Global Models of Collapsing Scalar Field: Endstate

From the middle of the last century, Hawking and Penrose’s theorems on geodesic incompleteness and the emergence of black hole candidates sparked interest in dynamic singularity formation. As interest in spacetime dynamics grew, it became evident that Schwarzschild or Kerr spacetimes could merely represent an asymptotic state, necessitating the consideration of richer solutions to understand relativistic collapse. Because of their more manageable geometry, researchers turned to spherically symmetric models, particularly focusing on collapse in dust clouds, radiation, and scalar fields.

Research on the collapse of scalar fields began in the late 1960s, and the mathematical problem was approached in a variety of ways, notably in a renowned series of papers by D. Christodoulou [1,2,3,4]. Christodoulou’s work laid the foundation for the theories by proving conditions leading to trapped surfaces and addressing instability with respect to initial data, which advanced the understanding of scalar field collapse and cosmic censorship.

Several years later, a new avenue emerged, spurred by connections to extended gravity theories and string theory. Interest grew in self-interacting models with a potential: indeed, the works cited above dealt at that point only with the free massless case, and refinements where potentials come into play were largely limited to the analysis of non-central singularities [5]. On the other hand, the new line of research, often within a cosmological context, studies homogeneous models with a Robertson–Walker (RW) spacetime background, which simplifies equations into an ODE system.

A wide array of potentials $V\left(\varphi \right)$ has been explored, investigating conditions leading to future singularities in a finite amount of RW time. One of the first and probably most important results in this realm is the work by Scott Foster [6], which introduced methods and techniques that influenced all the following advances on this topic. For further discussion, see [7,8,9] and references therein, including those in the present paper.

At the same time, the picture emerged where, under suitable conditions, the formation of the apparent horizon in this spacetime is such that one can build a model by gluing an RW interior with an external spherical metric. In this way, one obtains a global model where, starting from regular initial data, the apparent horizon does not form. In analogy to the example shown in [3] for non-homogeneous collapse, this example of naked singularity was examined to understand its genericity in terms of tunable parameters. By genericity, here, we mean the stability with respect to small perturbations of the parameters leading to a certain feature, e.g., the appearance of a naked singularity in the modality described above. This research intersected with efforts to identify quantum mechanisms for preventing singularity formation (see, e.g., the review [10]).

In this paper, we revisit the main results concerning open spatial topologies (specifically, when $\kappa =0$ or $\kappa =-1$) with the goal of employing the minimal necessary assumptions on the potential function $V\left(\varphi \right)$. While our focus is on elucidating the arguments underpinning their proofs, it is important to note that our scope does not encompass the vast literature on the subject. For instance, refs. [11,12,13] delve into potentials $V\left(\varphi \right)$ taking negative values on non-compact subsets of $\mathbb{R}$. Conditions on $V\left(\varphi \right)$ leading to generic singularity formation and to the generic formation of the apparent horizon will be derived.

The genericity results presented here refer to the stability with respect to the choice of the initial data for the dynamical system ruling these models, essentially derived from Einstein’s field equations where gravity is coupled with the scalar field self-interacting with $V\left(\varphi \right)$. In fact, one can imagine an analysis of these models where a varying parameter affects the functional dependence of a physical quantity, e.g., the energy density, on the scale factor of RW spacetime. In this way, an alternative and different notion of genericity is obtained, which can lead to a completely different interpretation; in [14,15,16], such situations are shown, where naked singularity formation is stable with respect to the choice of the parameter. The effect of altering this parameter is to simultaneously modify both the function $V\left(\varphi \right)$ and the initial data, thus obtaining a family of scalar field star models collapsing to a naked singularity. The theory we are discussing naturally applies to each member of this family, whose naked singularity formation is unstable to a perturbation of the initial data if the potential is held fixed. Noticeably enough, however, these naked singularities have been proved with numerical methods to be gravitationally strong [17].

The paper is organized as follows. Section 2 presents the interior RW metric and the equations ruling its dynamics. The asymptotic behavior of the variables in the open topologies $\kappa =0$ and $\kappa =-1$ are studied in Section 3 and Section 4, respectively. The construction of the global model is then given in Section 5, together with the main theorems on the genericity of singularity and horizon formation. The different notions of genericity, sketched above, are analyzed in full detail in Section 6, referencing the above-cited examples from the literature. Section 7 is devoted to the conclusions.

First of all, let us work in normalized units in such a way that the Einstein field equation will read as follows:

In order to build a global model, we will consider an interior matter where the energy–momentum tensor is given by where $V\left(\varphi \right)$ is a given potential that depends on a scalar field $\varphi$, a function defined on the spacetime interior manifold. We immediately observe that the potential $V\left(\varphi \right)$ can also embody the contribution of the cosmological constant $\Lambda$, via its redefinition $V⟶V+\Lambda$.

We suppose that the gradient of this function is timelike, allowing us to choose a comoving gauge such that the metric is given by a homogeneous and spherical RW model: where $\mathrm{d}{\Omega }^2$ is the line element of $S^2$ embedded in the Euclidean space ${\mathbb{R}}^3$, $a\left(t\right)$ is the scale factor of the homogeneous interior, and $\kappa$ is a parameter related to the (constant) curvature of the spatial part. As is well known, relevant cases are given by $\kappa =-1,0,+1$. The Einstein field equations then become: which implies the Bianchi identity $T_{;\nu }^{\mu \nu }=0$, now expressed as follows: By (3), solutions such that $\dot{\varphi }\left(t\right)\equiv 0$ correspond to an energy–momentum tensor given by $T_{\nu }^{\mu }=-V\left({\varphi }_0\right){\delta }_{\nu }^{\mu }$, hence to vacuum solutions with cosmological constant $\Lambda =V\left({\varphi }_0\right)$, i.e., to (anti-) de Sitter solutions. Apart from this particular case, the scalar field $\varphi$ is a solution to the Klein–Gordon equation: where □ is the D’Alambert operator induced by g.

Using the above remark, we will employ a normalization scheme widely used in the literature, introducing the new variables $x,w$ and z defined as follows:

Moreover, we employ a normalized line $\tau$: Using (4), (6) and (9) we obtain the system:

As a matter of fact, the above system is not free since, as we have seen in Remark 2, we have to take into account condition (3), which is guaranteed if we select the initial data satisfying it. This condition, when applied to the new set of unknowns, takes the following form, which allows x to be decoupled from the other unknowns in the system, assuming $V\left(\varphi \right)\ne 0$. This point will be further explored in subsequent discussions.

If $x\left(0\right)>0$ and $\kappa =0,-1$, then (13) implies that $x\left(\tau \right)$ is a decreasing and positive function, hence bounded for every $\tau \ge 0$. Postulating that $V\left(\varphi \right)$ is bounded from below, and as actually stipulated by Assumption 1, we deduce from (16) that w and z will also be bounded for every $\tau \ge 0$. The only unknown that can (and indeed does) attain unbounded values is $\varphi \left(\tau \right)$. To control the behavior at infinity of the scalar field, we introduce a new variable s, related to $\varphi$ by where $f:\mathbb{R}\to (-1,1)$ is a strictly increasing function of class $C^2$ such that the following two conditions hold: In other words, we “compactify” the variable $\varphi$, casting it into the set $(-1,1)$. With this new variable, the system reads as follows: where now the constraint takes the form:

The main drawback of this formulation is that it cannot be extended by continuity in $s=\pm 1$, due to the term $V^{\prime }\left(f^{-1}\left(s\right)\right)$ in (20). Indeed, while $f^{\prime }\left(f^{-1}\left(s\right)\right)$ can be extended by continuity at $s=\pm 1$ by setting its value to zero, $V^{\prime }\left(f^{-1}\left(s\right)\right)$ may possibly diverge. To overcome this issue, we define the function $u:(-1,1)\to \mathbb{R}$ as follows:

Exploiting the constraint (22), which is invariant by the flow of (18)–(21), we obtain the following dynamical system:

To extend the meaning of this system also at infinity and, in general, to obtain the main results of this paper, we introduce the following assumption.

Observe that the hypotheses made on $V\left(\varphi \right)$ allow the dynamical system (24)–(27) to be extended by continuity at $s=\pm 1$. Also observe that $u\left(s\right)$ is not well defined on the zeroes of the potential V, which, however, are contained in a compact subset of $(-1,1)$.

In this section, we revisit the case where $\kappa =0$, which was analyzed in [18], now employing the compactification scheme we previously introduced. This approach mirrors the methodology applied in [19] to the study of homogeneous perfect fluid collapse within $f\left(R\right)$ theories.

First, let us rewrite the system (24)–(27) in this particular case: where (22) now reads as follows:

Since (28)–(30) and (32) do not involve the unknown z, only the first three equations of the system should be considered. Moreover, (28) and (30) are independent from (29). However, it is important to notice that the described system is applicable only when $u\left(s\right)$ is well-defined, hence when we are sufficiently distant from the zeroes of $V\left(f^{-1}\left(s\right)\right)$. Should this not be the case, (30) must be replaced with the following equation: To overcome this problem, let us fix $\delta >0$ such that whose existence is granted by Assumption 1. As a consequence, (30) is well-defined whenever $\left|s\right|\in (1-\delta ,1)$. Let us choose a smooth function $\psi \left(s\right):[-1,1]\to [0,1]$ such that $\psi \left(s\right)=1$ if $\left|s\right|\le 1-\delta$ and $\psi \left(s\right)=0$ if $\left|s\right|\in (1-\delta /2,1]$. In particular, we have that $\psi \left(s\right)=0$ in a right neighborhood of $s=-1$ and in a left neighborhood of $s=1$, hence where $u\left(s\right)$ is well-defined, thanks to Assumption 1. Setting ${\theta }_1(s,w)$ and ${\theta }_2(s,x,w)$ the right-hand sides of (30) and (33) respectively, we have that both (30) and (33) can be substituted by the following equation: which, with a slight abuse of notation, is always well-defined.

Since we are interested in the asymptotic behavior of the system, we give the following result, recalling that “generically” refers to small perturbations of initial data.

According to the above theorem, generically, the study of limit points of trajectories within the set $M_0$ identifies the candidates for $\omega$-limit points of the system that we are considering. First, let us notice that all the points in $M_0$ such that $s\ne \pm 1$ are not equilibrium points. Indeed, these points should have $\left|w\right|=1$; hence, (28) implies that $s\left(\tau \right)$ is not constant. Hence, we can reduce our analysis on the neighborhoods of the extreme points where $\left|s\right|=1$, and we can work with (30). By reducing once again to the planar system given by (28) and (30), the linearization of the system at a point $(s,w)$ gives the following matrix:

Let us explore the possible cases corresponding to the equilibrium points of (28) and (30):

We summarize the above considerations in the following result.

For the reader’s convenience, we report here the system in its complete form, with $\kappa =-1$: where the constraint (invariant with respect to the flow of the system) reads as follows:

As for the case $\kappa =0$, near the zeros of the potential (43) should be replaced by

Setting ${\theta }_1(s,w,z)$ and ${\theta }_2(s,x,w,z)$ as the right-hand sides of (43) and (46), respectively, we can repeat the construction leading to Definition 1. For this case, we give the following definition.

We can now provide the counterpart of Theorem 1.

As performed for the case $\kappa =0$, the study of the stability of the equilibrium points for the system (41)–(44) in $M_{-1}$ gives us the candidates to $\omega$–limit points of the system. Since $M_{-1}$ is given by (49), we have the following cases:

Summarizing, by the above considerations, we obtain that Proposition 1 holds, even in this case; namely, if $u(-1)>-1$ or $u\left(1\right)<1$, then the scalar field generically diverges, meaning that $s^2\to 1$, with a velocity such that $w^2\to 1$ and $z^2\to 0$.

The above models can be considered the interior matter of a global model obtained performing a suitable junction with an exterior spacetime, thereby obtaining models of collapsing objects, whose endstate can be investigated.

A natural choice for the exterior is the so-called generalized Vaidya solution, which is the spacetime generated by a radiating object [22], and it reads as follows: where M is an arbitrary (positive) function. The matching is performed along a hypersurface $\Sigma =\{r=r_b\}$. As proved in [18,23], the junction conditions of Darmois–Israel between these two models require continuity of M at the junction hypersurface, and the additional condition $\frac{\partial M}{\partial U}=0$, again on the junction hypersurface.

Having established a global model, our next task is to investigate whether recollapse occurs within a finite amount of comoving time. Let us start by recalling that $\frac{\mathrm{d}t}{\mathrm{d}\tau }=-\frac{1}{\sqrt{6}h}=x/\sqrt{2}.$ The time of collapse is then given by On the other hand, (25) implies and therefore, we obtain

Let us consider, for example, the case $\kappa =0$. We have seen that, under suitable assumptions on $V\left(\varphi \right)$, generically $w^2\to 1$. By (55), we have $x\left(\tau \right)\sim e^{-\tau \sqrt{3/2}}$ and then the integral in (54) converges, so the solution collapses in a finite amount of comoving time.

There is also a nongeneric situation in the case of $\kappa =0$, where $w\to u\left(1\right)<1$ (a similar argument applies to $u(-1)>-1$). In this case, by (55), we infer $x\left(\tau \right)\sim e^{-\tau \sqrt{3/2}u{\left(1\right)}^2}$, so again, when $u\left(1\right)>0$, the integral converges and the singularity forms in a finite amount of time. Observe that, in case $w\to u\left(1\right)=0$, one can conceive situations where diverges. In this case, the integral (54) may also diverge, indicating a solution that indefinitely collapses, remaining regular eternally.

In the results stated above, genericity is meant with respect to the initial data of the dynamical system: even if there may exist solutions that do not develop a horizon, and therefore they collapse into a naked singularity, a small perturbation of its initial data will restore black hole formation.

On the other side, one can conceive the same model but prescribe the behavior of some physical quantity, e.g., the energy density, with respect to the scale factor, and see whether the endstate associated with a given prescription is stable with respect to the perturbation of the prescription itself. This analysis has been discussed—at least for the flat case $\kappa =0$—in the paper [23], where a function $\psi \left(a\right)$ is prescribed such that

In the general case, field Equations (3) and (4) yield the following expressions for ${\displaystyle {\left(\frac{\mathrm{d}\varphi }{\mathrm{d}a}\right)}^2}$ and $V=V\left(a\right)$ (see [23,24]):