Self-gravitating stellar collapse: explicit geodesics and path integration

We extend the work of Oppenheimer&Synder to model the gravitational collapse of a star to a black hole by including quantum mechanical effects. We first derive closed-form solutions for classical paths followed by a particle on the surface of the collapsing star in Schwarzschild and Kruskal coordinates for space-like, time-like and light-like geodesics. We next present an application of these paths to model the collapse of ultra-light dark matter particles, which necessitates incorporating quantum effects. To do so we treat a particle on the surface of the star as a wavepacket and integrate over all possible paths taken by the particle. The waveform is computed in Schwarzschild coordinates and found to exhibit an ingoing and an outgoing component, where the former contains the probability of collapse, while the latter contains the probability that the star will disperse. These calculations pave the way for investigating the possibility of quantum collapse that does not lead to black hole formation as well as for exploring the nature of the wavefunction inside r=2M.


A. Introduction
Black holes play a pivotal role in the evolution of the universe providing an important test laboratory for general relativity. Within classical general relativity, Oppenheimer & Synder modelled the gravitational collapse of star to a black hole by approximating the star with a uniform sphere of dust [1]. This model provides an analytic solution for stellar collapse that connects the Schwarzschild exterior of a star to a contracting Friedmann-Robertson-Walker (FRW) interior. Once the surface has passed within r = 2M , no internal pressures can halt the collapse and all configurations collapse to a point-like singularity at r = 0. The general features of this toy collapse model have been examined by many authors [2][3][4][5].
The inclusion of quantum effects is unavoidable both when determining whether a black hole will form and when pursuing an understanding of the final stages of the collapse. The smaller the mass of the particles forming the collapsing star, the more significant the quantum mechanical effects become. In particular, quantum treatment is necessary when the ratio of the action S toh is < ∼ 1, which in general may become true close to the singularity during the course of collapse [6]. The singularity formation implies the breakdown of classical general relativity. It is expected that general relativity unifies with quantum mechanics around Planck scale and that in a complete theory of quantum gravity singularities would naturally be removed. However, a satisfactory quantum gravity theory has yet to be developed.
In the absence of a full quantum theory of gravity, various models integrating quantum mechanical and gravitational physics exist. The class of such models which apply quantization procedures to general relativity operate with superspace, the configuration space of geometrodynamics. In superspace, a four-geometry is represented as a trajectory within space-like three geometries. Symmetry reduced models are known as minisuperspace models, and can give insight as to the physics in the strong curvature regime while remaining tractable for study. In minisuperspace models, trajectories are limited to a finite number of parameters describing constant t slices. This limitation corresponds to considering a finite dimensional sector of superspace. However, despite its shortcomings, it remains one of the few tractable models to date.
In this work, we consider the quantum mechanical collapse of a spherical symmetric sphere of dust in min-isuperspace where all spatial coordinates other than the radius of the star are frozen. We study the evolution of the surface of the star using the path integral formulation of Suen and Redmount [18], which is extended from a relativistic free particle to a particle constrained by non-trivial gravity. The initial wavefunction representing a particle on the surface of the star is taken to be Gaussian. We first consider a viable approximation to the Wheeler-DeWitt equation (the Schrödinger solution) for which we provide an analytic solution to the propagator in Schwarzschild coordinates, and compare it to a WKB approximation, which includes expansions around the classical paths. The path integral formulation includes all possible paths away and towards the centre of the star. Analytic solutions of all classical paths including space-like, time-like, and light-like paths are presented in Schwarzschild and Kruskal coordinates.
In Schwarzschild coordinates, we can follow the evolution of the surface of the star only until the formation of an apparent horizon due to the r = 2M coordinate singularity. We compute the propagator in a) analytical form when the wavefunction representing a particle on the surface of the star obeys the Schrödinger equation, which is an approximation for the Wheeler-DeWitt equation [19], and also provide b) a closed form solution of the propagator in the WKB approximation from which the time evolution of the wavefunction is determined by integrating about the classical paths. Our equations reduce to a free particle when the mass of the star is zero. In this limit, unlike in [18], we find that there is no disagreement between the wave-functions computed via the two propagators. Since the Wheeler-DeWitt equation becomes the Schrödinger equation when M → 0 [19], this convergence shows that the WKB approximation is accurate.
In Kruskal coordinates, one can continue to study the evolution of the star inside the apparent horizon (r < 2M ). We present analytic solutions for the classical paths that include time-reversing space-like paths, which allow for the possibility of extraction of information from inside the horizon to the outside. These paths are a starting point for future work to determine the Kruskal WKB wavefunction.
In Section I we describe the Oppenheimer-Snyder model. In Section II we detail the quantum treatment of dust collapse in Schwarzschild coordinates. This includes an overview of the classical action and paths, the Oppenheimer-Snyder limit, a heuristic analogy with the 1D vertical motion of a ball in a gravitational field, escape velocity, and the WKB approximation for the solution as a closed-form propagator. We conclude the section with a description first of the simplistic case of a free particle, where we show that the WKB converges to the Schrödinger solution at all times. In the more general case of a particle on the surface of a star, the two solutions converge towards one another at intermediate times. They are out of step at early and late times. For early times, the Schrödinger approximation is more exact, while for late times the WKB approximation is more accurate. This provides a scenario for testing higher order corrections to the Wheeler-DeWitt equation. Section III details the classical paths in Kruskal coordinates, which would be the starting point for future work that explores the collapse inside r = 2M . The conclusions follow in Section IV.

I. THE OPPENHEIMER-SNYDER MODEL
In General Relativity, a first approximation to the exterior space-time of any star, planet or black hole is a spherically symmetric space-time modelled by the Schwarzschild metric. This is a consequence of Birkhoff's theorem [3]. The Schwarzschild line element thus takes the usual form where Throughout this paper we use geometric units with G = c = 1. The Oppenheimer-Snyder (O-S) model follows the collapse of a star that is idealized as a dust sphere with uniform density and zero pressure from the perspective of an observer located on the surface of the star. The motion of the collapsing surface initially at radius r i can be parametrized by The star collapses to a singularity in a finite proper time. However, it takes an infinite Schwarzschild t to reach the apparent horizon at r = 2M and thus an external observer will never see the star passing its gravitational radius (r = 2M ).
In the classical O-S model, the stellar surface is considered to be initially at rest. However, a relativistic path integral quantum mechanical treatment includes configurations will all possible initial velocities. In this treatment particles are no longer point-like and the surface of the star is not localised. The position of a particle on the surface of the star is then approximated by a wavefunction.

II. QUANTUM TREATMENT IN SCHWARZSCHILD COORDINATES
We study the behaviour of the wavefunction of a particle on the surface of a collapsing star in the minisuperspace approach. The only spatial degree of freedom allowed is the radius r while the other degrees of freedom are frozen. The initial configuration is described by a Gaussian wavefunction centered far from r = 2M .

A. Classical Action and Radial Geodesics in Schwarzschild Coordinates
The relativistic action S and the Lagrangian L for this system are where τ is the proper time, andṙ = dr/dt. The Lagrangian is The momentum and the Hamiltonian is which reduces to the free particle Hamiltonian in the M → 0 limit. The equation of motion derived from the Euler-Lagrange equations is which can be solved implicitlẏ From this the classical paths are determined where The + sign represents motions from r i to r f > r i , and − sign represents the star collapsing from r i to r f < r i . The parameter c 1 can be used to separate the space-time regions      ds 2 > 0 ⇔ c 1 > 1 outside the light cone ds 2 = 0 ⇔ c 1 = 1 on the light cone ds 2 < 0 ⇔ c 1 < 1 inside the light cone (14) In the free particle case (M → 0 in Eq. (11)), c 1 =ṙ 2 represents the velocity squared of the particle.

The Oppenheimer-Snyder Limit
The Oppenheimer-Snyder assumes that the surface of the star is initially at rest. The initial velocity dr dt r=ri = 0.
From Eqs. (11), this corresponds to We can recover our equations of motion for this c 1 value from the O-S model. We proceed by taking the derivative of Eq. (3) and Eq. (5) w.r.t. η where we used that r = r i cos 2 η/2. We then divide dr/dη by dt/dη, and use r i = 2M (c 1 − 1)/c 1 from Eq. (15) to recover the equation of motion The negative sign − fits in with our convention for a collapsing star. While some paths will have low probability, the quantum mechanical treatment forces us to consider paths with all possible initial velocities.

B. Classical Paths in Schwarzschild Coordinates
The closed-form solution to Eq. (12) is written in Table I and Table II. To understand these paths we make an analogy with the vertical motion of a ball under gravity. This is shown schematically in Fig. 1. (a) r f < ri

Ball Analogy
Consider a ball traveling from point A to point B with B below A. It can either go down directly from A to B or it can go up from A, reach its highest point and then fall back down to B ( Fig. 1(a)). Analogously, if a ball has to travel from A to B with B above A, it can go directly from A to B or it can pass above B, reach its highest point above B and return down to B (Fig. 1(c)). Each of these paths takes a different amount of time to complete. For a fixed travel time, the two points A and B are connected by either a direct path or a turn-around path. At the turning point of a turn-around path (also its highest point) the speed of the particle is zero.
Consider a fixed time interval ∆t. Let B 1 be a point below A that the ball reaches when it has zero initial velocity. Then to reach any point B above B 1 , but still below A (A > B > B 1 ) in this same time interval, the ball must take a turn around path. It will first go up from A and then back down to B. Points below B 1 will be reached from A when the ball is thrown down with non-zero initial velocities.
Similarly, let B 2 be a point above A that the ball reaches with zero speed (at B 2 ) in a time interval ∆t. If B is a point above A but below B 2 ( B 2 > B > A ), the ball can reach there in this same interval ∆t by being given a greater initial velocity so that it turns around at some point above B 2 and comes back to B. Thus only turn-around paths are possible for points between A and B with B between A and B 2 . Table I contains the equations of motion used to determine r i for specific values of c 1 for a given time interval t f − t i and endpoint r f • c 1 = 1: the paths are light-like (ds 2 = 0) with r i = b 1 < r f and r i = b 6 > r f ,

Light-like, Time-like and Space-like Paths
This value of c 1 corresponds to the escape velocity dr/dt = (1 − 2M/r) 2M/r, where s = i, f, a. Since the star is represented by a wavefunction with a finite spread, the r i value is not fixed to one specific point as in the classical case. Table II is used to find c 1 for each r i value within regions of interval boundaries defined by Table I. Specifically, notice that the range b 3 → b 4 describes the turnaround paths with the turning point at r = r a (r a > r i > r f ). In this case, the paths from r i to r a correspond to the positive sign in Eq. (12), while the path from r a to r f corresponds to the negative sign (r f < r a ). Since dr/dt at r a = 0 implies x a = 0, The analysis of the motion of the particle on the surface of the star is analogous to that of a ball moving under gravity. The motion towards or away from r = 2M can be compared to the vertical motion of a ball under gravity when thrown straight down or up, respectively. Classically, the star will always collapse to a black hole. Quantum mechanically, we include all possible initial velocities of the surface of the star towards and away from 2M . The particle on the surface of the star is used to determine the trajectory of the surface of the star itself.

Escape Velocity
Continuing with the analogy of the ball falling under gravity, we introduce the concept of escape velocity. For a particle that is thrown up from the surface of the Earth, an initial upward velocity greater than or equal to approximately 11.2 km/sec ensures that the particle escapes Earth gravity. At this speed In our case, G = 1, R E is replaced by the initial radius of the star r i , and v i = dr/dτ . The kinetic energy of the particle on the surface of the star equals its potential energy when with where dr/dt =ṙ is given by Eq. (11). This results in whose solution is c 1 = 0. In terms of r a : This is consistent to having a turning point at infinity r = r a → ∞ (dr/dτ | ra = 0, x a = 0) as it should be for an escaped particle.
So, all paths with c 1 > 0 (r f > r i ) will escape. When c 1 > 0 (r f > r i ) an outgoing wave will keep moving out, while when c 1 < 0 the initially outgoing wave will come back inwards. This is consistent with our turn-around path being at c 1 = −2M/(r a − 2M ) < 0 with r = r a (finite r a > 2M ). Thus we expect the amplitude of the outgoing wave to decrease in time.
C. WKB Approximation: Closed-form Propagator, Numerical Wavefunction In order to study the wavefunction we use a WKB approximation of the propagator where S is the action associated with each path. The set of paths C include space-like, time-like and light-like paths.
The WKB approximation involves expanding about the classical action S cl to include paths that slightly deviate from the classical paths. The propagator is dominated by paths near the classical trajectory between (r i , t i ) and (r, t) and is approximated by the WKB expression [20] The WKB wave function is obtained from the integration of the propagator (28) The initial wavefunction describes a particle on the surface of a star of radius r i , which is taken to be a Gaussian centered about r c with r c >> 2M . The action from Eq. (6) is rewritten using Eqs. (11) and (13) as for direct paths connecting (r i , t i ) to (r f , t f ). The + sign corresponds to r f > r i and the − sign corresponds to r f < r i . Indirect paths connecting (r i , t i ) to (r f , t f ) through the turning point (r a , t a ) with r a > r i and r a > r f are described by the classical action The closed-form expressions for S cl and ∂ 2 S cl /(∂r i ∂r f ) are given in Tables III and IV. To simplify the expressions for S cl and ∂ 2 S cl /(∂r i ∂r f ) we have defined and also used The wavefunction is continuous across all regions outside r = 2M . The M → 0 limit corresponds to a relativistic free particle. This case was studied in detail in [18]. We revisit this case below and include updated calculations.

D. Free Particle
In the limit M → 0, Eqs. (6) -(9) reduce to the free particle equations with The wavefunction obeying the wave equation is with Fourier components φ(k, t) satisfying The solution of Eq. (36) is When ∆t = 0, we recover the original wavefunction, which is chosen to be a Gaussian centred around the origin, and so To determine N we normalise the wavefunction We first perform the x-integral and then the k-integral by using the definition of the δ-function where we have used that We obtain |N | 2 = m √ 2/(h √ π), which is time independent.
The wavefunction can be written in terms of a propagator that is given by [18] G(x, t; which reduces to δ(∆x) when ∆t = 0. In this simple case, the integral can be done exactly resulting in where K 1 is the modified Bessel function and λ ≡ ∆x 2 + (i∆t + ) 2 . The wavefunction can be written in terms of the propagator as The WKB propagator for the relativistic free particle [18] is and the resulting WKB wavefunction is given by In Fig. 2 it can be seen that the WKB approximation converges to the Schrödinger solution in all parts of the light cone. When no black hole is present, the Wheeler-DeWitt equation becomes the Schrödinger equation for which an exact solution can be found [18,19]. Redmount and Suen [18] performed the same comparison, and found some disagreement, which we now believe to be due to numerical error. The convergence of the wavefunction in the WKB approximation to the Schrödinger solution, shows that the approximation is highly accurate. We next extent this approach to the case of the quantum mechanical collapse of a dust ball.
E. Dust collapse: comparison of the Schrödinger and WKB approximations

Schrödinger Solution
We proceed to solve the Schrödinger equation for the quantum mechanical dust collapse in Schwarzschild coordinates. While in the free particle case the Schrödinger solution was exact, in this case it becomes an approximation. The exact Schrödinger solution is the zeroth order approximation of the Wheeler-DeWitt equation, whose corrections are included in Eq. (34) of [19].
As before, we start by expressing the wavefunction in terms of its Fourier transform which obeys the Schrödinger wave equation We then follow the same procedure as for the free particle (see Sec. II D and [18]), and re-write the Hamiltonian from Eq. (9) as where Applying the Hamiltonian The wave equation in terms of the Fourier component φ(k, t) is: with the solution is where N is a constant with respect to time. The whole wavefunction can then be written as where the propagator is given by The propagator can then be integrated to obtain where K 1 is the modified Bessel function and When r → 2M the propagator vanishes. Thus this solution is inaccurate at late times and cannot model the final stages of the collapse of the dust sphere. For numerical computations we scale all variables to be dimensionless. The free parameter in the above equations is mM/h.

WKB Approximation: Numerical Results
We integrate Eq. (28) numerically for the Gaussian wavefunction from Eq. (29) centered about r c = 30M . Classically, the star collapses to r = 2M in infinite Schwarzschild time. However, quantum mechanically there is a possibility for the star expanding and dispersing as well. Fig. 3 shows the time evolution from t = 0 to t = 45M of |Ψ(r f , t f )| 2 as a function of r f . An ingoing peak and an outgoing peak appear with the amplitude of the ingoing peak always remaining larger than that of the outgoing peak for the course of the evolution, which is expected for a collapsing star. We calculate the probability integral and observe that the lower the value of the mass combination mM/h, the higher the probability of dispersing. This is shown in Fig. 4(a) for mM/h = 0.25 and mM/h = 4.0. Fig. 4(b) and Fig. 4(c) show the ingoing and outgoing peaks of |Ψ(r, t f )| 2 for different values of mM/h at times of t f = 5M and t f = 10M , respectively. The speed of the collapse also increases slightly with the mass as does the value of the ingoing peak. This is consistent with our expectation that for higher particle mass the star is more classical.

Numerical Results for the Schrödinger Solution and Schrödinger vs WKB Comparison
Our paper takes the first steps towards testing the importance of the correction terms to the Schrödinger equation. Kiefer and Singh [19] suggest that the final fate of a black hole would be a situation in which these terms become significant. A star collapsing to a black hole is a way of producing such a scenario.  [19]) as the star approaches the apparent horizon. For higher masses the collapse occurs on a shorter timescale and so the breakdown occurs sooner. For lower masses the approximation holds slightly longer. Fig. 5(b) displays the comparison of the Schrödinger solution between the mM/h = 1 and mM/h = 0.25. Since the initial wavefunction is broader for smaller particle mass, the lower mass case collapses on a longer timescale. The difference in the outgoing and ingoing peaks becomes smaller for lower masses indicating the potential for a configuration that does not collapse if the particle have longer wavelength and behave more quantum mechanically. However, it is expected that the corrections to the Wheeler-DeWitt equation become more significant in that case. Fig. 6 compares the amplitude and the radial position of the ingoing and outgoing peaks for the Schrödinger and WKB approximations. Fig. 6(a) shows the time evolution of the maximum amplitude of the ingoing and outgoing peaks. As the star approaches r f = 2M , the ingoing peak amplitude for the WKB approximation increases developing a numerical singularity that indicates the formation of an apparent horizon, while the ingoing peak amplitude of the Schrödinger solution decreases since the propagator vanishes at r f → 2M as discussed before. In Fig. 6(b) it can be seen that the position of the ingoing peak of the WKB approximation drifts from the maximum ingoing |Ψ| 2 in the Schrödinger approximation, while position of the outgoing peaks agrees throughout the evolution.
In Fig. 7 we compare the WKB and Schrödinger solutions. It can be seen that the radial position of the ingoing peak drifts out of step at late times, while the outgoing peak continues to evolve at about the same position. The WKB solution collapses on a faster timescale. At later times, the drift of the ingoing peak increases for lower masses, which is consistent with the expectation that quantum effects become more significant and the correction terms to the Schrödinger equation become larger.

III. CLASSICAL PATHS IN KRUSKAL COORDINATES
The Schwarzschild coordinates allow us to explore the wavefunction of the star outside the r = 2M apparent horizon. At r = 2M a coordinate singularity with the Schwarzschild time t → ∞ appears. If we want to explore the collapse of the star up to the physical singularity at r = 0, we need to switch to a coordinate system that does not have a coordinate singularity at r = 2M . In this paper we restrict ourselves to deriving and discussing closed-form solutions to the equation of motion for the space-like, light-like and time-like paths in Kruskal coordinates. This will pave the wave to studying the WKB wavefunction inside r = 2M . We find that space-like geodesics have the interesting property that they can turn around outside r = 2M and move back in time.
Unlike the Schwarzschild t and r, which are time and space coordinates respectively outside r = 2M but switch roles for r < 2M , the Kruskal coordinate v is always a time coordinate and the coordinate u is always a space coordinate. The relation between the Schwarzschild r and t and the Kruskal variables u and v is given by and Thus the line element in Kruskal coordinates takes the form The singularity at r = 0 occurs at u 2 − v 2 = −1. In this paper we consider only the first quadrant u > 0 and v > 0. The horizon r = 2M occurs at u = v. Outside the horizon u > v, whereas inside the horizon u < v. The classical equation of motion can be derived from the Euler-Lagrange equations for the Lagrangian  The equation of motion can be reduced to where x r is given by Eq. (13). Direct paths from r i to r f > r i have the "+"-sign in the numerator and denominator with du/dv > 0. Direct paths from r i to r f with r f < r i will have the "−"-sign in the numerator and denominator. All time-like paths (c 1 < 1, du/dv < 1) that turn around will turn in space (u) reaching a du/dv = 0 point where the numerator of Eq. (65) vanishes. Similarly, all space-like paths (c 1 > 1, du/dv > 1 or du/dv < −1) that turn around will turn in time going through a dv/du = 0 point. The spacelike paths in Kruskal coordinates are characterized by u decreasing as v increases followed by v decreasing while u continues to decrease. These paths thus move back in time with the movement back in time beginning outside r = 2M .
Inside the horizon the space-like paths have turning points in r. From Eq. (65) it can be seen that x r (Eq. (13)) must be a real number in order for du/dv to be real. This occurs when c 1 ≥ −2M/(r −2M ) for all r. For any particular value of c 1 ≥ 1 we can find an r a ≤ 2M such that c 1 = −2M/(r a − 2M ). In order for x r to be real r > r a for all r. These paths enter the horizon at u = v while moving back in time (v increases, u decreases). They continue moving back in time and turn around inside the horizon, and return to r = 2M with v = u = 0. Light-like paths (c 1 = 1) can turn around at r a = 0. The question one must ask is whether these time travel paths can bring back information from inside the horizon to the outside. Clearly the classical paths cannot, but quantum mechanically, there could be such possibilities with paths close to the classical paths also being taken into consideration. However, there are other features of the space-like paths like the presence of caustics that will add complications to the WKB approach. Time-like paths (c 1 < 1) cannot turn inside the horizon.
We trace direct time-like, direct space-like, and indirect space-like trajectories of a particle on the surface of the star starting at a point r > 2M and follow the contraction of the star to points inside r = 2M . From a given point outside the horizon to a point inside the horizon which can be reached by a space-like path, there may be a direct space-like path and an indirect space like path (a path that goes to r a < r f before coming back to r f ) or two indirect space-like paths. Beyond a initial point u i > u i max there are no paths reaching the final point (u f , v f ). As u i approaches u i max the two indirect space-like paths come closer and closer together and merge when u i = u i max .
The light cone boundary values of 'u i ' for a given u f and v f can be determined by replacing t f − t i in Table  I by tanh −1 (v f /u f ) for r f > 2M and by tanh −1 (u f /v f ) for r f < 2M . Note that u f is a function of v f and r f and u i depends on v i , which is typically taken to be zero, and r i . Thus if one finds r i , this determines u i .
In Table V  It goes through P. As v increases from v = v i = 0, u initially decreases and then starts to increase outside the horizon after going through a point with du/dv = 0. It crosses the apparent horizon and reaches point P, which is inside the horizon.
Paths 3a and 3b represent space like paths, both of which originate at the same point (r i = 10.5M, u i = 27.60 at v i = 0) outside the apparent horizon and will reach P(u P = 9.995, v P = 10.0) where r f = r P ≈ 0.8M . Path 3a turns around at r a = 0.48M < r f after having passed through P, while Path 3b has turned around at r a = 0.3M (u a = 11.3, v a = 11.35) and then comes to point P on its way back to the apparent horizon where v = u = 0. These space-like paths start moving back in time outside the horizon after having passed through a point where dv/du = 0. While u continuously decreases, v initially increases and then decreases.
Paths 4a and 4b represent two space-like paths originating at the same point u i ≈ 46.3 (r i = 12.1M , in the region b 6 −b 7 . Both these paths arrive at P, after turning at points r a = 0.77147M < r P and r a = 0.79969M < r P respectively. These paths also pass through a point where dv/du = 0 outside the horizon. So, they also go back in time. Path 5 shows the single space-like path to P from u i ≈ 46.7 (r i = b 7 = 12.13M ). The two paths in the region r i = b 6 − b 7 come closer and closer and finally merge at b 7 . There are no real solutions beyond this point and hence no way to reach point P from r i > b 7 , u i > 46.7 at v i = 0. The features of the paths in Fig. 8(a) that have just been described can be seen more clearly in Fig. 8(b), which shows v as a function of r expanding the region the region inside the horizon. Extensions to this work to determine the Kruskal-WKB wavefunction will require an analysis of the caustic at r = b 7 to remove any potential divergences. The paths presented here would be a starting point for such future work.

IV. CONCLUSIONS
We have investigated the semi-classical collapse of a sphere of dust in a symmetry reduced minisuperspace model where the radius is the only spatial degree of freedom that is not frozen. This is the simplest model of stellar collapse that includes quantum mechanical effects. A particle on the surface of the star is no longer localised and is instead represented by a wavefunction. The initial wavefunction is taken to be a Gaussian that is centred far from r = 2M . We use Schwarzschild coordinates to derive closed form expressions for the propagator. The evolution of the wavefunction is studied in the WKB approximation and in the Schrödinger approximation of the Wheeler-DeWitt equation. Here we lay the foundation for the numerical comparison of the WKB approximation to the expansion of the Wheeler-DeWitt equation. In the free particle limit (M → 0), the Wheeler-DeWitt equation reduces to the Schrödinger equation. Numerically, we find that, in this case, the WKB approximation to the wavefunction converges to the Schrödinger solution, which shows that the WKB approximation is highly accurate.
For a particle on the surface of a star (stellar mass M = 0), the wavefunction exhibits an ingoing and an outgoing component. The ingoing part of the wavefunction contains the probability that the star will collapse, while outgoing part contains the probability of dispersion. This is unlike the classical case where the star simply collapses to a black hole. We obtain a better agreement between the WKB and Schrödinger solutions for the outgoing component of the wavefunction than for the ingoing component because the former is largely unaffected by the coordinate singularity at r = 2M . The Schrödinger and WKB solutions are out of step at early times when the Schrödinger wavefunction is more accurate. They come closer together at intermediate times, and fall out of step again at late times, when the Schrödinger approximation becomes more inaccurate failing to model the singularity formation and the WKB approximation is a better from which point P can be reached are displayed for each u i . No paths exist beyond r = b 7 , which is where the two roots merge into one.
approximation for the wavefunction. For lower particle mass, the star resists collapse longer and the probability that it disperses increases. Further work is needed (including the addition of higher order corrections to the Schrödinger solution) to determine the mass limit above which stellar configurations disperse instead of collapsing.
In order to study the quantum mechanical behaviour of the star, an integration on and around all classical paths is needed, which includes paths with all initial velocities. Schwarzschild coordinates permit the study of stellar collapse only outside r = 2M . We explicitly determine analytic solutions for classical space-like, light-like and time-like paths of a particle on the surface of a collapsing star in Schwarzschild and Kruskal geometries and present their individual features. All paths are parametrised by a factor c 1 , which when M = 0 reduces to the square of the velocity of the particle. case. This parameter determines the light cone boundaries with c 1 = 1 denoting a light-like path, c 1 > 1 representing space-like paths and c 1 < 1 parametrising time-like paths. We show that in the case of a star collapsing with zero initial velocity, our path equations reduce to the Oppenheimer-Schneider equations of motion.
We find that the time-like paths are unique. For a given time interval, a path between an initial and final point can be either direct or indirect, where it turns around in space. Thus some particles that initially move away from the star, can return and contribute to the collapse. On the other hand, space-like paths can turn around in time, and cannot turn in space. They also do not have to be unique when the final point lies inside r = 2M . In order to study the behaviour of the star for r < 2M , one cannot use Schwarzschild coordinates due to the coordinate singularity at r = 2M . The Schwarzschild r and t reverse from being space and time outside to being time and space coordinates inside 2M , respectively. The Kruskal coordinates are well behaved at r = 2M and can be used to study the evolution of the stellar collapse up to the physical singularity at r = 0.
In Kruskal coordinates, we observe space-like paths that move back in time outside r = 2M . Classically, no information can exit the black hole in any coordinates. However, by taking into account paths close to the classical paths, one might be able to extract information from inside the horizon. The analytical expressions for the classical paths in Kruskal coordinates are a first step towards deriving the wavefunction and modelling the collapse up to the singularity. We leave further exploration of quantum quantum collapse in Kruskal coordinates to future work.
In general, the dust collapse could be a testbed for corrections of the Schrödinger solution as predicted by Kiefer and Singh [19]. Our work is a starting point from which such numerical calculations could proceed. Furthermore, recent work by Dvali and Gomez [21] argues that black holes are a collection of Bose-Einstein condensates. Quantum effects are very important in this situation, and could add potentially new semi-classical corrections that can be investigated. Additionally, if somehow black holes accrete dark matter, the paucity of black holes immediately above a certain mass could be linked to the presence of dark matter halos that the black holes can feed on. Black holes below this mass would be too small to capture the Bose-condensed dark matter halos.