r)- (2.13) This gives us a single first-order ordinary differential equation for R (r). The solution so obtained behaves like R(r)=Q+e r', as ~~oo. We can then use this solution to check that the other coefficient functions, to leading order, are well behaved for all finite values of ~.

We can continue this procedure perturbatively, to verify that the coefficients in the expansion in powers of n are smooth functions of ~. Of course, this demonstration of a smooth perturbation expansion around the shell, does not guarantee the existence of an everywhere smooth solu-tion. We continue to search for a sensible ansatz that will enable us to demonstrate explicitly the existence of a smooth collapsing solution, but we feel confident that such a solution exists. The collapsing solution that we have described, begins as a dimple on Aat space. At any finite time after its for- mation, it will have the geometry shown in Fig. 2. We will refer to such an object as a finite volume cornu- copion. It is a solution of the field equations that is static over most of space. The time dependence occurs only in the tip of the horn. The quantum gravitational effects in regions A and B are distinct, and confusing them is a source of misunderstanding. Notice that a generic spacetime region in A is spacelike separated and in general very distant from region B. By locality, there is no reason to expect these two regions to influence one another.

The quantum gravitational physical process happening at these two regions must be considered separately.

III. THE A REGION: TRANSITIONING ACROSS THE SINGULARITY

To study the A region, let us focus on an arbitrary finite portion of the collapsing interior tube. As we approach the singularity, the Schwarzschild radius r s , which is a temporal coordinate inside the hole, decreases and the curvature increases. When the curvature approaches Planckian values, the classical approximation becomes unreliable. Quantum gravity effects are expected to bound the curvature [8-11, 13-19, 22-24, 27, 29, 64, 65]. Let us see what a bound on the curvature can yield. Following [START_REF] Rovelli | How Information Crosses Schwarzschild's Central Singularity[END_REF], consider the line element

ds 2 = - 4(τ 2 + l) 2 2m -τ 2 dτ 2 + 2m -τ 2 τ 2 + l dx 2 +(τ 2 +l) 2 dΩ 2 , (4)
where l m. This line element defines a genuine Riemannian spacetime, with no divergences and no singularities. Curvature is bounded. For instance, the Kretschmann invariant K ≡ R µνρσ R µνρσ is easily computed to be

K(τ ) ≈ 9 l 2 -24 lτ 2 + 48 τ 4 (l + τ 2 ) 8 m 2 (5)
in the large mass limit, which has the finite maximum

K(0) ≈ 9 m 2 l 6 . ( 6
)
For all the values of τ where l τ 2 < 2m the line element is well approximated by taking l = 0 which gives

ds 2 = - 4τ 4 2m -τ 2 dτ 2 + 2m -τ 2 τ 2 dx 2 + τ 4 dΩ 2 . (7)
For τ < 0, this is the Schwarzschild metric inside the black hole, as can be readily seen going to Schwarzschild coordinates

t s = x, and r s = τ 2 . ( 8
)
For τ > 0, this is the Schwarzschild metric inside a white hole. Thus the metric (4) represents a continuous transition of the geometry of a black hole into the geometry of a white hole, across a region of Planckian, but bounded curvature.

Geometrically, τ = constant (space-like) surfaces foliate the interior of a black hole. Each of these surfaces has the topology S 2 × R, namely is a long cylinder. As time passes, the radial size of the cylinder shrinks while the axis of the cylinder gets stretched. Around τ = 0 the cylinder reaches a minimal size, and then smoothly bounces back and starts increasing its radial size and shrinking its length. The cylinder never reaches zero size but bounces at a small finite radius l. The Ricci tensor vanishes up to terms O(l/m).

The resulting geometry is depicted in Figure 3. The region around τ = 0 is the smoothing of the central black hole singularity at r s = 0. This geometry can be given a simple physical interpretation. General relativity is not reliable at high curvature, because of quantum gravity. Therefore the "prediction" of the singularity by the classical theory has no ground. High curvature induces quantum particle creation, including gravitons, and these can have an effective energy momentum tensor that back-reacts on the classical geometry, modifying its evolution. Since the energy momentum tensor of these quantum particles can violate energy conditions (Hawking radiation does), the evolution is not constrained by Penrose's singularity theorem. Equivalently, we can assume that the expectation value of the gravitational field will satisfy modified effective quantum corrections that alter the classical evolution. The expected scale of the correction is the Planck scale. As long as l m the correction to the classical theory is negligible in all regions of small curvature; as we approach the high-curvature region the curvature is suppressed with respect to the classical evolution, and the geometry continues smoothly past τ = 0.

One may be tempted to take l to be Planckian l P l = G/c 3 ∼ √ , but this would be wrong. The value of l can be estimated from the requirement that the curvature is bounded at the Planck scale, K(0) ∼ 1/ 2 . Using this in [START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF] gives

l ∼ (m ) 1 3 , (9)
or, restoring for a moment physical units

l ∼ l P l m m P l 1 3 , (10)
which is much larger than the Planck length when m m P l [START_REF] Rovelli | Planck stars[END_REF]. The three-geometry inside the hole at the transition time is

ds 2 3 = 2m l dx 2 + l 2 dΩ 2 . ( 11
)
The volume of the "Planck star" [START_REF] Rovelli | Planck stars[END_REF], namely the minimal radius surface is

V = 4πl 2 2m l (x max -x min ). ( 12
)
The range of x is determined by the lifetime of the hole from the collapse to the onset of region B, as x = t s . If region B is at the end of the Hawking evaporation, then (x max -x min ) ∼ m 3 / and from Eq. ( 9), l ∼ (m ) 1/3 , leading to an internal volume at crossover that scales as

V ∼ m 4 / √ . ( 13
)
We observe that in the classical limit the interior volume diverges, but quantum effects make it finite.

The l → 0 limit of the line element (4) defines a metric space which is a Riemannian manifold almost everywhere and which can be taken as a solution of the Einstein's equations that is not everywhere a Riemannian manifold [START_REF] Rovelli | How Information Crosses Schwarzschild's Central Singularity[END_REF]. Geodesics of this solution crossing the singularity are studied in [START_REF] Rovelli | How Information Crosses Schwarzschild's Central Singularity[END_REF]: they are well behaved at τ = 0 and they cross the singularity in a finite proper time. The possibility of this natural continuation of the Einstein equations across the central singularity of the Schwarzschild metric has been noticed repeatedly by many authors. To the best of our knowledge it was first noticed by Synge in the fifties [START_REF] Synge | The Gravitational Field of a Particle[END_REF] and rediscovered by Peeters, Schweigert and van Holten in the nineties [START_REF] Peeters | Extended geometry of black holes[END_REF]. A similar observation has recently been made in the context of cosmology in [START_REF] Koslowski | Through the Big Bang[END_REF].

As we shall see in the next section, what the → 0 limit does is to confine the transition inside an event horizon, making it invisible from the exterior. Reciprocally, the effect of turning on is to de-confine the interior of the hole.

IV. THE TRANSITION AND THE GLOBAL STRUCTURE

The physics of the B region concerns gravitational quantum phenomena that can happen around the horizon after a sufficiently long time. The Hawking radiation provides the upper bound ∼ m 3 o / for this time. After this time the classical theory does not work anymore. Before studying the details of the B region, let us consider what we have so far. The spacetime diagram utilized to discuss the black hole evaporation is often drawn as in the left panel of Figure 4. What happens in the circular shaded region? What physics determines it? This diagram rests on an unphysical assumption: that the Hawking process proceeds beyond the Planck curvature at the horizon and pinches off the large interior of the black hole from the rest of spacetime. This assumption uses quantum field theory on curved spacetimes beyond its regime of validity. Without a physical mechanism for the pinching off, this scenario is unrealistic.

Spacetime diagrams representing the possible formation and full evaporation of a black hole more realistically abound in the literature [8-11, 13-19, 22-24, 29] and they are all similar. In particular, it is shown in [START_REF] Haggard | Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling[END_REF][START_REF] Lorenzo | Improved black hole fireworks: Asymmetric black-hole-to-white-hole tunneling scenario[END_REF] that the spacetime represented in the right panel of Figure 4, can be an exact solution of the Einstein equations, except for the two regions A and B, but including regions within the horizons.

If the quantum effects in the region A are simply the crossing described in the previous section, this deter-mines the geometry of the region past it, and shows that the entire problem of the end of a black hole reduces to the quantum transition in the region B.

The important point is that there are two regions inside horizons: one below and one above the central singularity. That is, the black hole does not simply pop out of existence: it tunnels into a region that is screened inside an (anti-trapping) horizon. Since it is anti-trapped, this region is actually the interior of a white hole. Thus, black holes die by tunneling into white holes.

Unlike for the case of the left panel of Figure 4, now running the time evolution backwards makes sense: the central singularity is screened by an horizon ('time reversed cosmic censorship') and the overall backward evolution behaves qualitatively (not necessarily quantitively, as initial conditions may differ) like the time-forward one.

Since we have the explicit metric across the central singularity, we know the features of the resulting white hole. The main consequence is that its interior is what results from the transition described in the above section: namely a white hole born possibly with a small horizon area, but in any case with a very large interior volume, inherited from the black hole that generated it.

If the original black hole is an old hole that started out with a large mass m o , then its interior is a very long tube. Continuity of the size of the tube in the transition across the singularity, results in a white hole formed by the bounce, which initially also consists of a very long interior tube, as in Figure 5. Subsequent evolution shortens it (because the time evolution of a white hole is the time reversal of that of a black hole), but this process can take a long time. Remarkably, this process results in a white hole that has a small Planckian mass and a long life determined by how old the parent black hole was. In other words, the outcome of the end of a black hole evaporation is a long-lived remnant. The time scales of the process can be labelled as in Figure 5. We call v o the advanced time of the collapse, v -and v + the advanced time of the onset and end of the quantum transition, u o the retarded time of the final disappearance of the white hole, and u -and u + the retarded times of the onset and end of the quantum transition. The black hole lifetime is

τ bh = v --v o . (14)
The white hole lifetime is

τ wh = u o -u + . ( 15
)
And we assume that the duration of the quantum transition of the B region satisfies u + -u -= v + -v -≡ ∆τ . Disregarding Hawking evaporation, a metric describing this process outside the B region can be written explicitly by cutting and pasting the extended Schwarzschild solution, following [START_REF] Haggard | Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling[END_REF]. This is illustrated in Figure 6: two Kruskal spacetimes are glued across the singularity as described in the previous section and the shaded region is the metric of the portion of spacetime outside a collapsing shell (here chosen to be null). While the location of the A region is determined by the classical theory, the location of the B region, instead, is determined by quantum theory. The B process is indeed a typical quantum tunneling process: it has a long lifetime. A priori, the value of τ bh is determined probabilistically by quantum theory. As in conventional tunneling, in a stationary situation (when the horizon area varies slowly), we expect the probability p per unit time for the tunneling to happen to be time independent. This implies that the normalised probability P (t) that the tunneling happens between times t and t + dt is governed by dP (t)/dt = -pP (t), namely is

P (t) = 1 τ bh e -t τ bh , (16)
which is normalised ( ∞ 0 P (t)dt = 1) and where τ bh satisfies

τ bh = 1/p. ( 17
)
We note parenthetically that the quantum spread in the lifetime can be a source of apparent unitarity violation, for the following reason. In conventional nuclear decay, a tunneling phenomenon, the quantum indetermination in the decay time is of the same order as the lifetime. The unitary evolution of the state of a particle trapped in the nucleus is such that the state slowly leaks out, spreading it over a vast region. A Geiger counter has a small probability of detecting a particle at the time where it happens to be. Once the detection happens, there is an apparent violation of unitarity. (In the Copenhagen language the Geiger counter measures the state, causing it to collapse, loosing information. In the Many Worlds language, the state splits into a continuum of branches that decohere and the information of a single branch is less than the initial total information.) In either case, the evolution of the quantum state from the nucleus to a given Geiger counter detection is not unitary; unitarity is recovered by taking into account the full spread of different detection times. The same must be true for the tunneling that disrupts the black hole. If tunneling will happen at a time t, unitarity can only be recovered by taking into account the full quantum spread of the tunneling time, which is to say: over different future goemetries. The quantum state is actually given by a quantum superposition of a continuum of spacetimes as in Figure 5, each with a different value of v -and v + . We shall not further pursue here the analysis of this apparent source of unitarity, but we indicate it for future reference.

V. THE B REGION: THE HORIZON AT THE TRANSITION

The geometry surrounding the transition in the B region is depicted in detail in Figure 7. The metric of the entire neighbourhood of the B region is an extended Schwarzschild metric. can therefore be written in null Kruskal coordinates

ds 2 = - 32m 3 r e -r 2m dudv + r 2 dΩ 2 , ( 18
)
where

1 - 2m e r 2m = uv. ( 19
)
On the two horizons we have respectively v = 0 and u = 0, and separate regions where u and v have different signs as in the right panel of Figure 7. Notice the rapid change of the value of the radius across the B region, which yields a rapid variation of the metric components in [START_REF] Hossenfelder | Conservative solutions to the black hole information problem[END_REF].

To fix the region B, we need to specify more precisely its boundary, which we have not done so far. It is possible to do so by identifying it with the diamond (in the 2d diagram) defined by two points P + and P -with coordinates v ± , u ± both outside the horizon, at the same radius r P , and at opposite timelike distance from the bounce time, see Figure 8. The same radius r P implies

v + u + = v -u -≡ 1 - r P 2m e r P 2m . ( 20
)
The same time from the horizon implies that the light lines u = u -and v = v + cross on t s = 0, or u + v = 0, hence

u -= -v + . ( 21
)
This crossing point is the outermost reach of the quantum region, with radius r m determined by

v + u -≡ 1 - r m 2m e rm 2m . ( 22
)
The region is then entirely specified by two parameters. We can take them to be r P and ∆τ = v + -v -∼ u + -u -.

The first characterizes the radius at which the quantum transition starts. The second its duration. (Strictly speaking, we could also have v + -v -and u + -u -of different orders of magnitude, but we do not explore this possibility here.) There are indications about both metric scales in the literature. In [START_REF] Haggard | Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling[END_REF][START_REF] Haggard | Quantum Gravity Effects around Sagittarius A*[END_REF], arguments where given for r P ∼ 7/3 m. Following [START_REF] Christodoulou | Planck star tunneling time: An astrophysically relevant observable from background-free quantum gravity[END_REF], the duration of the transition has been called "crossing time" and computed by Christodoulou and D'Ambrosio in [START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF][START_REF] Christodoulou | Geometry Transition in Covariant Loop Quantum Gravity[END_REF] using Loop Quantum Gravity: the result is ∆τ ∼ m, which can be taken as a confirmation of earlier results [START_REF] Ambrus | Quantum superposition principle and gravitational collapse: Scattering times for spherical shells[END_REF][START_REF] Barceló | The lifetime problem of evaporating black holes: mutiny or resignation[END_REF][START_REF] Barceló | Black holes turn white fast, otherwise stay black: no half measures[END_REF] obtained with other methods. The two crucial remaining parameters are the black hole and the white hole lifetimes, τ bh and τ wh .

The result in [START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF] indicates also that p, the probability of tunneling per unit time, is suppressed exponentially by a factor e -m 2 / . Here m is not the initial mass m o of the black hole at the time of its formation, rather, it is the mass of the black hole at the decay time. This is in accord with the semiclassical estimate that tunneling is suppressed as in ( 1) and ( 2). As mentioned in the introduction, because of Hawking evaporation, the mass of the black hole shrinks to Planckian values in a time of order m 3 o / , where the probability density becomes of order unit, giving

τ bh ∼ m 3 o / (23)
and

∆τ ∼ √ . ( 24
)
We conclude that region B has a Planckian size. We notice parenthetically that the value of p above is at odds with the arguments given in [START_REF] Haggard | Black hole fireworks: quantum-gravity effects outside the horizon spark black to white hole tunneling[END_REF] for a shorter lifetime

τ bh ∼ m 2 o /
√ . This might be because the analysis in [START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF] captures the dynamics of only a few of the relevant degrees of freedom, but we do not consider this possibility here. The entire range of possibilities for the black to white transition lifetime, m 2 o / √ ≤ τ bh ≤ m 3 o / , may have phenomenological consequences, which have been explored in [START_REF] Barrau | Planck star phenomenology[END_REF][START_REF] Barrau | Fast radio bursts and white hole signals[END_REF][START_REF] Barrau | Phenomenology of bouncing black holes in quantum gravity: a closer look[END_REF][START_REF] Barrau | Bouncing black holes in quantum gravity and the Fermi gamma-ray excess[END_REF][START_REF] Rovelli | Planck stars as observational probes of quantum gravity[END_REF]. (On hypothetical white hole observations see also [START_REF] Retter | The revival of white holes as Small Bangs[END_REF]).

VI. INTERIOR VOLUME AND PURIFICATION TIME

Consider a quantum field living on the background geometry described above. Near the black hole horizon there is production of Hawking radiation. Its backreaction on the geometry gradually decreases the area of the horizon. This, in turn, increases the transition probability to a white hole. After a time τ bh ∼ m 3 o / , the area of the black hole reaches the Planckian scale A bh (final) ∼ , and the transition probability becomes of order unity. The volume of the transition surface is huge.

To compute it with precision, we should compute the back-reaction of the inside component of the Hawking radiation, which gradually decreases the value of m as the coordinate x increases. Intuitively, the inside components of the Hawking pairs fall toward the singularity, decreasing m. Since most of the decrease is at the end of the process, we may approximate the full interior of the hole with that of a Schwarzschild solution of mass m o and the first order estimate of the inside volume should not be affected by this process. Thus we may assume that the volume at the transition has the same order as the one derived in Eq. ( 13), namely

V bh (final) ∼ √ m o τ bh ∼ m 4 o / √ . (25)
Using the same logic in the future of the transition, we approximate the inside metric of the white hole with that of a Schwarzschild solution of Planckian mass, since in the future of the singularity, the metric is again of Kruskal type, but now for a white hole of Plankian mass.

The last parameter to estimate is the lifetime τ wh = u 0 -u + of the white hole produced by the transition. To do so, we can assume that the internal volume is conserved in the quantum transition. The volume of the region of Planckian curvature inside the white hole horizon is then

V wh (u) ∼ l 2 m l τ wh , (26)
where now l ∼ m ∼ √ , and therefore

V wh (initial) ∼ τ wh . (27)
Gluing the geometry on the past of the singularity to the geometry on the future side requires that the two volumes match, namely that ( 26) matches ( 13) and this gives

τ wh ∼ m 4 o / 3/2 . ( 28
)
This shows that the Planck-mass white hole is a longlived remnant [START_REF] Christodoulou | Volume inside old black holes[END_REF].

With these results, we can address the black hole information paradox. The Hawking radiation reaches future infinity before u -, and is described by a mixed state with an entropy of order m 2 o / . This must be purified by correlations with field excitations inside the hole. In spite of the smallness of the mass of the hole, the large internal volume ( 25) is sufficient to host these excitations [START_REF] Rovelli | Black holes have more states than those giving the Bekenstein-Hawking entropy: a simple argument[END_REF]. This addresses the requirement (a) of the introduction, namely that there is a large information capacity.

To release this entropy, the remnant must be longlived. During this time, any internal information that was trapped by the black hole horizon can leak out. Intuitively, the interior member of a Hawking pair can now escape and purify the exterior quantum state. The long lifetime of the white hole allows this information to escape in the form of very low frequency particles, thus respecting bounds on the maximal entropy contained in a given volume with given energy.

The lower bound imposed by unitarity and energy con- [START_REF] Preskill | Do Black Holes Destroy Information?[END_REF][START_REF] Marolf | The Black Hole information problem: past, present, and future[END_REF][START_REF] Bianchi | Entanglement entropy production in gravitational collapse: covariant regularization and solvable models[END_REF] and this is precisely the white hole lifetime (28) deduced above; hence we see that they satisfy the requirement (b) of the introduction. Therefore white holes realize precisely the long-lived remnant scenario for the end of the black hole evaporation that was conjectured and discussed mostly in the 1990's [START_REF] Giddings | Black holes and massive remnants[END_REF][START_REF] Giddings | Constraints on black hole remnants[END_REF][START_REF] Banks | Classical and quantum production of cornucopions at energies below 1018 GeV[END_REF][START_REF] Banks | Lectures on black holes and information loss[END_REF][START_REF] Banks | Are horned particles the end point of Hawking evaporation?[END_REF][START_REF] Giddings | Dynamics of extremal black holes[END_REF][START_REF] Banks | Black hole remnants and the information puzzle[END_REF][START_REF] Giddings | Constraints on black hole remnants[END_REF].

siderations is τ R ∼ m 4 o / 3/2
The last issue we should discuss is stability. Generically, white holes are known to be unstable under perturbations (see for instance Chapter 15 in [START_REF] Frolov | Black Hole Physics: Basic Concepts and New Developments[END_REF] and references therein). The instability arises because modes of short-wavelength are exponentially blue-shifted along the white hole horizon. In the present case, however, we have a Planck-size white hole. To run this argument for instability in the case of a planckian white hole, it is necessary to consider transplanckian perturbations. Assuming no transplanckian perturbations to exist, there are no instabilities to be considered. This addresses the requirement (c). Alternatively: a white hole is unstable because it may re-collapse into a black hole with similar mass; therefore a Planck size white hole can at most re-collapse into a Planck size black hole; but this has probability of order unity to tunnel back into a white hole in a Planck time.

Therefore the proposed scenario addresses the consistency requirements (a), (b), and (c) for the solution of the information-loss paradox and provides an effective geometry for the end-point of black hole evaporation: a long-lived Planck-mass white hole.

VII. ON WHITE HOLES

Notice that from the outside, a white hole is indistinguishable from a black hole. This is obvious from the existence of the Kruskal spacetime, where the same region of spacetime (region I) describes both the exterior of a black hole and the exterior of a white hole. For r s > 2m, the conventional Schwarzschild line element describes equally well a black hole exterior and a white hole exterior. The difference is only what happens at r = 2m.

The only locally salient difference between a white and a black hole is that if we add some generic perturbation or matter on a given constant t s surface, in (the Schwarzschild coordinate description of) a black hole we see matter falling towards the center and accumulating around the horizon. While in (the Schwarzschild coordinate description of) a white hole we see matter accumulated around the horizon in the past, moving away from the center. Therefore the distinction is only one of "naturalness" of initial conditions: a black hole has "special" boundary conditions in the future, a white hole has "special" boundary conditions in the past. This difference can be described physically also as follows: if we look at a black hole (for instance when the Event Horizon Telescope [START_REF] Doeleman | Imaging an Event Horizon: submm-VLBI of a Super Massive Black Hole[END_REF] examines Sagittarius A*), we see a black disk. This means that generic initial conditions on past null infinity give rise on future null infinity to a black spot with minimal incoming radiation: a "special" configuration in the future sky. By time reversal symmetry, the opposite is true for a white hole; generic initial conditions on future null infinity require a black spot with minimal incoming radiation from past null infinity: a "special" configuration in the past.

We close this section by briefly discussing the "no transition principle" considered by Engelhardt and Horowitz in [START_REF] Engelhardt | Holographic consequences of a no transmission principle[END_REF]. By assuming "holographic" unitarity at infinity and observing that consequently information cannot leak out from the spacetime enclosed by a single asymptotic region, these authors rule out a number of potential scenarios, including the possibility of resolving generic singularities inside black holes. Remarkably, the scenario described here circumvents the no transition principle and permits singularity resolution in the bulk: the reason is that this singularity is confined in a finite spacetime region and does not alter the global causal structure.

VIII. ON REMNANTS

The long-lived remnant scenario provides a satisfactory solution to the black-hole information paradox. The main reason for which it was largely discarded was the fact that remnants appeared to be exotic objects extraneous to known physics. Here we have shown that they are not: white holes are well known solutions of the Einstein equations and they provide a concrete model for long-lived remnants.

Two other arguments made long-lived remnants unpopular: Page's version of the information paradox; and the fact that if remnants existed they would easily be produced in accelerators. Neither of these arguments applies to the long-lived remnant scenario of this paper. We discuss them below.

In its interactions with its surroundings, a black hole with horizon area A behaves thermally as a system with entropy S bh = A/4 . This is a fact supported by a large number of convincing arguments and continues to hold for the dynamical horizons we consider here. The Bekenstein-Hawking entropy provides a good notion of entropy that satisfies Bekenstein's generalized second law, in the approximation in which we can treat the horizon as an event horizon. In the white hole remnant scenario this is a good approximation for a long time, but fails at the Planck scale when the black hole transitions to a white hole.

Let us assume for the moment that these facts imply the following hypothesis (see for instance [START_REF] Marolf | The Black Hole information problem: past, present, and future[END_REF]) (H) The total number of available states for a quantum system living on the internal spatial slice Σ i of Figure 1 is N bh = e S bh = e A/4 . Then, as noticed by Page [START_REF] Page | Information in black hole radiation[END_REF], we have immediately an information paradox regardless of what happens at the end of the evaporation. The reason is that the entropy of the Hawking radiation grows with time. It is natural to interpret this entropy as correlation entropy with the Hawking quanta that have fallen inside the hole, but for this to happen there must be a sufficient number of available states inside the hole. If hypothesis (H) above is true, then this cannot be, because as the area of the horizon decreases with time, the number of available internal states decreases and becomes insufficient to purify the Hawking radiation. The time at which the entropy surpasses the area is known as the Page time. This has lead many to hypothesize that the Hawking radiation is already purifying itself by the Page time: a consequence of this idea is the firewall scenario [START_REF] Almheiri | Black holes: Complementarity or firewalls?[END_REF].

The hypothesis (H) does not apply to the white-hole remnants. As argued in [START_REF] Rovelli | Black holes have more states than those giving the Bekenstein-Hawking entropy: a simple argument[END_REF], growing interior volumes together with the existence of local observables implies that the number of internal states grows with time instead of decreasing as stated in (H). This is not in contradiction with the fact that a black hole behaves thermally in its interactions with its surroundings as a system with entropy S = A/4 . The reason is that "entropy" is not an absolute concept and the notion of entropy must be qualified. Any definition of "entropy" relies on a coarse graining, namely on ignoring some variables: these could be microscopic variables, as in the statistical mechanical notion of entropy, or the variables of a subsystem over which we trace, as in the von Neumann entropy. The Bekenstein-Hawking entropy correctly describes the thermal interactions of the hole with its surroundings, because the boundary is an outgoing null surface and S bh counts the number of states that can be distinguished from the exterior; but this is not the number of states that can be distinguished by local quantum field operators on Σ i [START_REF] Rovelli | Black holes have more states than those giving the Bekenstein-Hawking entropy: a simple argument[END_REF]. See also [START_REF] Giddings | Statistical physics of black holes as quantum-mechanical systems[END_REF].

Therefore there is no reason for the Hawking radiation to purify itself by the Page time. This point has been stressed by Unruh and Wald in their discussion of the evaporation process on the spacetime pictured in the left panel of Figure 4, see e.g. [START_REF] Unruh | Information Loss[END_REF]. Our scenario differs from Unruh and Wald's in that the white hole transition allows the Hawking partners that fell into the black hole to emerge later and purify the state. They emerge slowly, over a time of order m 4 o / 3/2 , in a manner consistent with the long life of the white hole established here.

The second standard argument against remnants is that, if they existed, it would be easy to produce them. This argument assumes that a remnant has a small boundary area and little energy, but can have a very large number of states. The large number of states would contribute a large phase-space volume factor in any scattering process, making the production of these objects in scattering processes highly probable. Actually, since in principle these remnants could have an arbitrarily large number of states, their phase-space volume factor would be infinite, and hence they would be produced spontaneously everywhere.

This argument does not apply to white holes. The reason is that a white hole is screened by an anti-trapping horizon: the only way to produce it is through quantum gravity tunneling from a black hole! Even more, to produce a Planck mass white hole with a large interior volume, we must first produce a large black hole and let it evaporate for a long time. Therefore the threshold to access the full phase-space volume of white holes is high. A related argument is in [START_REF] Banks | Classical and quantum production of cornucopions at energies below 1018 GeV[END_REF], based on the fact that infinite production rate is prevented by locality. In [START_REF] Giddings | Constraints on black hole remnants[END_REF] Giddings questions this point treating remnants as particles of an effective field theory; the field theory, however, may be a good approximation of such a highly non-local structure as a large white hole only in the approximation where the large number of internal states is not seen. See also [START_REF] Banks | Lectures on black holes and information loss[END_REF].

IX. CONCLUSION

As a black hole evaporates, the probability to tunnel into a white hole increases. The suppression factor for this tunneling process is of order e -m 2 /m 2 P l . Before reaching sub-Planckian size, the probability ceases to be suppressed and the black hole tunnels into a white hole.

Old black holes have a large volume. Quantum gravitational tunneling results in a Planck-mass white hole that also has a large interior volume. The white hole is long-lived because it takes awhile for its finite, but large, interior to become visible from infinity.

The geometry outside the black to white hole transition is described by a single asymptotically-flat spacetime. The Einstein equations are violated in two regions: The Planck-curvature region A, for which we have given an effective metric that smoothes out of the singularity; and the tunneling region B, whose size and decay probability can be computed [START_REF] Christodoulou | Characteristic Time Scales for the Geometry Transition of a Black Hole to a White Hole from Spinfoams[END_REF]. These ingredients combine to give a white hole remnant scenario.

This scenario provides a way to address the information problem. We distinguish two ways of encoding information, the first associated with the small area of the horizon and the second associated to the remnant's interior. The Bekenstein-Hawking entropy S bh = A/4 is encoded on the horizon and counts states that can only be distinguished from outside. On the other hand, a white hole resulting from a quantum gravity transition has a large volume that is available to encode substantial information even when the horizon area is small. The white hole scenario's apparent horizon, in contrast to an event horizon, allows for information to be released. The long-lived white hole releases this information slowly and purifies the Hawking radiation emitted during evaporation. Quantum gravity resolves the information problem.

-CR thanks Ted Jacobson, Steve Giddings, Gary Horowitz, Steve Carlip, and Claus Kiefer for very useful exchanges during the preparation of this work. EB and HMH thank Tommaso De Lorenzo for discussion of time scales. EB thanks Abhay Ashtekar for discussion of remnants. HMH thanks the CPT for warm hospitality and support, Bard College for extended support to visit the CPT with students, and the Perimeter Institute for Theoretical Physics for generous sabbatical support. MC acknowledges support from the SM Center for Space, Time and the Quantum and the Leventis Educational Grants Scheme. This work is supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

FIG. 1 .

 1 FIG. 1. Conformal diagram of a classical black hole. The dashed line is the horizon. The dotted line is a Cauchy surface Σ. In regions A and B we expect (distinct) quantum gravitational effects and classical GR is unreliable.

FIG. 2 .

 2 FIG.2. Instantaneous snapshot of a collapsing cornucopion.

FIG. 2 .

 2 FIG.2. The interior geometry of an old black hole: a very long thin tube, whose length increases and whose radius decreases with time. Notice it is finite, unlikely the Einstein-Rosen bridge.

FIG. 3 .

 3 FIG. 3. The transition across the A region.

FIG. 4 .

 4 FIG. 4.Left: A commonly drawn diagram for black hole evaporation that we argue against. Right: A black-to-white hole transition. The dashed lines are the horizons.

FIG. 5 .

 5 FIG.5. Black hole bounce, with a sketch of the inside geometries, before and after the quantum-gravitational transition.

FIG. 6 .

 6 FIG. 6. Left: Two Kruskal spacetimes are glued at the singularity. The grey region is the metric of a black to white hole transition outside a collapsing and the exploding null shell. Right: The corresponding regions in the physical spacetime.

FIG. 7 .

 7 FIG. 7. The B region. Left: Surfaces of equal Schwarzschild radius are depicted. Right: The signs of the null Kruskal coordinates around B.

FIG. 8 .

 8 FIG. 8. The B transition region.

  2. Instantaneous snapshot of a collapsing cornucopion.

	9Appendix C.	2	(2.11)	quation	the rest	to the	(2.10)	for the	or com-	there is	s tensor	em now	(2.9)	(2.8)	ts have	etric is	rmined	(2.7)	(2.6)	panding	.	xtremal	s with	given in	(2.5)

  2. Instantaneous snapshot of a collapsing cornucopion.

	11)	n	st	he	10)	he	m-	is	or	w	.9)	.8)	ve	is	d	.7)	.6)
	9Appendix C.


Camila Morais Canellas
email: camila.canellas@univ-nantes.fr

Colin De

La Higuera

Teacher's Time is Valuable 1

Keywords: OER adoption, teachers' time, technology, pedagogy, workload

Consistent adoption adoption of Open Educational Resources (OER) depends on a number of factors. These may correspond to business, legal, policy, academic or technology issues. Among the vast number of elements making deployment harder, one is often underestimated: an increase on the demand for teacher's time. Some solutions to motivate and/or gratify teachers that adopt OER have been proposed over the years and used successfully among the first adopters. Unfortunately, such solutions are not always possible in several contexts. In this paper, we propose that a key concern of research on OER technologies should focus on saving teacher's time when creating pedagogical resources.

Introduction

When deploying Open Educational Resources (OER) initiatives it is often the case that the project stalls because of having failed to foresee that the excellent teacher, whose work the OER is supposed to be based upon, does not have the time nor the energy to brush2 her slides, to reengineer her course, to add the meta-data, to make herself available to answer the forum or prepare the additional material without which the courseware is not what the promoters were expecting.

A number of studies have already pointed out that the lack of teacher's involvement is an important (if not the single most important) issue to start the adoption of OER. In certain contexts, the institution uses an incentive-based model in order to encourage the teachers to accept the extra workload. In under-resourced systems, or in places where newcomers to these questions are failing to understand the different issues, this may not be possible. In this work, we analyse the problem, the proposed alternatives and argue in favour of deploying technology in a way that enhances the learning (and teaching) experience, whilst avoiding burdening the teacher with a heavier workload.

Problem Statement 2.1. A List of Issues for OER

Projects regarding the adoption of OER face a number of issues related to its attributes and decision points as described in the framework proposed by [START_REF] Stacey | A Dialogue on Open Educational Resources and Social Authoring Models[END_REF]. The issues are grouped under the following categories: business, legal, policy, academic, technology.

Moreover, in many countries the law has more or less broad definitions of fair use and educational use. Such a circumstance often means that teachers are used to incorporate in their pedagogical resources some materials of which they are not authors, but that they believe they can use if in a pedagogical environment. Nonetheless, when creating/adapting this material to be published as OER, this "exception" no longer applies and they face the need to brush the material3 .

Frequently, some of these key attributes (or the combination of them) may end up being more time consuming than expected, especially regarding teacher's time.

It was also noted that the question of effort is a crucial one when the teacher's perception of OER production was surveyed: teachers do not perceive OER as a means to save time (Masterman, Wild, White, & Manton, 2011, p. 138).

About Teacher's Time

Therefore, the issues related to the adoption of OER and the fact of having her work made public will have consequences regarding the work of the teacher:

• She will receive extra pressure because her material is going to be made public -perhaps due to the fact that her course is going to be filmed; • The teacher is expected to help render her material impeccable. In some cases she will be asked to sign agreements she does not really understand (copyright, image forms); • The teacher may be asked to enrich her contents, by providing additional texts, updating her lectures, adding a forum, some quizzes and exams, in order to better adapt the material to a larger public.

The above has also been noted by several authors, including in the work by [START_REF] Masterman | The impact of OER on teaching and learning in UK universities: implications for Learning Design[END_REF] already mentioned. In fact, "time" is mentioned as the most significant barrier for not adopting OER by two thirds of the respondents of an OECD questionnaire (Hylén, 2005, p. 4).

Why Is This Not Always Taken into Account?

We can conclude that the process of adopting OER is not easy, automatic or effortless. Still, why is it usually seen as so?

Possibly, the common confusion between free and open has a role in this misconception. The result of this work, the resource itself, may be open and free, but not necessarily the work itself.

As stated before, the effort is substantial.

Another probable reason lies in the view that, being seen as technology, OER are supposed to save money and time. When a clear policy for adoption of OER is absent, it is difficult for the decision makers to accept to spend more money on this.

Extra Funding, Direct Rewards to the Teacher

Sometimes the institution is conscious that there is extra effort, that it needs this effort (because the institution has a visibility strategy, typically), and it is prepared to directly reward the teacher.

The following types of incentives have been identified by [START_REF] West | OER Incentive Models Summaries[END_REF]: large lump sum stipends, incremented stipends, gifts instead of money, pay for creation of OER, sabbaticals and other existing institutional incentive plans.

It should be noted that, in institutions that chose these incentives to get the ball rolling a few years ago, the tendency is now that enough motivation has been generated and these are no longer needed.

Promotion of Teaching Quality

In other cases, the prize may be differed: the institution will attempt to promote quality of teaching as it promotes quality of research. These promotion efforts are visible: the teacher will know that making the extra effort will result in some form of prestige or of a differed prize. The establishment of a credible academic reward system that includes OER was pointed out as one of the most important policy issue that leads to a large scale adoption in an Australian study (Hylén, 2005, p. 6).

Adding Support to the Teacher's Tasks

In this case, rather than rewarding or recognizing the work made by the teacher, the organization encourages her work by creating services that will support her. Staff that is specialized in the juridical questions regarding openness or grants that can be used to accomplish the task are some examples. This not only can save the teacher's time but also somehow demonstrates the importance of the work for the institution.

Convincing the Staff that Practices Have Changed and Adaptation is Needed

Finally, the argument used by decision makers can be that no intervention is required, as progress is just the natural sense of history, and that teachers should adapt to new technologies just because it is their job to do so. Following this view, one should not regard the creation of OER as something exceptional, worth a specific prize. The point of view is defended by two very different types of decision makers: those that no longer teach or those who are currently promoting the way teaching is (or should be) done and are possibly over-enthusiastic and sometimes fail to see that other teachers are not always so keen or have less time.

The Role of Technology

Arguments in favour of technology are often that it makes life simpler. Yet in the case of teaching, technology has been introduced to provide the teacher with tools she did not have before, allowing her to do her job much better, provided she uses these tools. However, it often represented an increase cost regarding time.

Accordingly, it is possible to distinguish two types of technologies proposed to teachers: 1. Time-saving technologies: as an example provided by [START_REF] Cuban | Techno-Reformers and Classroom Teachers[END_REF], mimeograph machines and projectors were quickly and largely adopted by teachers around the world.

2. Time-consuming technologies: computers and television, also cited by [START_REF] Cuban | Techno-Reformers and Classroom Teachers[END_REF], represent technologies that may enhance the quality and possibilities for the teachers' practices, but usually demanded more time and were not as well adopted in classrooms compared with time-saving technologies.

In fact, research has shown that having technology available does not mean that teachers will adopt it. The same occurs regarding OER adoption (Schuwer, Kreijns, & Vermeulen, 2014, p. 95).

Two Alternatives

We see two alternatives to the problem of teacher's time when adopting OER: the introduction of (or the publicity given to) time-saving technologies and/or a scenario where the teacher can do better with less work, or at least with as much work as before. Some examples: 1. Time-saving technology: a typical heavily time consuming task for the teacher is the production of metadata. This has been identified, for instance, as a blocking point in the Wikiwijs project [START_REF] Schuwer | Wikiwijs: An unexpected journey and the lessons learned towards OER[END_REF]. As a recurring theme for sessions in the past OCW Conferences, the importance and difficulty of the question is clear. On the other hand, natural language processing (NLP) research is able, today, to provide automatic transcriptions for videos, summarizations, and identification of names entities4 . 2. A pedagogical scenario using technology to improve the results, without adding much time to the teacher's agenda is currently studied at University of Nantes. In the Informatics for beginners course, the teachers have to face a number of problems:

• A large number of students, and hence of groups to be run in parallel;

• The necessity to propose innovative courses, not just one that can be taught by the different instructors.

The choice which has been made is that of a first part including lectures on a wide range of topics (bio-informatics, natural language processing, distributed algorithms, cryptology and social networks); and a second part consisting of a programming course which makes use of the general topics addressed in the previous lectures.

One of the problems is that a lot of information is passed to the students, and they need time to digest it. It has therefore been proposed to record the classes and make the videos available as OER. In order to make things more interesting, we considered pedagogical scenarios allowing the students to interact better with this material. However, in most of the envisaged scenarios, the teacher's cooperation was needed and the ideas were too demanding of her time, for which we did not have any support.

We are therefore proposing to do the following:

• The teacher proposes 8 to 10 quiz questions. It should be said that she has to do this anyways for the examination; • These quizzes are added to the video and synchronized with the moment the topic is addressed by the teacher; • The students, when viewing the video, can not only answer a quiz (and know if they were right -self assessment), but also rate the quiz and even propose better quizzes (learn by teaching); • In order to avoid overcharging the video with quizzes, the system automatically updates the list of the better quizzes proposed so far, following the evaluation made by the students.

In this way, the intervention of the teacher is kept as small as possible: she is not asked to do more than she would do anyhow, and we hope to end with a better material.

Conclusion

Whereas technology has usually been deployed to answer necessities identified by the teacher, and often by over-enthusiastic pedagogues who will be identifying more complex and interesting learning scenarios, we feel that (at least part of) technology should be introduced in order to help the teacher do better for an (almost) identical cost regarding time.

Technology is now being identified as a key factor to success in open education as shown by:

• We believe and expect new ideas and tools in this direction in the future.

  The creation of the UNESCO Chair on Open Technologies for Open Educational Resources and Open Learning 5 ; • The Opening up Slovenia initiative 6 : initiatives to help promote open education in Slovenia are heavily backed by research; • The creation of partnerships based on research in tools based on machine learning, as suggested by the Knowledge 4 All Foundation 7 .

This work has received a French government support granted to the COMIN Labs excellence laboratory and managed by the National Research Agency in the "Investing for the Futures" program ANR-JO-LABX-07-0J.

Brushing the slides consists in making them free of any material inconsistent with the chosen licence.

The Usual SolutionsOverall, the effort the teacher has to make, once she has agreed to adopt OER, is quite substantial. On the other hand, what will the employer, or the team running the open education scheme, propose to leverage this effort? 3 Cf. http://www.jisc.ac.uk/publications/programmerelated/2013/Openeducationalresources.aspx, section Open Licensing and http://poerup.referata.com/wiki/France#Copyright_in_education.

There are a number of conferences on these topics. The webpage of the Association for Computational Linguistics is a good starting point: http://www.aclweb.org/

http://www.ouslovenia.net/project/unesco-chair-slovenia/

http://www.ouslovenia.net/

http://www.k4all.org/