Models of cosmological black holes

Introduction.— More than a hundred astrophysical black holes (ABHs) — dark, massive, ultra-compact objects — have been identified [1, 2, 3, 4]. All observations are well-described by the classical Kerr solution to Einstein-Hilbert gravity [2, 4, 5]. This consistency is not yet a demonstration of key black hole features, such as light trapping. Moreover, a literal identification of ABHs with classical black hole solutions requires addressing singularities and unresolved semiclassical issues. This tension motivates development of the alternative ABH models, many of which are observationally viable [6, 5].

A rough classification scheme differentiates ABH models with horizons from their horizonless counterparts. The event horizon, a null surface that causally disconnects the black hole’s interior from the outside world [7, 8], is the defining feature of a mathematical black hole (MBH) [9]. Physical black holes (PBHs) [9] are trapped spacetime regions, potentially transient and not always overlapping with MBHs [10]. In contrast, various models describe ABHs without introducing horizons [5]. Distinguishing among ABH models remains one of the central problems in black hole physics [6].

In fact, we should distinguish three possibilities [10, 11]. From a distant observer’s point of view, gravitational collapse beyond neutron star density can result in either: (i) Ongoing collapse, with a horizon as an asymptotic ($t\to\infty$) concept. Under broad conditions (the energy conditions discussed below), MBHs are gravitational collapse’s asymptotic states [7, 8]; (ii) Formation of a transient or stable object, where a suitably-defined deviation from $r_{\textsl{g}}$ reaches a minimum $\varepsilon>0$ [5]. Black hole mimickers fall into this category; (iii) Formation of an apparent horizon in finite $t_{\text{f}}$. PBHs, subject to this requirement, fall into this category [10, 12]. The same alternatives arise in mergers.

Here, we investigate the consequences of accepting option (iii). After describing PBHs, we analyse their direct coupling with cosmological dynamics, identify a viable black-to-white hole transition scenarios, and present the obstacles in forming traversable wormholes. It is noteworthy that, like most conceptual and observational black hole discussions, our analysis is framed within the semiclassical framework. Thus, our conclusions serve both as a list of PBH features, and as indicators of potential breakdown of semiclassical gravity on a macroscopic scale.

Semiclassical black and white holes. — Our formalism is based on two premises. First, the spacetime geometry can be described using a metric $\textsl{g}_{\mu\nu}$ with classical concepts of horizons, trajectories, and the equations that describe them remaining valid. Second, we assume the metric $\textsl{g}_{\mu\nu}$ is a solution to the semiclassical Einstein equations [13, 14, 15]

$$G_{\mu\nu}=8\pi{\langle}\hat{T}_{\mu\nu}{\rangle}_{\omega}\ ,$$

(1)

where the left hand side is the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\tfrac{1}{2}R\textsl{g}_{\mu\nu}$, and the right hand side is the effective energy-momentum tensor (EMT), ${\langle}\hat{T}_{\mu\nu}{\rangle}_{\omega}\equiv T_{\mu\nu}$. It includes the renormalized expectation value of all matter fields, higher-order terms arising from its regularisation, and possible contributions arising from modifications to the Einstein-Hilbert gravity or a cosmological constant $\Lambda$. In the subsequent analysis we do not use any specific property of the state $\omega$ and do not separate the matter EMT into the collapsing matter and (perturbatively-obtained) quantum excitations.

In discussing PBH properties [10], we apply the weakest form of the cosmic censorship and require absence of scalar curvature singularities at the apparent horizon [7]. Unlike the teleological event horizon, the apparent horizon is in principle observable [16]. We express this observability by requiring a finite formation time according to the clocks of distant observers [17, 10].

Most of our discussion is restricted to spherical symmetry, where the two above assumptions lead to the exhaustive classification of allowed metrics and to clarification of some key features of the black hole formation and the near-horizon geometry. We will also use Kerr–Vaidya black holes as the simplest example of axially-symmetric dynamical models [18, 19].

A general spherically symmetric metric in Schwarzschild coordinates ($r$ is the areal radius) is given by

$$ds^{2}=-e^{2h(t,r)}f(t,r)dt^{2}+f(t,r)^{-1}dr^{2}+r^{2}d\Omega_{2},$$

(2)

while using the advanced null coordinate $v$ results in the form

$$ds^{2}=-e^{2h_{+}(v,r)}f_{+}(v,r)dv^{2}+2e^{h_{+}(v,r)}dvdr+r^{2}d\Omega_{2}\ .$$

(3)

The function $f$ is coordinate-independent, i.e. $f(t,r)\equiv f_{+}\big{(}v(t,r),r\big{)}$ and in what follows we omit the subscript. It is conveniently represented via the Misner–Sharp–Hernandez (MSH) mass $M_{\mathrm{M}}\equiv C_{\mathrm{M}}/2$ [20] as

$$f=1-\frac{C_{\mathrm{M}}(t,r)}{r}=1-\frac{C_{+}^{\mathrm{M}}(v,r)}{r}={\partial}_{\mu}r{\partial}^{\mu}r\ ,$$

(4)

The functions $h$ and $h_{+}$ play the role of integrating factors in the coordinate transformation

$$dt=e^{-h}(e^{h_{+}}dv-f^{-1}dr)\ .$$

(5)

For example, the Schwarzschild metric corresponds to $h\equiv 0$, $C_{\mathrm{M}}\equiv r_{\textsl{g}}=\mathrm{const}$, and $v=t+r_{*}$, where $r_{*}$ is the tortoise coordinate [8, 7], while the de Sitter metric in the static patch is given by $h\equiv 0$, $C_{\mathrm{M}}=H^{2}r^{3}$, where $H$ is the constant Hubble parameter [20, 21].

In a cosmological setting (such as depicted on Fig. 1), we assume that a separation of scales exists between geometric features associated with the black hole and those of the large-scale universe [12]. In this case, the apparent horizons of the PBH is given by the real roots of $f(t,r)=0$ in the near-region  [10]. The largest such root is the Schwarzschild radius $r_{\textsl{g}}$ that represents the location of the outer black hole horizon. Invariance of the MSH mass implies

$$r_{\textsl{g}}(t)=r_{+}\big{(}v(t,r_{\textsl{g}}(t)\big{)}\ ,$$

(6)

where $r_{+}$ is the equivalent root of $f(v,r)$. Unlike the globally defined event horizon, the apparent horizon is foliation-dependent. However, it is invariantly defined in all foliations that respect spherical symmetry [22] which will be used in the following.

In preparation for the study of a PBH as an inhomogeneity on the cosmological background, described by $\Lambda$–Cold Dark Matter [23], we set

$$T_{\mu\nu}=T_{\mu\nu}^{\mathrm{mat}}-\Lambda\textsl{g}_{\mu\nu}/8\pi\ ,$$

(7)

separating the EMT into the matter and the cosmological constant parts, respectively. Both the Einstein equations and the regularity requirements are most conveniently written if we introduce the effective EMT components

	$\displaystyle\tau{{}_{t}}\vcentcolon=$	$\displaystyle e^{-2h}{T}_{tt}=\tau_{t}^{\mathrm{mat}}+\Lambda f/8\pi\ ,$		(8)
	$\displaystyle\qquad{\tau}{{}^{r}}\vcentcolon=$	$\displaystyle T^{rr}=\tau^{r}_{\mathrm{mat}}-\Lambda f/8\pi\ ,$		(9)
	$\displaystyle\tau{{}_{t}^{r}}\vcentcolon=$	$\displaystyle e^{-h}{T}{{}_{t}^{r}}=\tau_{t}^{r\mathrm{mat}}.$		(10)

The Einstein equations for the components $G_{tt}$, ${G}{{}_{t}^{r}}$, and ${G}^{rr}$ are then, respectively

	$\displaystyle\partial_{r}C_{\mathrm{M}}=8\pi r^{2}{\tau}{{}_{t}}^{\mathrm{mat}}/f+\Lambda r^{2}\ ,$		(11)
	$\displaystyle\partial_{t}C_{\mathrm{M}}=8\pi r^{2}e^{h}\tau_{t}^{r\,\mathrm{mat}}\ ,$		(12)
	$\displaystyle\partial_{r}h=4\pi r\left(\tau_{t}^{\mathrm{mat}}+\tau^{r}_{\mathrm{mat}}\right)/f^{2}\ .$		(13)

Hence the most general metric that describes the cosmological embedding of a spherical inhomogeneity into spatially flat de Sitter spacetime is given by Eq. (2) with

$$C_{\mathrm{M}}=C(t,r)+H^{2}r^{3}\ ,$$

(14)

where $C$ (formally) satisfies the Einstein equations without the cosmological constant and $H=\sqrt{\Lambda/3}$.

To enforce the regularity condition it is sufficient to ensure that only the curvature scalars $R$ and $R_{\mu\nu}R^{\mu\nu}$ are finite [10]. Moreover, the cosmological constant drops out of the key equations, and the analysis is performed along the usual lines of the self-consistent approach [10] (see Supplementary Material). As a result, $C$ and $C_{\mathrm{M}}$ have the same structure [12] as $r\to r_{\textsl{g}}$. There are two admissible classes of the near-horizon solutions that are distinguished by the scaling of the effective EMT components as $f^{k}$, $k=0,1$ when $r\to r_{\textsl{g}}$.

The three components scale for the generic ($k=0$) solution as

$$\tau_{t},\tau^{r}\to-\Upsilon^{2},\qquad\tau_{t}^{r}\to\pm\Upsilon^{2}\ ,$$

(15)

for some $\Upsilon(t)$. The two metric functions are

$$C=r_{\textsl{g}}-4\sqrt{\pi}r_{\textsl{g}}^{3/2}\Upsilon\sqrt{x}+\mathcal{O}(x),\qquad h=-\frac{1}{2}\ln{\frac{x}{\xi}}+\mathcal{O}(\sqrt{x}),$$

(16)

where $x\vcentcolon=r-r_{\textsl{g}}(t)$, and the function $\xi(t)$ is determined by the choice of the time variable.

Consistency of the Einstein equations requires that

$$\frac{dr_{\textsl{g}}}{dt}{\equiv r^{\prime}_{\textsl{g}}}=\pm 4\Upsilon\sqrt{\pi r_{\textsl{g}}\xi},$$

(17)

where the plus (minus) sign corresponds to expansion (contraction) of the Schwarzschild radius. The case of $r_{\textsl{g}}^{\prime}<0$ is most conveniently described using the advanced null coordinate $v$. Evaluation of the expansions of null geodesic congruences [7, 20] identifies the domain $f<0$ as a trapped region, and thus a PBH. The case of $r_{\textsl{g}}^{\prime}>0$ is most conveniently described using the retarded null coordinate $u$. In this case the domain $f<0$ is the anti-trapped region, and we call the resulting entity a physical white hole (PWH) [10].

The function $\Upsilon^{2}(t)>0$ determines the energy density at the apparent horizon, and the higher-order terms are matched with higher-order terms in the EMT expansion [10]. Both solutions violate the null energy condition (NEC) [7, 24], i.e. there are null vectors $k^{\mu}$ such that $T_{\mu\nu}k^{\mu}k^{\nu}<0$. This is consistent with the result [7, 8] that the apparent horizon is not ‘visible’ to a distant observer unless the NEC is violated.

In both cases, the hypersurfaces $r=r_{\textsl{g}}$ are timelike. Therefore, both null (as indicated by Eq. (17)) and sufficiently fast massive test particles can enter a PBH in finite time $t$. The same applies to a PWH, although the analysis is somewhat more complex [19].

Regular black holes (RBHs) are non-singular PBHs. As such, they must have an inner horizon, which can be either timelike or null [25]. Analysis that is similar to the above shows that it is then a timelike hypersurface and the NEC is satisfied in its vicinity [26, 27]. The horizons do not joint smoothly, enabling the foliation-independent characterisation of the inner and outer segments [12, 27].

Two properties are crucial for the following. First, after formation of a spherically symmetric PBH, its outer apparent horizon can only contract, meaning the PBH mass decreases. Conversely, a PWH can only grow. In axially symmetric models like Kerr–Vaidya black holes, the situation is more complex. Expansion of the apparent horizon in $(v,r)$ coordinates and contraction of the anti-trapping horizon in $(u,r)$ coordinates are not obviously excluded. Nevertheless, Ref. [19] identifies two issues. During a black hole’s growth, signals from the apparent horizon cannot cross the hypersurface $r=M(v)+\sqrt{M^{2}(v)-a^{2}}$, which for expanding black holes is spacelike and screens the apparent horizon. Some white hole spacetimes with decreasing $M(u)$ are timelike geodesically incomplete.

Second, the equation of state near the apparent or anti-trapping horizon is $\rho=p$. For small angular momentum values, this applies to Kerr–Vaidya black holes. At the outer horizon, both energy density and pressure are negative, while at the inner horizon, they are positive. Therefore, the PBH equation of state is more exotic than the de Sitter vacuum $\rho=-p>0$ that is used as a core in a variety of models [5].

Cosmological coupling of black holes.— Within sub-percent precision the Universe is described at cosmological scales by spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric [23, 28]

$$ds^{2}=-d\tau^{2}+a^{2}(\tau)(d\chi^{2}+\chi^{2}d\Omega_{2})\ ,$$

(18)

where $\chi$ is the comoving distance, $\tau$ the cosmic standard time, and $a(\tau)$ the scale factor. The Hubble parameter $H\vcentcolon=\dot{a}/a$. In the present $\Lambda$-dominated epoch this geometry is well-approximated by the spatial flat de Sitter metric with $a(\tau)=H^{-1}\exp(H\tau)$.

Smaller scale inhomogenuities, including embedding of the black holes in this cosmological background [20, 21, 29] are active research subjects. In particular, the question whether black holes are affected by the large-scale dynamics of the cosmological background [30] was recently revisited. It was proposed in Ref. [31] that the black hole masses vary with the scale factor according to

$$M(a)=M(a_{\mathrm{in}})\left(\frac{a}{a_{\mathrm{in}}}\right)^{q}\ ,$$

(19)

where $a_{\mathrm{in}}<a$ is the scale factor at which the object becomes cosmologically coupled, and $a$ is its current value.

Ref. [32] then reported a value $q=3.11^{+1.19}_{-1.33}$, and a strong confidence in excluding $q=0$. This growth is immediately consistent with a cosmological vacuum equation of state $\rho=-p$ and thus black holes are a source of the dark energy. These conclusions, however, were criticised on both observational [33, 34, 35, 36] and theoretical [37, 30, 36, 38, 39] grounds.

The known explicit embedding models (Schwarzschild–Kottler–de Sitter, McVittie [20, 21] and Kerr–de Sitter [29] show no cosmological coupling [30, 39]. Analysis of general scenarios typically focuses on coupling of the cosmological background to local, spherically symmetric objects, since the black hole angular momentum effects are expected to be negligible on cosmological scales [30]. Despite ambiguity of black hole definitions (MBH and PBH are just two of them, see Ref. [40] for the discussion), in spherical symmetry a quasi-local MSH mass, that coincides with the Hawking–Hayward mass, is the preferred quantity [30, 36]. Moreover, it is directly tied to the notion of a PBH, as $M_{\mathrm{M}}\equiv r_{\textsl{g}}/2$. Hence Eq. (19) implies

$$\frac{dr_{\textsl{g}}}{da}=\frac{\dot{r}_{\textsl{g}}}{\dot{a}}=q\frac{r_{\textsl{g}}}{a},$$

(20)

where derivatives both of the Schwarzschild radius and the scale factor are taken with respect to the cosmological time.

As a result

$$\frac{dr_{\textsl{g}}}{da}=\frac{\dot{r}_{\textsl{g}}}{aH}\approx\frac{r^{\prime}_{\textsl{g}}}{r_{\textsl{g}}\,H}\frac{r_{\textsl{g}}}{a},$$

(21)

where we assumed that the distant observer is in the asymptotically de Sitter region but still far from the cosmological horizon, i.e. $r_{\textsl{g}}\ll r\ll H^{-1}$ and in this case $dt\approx d\tau$. Eq. (17) implies that $dr_{\textsl{g}}/da$ is then negative. So long as the evaporation does not dramatically deviate from the Hawking–Page form [8] $r_{\textsl{g}}^{\prime}\propto r_{\textsl{g}}^{-2}$, at the current epoch $1\gg|q|\sim 0$. On the other hand, for the primordial black holes at the end of inflation and/or early radiation-dominated epochs [23, 41, 42] (that are treated as PBHs), the cosmological coupling can be strong.

Black-to-white hole bounce.— The idea that quantum effects can prevent black hole singularity from forming can be realised in several ways. For example, reformulationg the one-loop corrections as effectively modifying the Langrangian to fourth order, and collapse of a thin null shell results in transient, even if long-lived, trapped region [43]. Loop quantum gravity (LQG) corrections [44, 45], similarly to their effect in quantum cosmology, modify the collapse (for example, by introducing additional terms the classical Oppenheimer–Snyder collapse [45, 46] leading to the singularity avoidance and transient trapped region.

Over the past decade, there has been a resurgence in interest in white hole solutions, once considered less physically relevant predictions of general relativity [7, 8, 20]. Black-to-white hole transition scenarios [46, 47, 48, 49, 50, 52, 51, 53, 54, 55] have elevated their status. Two versions are discussed [52, 53, 55]: “fireworks” [48] and “Planck stars” [49], both modeled as tunneling processes with differing characteristic times $t_{\mathrm{T}}$. In the former, tunneling probability is of the order $\exp{(-M^{2}/m_{\mathrm{Pl}}^{2})}$, thus $t_{\mathrm{T}}\sim M^{2}$, with $M=r_{\textsl{g}}/2$ as the pre-transition black hole mass and $m_{\mathrm{Pl}}$ the Planck mass. Since the Hawking–Page evaporation time scales as $r_{\textsl{g}}^{3}$, black and white holes are macroscopic, and Hawking radiation effects are negligible. Recently, an LQG-inspired modification of the Oppenheimer-Snyder model was described [55], featuring a single asymptotic region with a complete, singularity-free Lorentzian metric that satisfies Einstein equations up to the tunneling region. In the latter scenario, $t_{\mathrm{T}}\sim M^{3}$, with tunneling occurring after the black hole shrinks to about $m_{\mathrm{Pl}}$. This potentially long-lived Planck-scale remnant is suggested as a dark matter component [51].

Regardless of scenario specifics, it is agreed that the transition occurs in a regime where semiclassical physics breaks down [46, 47]. However, significant parts of both black and white hole domains are described semiclassically. Depending on their specifics, bounce scenarios may involve spacelike and timelike horizon segments, as well as a constant final white hole mass. Our approach limits these possibilities. Fig. (2) shows a modified semiclassical Penrose diagram of the black-to-white hole transition with a single asymptotic region based on this model. Within the validity domain of semiclassical gravity, objects before and after the bounce are PBH and PWH, respectively. While agnostic about the actual bounce mechanism, the timelike nature of PBH’s inner and outer apparent horizons determines the pre-bounce structure. As only $r^{\prime}_{\textsl{g}}>0$ white hole solutions are allowed, growth termination requires closing the anti-trapped region, leading to a massive object. Its subsequent evolution is undefined in our model, but it is momentarily singularity-free and contains neither trapped nor anti-trapped regions.

Wormholes.— A traversable wormhole (TWH) is a hypothetical shortcut in spacetime that connects two distant regions of our Universe or two separate Universes [56, 57]. As the NEC violation [63, 57] and closed timelike curves are their unavoidable attributes, their primary use was in the realm of science fiction. However, in the last two decades they are considered as another class of ABH models [5, 58, 59, 60, 61] their potential observational signatures are investigated [5, 59].

The original wormhole solutions were characterized using the embedding diagrams and explicit description of the two spatial sheets that are connected at the wormhole’s throat. Wormhole metrics are often written using the Schwarzschild coordinates on each sheet,

$$ds^{2}=-e^{\Phi}dt^{2}+\frac{dr^{2}}{1-C/r}+r^{2}d\Omega_{2},$$

(22)

where $\Phi$ is a regular function and $r\geqslant b_{0}>0$, where $b_{0}$ is the (largest) root of $C(t,r)=r$ [57] that defines the wormhole throat. The invariant characterization of the throat that is valid for generic wormholes identifies it as an outer marginal trapped surface subject to additional conditions [63, 64, 65], allowing to use our formalism [66].

Consider a dynamical generalisation [62] of the Simpson–Visser metric [60] that is obtained by rewriting it in the retarded or advanced coordinates $w=u,v$ and allowing the mass paramater to depend it,

	$\displaystyle ds^{2}=$	$\displaystyle-\left(1-\frac{2M(w)}{\sqrt{\eta^{2}+a^{2}}}\right)dw^{2}-2\epsilon dwd\eta$
		$\displaystyle+(\eta^{2}+a^{2})d\Omega_{2}\ .$		(23)

Here $a\geqslant 0$ is a parameter, $-\infty<\eta<+\infty$, and $\epsilon=\pm 1$ for $w=u,v$, respectively. As $r=\sqrt{\eta^{2}+a^{2}}\geqslant a$, the MSH mass is $C_{\mathrm{M}}=2M+a^{2}(r-2M)/r^{2}$.

This metric represents the Schwarzschild spacetime for $a=0$, a regular black or white hole (also known as a black bounce) for $a<2M$, and a traversable wormhole for $a>2M$ [62]. In the former case, the Schwarzschild radius $r_{\epsilon}(w)=a\equiv b_{0}$ marks the wormhole throat, while for a regular black or white hole, $r_{\epsilon}(w)=2M(w)>a$. Notably, in the static case, the solution falls into the $k=1$ class, with all dynamic cases belonging to $k=0$.

TWH analysis parallels RBH’s. Irrespective of quantum gravitational effects during its formation, it is presumed that from some stage it is describable by semiclassical gravity. Finite formation time from a distant observer’s perspective and absence of curvature singularities are now part of the basic traversability requirements [56]. Ref. [66] demonstrates that dynamical wormholes do not lead to standard Ellis–Morris–Thorne [56] or Simpson–Visser [60] static solutions, while allowed static limits are non-traversable and/or violate the bounds on the NEC violation [24].

Modification of the construction of Ref. [62] allows to bypass this no-go result but faces another obstacle. If the process starts with the metric (23) describing a PBH with $r_{+}(v)=2M(v)>a$, then when the decreasing Schwarzschild radius reaches down to $a$ the object becomes an one-way wormhole [60]. Then, if the mass function $M(v)<a/2$ continues to decrease, the solution describes a TWH with the throat at $b_{0}=a$. However, if the loss of mass is driven by the Hawking radiation, and its temperature is proportional to $\kappa_{\mathrm{K}}$, the Hayward–Kodama surface gravity [67] (see Supplementary Material for details), then as $r_{+}=2M(v)\to a$ the temperature approaches zero, as $\kappa_{\mathrm{K}}\to 0$ [17].

Discussion.— Proving the adequacy of a particular ABH model includes disproving its alternatives. A useful step in this direction is constraining their properties, and the results of this work provide such constraints. PBHs, whether or not they possess an event horizon, do not cosmologically couple, likely from the radiation-dominated era. Theoretically, models that depict RBHs with de Sitter cores must reconcile with the equation of state $p=\rho$ near both the inner and outer horizons.

Additional models for singularity avoidance have been proposed. We demonstrate that black-to-white hole transition aligns with semiclassical gravity, but a macroscopic stable remnant, be it a black or white hole, does not. However, the rate of change in its mass remains unconstrained by these general considerations. Beyond the absence of a known mechanism for producing wormholes, their semiclassical existence is restricted by requiring each mouth to develop from an evaporating black hole.

PBHs exhibit remarkable properties distinguishing them from conventional black holes. Translating these properties into observational signatures will enable answering the question of their role in the physical Universe.

Acknowledgments.— PKD, SM, and IS are supported by an International Macquarie University Research Excellence Scholarship. The work of DRT is supported by the ARC Discovery project Grant No.DP210101279. FS is funded by the ARC Discovery project Grant No. DP210101279.