Firewalls from wormholes in higher genus

An old black hole can tunnel into a white hole/ firewall by emitting large baby universes. This phenomenon was investigated in Jackiw-Teitelboim (JT) gravity for genus one. In this paper, the focus is on higher genus corresponding to emitting more than one baby universe ($n>1$). The probability of encountering a firewall or tunneling into a white hole after emitting $n$ baby universes is proportional to $e^{-2nS(E)}e^{4 \pi \sqrt{E}(n-1)}E^{2n^2-n-9/2}t^{4n^2-2n-5}$, where $t$ is the age of the black hole, and $S$ and $E$ represent the entropy and energy of the black hole, respectively.


Introduction
It has been argued that large AdS black holes in late times must possess some structures at the horizon, often referred to as a fuzzball or firewall [1,2,3,4,5,6,7,8,9,10].Firewalls exhibit similar characteristics to white holes.White holes may not be common in nature, but they are just as abundant as black holes in the Hilbert space that describes systems with entropy S. In the subsequent part of this section, we will review the resemblance between a firewall and a white hole and the phenomenon of tunneling a black hole into a white hole.

Firewall and white hole
One can distinguish between two types of black hole horizons: transparent and opaque.A transparent horizon can be crossed freely by an in-falling observer, while an opaque horizon does not allow such passage [11].In classical gravity, there are two criteria to support the notion that certain black holes may have opaque horizons.The expansion criterion states that black hole horizons are transparent when the interior geometry expands, (see figure 1).The Page criterion suggests that black holes formed from non-singular Cauchy data, with no past singularities, have transparent horizons.Based on these criteria, an object assumed to be a black hole with a non-transparent horizon could be a white hole.Because the white hole has a contracting interior geometry, making it impossible to enter its horizon.White holes are essentially geometries formed from singular Cauchy data, and their horizons are not transparent when past singularities are present [12].The expansion and Page criteria are based on classical geometric terms and do not In region I, located behind the black hole horizon, the spacetime exhibits expansion.As a result, the black hole horizon is transparent.On the other hand, in region II, behind the white hole horizon, the spacetime undergoes contraction, and the horizon is opaque.directly consider properties of the quantum state.Additionally, there exists a third criterion that solely focuses on the dual quantum state.According to this criterion, the black hole horizon will remain transparent as long as the complexity of the dual state continues to increase [11].This criterion can be related to the expansion criterion by the complexity=volume conjecture: Here, V is the volume of the Einstein-Rosen bridge (ERB), G represents Newton's gravitational constant, and l is a specific length scale associated with the geometry, typically chosen to be the AdS radius of curvature or the Schwarzschild radius.Another potential explanation for opacity is the presence of a firewall that effectively seals off the interior geometry [13].It was demonstrated that shock waves can generate firewalls [14].The volume of the wormhole (ERB) as a function of the times of the left and right boundaries (t L , t R ) in the presence of a firewall, which arises from a shock wave at time t w is [15]: Here, t * is the scrambling time.Assuming t L to be fixed and sufficiently large for simplicity, then V(t R ), or equivalently the complexity of the perturbed state is a decreasing function of t R for t R < |t w |−t * .Consequently, the horizon is deemed to be opaque.However, for times when t R ≫ |t w |−t * , the wormhole grows linearly with time, V(t R , t L ) ∼ t R , and the horizon becomes transparent.As a result, if the horizon of the black hole becomes opaque, it could indicate the presence of a firewall at the horizon or that the black hole has tunneled into a white hole.

Firewalls from wormholes
In classical general relativity wormhole (ERB) grows forever.According to this point, it was introduced a different version of the information paradox that applies to large, eternal black holes in the context of AdS/CFT [16,17,18,19]. 1 Let us consider a thermofield double state which is dual to two-sided eternal AdS black hole.In such a system, we would like to understand the long-time behavior of correlation functions.In the gravity side, two point function between opposite sides behaves like: where ℓ(t) is the length of the wormhole at time t.The cause of the decreasing correlation function is the expanding wormhole that separates the two sides.Using the extrapolate dictionary: O R,L (t) = lim r→∞ r −∆ ϕ R,L (t, r), (1.4) the correlation function on the CFT side becomes: e −βE i +i(E i −E j )t |O| ij (1.5)In the CFT, any perturbation of the thermal state (suppose O L (0) is a perturbation on thermofield double state) remains a perturbation of the thermal state forever.Although it undergoes scrambling and appears to thermalize, the initial perturbation is never completely forgotten.In fact, the corrections to the thermal state are finite and suppressed by the entropy for t ≫ β.Therefore, we have: So, at very late times, gravity forgets the initial perturbation, while a unitary CFT does not: This reveals where the gravity derivation deviates.Also, if we accept the complexity=volume conjecture, we will run into trouble.The problem arises from the limited number of available mutually orthogonal states, which is of the order of exp(S) (where S is the entropy of the black hole).After an exponential time, we exhaust all orthogonal states, leading to a saturation of complexity, while the volume of the wormhole continues to grow indefinitely.In other words, we expect that the length of the wormhole to stop growing when t ∼ exp(S), while classical geometry does not show this behavior.
In the context of JT gravity, it has been demonstrated that the length of wormholes can be diminished through the process of emitting baby universes [24]. 2 The emission of a baby universe can have different consequences depending on its size.As we will see later; in the case where the size of the emitted baby universe is larger than the age of the black hole, it can lead to the formation of a firewall or cause the black hole to undergo a tunneling process, transforming it into a white hole.Stanford and Yang computed the probability of emitting a single baby universe at late times as [32]: where and the subscript one corresponds to genus, representing the emission of a single baby universe.This paper aims to extend their work by considering higher genus scenarios.To achieve this, Section 2 provides a review of the JT gravity wave function.Section 3 focuses on the computations of the probabilities of finding a firewall and smooth geometry through the emission of two baby universes.In Section 4, the probability of encountering a firewall for genus two is calculated using an alternative approach.Section 5 extends the calculations to include an arbitrary number of emitting baby universes.Finally, concluding remarks are presented in Section 6.

JT gravity wave function setup
In Euclidean JT gravity, to compute partition function one should integrate over surfaces of constant negative curvature (usually chosen to be R = −2).These surfaces can be decomposed into "trumpets" with asymptotic and geodesic boundaries and surfaces with g handles and a certain number of geodesic boundaries (which are glued to the trumpets).The partition function of a geometry with one asymptotic boundary, n geodesic boundaries (referred to as baby universes), and genus g, can be written as [33,34]: here, χ and S 0 represent the Euler characteristic and the ground state entropy, respectively.Z trumpet refers to the trumpet partition function, and V g,n (b, a) represents the Weil-Petersson volume of the moduli space of hyperbolic Riemann surfaces with genus g and n + 1 geodesic boundaries of lengths (b, a 1 , a 2 , ..., a n ).In the following for simplicity, we set g = 0 (see figure 2).The aforementioned partition function can be written differently, as follows: where ρ (E, a) denotes the density of eigenvalues.To derive the expression of ρ (E, a), one can use the substitution of the trumpet partition function Z trumpet (β, b) defined by: into the relation (2.1) as follows: The expression for ρ (E, a) is obtained by comparing equations (2.4) and (2.2).The resulting expression is: (2.5) The partition function (2.1) can be alternatively expressed as the gluing of two wave functions of Hartle-Hawking states with a specific amplitude, as follows: where the amplitude ⟨ℓ, a|ℓ 1 ⟩ is given by: The expression ⟨ℓ, a|ℓ 1 ⟩ represents the amplitude for a wormhole with length ℓ 1 to emit n baby universes and transition into another wormhole with length ℓ ( see figure 3).For latter calculations, it would be useful to replace ρ (E, a) from expression (2.5) into (2.7),so one gets: In the framework of JT gravity, the wave functions ⟨ℓ|E⟩ and the density of states ρ (E) are defined as [35,36,37,38]: (2.9) They satisfy the orthogonality and completeness relations:

Probability distributions in genus 2
In this section, we investigate a bulk topology with two handles.The goal is to calculate the probability distribution for the length of a spatial slice connecting two boundary points as shown in figure 4. The focus is on the scenario where S and t are large.The probability distribution can be written as: The notation P represents an un-normalized probability distribution.To achieve proper normalization, we must divide P by the partition function.The subscript two refers to the genus.The inverse Laplace transformation yields the fixed-energy version of the probability distribution as follows: From (2.8) one can get: where V 0,3 (b, a 1 , a 2 ) = 1 [39].By substituting (3.3) into (3.2) and integrating over ℓ 1 and ℓ 2 , the following equation is obtained: By substituting inner product ⟨E 1,2 | β 2 + it⟩ = e (−β/2+it)E 1,2 we have: The integral over β sets E 1 + E 2 = 2E.By performing a change of variables: expression (3.5) reduces to: Notice that the explicit value of ρ trumpet is replaced from (2.3).Using the following integral: and considering the conditions b ∞ , E ≫ 1, after setting k = 1, the relation (3.7) can be approximated as: (3.9) Our calculation is restricted to the semi-classical regime.In this regime, E is large and the wave functions ⟨E|ℓ⟩ are approximated as: (3.10) Also in this limit one should set Therefore, the probability distribution (3.9) in semi-classical regime becomes: where ℓ t = 2t √ E and some unimportant 2 log(2E) terms were omitted.The integration over ω yields: (3.12) In the expression ∆ (b 1 , b 2 , ℓ t ), the absence of ℓ in the delta functions indicates that these terms do not impose any constraint between the length of the wormhole and the baby universes.Although As a result, within the probability expression (3.12),only three terms are relevant for our specific purposes, leading to a simplified expression as presented below: These three terms correspond to three different types of geometry that appeared in [32].In the late-time limit, the lengths ℓ 1 and ℓ 2 will be significantly large.Furthermore, according to equation (3.8), both b 1 and b 2 (or b) are large, and for most of the support of the probability distribution, a 1 , a 2 , and ℓ are expected to be large.In accordance with the Gauss-Bonnet theorem, the area of the region corresponding to tunneling amplitudes remains small, allowing us to approximate the geometry as thin strips (see subsection 4.2).By employing the thin strip approximation, the length b can be expressed in terms of the lengths of the baby universes a 1 and a 2 , as well as their common region λ, as follows: Using the above relation, we can establish a connection between the lengths of the baby universes and the length of the wormhole.
Integration over baby universes is crucial in our computation, as it relates to the fate of the infalling observer.It seems possible to choose the correct interior using a very simple criterion: it should be ensured that the spatial slice does not have shortcuts, meaning that distant points along the spatial slice or wormhole should not be close together in the spacetime geometry or bulk.This criterion is designed to avoid overcounting geodesics that cover portions of the same fundamental spatial slice more than once.This criterion will be discussed further in the following.

Firewall-free geometry
The wormhole of the geometry corresponding to ℓ = ℓ t − b = 2 √ Et − b is expanding with time, so our black hole has a smooth horizon.The thin strip diagram in figure 5 clearly shows that points on the ℓ slice are not close to each other, regardless of the assigned values of a 1 , a 2 , twists s 1 , and s 2 .Thus, the "no shortcut" criterion does not impose any restrictions on the moduli space.Therefore, twists s 1 and s 2 should be integrated over the entire fundamental region 0 < s 1 < a 1 and 0 < s 2 < a 2 .The integration ranges for a 1 and a 2 extend from 0 to some upper limits, where the upper limits are constrained by the condition that ℓ = ℓ t + 2λ − a 1 − a 2 must be positive.This constraint can be enforced by incorporating a theta function and setting the upper limit of integration to ∞.So, the probability of a firewall-free geometry with a wormhole length of ℓ and a shared region λ between baby universes is (3.17)

Firewall geometry first method
To calculate the probability of finding a firewall, one should consider the geometry in which ℓ shrinks over time.In this case, the delta functions that establish the relationship ℓ = b − 2 √ Et can potentially contribute to the firewall probability.In this particular geometry, the wormhole is overlapped by segments of both baby universes, denoted as a 1 and a 2 in figure 6.The portion of the wormhole overlapping with baby universe a 1 is labeled as γ, while the overlapping region with baby universe a 2 is referred to as ℓ − γ := γ.The shared region between the two baby universes is represented by λ.In this case, the "no shortcut" condition affects the moduli space.To see this point, let us assume initially that twists satisfy 0 < s 1 < γ and 0 < s 2 < γ.Then for example the four points marked with "×" in the left hand side of the figure 7 will be identified with each other, respectively.In this case, there exists a more fundamental geodesic ℓ ′ = γ ′ + γ′ , which serves as a better candidate for the correct spatial slice.This geodesic indicated with thick black line in the right hand side of the figure 7.In other words, the mapping class group The geometry characterized by ℓ = b − ℓ t , corresponds to a contracting branch of the wave functions (firewall).In this configuration, both baby universes share a section with the wormhole ℓ, denoted as γ for baby universe a 1 (and γ for a 2 ).The region between the two baby universes is represented by λ.
element that transforms the left sketch into the right in figure 7 can be understood, in the thin strip approximation, as mapping: As this process is continued, we will reach a point where γ + λ < s 1 and γ + λ < s 2 , and there will be no shortcuts.Additionally, after applying the mapping class group, we expect that a ′ 1 > γ and a ′ 2 > γ.Consequently, the constraints on the twists are γ + λ < s 1 < a 1 − γ and ℓ − γ + λ < s 2 < a 2 − ℓ + γ.These constraints can be written as: Using the delta function that sets ℓ = a 1 + a 2 − 2λ − ℓ t , one can rewrite the constraint (3.20) as: To ensure the positivity of ℓ, it is necessary that Points marked with × are identified.The geodesic ℓ ′ , depicted as a thick black line, might be a more suitable candidate for the correct spatial slice, as the red line geodesic ℓ is essentially a repetition of ℓ ′ .In other words, left picture shows that points along the geodesic ℓ that appear distant are, in fact, close to each other within the entire geometry or bulk.
So, the probability of emitting two baby universes, leading to a firewall, with the wormhole length ℓ, and both baby universes having a common region λ, is: Note that, the constraints from inequalities (3.19), (3.20) and (3.22) are enforced through the use of theta functions.

Negative probability contribution
The wormhole in the geometry corresponding to ℓ = ℓ t + b is also expanding.Within this specific class of thin-strip geometry, illustrated in figure 8, any values of the twists s 1 and s 2 induce nonlocal identifications along the ℓ geodesic.Another way to say this is that the ℓ geodesic closely traces the paths of either the ℓ 1 or ℓ 2 geodesics, exhibiting a slight deviation around the b 1 or b 2 cycles.This behavior suggests that the correct interior is defined by the ℓ 1 or ℓ 2 geodesics [32].Calculating the total probability within this region poses difficulties due to the small size of the closed geodesic homologous to the asymptotic boundary, referred to as b in figure 10.Additionally, the straightforward gluing procedure mentioned earlier does not include the mapping class group restriction on the corresponding twist parameter τ .In the next section, we demonstrate that the small b region contributes negatively to the probability.

Firewall geometry second method
The surface of interest is a trumpet attached to a double torus, which has a single geodesic boundary with a length of b, as shown in figure 9.The flat Weil-Petersson measure db 2 dτ 2 db 1 dτ 1 da 2 ds 2 da 1 ds 1 is the only factor related to the handle parts of the spacetime.As we observe later, at long times, the most important contribution comes from double torus geometries with large b.Therefore, it is expected that the volume in the large b limit is proportional to (see relation (4.28)): However, our goal is not to calculate the partition function of the double handle disk.Instead, we are interested in the fate of the infalling observer.To achieve this, the inclusion of a delta function into the partition function is required.This delta function is employed to choose the ℓ slice that defines the correct interior.So one can write: where the ℓ(moduli) is the length of the geodesic labeled ℓ in the figure 9. 4.1 Five-holed sphere with a 1 + a 2 > ℓ t Let us begin by considering the geometry, which consists of a trumpet with a closed geodesic of length b and a five holed sphere, as shown in figure 10.The length distribution p(ℓ, a) is described by the following expression: The constraint that ℓ is held fixed.
To fix other parts of the geometry. .
In comparison to the case of a three-holed sphere examined in [32], expression (4.3) includes two additional delta functions.For the three-holed sphere, the ℓ (moduli) can be expressed as a function of two holes or baby universes [32].In other words, in the case of a three-holed sphere, the insertion of a delta function in the partition function restricts us to choose a geometry where the wormhole's length is determined by the lengths of baby universes and ℓ t .However, in the case of a five-holed sphere, the length of the wormhole can be expressed as a function of geodesics b 1 and b 2 , as computed in Appendix A, and this length is independent of geodesic boundaries or baby universes.Hence, to choose a geometry in the partition function where baby universes determine the wormhole, it is necessary to insert two more delta functions.These delta functions fix the lengths of b 1 and b 2 in terms of baby universes a 1 and a 2 .Note that the presence of "E" in the expression (4.3) is related to dimensional analysis.In the trumpet region, one can take the metric: of wave function to get: The twist τ was integrated from zero to b/2 instead of b because of the π rotation symmetry of the five-holed sphere.By slightly smearing over E, the contribution from b = ∞ can be eliminated, resulting in the following: and the integration over λ will give: The above result will be used for the contribution of small b, and, similar to the three-holed sphere answer, it is also negative.For the negative terms, it is crucial to consider the appropriate spatial slice, which should be either the ℓ 1 or ℓ 2 slice, not ℓ itself.Therefore, the no shortcut condition for ℓ should not be imposed [32].For simplicity, the calculations in the following will focus on large b region with b as following: where x 12 = x 1 − x 2 , and σ+ = σ1 + σ2 is the sum of the regularized σ variables e σ = ϵe σ .The measure in the new coordinates is dℓ 1 dℓ 2 db = 2dx 1,2 db dσ + , so: The pieces that depend on β can be extracted, and the integral can be performed as following: 2 by using (3.6) and (3.10) one can get: after simplifying and omitting unimportant terms, the above relation becomes: The factor ⟨ℓ 2 , b|ℓ 1 ⟩ in the integrand of (4.9) is given by: and in the first approximation at large b, Bessel function becomes: Replacing (4.12) and (4.14) into (4.9)results in: and by substituting ℓ (moduli) from (A.3), (4.15) becomes: (4.16) To carry out the integration over x 1 and x 2 , one can expand the δ function in powers of log cosh x i 2 : Only the third term of expansion (4.17) contributes to the integral (4.16) and other terms become zero after integrating over x 1 and x 2 .Using the integral: and the expansion (4.17), (4.16) becomes: In the above expression, the terms from the b ⋆ limit were dropped.It would be more convenient to approximate the Bessel function as a delta function: Therefore, (4.19) simplifies to: where in the last step, γ 2 was replaced from (A.1).Due to the condition ℓ t ≫ 1, and both b 1 and b 2 being significantly larger than ℓ t , we can approximate the above expression as follows: after performing the integration, the expression above yields: this corresponds to the delta function in (3.15), which characterizes the firewall geometry.How-ever, the current method clarifies that additional terms (4.7) originated from the small b region.At this stage, we need to integrate over a 1 , a 2 , s 1 , and s 2 , ensuring adherence to the no shortcut rule for the ℓ geodesic.Therefore, we delve into the details of the geometry of a disk with two handles (a double torus with one geodesic boundary) in the following.

Double handle disk
Let us consider a more general surface with geodesic boundaries, in the limit that the geodesic boundaries are long.The Gauss-Bonnet theorem, (4.24) makes a connection between the integrals of the curvature of the bulk and the integral of the curvature along the boundary of the surface.In the case we are interested in, the bulk curvature is constant (R = −2), and the extrinsic curvature of the boundaries is zero.So, the total area of the surface is connected to the Euler characteristic as: In the limit where the boundaries become long (Airy limit), considering that the area is fixed, the surface must become thin.This implies that any segment of the boundary approaches closely to another segment of the boundary.It can be helpful to maintain the intuitive notion that the hyperbolic surface with long boundaries resembles the ribbon graph or strip diagram [40,41,42].Figure 11 represents the ribbon graph of five holed sphere.Also, figure 12 represents the ribbon graph of a double handle disk.This ribbon graph is constructed by gluing geodesics of the same length together in figure 11.The resulting strip geometry can be depicted as a trivalent band structure, assembled from nine strips of lengths y 1 , y 2 , . . ., y 9 , and connected with two twists, resulting in a geometry with a single boundary.Sometimes, the actual surface is represented solely by a trivalent graph, where the edges are assigned lengths, and the bulk geometry is not depicted.This graph exhibits E = 6g − 6 + 3n edges and V = 4g − 4 + 2n vertices.In other words, a graph with E edges and V vertices is associated to a ribbon graph with genus g and n boundaries through the following relation: By thickening the graph one can obtain the ribbon graph.For a double handle disk (with g = 2 and n = 1), the graph has nine edges and six vertices, as depicted in figure 13.The application of the thin strip limit simplifies the decomposition of the moduli space related to hyperbolic surfaces.Within this thin strip context, the moduli space can be expressed as a summation over trivalent ribbon graphs.This summation is coupled with an integral over the lengths of the edges constituting these graphs, with the condition that the boundaries have lengths a i .The resulting expression is given by: Here, Γ g,n ∈ Γ, where Γ is the set of trivalent ribbon graphs with genus g and n boundaries.These graphs are constructed from E = 6g − 6 + 3n edges and V = 4g − 4 + 2n trivalent vertices.In the above expression y k is the length of edge k, and n i ∈ {0, 1, 2} denotes the number of sides of edge k that belong to boundary i.
For a double handle disk, the volume of the moduli space is [43]: in the limit of large b, it approximates to:

.29)
A portion of the moduli space volume, corresponding to the ribbon graph depicted in figure 12, can be calculated using equation (4.27).The length of the boundary of double handle disk according to figure 12 is: for more explanation, one can begin from an arbitrary point on the thick black line and follow it.After tracing both sides of each edge, one will eventually return to the starting point.So, the contribution of this ribbon graph for the volume of the moduli space of double torus with one boundary is: After the preparations to establish the connection between the graph geometries and ordinary hyperbolic geometry, we will proceed to perform the integrals da 2 ds 2 da 1 ds 1 in equation (4.2).According to equations (4.1) and (4.31), the y i 's can be represented as functions of As shown in figure 9, there are four distinct closed geodesics, where none of them intersect with ℓ. Figure 14 shows geodesics a 1 and a 2 , which do not intersect each other, nor do they intersect the ℓ geodesic depicted in figure 12.By cutting the ribbon graph along these curves, one can obtain figure 11.It is evident that the a 1 geodesic goes once through the y 1 , y 2 , y 4 , and y 6 edges, while it passes through the y 3 edge twice, leading to: The identification of the a 1 geodesics and a 2 geodesics in a five-holed sphere can be done up to twists s 1 and s 2 , respectively (see figure 15).One can specify twist s 1 (s 2 ) by stating that the A 1 (A 2 ) point is identified with the B 1 (B 2 ) point, where s 1 (s 2 ) represents the distance between the B 1 (B 2 ) point and the "mirror image" of the A 1 (A 2 ) point, denoted as A ′ 1 (A ′ 2 ).As shown in figure 14, the a 1 (a 2 ) geodesic passes through the very thin y 3 (y 7 ) edge twice.In this edge, the a 1 (a 2 ) geodesic does not intersect itself but closely follows a nearly identical path.By examining figure 15, it becomes evident that within this retraced section of the a 1 (a 2 ) geodesic, a specific point, , is becoming very close to another point, B 1 (B 2 ), located at a distance of s 1 (s 2 ) along the a 1 (a 2 ) geodesic.So, to determine s 1 (s 2 ) in terms of the y i 's, the distance along the a 1 (a 2 ) geodesic at which pairs of points are brought close must be found.The two indicated points on the a 1 (a 2 ) geodesic are close in the bulk, but to travel from one to the other along the a 1 (a 2 ) geodesic, the indicated path must be followed, which has a length of y 1 + y 3 + y 5 (y 4 + y 5 + y 7 + y 9 ).So we have: Figure 14: The blue and green curves correspond to a 1 and a 2 geodesics, respectively.A ′ 1 (A ′ 2 ) and B 1 (B 2 ) represent typical points that are far apart along the a 1 (a 2 ) geodesic but are close within the bulk geometry.The distance they are situated along the a 1 (a 2 ) characterizes the twist s 1 (s 2 ).
is the mirror image of A 1 (A 2 ) point and it is very close to it.The twist and In addition to a 1 and a 2 geodesics, there are two more simple closed curves, b 1 and b 2 .These curves do not intersect with each other as well as ℓ, a 1 and a 2 geodesics.However, certain segments of these curves overlap with segments of the a 1 and a 2 geodesics.According to figures 16 and 17, the lengths of these curves are: By applying the same procedure used to determine the twists of a 1 and a 2 geodesics, we can determine the twists of these curves as follows:  Equation (4.30) along with eight equations from (4.32) to (4.39) can be combined to obtain: Now, we would like to enforce the condition that ℓ geodesic should be free of shortcuts.This implies that on the trivalent ribbon geometry, segments of the ℓ geodesic should not overlap.So, according to figure 12 we require: and From relations (4.41) and (4.42) one can write: Assuming the replacement of the |x| functions in the strip approximation with their corresponding 2 log cosh(x/2) functions, and subsequently employing (A.3), along with the delta functions in (4.12) and (4.20), we have: Given that all y i 's are greater than 0, we can write the following inequality: which leads to: Furthermore, it can be observed that: and using the delta function, setting ℓ = a 1 + a 2 − 2λ − ℓ t , we have: By integrating the five-holed sphere answer (4.23), over the moduli space restricted by relations (4.44), (4.46) and (4.48) as follows: expression (3.23) is derived.

Probability distributions in higher genus
In this section, the previous calculation will be expanded to include the emission of n baby universes.The volume V g,n+1 (b, a) in (2.8) is a symmetric polynomial function in b 2 , a 2 1 , a 2 2 , . . ., a 2 n of degree 3g − 3 + n + 1, and can be written as [39,43]: Simplifying by setting g = 0, we approximate relation (5.1) for b ≫ 1 as: where The equation: ) represents the probability of emitting n baby universes (n ≥ 2) and obtaining a wormhole with length ℓ.This expression can be derived using the same steps that were taken to obtain equation (3.9).In this case, equation (3.3) needs to be modified by substituting (5.2), and in equation (3.8), k should be replaced by "2n − 3".It is worth mentioning that, dads = n j=1 da j n j=1 ds j .The integration over ω in equation (5.4) yields the following result: which is similar to equation (3.15).

Firewall-free geometries
The probability of finding a wormhole with a length of ℓ in a firewall-free geometry, after the emission of more than two baby universes is: Using the thin strip approximation (b 1 +b 2 )/2 has been replaced with n j=1 a j −2λ, as illustrated in figure 18.The λ corresponds to the region between all baby universes.In this geometry, similar to the geometry described in subsection (3.1), the no shortcut criterion does not impose any restrictions on the moduli space.Therefore, the domain of each twist is 0 < s j < a j .Due to the presence of the delta function, we can assume that the domain of each baby universe is within the range of 0 < a j < ∞.By expressing F 2 (a) in the following form: where: the expression (5.6) after integrating over all twists becomes: where P n (α,β) (ℓ) is defined as: (5.10) It is convenient to multiply (5.10) by a factor "1" written using the definition of gamma function, Figure 18: The geometry corresponding to ℓ = ℓ t − b, approximated with a ribbon graph, features a growing wormhole that emits n baby universes within the black hole regions.
and perform a change of variables to q k = ζa k : (5.12) By integrating over ζ, the delta function is eliminated, resulting in the following expression: (5.13) Now, the definition of the gamma function can be used to obtain the following expression: Figure 19: The geometry, represented by ℓ = b − ℓ t and approximated with a ribbon graph, features a diminishing wormhole within the white hole regions, leading to the emergence of a firewall.This geometric configuration includes the emission of n baby universes, with some of them sharing a common section with the wormhole.
criterion imposes restrictions on the moduli space.As discussed in subsection 3.2, the existence of overlapping portions between the baby universes and wormholes is crucial for the application of the no shortcut criterion.Let us assume that each baby universe has a parameter γ i representing its common share with the wormhole, as indicated in figure 19.The domain of γ i is (0, ℓ) with the condition: n j=1 γ j = ℓ.After imposing the no shortcut criterion, the twist s i of the i-th baby universe is constrained within the range γ i + λ i + λ i+1 < s i < a i − γ i , where λ i is a common region between baby universe a i and a i+1 .So, one can expect that: (5.22) Therefore, the relation (5.21) after imposing constraints (19) takes the following form: (5.23) After the change of variables a k → ℓ t q k and s k → ℓ t b k one can express the relation (5.23) as following: where K(α,β) (λ/ℓ t , ℓ/ℓ t ) is a polynomial of λ/ℓ t and ℓ/ℓ t . 5If the highest power of ℓ t is chosen in equation (5.20), i.e. α, β = n − 2, the probability of encountering a firewall with a wormhole of length ℓ is given by: Here Kn (ℓ/ℓ t ) is: (5.26) Notice the no shortcut condition imposes no restrictions on λ and in the above relation, we assumed that λ varies in the region 0 < λ < ℓ t .After integrating over ℓ, the normalized probability of encountering a firewall after emitting n baby universes is: and the Jn is a numerical coefficient.

"2n + 1"-holed sphere
Now, we will explore the additional terms that arise from the small b region.Let us consider the geometry, which consists of a trumpet with a closed geodesic of length b and "2n + 1"-holed sphere with n i=1 a i > ℓ t , as shown in figure 20.The length probability distribution is described by the following expression: (5.28) In the above relation, V 0,n+1 (b 1 , a) and V 0,n+1 (b 2 , a) denote the volumes of moduli spaces corresponding to "n + 1"-holed spheres attached to geodesics b 1 and b 2 , respectively.In the expression (5.28), analogous to the case of the five-holed sphere given by (4.(5.2) to get: which is also negative.By following the same procedure described in section 4, the expression (5.28) leads to: this corresponds to the delta function in (5.5), which characterizes the firewall geometry.

Discussion
An old black hole may emit baby universes and undergo a tunneling process, transforming into a white hole with a firewall at its horizon.The probability of this transition, which involves the emission of a single baby universe corresponding to a genus one surface, was computed in [32].This probability increases with time and reaches order one when the age of the black hole approaches e S(E) .Furthermore, it was demonstrated that the probability of having a smooth horizon is equal to the probability of encountering a firewall.This observation was not apparent during intermediate stages since the computation of P 1,smooth did not involve any mapping class group issues.In this note, an attempt was made to extend these calculations to higher genus scenarios.The results show that the probability distributions of finding firewall and smooth horizon, after emitting n baby universes, exhibit a similar behavior: up to a numerical coefficient.In our calculations, we employed the thin strip approximation and, using the "no shortcut" condition as defined in [32], we constrained the moduli space of higher genus geometries.However, it appears that this condition alone does not determine all parameters, requiring additional physical constraints.By precisely defining the domain of the moduli space, particularly by specifying constraints on the region between baby universes denoted as "λ", it may be feasible to determine numerical coefficients more accurately.For genus two, the requirement that P 2,firewall (t) = P 2,smooth (t) implies that 0 < λ < ℓ t in smooth geometry and 0 < λ < ℓ t /2 in firewall geometry.Strictly speaking, the probability of having a firewall-free geometry with a wormhole length of ℓ is given by: The probability of finding a firewall-free geometry is: when multiplied by the normalization factor e −S(E) , as defined in equation (1.9), one can obtain: The probability of encountering a firewall with a wormhole of length ℓ is: and the probability of finding a firewall geometry is: After normalization we obtain: The calculation of the probability of tunneling is reminiscent of the spectral form factor, |⟨β + it ′ |β + it⟩| 2 , which is an important tool for the geometrical interpretation of the discrete energy spectrum of black holes.After some initial non-universal decay, the spectral form factor has a linearly growing ramp for a long amount of time 2πe S(E) and then it flattens out into a plateau.This late time behavior of the spectral form factor is universal and depends on density of states, ρ(E) ≈ e S(E) , and symmetry classification.Cylindrical topology gives the linear ramp and there is no perturbative correction to it and quadratic answer for the probability of finding firewall geometry for genus one arises from its integration.Plateau is non-perturbation in genus expansion, although there have been some attempts to find a perturbative approach to the latetime plateau [44,45,46], see also [47].So, an intriguing question would be to find a convergent sum over the genera of firewall geometries.
From the second method of computing the probability of encountering a firewall geometry for genus two, similar to the case with firewall geometry of genus one, an additional negative term at small b was observed.These terms are expected to contribute something proportional to δ (ℓ − ℓ t ), with the coefficient being determined by the requirement that the total probability equals one ( ℓt 0 P (ℓ)dℓ = 1).A detailed analysis of the negative terms was not carried out due to the belief that these terms effectively subtract a probability mass equivalent to that of the positive terms from the disk answer.To provide a specific formula, after combining the disk answer (P disk (ℓ) = δ(ℓ − ℓ t )) with the contributions from the firewall and smooth geometries, the probability distribution for the physical length should take the following form: The first term signifies the contribution of the disk, including possible negative contributions from the handle disk and double handle disk.The second term includes P 1,firewall and an identical term for P 1,smooth , along with P 2,firewall and P 2,smooth , i.e. (6.3) and (6.5).Using the relation (6.8), the complexity of the dual state can be computed as the expectation value of the wormhole length.In [48,49], complexity was calculated by determining the expectation value of all possible geodesics without applying the no shortcut criterion.It was observed that, following a period of linear growth at early times, the complexity saturates at late times.If linear growth is considered an essential feature of complexity, it suggests that the no shortcut criterion may be disregarded for computing complexity [50].It is worth noting that the volume of wormholes in all classical geometries does not exhibit linear growth forever.For instance, in multi-black hole geometries [51], it does not follow a linear growth pattern and also saturates at late times [52].It would be interesting to examine these geometries and study the tunneling process.Now we calculate the length of γ 2 .Referring to figure 22, we choose the pair of pants on the five-holed sphere with geodesic boundaries (b, b 1 , b 2 ) that contains the geodesic γ 2 .After cutting this pair of pants so that each geodesic boundary is divided in half, two hexagons are obtained, with one of them represented in figure 23.The geodesic with a length of γ 2 /2 bisects this hexagon, creating two pentagons.In the upper and lower pentagons, the lengths of a side are denoted by α and β respectively, and their sum equals b/2.By a standard hyperbolic trigonometric formula for pentagons (for example, consider the references [53,54]) we have the following relations: and by combining these equations, we arrive at equation (A.1).

B Integrals with step functions
To perform integrals in (5.23), let us consider the following integral:

Figure 1 :
Figure1: Penrose diagram of eternal AdS black/white hole.In region I, located behind the black hole horizon, the spacetime exhibits expansion.As a result, the black hole horizon is transparent.On the other hand, in region II, behind the white hole horizon, the spacetime undergoes contraction, and the horizon is opaque.

Figure 2 :
Figure 2: A Riemann surface with genus g = 0 and an asymptotically AdS boundary β, along with n geodesic boundaries or baby universes (a 1 , a 2 , . . ., a n ).The loop b, minimized within its homology class, is homologous to β.This geometric structure can be reconstructed by joining a trumpet, marked in orange, and a multi-holed sphere, shown in gray, along the geodesic b.

Figure 3 :
Figure 3: The amplitude ⟨ℓ, a|ℓ 1 ⟩ is illustrated, representing a wormhole with length ℓ 1 emits n baby universes and transforms into another wormhole with length ℓ.The colored region is referred to as the tunneling amplitude region.

1 bFigure 4 :
Figure 4: The configuration with Z 2 symmetry represents the length of the spatial slice connecting two boundary points, accompanied by two baby universes a 1 and a 2 .The closed geodesic b is homologous to the asymptotic boundary.The colored region indicates the tunneling amplitude region.
the delta functions in the expression ∆ (b 1 , b 2 , ℓ, ℓ t ) establish relations between b 1 , b 2 , ℓ, and ℓ t , its terms do not affect the amplitude.The first term in the expression ∆ (b 1 , b 2 , ℓ, ℓ t ) is zero due to the conditions that b 1 , b 2 , and ℓ t are greater than zero, and ℓ is greater than or equal to zero.Similarly, the other terms in the expression ∆ (b 1 , b 2 , ℓ, ℓ t ) do not contribute because they would lead to non-physical consequences b 1 = b 2 ± 2ℓ ± 2ℓ t , while it is expected that b 1 and b 2 are approximately equal (b 1 ≈ b 2 := b), as indicated in figure4the probability amplitude geometry has Z 2 symmetry.In other words, after integrating over E with smooth window functions, it becomes evident that b 1 = b 2 .

Figure 5 :
Figure 5: The geometry that corresponds to ℓ = ℓ t − b depicts a growing wormhole within the black hole regions, resulting in a transparent horizon.The parameter λ represents the length of the region between two baby universes.

Figure 7 :
Figure 7: The left and right geometries can be related by an action of the mapping class group.Points marked with × are identified.The geodesic ℓ ′ , depicted as a thick black line, might be a more suitable candidate for the correct spatial slice, as the red line geodesic ℓ is essentially a repetition of ℓ ′ .In other words, left picture shows that points along the geodesic ℓ that appear distant are, in fact, close to each other within the entire geometry or bulk.

Figure 8 :
Figure 8: In the geometry associated with ℓ = ℓ t + b, any value of the twists s 1 and s 2 induces a nonlocal identification along the ℓ geodesic.The ℓ geodesic closely tracks either the ℓ 1 or ℓ 2 geodesics, deviating slightly around the b 1 or b 2 cycles.

Figure 9 :
Figure 9: A trumpet attached to a double handle disk, which has a single geodesic boundary with a length of b.The ℓ is the spatial slice connecting the two boundary points.

. 4 ) 1 Figure 10 :
Figure 10: A five-holed sphere attached to a trumpet.The endpoints of the geodesic ℓ are at the boundary of the trumpet region.

2 bFigure 11 : 1 Figure 12 :
Figure 11: The Ribbon graph of a five-holed.The red line depicts the portion of the wormhole located within the five holed sphere.

3 Figure 13 :
Figure 13: The graph corresponding to the ribbon graph in figure 11.This graph can be embedded within the ribbon graph, or, by thickening the graph, one can obtain the associated ribbon graph.

3 Figure 17 :
Figure 17: The geodesic b 2 is depicted by the purple curve.

Figure 20 :
Figure 20: A trumpet glued to a 2n + 1-holed sphere.The endpoints of the geodesic ℓ are at the boundary of the trumpet region.

1 Figure 22 :
Figure 22: In five holed sphere, the γ 2 is a portion of geodesic γ which is perpendicular to b.

2 Figure 23 :
Figure 23: Two hexagons can be obtained by symmetrically cutting a pair of pants with geodesic boundaries (b, b 1 , b 2 ).