Transition to light-like trajectories in thin shell dynamics

It was recently shown that a massive thin shell that is sandwiched between a flat interior and an exterior geometry given by the outgoing Vaidya metric becomes null in a finite proper time. We investigate this transition for a general spherically-symmetric metric outside the shell and find that it occurs generically. Once the shell is null its persistence on a null trajectory can be ensured by several mechanisms that we describe. Using the outgoing Vaidya metric as an example we show that if a dust shell acquires surface pressure on its transition to a null trajectory it can evade the Schwarzschild radius through its collapse. Alternatively, the pressureless collapse may continue if the exterior geometry acquires a more general form.


I. INTRODUCTION
Hypersurfaces of discontinuity are idealizations of narrow transitional regions between spacetime domains with different physical properties. The thin shell formalism [1][2][3] makes this idealization consistent by prescribing joining rules for the solutions of the Einstein equations on both sides of the hypersurface. These rules -junction conditions -determine dynamics of the shell. The resulting joined geometry is a solution of the Einstein equations with an additional distributional stress-energy tensor that is concentrated on the hypersurface.
Thin shell formalism plays a role in studies of cosmological phase transitions [4], impulsive gravitational waves [5], gravastars and other non-singular substitutes of black holes [6], traversable wormholes [7], and gravitationally-induced decoherence [8]. A massive thin shell separating a flat interior from a curved exterior spacetime region provides the simplest model of collapse. Classically the exterior spherical geometry is described by a Schwarzschild metric and the shell collapses into a black hole in finite proper time.
Such models has also been used in investigations of collapse-induced radiation [9,10] anticipated before formation of the event horizon. The basic idea is that the process of gravitational collapse excites fields in the spacetime, giving rise to asymptotically thermal radiation [9]. We shall refer to this as pre-Hawking radiation [11]. The consequences it might have for black hole formation and the information paradox have been a subject of interest in recent years [9,10,[12][13][14][15][16][17][18][19]. While it has been argued that such effects are too small to prevent the formation of an event horizon [10,19], others contend that such approximations are not reliable and that horizons may not form if pre-Hawking radiation is properly taken into account [13,16]. Indeed, it has been posited that this should be a generic feature of quantum gravity, with the black hole interior and accompanying singularity replaced with a genuine quantum geometry where the notion of event horizon ceases to be useful [20].
A number of researchers have argued [12][13][14][15] that there are two options for the evolution of a thin shell in a spacetime with pre-Hawking radiation. One possibility is that an event horizon never forms: either the shell does not cross its Schwarzschild radius r g before complete evaporation or a manifest breakdown of semiclassicial dynamics, such as for-mation of a Planck-scale remnant [22], violation of the adiabatic condition [23], or formation of some quantum geometry [20] occurs. The other alternative is that evaporation stops, forever preventing a distant observer at late times from detecting Hawking radiation. An outgoing Vaidya metric [24] is often used as an example of the exterior geometry of this process despite its known limitations [25].
This result is based on an implicit assumption that through their evolution a massive shell remains timelike and a massless shell remains null. However, it was recently demonstrated by Chen, Unruh, Wu and Yeom (CUWY) [26] that this assumption is unwarranted. Indeed, it was shown that if the exterior metric outside is rigorously Vaidya, a massive dust shell sheds its rest mass in finite proper time (while still outside its Schwarzschild radius), becoming null. It was further argued that if the evaporation continues the shell becomes superluminal. The choice is evidently between eventual tachyonic behaviour or switching off the radiation. In the latter case the subsequent development is classical and the shell crosses r g at a finite value of a suitable affine parameter [26].
Motivated by the goal of understanding the limits of validity of the semiclassical approximation in the context of gravitational collapse, we focus here on the dynamics of thin shells in spacetimes that model evaporation due to pre-Hawking radiation. The detailed description of the basic assumptions of this approximation and their application to thin shells is given in [14]. In practical terms, the standard curvature terms of the left hand side of the Einstein equations are equated to the expectation value of the renormalized stress-energy tensor. We assume its existence and consistency, but make no assumptions beyond that of spherical symmetry and certain regularity conditions that are described below. We leave aside the conceptual implications of radiation suppression and/or horizon avoidance [14,21]. The question of the origin of the pre-Hawking quanta is open, as it has been posited that this takes place at or near the surface of the collapsing body [13] or within a region ∆r ∼ r g outside the horizon [27].
The flat geometry inside is given in the outgoing Eddington-Finkelstein (EF) coordinates, where u − = t−r, and the most general spherically-symmetric geometry outside is extending the set-up of the previous studies [14,26]. In the following we omit the subscript " + " from the exterior quantities when it does not lead to confusion. Here where in the Schwarzschild geometry C = 2M = r g .
To represent evaporation we assume that ∂ u C ≤ 0, and the evaporation stops at some u * either at f ≡ 1 or with some finite value of the mass function C(r) > 0. The null coordinates u ± are distinct and the relation between them is determined by the first junction condition, while the radial coordinate is continuous across the surface [3].
We consider the transition of a massive evaporating shell to a null trajectory and investigate the circumstances under which such a shell continues along a null trajectory. We find this does not take place only if h(u, r) ≡ 0 and the absence of surface tension or pressure is imposed on the shell when it becomes null. Provided that the metric at the Schwarzchild radius is regular -specifically, the function h u, r g (u) is finite, we show by reworking the arguments of [15] that the subsequent null trajectory will never cross the ever-shrinking r g (u). In this sense the massive-to-null-to-superluminal case considered by CUWY is exceptional. We discuss the physical implications of the various possible cases in the concluding section of our paper, noting that which, if any, scenario is realized can be decided only by performing explicit analysis of the pre-Hawking radiation.
To simplify the notation in the following we use w := u − and refer the quantities on the shell Σ by capital letters, such as R := r |Σ , F := f (U, R). The jump of some quantity A across the shell is [A] := A| Σ+ − A| Σ− . All derivatives are explicitly indicated by subscripts, as in A R = ∂ R A(U, R). The total proper time derivative dA/dτ is denoted asȦ, and the total derivative over some parameter λ is A λ := A R R λ + A U U λ .

II. TRANSITION TO MASSLESS SHELL
The metric across the two domains can be represented as the continuous distributional tensor [3] using the set of special coordinatesx µ = (w, z, θ, φ). Here Θ(z) is the step function and the interior and exterior metricsḡ ± (x) are continuously joined at z = 0. Mathematically equivalent and sometimes easier to implement approach is the thin shell formalism. We will use it in most of our analysis, both for the consistency with [14,15] and because it makes structure of the distributional stress-energy tensor more transparent.
The explicit form of the interpolating metricḡ µν when the exterior geometry that is modelled by the outgoing Vaidya metric is given in [26]. We treat a general sphericallysymmetric exterior geometry and provide the resulting metric g µν in Appendix A.
In discussing the timelike-to-null transition it is particularly convenient to have a unified description that is applicable to both types of shells [28]. Unlike the proper time τ that diverges at the transition to the null trajectory, two additional parameterizations are regular there. When discussing the null shell it is convenient to use while the Minkowski retarded coordinate w is used in calculations that involve the interpolating metric. We will primarily use the thin shell formalism in τ and λ parameterizations. We first consider an initially massive thin shell Σ and assume that the exterior geometry is described by the outgoing Vaidya metric (eq. (2) withf (u, r) = 1 − C(u)/r, for some decreasing function C(u), h(u, r) = 0). While the shell is timelike its four-velocity is given by where The first junction condition identifies the induced metric on the two sides of Σ, where F = f (U, R). Similarly, the conditionk µ ≡k µ − =k µ + (see Appendix A) holds both for a massive shell and in the lightlike regime, where k 2 ± = 0. For a massive shell we also haveU that approximately becomesU ≈ −2Ṙ/F for large −Ṙ, and for the null shell An important auxiliary quantity for a massive shell is the outward pointing (unit) spacelike normal, and as the shell approaches the null trajectory n µ → −k µ . Bothk µ andn µ are continuous across the shell (when written in the interpolating coordinatesx µ ). The second junction condition relates the jump in extrinsic curvature to the surface stress-energy tensor. Here we use the surface coordinates y a , a = 1, 2, 3, the shell is given via parametric expressions x µ ± (y), and e µ a = ∂x µ /∂y a . In this case the optimal choise of the surface coordinates is (τ, Θ := θ| Σ , Φ := φ| Σ ).
Assuming a general relationship between the proper mass density (that is related to the shell's rest mass via m 0 = 4πR 2 σ) and the tension/pressure p(σ), we obtain equations that govern their evolution, and The details of the derivation can be found in [14]. The system can be solved forR providing the basis for numerical integration. In the limit of largeṘ the pressure is negligible, and the asymptotic expression becomes [14] R where we defined the gap between the shell and the Schwarzschild radius, and the second equality in Eq. (16) holds for X ≪ C.
To illustrate the timelike-to-null transition we set p(σ) = 0 and following CUWY adapt the law The results of numerical integration of (14) with p = 0 are presented in Fig. 1; using the same initial conditions as in [26] we obtain the same result. Henceforth we use while We establish the divergence inṘ at finite proper time by considering the Taylor series for R (and thusṘ) at some regular point τ and showing that its radius of convergence goes to zero as τ increases. The third and higher derivatives R (r) (τ ) are calculated from the knowledge of the coordinates R(τ ) and U (τ ), velocityṘ and the function C(U ). We estimate the radius of convergence of the series using Eqs. (16) (since for large values of |Ṙ| it is the dominant contribution to the derivative ofṘ) and (18). In the leading term the (r + 1)-th derivative pulls down the exponent fromṘ (which equals to 3r − 2), increases the power ofṘ by 3 = −1 + 4 and increases the power of x in the denominator by 2, when we substituteR. As a result, the leading term in the derivatives scales as and the r-th coefficient in the Taylor series is Then the radius of convergence ̺ is which goes to zero with decreasing C and X and increasing |Ṙ|.
On the other hand, if the shell is still timelike the gap X begins to increase after reaching approximately X ≈ ǫ * = α/C ( [14]; in Section IV we revisit this estimate taking into account the timelike-to-null transition). Since the acceleration is negative and increasing in absolute value the transition to a null trajectory can occur only for X > 0, i.e. outside the Schwarzschild radius. While the above result does not allow identification of the transition point τ 0 , it shows that it exists. Lightlike matter has a vanishing rest mass. Eq. (15) ensures that when the shell becomes null [i.e.Ṙ → −∞], its surface density goes to zero, causing the rest mass m 0 = 4πσR 2 to vanish. The rate of shedding of the rest mass at large velocities iṡ where only the term proportional toR < 0, approximated by using Eq. (16), appreciably contributes to the final result. We see that this rate is much higher than the rate of decrease of C. Indeed, for macroscopic shells C ≫ 1 the fraction of the gravitational mass lost before the transition to the null trajectory goes to zero slover than 1/C (Appendix B).
In Appendix C we show that such transitions happen for a general spherically-symmetric exterior metric.

III. PRESERVING THE NULL CONDITION
For the shell becoming null at λ = λ 0 we now investigate the conditions necessary to keep it null in a general spherically symmetric metric (2). First we recall a few properties of the surface stress-energy tensor in the null case. Since the normal n µ "declines into tangency with Σ" [28] an alternative auxiliary vector is used, defined by on both sides of the shell. When the shell becomes null at some λ = λ 0 Analogously to the vectork µ it is continuous at Σ, [N µ ] = 0.
On the null hypersurface orthogonal and tangent to the 4velocity of the null shell (that is traversed by the shell if it remains null) we install the coordinates y a that consist of λ ≥ λ 0 , possibly non-affine parameter of the hypersurface generators, and the transversal y A that are most conveniently taken to be (Θ, Φ). From the bulk point of view the shell is described by parametric relations x µ (y a ), and the set of three tangent vector fields is formed by two spacelike vectors transverse to it that are also continuous across the surface, and the null vector e µ λ := k µ . The transverse inner product σ AB := e µ A e Bµ is continuous as well. For a spherical shell it is σ AB dy A dy B = λ 2 (dΘ 2 + sin 2 ΘdΦ 2 ).
The surface stress-energy tensor of a massless shell depends on the observer. However, all such objects are derived from where ς is the shell's surface energy density, p is an isotropic surface pressure, and j A is surface current that is zero in the spherical case. From the extrinsic perspective the three above parameters are related to the discontinuity of the derivative of the continuous interpolating metric via γ µν := [ḡ µν,α ]N α where the calculation is performed using the coordinatesx µ . From the intrinsic perspective the key quantity is a transverse curvature C ab , Density, pressure and current are obtained from its discontinuity, [C ab ] = 1 2 γ ab e µ a e ν b . A straightforward calculation identifies while the pressure is directly connected to the preservation of the null condition k µ k µ = 0, where κ = C λλ = −N µ k µ ;ν k ν measures the failure of λ to be the affine parameter, with acceleration a µ ± := k µ ±;ν k ν ± being on either side of the shell. Note that unlike its massive counterpart, a massless shell moves on a geodesic, possibly nonaffinely parameterized. The shell that becomes null at λ = λ 0 and continues as as null for λ > λ 0 should satisfy Eq. (34) already at λ = λ 0 . At this stage two options are possible. One is that the spacetime outside the shell is still described by the outgoing Vaidya metric. The other is that the form of the metric changes at the null transition. We shall explore each in turn.
Suppose the spacetime outside the shell retains its outgoing Vaidya form. The shell will continue on a null trajectory for λ > λ 0 , provided it acquires a surface pressure. Indeed, the acceleration of the shell expressed in the outside and the inside coordinates is respectively, where we suppressed the trivial angular components. Then from Eqs. (7) and (35) it follows that resulting in the surface pressure upon using (33). Furthermore, (as we shall see in Sec. IV) R ≥ C + ǫ * , and so the pressure is finite for all finite values of C U . The shell moves on a null geodesic with a consistent value of the second derivative U λλ ≡ d(2/F )/dλ for λ ≥ λ 0 . Alternatively, one could impose the requirement p ≡ 0 as was done by CUWY [26]. A combination of the null shell property W λ = 2 (Eq. (A16)) and then ensures that the only consistent solution for λ > λ 0 that is not superluminal is In other words, the pre-Hawking radiation cuts off and (as C u must drop to zero at λ = λ 0 ) the metric has a discontinuity in the first derivative, reducing it to the Schwarzschild metric expressed in Eddington-Finkelstein coordinates. We discuss the superluminal solution in Appendix E.
We therefore see that it is impossible to have both a pressureless null shell and a shrinking Schwarzschild radius dr g /du < 0 if the exterior metric is the outgoing Vaidya.
The surface stress-energy tensor of Eq. (30) satisfies the weak energy condition for positive values of ς and p. Indeed, for an arbitrary timelike vector t µ = (u,ṙ,θ,φ) we have The transverse pressure p, however, may not be a benign feature of the solution. Having normal matter is not sufficient to rule out the superluminal propagation of disturbances, i.e. to guarantee that the speed of sound is less then the speed of light [29]. For the most part the shell is super-stiff, i.e., p > ς. Note that motion of the shell determines the surface quantities (p, ς) via the junction conditions; there is no equation of state and thus no well-defined speed of sound.
This leads us to the other option: the metric has a different discontinuity at the null transition allowing both zero pressure and a shrinking r g . For a general metric (2) the two first components of Eq. (34) become where H = h(U, R). Given the functions f and h this pair of equations yields κ and U λλ . The shells continues on a null trajectory, with the velocity U λ = 2e −H /F in a general metric of Eq. (2) satisfying Since κ − ≡ 0 it is enough to require κ = 0 to ensure that the shell remains pressureless while continuing to move on a null geodesic. In this case Eq. (42) will serve as a constraint on the exterior metric: H R = −2e −H F U /F 2 . The evaporation continues with F U < 0. There is a discontinuity in the derivative of the metric, h U (λ 0 ), R(λ 0 ) = 0, ∂ R h U (λ 0 ), R(λ 0 ) = −2F U /F 2 , and for λ > λ 0 the metric is not of the Vaidya form, but rather of the form (2).
The question concerning which scenario is actually realized can be answered only through the detailed studies that involve matching of the bulk stress-energy tensor that results in a selfconsistent analysis of evaporation.

IV. HORIZON AVOIDANCE
We have seen that the shell, once it becomes null, continues as such. Here we show how the event horizon is avoided if we still model the exterior geometry by the outgoing Vaidya metric. In this case the shell must have a surface pressure (37). A general analysis, including comparison of the evolution described in different coordinate systems will be presented elsewhere.
The key quantity is the gap where the last equality holds only for the Vaidya metric. This quantity can be viewed as either function of w or λ, via the relationships R(w), U (w), or R(λ), U (λ) respectively. Evaluating its derivative over, e.g., λ, we have As a result the gap decreases only until X ≈ ǫ * := 2C|C U |, and crossing of the Schwarzschild radius is possible only if the evaporation completely stops. A detailed evaluation of the approach to ǫ * is given in Appendix B.

V. DISCUSSION
If gravitational collapse is accompanied by emission of pre-Hawking radiation (that does not get cut off) then initially massive thin shells shed their rest mass and become null. This is a generic property of the semiclassical model of a massive spherical collapsing body as a thin shell that is sandwiched between flat Minkowski spacetime and a generic sphericallysymmetric spacetime that is self-consistently generated via emission of radiation. Once the shell becomes null there are several options, that are best illustrated by the evolution of a massive dust shell with the exterior geometry given by the outgoing Vaidya metric 1. The pre-Hawking radiation halts at or before the null transition. The metric retains its Vaidya form but has derivative discontinuity, though this perhaps could be ameliorated if the process halts sufficiently smoothly. Collapse to a black hole proceeds classically on a null geodesic (as a hypersurface separating Minkowski and Schwarzschild geometries).
2. The metric retains its Vaidya form and the shell remains pressureless, in which case it must become superluminal. This option can be discarded as unphysical [26].
3. The shell acquires surface pressure discontinuously (though one could consider modelling this as a smooth but rapid transition) and propagates on a null geodesic (as a hypersurface separating Minkowski and outgoing Vaidya geometries). In this case an horizon does not form, as shown in section IV.
4. At the opposite extreme the shell remains pressureless, for a part or the entire duration of its evolution.
There is a derivative discontinuity in the metric, but the exterior Vaidya form is not retained.The shell propagates on a null geodesic (as a hypersurface separating Minkowski and a generalized outgoing Vaidya geometries). It is possible to show by modifying the analysis of [15] along the lines of Sec. IV that the shell does not cross its Schwarzschild radius. A more plausible option is a combination of some surface pressure and h(u, r) = 0, ensuring subliminal propagation of density perturbations.
It is clear that pre-Hawking radiation can modify thin shell collapse in a variety of ways. In particular, it can lead to horizon avoidance or to evaporation suppression. Whether such avoidance or suppression are universal features can be determined only from a more explicit analysis of the coupled matter-gravity systems.

Appendix A: Interpolating metric
Extending the construction of [26] we adapt the coordinatesx µ = (w, z, θ, φ), where w := u − and r =: We abbreviate h U (w), R(w) as H(w). For general exterior point (u, r) the coordinates (w, z) are obtained by identifying u w ≡ U w . Equivalently, after finding the radial coordinate of the shell at the moment of the retarded time w, one obtains the equation R − (w) = R + (U ) ≡ R + (u) that can be solved for u(w). We explicitly use the subscripts indicating the spacetime domain because even if the radial coordinate is continuous, the functional dependence on the relevant retarded time is different in each region. In addition we note that We will need the explicit form of the Jacobian on the shell: The first junction condition in the form and the requirement that both u − and u + increase together result in the explicit expression The tangent vector is continuous in the coordinatesx µ across the shell. Both becomek where we used Eqs. (A1), (A2) and (A3). In these coordinates the metric inside the shell is written as Outside the shell we have and where As a result the metric is wheref = 1 − C u(w, z), r(w, z) r(w, z) ,h(w, z) = h u(w, z), r(w, z) .

(A13)
While the shell is timelike the normalization v µ v µ = −1 implieṡ resulting inẆ where the approximate equality holds for the large values of |Ṙ|. When the shell becomes null the first junction condition leads to The interior metric becomes and the exterior metric simplifies to with r(w, z) = R(w) + zF (w).

(A19)
Appendix B: Estimate of gravitational mass loss and the closest approach to the Schwarzschild radius First we show that from the moment the evaporation becomes important (or switched-on in the numerical simulation) and until the shell loses all of its rest mass only a relatively small fraction of the Bondi-Sachs mass C/2 evaporates. Equivalently, the elapsed interval of the EF coordinate u is much smaller then the evaporation time, ∆U ≪ u E . If the evaporation is governed by Eq. (18) then and where the evaporation time is given by u E = C 3 0 /3α. For C 0 ≫ 1 we can assume C = C 0 up to the transition as the first approximation. In this case using Eqs. (5) and (10) we have where we also assume that X ≪ C. The integration from the "initial" R i (a quantity sharply defined in the simulation and approximately in, e.g., adiabatic approximation) to the radial coordinate R 0 where the shell becomes null results in The second equality is obtained by assuming that X 0 = ǫ * . Numerical simulations indicates the actual value is different by a factor 2-10, but this precision is sufficient for our estimate. Substituting this result into Eq. (B1) we find that the relative reduction of the rest mass is We now provide a better estimate of the mass loss that also demonstrates how the shell radius R approaches the Schwarzschild radius. We assume that X ≪ C, but take into account Eq. (B1). Hence the approach of the shell to the Schwarzschild radius is governed by the equation The first equation is exact for the null shell. Solution of the approximate equation is where Erf(z) is the error function, and L is determined by the initial conditions. It allows us to find the minimal gap X between a (massive or null) shell and its Schwarzschild radius. Setting U (λ 0 ) = 0 and λ 0 = −C 0 − X 0 , and approximating the error function of a large argument as 1, we find that where we suppressed the exponentially small correction terms, and for C 0 ≥ C ≫ √ α we find where the last equality holds for C 0 ≫ C, in agrement with the discussion in Section IV. Note that the condition C 2 /α ≫ 1 corresponds to the adiabatic condition of [23].
where θ is expansion of the geodesic congruences. In our case κ − ≡ 0, for an incoming spherical null shell θ = −2/R, and the non-zero components of the Einstein tensor outside the shell are G ur = e h1 ∂ r C r 2 , (D3) A different method is based on the direct use of the discontinuity γ µνk µkν , γ µν := [ḡ µν,α ]N α .