Particle creation in gravitational collapse to a horizonless compact object

Black holes (BHs) play a central role in physics. However, gathering observational evidence for their existence is a notoriously difficult task. Current strategies to quantify the evidence for BHs all boil down to looking for signs of highly compact, horizonless bodies. Here, we study particle creation by objects which collapse to form ultra-compact configurations, with surface at an areal radius $R=R_{f}$ satisfying $1-(2M/R_{f})= \epsilon^{2}\ll 1$ with $M$ the object mass. We assume that gravitational collapse proceeds in a `standard' manner until $R=R_{f}+2M \epsilon^{2\beta}$, where $\beta>0$, and then slows down to form a static object of radius $R_{f}$. In the standard collapsing phase, Hawking-like thermal radiation is emitted, which is as strong as the Hawking radiation of a BH with the same mass but lasts only for $\sim 40~(M/M_{\odot})[44+\ln (10^{-19}/\epsilon)]~\mu \mbox{s}$. Thereafter, in a very large class of models, there exist two bursts of radiation separated by a very long dormant stage. The first burst occurs at the end of the transient Hawking radiation, and is followed by a quiescent stage which lasts for $\sim 6\times 10^{6}~(\epsilon/10^{-19})^{-1}(M/M_{\odot})~\mbox{yr}$. Afterwards, the second burst is triggered, after which there is no more particle production and the star is forever dark. In a model with $\beta=1$, both the first and second bursts outpower the transient Hawking radiation by a factor $\sim 10^{38}(\epsilon/10^{-19})^{-2}$.


I. INTRODUCTION AND SUMMARY
It is generally accepted that black holes (BHs) can be and have been found in various astrophysical systems, such as X-ray binaries, galactic nuclei and binary systems sourcing gravitational waves. These systems all contain dark, compact and massive objects whose properties are all consistent with the BH paradigm. However, BHs are defined by the existence of an event horizon, which is the boundary of the causal past of future null infinity.
Thus, sufficiently compact bodies can mimic BHs at a classical level. Given the crucial role of horizons in a number of fundamental issues, quantifying the evidence for BHs is as important as quantifying, say, the level to which the equivalence principle is satisfied [1][2][3][5][6][7][8][9].
A natural strategy to test the BH paradigm is to look for smoking-gun imprints of horizonless bodies. The number of proposals for ultra-compact horizonless objects is large and growing (see Ref. [2] for a review). The exterior of such (static) objects is described by the same Schwarzschild geometry as that of a non-spinning BH. Thus, as we stressed already, it is challenging to find evidence of a surface using classical electromagnetic or gravitational waves [1-3, 5-10, 15-18].
Classical physics predicts measurable differences between ultra-compact horizonless stars and BHs, but these may either be inaccessible to observers far away, or simply take too long to affect our detectors. However, there is a semi-classical effect which is, seemingly, particular to BH geometries: Hawking radiation. In fact, when quantum effects are included at a semiclassical level, particles are created and emitted by BHs, and the spectrum of the radiation is thermal, like that of a black body [19][20][21] . In Refs. [22][23][24][25][26], quantum particle creation by a collapsing object and its semiclassical effect on the formation of an apparent horizon have been discussed, based on quantum field theory in curved spacetime, in a very general context. Quantum particle creation in horizonless gravitational collapse has been also discussed in the context of naked singularity formation [28][29][30][31][32][33][34].
The organization and summary of this work is the following. In Section II, we review quantum particle creation in spherically symmetric spacetimes. In Section III, we expand on our toy model of a collapsing spacetime by pasting a Minkowski and Schwarzschild spacetimes with a timelike shell. In Section IV, we review how the present formalism can be used to recover a constant particle radiation by BHs, i.e., the Hawking radiation, with an emphasis on transient thermal radiation in the absence of horizon formation. In Section V, we introduce a collapse model with a null shell to a horizonless compact object, yielding delta-functional divergent emissions both at the end of the transient Hawking radiation and at the end of the long dormant stage. In Section VI, we construct a collapse model with a timelike shell to a horizonless compact object, show the couple of finite bursts of radiation as a common feature in a broad class of models and present the temporal change of radiation for specific models. Section VII is devoted to discussion. We use units in which G = c = = 1. Consider standard radial null coordinates u and v in the asymptotic region in spherical symmetry, and a pair of ingoing and outgoing null rays, v = v in and u = u out , respectively, which are connected at the regular centre r = 0 with each other, as in Fig. 1. We call such related outgoing and ingoing null rays a null-ray pair. The mapping function G is defined as v in = G(u out ). Note that u can be identified with the observer's time at infinity. Following Refs. [22,24], we define which is physically interpreted as the growth rate of redshift of the outgoing photon with respect to the ingoing photon as a function of the retarded time u out . The function κ(u) determines the radiation power at future null infinity in the geometrical optics approximation, through [21,27] with δ = 1 and 0 for minimally and conformally coupled massless scalars, respectively. The second term in parentheses does not contribute to the integrated radiated energy because it is a total derivative; hence, we will mainly concentrate on the first term, i.e., that for the conformally coupled scalar field.
If the function κ(u) satisfies the adiabatic condition then the spectrum of outgoing particles at u = u * can be regarded as Planckian with where κ(u * ) > 0 is assumed.

III. SPHERICAL SHELL IN VACUUM
Our model is a spherically symmetric vacuum spacetime with a shell. The areal radius of a timelike shell is given by r = R(τ ), where τ is the proper time for the observer at rest on the shell. The induced metric on the timelike world tube Σ is given by where dΩ 2 = dθ 2 + sin 2 θdφ 2 is the metric on a unit sphere. The interior is described by the Minkowski metric The null coordinates in the interior are U = T − r and V = T + r. The exterior is given by the Schwarzschild metric The standard null coordinates are given by u = t − r * and v = t + r * , where r * := r + 2M ln [(r/2M) − 1]. The junction condition for the first fundamental form givesṫ andṪ , where the dot denotes the derivative with respect to τ . This givesU andV andu andv.
The explicit expressions for them are relegated to Appendix A.
Since V = V in and U = U out are related through V in = U out at the centre r = 0, we find

5)
A out = A(τ out (u)) and so on, and τ out (u) and τ in (u) are the values of τ when the outgoing and ingoing null rays cross the shell, respectively. Further, we can obtain the expression for κ(u) as follows: where (3.7) The first and second terms on the right-hand side of Eq. (3.6) can be regarded as the contributions from the shell at τ = τ out and τ = τ in , respectively. We can obtain the explicit expressions for A, B, C and D in terms of R,Ṙ andR, which are relegated to Appendix A.
For reference, if the shell is marginally bound and made of dust, then the junction condition for the second fundamental form giveṡ However, we will not assume any equation of state for the surface energy density and pressure on the shell. Instead, we specify the dynamics of the shell. The evolution of the surface energy density and pressure will be then determined by the junction condition for the second fundamental form. This freedom has a price: our simplistic model may contain unphysical matter content with exotic equation of state. We should stress that our purpose here is not to produce alternatives to BHs; rather, we are interested in understanding possible consequences of failing to produce horizons. This programme, if successful, then allows us to quantify in a better way the evidence for BHs and to strengthen the BH paradigm.

IV. PARTICLE CREATION IN STANDARD-COLLAPSE PHASE
Conventionally, to derive the Hawking radiation, the expansion of R(τ ) with respect to τ at the entry into the horizon τ = τ H has often been assumed [21]. However, such an assumption seems to prescribe the behaviour of the shell at an event which is not in the causal past of the observer. Here we show that the expandability at τ = τ H is unnecessary and, hence, that the (temporarily) thermal radiation does not need any horizon.
Instead, observing the dust-shell collapse described by Eq. (3.8), we assume that the standard collapse is divided to the following two phases.
We assume We can additionally assume that the shell is initially static at some radius R i .
We assume The functions A, B, C and D take expressions A i , B i , C i and D i for phase i. The explicit expressions are relegated to Appendix B. The transition between the above two regimes occurs at τ = τ 0 when R = 4M. This scenario of standard collapse is then consistent with the dust-shell collapse.
Denoting the outgoing null ray in the Schwarzschild region which leaves the shell at τ = τ 0 with u = u 0 , we can obtain the expression for G ′ (u) and κ(u) separately for u < u 0 and u > u 0 . To do this, it is a key to determine when the outgoing null ray crosses the shell outwardly and when the ingoing null ray, which is a counterpart of the outgoing null ray in the pair, crosses the shell inwardly. If the outgoing null ray crosses the shell outwardly in phase i at τ = τ out and the ingoing null ray crosses the shell inwardly in phase j at τ = τ in , we classify the null-ray pair as (i, j). For the null-ray pair of class (i, j), G ′ and κ are given by respectively, where we use the notation A i,out (u) = A i (τ out (u)) and so on. Then, we can find that there are two radiation stages.
Here, we discuss the radiation for u > u 0 . Using Eq. (2.2), we obtain originates from the behaviour of the shell in the late-collapse phase at τ = τ out .
Equations (2.1) and (2.4) give temporarily thermal radiation with temperature where we can easily see that the adiabatic condition (2.3) is also satisfied. Since no horizon has formed yet, this means that transient Hawking radiation does not need any horizon.
If the late-collapse phase continues up until R ≃ 2M(1 + ǫ 2 ), then the transient Hawking radiation arises and lasts for ∆u ≃ 4M ln ǫ −2 , which can be seen from Eq. (B7). Therefore, the radiated energy through this transient Hawking radiation is given by . (4.8) In the limit ǫ → 0, the Hawking radiation continues eternally and the energy radiated goes to infinity. ingoing null shell at u = u 0 and again becomes static with R = R f at u = u 1 . The ingoing null shell is extended to the Minkowski region with an ingoing null ray, which is denoted by a blue dashed line, and reflected to an outgoing null ray which passes the shell outwardly to the Schwarzschild region, which is denoted with a red line labelled u =ũ 1 . This model was introduced in Ref. [35].
We now review and re-analyse an exact collapse model with a null shell, which can result in a static compact star with radius slightly larger than 2M. The schematic diagram of this model -introduced in Ref. [35] -is shown in Fig. 2, and consists of three phases. Note that these phases are different from those in the timelike-shell model.
Initially, the shell is static with R = R i .
At u = u 0 , the shell suddenly turns ingoing null with V = 0. Since the shell is also given by v =const., we find When the shell reaches the radius R f := 2M/(1 − ǫ 2 ) at u = u 1 , it stops and becomes static again, where u 1 is determined by We treat ǫ as a constant free parameter satisfying 0 < ǫ < 1.
We also defineũ 1 such that the ingoing null shell V = 0 is extended with an ingoing null ray to the centre r = 0 in the Minkowski region, being reflected to the outgoing null ray and going through the shell to an outgoing null ray u =ũ 1 in the Schwarzschild region. We can The functions G ′ (u) and κ(u) are calculated as follows: • u < u 0 All null-ray pairs are classified as (0, 0), for which we have G ′ (u) = 1 and κ(u) = 0.
The radiation for u 0 < u < u 1 can be regarded as temporarily thermal with temperature kT = κ(u)/(2π) = M/(2πR 2 out ). Therefore, kT ≃ 1/(8πM) for 1 − 2M/R out ≪ 1. This is transient Hawking radiation. In this model, we have bursts of radiation at u = u 0 , u 1 andũ 1 because G ′ changes discontinuously then and κ is given by Eq. (2.1). The discontinuities in (− ln G ′ ), which we denote with ∆(− ln G ′ ), are given as follows: To have a smooth process and extract meaningful physics, we propose a collapse model of a timelike shell with finiteR andṘ. This model consists of five phases: an early-collapse phase, late-collapse phase, early-braking phase, late-braking phase and final static phase.
• Phase 1, a late-collapse phase: This phase is also identical to that in standard collapse discussed in Sec. IV, i.e., • Phase 2, an early-braking phase: We assume that at τ = τ 1 or R = R b , the shell begins to brake. For τ 1 < τ < τ 2 , we assume the following inequality: • Phase 3, a late-braking phase: We assume that at τ = τ 2 , when R = R 2 , the following equality holds: For τ 2 < τ < τ 3 , the following inequality holds: 3) The radius of the shell approaches the final value R f .
Later on, the shell is completely static.
For later convenience, as is seen in Fig. 4, we label as u = u 0 , u 1 , u 2 and u 3 those outgoing null rays in the Schwarzschild region which leave the shell outwardly at τ = τ 0 , τ 1 , τ 2 and τ 3 , respectively. We use u =ũ 1 ,ũ 2 ,ũ 3 for those outgoing null rays which are traced back through the centre to ingoing null rays and reach the shell at τ = τ 1 , τ 2 , τ 3 , respectively.

B. Post-Hawking burst
We find that emission of bursts of radiation both at the end of the transient Hawking radiation and at the end of a long dormant stage is a general feature of quantum particle creation in setups leading to a compact horizonless object. Here, we briefly describe this phenomenon.
For u 1 < u < u 2 , the observer receives the outgoing null ray which left the shell outwardly in the braking phase, and which can be traced back to the ingoing null ray which crosses the shell inwardly in the standard-collapse phase. From Appendix B, κ(u) is estimated as for u 1 < u < u 2 and u 2 < u < u 3 , respectively. Note that the factor (1 − 2M R )/Ṙ 2 is generally an increasing function for u 1 < u < u 2 , which is much smaller than unity at u = u 1 , unity at u = u 2 , and diverging at u = u 3 . In the above expressions, the second term can be regarded as the transient Hawking radiation, which keeps constant for u 1 < u < u 2 and decays for u 2 < u < u 3 . This implies that u 2 (or τ 2 ) plays a clear physical role: it triggers the decay of the transient Hawking radiation. On the other hand, the first term is negative and dominates the second term ifR 1/(2M) for u 2 u < u 3 . The emission due to the first term completely ends at u = u 3 . This gives a burst of radiation at the end of the transient Hawking radiation around at u = u 2 , which we call a post-Hawking burst. This particle creation occurs due to the braking of the shell at τ = τ out . The details of the burst depend on the specific behaviour of the shell in the braking phase.
C. Late-time burst from a static star Next, we consider the intervalũ 1 < u <ũ 3 , when the ingoing null ray crosses the shell inwardly in the braking phase and the outgoing null ray crosses the shell outwardly in the final static phase. In this case, from Appendix B, κ is negative and estimated as forũ 1 < u <ũ 2 andũ 2 < u <ũ 3 , respectively. Therefore, ifR 1/(2M) at τ = τ 2 in the braking phase, the first term in the above expressions dominates κ(u) at u =ũ 2 and, hence, Whether or not the deceleration is effective in particle creation, the observation of the burst is delayed from the direct observation of the deceleration at u = u 2 byũ 2 −u 2 ≃ 4M/ǫ.

D. Time dependence of particle creation
We will discuss the whole temporal change of radiation for specific models below.
However, this cannot be interpreted as a Planckian distribution with negative temperature: the stationary phase approximation or saddle point approximation to derive the Planck distribution [22,24] is simply not applicable 1 . The radiated energy during the burst is calculated to This is approximately equal to energy radiated through the transient Hawking radiation for β = 1/2, while this dominates the latter for β > 1/2. The temporal dependences of particle emission are summarised for β = 1/2 and β = 1 in Fig. 5. The shell begins braking at R = R b = R f + 2M ǫ 2β for (a) β = 1/2 and (b) β = 1. We here neglect the power of the order of ǫ 2 P H .

Model B: constant-deceleration model
Next we consider a technically simpler model, where the deceleration a of the shell is where |Ṙ b | = O(1). We can naturally assume a ≫ 1/(4M). Therefore, It is interesting to see the limit β → ∞ or R b − R f → 0 while 1 − (2M/R f ) = ǫ 2 is fixed in both models A and B. In this limit, the power becomes stronger and stronger, the time width becomes shorter and shorter and the energy radiated becomes more and more in both the post-Hawking and late-time bursts, while the duration of the dormant stage in between is unchanged. Thus, we can reproduce the last two delta-functional bursts in the null-shell model in Sec. V.

VII. DISCUSSION
It is important to compare our result with previous results in similar setups. In Refs. [35,36], a timelike-shell model was also used, the end state of which is a static shell with radius slightly larger than 2M. However, instead of prescribing the shell dynamics, the function G(u) was assumed directly to satisfy the expected qualitative asymptotic properties and changes in a time scale of the order of M. Figures 7 and 9 in Refs. [35,36] indicate that the width of the late-time burst is several tens of M and the power is bounded by that of the Hawking radiation P H . It was also observed that the width of the burst increases for smaller ǫ. As seen in Sec. VI D, these features correspond to our model A with β = 1/2. On the other hand, we can argue that the physically natural scenario corresponds to model A with β = 1 from the argument that the shell begins to brake when and settles down to R = R f = R H + R H ǫ 2 , if the there is a unique characteristic scale which controls both the braking and the freeze-in of the shell and that the force onto the shell is It is interesting to estimate the quantities which appear here using astrophysical values. We have shown that the duration of the dormant stage is ∼ 4M/ǫ. Physically, the 4M factor is simply the proper time along the shell for a null ray to cross its diameter, when the shell is sufficiently close to 2M. The factor 1/ǫ comes from the redshift factor between the proper time of the almost static shell and the observer time.
The particle production process is characterized by different stages, after what we termed the "standard" collapse phase. This large number of particle production stages is due to the different classes of null-ray pairs that govern quantum particle creation. We can summarise the correspondence as follows: braking at τ = τ out and standard collapse at τ = τ in contribute to the post-Hawking burst, the final static phase at τ = τ out and standard collapse at τ = τ in produce the dormant stage, whereas the final static phase at τ = τ out and braking at τ = τ in give the late-time burst.
We have applied the geometrical optics approximation in the entire treatment. This is valid for s-waves and for sufficiently high frequencies. On the other hand, the reflection of waves by the shell and the geometry is completely neglected. This implies that if we relax this approximation, we will obtain not only the post-Hawking and late-time bursts but also echoes in particle creation due to the reflections of waves (cf. Refs. [2,3,5]). The details of this process require further calculations.
Finally, we have prescribed the shell dynamics in this paper, but postpone a discussion about the matter content of the shell which enables such an unusual time evolution. We expect that some energy conditions must be violated. The physical significance of such violations is not completely clear. However, we take this opportunity to once more stress that one of the main goals of this work is to look for distinctive features of horizonless objects as a way to strengthen the black hole paradigm.

Appendix A: Expressions for a timelike-shell model
The junction condition for the first fundamental form giveṡ (A1) The relation between the null coordinates and the proper time of the shell is given bẏ From Eqs. (A2) and (A3), we can write down the explicit expression for A and B in terms of R as follows: From Eqs. (A3) and (A4), we can write down the expression for C and D in terms of R as follows:

Appendix B: Expressions for a timelike-shell model in different regimes
To estimate the functions A, B, C and D, we are interested in the following phases: 0.

Appendix C: Time intervals
Since R = 4M at u = u 0 and R = R b = R f + 2Mǫ 2β at u = u 1 , Eq. (B7) implies Eqs. (A2) and (A3) imply that the intervals in terms of u are given as follows: where we have usedτ 2 − τ 2 ≃τ 3 − τ 3 ≃ 4M. The above relations do not depend on the details of the model.