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Interior design of a two-dimensional semiclassic black hole

We look into the inner structure of a two-dimensional dilatonic evaporating black hole. We establish and employ the homogenous approximation for the black-hole interior. The field equations admit two types of singularities, and their local asymptotic structure is investigated. One of these singularities is found to develop, as a spacelike singularity, inside the black hole. We then study the internal structure of the evaporating black hole from the horizon to the singularity.


INTRODUCTION
The discovery of Hawking radiation and the blackhole (BH) evaporation phenomenon [1] raised several outstanding open questions and intriguing puzzles. One such problem which attracted a lot of attention is the information puzzle: Simple thought experiments suggest that in the process of black-hole formation and evaporation, a pure initial quantum state will evolve into a mixed state, and consequently part of the information encoded in the initial state will get lost.
Apparently this problem of information loss is intimately related to another conceptual problem in blackhole physics: The formation of a spacetime singularity inside the BH. Pictorially speaking, a portion of the initial information propagates to the singularity and disappears there. It is widely anticipated, however, that the formation of spacetime singularities is a mere artifact of the classical (and semiclassical) theory, but Quantum Gravity will eventually resolve the black-hole singularities, and presumably solve the information puzzle as well.
In 1992, Callan, Giddings, Harvey, and Strominger (CGHS) [2] proposed a simplified framework for investigating black-hole formation and evaporation. They introduced a two-dimensional toy-model with gravity and matter fields coupled to a dilaton scalar field. At the classical level this model admits a one-parameter family of static black-hole solutions (parameterized by their mass). When semiclassical corrections are added, the two-dimensional black hole emits thermal radiation and consequently evaporates. Based on the trace anomaly, CGHS provided simple explicit expressions for the semiclassical contribution to the energy-momentum tensor. Consequence the semiclassical dynamics may be formulated as a closed (constrained) system of second-order partial differential equations. It thus provides a simple framework for exploring various aspects of black-hole evaporation, and particularly the information puzzle [3].
Originally it was hoped [2] that the CGHS evaporating BH will be free of any singularities. However, Russo, Susskind and Thorlacius [4] soon found that a singularity inevitably develops inside the CGHS BH, at a certain value of the dilaton filed. Thus, the CGHS formalism does not resolve the information puzzle at the semiclassical level. One may still hope, however, that when the model is fully quantized the singularity (and the information puzzle) will be resolved. This approach was pursued by Ashtekar, Taveras, and Varadarajan [5], who formulated the quantum-field analog of the two-dimensional CGHS model. In this set-up, the original CGHS variables are replaced by quantum operators. Spacetime evolution is then described by a system of operator partial differential equations. The exact solution to these operator equations is not known. Nevertheless, some approximate solutions were constructed [5], illuminating certain aspects of the problem of black-hole evaporation, and suggesting that the singularity will indeed be resolved in the quantized theory.
The above discussion signifies the inevitable spacetime singularity as a key feature of the CGHS semiclassical BH spacetime. However, the detailed structure and properties of this singularity have not been investigated so far to the best of our knowledge. The main purpose of this manuscript is to present such a detailed analysis of the structure of that singularity. The motivation for this investigation is obvious from the discussion above: First, since this singularity plays such a crucial role in the information-loss puzzle, it will be useful to better understand its properties. For example, one would like to know how strong this singularity is, and what are the prospects for extending semiclassical spacetime beyond it. Second, understanding the asymptotic behavior of the various fields at the singularity may provide a useful starting point for exploring how quantum treatments (like that of Ref. [5] for example) may resolve the singularity.
Although the main objective of this paper is the asymptotic behavior near the singularity, we also analyze here the internal structure of the evaporating semiclassical BH in the entire range from the horizon to the singularity. We do this by constructing approximate solutions in various domains of the BH interior, and then matching these solutions at their respective overlap regions. Understanding the entire BH internal structure is interesting on its own right, but is also important for full determination of the singularity structure: When the latter is derived by a purely local analysis, one obtains a family of local asymptotic solutions which depend on certain free parameters (or free functions, depending on the context). The value of these parameters in an actual BH solution needs to be determined by matching to initial conditions (e.g. at the horizon). The full internal solution, from the horizon to the singularity, is required for obtaining the right values of these parameters.
As long as the evaporating black hole is macroscopic, the semiclassical effects are very weak in a local dynamical sense. This allows one to employ the adiabatic approximation. Namely, any local region of spacetime may be well approximated by a certain classical CGHS BH solution (with a certain mass parameter). The classical BH solutions are static outside the horizon but homogeneous inside it, implying that the interior of a macroscopic evaporating BH is (locally) approximately homogeneous. This homogeneous approximation is a key element in our investigation. It greatly simplifies the analysis, as the field equations now reduce to ordinary differential equations.
In Sec. II we present the CGHS action and field equations, and re-formulate them in new variables. Sec. III is devoted to the homogenous approximation. We explore its main properties and its domain of validity. In Sec. IV we analyze the singularities of the homogenous field equations, and obtain the local asymptotic behavior of the solution near these singularities. In Sec. V we construct an approximate solution for the fields inside the evaporating BH, in the entire range from the horizon to the spacelike singularity.
In Sec. VI we present simple effective Lagrangian and Hamiltonian which yields the local asymptotic dynamics near the spacelike singularity. The construction of such effective Lagrangian and Hamiltonian is motivated by the idea, that perhaps it will be possible to employ simple quantum-mechanical considerations to resolve the singularity (and thereby to explore the possible extension of spacetime beyond it). Although such a quantummechanical treatment is beyond the scope of the present paper, we nevertheless take here the preparatory step (carried out entirely within the semiclassical framework) of constructing the effective near-singularity Lagrangian and Hamiltonian. Finally, in the last section we summarize our main results and discuss their significance.

THE MODEL AND FIELD EQUATIONS
We begin by presenting the two-dimensional semiclassical CGHS model [2]. Beside the dilaton φ, this model contains a large number N >> 1 of identical scalar matter fields f i , and a cosmological constant λ 2 . We express the metric in double-null coordinates u, v (the "conformal gauge"), namely ds 2 = −e 2ρ dudv. The action then . (1) The last term in the action expresses the semiclassical effects. The Einstein equations take the form: encode the information about the initial quantum state, which in turn determines the semiclassical fluxes. The dilaton and matter equations are: Throughout this paper we set λ = 1. This choice is equivalent to the change of variable ρ ′ = ρ + ln(λ), which does not affect the field equations otherwise. We also set f i = 0, as we are dealing here with the evaporation rather than formation of the BH.
Following Ref. [6], we define new variables: R ≡ e −2φ , S ≡ 2(ρ − φ). A straightforward substitution of the new variables in Eqs. (2,5) yields: is to be substituted, and K ≡ N/12 expresses the magnitude of the quantum effects. The constraint equations become: where hereafter w stands for either u or v. The semiclassical energy fluxesT ww along both null directions are given by:T It is useful to re-express the system of evolution equations (6) in its standard form, in which R, uv and S, uv are explicitly given in terms of lower-order derivatives: This form makes it obvious that the evolution equations become singular when R = K. This singularity, which in the original variables takes place at φ = − 1 2 ln( N 12 ), was already noticed preivously [4]. Below we shall explore in some detail the homogeneous variant of this singularity.
The semi-classical equations reduce to the classical ones by setting K = 0: S, uv = 0, and the constraint equations: This set of equations admits a one-parameter family of solutions (up to gauge transformations), which is the dilatonic two-dimensional analog of the standard Schwarzschild solution. Each member of this family describes a black-hole space-time, which (like its standard four-dimensional counterpart) is static outside the BH and homogeneous inside it. Throughout this paper we shall simply refer to this class of solutions as the "Schwarzschild solution" (despite a slight abuse of standard terminology). We shall focus on the internal part of the BH. In Eddington-like double-null coordinates the internal solution takes the form where M is the (dilatonic, 2-dimensional) Schwarzschild mass parameter.

HOMOGENOUS SET-UP Justification and domain of validity
We shall now restrict our attention to the homogeneous solutions of the semiclassical field equations inside the evaporating BH, namely, solutions which only depend on t, where we define We first need to discuss the justification for this homogeneous approximation and its domain of validity. The classical interior solution (11) is obviously homogeneous as it depends on v + u solely. In an evaporating BH, however, the semiclassical effects spoil the exact homogeneity, as manifested by the drift in the BH mass. Yet, as long as the BH is macroscopic (M >> K), this drift is very slow. Indeed it undoubtedly has a dramatic effect over the long evaporation time-scale-the BH eventually disappears (or at least becomes microscopic) after all. Yet, from a local point of view this drift is of negligibly small rate. Thus, over a typical dynamical time/length scale (say, ∆v, ∆u of order ∼ 10), the change in M is δM << M and may be neglected, which naturally leads to the homogeneous approximation. The local, relative magnitude of the semiclassical terms is of order K/R, as can be seen in the evolution equations (9). As long as the BH is macroscopic (M >> K), and as long as we are dealing with the portion R >> K of the BH space-time, we may apply the adiabatic point of view to the evaporation process: Namely, in each region of space-time the solution is approximately Schwarzschild, with a local effective mass parameter M which slowly drifts in v and/or u. We shall refer to this slowly varying function M (v, u) as the effective mass of the evaporating BH. (In fact, as long as the interior of the BH is considered, the effective mass depends only on v. [11]) The adiabatic approximation, wherever applicable, automatically implies approximate homogeneity inside the BH-simply because the Schwarzschild interior solution is homogeneous.
It should be emphasized, however, that the adiabatic approximation does not apply in the small-R region near the singularity, where R becomes comparable to K. In this region there are strong local semiclassical effects, and the local solution is very different from Schwarzschild (as demonstrated in the next section). We argue, however, that the homogeneous approximation is still valid in this small-R domain. Generally speaking, this follows from causality: Since the small-R region is located at the causal future of the moderate-R interior region, it simply inherits the approximate homogeneity of the latter.
To be more specific, let us pick a point inside the evaporating BH, in a region where the effective mass is still macroscopic, M (v) ≡ M 0 >> K. This point is picked at a certain R value, R = R 0 , which is >> K -say R 0 = M 0 /2 (recall that the horizon is located at R ≈ M ). Let us denote the coordinates of this point by (v 0 , u 0 ), and let t 0 ≡ v 0 + u 0 , x 0 ≡ v 0 − u 0 . Let Σ denote an initial surface which is the line t = t 0 restricted to the range x 0 ≤ x ≤ x 0 + ∆x, with ∆x taken to be ∼ 10 (say). This construction is illustrated in Fig.1. We shall now consider two cases, namely two slightly different solutions: In case (i), we specify on Σ initial data which exactly correspond to the classical Schwarzschild solution with mass M (v 0 ) ≡ M 0 . Since these data are independent of x by construction, the evolving solution will be precisely homogeneous, throughout the domain of dependence D + (Σ). Provided that ∆x is taken to be large enough, D + (Σ) will reach the small-R space-like singularity (more precisely, the R = K singularity which we explore in the next section). To this end ∆x only need to exceed the span of t from t 0 to the singularity (of the homogeneous solution). This span is of order 1 (for instance, for R 0 = M 0 /2 this t-span is ln(1/2) in the classical solution, and approximately the same number in the semiclassical solution). Hence taking ∆x ∼ 10 (say) guaranties that D + (Σ) will include a portion of the small-R singularity (as illustrated in Fig.1). In case (ii), we specify on Σ the initial data which correspond to the actual evaporating BH solution. Since both M 0 and R 0 are >> K, the adiabatic approximation applies, and the solution in the neighborhood of (v 0 , u 0 ) is approximately Schwarzschild. Therefore, we may regard the actual initial data (ii) as the homogeneous, Schwarzschild data of case (i) plus a small perturbation. This initial perturbation will evolve into a small perturbation throughout the interior of D + (Σ). The relative magnitude of this perturbation will be controlled by the small parameter K/M 0 , and for sufficiently small value of this parameter we may neglect the perturbation at the leading order.
The smallness of the perturbations in the interior of D + (Σ) is then guaranteed by standard stability theorems for nonlinear hyperbolic systems. One may be concerned, however, about the effect of the perturbation in the immediate neighborhood of the small-R singularity, and in particular on the very structure of the latter. A close examination, which is beyond the scope of the present paper, reveals that this singularity is locally stable to inhomogeneities. (That is, an inhomogeneous variant of the R = K singularity exists [7], and this inhomogeneous singularity is generic, in the sense that it depends on four arbitrary functions of x [12]. ) We conclude that in the macroscopic-mass domain (M 0 >> K) the homogeneous solution provides a good approximation not only in the domain R >> K, but also up to the small-R singularity. This was also verified numerically by directly integrating the field equations (6) with initial data corresponding to a BH which forms by a collapsing null shell and subsequently evaporates [8]. In particular it was numerically verified that the structure of the evolving spacelike singularity is well described by the homogeneous R = K singularity (described in the next section).
In the above construction we considered for concreteness an initial hypersurface Σ located at R = M 0 /2. We could have started instead at any other R = R 0 initial value which satisfied K << R 0 < M 0 . The evolving homogeneous solution depends very weakly on R 0 , because of the local smallness of the semiclassical corrections at R >> K. For concreteness, and in order to avoid the arbitrariness associated with the extra parameter R 0 , in the analysis below we shall consider the homogeneous solution obtained at the limit R 0 → M 0 . The solution obtained in this way may be thought of as the (horizonregular) semiclassical counterpart of the homogeneous interior Schwarzschild solution. [13] It carries (besides K) a single parameter M 0 , representing the BH's remaining mass at the epoch of interest. In the rest of the paper we shall simply refer to this parameter as M for brevity.

Homogenous field equations
In the homogeneous framework the evolution equations (9) becomeR where an over-dot denotes differentiation with respect to t. The constraint equation (7) now reads whereT ≡T vv =T uu is given bŷ andẑ ≡ẑ v =ẑ u . Note that the equalitiesT vv =T uu andẑ v =ẑ u are direct consequences of the homogeneous set-up. The latter equality also implies thatẑ must be a constant. (By homogeneityẑ could at most depend on t, but then dẑ/dt =ẑ ,v =ẑ u,v = 0.) The Schwarzschild solution inside the BH may be expressed in the explicitly homogeneous form In particular it satisfies the relationṡ which will be useful in the analysis below. Note also the classical homogeneous constraint equationR −ṘṠ = 0.

SINGULARITIES
The semiclassical equations are singular at the points R = 0 and R = K, as can be seen from Eqs. (12,13). In this section we will look at the asymptotic behavior near these two special R values.
Since we set the initial conditions for the homogeneous solution at R >> K, the first singularity to be encountered is R = K. As it turns out, this singularity is characterized by the divergence ofṘ andṠ, while R and S are finite.
We therefore assume that in the right-hand side of Eqs. (12,13) we can neglect the terms proportional to e S compared to those ∝Ṙ 2 , obtaining at leading order Eq. (18) constitutes a closed equation for R, which is easily solved: Then Eq. (19) is solved to yield The solution depends on four free parameters B, B 2 , B 3 , t 0 as required, hence this is a locally-generic asymptotic solution. We see that the variables R, S are continuous at R = K, butṘ diverges as |t − t 0 | −1/3 (and the same forṠ). This behavior, which is numerically demonstrated in Fig. 2, justifies our preassumption that e S is indeed negligible compare toṘ 2 .
In the second singular point R = 0, we expand Eqs. (12,13) near R = 0 under the same assumption as before, namely that the terms proportional to e S may be ignored (this requires that e S <<Ṙ 2 /R, which will again be justified a posteriori). We obtain These equations are easily solved: and where C 1 , C 2 , C 3 , t 0 are four arbitrary parameters. We find that the present situation differs from the R = K singularity, because nowṘ (like R) is finite and S (likeṠ) diverges. Nevertheless, our preassumption e S <<Ṙ 2 /R is still justified, because e S vanishes as (t − t 0 ) 2 whereaṡ R 2 /R approaches a constant. It is quite surprising to find that although the evolution equations are singular at R = 0, the evolving solution for both R and e S is regular there. In particular, the metric function ρ = (S − ln R)/2 is finite at R = 0. Note, however, that the dilaton φ diverges there.
Our homogeneous solution starts at R = R 0 >> K, and R decreases until the R = K singularity is reached. The divergence ofṘ andṠ at R = K results in an inability to provide a unique prediction for the evolution of the fields beyond that point. Therefore the R = 0 singularity and its neighborhood are beyond our domain of prediction. In the rest of this paper we shall concentrate on the R = K singularity, as well as on the global behavior in the domain K < R < M .

CONSTRUCTING THE INTERIOR HOMOGENEOUS SOLUTION
So far, we dealt with the local structure of the fields near the singularity. Our ultimate goal, however, is to understand the global behavior of the fields inside the BH. To this end, we need to follow the evolution of R(t) and S(t) from the initial hypersurface (say at R ≈ M ) up to the singularity at R = K. We shall do this by designing a couple of analytical approximations, the union of which cover the entire domain K < R < M . We shall also augment the analytic approximations by direct numerical integration of the homogeneous evolution equations (12,13). We shall start by specifying the initial conditions, which are required for both the numerical and analytical solutions.

Setting the initial conditions
The required initial conditions for the homogeneous solution are the four functions R,Ṙ, S,Ṡ, all set at a certain initial moment t 0 . The basic strategy of setting the initial data follows from the discussion in section . Suppose that we want to explore the interior of the evaporating BH at a stage (i.e. v value) where its remaining mass is M (with M >> K). Then we take the initial conditions to be those corresponding to the classical internal Schwarzschild solution with the same mass parameter M , at a hypersurface R = const ≡ R 0 , at the limit R 0 → M . From Eq. (16) this amounts to setting R = M ,Ṙ = 0, S = 1, and S = t 0 → −∞. In actual numerical implementations we pick a large negative value of t 0 (say, t 0 = −30) and set S = t 0 . Note that in such a horizonlimit setup of initial conditions the choiceṘ = 0 is crucial for regularity. Any other choice of initialṘ = 0 at t → −∞ will lead to a solution which lacks a regular horizon, and is hence inappropriate for approximating the spacetime of an evaporating BH. [14] For this initial-value setup, the constraint equation (14) implies thatT vanishes at the horizon limit t → −∞. Eq. (15) then yieldsẑ = 1/4. [15] Approximate solutions in the different regimes We need to evolve the fields R, S from the initial hypersurface, which we set to be at R → M , up to the singularity at R = K (recall we assume M >> K). We notice three important domains in this overall range : (i) the macroscopic domain R >> K. (ii) very close to the R = K singularity, and (iii) the domain K < R << M [Note that (iii) is an extension of (ii), and it overlaps with (i).] We shall now discuss the approximate solution in each of these domains.

The macroscopic (or classical) domain
It is easy to see that in the domain R >> K the quantum contribution to the right-hand side of Eq. (12) is negligible compared to the classical contribution (K = 0). The same applies to Eq. (13). Therefore, in this domain the semiclassical solution may be well approximated by the classical solution (16). (For numerical verification see Fig. 4

below.)
Near the singularity This domain is characterized by the inequality R − K << K. The approximate solution in this region was constructed in the previous section, Eqs. (20,21). This solution is characterized by four parameters, of which the most significant one is B.
The small-R approximation (R << M ) We seek an intermediate approximate solution, in the domain where R is << M , yet it is not quite close to K (say, R ∼ 3K). Formulating the desired approximation, that will apply both near the singularity and in the portion R << M of the classical domain, requires understanding the common basis of these two approximations. Consider the expression given in the right-hand side of Eq. (12) forR. In the near-singularity approximation the first term in the right-hand side (which is ∝ e S ) is negligible compared to the second one sinceṘ diverges, whereas S(t) is bounded as seen in Eq. (21). One observes, however, that this term remains unimportant even in the portion R << M of the classical domain. In this regime, bothṘ and −e S are given approximately by R − M ∼ = −M , see Eq. (17), hence e S /Ṙ 2 scales as 1/M . Therefore, if we fix R and increase M , the first term in the right-hand side of Eq. (12) becomes negligible-just like in the near-singularity approximation. Omitting this term we obtain:R We shall refer to this approximation as the small-R approximation. Note that the near-singularity approximation may be obtained from the small-R approximation by substituting R ∼ = K, as may be seen by comparing Eqs.
(26) and (18). Solving Eq. 26 forṘ we obtaiṅ where A is a free parameter. One more simple integration yields an expression for t(r), which we do not need however.
In a particular evaporating-BH space-time, the parameter The small-R approximation was established above by justifying it in both edges of its domain of validity (namely at R ∼ = K and at K << R << M ). We still need to demonstrate its validity in the region in between, where R is of order a few times K. To this end we note from Eqs. (27,28), that for fixed R and K,Ṙ scales as M (just like e S ). Therefore, in the right-hand side of Eq. (12) the second term scales as M 2 , whereas the first term scales as M and can therefore be neglected at the large-M limit (M >> K). This justifies the small-R approximation a posteriori, in its entire domain K < R << M .
The validity of the small-R approximation, and its effectiveness compared to the near-singularity approximation, is demonstrated numerically in Fig. 3.

EFFECTIVE LAGRANGIAN AND HAMILTONIAN FOR THE NEAR-SINGULARITY REGION
The occurrence of a spacelike singularity at R = K makes it difficult (if not impossible) to explore the evolution of physics-and spacetime-beyond R = K by semiclassical methods. In Ref. [5] it was suggested that quantum evolution will still be regular even at the (would-be) semiclassical singularity. This motivates one to study the evolution of R and S in the neighborhood of the singularity in a quantum-mechanical framework. One of the simplest options would be to address the quantum problem at the "mini-superspace" level: Since at the semiclassical level the homogeneous solution provides a good approximation to the actual spacetime evolution (as long as M >> K), one may drop all spatial derivatives and analyze the simpler problem in which R and S depend on t solely. Furthermore, since the critical stage of evolution is the transition across the R = K singularity, it may be sufficient (at least as a first step) to analyze the problem at the leading order in R − K.
This problem of quantum evolution is certainly beyond the scope of the present paper. Nevertheless, we do take here the preparatory steps which are to be implemented already at the semiclassical level: Namely, the formulation of an effective Lagrangian (and subsequently an effective Hamiltonian) that describes the evolution of R(t) and S(t) near R = K. We therefore seek a Lagrangian L(R, S,Ṙ,Ṡ) that will recover the evolution equations (18,19) near R = K.
Since Eq. (18) constitutes a closed equation of motion for R(t), we start by constructing an effective Lagrangian L(R,Ṙ) for this equation. This turns out to be as one can easily verify by applying to it the Euler-Lagrange equation. The momentum conjugate to R is This yields the effective Hamiltonian for the R-field: [16]. Next we treat the combined system of R and S (still in the homogeneous framework and at the leading order in R − K). To this end we find it convenient to replace S by the new variable Z = R + KS. From Eqs. (18,19) it follows thatZ = 0. Hence Z(t) corresponds to a free one-dimensional motion, with L Z = 1 2Ż 2 , conjugate momentum P Z =Ż, and Hamiltonian H Z = 1 2 P 2 Z . The overall near-singularity Lagrangian and Hamiltonian are L = L R + L Z and H = H R + H Z .

DISCUSSION
In this manuscript we primarily discussed two closely related issues. The first is the structure and main properties of the spacelike singularity which forms inside a CGHS [2] evaporating BH. The second is the structure of the fields (spacetime and dilaton) in the entire range of the BH interior, from the horizon up to the singularity.
As a first step we established the homogenous approximation, a central tool which allowed us to analyze the above issues by means of ordinary differential equations. This approximation is valid as long as the BH mass is macroscopic.
We find that the field equations admit two singular points: at R = 0 and R = K. Using the homogenous approximation, we wrote down the asymptotic form of the field equations near each of these singularities [Eqs. (18,19) and (22,23)] and solved them exactly, obtaining the asymptotic solutions (20,21) and (24,25) at the two singularities. In the scenario of BH formation, R starts at large values and then decreases, so the R = K singularity will be reached first. It is therefore not clear whether the R = 0 singularity will ever form. Hence, in the rest of our analysis we concentrate on the R = K singularity.
Next we explored the evolution of the fields inside the BH from the horizon up to the singularity. By combining several approximations, we composed a global approximation forṘ as a function of R, given in Eq. (30). This expression can be further integrated in a straightforward manner to yield an expression for R(t) (though in a functionally-implicit form). It should also be possible to use this expression in order to construct an approximate global solution for S(t), but this requires some technical work which is beyond the scope of this manuscript [17].
The overall approximate expression (30) forṘ(R) allows us to determine the parameter B of Eq. (20), which was left as a free parameter by the local analysis near R = K. This parameter is found to depend on the BH's effective mass M , through Eq. (29).
Our analysis made an extensive use of the local approximate homogeneity which characterizes the R = K singularity. One should bare in mind, however, that the overall, large-scale, structure of the singularity is inhomogeneous after all. The parameter B, which characterizes the local singularity structure, slowly drifts along the R = K singularity line of Fig.1. This is explicitly seen by substituting M = M (v) in Eq. (29). (A similar drift may also apply to the other local parameters B 2 , B 3 , t 0 .) This qualitative picture holds as long as the remaining BH mass is macroscopic, namely M (v) >> K.
From the asymptotic solution (20,21), one immediately observes that R and S are continuous at R = K, yet their derivatives diverge there. The same applies to ρ. Thus, the metric tensor is continuous and non-singular, but its derivative diverges. The R = K singularity may be clas-sified as deformationally-weak [9], [10]: An extended body will only experience a finite tidal deformation as it approaches the singularity.
We briefly discuss here the possibility of extending semiclassical spacetime beyond the R = K singularity. From the discussion above it is obvious that a differentiable extended spacetime is ruled out, because differentiability already breaks down on the approach to R = K from the past. On the other hand, continuous extensions do exist. In fact, there is a two-parameter family of such continuous extensions (characterized by the values of B and B 2 beyond R = K). Thus, in our view, the problem of extending the semiclassical solution beyond the singularity is primarily a problem of ambiguity: A priori it is not clear which of the infinite possible extensions will be chosen by Nature (if any). We expect that a fully-quantized treatment (like [5] for example) should provide the answer to the extension problem: First, it should tell us if a semiclassical phase at all develops beyond the R = K singularity, and second, if such a semiclassical extension indeed develops, it should determine which semiclassical branch is selected.
Finally we constructed an effective Lagrangian and Hamiltonian near the R = K singularity. Such an effective Hamiltonian may be a useful input for future quantum-mechanical treatments of the behavior near the singularity.
This research was supported by the Israel Science Foundation (grant no. 1346/07)