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 and Synder modeled the gravitational collapse of star to a black hole by approximating the star with a uniform sphere of dust (hereafter O-S model) (Oppenheimer and Snyder, 1939). 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 (Vaidya, 1951; Misner et al., 1973; Singh and Witten, 1997; Goswami and Joshi, 2004; Joshi, 2007).

In the standard formalism from Misner et al. (1973), the stellar surface is considered to be initially at rest. Here we consider configurations with all possible initial velocities. We derive closed-form solutions for the equations of motion in Schwarzschild and Kruskal coordinates for space-like, time-like, and light-like geodesics.

As an example application of our closed-form solutions for the classical O-S model with non-zero initial velocities, we consider the macroscopic collapse of a spherically symmetric sphere of dust composed of ultra-light particles. To approximate quantum effects, the radius of the star is approximated by a Gaussian wavepacket that is initially centered far from r = 2M. Its evolution is then followed via a simple path integral approach that extends the results of Redmount and Suen (1993) from a relativistic free particle to a particle constrained by non-trivial gravity. Gravitational collapse incorporating approximate treatments of quantum mechanics has been considered by a variety of authors (Hájíček et al., 1992; Hawkins, 1994; Casadio and Venturi, 1996; Ansoldi et al., 1997; Berezin, 1997; Zloshchastiev, 1998; Alberghi et al., 1999; Vaz and Witten, 2001; Corichi et al., 2002; Ortíz and Ryan, 2007). They involve non-equivalent ways of quantization that often produce physically different results (Dolgov and Khriplovich, 1997). Our path-integral approach approach is simpler, and can be easily compared to the assumption that the particle obeys the relativistic Schrödinger equation.

In the classical model a star is idealized as a collapsing self-gravitating dust sphere of uniform density and zero pressure where the constituent particles have the attributes of classical dust: each particle is assumed to be infinitesimal in size and to interact only gravitationally with other matter. The inclusion of quantum mechanical effects lifts some of these assumptions allowing for the possibility that some configurations will not collapse to black holes but will disperse or even form stable new configurations. Quantitatively, quantum treatment is necessary in macroscopic stellar collapse when the action S is of the order ℏ (Narlikar, 1977). In our case, the S/ℏ ~ 1 condition corresponds to mM ~ ℏ. For macroscopic black holes of masses M=1M⊙-109M⊙, this implies that quantum treatment is necessary for ultra-light constituent particles of m ~ 10⁻¹⁰ − 10⁻¹⁹ eV. In nature, such ultralight particle could be dark matter. Dark matter comes close to the attributes of classical dust, and some dark matter clouds may be dense enough to collapse to black holes. Particles as light as 10⁻²² − 10⁻²³ eV have been proposed as constituents of dark matter halos (Hu et al., 2000; Lesgourgues et al., 2002; Lundgren et al., 2010; Khmelnitsky and Rubakov, 2014).

To approximate quantum effects of the collapse of such a halo comprised of ultra-light particles, we start with an initial wavefunction that represents the position of the particle on the stellar surface, and is at first far from 2M. To study the propagation of the wavefunction, we integrate over all possible paths taken by the particle. At a given time, the outgoing wavefunction comprises the probability that the star disperses, and the ingoing wavefunction the probability that it collapses. We compute the propagator in analytical form for a particle on the surface of the star in the WKB approximation, and compare this to the limited assumption that the particle obeys the relativistic Schrödinger equation. Our equations reduce to a free particle case when the mass of the star is zero; then the solution to the relativistic Schrödinger equation is the exact representation of the wave function (Redmount and Suen, 1993). In the more general case of a particle on the surface of a star, this representation is no longer correct. However, the comparison is instructive. As expected, we observed that the WKB and relativistic Schrödinger approximations are out of step at early and late times, and appear to converge toward one another at intermediate times.

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. In Kruskal coordinates, one can continue to study the evolution of the star inside the apparent horizon (r < 2M). The paths we have derived here include time-reversing space-like paths, which allow for the possibility of extraction of information from inside the horizon to the outside. These paths turn inside the horizon r > 0, and head toward r = 2M. In future work, explicit geodesic equations in Kruskal coordinates may be used as a stepping stone to model behavior inside r = 2M.

The rest of the paper is structured as follows: in Section 2 we describe the Oppenheimer–Snyder model. Section 3 includes an overview of the classical action and paths, a computation of the Oppenheimer–Snyder limit and the derivation of the classical paths in Schwarzschild coordinates. Section 4 details the classical paths in Kruskal coordinates, which comprise a starting point for future work that explores the collapse inside r = 2M. As an example application of the closed form geodesics, Section 5 describes the quantum treatment of the dust collapse applicable to ultra-light particles in Schwarzschild coordinates. The conclusions follow in Section 6.

In General Relativity, a first approximation to the exterior space-time of any star, planet or black hole is a spherically symmetric space-time modeled by the Schwarzschild metric. This is a consequence of Birkhoff's theorem (Misner et al., 1973). 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

These equations correspond to Equations (32.10a–32.10c) of Misner et al. (1973). The star collapses to a singularity in 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 Misner et al. (1973) the stellar surface is considered to be initially at rest. Here we integrate the equations of motion for configurations with all possible initial velocities.

The relativistic action S and the Lagrangian L for this system are

where τ is the proper time, and r˙ = 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 integrated to

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 c₁ can be used to separate the space-time regions

In the free particle case (M → 0 in Equation 11), c1=r˙2 represents the velocity squared of the particle.

Our study is the first to consider space-like paths in Kruskal coordinates for a particle on the surface of a collapsing star. They have not been explicitly considered before because they are believed to be of low probability. We conjecture that they play an important role in quantum mechanical collapse in the same way the probability of passing through the potential barier is important in tunneling. Classically, it can never happen, and yet this probability cannot be ignored in quantum mechanics.

In this section we derive and discuss the closed-form solutions to the equation of motion for the space-like, light-like and time-like paths in Kruskal coordinates. 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

with

and

Thus the line element in Kruskal coordinates takes the form

The singularity at r = 0 occurs at u² − v² = −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

Note that here u˙=du/dv.

The equation of motion can be reduced to

where x_r is given by Equation (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.

The classical paths describing the evolution of the star outside r = 2M have been discussed in Sec. 3.2 in Schwarzschild coordinates. It was shown that there were direct space-like and light-like paths as well as direct and indirect time-like paths. Any two points in (r, t) were connected by a unique classical path.

What is new in Kruskal space-time are the turning points in the u − v plane for all space-like, light-like and time-like paths. Spacelike paths (c₁ > 1, |du/dv| > 1) turn in time at points when dv/du = 0, whereas timelike paths (c₁ < 1, |du/dv| < 1) turn in space when du/dv = 0. Additionally, there are indirect space-like and light-like paths that have turning points in r inside the apparent horizon (here we refer to paths as being indirect when they turn in r; the light-like paths turn at r = 0). In contrast to the time-like paths, the in-going space-like paths are not unique. A point outside r = 2M might be connected to a point inside the apparent horizon via a direct and an indirect space-like path or via two indirect space-like paths.

We determine the regions (Table 3) and the light-cone boundaries (Table 4) that delimit the different kinds of paths that start at (u_i, v_i = 0) outside the horizon and reach (u_f, v_f) inside the apparent horizon. Outside the horizon, 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 2 by tanh-1(vf/uf). The boundary values of r_i (v_i = 0) with r_f < 2M, are calculated from the equations in Table 4. 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. The point ri=b4′ corresponds to x_i = 0 with c1=-2M/(b4′-2M). The c₁ = 0 and c₁ = 1 (lightlike) paths that reach points (u_f, v_f) inside the horizon are used to determine the r_i values b5′ and b56′, respectively. The ri=b6′ value corresponds to x_f = 0 with c₁ = −2M/(r_f − 2M) (turning point at r = r_f < 2M, space-like path, c₁ > 1). The value of ri=b7′ is to be determined by using c₁ = −2M/(r_a − 2M) (x_a = 0, r_a < r_f < 2M) in the equation of motion. The two unknowns b7′ and r_a are determined by using the equation of motion and its partial derivative with respect to c₁. Both equations are given in Table 4. In the region outside b7′ (ri>b7′), there exist no classical paths to (u_f, v_f).

In Figures 2A,B we trace direct time-like, direct space-like, and indirect space-like trajectories that start at v_i = 0 and pass through a fixed point P(u_P = 9.995, v_P = 10.0) with r_P ≈ 0.8M, which is located inside the apparent horizon. Path 1 and path 2 are time-like, and originate at different points outside r = 2M. Path 1 (u_i = 2.59, c₁ = −1.05) lies in the b4′-b5′ region of Table 3. Path 2 (u_i = 16.15, c₁ = 0.8) lies in the b5′-b56′ region, which contains time-like paths with a turning point in the u − v plane (du/dv = 0) that lies outside the apparent horizon.

The next region is b56′-b6′, where each point outside the horizon is connected to points inside the horizon by two space-like paths: one direct and one indirect. Path 3a and path 3b originate at the same u_i = 27.6 (r_i = 10.5M). Path 3a reaches P directly. However, after having passed through P, it turns at r_a = 0.48M before reaching the horizon like all space-like paths. Path 3b is indirect turning at r_a = 0.3M (u_a = 11.3, v_a = 11.35), and then reaching P on its way back toward the apparent horizon. Both paths also have a v − u turning point outside the apparent horizon where dv/du = 0. After having passed through its v − u turning point, each path moves back in time (the space coordinate u continuously decreases, while the time coordinate v increases up to the turning point and then decreases).

In the ri=b6′-b7′ region there are two indirect paths that connect the same point outside the horizon to a point inside r = 2M. Paths 4a and 4b connect the point of u_i = 46.3 to P. They each have a dv/du = 0 turning point outside the horizon, and also turn in r inside the horizon before reaching P. Path 4a turns at r_a = 0.77147 < r_P and path 4b turns at r_a = 0.79969 < r_P. It can be seen that the paths are very close together. As u_i increases, the paths in this region become progressively closer until they merge at ri=b7′. Path 5 shows the single indirect path that originates at u_i = 46.7 (ri=b7′=12.13M). Beyond this point, there are no real solutions and hence no way to reach point P.

Figure 2C displays the c₁ values of space-like paths reaching point P (u_P = 9.995, v_P = 10.0) from u_i ≥ 46.1 at v_i = 0. At first, for each u_i in the figure there are two c₁ values with which the final point P can be reached. The figure clearly shows the roots (c₁ values) coming closer and closer together until they merge at the caustic point r=b7′. Beyond this point there are no paths that reach point P. Extensions to this work to determine the Kruskal-WKB wavefunction will require an analysis of the caustic at r=b7′ to remove any potential divergences.

The indirect paths connecting (u_i, v_i) with r_i > 2M to (u_P, v_P) with r_P < 2M penetrate the horizon deeper than r_P turning at r_a < r_P before reaching r_P (e.g., see the green path in Figure 2A). After turning, all paths must continue toward the apparent horizon reaching it at u = v = 0. These paths move back in time from a point outside the horizon, where dv/du = 0, after having traveled to that point forward in Kruskal time from (u_i, v_i = 0) with r_i > 2M.

As discussed before, the turning points in r correspond to x_r = 0 (in the Kruskal paths x_r appears in Equation 58). They occur at r = r_a when x_a = 0, where c₁ = − 2M/(r_a − 2M). Since c₁ > 1 for space-like paths, clearly r_a < 2M (the turning points are inside the horizon). For any r on a path with r_a as the turning point, x_r can be written as 2Mr(r-ra)/(2M-ra). To ensure the non-negativity of the term under the square root all points on this path must have r > r_a. By the same token, for time-like paths r_a > 2M and r < r_a. This case was described in the Schwarzschild analysis.

All space-like paths that reach points inside the horizon subsequently head toward the horizon r = 2M (u = v). Classically, they end at u_f = v_f = 0. However, quantum mechanically there may be paths close to the classical path that bring information from within the black hole horizon to the outside.

We consider a collapsing cloud of ultra-light particles with mM ~ ℏ. Such particles are not localized and thus neither is the surface of the star. The position of a particle on the surface of the star is then approximated by a wavefunction. In a relativistic path integral approach, this wavefunction can be computed in the WKB approximation via an integral over the classical action (Schulman, 2005). It is necessary to include configurations will all possible initial velocities. For each point (r_f, t_f) all classical paths derived above are needed. In this paper, we only perform the WKB analysis in Schwarzschild coordinates, where we can only use the paths to r_f > 2M.