End of a black hole’s evaporation

The End of a Black Hole’s Evaporation – Part I Fabio D’Ambrosioa , Marios Christodouloub,c , Pierre Martin-Dussaudd,e , Carlo Rovellie,f,g and Farshid Soltanih,l a Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, CH b Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong c Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, United Kingdom d Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, Pennsylvania 16802, USA e Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France f Perimeter Institute, 31 Caroline Street N, Waterloo ON, N2L2Y5, Canada g The Rotman Institute of Philosophy, 1151 Richmond St. N London N6A5B7, Canada h Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada and l Dipartimento di Fisica, Università La Sapienza, I-00185 Roma, EU (Dated: February 9, 2021) At the end of the Hawking evaporation the horizon of a black hole enters a physical region where quantum gravity cannot be neglected. The physics of this region has not been much explored. We arXiv:2009.05016v2 [gr-qc] 5 Feb 2021 characterise its physics and introduce a technique to study it. I. INTRODUCTION In a spacetime formed by gravitationally collapsed mat- ter, there are three distinct regions in which curvature becomes Planckian. We expect the approximation de- fined by quantum field theory interacting with classical general relativity to break down in all three of them. The physics of these regions is quite different. The three regions are illustrated in the Carter-Penrose causal diagram of Figure 1. The dark grey area is the re- gion where quantum gravity cannot be neglected and the diagram itself becomes unreliable. The light grey area is the collapsing matter and the dashed line is the (trap- ping) horizon (the event horizon is not determined by classical physics). The three physically distinct regions where curvature becomes Planckian are: 1. Region C, in the future of the event c in the di- agram, which is directly affected by the collapsing Figure 1: The three regions of a black hole spacetime where matter reaching Planckian density. quantum gravity becomes relevant. In the dark grey region quantum gravity cannot be neglected and the diagram itself 2. Region B, in the future of the event b in the di- becomes unreliable. The future of the locations a, b and c en- agram, which is affected by the horizon reaching counter different quantum gravity phenomena depending, re- spectively, on the presence of the collapsing matter (C), the Planckian size because of Hawking’s evaporation. horizon (B), or neither (A). 3. Region A, in the future of any location like a (that is a generic event in the dark grey area distant from line element is the events b and c) in the diagram, where the cur- vature becomes Planckian but the classical evolu- −1 2 2Gm 2 2Gm tion to the singularity is not causally connected to ds = − 1 − dt + 1 − dr2 r r the collapsing matter or to the horizon. + r2 dθ2 + sin2 θ dφ2 (1) The physical distance between these regions depends We can take the three locations a, b and c to be at the on the age of the black hole at the time when its horizon same fixed values of θ, φ, r and at three different values reaches the quantum region. This age depends in turn ta , tb , tc of the t coordinate. The proper distance dl along on the overall mass of the black hole before being shrunk a line of constant θ, φ, r, namely a nearly horizontal line by Hawking evaporation. in the causal diagram, is given by the line element To give a rough estimate of these distances we consider r for simplicity the interior of a Schwarzschild black hole. 2Gm (Most of the evaporation takes place at late times.) The dl = dt −1 . (2) r 2 The quantity dl becomes large approaching the quantum gravitational dark grey region of Figure 1. Curvature scalars behave as ∼ m/r3 and hence become Planckian at r/LP l ∼ (m/MP l )1/3 where LP l and MP l are the Planck length and the Plank mass, giving √ dl ∼ 2 (m/MP l )1/3 dt (3) near the quantum gravitational dark grey region of Fig- ure 1. For a stellar mass (m ∼ M ∼ 1038 MP l ) black hole, if no further mass enters the horizon, the end of the Hawking evaporation is at tb − tc ∼ (M /MP l )3 LP l , hence the distance between b and c is 10 M 3 L ∼ LP l ∼ 1075 light years, (4) MP l which is huge. That is: the locations b and c are ex- tremely distant from each other [1–4]. This is a rough Figure 2: Carter-Penrose causal diagram of the black to white estimate, but the conclusion is general: the distance be- transition. tween b and c, which is to say the ‘depth’ of the black hole, is huge, for an old black hole. Notice that what makes this distance large is not the smallness of the r entire effect of quantum gravity is a slight violation of the coordinate considered (which is not Planckian): rather, Einstein equations in the high curvature region, which it is the long lifetime of the black hole that builds up the prevents curvature to diverge and allows spacetime to length. continue. This possibility was noticed long ago, already It is worthwhile pausing to ponder this fact: near the in the fifties, by Synge [16]. end of the Hawking evaporation of an isolated stellar- Following in particular [17–20], we assume here that size black hole, the collapsing matter entering the quan- the anti-trapped region is bounded by a future horizon tum region is at a —spatial, not temporal!— distance of that connects it to the region external to the black hole, a 1075 light years from the horizon. Locality demands the surprising possibility first noticed in [17]. The full space- physics of spatially widely separated phenomena to be time has therefore the causal structure depicted in the independent. It is reasonable to expect quantum gravity Carter-Penrose diagram in Figure 2. to affect the causal structure of spacetime, but in small The C region is where the collapsing matter itself fluctuations, not by suddenly causally connecting events reaches the quantum gravity regime. It is called the that are extremely far apart. ‘Planck star’ phase of the collapsing matter [21]. Here, It follows that the physics of each of the three regions following [21], we simply assume that some form of mat- A, B and C can be studied independently from the oth- ter bounce compatible with this scenario happens. ers (until something brings them in causal contact). More In this paper we focus on the physics of the B re- precisely, the physics of the A region can be studied in- gion, namely on the events near the end of the Hawking dependently from what happens at B or C; while these evaporation of the black hole. This is the region where depend on the physics of the A region, since this bounds the trapping horizon tunnels into an anti-trapping hori- the horizon and the collapsing matter. zon. Covariant Loop Quantum Gravity (LQG) [22] can Since the physics of the A region neither depends on be utilised to study the region around the classical singu- the collapsing star nor on the shrinking horizon, it can larity using the spinfoam formalism [23]. The transition be studied in the context of an eternal black hole. This amplitude for the entire quantum region (the whole dark setting gets rid of the collapsing matter, and allows is grey region in Figure 1) was first roughly estimated using to neglect the Hawking radiation, whose back-reaction LQG in [24, 25]. Here we use a similar technique to begin shrinks the area of the horizon until it enters the quantum a more refined study of the B region only. region. There is an extensive recent literature on the In particular, in section II we compute the classical possible scenarios for the physics of the A region. A intrinsic and extrinsic geometry of a boundary of the B much studied possibility is that spacetime continues on region in terms of a small number of parameters charac- the future of the would-be singularity, namely on the terising the spacetime and the transition. The quantum future of the dark grey region of Figure 1, into an anti- transition amplitude that describes the B region is going trapped region, namely a region with the metric of a to be a function of these parameters. Furthermore, in white hole [5–15]. view of the spinfoam transition amplitude calculations, Here we take this possibility as an assumption. This in section III we introduce and study a triangulation of seems by far the most plausible scenario, the one which is the boundary of B and its discrete geometry. In a forth- more coherent with the physics that we know. In fact, the coming companion paper we introduce a full discretisa- 3 2 K Κ2 tion of the B region compatible with the triangulation of its boundary introduced in section III and we use it τ to explicitly write the transition amplitude for the phe- nomenon in terms of LQG (spinfoam) techniques. K2 The main result of this paper is the identification of the τ four parameters which characterise the quantum transi- τ tion at the B region and the definition of the correspond- Figure 3: The bounded curvature scalar (7). ing transition amplitude as a function of these parame- ters. The actual computation of this amplitude will be addressed in the forthcoming companion paper. Up to terms of order O (l/m), the curvature scalar K 2 ∼ Rµνρσ Rµνρσ , which is plotted in Figure 3, is 9 l2 − 24 lτ 2 + 48 τ 4 2 II. THE BOUNDARY OF THE B REGION K 2 (τ ) = m . (7) (l + τ 2 )8 To study the B region, we restrict for simplicity to the It has finite maximum value spherically symmetric case and we assume the rest of spacetime to be classical. For the most part, this is a 9 m2 good approximation, since the curvature is below Planck- K 2 (0) = . (8) l6 ian values and quantum effects are likely to be negligible. This is however not true for the boundary between the B The Ricci tensor vanishes up to terms of order O(l/m). and the A region. We therefore simplify the problem by The space-like surfaces τ = constant can be used to describing the A region with an effective classical metric, foliate the interior of both the black and the white hole. as in [19]. Each of these surfaces has the topology S 2 ×R. Suppress- An effective metric for the entire spacetime that takes ing one angular coordinate, they can be depicted as long into account the effect of Hawking radiation was studied cylinders of different√radii and heights. In the interior in [26]. Here we further simplify this scenario by dis- black hole region (− 2m < τ < 0), as τ increases, the regarding the presence of Hawking radiation in the last radial size of the cylinder shrinks while the axis of the phases of the evaporation, hence around the B region, in cylinder gets stretched. At τ = 0 the cylinder reaches a spite of the radiation being strong in this region. We do minimal width and maximal length, and then smoothly not know how good this approximation is. We assume bounces back and starts increasing its radial size and that the black hole has already evaporated to a small size shrinking its length as√ τ increases in the interior white and we take the metric around the B region to be well hole region (0 < τ < 2m). The cylinder inside the hole approximated by a Schwarzschild metric, up to quantum never reaches arbitrary small sizes (the singularity), but corrections in the A region. it rather bounces at a small finite radius l. The effective geometry of the A region continuing from The value of l can be roughly fixed by the require- the trapped to the anti-trapped region can be described ment that K(0) ∼ 1 (in Planck units), which gives by the line element [19] l ∼ m1/3 . This means that the bounce happens at a larger scale than the Planck one. The limit l → 0 is sim- 4(τ 2 + l)2 2 2m − τ 2 2 ply the joining of a Schwarzschild black hole interior and ds2l = − dτ + 2 dx +(τ 2 +l)2 dΩ2 , (5) a Schwarzschild white hole interior through the singular- 2m − τ 2 τ +l ity. This is not a Riemann space — it is analogous to a where dΩ2 is the metric of the 2-sphere, l m √ is an double cone: a space with a singular region of measure intrinsic parameter of the effective metric and − 2m < zero — but it is a rather well behaved metric manifold, √ τ < 2m. This line element defines a genuine pseudo- where geodesics can be defined and studied [19]. Riemannian space, with no divergences and no singular- The presence of a minimum finite radius l in the A ities ∀ l 6= 0. region has far-reaching consequences for the physics of In the limit l → 0, the metric locally converges to the B region. the √ interior Schwarzschild metric for a black hole in − 2m < τ < 0 and to the √ interior Schwarzschild metric for a white hole in 0 < τ < 2m. This can be easily seen A. Choice of the boundary by performing the following change of coordinates: The idea to define a boundary for the B region is to r = τ2 and t = x , (6) first surround it in the causal diagram with a diamond shaped null surface Σ (see Figure 2), that is a diamond where r and t are the usual Schwarzschild coordinates. null surface times a sphere in spacetime, and then, since In this limit, τ = 0 becomes the singularity separating an appropriate boundary for computing transition am- the trapped from the anti-trapped region. For l 6= 0 the plitudes must be spacelike, to slightly deform Σ into a curvature remains instead bounded. spacelike surface. This surface is the Heisenberg cut we 4 Schwarzschild radius r− = l. The null past diamond boundary is taken to be the union of the outgoing past S− light cone of Sout and of the ingoing past light cone of Sin from their intersection upward; see Figure 4. p Note that the presence of a minimum finite radius l Σ − S+ in the A region fixes the value r− of the Schwarzschild p Σ + radius of the point Sin . Furthermore, being the Sp tL = cst Schwarzschild radius r a temporal coordinate inside the v+ black hole, while the radius r+ of Sout is a measure of the spatial coordinate distance of Sout from the horizon, v− the radius r− = l must not be interpreted as a measure of the spatial coordinate distance of Sin from the horizon but as the minimal internal radius reached by the black Figure 4: The past portion of the boundary surface. hole in region A. To simplify the notation, in the following we replace the labels out and in with the index ± = {+, −} ≡ choose, namely the surface we shall take as the boundary {out, in}, e.g. S+ ≡ Sout and S− ≡ Sin . between the quantum and the classical regions. Notice Next, we define the spacelike past boundary Σp by that in quantum gravity, the Heisenberg cut is also a slightly deforming the null past diamond boundary while spacetime boundary (see [27], section 5.6.4). keeping fixed S+ and S− . A convenient choice of defor- We want now to concretely specify Σ and compute mation is the following one. Consider the surface Σp− of its intrinsic and extrinsic geometry. Since it has been constant Lemaı̂tre time coordinate [28, 29] assumed that the dissipative irreversible physics of the √ p Hawking radiation is over at this point, the B region r/2m − 1 must be time-reversal invariant. The surface Σ can con- tL = t + 2 2mr + 2m ln p , (12) r/2m + 1 sequently be seen as the union of two surfaces, a past one Σp and a future one Σf , equal up to time reflec- passing by S− and the surface Σp+ defined by tion, Σ = Σp ∪ Σf . Here, the labels p and f stand for v − βr = const, (13) past and future, and later on we shall also use the index t = {p, f } (hence Σt ) where t stand for time. Accord- passing by S+ , for some constant β ∈ R. Let S p be ingly, we only need to study the past boundary Σp , as their intersection; see Figure 4. We choose the spacelike the future boundary Σf is determined by symmetry. past boundary Σp to be the union of the portion of Σp− The metric around the B region is assumed to be well between S p and S− and the portion of Σ+ between S p approximated by the Schwarzschild metric up to quan- and S+ . The parameter β can be fixed by requiring the tum corrections in the A region. The past boundary is continuity of the normal to Σp at S p . then contained in the external and in the black hole re- The spacelike future boundary surface Σf is defined to gions of a Kruskal diagram representing Schwarzschild be the time-reversal of the surface Σp and the full space- spacetime. Since both regions are covered by the ingo- like boundary surface Σ is then partitioned in the four ing Eddington-Finkelstein coordinates, we can use these components Σp+ , Σp− , Σf+ and Σf− ; see Figure 5. The coordinates to define Σp . The line element in these coor- Carter-Penrose diagram of the B region consists of two dinates reads separate portions of the Kruskal diagram which are ap- propriately joined. This is the ‘cutting and pasting’ used 2m ds2 = − 1 − dv 2 + 2dr dv + r2 dΩ2 . (9) in [17] in order to write for the first time a metric for the r black-to-white transition. The Schwarzschild time coordinate t is related to the in- We now need to determine the intrinsic and the extrin- going Eddington–Finkelstein coordinates by sic geometry of Σ. r t = v − r∗ = v − r − 2m ln −1 , (10) 2m B. Intrinsic geometry or The intrinsic geometry of Σp+ is obtained by differentiat- dr ing its defining equation (equation (13)), dt = dv − . (11) 1 − 2m r dv = βdr, (14) The null past diamond boundary can be defined in and inserting the result in the line element in equa- the Kruskal diagram as follows. Let Sout be a point (a tion (9). This gives two-sphere in spacetime) outside the horizon at advanced time v+ and Schwarzschild time t = 0. Let Sin be a 2 2m ds+ = β 2 − β 1 − dr2 + r2 dΩ2 . (15) point inside the horizon at advanced time v− < v+ and r 5 To deal with the extrinsic curvature of the surfaces Σp± it is easier to express them as systems of parametric equa- tions xµ± = xµ± (y±a a ), where y± are some parameters which serves as intrinsic coordinates to the surfaces. Given a generic surface defined by the system of parametric equa- tions xµ = xµ (y a ) for some y a , the tangent 1-form to the surface eµa is given by ∂xµ eµa = (22) ∂y a and the extrinsic curvature tensor kab of the surface reads Figure 5: Carter-Penrose diagram of the B region with the kab = eµa eνb ∇µ nν . (23) surface Σ and its components highlighted. ± Let kab be the extrinsic curvature of Σp± . Then, a straightforward calculation gives To find the intrinsic geometry of Σp− , we rewrite the √ r explicit expression of the Lemaı̂tre time coordinate in − − m 2 a k ≡ kab dx dx =b dr − 2mr dΩ2 (24) equation (12) in terms of the (v, r) coordinates. Then we 2r3 differentiate it, finding that on a constant tl surface the following relation is satisfied: and dr + mβ 3/2 (r(3 − β) + 2mβ) 2 dv = . (16) k + ≡ kab dxa dxb = p dr r5 (r(2 − β) + 2mβ) p 1 + 2m/r (25) r(1 − β) + 2mβ 2 Using this relation in the line element in equation (9), we −p dΩ . obtain that the line element resulting from the intrinsic β(2 − (1 − 2m/r)β) metric of the Σp− surface is This completes the computation of the geometry of the ds2− 2 2 = dr + r dΩ . 2 (17) boundary of B. This geometry is entirely determined by four parameters: the mass m, the Schwarzschild radii r± That is, Σp− is intrinsically flat. of the spheres S± , which by construction satisfy r− < 2m < r+ , (26) C. Extrinsic geometry and the retarded time v = v+ − v−. The physical inter- Next, we want to determine the extrinsic geometry of Σ. pretation of these four parameters is transparent. The Since the two surfaces Σp+ and Σp− are both defined mass m is the mass of the black hole when the black- by constraint equations of the form C = 0, it is easy to to-white transition happens; the retarded time v is the compute their normal 1-forms using external (asymptotic) time it takes for the transition to happen; the radius r+ is the minimal external radius ∂µ C where we assume the classical approximation to hold; nµ = − . (18) |∂ ν C∂ν C|1/2 the radius r− is the minimal internal radius reached by the black hole interior in region A. When m and r± are In Schwarzschild coordinates, the normals to the surfaces fixed, the value of v can be equivalently determined by Σp− and Σp+ are then given by fixing β or rS p . These are the only parameters describing √ ! the quantum transition. − 2mr nµ = −1, − , 0, 0 , (19) Quantum gravity should determine a transition am- r − 2m plitude W for the process as a function of these four parameters −1 −1, β − 1 − 2m r , 0, 0 W = W (m, r± , v). (27) n+ µ = 1/2 . (20) β β − 2 − 2mβ r In Planck units, the four parameters can be seen as di- mensionless. We expect the specific details of the chosen Demanding that the normals match on S p , uniquely fixes Σ not to matter, as they can be absorbed in a shift of the the value of β: Heisenberg cut (as long as it does not enter the quantum 1 region). β= q . (21) The task of the forthcoming companion paper is to 2m 1+ rS p write an explicit expression for the function W (m, r± , v) 6 (a) (b) (c) Figure 6: The triangulation of Σt . The brown tetrahedron t− is inscribed into the larger violet tetrahedron t+ . The blue segments connect vertices of the two tetrahedra radially. (d) (e) (f) using the covariant LQG transition amplitudes. These are given in an expansion in number of degrees of free- Figure 7: All images represent the triangulation of Σt , but dom. At finite order, the amplitudes are defined for spe- with different tetrahedra highlighted. In (a) no tetrahedron cific discretisations of the geometry. Below we define a is highlighted (the tetrahedron t− is in brown to remind that it first order discretisation of Σ in the form of a triangu- t is hollowed inside); in (b) the four T−a are highlighted; in (c) lation. As we shall see in the companion paper, this t three T+a out of four are highlighted; in (d) three Tabt out of six triangulation can in fact be seen as the boundary of a are highlighted; in (e) the remaining three Tabt are highlighted; t t t cellular decomposition of the B region. in (f) two T−a , two T+b and two Tcd are highlighted. III. THE TRIANGULATION OF Σ tetrahedron t+ (which coincides with the centroid of the tetrahedron t− ) and the centroid of the face `+a . Then, The topology of the B region is S 2 × [0, 1] × [0, 1] and the each vertex v+a of t+ is linked to the three vertices of the topology of its boundary ∂B = Σ = Σp ∪ Σf is S 2 × S 1 . triangle `−a (Figures 6 and 7(a)), creating 14 tetrahedra We can identify two symmetries of the geometry of Σ in total. t and one symmetry of its topology: We call T+a (violet in Figure 7(b)) the tetrahedron t having `+a as one of its faces and T−a (brown in Fig- • The Z2 time reversal symmetry that exchanges p ure 7(c)) the tetrahedron having `−a as one of its faces. and f . Each of the six remaining tetrahedra (blue in Figures 7(d) t and 7(e)) is bounded by two of the T+a tetrahedra and • The SO(3) symmetry inherited by the spherical t two of the T−a tetrahedra. Noting that the labels given to symmetry of the overall geometry. t t the T+a and T−a tetrahedra are such that each of the six • A Z2 symmetry that exchanges the internal (min- remaining tetrahedra is bounded by a set of tetrahedra t t t t imal radius) sphere S− and the external (maximal T+b , T+c , T−d and T−e , with b 6= c 6= d 6= e, we can then t t radius) sphere S+ . This is a symmetry of the topol- label the six remaining tetrahedra as Tbc ≡ Tcb , where t t t ogy, but not of the geometry, since S− and S+ have the labels b and c refer to T+b and T+c . Clearly, from Tbc t different size. one can readily trace back the other two tetrahedra T−d t and T−e . To find a triangulation of Σ we discretise the two The full triangulation of Σ is constructed identifying spheres S− and S+ into regular tetrahedra. This replaces each `±a face of Σp with the `±a face of Σf . This com- the continuous SO(3) symmetry with the discrete sym- pletely defines the triangulation of Σ. metries of a tetrahedron. In particular, we discretise each The complication of the triangulation chosen is due to of the two spheres S± in terms of a tetrahedron t± . We the non trivial topology of Σ and from the computational label the four vertices of each tetrahedron as v±a where opportunity of choosing a triangulation that respects the a = 1, 2, 3, 4; and the triangles bounding the tetrahedra symmetries of the problem. as `±a , where the triangle `±a is opposite to the ver- tex v±a . Thanks to the Z2 time reversal symmetry, the trian- gulations describing Σp and Σf must be topologically A. The dual of the triangulation equivalent. For this reason, the same construction can be applied to both. A convenient triangulation for Σt , In covariant LQG one works with the dual of a cellular illustrated in Figures 6 and 7, is the following one. The decomposition of a spacetime region. More precisely, the placement of the smaller tetrahedron t− inside the big- spinfoam that captures the discretised degrees of freedom ger tetrahedron t+ , which can be chosen arbitrarily, is of the geometry is supported by the 2-skeleton of the taken to be as in Figures 6 and 7(a), such that the ver- dual of the cellular decomposition. The boundary of the tex v−a lies on the segment linking the centroid of the spinfoam is the boundary spin-network, which is dual to 7 the boundary triangulation. The graph ΓΣ of the spin- • the 24 links `t(−a)(bc) (12 for each t), with a 6= b 6= c, network is the two-skeleton of the dual of the boundary dual to the internal triangles separating the bound- triangulation. t ary tetrahedra T−a (brown) from the internal tetra- The spin-network graph ΓΣ for the triangulation we t hedra Tbc (blue). have constructed, is illustrated in Figure 8. Each circle is a node of the spin-network, and represents a tetrahe- The geometrical data that characterise the discretised dron, and each link joining two nodes represent a triangle geometry (and define coherent spin-network states) are separating two tetrahedra. (Intersections of links in this the areas of the triangles and the angles between tetra- two-dimensional graph representation have no meaning.) hedra at these triangles. Hence, the relevant boundary Since the information carried by the graph of a spin- data for the calculation are: network is only in its topology, as long as the latter re- mains unchanged, the graph can be deformed at will. • the 2 areas a± of the internal and the external Although the graphical representation of the dual graph sphere S± , which determine the areas associated ΓΣ in Figure 8 is completely fine to represent the topo- to the links `±a ; logical information of the spin-network, it is not the best choice to manifestly represent all of its symmetries. A • the 2 areas A± of the triangles dual to the `t(±a)(bc) more symmetrical representation is the one in Figure 9. links; Although the graph ΓΣ is quite complicated, thanks to the symmetries of the problem it has only two kinds of • the 2 (thin) angles k± between Σp± and Σf± at the t internal and external sphere, which determine di- nodes that are topologically distinct: the T±a nodes and the Tabt nodes. The symmetries act by permuting the a rectly the angles associated to the links `±a ; indices and exchanging p with f or + with −. Geomet- t rically, the T+a t nodes differ from the T−a ones, as the • the two (thick) angles K± that depend on the ex- last symmetry is not geometrical. For the same reasons, trinsic curvature of Σ± and that are associated to there are only four kind of links up to geometrical sym- the triangles dual to the `t(±a)(bc) links; the angles metries (two kind up to topological symmetries). These in Σp have the opposite sign of the angles in Σf . correspond to: The extrinsic coherent state ψa± ,k± ,A± ,K± on • the 4 links `+a dual to triangles forming the discre- the graph ΓΣ defined by the geometrical data tised sphere S+ ; (a± , k± , A± , K± ) represents the incoming and out- going quantum states that correspond to the external • the 4 links `−a dual to triangles forming the discre- classical geometry [22]. The LQG transition amplitude tised sphere S− ; together with the `+a links they between coherent states will be a function of eight real connect Σp with Σf (they are the vertical links in numbers, with rather clear geometrical interpretation: the second panel of Figure 9); • the 24 links `t(+a)(bc) (12 for each t), with b = W (a± , k± , A± , K± ) = W (ψa± ,k± ,A± ,K± ) , (28) a, c 6= a, dual to the internal triangles separat- t where W (ψ) for an arbitrary state ψ in the boundary ing the boundary tetrahedra T+a (violet) from the t quantum state is defined in [22]. internal tetrahedra Tbc (blue); In turn, these eight numbers cn = (a± , k± , A± , K± ) depend on the geometry of Σ described in the previ- ous section. Hence, they depend on the four parameters m, r± , and v defined above. This defines the amplitude Figure 9: The left figure portrays a two-dimensional repre- sentation of the dual of the triangulation of Σt and the right figure portrays a three-dimensional representation of the dual ΓΣ of the full triangulation of Σ, with labels omitted; one can Figure 8: Two-dimensional graph ΓΣ of the spin-network of Σ. easily label the right figure reading the different labels from The circles are nodes (dual to tetrahedra) and the segments the left figure and using t = p for the bottom and t = f for are links (dual to the triangles). the top. 8 for the black-to-white hole transition as a function of by equation (32) and (33). The volume VΣp is given by these parameters: Z q VΣp = d3 x | det g (3) | Σt W (m, r± , v) = W (cn (m, r± , v)) . (29) Z s 2 2m = dr dθ dφ r | sin θ| β 2 − β 1 − Our last task is to compute the functions cn (m, r± , v). Σt+ r Z + dr dθ dφ r2 | sin θ| Σt− s B. Discrete geometrical data Z 2m 2 = 4π dr r β 2−β 1− Σt+ r There is no unique or right way to assign discrete ge- 4π 3 3 ometrical data to the graph ΓΣ starting from the con- + r p − r− . 3 S tinuous geometry of Σ. Each choice defines a different approximation of the continuous geometry and it has its The integral over Σp+ can be computed explicitly (with own strengths and its own weaknesses. In this section computer algebra) in the case in which β is fixed by the we will introduce a convenient set of discrete geometrical continuity at S p . We do not give the explicit expression data approximating the continuous geometry of Σ. We here. The volume VΣαp is instead given by will discuss the discrete geometry of the triangulation Z q Z approximating Σp . The discrete geometry of the trian- α VΣp = 3 (3) d x | det gα | = dr dθ dφ α r2 | sin θ| gulation approximating Σf is simply related to the first Σt Σt one by a time reversal transformation. 4πα 3 3 = r+ − r− . The area of the spheres S± is directly determined by 3 the radii r± . Since the four triangles `±a bounding the We fix the value of α by requiring that VΣαp = VΣp . This tetrahedra t± that discretise the spheres S± are equal by explicitly gives the value of α as a function of the space- symmetry, we take their area a± to be one fourth of the time free parameters (m, r+ , r− , β). area of the spheres, that is We can now assign discrete geometrical data to the triangulation starting from the continuous instrinsic ge- 2 ometry in equation (33). Let us first consider the α = 1 a± = πr± . (30) case. When α = 1 the line element describes flat space The side length b± of the tetrahedron t± is then trivially and the intrinsic discrete geometry of the triangulation given by is completely determined by the side lengths b± . Let t t dev ± , def ± and e ht± be respectively the distance between s the centroid and a vertex of the tetrahedron t± , the dis- 4π tance between the centroid and a face of the tetrahedron b± = √ r± . (31) 3 t± and the height of the tetrahedron t± . Basic geometry shows that r r The line element ds2 on Σp can be written as t± 3 t± 1 2 dv = e b± , df = √ b± , ht± = e e b± . (34) 8 24 3 ds2 = f 2 (r) dr2 + r2 dΩ2 , (32) p hT± of a tetrahedron T±a The height e (they are all equal by symmetry) relative to the base `±a can be expressed where as t t 2m hT+ = def + − devt− e hT− = devt+ − def − . and e (35) 2 f (r) = β 2 − β 1 − r It is then possible to define the volumes of all the tetra- p hedra, except the Tab ones, as on Σp+ (see equation (15)) and f 2 (r) = 1 on Σp− (see 1 equation (17)). We approximate this line element as VeX = ehX aX , (36) 3 ds2 = α2 dr2 + r2 dΩ2 , (33) where X = {t+ , t− , T+ , T− }, at+ = aT+ = a+ and at− = p aT− = a− . Finally, the volume VeT of a Tab tetrahedron where α is a constant that needs to be determined. In (they are all equal by symmetry) is given by order to do so, let VΣp and VΣαp be the volume of the 1 e surface Σp whose intrinsic geometry is given respectively VeT = Vt+ − Vet− − 4VeT+ − 4VeT− . (37) 6 9 This completely determines the intrinsic discrete geome- try of the triangulation in the flat case in terms of b± . When α 6= 1 the line element in equation (33) describes a three-dimensional cone. We are only interested in the region of the cone in which r ∈ [r− , r+ ]. Hence, in this approximation, Σp turns out to be locally flat. However, it cannot be embedded in a flat three-dimensional space in the same way in which a two-dimensional cone cannot Figure 10: Definition of k+ be embedded in a flat two-dimensional space. Let us focus on the three-dimensional curved geometry defined by the line element in equation (33). The conse- So quence of the presence of α is simply a stretching of the s 2 α2 radial lengths with respect to the geometry discussed in 2 r∓ 2 A± = πr± 1−3 + (47) the α = 1 case while the tangential lengths remain fixed. 18 r± 3 In analogy with the α = 1 case we can then define the following quantities: With α fixed by the value of the total volume of Σp , r equation (47) explicitly gives the value of the areas A± t± t± 3 in terms of the spacetime free parameters (m, r+ , r− , β). dv = α dv = e α b± , (38) 8 Let us now focus on the extrinsic discrete geometry. The angles k± , which are represented in Figure 10, are t t 1 defined as df ± = α def ± = √ α b± , (39) 24 def cos k± = g µν n±f ±p r µ nν S . (48) ± 2 ht± = α eht± = α b± , (40) 3 It is then straightforward to find 2 t 1 + (1 − 2m/r+ ) β − 1 hT+ = α def + − devt− , hT+ = α e (41) cos k+ = (49) |β(β − 2 − 2mβ/r+ )|(1 − 2m/r+ ) t hT− = α devt+ − def − . hT− = α e (42) and p The volumes of all the tetrahedra, except the Tab ones, 1 + 2m/r− cos k− = . (50) can be then written as 1 − 2m/r− 1 1 The angles K± bear the extrinsic curvature of Σ± . We VX = hX aX = α e hX aX = α VeX , (43) 3 3 choose to define them as the average of the extrinsic cur- t vature, shared over the 12 triangles l(±a)(bc) : where X = {t+ , t− , T+ , T− }, at+ = aT+ = a+ and at− = aT− = a− . Furthermore, since the curved counterpart of Z 1 equation (37) must still be valid, we can write K± = kaa . (51) 12 Σ± 1 VT = Vt − Vt− − 4VT+ − 4VT− We have 6 + α e (44) = Vt+ − Vet− − 4VeT+ − 4VeT− 2m mβ 3/2 (r(3 − β) + 2mβ) 6 (k + )aa = 1− p = α VeT . r r5 (r(2 − β) + 2mβ) 2 r(1 − β) + 2mβ The intrinsic discrete geometry of the triangulation is − 2p (52) thus completely determined in terms of b± and α. r β(2 − (1 − 2m/r)β) We are interested in the value of the areas A± of the and triangles dual to the `p(±a)(bc) links. These values are r given by m 2m (k − )aa =− 3+ . (53) 1 2r3 r A± = b± h± , (45) 2 For the time being, we leave the integral in equation (51) where h± is the height of the triangles dual to `p(±a)(bc) unsolved. relative to b± . Basic geometry shows that Hence, we have found analytic expressions for the four r areas a± and A± and the four angles k± and K± as func- 1 tions of the four parameters m, r± and β (the parameter h± = h2T± + b2± . (46) β can equivalently be traded for rS p or v). 12 10 IV. CONCLUSIONS ternative scenarios on the end of the life of a black hole. The above construction defines the black-to-white hole transition amplitude W (m, r± , v) as a function of the • We have taken a number of approximations which physical parameters (m, r± , v) that characterise the tran- we do not control. Physical intuition suggests that sition and in terms of covariant LQG transition ampli- the approximation given by disregarding a direct tudes. A number of questions, which we list here, remain effect of Hawking radiation in the last phases of open. the evaporation, besides having already shrunk the horizon, may be of particular interest to check. • To compute the amplitude W (ψa± ,k± ,A± ,K± ) to the first relevant order, we need to find a spinfoam • The black-to-white hole transition may have impor- bounded by ΓΣ . This will be done in a forthcoming tant astrophysical and cosmological implications. companion paper. White hole produced by the transition of Planck size holes may be stable [34] and form a component • The amplitude is then given by a complicated of dark matter. Alternatively, if the transition can multiple group integral, which is hard to study. happen at larger black hole masses, it may be re- Asymptotic techniques, and in particular those re- lated to cosmic rays and fast radio bursts [35–37]. cently developed in [30] are likely to be essential for A control on the amplitude of this transition should this. Alternatively, a numerical approach, follow- help to shed light on these possibilities. ing [31, 32] may provide insights in the amplitude. • The question of eventual infrared divergences in the *** amplitudes and, eventually, how to deal with them, needs to be addressed. • To compute probabilities from amplitudes we have Acknowledgments to address the problem of the normalisation. This can be solved using the techniques developed in the We thank Alejandro Perez for useful comments. This general boundary formulation of quantum gravity. work was made possible through the support of the FQXi See in particular [33]. Obviously the probabilities Grant FQXi-RFP-1818 and of the ID# 61466 grant from computed give the relative likelihood of a transi- the John Templeton Foundation, as part of the The tion within the space of the parameter considered, Quantum Information Structure of Spacetime (QISS) and not the relative probability with respect to al- Project (qiss.fr). [1] A. Perez, “No firewalls in quantum gravity: The role of Gravity,” Advances in High Energy Physics 2008 (Nov, discreteness of quantum geometry in resolving the 2008) 1–12, arXiv:gr-qc/0611043. information loss paradox,” Classical and Quantum [9] R. Gambini and J. Pullin, “Black holes in loop quantum Gravity 32 no. 8, (2015) , arXiv:1410.7062. gravity: the complete space- time,” Phys. Rev. Lett. [2] M. Christodoulou and C. Rovelli, “How big is a black 101 (2008) 161301, arXiv:0805.1187. hole?,” Physical Review D 91 (2015) 064046, [10] A. Corichi and P. Singh, “Loop quantization of the arXiv:1411.2854. Schwarzschild interior revisited,” Classical and [3] I. Bengtsson and E. Jakobsson, “Black holes: Their Quantum Gravity 33 no. 5, (Jun, 2016) , large interiors,” Mod. Phys. Lett. A 30 (2015) 1550103, arXiv:1506.08015. arXiv:1502.0190. [11] A. Ashtekar, J. Olmedo, and P. Singh, “Quantum [4] M. Christodoulou and T. De Lorenzo, “Volume inside Transfiguration of Kruskal Black Holes,” Physical old black holes,” Physical Review D 94 (2016) 104002, Review Letters 121 no. 24, (Jun, 2018) , arXiv:1604.07222. arXiv:1806.00648. [5] L. Modesto, “Disappearance of the black hole [12] K. Clements, “Black Hole to White Hole Quantum singularity in loop quantum gravity,” Physical Review Tunnelling,” undefined (2019) . D - Particles, Fields, Gravitation and Cosmology 70 https://www.semanticscholar.org/paper/ no. 12, (2004) 5, arXiv:gr-qc/0407097v2. Black-Hole-to-White-Hole-Quantum-Tunnelling-Clements/ [6] A. Ashtekar and M. Bojowald, “Quantum geometry and 6f62a4cadc51a3e60ad9d161de90e2207e1d47c7. the Schwarzschild singularity,” Classical and Quantum [13] N. Bodendorfer, F. M. Mele, and J. Münch, “Mass and Gravity 23 no. 2, (2006) 391–411, Horizon Dirac Observables in Effective Models of arXiv:gr-qc/0509075. Quantum Black-to-White Hole Transition,” [7] L. Modesto, “Loop quantum black hole,” Classical and arXiv:1912.00774. Quantum Gravity 23 no. 18, (Sep, 2006) 5587–5601, [14] A. Ashtekar and J. Olmedo, “Properties of a recent arXiv:gr-qc/0509078. quantum extension of the Kruskal geometry,” [8] L. Modesto, “Black Hole Interior from Loop Quantum arXiv:2005.02309. 11 [15] R. Gambini, J. Olmedo, and J. Pullin, “Spherically Black-to-White Hole,” Classical and Quantum Gravity symmetric loop quantum gravity: analysis of improved (May, 2019) , arXiv:1905.07251. dynamics,” arXiv:2006.01513. [27] C. Rovelli, Quantum Gravity. Cambridge University [16] J. L. Synge, “The Gravitational Field of a Particle,” Press, 2004. Proc. Irish Acad. A53 (1950) 83–114. [28] G. Lemaı̂tre, “L’Univers en expansion,” Annales de la [17] H. M. Haggard and C. Rovelli, “Quantum-gravity Société Scientifique de Bruxelles A53 (1933) 51–85. effects outside the horizon spark black to white hole [29] M. Blau, Lecture Notes in General Relativity. tunneling,” Physical Review D 92 no. 10, (2015) http://www.blau.itp.unibe.ch/Lecturenotes.htms. 104020, arXiv:1407.0989. [30] P. Dona and S. Speziale, “Asymptotics of lowest unitary [18] T. De Lorenzo and A. Perez, “Improved black hole SL(2,C) invariants on graphs,” arXiv:2007.09089. fireworks: Asymmetric black-hole-to-white-hole [31] P. Donà and G. Sarno, “Numerical methods for EPRL tunneling scenario,” Physical Review D 93 (2016) spin foam transition amplitudes and Lorentzian 124018, arXiv:1512.04566. recoupling theory,” General Relativity and Gravitation [19] F. D’Ambrosio and C. Rovelli, “How information 50 no. 10, (2018) , arXiv:1807.03066. crosses Schwarzschild’s central singularity,” Classical [32] P. Donà, M. Fanizza, G. Sarno, and S. Speziale, and Quantum Gravity 35 no. 21, (Mar, 2018) , “Numerical study of the Lorentzian arXiv:1803.05015. Engle-Pereira-Rovelli-Livine spin foam amplitude,” [20] E. Bianchi, M. Christodoulou, F. D’Ambrosio, H. M. Physical Review D 100 no. 10, (Mar, 2019) , Haggard, and C. Rovelli, “White holes as remnants: A arXiv:1903.12624. surprising scenario for the end of a black hole,” [33] R. Oeckl, “A predictive framework for quantum gravity Classical and Quantum Gravity 35 (2018) 225003, and black hole to white hole transition,” Physics arXiv:1802.04264. Letters, Section A: General, Atomic and Solid State [21] C. Rovelli and F. Vidotto, “Planck stars,” Int. J. Mod. Physics 382 no. 37, (Apr, 2018) 2622–2625, Phys. D 23 (2014) 1442026, arXiv:1401.6562. arXiv:1804.02428v1. [22] C. Rovelli and F. Vidotto, Covariant loop quantum [34] C. Rovelli and F. Vidotto, “Small black/white hole gravity: An elementary introduction to quantum gravity stability and dark matter,” Universe 4 no. 11, (Nov, and spinfoam theory. Cambridge Univeristity Press, 2018) 127, arXiv:1805.03872. 2015. [35] A. Barrau, C. Rovelli, and F. Vidotto, “Fast radio [23] A. Perez, “The Spin-Foam Approach to Quantum bursts and white hole signals,” Physical Review D 90 Gravity,” Living Reviews in Relativity 16 (2013) , no. 12, (2014) 127503, arXiv:1409.4031. arXiv:1205.2019. [36] A. Barrau, B. Bolliet, F. Vidotto, and C. Weimer, [24] M. Christodoulou, C. Rovelli, S. Speziale, and “Phenomenology of bouncing black holes in quantum I. Vilensky, “Planck star tunneling time: An gravity: A closer look,” Journal of Cosmology and astrophysically relevant observable from Astroparticle Physics 2016 no. 2, (2016) 022–022, background-free quantum gravity,” Physical Review D arXiv:1507.05424. 94 (2016) 084035, arXiv:1605.05268. [37] A. Barrau, B. Bolliet, M. Schutten, and F. Vidotto, [25] M. Christodoulou and F. D’Ambrosio, “Characteristic “Bouncing black holes in quantum gravity and the Time Scales for the Geometry Transition of a Black Fermi gamma-ray excess,” Physics Letters B 772 (2017) Hole to a White Hole from Spinfoams,” 58–62, arXiv:1606.08031. arXiv:1801.03027. [26] P. Martin-Dussaud and C. Rovelli, “Evaporating