# Dust collapse in asymptotic safety: a path to regular black holes

Alfio Bonanno<EMAIL_ADDRESS>INAF, Osservatorio Astrofisico di
Catania, via S.Sofia 78, I-95123 Catania, Italy  INFN, Sezione di Catania, via
S.Sofia 64, I-95123 Catania, Italy  Daniele Malafarina
<EMAIL_ADDRESS>Department of Physics, Nazarbayev University,
Kabanbay Batyr 53, 010000 Astana, Kazakhstan  Antonio Panassiti
<EMAIL_ADDRESS>INAF, Osservatorio Astrofisico di Catania, via
S.Sofia 78, I-95123 Catania, Italy  INFN, Sezione di Catania, via S.Sofia 64,
I-95123 Catania, Italy  Dipartimento di Fisica e Astronomia “Ettore Majorana”,
Università di Catania, Via S. Sofia 64, 95123, Catania, Italy

###### Abstract

Regular black hole spacetimes are obtained from an effective Lagrangian for
Quantum Einstein Gravity. The interior matter is modeled as a dust fluid,
which interacts with the geometry through a multiplicative coupling function
denoted as $\chi$. The specific functional form of $\chi$ is deduced from
Asymptotically Safe gravity, under the key assumption that the Reuter fixed
point remains minimally affected by the presence of matter. As a consequence
the gravitational coupling vanishes at high energies. The static exterior
geometry of the black hole is entirely determined by the junction conditions
at the boundary surface. Consequently, the resulting global spacetime geometry
remains devoid of singularities at all times. This result offers a novel
perspective on regular black holes in Asymptotically Safe gravity.

In the realm of general relativity, black holes (BH) are fascinating objects
characterized by spacetime singularities concealed within an event horizon
Hawking and Penrose (1970). The occurrence of singularities makes a compelling
case for the study of models beyond general relativity, where spacetime
remains geodesically complete. One prominent approach to achieve this involves
replacing the singularity with a regular patch of de Sitter space Frolov et
al. (1989), an old concept that has garnered renewed attention in recent
times. Much of the existing research in the literature derives regular black
hole geometries through modifications of the static Misner-Sharp mass with the
aim of rapid convergence to zero at small distances. This is the case for
static regular BH metrics like the Poisson-Israel model Poisson and Israel
(1988), the Asymptotically Safe (AS) model Bonanno and Reuter (2000), the
Dymnikova regular black hole model Dymnikova (1992); Platania (2019), or the
Hayward metric Hayward (2006), to name a few (see Bambi (2023) for an extended
review). However, endeavors to obtain these solutions from an underlying
theory have encountered several challenges, and a consensus on embedding
regular black hole solutions within a more general gravity theory remains
elusive Giacchini et al. (2021); Knorr and Platania (2022).

An alternative route to obtaining modifications of the classical BH solutions
is by matching a non-singular homogeneous interior model, describing
collapsing matter, to a static exterior black hole solution. This method has
received considerable attention in recent years. The minisuperspace
approximation in Loop Quantum Cosmology leads to an effective Friedman
equation with repulsive (i.e., negative) gravity at high densities, resulting
in a bouncing interior model Hossenfelder et al. (2010); Bojowald et al.
(2005); Bambi et al. (2013); Lewandowski et al. (2023). Conversely, assuming a
Renormalization Group (RG) improved regular exterior allows for obtaining a
singularity-free collapsing dust model as the interior solution Torres (2014).
Other methods obtain similar results relying on different approaches to
quantization (see for example Husain et al. (2022); Calmet et al. (2022);
Giesel et al. (2021)). However, similarly to the above mentioned static
solutions, deducing these models from an effective Lagrangian remains
challenging. On the other hand numerical simulations of the formation of
regular BH are often limited to a 2D dilaton gravity model Kunstatter et al.
(2016); Biasi et al. (2022) and the generalization to a 4D model is
problematic unless a physically motivated Lagrangian formulation is found.

Within this study, we present a resolution to this quandary by extending an
initial idea by Markov and Mukanov Markov and Mukhanov (1985). Our approach
involves formulating gravity’s antiscreening behavior in ultraplanckian energy
domains Polyakov (1993) through the inclusion of a multiplicative coupling
with the matter Lagrangian. The structure of this coupling is guided by the
Reuter fixed point of AS gravity Reuter (1998). Remarkably both the energy-
momentum and the effective energy-momentum tensors are conserved in this
theory, in contrast to what happens in most approaches mentioned above.
Notably, under conditions of low energy, our model seamlessly recovers the
equations of standard general relativity. Also, over extensive distances, the
solution for black holes bears resemblance to the Schwarzschild solution.

To be more specific, let us consider a matter fluid with a proper density
$\epsilon$, characterized by a 4-velocity $u^{\mu}$ such that
$u_{\mu}u^{\mu}=-1$, and a rest-mass density $\rho$. The mass continuity
equation is expressed as $(\rho u^{\mu})_{;\mu}=0$, and for a non-dissipative
fluid, the relative variations of density are related as
$\delta\rho/\rho=\delta\epsilon/(p(\epsilon)+\epsilon)$.

Following the approach in Markov and Mukhanov (1985), we introduce the action
for our system as follows:

$S=\frac{1}{16\pi G_{N}}\int
d^{4}x\sqrt{-g}\left[R+2\chi(\epsilon)\mathcal{L}\right].$ (1)

Here, $\mathcal{L}=-\epsilon$ represents the matter Lagrangian, and the
function $\chi=\chi(\epsilon)$ serves as a multiplicative gravity-matter
coupling with the important property $\chi(\epsilon=0)=8\pi G_{N}$. The metric
variation of the matter part of the Lagrangian yields

$\frac{1}{\sqrt{-g}}\,\delta\left(2\,\sqrt{-g}\,\chi\,\epsilon\right)=2\frac{\partial(\chi\epsilon)}{{\partial\epsilon}}\delta\epsilon-\chi\,\epsilon\,g_{\mu\nu}\,\delta
g^{\mu\nu}\ .$ (2)

Note that the variation of $\rho$ under a change of the metric is given by
Groen and Hervik (2007):

$\delta\rho=\frac{\rho}{2}(g_{\mu\nu}+u_{\mu}u_{\nu})\delta g^{\mu\nu}.$ (3)

As a result, the total variation of the action (1) leads to the following
field equations:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{\partial(\chi\epsilon)}{\partial\epsilon}T_{\mu\nu}+\frac{\partial\chi}{\partial\epsilon}\epsilon^{2}g_{\mu\nu}\equiv
T_{\mu\nu}^{\rm eff},$ (4)

where

$8\pi
G(\epsilon)=\frac{\partial(\chi\epsilon)}{\partial\epsilon},\quad\Lambda(\epsilon)=-\frac{\partial\chi}{\partial\epsilon}\epsilon^{2},$
(5)

represent the effective Newton constant and cosmological constant,
respectively. Here,
$T_{\mu\nu}=(\epsilon+p(\epsilon))u_{\mu}u_{\nu}+pg_{\mu\nu}$ is conserved.
For spherically homogeneous collapse the metric functions in the diagonal line
element in co-moving coordinates $\\{t,r,\theta,\phi\\}$ are
$g_{00}=-e^{2\nu(r,t)}$, $g_{11}=e^{2\psi(r,t)}$, and $g_{22}=C^{2}(r,t)$.
Then we can express the field equations (4) as follows

$\displaystyle\frac{F_{\rm eff}^{\prime}}{C^{2}C^{\prime}}=8\pi
G(\epsilon)\epsilon+\Lambda(\epsilon)=\chi(\epsilon)\epsilon\equiv\epsilon_{\rm
eff}$ (6) $\displaystyle-\frac{\dot{F}_{\rm eff}}{C^{2}\dot{C}}=8\pi
G(\epsilon)p-\Lambda(\epsilon)\equiv p_{\rm eff},$ (7)

and

$\dot{C}^{\prime}=\dot{C}\nu^{\prime}+C^{\prime}\dot{\psi}.$ (8)

Note that $\epsilon_{\rm eff}>0$ always, while $p_{\rm eff}$ can become
negative. Dotted quantities represent derivatives with respect to the comoving
time ($t$), while prime denotes derivatives with respect to the comoving
radial coordinate ($r$). The function $F_{\rm eff}$ represents the effective
Misner-Sharp mass of the system, which can be defined in analogy with the
Schwarzschild mass $M$ from $1-F_{\rm
eff}/C=g_{\mu\nu}\nabla^{\mu}C\nabla^{\nu}C$ Misner and Sharp (1964), and it
is given by:

$F_{\rm eff}=C(1-C^{\prime 2}e^{-2\psi}+\dot{C}^{2}e^{-2\nu}).$ (9)

Additionally, the Bianchi identity takes the form:

$\nu^{\prime}=-\frac{p_{\rm eff}^{\prime}}{\epsilon_{\rm eff}+p_{\rm eff}}.$
(10)

In the case of a homogeneous perfect fluid, we have $\epsilon(t)$ and $p(t)$,
which in turn imply $\epsilon_{\rm eff}(t)$ and $p_{\rm eff}(t)$. As a
consequence of the Bianchi identity, we can set $\nu=0$ and integrate equation
(8) to obtain $e^{2\psi}=C^{\prime 2}/(1-Kr^{2})$, where $K$ is the
integration constant related to the curvature of the 3-space. The line element
can then be written as:

$ds^{2}=-dt^{2}+\frac{C^{\prime 2}}{1-Kr^{2}}dr^{2}+C^{2}d\Omega^{2},$ (11)

where $d\Omega^{2}$ represents the metric on the unit 2-sphere. By rescaling
the area-radius function $C$ using the adimensional scale factor $a$ according
to $C=ra$, the energy-momentum conservation equation gives

$d\epsilon+3(p(\epsilon)+\epsilon)d\ln a=0.$ (12)

Following Malafarina (2023) we rescale also the effective Misner-Sharp mass as
$F_{\rm eff}=r^{3}m_{\rm eff}$. We can then rewrite equations (6) and (7) as
follows:

$\epsilon_{\rm eff}=\frac{3m_{\rm eff}}{a^{3}},\quad\quad p_{\rm
eff}=-\frac{m_{{\rm eff},a}}{a^{2}},$ (13)

where by $X_{,a}$ we indicate derivatives of $X$ with respect to $a$, and
equation (9) becomes:

$m_{\rm eff}=a(\dot{a}^{2}+K).$ (14)

Here, $m_{\rm eff}(a)=-aV(a)$, where the potential $V(a)$ reads

$V(a)=-\frac{a^{2}}{3}\int_{0}^{\epsilon(a)}G(s)ds.$ (15)

In principle, if we know $G(\epsilon)$ from a fundamental theory, from (5) is
possible to determine $\chi$ and close the system.

In this work, we assume that the behavior of $G$ as a function of the energy
scale is governed by a renormalization group trajectory close to the
ultraviolet (UV) fixed point of the AS program Weinberg (1980); Reuter (1998);
Eichhorn (2019); Pawlowski and Reichert (2021); Reichert (2020); Reuter and
Saueressig (2019); Percacci (2017); Bonanno et al. (2020a). Specifically, we
adopt the approximate running of $G$ as proposed in Bonanno et al. (2022) in
the limit of Quantum Einstein Gravity

$G(k)=\frac{G_{N}}{1+G_{N}k^{2}/g_{*}},$ (16)

where $k$ represents the IR regulator scale, and $g_{\ast}=570\pi/833$ is the
UV fixed point. To connect $k$ to $\epsilon$, we require a prescription. The
analysis in Bonanno et al. (2022) demonstrates that the UV scaling of the
physical Newton constant $G_{N}(q)$, where $q$ is the external momentum, is in
essence the same as predicted by the renormalized coupling $G(k)$. The only
distinction lies in the crossover scale, a non-universal feature of the flow
(see particularly Fig.4 in Bonanno et al. (2022)). Hence, identifying the
cutoff scale $k$ with the characteristic energy scale of the system should
capture the qualitative features of the physical flow Bonanno et al. (2020b)
at distances $d\sim 1/k$, consistent with early applications of the AS
scenario. However, it is essential to acknowledge that the key assumption
underlying equation (16) is that the presence of matter does not significantly
deform the flow and compromise the fixed point Donà et al. (2014). This
assumption is crucial in maintaining the integrity of the renormalization
group trajectory and ensuring its consistency in the presence of matter.

We now consider the scenario of dust collapse, therefore taking $p=0$,
$\epsilon\propto a^{-3}$, and $\rho=\epsilon$. In accordance with the
rationale presented in Bonanno and Reuter (2000), we interpret the variable
$d$ as the proper distance. As as consequence we find the relationship:

$d\sim\frac{r^{3/2}}{\sqrt{m}}\sim\frac{1}{\sqrt{\epsilon}},$ (17)

where $r$ is the radial distance. We thus obtain the following expression for
the behavior of $G(\epsilon)$:

$G(\epsilon)=\frac{G_{N}}{1+\xi\epsilon},$ (18)

where we introduce the dimensionful scale $\xi$, and we include the pure
number $g_{\ast}$ in the definition of $\xi$. It is important to note that, in
general, we would expect $\xi\sim 1/m_{\rm pl}^{4}$, but currently, there is
no clear method to determine $\xi$ from first principles, and this parameter
should be constrained from observations. Setting $8\pi G_{N}=1$, we obtain:

$\chi(\epsilon)=\frac{\log(1+\xi\epsilon)}{\xi\epsilon},\quad\Lambda(\epsilon)=\frac{\log(1+\xi\epsilon)}{\xi}-\frac{\epsilon}{1+\xi\epsilon}.$
(19)

Importantly, in the classical limit, achieved for $\xi\rightarrow 0$, we
recover $\chi=1$ and $\Lambda=0$, as expected. Figure 1 illustrates the
potential $V(a)$ for the running coupling defined in equation (18), as
compared to the Oppenheimer-Snyder-Datt (OSD) case.

Figure 1: The thick line shows the potential $V(a)$ for dust collapse in the
AS model with running $G$ given by equation (18). For comparison the OSD model
with $8\pi G_{N}=1$ is shown as the dotted line. The parameters are fixed for
illustrative purposes to $m_{0}=1$ and $\xi=0.001$.

Reducing our formalism to the OSD model Oppenheimer and Snyder (1939); Datt
(1938), which describes non-interacting dust particles with $p=0$, we have
$T_{\mu\nu}=\epsilon u_{\mu}u_{\nu}$, $\epsilon=3m_{0}/a^{3}$ and $m_{\rm
eff}\rightarrow m_{0}$. Equation (15) yields $V(a)=-m_{0}/a$ recovering the
usual equation for the scale factor of homogeneous dust:

$\dot{a}=-\sqrt{\frac{m_{0}}{a}-K}.$ (20)

It is worth noting that bound collapse is obtained for $K>0$, while the
marginally bound case has $K=0$. Interestingly, to achieve singularity
resolution at the end of the collapse, the usual energy conditions must be
violated. In many models available in the literature, the growth of negative
effective pressures leads to repulsive effects that halt the collapse.
However, in our case, such a term is due to the running cosmological constant,
which generates repulsive effects, allowing the collapse to come to a halt.
For $\xi\neq 0$ and (18), we obtain:

$\dot{a}=-\sqrt{\frac{\log(1+3m_{0}\xi/a^{3})}{3\xi}a^{2}-K}.$ (21)

The solution to this equation for marginally bound collapse ($K=0$) is shown
in Figure 2 and compared with the OSD model and the semi-classical collapse
model developed in Malafarina and Toshmatov (2022). At large times, the scale
factor behaves as:

$a(t)\sim e^{-t^{2}/4\xi},\quad t\rightarrow\infty,$ (22)

indicating that $a=0$ is never reached at any finite time, and the spacetime
is geodesically complete. As $t$ approaches infinity, the scale factor tends
to diminish exponentially, but it never reaches zero, ensuring the avoidance
of singularities within any finite time frame.

Figure 2: The thick line shows the scale factor $a(t)$ for marginally bound
($K=0$) dust collapse in the AS model with running gravitation and
cosmological constant solution of equation (21). For comparison the OSD
collapse model is shown as the dotted line and the semi-classical collapse
leading to a regular black hole developed in Malafarina and Toshmatov (2022)
is shown as the dashed line. The parameters are fixed for illustrative
purposes to $m_{0}=1$ and $\xi=0.01$.

To implement the matching of the collapsing matter cloud described above with
a suitable exterior, we employ the formalism developed by Israel in Israel
(1966), which was further refined by Senovilla and others in Fayos et al.
(1992, 1996). We consider the matching across a comoving boundary $r=r_{b}$ in
the interior, which corresponds to a collapsing boundary
$C_{b}(t)=C(t,r_{b})=r_{b}a(t)$. The induced metric on the matching surface
$\Sigma$ in comoving coordinates can be expressed as:

$ds^{2}_{\Sigma}=-dt^{2}+r_{b}^{2}a^{2}d\Omega^{2}.$ (23)

For the exterior, we consider a generic static and spherically symmetric line
element in $\\{T,R,\theta,\phi\\}$ coordinates written as:

$ds^{2}=-f(R)dT^{2}+\frac{1}{f(R)}dR^{2}+R^{2}d\Omega^{2},$ (24)

where $f=1-2M(R)/R$, and we assume a continuous matching between the two
geometries. The continuity condition uniquely determines the form of $M(R)$ in
the exterior. Specifically, if the collapsing boundary is parametrized by
$R=R_{b}(T)$, the induced metric on the boundary becomes:

$ds^{2}_{\Sigma}=-\left[f(R_{b})-f(R_{b})^{-1}\left(\frac{dR_{b}}{dT}\right)^{2}\right]dT^{2}+R_{b}^{2}d\Omega^{2}.$
(25)

The matching conditions for the metric functions on the boundary surface
$\Sigma$ immediately provide the relation between $t$ and $T$ on $\Sigma$ and
the condition $R_{b}(T(t))=r_{b}a(t)$. The second fundamental form for the
interior metric in comoving coordinates is

$K_{tt}^{-}=0,\quad K_{\theta\theta}^{-}=r_{b}a\sqrt{1-Kr_{b}^{2}}.$ (26)

From the extrinsic curvature on the exterior we obtain

$K_{tt}^{+}=-\frac{1}{2}\frac{2\ddot{R}_{b}+f_{,R}(R_{b})}{\Delta(R_{b})},\quad
K_{\theta\theta}^{+}=R_{b}\Delta(R_{b}),$ (27)

with $\Delta(R_{b})=\sqrt{1-2M(R_{b})/R_{b}+\dot{R}_{b}^{2}}$, so that on
imposing

$[K_{tt}]=K_{tt}^{+}-K_{tt}^{-}=0,\quad\left[K_{\theta\theta}\right]=K_{\theta\theta}^{+}-K_{\theta\theta}^{-}=0$
(28)

the functional form of $M(R)$ can be obtained. Finally, we arrive at the most
important result of our investigation:

$M(R)=\frac{R^{3}}{6\xi}\log\left(1+\frac{6M_{0}\xi}{R^{3}}\right),$ (29)

where the matching implies $m_{0}r_{b}^{3}=2M_{0}$. This expression describes
the Misner-Sharp mass pertaining to the static exterior, originating from an
interior undergoing gravitational collapse. The dynamics of this interior are
guided by an effective Lagrangian that incorporates the Asymptotically Safe
(AS) nature of gravitational interaction at Planckian energy scales.

Importantly, the classical limit is recovered for $\xi\rightarrow 0$ (or
equivalently for $R\rightarrow\infty$), leading to:

$M(R)=M_{0}-\frac{3M_{0}^{2}\xi}{R^{3}}+\frac{12M_{0}^{3}\xi^{2}}{R^{6}}+O(\xi^{3}),$
(30)

as expected. In the low-energy limit, the Schwarzschild solution is regained.
Notably, in the small $R$ regime, $M(R)$ behaves like

$M(R)=\frac{1}{6\xi}R^{3}\log\left(\frac{6M_{0}\xi}{R^{3}}\right)+\frac{R^{6}}{36M_{0}\xi^{2}}+O\left(R^{7}\right)$
(31)

moreover, as $R\geq R_{b}=r_{b}a(t)$ and $a(t)>0$ always, our solution remains
everywhere regular, avoiding any singularities. This result is of significant
importance as it demonstrates the compatibility of the collapsing matter
interior with our AS effective Lagrangian model in producing a regular black
hole exterior.

Figure 3: The behavior the metric function $f(R)$ for different values of the
parameter $\xi$ for $M_{0}=1$. For $\xi<\xi_{\rm cr}$ there are an inner and
an outer horizon, corresponding to the two solutions of $f(R)=0$ (dashed curve
for $\xi=0.1$) and for $\xi=\xi_{\rm cr}\approx 0.45$ there is only one
horizon (thick curve). For $\xi>\xi_{\rm cr}$ the two horizons disappear and
one is left with a scalar remnant (dotted curve for $\xi=0.6$).

The horizon’s position is determined by solving the transcendental equation
$f(R)=0$, which can yield intriguing results. Notably, for any specified value
of $M_{0}$, a critical threshold $\xi_{\rm cr}$ exists. When $\xi<\xi_{\rm
cr}$, an event horizon becomes evident, accompanied by an inner horizon at
smaller values of the radial coordinate. However, for $\xi>\xi_{\rm cr}$, a
scalar remnant emerges, as illustrated in Figure 3 for $M_{0}=1$.

We expect for the global solution, that the evolution of the causal structure
in the interior matches the formation of the causal structure for the exterior
geometry. The condition for the formation of trapped surfaces in the interior
may be obtained from

$1-\frac{F_{\rm eff}}{C}=1-r^{2}(\dot{a}^{2}+K)=0,$ (32)

which implicitly gives the curve $r_{\rm ah}(t)$ describing the comoving time
$t$ at which the shell $r$ becomes trapped:

$r_{\rm ah}(t)=\frac{1}{\sqrt{\dot{a}^{2}+K}}.$ (33)

It is clear that in the OSD case as $\dot{a}$ diverges the apparent horizon
curve tends to zero. On the other hand in the AS case we have that $\dot{a}$
goes to zero asymptotically and therefore $r_{\rm ah}\rightarrow 1/\sqrt{K}$.
In the marginally bound case, for any given $\xi$ we have that $r_{\rm ah}$
reaches a minimum value $r_{\rm min}$ and then grows to infinity. Then there
are values of the boundary $r_{b}>r_{\rm min}$ that lead to $r_{ah}$ crossing
the boundary twice thus creating two horizons in the exterior. Accordingly
there is a critical value $r_{b}=r_{\rm min}$ for which only one horizon
exists and for values smaller than $r_{\rm min}$ no horizon forms throughout
collapse. The apparent horizon curve is shown in Figure 4.

Figure 4: The thick line shows the apparent horizon $r_{\rm ah}(t)$ for
marginally bound dust collapse in the Asymptotically Safe model with running
$G$, the dotted line shows $r_{\rm ah}(t)$ for the OSD model with $8\pi
G_{N}=1$. Notice that in the AS case the apparent horizon reaches a minimum
value $r_{\rm min}$, as shown by the thin horizontal line, and then grows.
This means that for a boundary $r_{b}={\rm const}.$ taken at $r_{b}<r_{\rm
min}$ the whole collapse process is not covered by any horizon. The parameters
are fixed for illustrative purposes to $m_{0}=1$ and $\xi=0.01$.

Our solution represents a significant alternative to present models of regular
black holes. It is built upon the assumption that black hole solutions
observed in Nature are sourced by a matter interior whose evolution is non-
singular due to the antiscreening of the gravitational constant at small
distances Polyakov (1993), according to a specific renormalization group
trajectory terminating at the Reuter fixed point of AS gravity. This mechanism
is implemented using an effective Lagrangian that incorporates a
multiplicative coupling with the matter component. Although in this work, we
considered an idealized model of matter consisting of a pressureless fluid,
our framework can be generalized to incorporate more realistic equations of
state and more accurate RG trajectories, providing a consistent description of
the matter component. We intend to address these issues in future
investigations.

## Acknowledgement

DM would like to thank Catania Astrophysical Observatory - INAF- for warm
hospitality during the preparation of the manuscript. DM acknowledges support
from Nazarbayev University Faculty Development Competitive Research Grant No.
11022021FD2926.

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