## JHEP04(2016)147

Published for SISSA by Springer

Received: April 1, 2016 Accepted: April 13, 2016 Published: April 22, 2016

### Black supernovae and black holes in non-local gravity

Cosimo Bambi,^{a,b} Daniele Malafarina^{c} and Leonardo Modesto^{a}

aCenter for Field Theory and Particle Physics and Department of Physics, Fudan University, 200433 Shanghai, China

bTheoretical Astrophysics, Eberhard-Karls Universit¨at T¨ubingen, 72076 T¨ubingen, Germany

cDepartment of Physics, Nazarbayev University, 010000 Astana, Kazakhstan

E-mail: [email protected],[email protected], [email protected]

Abstract: In a previous paper, we studied the interior solution of a collapsing body in a non-local theory of gravity super-renormalizable at the quantum level. We found that the classical singularity is replaced by a bounce, after which the body starts expanding.

A black hole, strictly speaking, never forms. The gravitational collapse does not create an event horizon but only an apparent one for a ﬁnite time. In this paper, we solve the equations of motion assuming that the exterior solution is static. With such an assumption, we are able to reconstruct the solution in the whole spacetime, namely in both the exterior and interior regions. Now the gravitational collapse creates an event horizon in a ﬁnite comoving time, but the central singularity is approached in an inﬁnite time. We argue that these black holes should be unstable, providing a link between the scenarios with and without black holes. Indeed, we ﬁnd a non catastrophic ghost-instability of the metric in the exterior region. Interestingly, under certain conditions, the lifetime of our black holes exactly scales as the Hawking evaporation time.

Keywords: Black Holes, Models of Quantum Gravity, Spacetime Singularities ArXiv ePrint: 1603.09592

## JHEP04(2016)147

Contents

1 Introduction 1

2 Gravitational collapse in Einstein gravity 2

3 Black supernovae 4

4 Black holes 8

4.1 Exterior solution 8

4.2 Interior solution 9

4.3 From in to out making use of the boundary conditions 11 5 Coexistence of the two scenarios and Hawking evaporation 12

6 Conclusions 16

1 Introduction

In Einstein gravity and under a set of physically reasonable assumptions, the complete gravitational collapse of a body creates a spacetime singularity and the ﬁnal product is a black hole. The simplest example is the Oppenheimer-Snyder (OS) model, which describes the collapse of a homogeneous and spherically symmetric cloud of dust [1]. However, it is often believed that the spacetime singularities created in a collapse are a symptom of the breakdown of the classical theory and they can be removed by quantum gravity eﬀects.

Alternatively, we can assume that spacetime singularities are resolved by employing a new action principle for classical gravity. However, the equations of motion of the new theory are typically quite diﬃcult to solve. One can thus attempt to study toy-models, which can hopefully capture the fundamental features of the full theory. With a similar approach, one usually ﬁnds that the formation of a singularity is replaced by a bounce, after which the collapsing matter starts expanding [2–23].

Even in simple models, it is usually quite diﬃcult to ﬁnd a global solution that covers the whole spacetime. Nevertheless, on the basis of general arguments, we can conclude that there are two plausible scenarios. One possibility is that the bounce generates a baby universe inside the black hole [24]. This kind of scenario can generally be obtained analyti- cally with a cut-and-paste technique, in which the singularity is removed and the spacetime is sewed to a new non-singular manifold describing an expanding baby universe. However, such a procedure seems to work only in very simple examples: the matching requires the continuity of the ﬁrst and of the second fundamental forms across some hypersurface, which is not always possible because of the absence of a suﬃcient number of free parameters. In the second scenario, a black hole does not form. The gravitational collapse only creates a

## JHEP04(2016)147

temporary trapped surface, which looks like an event horizon for a ﬁnite time (which may, however, be very long for a far-away observer). Such a possibility has recently attracted a lot of interest because of a paper by Hawking [25], but actually it was proposed a long time ago by Frolov and Vilkovisky [2,3], and was recently rediscovered by several groups [4–17], following diﬀerent approaches and within diﬀerent models.

The aim of this paper is to go ahead in the investigation of this topic. Following ref. [15], we start from a model for the exterior vacuum spacetime. We assume that the exterior metric is static, and we solve our eﬀective equations of motion (EOM) for the non- local gravitational theory. With an ansatz for the interior solution, we are able to do the matching and eventually to obtain a solution for the whole spacetime. The result of this procedure is the formation of a black hole, characterized by a Cauchy internal horizon and an event horizon. More importantly, there is no bounce. The collapsing object approaches a singular state in an inﬁnite time. It seems thus that the properties of the exterior solution, which could in principle be derived by the underlying fundamental theory, play a major role in the fate of the collapse. However, our exterior spacetime metric appears to be unstable because of the presence of a massive ghost. The latter can likely cause the destruction of the black hole, but the timescale is extremely long for a stellar-mass object. We thus argue that, once again, a true event horizon may never be created.

The content of the paper is as follows. In section2, we brieﬂy review the gravitational collapse of a spherically symmetric cloud in classical general relativity. In sections 3, we summarize the bouncing solutions (black supernovae) in weakly non-local theories of gravity found in [8, 12]. Moreover, we provide the correct spacetime structure missed in the previous papers. In section 4, we follow the approach of ref. [15] and we construct the interior metric on the base of an external black hole metric [26] that captures all the features of the approximate solutions in non-local gravitational theories [27]. In section5, we provide a (in-)stability mechanism to reconcile the contradictory outcome of the previous sections. Indeed, the black hole metric shows a ghost instability which makes the black hole lifetime ﬁnite, but very long due to the non-locality scale. Summary and conclusions are reported in section6.

Throughout the paper, we use units in with c = ~ = 1, while we explicitly show
Newton’s gravitational constant G_{N}.

2 Gravitational collapse in Einstein gravity

In the case of spherical symmetry, we can always write the line element in the comoving frame as

ds^{2} =−e^{2ν}dt^{2}+R^{′}^{2}

Y dr^{2}+R^{2}dΩ^{2} , (2.1)

wheredΩ^{2} represents the metric on the unit 2-sphere. The metric functionsν(r, t),Y(r, t),
andR(r, t) must be determined by solving the Einstein equations for a given matter distri-
bution. We note thatR(r, t) represents the collapsing areal coordinate, while the comoving
radiusr is a coordinate “attached” to the collapsing ﬂuid. The energy momentum tensor

## JHEP04(2016)147

in comoving coordinates takes diagonal form and for a matter ﬂuid source can be written
asT_{µ}^{ν} = diag{−ρ, p_{r}, p_{θ}, p_{θ}}. With this set-up, the Einstein equations become

ρ = F^{′}

4πR^{2}R^{′} , (2.2)

p_{r} = − F˙

4πR^{2}R˙ , (2.3)

ν^{′} = 2p_{θ}−p_{r}
ρ+p_{r}

R^{′}

R − p^{′}_{r}

ρ+p_{r} , (2.4)

Y˙ = 2ν^{′}R˙

R^{′} Y , (2.5)

where ^{′} indicates the derivative with respect to r, while ˙ the one with respect to t. The
functionF is the Misner-Sharp mass of the system and is deﬁned by (please note that there
is a diﬀerence of a factor 2GN in our deﬁnition ofF with respect our previous papers [8–12])
2G_{N}F =R(1−g_{µν}∇^{µ}R∇^{ν}R). (2.6)
It is easy to see thatF plays the same role as the mass parameterMsin the Schwarzschild
metric and represents the amount of gravitating matter within the shellrat the timet[28].

Using the metric (2.1), F can be written as
2G_{N}F =R

1−Y +e^{−}^{2ν}R˙^{2}

. (2.7)

We immediately see that these equations can be considerably simpliﬁed if the matter
source satisﬁespr =p_{θ} andp^{′}_{r} = 0. In this case, we haveν^{′}= 0, from which we getν =ν(t)
and, by a suitable redeﬁnition of the time gauge, we can setν= 0. Eq. (2.5) becomes ˙Y = 0,
which can be integrated to giveY =Y(r) = 1 +f(r). A cloud composed of non interacting
particles has pr = p_{θ} = 0 and satisﬁes the conditions above. This is the so called dust
collapse and was ﬁrst investigated in the case of a homogeneous density distribution in [1].

From eq. (2.3), we see that in the case of dust F = F(r) and therefore the amount of
matter enclosed within the shell r is conserved. This means that there is no inﬂow or
outﬂow of matter at any radius during the process of collapse. As a consequence, there is
no ﬂux of matter through the boundary of the star as well. Therefore, by setting the outer
boundary of the cloud at the comoving radiusr =r_{b}, which corresponds to the shrinking
physical area-radiusR_{b}(t) =R(r_{b}, t), we see that we can always perform the matching with
an exterior Schwarzschild spacetime with mass parameterMs=F(r_{b}) [29–33].

Once we substituteν and Y for dust in the deﬁnition of the Misner-Sharp mass given by eq. (2.7), we obtain the equation of motion for the system

R˙ =−

r2G_{N}F

R +f . (2.8)

The free functionf coming from the integration of eq. (2.5) is related to the initial velocity of the infalling particles. If the cloud had no boundary and extended to inﬁnity, then the velocity of particles at inﬁnity would be given by limr→∞f(r). This allows us to distinguish

## JHEP04(2016)147

three cases. Unbound collapse happens when particles have positive velocity at inﬁnity.

Marginally bound collapse happens when particles have zero velocity at inﬁnity. Bound collapse happens when particles reach zero velocity at a ﬁnite radius.

There is a gauge degree of freedom given by setting the value of the area-radius R at
the initial time. This sets the initial scale of the system but does not aﬀect the physics of
the collapse. We can choose the initial scaling in such a way that at the initial timet_{i}= 0
we have R(r,0) = r and introduce a dimensionless scale factor a(r, t) such that R = ra.

Then the whole set of the Einstein equations can be rewritten in this gauge once we deﬁne two functions,µ(r) and b(r), such that

F =r^{3}µ , f =r^{2}b . (2.9)

The equation of motion (2.8) is immediately rewritten as

˙ a=−

r2G_{N}µ

a +b . (2.10)

As a consequence of the above choice, we see that the regularity of the initial data at the center follows directly from the ﬁniteness ofµandb. This choice makes also the appearance of the singularity more manifest, since the energy density becomes

ρ= 3µ+rµ^{′}

4πa^{2}(a+ra^{′}) , (2.11)

which diverges for a= 0 and is clearly ﬁnite at the initial time when a = 1. As we can
see, the homogeneous dust collapse model is obtained easily by setting µand b to be con-
stant, namelyµ=µ_{0} andb=b_{0}. In this case, marginally bound collapse is simply given by
b0 = 0. Consideringµand/orbas functions ofr, one gets an inhomogeneous density proﬁle,
which corresponds to the so called Lema`ıtre-Tolman-Bondi model (LTB) [34–36]. In both
the homogeneous and inhomogeneous case, the collapse ends with the production of a grav-
itationally strong, shell-focusing singularity. The singularity is hidden behind the horizon
in the OS model, while it may be visible to far-away observers in the LTB model [37–41].

3 Black supernovae

While most of the bouncing solutions are based on toy-models [9–11, 14–17, 25], or at best on theories non renormalizable at the quantum level [2], in refs. [8, 12] we found the bounce in a family of asymptotically free weakly non-local theories of gravity. These theories are unitary, super-renormalizable or ﬁnite at the quantum level, and there are no extra degrees of freedom (ghosts or tachyons) expanding around the ﬂat spacetime (for the details, see refs. [8,12]). The simplest classical Lagrangian for these super-renormaliable theories reads [42–55]

S_{g} = 2
κ^{2}

Z

d^{4}xp

|g|

R+G_{µν}e^{H(}^{−}^{/Λ}^{2}^{)}−1
R^{µν}

, (3.1)

where G_{µν} is the Einstein tensor and κ^{2} = 32πG_{N}. All the non-polynomiality is in the
form factor exp H(−/Λ^{2}), which must be an entire function. Λ is the non-locality or

## JHEP04(2016)147

quasi-polynomiality scale. The natural value of Λ is of order the Planck mass and in this case all the observational constraints are satisﬁed. The theory is uniquely speciﬁed once the form factor is ﬁxed, because the latter does not receive any renormalization: the ultraviolet theory is dominated by the bare action (that is, the counterterms are negligible). In this class of theories, we only have the graviton pole. Since exp H(−/Λ) is an entire function without zeros or poles in the whole complex plane, at perturbative level there are no ghosts and no tachyons independently of the number of time derivatives present in the action.

Let us now consider the gravitational collapse in the class of theories given by eq. (3.1).

In particular we look for approximate solutions for the interior of a collapsing body. The
scale factor a(t) is determined via the propagator approach [2, 8, 12, 56–58] or the lin-
earized equations of motion in the way we are going to describe. We consider a Friedman-
Robertson-Walker (FRW) cosmological model since we can easily export the result to the
gravitational collapse by inverting the time direction. We start writing the FRW metric as
a ﬂat Minkowski background plus a ﬂuctuationh_{µν},

gµν =ηµν+κ hµν, ds^{2} =−dt^{2}+a(t)^{2}dx^{i}dx^{j}δij, (3.2)
whereη_{µν} = diag(−1,1,1,1). The conformal scale factor a(t) and the ﬂuctuationh_{µν}(t, ~x)
are related by the following relations:

a^{2}(t) = 1−κh(t), (3.3)

h(t=t_{0}) = 0,
g_{µν}(t=t_{0}) = η_{µν},

h_{µν}(t, ~x) = −h(t) diag(0, δ_{ij})≡ −h(t)Iµν. (3.4)
After a gauge transformation, we can rewrite the ﬂuctuation in the usual harmonic gauge,
in which the propagator is evaluated, namely

h_{µν}(x)→h^{′}_{µν}(x) = h_{µν}(x) +∂_{µ}ξ_{ν} +∂_{ν}ξ_{µ},
ξ_{µ}(t) = 3κ

2 Z t

0

h(t^{′})dt^{′},0,0,0

. (3.5)

The ﬂuctuation in the harmonic gauge reads

h^{′}_{µν}(t, ~x) =h(t) diag(3,−δij), h^{′}_{µ}^{µ}(t, ~x) =−6h(t). (3.6)
We can then switch to the standard gravitational “barred” ﬁeld ¯h^{′}_{µν} deﬁned by

¯h^{′}_{µν} =h^{′}_{µν}−1

2η_{µν}h^{′}_{λ}^{λ} = 2h(t)Iµν, (3.7)
satisfying ∂^{µ}¯h^{′}_{µν} = 0. The Fourier transform of ¯h^{′}_{µν} is

˜¯

h^{′}_{µν}(E, ~p) = 2˜h(E)(2π)^{3}δ^{3}(~p)I^{µν}. (3.8)

## JHEP04(2016)147

For the generic case of a perfect ﬂuid with equation of statep=ωρ, the scale factor for the homogeneous and spherically symmetric gravitational collapse (or cosmological metric) is (for ω6=−1)

a(t) =
t
t_{0}

2 3(ω+1)

, (3.9)

where nowt= 0 is the time of the formation of the singularity.

We can thus compute the Fourier transform ˜h(E) deﬁned in (3.8). Forω6=−1, we have

˜h(E) = 2πδ(E)

κ +

2Γ

4

3(ω+1) + 1 sin

π

2 4

3(ω+1)

κt

4 3(ω+1)

0 |E|^{3(ω+1)}^{4} ^{+1}

. (3.10)

In the case of radiation and dust, we have

˜h(E) = 2πδ(E)

κ + 2

κt_{0}E^{2}, (radiation) (3.11)

˜h(E) = 2πδ(E)

κ + 4Γ(^{4}_{3})

√3κt^{4/3}_{0} |E|^{7/3}, (dust). (3.12)
Since the theory is asymptotically free, we can get a good approximation of the solution
from the linear EOM of the non-local theory. In particular, given the energy tensor, we can
extract the relation between the Einstein solution and the non-local solution comparing
the following two equations,

¯h^{′}_{µν} = 8πGNTµν, e^{H()}¯h^{′}_{µν}^{nl}= 8πGNTµν, (3.13)
where here ¯h^{′}_{µν} is the solution of the linearized Einstein EOM, while ¯h^{′}_{µν}^{nl} is the solution
of the linearized non-local EOM. Therefore, the relation between the two gravitational
perturbations is:

e^{H}^{()}¯h^{′}_{µν}^{nl}= ¯h^{′}_{µν}. (3.14)
In Fourier transform, the above relation reads

˜¯

h^{′}_{µν}^{nl}(k) =e^{−}^{H(k}^{2}^{)}˜¯h^{′}_{µν}(k), (3.15)
or, for our homogeneous case,

˜h^{nl}(E) =e^{−}^{H(E}^{2}^{)}˜h(E). (3.16)
Considering the gravitational collapse for an homogeneous and spherically symmetric cloud
and evaluating the anti-Fourier transform of (3.15), we ﬁnd the solution forh(t) and then
the scale factora(t) (3.3). Everything in this section can be applied to the FRW cosmology
as well as to the gravitational collapse. The solution for the gravitational collapse scenario

## JHEP04(2016)147

is obtained by replacingtwith−t+t0. For instance, in the radiation case and for the form
factor exp(−/Λ^{2}), the result is [8]

a^{2}(t) = 2e^{−}^{1}^{4}^{Λ}^{2}^{(t}^{−}^{t}^{0}^{)}^{2}
Λ√

π t_{0} +

(t_{0}−t) erf

Λ(t0−t) 2

t0 , (3.17)

where erf(z) = 2R_{z}

0 exp(−t^{2})dt/√

π. The classical singularity is now replaced by a bounce
at t = t_{0}, after which the cloud starts expanding (hence the name black supernova).

For dust, we ﬁnd a very similar solution [8]. The resulting proﬁle for a(t) is slightly
diﬀerent if we consider consistent form factor in Minkowski signature [59], namely exp(^{N}),
whereN is an even integer. It is a general feature of these theories that the gravitational
interaction is switched oﬀ at high energies, namely the theories are asymptotically free. In
our framework, the asymptotic freedom is due to a higher derivative form factor, which
makes gravity repulsive at very small distances. In terms of an eﬀective picture in which
gravity is supposed to be described by the Einstein-Hilbert theory and new physics is
absorbed into the matter sector, the bounce comes from the conservation of the (eﬀective)
energy-momentum tensor: matter is transformed into a state withρeff+peff <0, which is
unstable and therefore the bounce is the only available possibility.

The bounce seems thus to be unavoidable in this class of theories. If we exclude the possibility of the creation of a baby universe, motivated by the problems mentioned in the introduction, a black hole, in the strict mathematical sense of the deﬁnition, never forms.

Gravitational collapse only produces a trapped surface lasting for a ﬁnite time. No Cauchy and event horizon are formed. Since an apparent horizon cannot be destroyed from the inside, at least if we do not invoke exotic mechanisms like super-luminal motion, it must be destroyed from the outside. We thus argue that the solution outside the horizon cannot be static but must belong to the radiating Vaidya family. We can think of it as an eﬀective negative energy ﬂux destroying the horizon from the exterior. For a large black hole, we do not expect signiﬁcant deviations from standard general relativity at the horizon (the value of scalar quantities like the Kretschmann invariant is much smaller than the Planck scale) and therefore the process is expected to be very slow. In other words, we recover the classical picture of an almost classical black hole and we can realize that the object is not a black hole only if the observation of a far-away observer lasts for a very long time.

In summary, with the approach employed in ref. [8, 12] we start with a well-deﬁned and consistent theory of gravity for the interior solution and we ﬁnd that the bounce is unavoidable. On this basis, we can guess the exterior behavior. Figure 1 shows the Finkelstein diagram of the collapse. Figure 2 shows instead the corresponding Penrose diagram. We note that the latter corrects current diagrams presented in the literature.

There is more likely only one trapped surface (not two), because gravity is switched oﬀ only inside the cloud of matter. The apparent horizon propagating inward from the cloud surface may either coincide with the cloud surface at the moment of the bounce (left panel in ﬁgure 1) or be in the exterior region (right panel). The actual situation may depend on the gravity theory. In our case we do not know because we are only able to solve the interior solution, so we cannot make predictions about the exterior region. The right panel

## JHEP04(2016)147

Figure 1. Finkelstein diagram of the black supernova scenario. The two panels diﬀer for the position of the Cauchy horizon with respect to the boundary of the cloud. However, the spacetime structure of the gravitational collapse has a universal feature characterize by the formation of a trapped surface without any ﬁnal black hole state.

in ﬁgure1may be motivated by the fact that the static black hole solutions in these theories have indeed an internal Cauchy horizon [12]. For a ﬁnite observational time, the trapped surface ﬁrst behaves as a black hole (left bottom side of the trapped surface in ﬁgure 2) and then as a while hole (left top side) [14].

4 Black holes

In this section, we employ a semi-classical picture in which deviations from the classical
theory are encoded in an eﬀective Newton gravitational constant. G_{N} is replaced by a
function G of the radial coordinate, which is used to reproduce the eﬀects of (3.1) or a
generic quantum eﬀective action for gravity [15,60]. To this aim we start from the exterior
solution and we reconstruct the interior.

4.1 Exterior solution

As done in [15,16], we assume that the exterior metric is a generalization of the classical Schwarzschild solution. The line element can be written as

ds^{2}=−

1−2G(x)M_{s}
x

dt^{2}+

1−2G(x)M_{s}
x

^{−}1

dx^{2}+x^{2}dΩ^{2} , (4.1)
where x is the radial coordinate in the exterior spacetime. In super-renormalizable/ﬁnite
theories of gravity, spherically symmetric exact black hole solutions can be written in this
form [12, 27]. Notice the following key point: we are assuming that the exterior vacuum
metric is static, as it is true in general relativity thanks to the Birkhoﬀ theorem. A
prototype of G(x) that captures all the important and universal features in these theories
has the following form

G(x) = x^{3}G_{N}

x^{3}+L^{3} , (4.2)

## JHEP04(2016)147

I^{+}

I^{−}

r= 0 i^{0}

i^{+}

i^{−}
rb

Figure 2. Penrose diagram of the black supernova scenario. There is a single trapped surface, which behaves for a ﬁnite time ﬁrst as a black hole and then as a white hole. See the text for more details.

where L is a new scale and it is natural to expect it to be of order the Planck length,
namely L ≈ L_{Pl} = G^{1/2}_{N} . Of course, eq. (4.1) is not a vacuum solution of the Einstein
equations. If we impose the latter, we ﬁnd an eﬀective, or “unphysical”, matter source for
the spacetime in the form of an energy-momentum tensor for a ﬂuid with eﬀective density
and pressures given by

ρ^{ext} =−p^{ext}_{r} = M_{s}G_{,x}

4πG_{N}x^{2} , p^{ext}_{θ} =−M_{s}G_{,xx}

8πG_{N}x . (4.3)

New physics is encoded inG(x), but one could have equivalently absorbed everything in a variable mass parameterM(x), as done in [12,27]. In the next subsection, the line element in (4.1) will be matched to a suitable interior in the form of (2.1) through a 3-dimensional hypersurface Σ describing the boundary of the collapsing cloud.

4.2 Interior solution

The use of a non-constant G in the interior will aﬀect the energy-momentum tensor by
introducing some eﬀective terms in the density and in the pressures. If Σ is the comoving
boundary hypersurface, then continuity of g_{θθ} and g_{φφ} implies that R(r, t)|^{Σ} =R(r_{b}, t) =

## JHEP04(2016)147

x_{b}(τ). We can then take the functionG(x) from the exterior and obtain the corresponding
G(R) in the interior through the matching conditions. Standard matching conditions imply
continuity of the ﬁrst and second fundamental forms across Σ [29–33], namely the metric
coeﬃcients on the induced metric and the rate of change of the unit normal to Σ must be
the same on both sides. With the exterior metric given in eq. (4.1), the matching conditions
across Σ imply that the density and the pressures in the interior take the form

ρ= G(R)F^{′}

4πG_{N}R^{2}R^{′} + F G,R

4πG_{N}R^{2} , p_{r} =− F G,R

4πG_{N}R^{2} , p_{θ} =−F G,RR

8πG_{N}R − F^{′}G,R

8πG_{N}RR^{′} , (4.4)
which reduce to the usual Einstein equations for dust in the case G = G_{N} is constant.

From these equations and eq. (2.4), we ﬁnd that ν^{′} = 0, and therefore the metric in the
interior region still satisﬁes the same condition as the classical dust case. The line element
can then be taken as

ds^{2} =−dτ^{2}+R^{′}(r, τ)^{2}

1 +f(r)dr^{2}+R^{2}(r, τ)dΩ^{2} . (4.5)
This is the usual LTB spacetime describing the collapse of a dust cloud, where now the
energy-momentum tensor is the sum of the classical dust energy momentum-tensor and an
eﬀective contribution coming from the fact thatGis not constant. The equation of motion
for the system becomes

R˙^{2} = 2G(R)F(r)

R +f(r). (4.6)

At this point, we have to specify the expression of G(R) for the interior. As an example, for the sake of simplicity we consider a modiﬁed Hayward metric [26] that gives an equation (4.6) independent on the coordinate r, namely

G(R) = R^{3}GN

R^{3}+G_{N}F(r)L^{2}_{Pl} . (4.7)
In the simplest case of a homogeneous cloud, F(r) =µ0r^{3} withµ0 constant. Therefore

G(a) = a^{3}G_{N}

a^{3}+G_{N}µ_{0}L^{2}_{Pl}, (4.8)

which is independent on the radial coordinater. With the further assumption of marginally bound collapse, namely f = 0, eq. (4.6) becomes

˙
a^{2}

a^{2} = 2G_{N}µ_{0}

a^{3}+G_{N}µ0L^{2}_{Pl} . (4.9)

Eq. (4.9) can be integrate from ato 1, namely
p2G_{N}µ_{0}t= 2

3 −p

a^{3}+c+√

ctanh^{−}^{1}

ra^{3}+c
c

! +√

c+1−√

ctanh^{−}^{1}
r1

c + 1

!!

, (4.10)

## JHEP04(2016)147

Figure 3. Behavior of the scale factor a(t) in the black hole scenario. The singular state with a= 0 is approached in an inﬁnite time. Therefore, a black hole forms presenting a Cauchy horizon and an event horizon.

wherec=G_{N}µ_{0}L^{2}_{Pl}. The classical solution can be recovered in the limitc→0
p2G_{N}µ_{0}t= 2

3

1−a^{3/2}

. (4.11)

The behavior of the scale factor is shown in ﬁgure 3 (solid line). The singular state a = 0 is approached in an inﬁnite time. For comparison, ﬁgure 3 also shows the case of general relativity (dashed line) whose analytic expression is given in (4.11). In the GR case a= 0 is reached in a ﬁnite time. The Finkelstein diagram of this collapse is shown in ﬁgure 4. It is clear that in this scenario we have a real black hole with a Cauchy horizon and an event horizon.

4.3 From in to out making use of the boundary conditions

The gravitational collapse and the cosmological solutions previously obtained in the asymp- totic free limit of the weakly non-local theories are all consistent with a general eﬀective FRW equation for the interior matter bouncing. This is a universal property of super- renormalizable asymptotically free gravitational theories including the recent proposed Lee-Wick gravities [61–63]. The simplest eﬀective FRW equation compatible with the general feature discussed in section3 reads

H^{2} = a˙^{2}

a^{2} = 8πG_{N}
3

1− ρ

ρ_{c}

or a˙^{2}

2 = 4πG_{N}
3 ρ_{0}

a^{3}−a^{3}_{c}
a^{3}

1

a . (4.12) Here we only consider the homogeneous interior. Applying again the “Torres” procedure to reconstruct the metric in the vacuum from the metric in the matter region, we get

## JHEP04(2016)147

Figure 4. Finkelstein diagram of the black hole scenario. See the text for more details.

the exterior spacetime imposing that the boundary conditions of the previous sections are satisﬁed. Comparing the interior FRW equation (4.12) with (4.6), we can derive the eﬀective scaling of the Newton constant with the radial coordinate, namely

G(x) = x^{3}−l^{3}_{Λ}

x^{3} , (4.13)

wherexis the radial coordinate. The exterior Schwarzschild spacetime is again (4.1). The
metric is singular in x = 0, but our derivation is correct only for x > x_{bounce} = l_{Λ}, and
x_{bounce} is a ﬁnite positive value. Therefore the metric (4.1) with (4.13) is only valid for
x > lΛ. The Cauchy and event horizons, if any, are located where the function g00(r)
vanish. For diﬀerent values of the mass M we can have two roots, two coincident roots,
or zero roots. Therefore, we here provide a justiﬁcation for the diagrams in section 3
that are only correct whether the metric in the external region present a Cauchy horizon.

Nevertheless, this is the spacetime structure of any astrophysical object with M ≫ M_{Pl}
and then the metric in this subsection, by construction, is compatible with the internal
matter bounce. For completion, the Kretschmann invariant is

RµνρσR^{µνρσ} = 48G^{2}_{N}M^{2} 39l^{6}_{Λ}−10l^{3}_{Λ}r^{3}+r^{6}

r^{12} . (4.14)

5 Coexistence of the two scenarios and Hawking evaporation

The bouncing (black supernova) and the non-bouncing (singularity free black hole) solu- tions seem two diﬀerent scenarios emerging from the same theory. In our class of weakly non-local gravities (3.1) and in many other frameworks [2, 9–11, 14–17], the bounce ap- pears to be unavoidable. However, we do not have the metric of the whole spacetime under control. If we make the reasonable assumption that the exterior vacuum solution is static, we end up with a regular black hole. The ﬁnal product of the collapse would thus depend on whether we reconstruct the external spacetime (imposing the boundary conditions for the continuity of the metric and its ﬁrst derivatives) from the approximate solution inside

## JHEP04(2016)147

the matter (section 3) or the matter interior spacetime from the static metric outside the collapsing body. While at the moment we cannot completely exclude the coexistence of both the dynamics, we would like to provide another possibility.

In this section we provide a mechanism to reconcile the two scenarios based on the stability analysis of the spacetime outside the matter region.

As we have already pointed out in ref. [12], it is quite mysterious that in our class of weakly non-local theories of gravity (3.1) we can ﬁnd the bouncing solution when we consider the gravitational collapse of a spherically symmetric cloud of matter and, on the other hand, regular black hole (approximate) solutions when we consider the static case.

It is possible that all these regular black hole spacetimes are not stable and that their instability provides a link between the bouncing and non-bouncing scenarios.

The black hole solutions are indeed characterized by a de Sitter core, in which the eﬀective cosmological constant is proportional to the mass of the collapsing object [12].

From an analysis of the propagator, we can infer that there is a ghost-like pole, namely
the spacetime is unstable. We can thus expect that the black hole decays into another
black hole state with a de-Sitter core with a smaller eﬀective cosmological constant in
one or more steps through metastable conﬁgurations. The process should end when the
eﬀective cosmological constant is of the order of our non-local scale Λ, likely close to the
Planck mass M_{Pl} if we identify the two scales in the theory. A solution with a de Sitter
core proportional toM_{Pl} is not a black hole but a “particle” with a sub-Planck mass and
without Cauchy and event horizons. Even if we do not know the intermedia states, the
stability analysis may suggest that the black supernova and regular black hole scenarios
are two faces of the same coin. In this way we also provide a reasonable justiﬁcation for
the well known instability of the Cauchy horizon. In our picture, the Cauchy horizon is
just a sector of the close trapped surface, which of course do not extend to inﬁnity. In all
the approximate black hole solutions studied in the past [12, 27], three possible diﬀerent
spacetime structures were presented depending on the value of the mass: with two event
horizons, with two coincident horizons (extremal black hole case), and without any event
horizon (Planck mass particle). However, the correct way to interpret such spacetimes is
not as unstable black hole because of the Cauchy horizon, but as diﬀerent phases of the
collapse and bounce (black supernova).

Let us now expand on the ghost-instability. While a spacetime with a ghost-instability compatible with the optical theorem in general does not exist at all [64], because its decay time is not small but exactly zero, this is not true for weakly non-local theories [65], and our class of theories (3.1) belongs to this group. It is crucial to notice that the singularity-free black hole metrics always show a de Sitter core with a huge eﬀective cosmological constant, namely

Λ_{eff} ≈M G_{N}Λ^{3}, (5.1)

where M is the mass of the body. Therefore, we can easily calculate the second variation of the action (3.1) for the tensor perturbations around the de Sitter spacetime, namely

gµν = ¯gµν+hµν (5.2)

## JHEP04(2016)147

Figure 5. Plot of the inverse propagator (5.5) for 8H^{2}/Λ^{2} = 1,10,25. The lowest dashed curve
corresponds to the local two derivative case, namely Λ→+∞andP^{−1}∝x−1. Here we used the
following form factor: H(z) =^{1}_{2} log z^{4}

+ Γ 0, z^{4}
+γ^{E}

.

where ¯g_{µν} is the de Sitter metric. Here, we also use the parametrization

ds^{2} =−dt^{2}+ exp(2Ht)d~x^{2}, (5.3)

where 8H^{2} = 8Λ_{eff}/3. Moreover, the non-vanishing components for the tensor perturba-
tions are purely spatial, h^{0}_{µ}= 0, and satisfy the usual transverse and traceless conditions:

h^{i}_{i} = 0,∂_{i}h^{i}_{j} = 0. This computation was done for the ﬁrst time in the paper [66] without
introducing any cosmological constant in the action. The ﬁnal result for the variation of
the action reads

δS_{g} = 2κ^{−}_{4}^{2}
Z

d^{4}xp

|g¯|1

4h_{ij}[(−8H^{2}) + (−2H^{2})γ()(−2H^{2})]h^{ij},
γ() = e^{H(}^{−/Λ}^{2}^{)}−1

. (5.4)

From the deﬁnition = −8H^{2}q^{2} = 8H^{2}x (here we introduced a basis of eigenfunctions
h^{(q)}_{ij} for theoperator, with dimensionless momentum eigenvalues−q^{2}), the inverse prop-
agator is

P^{−}^{1}(x)

4H^{2}κ^{−}_{4}^{2} =x−1 +

x−1 4

e^{H(8H}^{2}^{x/Λ}^{2}^{)}−1
x

x− 1

4

. (5.5)

Notice that for the class of form factors we are considering here, H(z) = H(−z). If Λ_{eff} is
large with respect to Λ, we ﬁnd three poles, see ﬁgure5. The second pole in the ﬁgure 5
corresponds to a ghost particle. The outcome of this analysis is a ghost instability of
the approximate black hole solution. However, in a non-local theory the instability is not
catastrophic and can be estimated [65,67]. Let us to consider the vacuum decay (in our

## JHEP04(2016)147

case the black hole spacetime or actually the de Sitter spacetime) into a ghost particle and two normal gravitons [65–67], BH→g, h, h. The decay probability per unit of volume and unit of time reads

Γ_{BH}→g,h,h= w

V T = Λ^{6}

M_{Pl}^{2} e^{−}^{H(8H}^{2}^{x}^{0}^{/Λ}^{2}^{)}, (5.6)
where x_{0} is the ghost-like root in ﬁgure 5 and is obtained expanding the action near the
ghost-pole. For the case of simplicity, here we assume Λ =M_{Pl}. Therefore the lifetime is

τ_{BH}→g,h,h = 1

Γ_{BH}→g,h,hV = 1

V M_{Pl}^{4} e^{H(8H}^{2}^{x}^{0}^{/Λ}^{2}^{)}. (5.7)
The above decaying time is ﬁnite and actually very long because the eﬀective cosmological
constant is proportional to the mass of the black hole [65], namley

τ_{BH}→g,h,h= 1

V M_{Pl}^{4} e^{H(8M x}^{0}^{/M}^{Pl}^{)}. (5.8)

If we consider an astrophysical object, M is of order the Solar mass or more. The result is that the lifetime the all the processes of collapse, bounce and explosion take a very long time. The same exponential factor can be inferred from the ghost-instability pre- sented in [66], replacing the Lorentz-violating scale with the scale of non-locality in the theory (3.1).

We now explicitly consider a class of form factors compatible with super-renormalizability and asymptotic polynomiality, namely

e^{H(z)}=e^{1}^{2}(^{γ}E+Γ(0,z^{2(γ+1)})+logz^{2(γ+1)}), (5.9)
whereby the decay time in the large mass limit simpliﬁes to

τ_{BH}→g,h,h∝ 1
V M_{Pl}^{4}

M
M_{Pl}

γ+1

(5.10)
Taking V = 1/M_{Pl} and γ = 2, we exactly reproduces the Hawking result

τ_{BH}→g,h,h∝ 1
MPl

M MPl

3

. (5.11)

It is quite impressive that the minimal super-renormalizable theory (the one for γ = 2) embodies the Hawking evaporation process through the instability of the vacuum.

Summarizing this section, we have shown that in a large class of weakly non-local gravitational theories any (approximate) black hole solution presenting a de Sitter core nearr= 0 is unstable due to the presence of a ghost instability. However, in these theories this is not a catastrophe because of the non-locality scale. Therefore, the collapse of a cloud always produces a black supernova and never ends up with a black hole. Moreover, for the simplest range of theories compatible with super-renormalizability, the bouncing time perfectly agrees with the Hawking evaporation time. Despite this feature is not universal, it is impressive that it is a distinction of the minimal theory consistent at the quantum level.

## JHEP04(2016)147

6 Conclusions

In ref. [8, 12], we studied the gravitational collapse of a spherically symmetric cloud in a class of weakly non-local theories of gravity that are a ﬁeld theory proposal for a consistent theory of quantum gravity [42,44,45,47–51]. However, in [8,12] we only derived an approx- imate solution for the interior, while the external spacetime was completely conjectured, as we were not able to ﬁnd a metric for the whole spacetime. Nevertheless, we found a new picture for the gravitational collapse with the classical singularity replaced by a bounce, after which the collapsing body starts expanding. We inferred that black holes — in the mathematical sense of regions covered by an event horizon — do not form. The collapse only creates a temporary trapped surface, which can be interpreted as an event horizon only for a timescale shorter than the whole physical process. However, the latter might be extremely long for a stellar-mass object observed by a far-away observer. Our result is in agreement with those of other groups obtained with diﬀerent approaches [2–7,14–17].

In this paper, we have adopted a diﬀerent approach to get an approximate solution for the whole spacetime. Following the idea in [15], we have started from the exterior region and assumed that the spacetime is static outside the matter. This is possible in classical general relativity as a consequence of the Birkhoﬀ theorem, and it may be correct here as well. Such an assumption seems to play a crucial role in the ﬁnal fate of the collapse.

The approximate vacuum solution has two universal features: the spacetime nearr= 0 is well approximated by the de Sitter metric and the global structure show up an event horizon as well as a Cauchy internal horizon. If the mass is comparable to the Planck mass, there are no horizons at all. It is clear that in a dynamical evolution of the black hole the Cauchy horizon instability is not a problem because it is just the internal part of o globally simply-connected trapped surface. These black holes are just like photo shoots of a non static but evolving black hole (where by evolution we mean the dynamics of the black hole mass).

After imposing the boundary conditions, we have reconstructed the interior matter metric that, in contrast to previous results reminded in the ﬁrst part of the paper, does not show the expected bounce. On the contrary, there is an event horizon and a black hole does form. However, we have proved that the exterior metric is actually unstable due to the presence of a ghost-like pole in the propagator. The instability here is not catastrophic because of the non-locality scale that actually allowed us to estimate the lifetime of the system (5.7). It is quite remarkable that for the minimal super-renormalizable theory, the black hole lifetime is identical to the Hawking evaporation time (5.11).

Acknowledgments

C.B. acknowledges support from the NSFC grants No. 11305038 and No. U1531117, the Thousand Young Talents Program, and the Alexander von Humboldt Foundation.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## JHEP04(2016)147

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