11institutetext: Luca Buoninfante 22institutetext: Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
Stockholm, SE-106 91, Sweden, 22email<EMAIL_ADDRESS>33institutetext:
Breno L. Giacchini 44institutetext: Department of Physics, Southern University
of Science and Technology, Shenzhen 518055, China, 44email:
<EMAIL_ADDRESS>55institutetext: Tibério de Paula Netto 66institutetext:
Department of Physics, Southern University of Science and Technology, Shenzhen
518055, China 66email<EMAIL_ADDRESS>
# Black holes in non-local gravity

Luca Buoninfante  Breno L. Giacchini and Tibério de Paula Netto

###### Abstract

In this chapter we present a status report of black hole-like solutions in
non-local theories of gravity in which the Lagrangians are at least quadratic
in curvature and contain specific non-polynomial (i.e., non-local) operators.
In the absence of exact black hole solutions valid in the whole spacetime,
most of the literature on this topic focus on approximate and simplified
equations of motion, which could provide insights on the full non-linear
solutions. Therefore, the largest part of this chapter is devoted to the
linear approximation. We present results on stationary metric solutions
(including both static and rotating cases) and dynamical spacetimes describing
the formation of non-rotating mini black holes by the collapse of null shells.
Non-local effects can regularize the curvature singularities in both scenarios
and, in the dynamical case, there exists a mass gap below which the formation
of an apparent horizon can be avoided. In the final part we discuss
interesting attempts towards finding non-linear black hole solutions in non-
local gravity. Throughout this chapter, instead of focusing on a particular
non-local model, we present results valid for large classes of theories (to a
feasible extent). This more general approach allows the comparison of
similarities and differences of the various types of non-local gravity models.

## Keywords

Black holes, non-local gravity, ghost-free gravity, curvature singularity,
regular solutions, non-locality, infinite derivatives

##  1 Introduction

Einstein’s general relativity (GR) has been the most successful theory to
describe classical aspects of gravity so far as many of its predictions have
been confirmed to high precision Will:2018bme . Despite its great
achievements, there are still unsolved problems suggesting that GR can only be
used as an effective field theory of gravitational interaction, which works
very well at low energy but breaks down in the ultraviolet (UV) regime. In
fact, at the classical level GR is plagued by the presence of black-hole and
cosmological singularities Hawking:1973uf , while at the quantum level the
theory is perturbatively non-renormalizable tHooft:1974toh ; Goroff:1985sz .

A natural way to address these issues is to extend GR by adding higher-
curvature terms to the Einstein–Hilbert action. Remarkably, a theory of
gravity quadratic in curvature can be perturbatively renormalizable in four
dimensions Stelle77 . However, this higher-derivative theory turns out to be
pathological, because of a massive spin-$2$ ghost-like degree of freedom that
causes instabilities and breaks the $S$-matrix unitarity.

This type of unhealthy degree of freedom can be avoided if the action is _non-
polynomial_ in field derivatives. Indeed, by considering quadratic-curvature
terms that contain specific infinite-derivative operators one can prevent the
appearance of extra pathological modes in the physical spectrum, and still
have an improved propagator in the UV, making it possible to formulate
theories of gravity that are ghost-free and renormalizable Krasnikov ; Kuzmin
; Tomboulis:1997gg ; Biswas:2005qr ; Modesto:2011kw ; Biswas:2011ar . The
presence of non-polynomial differential operators renders the gravitational
action _non-local_.

In this chapter, we discuss black-hole solutions and alternatives in the
context of non-local theories of gravity that are UV-modifications of GR. The
presentation has the form of a status report on this topic and it reviews the
most relevant results obtained so far, providing the essential details of both
conceptual and computational aspects. We emphasize that, owing to the
difficulties involved, a rigorous study of exact black-hole solutions in these
models is still lacking. Therefore, the results compiled here mainly consist
of approximate solutions valid in certain regimes, that might be useful to
understand the behaviour of the full non-linear ones.

The chapter is organized as follows:

Sec. 2:

Since most of the considerations are carried out in the linear regime, in this
section we briefly review the linearised formulation of ghost-free non-local
theories of gravity, and show several interesting models that have been
intensively studied in the literature.

Sec. 3:

We discuss linearised stationary solutions in two situations, namely, static
and rotating cases. We show that non-locality can regularize both point-like
and ring-like singularities, and make several remarks about the physical
implications. In what concerns the resolution of point-like singularities, we
show that this regularization can occur at different levels (i.e., potential,
curvature invariants and curvature-derivative invariants), depending on the
type of form factor used.

Sec. 4:

The dynamical solution of mini black hole formation by the collapse of null
shells is discussed. We consider the cases of thin and thick shells. In both
cases there exists a mass gap below which the formation of an apparent horizon
can be prevented, while the curvature singularities can also be regularized if
the imploding shell has some non-vanishing thickness.

Sec. 5:

We discuss the difficulties introduced by non-linearities in the field
equations and review an interesting attempt towards understanding the effects
of non-locality at the non-linear level. By working in a simplified situation,
it is possible to obtain regular black hole solutions with a mass gap and with
multiple horizons.

Sec. 6:

We present our concluding remarks.

Throughout this chapter we use the mostly plus metric convention, with the
Minkowski metric $\eta_{\mu\nu}=\,\mbox{diag}\,(-1,+1,+1,+1)$. The Riemann
curvature tensor is defined by

$\displaystyle
R^{\alpha}{}_{\beta\mu\nu}\,=\,\partial_{\mu}\Gamma^{\alpha}_{\beta\nu}-\partial_{\nu}\Gamma^{\alpha}_{\beta\mu}+\Gamma^{\alpha}_{\mu\tau}\,\Gamma^{\tau}_{\beta\nu}-\Gamma^{\alpha}_{\nu\tau}\,\Gamma^{\tau}_{\beta\mu}\,,$
(1)

while $R_{\mu\nu}=R^{\alpha}{}_{\mu\alpha\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$
are, respectively, the Ricci tensor and the scalar curvature. Also, we adopt
the unit system such that $c=1$ and $\hslash=1$.

##  2 Non-local theories of gravity

Einstein’s GR is described in terms of a very simple Lagrangian which is
linear in curvature. A natural way of modifying the theory in the UV regime is
to generalise the Einstein–Hilbert action through the inclusion of higher-
order curvature invariants which contain higher derivatives. Local higher-
derivative theories of gravity are usually considered pathological because of
the ghost-like degrees of freedom that occur in the spectrum, in the
conventional quantum field theory framework. However, as we discuss in this
section, ghosts can be avoided if the principle of locality is given up.

Let us consider the generic action quadratic in curvature,

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int{\rm
d}^{4}x\sqrt{-g}\,\Big{\\{}R+\frac{1}{2}[\,R\mathcal{F}_{1}(\Box)R+R_{\mu\nu}\mathcal{F}_{2}(\Box)R^{\mu\nu}$
(2) $\displaystyle\hskip
17.07182pt+R_{\mu\nu\rho\sigma}\mathcal{F}_{3}(\Box)R^{\mu\nu\rho\sigma}\,]\Big{\\}}\,,$

where $\kappa=\sqrt{8\pi G},$ $G$ is Newton’s constant and the form factors
$\mathcal{F}_{i}(\Box)$ are non-local, i.e., non-polynomial functions of the
d’Alembertian $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}.$ In particular, we
assume that $\mathcal{F}_{i}(z)$ are analytic functions around $z=0$, in order
to guarantee a smooth and consistent infrared limit. Therefore, expanding them
in Taylor series,

$\displaystyle\mathcal{F}_{i}(\Box)=\sum\limits_{n=0}^{\infty}f_{i,n}\Box^{n}\,,\qquad
i=1,2,3,$ (3)

and using the identity (see, e.g., Deser:1986xr ; AsoreyLopezShapiro )

$R_{\mu\nu\rho\sigma}\Box^{n}R^{\mu\nu\rho\sigma}=4R_{\mu\nu}\Box^{n}R^{\mu\nu}-R\Box^{n}R+O(R^{3}_{\ldots})+{\rm
div}\,,$ (4)

the action (2) can be rewritten as

$S=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{-g}\,\Big{\\{}R+\frac{1}{2}\left[\,R{F}_{1}(\Box)R+R_{\mu\nu}{F}_{2}(\Box)R^{\mu\nu}+O(R^{3}_{\ldots})\,\right]\Big{\\}},$
(5)

where

${F}_{1}(\Box)=\mathcal{F}_{1}(\Box)-\mathcal{F}_{3}(\Box)\,,\quad\quad{F}_{2}(\Box)=\mathcal{F}_{2}(\Box)+4\mathcal{F}_{3}(\Box)\,,$
(6)

and we have omitted boundary terms.

Since most part of this chapter is based on the linearised equations of
motion, which come from the second-order metric perturbations around the
Minkowski spacetime at action level, we neglect the cubic-curvature terms
$O(R^{3}_{\ldots})$ in (5) (or, in other words, the Riemann-squared term
$R_{\mu\nu\rho\sigma}{\cal F}_{3}(\Box)R^{\mu\nu\rho\sigma}$). Therefore, in
what follows we analyse the gravitational action

$S=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\sqrt{-g}\,\Big{\\{}R+\frac{1}{2}\left[\,R{F}_{1}(\Box)R+R_{\mu\nu}{F}_{2}(\Box)R^{\mu\nu}\,\right]\Big{\\}}\,.$
(7)

###  2.1 Linearised theory

In the weak gravitational field approximation, we consider linear
perturbations around the Minkowski background,

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}\,,$ (8)

and expand the action (7) up to order $O(h^{2}_{\ldots})$ Modesto:2011kw ;
Biswas:2011ar ; hedao , namely,

$\displaystyle S^{(2)}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\int{\rm
d}^{4}x\,\Big{\\{}\frac{1}{2}h_{\mu\nu}a(\Box)\Box
h^{\mu\nu}-h_{\mu}^{\sigma}a(\Box)\partial_{\sigma}\partial_{\nu}h^{\mu\nu}+hc(\Box)\partial_{\mu}\partial_{\nu}h^{\mu\nu}$
(9) $\displaystyle\vspace{1cm}-\frac{1}{2}hc(\Box)\Box
h+\frac{1}{2}h^{\lambda\sigma}\frac{a(\Box)-c(\Box)}{\Box}\partial_{\lambda}\partial_{\sigma}\partial_{\mu}\partial_{\nu}h^{\mu\nu}\Big{\\}}\,,$

where

$\displaystyle a(\Box)$ $\displaystyle\equiv$ $\displaystyle
1+\frac{1}{2}{F}_{2}(\Box)\Box,$ (10) $\displaystyle c(\Box)$
$\displaystyle\equiv$ $\displaystyle
1-2{F}_{1}(\Box)\Box-\frac{1}{2}{F}_{2}(\Box)\Box,$ (11)

and $h\equiv\eta_{\mu\nu}h^{\mu\nu}$ is the trace of the field perturbation.
From Eq. (9) we can derive the linearised field equations,

$\displaystyle a(\Box)\left(\Box
h_{\mu\nu}-\partial_{\sigma}\partial_{\nu}h_{\mu}^{\sigma}-\partial_{\sigma}\partial_{\mu}h_{\nu}^{\sigma}\right)\displaystyle+c(\Box)\left(\eta_{\mu\nu}\partial_{\rho}\partial_{\sigma}h^{\rho\sigma}+\partial_{\mu}\partial_{\nu}h-\eta_{\mu\nu}\Box
h\right)$ $\displaystyle\hskip
62.59596pt+\frac{a(\Box)-c(\Box)}{\Box}\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}h^{\rho\sigma}=-2\kappa
T_{\mu\nu}\,,$ (12)

where the coupling to matter is introduced through the energy-momentum tensor

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{m}}}{\delta
g^{\mu\nu}}\simeq\frac{2}{\kappa}\frac{\delta S_{\text{m}}}{\delta
h^{\mu\nu}}\,.$ (13)

In Eq. (13) $S_{\text{m}}$ is the matter action. The energy-momentum tensor
satisfies the conservation law $\partial_{\mu}T^{\mu\nu}=0,$ consistently with
the Bianchi identity.

###  2.2 Propagator

The propagator around the Minkowski background is computed by inverting the
kinetic operator in the Lagrangian (9). Its gauge-fixing independent part
reads Tomboulis:1997gg ; Modesto:2011kw ; Biswas:2011ar ; Accioly:2002tz

$\Pi_{\mu\nu\rho\sigma}(k)=\frac{{P}^{(2)}_{\mu\nu\rho\sigma}}{k^{2}f_{2}(-k^{2})}-\frac{{P}^{(0-s)}_{\mu\nu\rho\sigma}}{2k^{2}f_{0}(-k^{2})}\,,$
(14)

where

$f_{2}(-k^{2})\equiv a(-k^{2})\,,\qquad
f_{0}(-k^{2})\equiv\frac{3c(-k^{2})-a(-k^{2})}{2}\,,$ (15)

and ${P}^{(2,0-s)}_{\mu\nu\rho\sigma}$ are the Barnes–Rivers operators that
project the metric fluctuation $h_{\mu\nu}$ along its gauge-independent
spin-$2$ and spin-$0$ components Barnes ; Rivers ; VanNieuwenhuizen:1973fi .

In general, if the functions $f_{0,2}(-k^{2})$ are polynomials (or,
equivalently, if the form factors ${F}_{i}(\Box)$ are polynomials) the
propagator (14) possesses, besides the massless graviton, extra massive
degrees of freedom that can cause instabilities. This happens because some of
these modes are associated with real poles with negative residue, i.e., they
are unhealthy ghosts.

However, in the framework of non-local models it is possible to construct
classes of ghost-free theories by imposing some requirements on the form
factors. Indeed, if

$\displaystyle f_{s}(-k^{2})=e^{\gamma_{s}(-k^{2})}\,,\qquad s=0,2\,,$ (16)

where $\gamma_{2}(-k^{2})$ and $\gamma_{0}(-k^{2})$ are entire functions, the
propagator only has the pole at $k^{2}=0$ (corresponding to the graviton) and
no ghost-like degree of freedom. It is worth mentioning that, given the sign
difference between the spin-$2$ and spin-$0$ parts of the propagator (14), it
is possible to introduce an additional healthy massive scalar in the theory,
which has important applications in cosmology Biswas:2010zk ; Biswas:2011ar ;
Koshelev:2016xqb ; Koshelev:2020xby ; Koshelev:2020foq ; Koshelev:2017tvv .

From Eq. (16) we can also obtain the corresponding relations for the form
factors,

$\displaystyle{F}_{2}(\Box)=2\frac{e^{\gamma_{2}(\Box)}-1}{\Box}\,,\qquad{F}_{1}(\Box)=-\frac{1}{3}\Big{[}\frac{e^{\gamma_{0}(\Box)}-1}{\Box}+F_{2}(\Box)\Big{]}\,.$
(17)

Let us emphasize that the key role is played by the functions $\gamma_{2}$ and
$\gamma_{0}$, which are entire; thus, the exponentials do not introduce any
unhealthy pole. Therefore, non-locality can help to resolve the ghost problem
in higher-derivative gravity by means of the introduction of non-polynomial
(infinite-derivative) differential operators in the action.

Finally, some simplifications can be achieved for the particular case in which
$\gamma_{2}=\gamma_{0}\equiv\gamma$. In this situation the following condition
holds true:

$f_{2}(\Box)=f_{0}(\Box)\quad\Longrightarrow\quad{F}_{1}(\Box)=-\frac{1}{2}{F}_{2}(\Box)=\frac{1-e^{\gamma(\Box)}}{\Box}\,.$
(18)

Then, the non-local gravity action corresponding to the choice (18) is given
by

$S=\frac{1}{2\kappa^{2}}\int{\rm
d}^{4}x\sqrt{-g}\,\Big{[}R+G_{\mu\nu}\frac{e^{\gamma(\Box)}-1}{\Box}R^{\mu\nu}\Big{]}\,,$
(19)

where $\,G_{\mu\nu}=R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R\,$ is the Einstein
tensor.

###  2.3 Examples of form factors

Here we review some explicit choices for the entire functions
$\gamma_{s}(\Box)$ that have been intensively studied in the literature.

  1. 1.

Gaussian form factor. This is the simplest possible choice, corresponding to
the entire functions $\gamma_{s}(-k^{2})$ that are monomials of degree one
Biswas:2005qr ; Modesto:2011kw ; Biswas:2011ar :

$\gamma_{s}(-k^{2})=\frac{k^{2}}{\mu_{s}^{2}}\,,\qquad s=2,0\,,$ (20)

where $\mu_{s}$ represents the physical energy scale at which non-local
effects are expected to become important.

  2. 2.

Generic exponential form factor. One can generalise the previous example to a
generic monomial of degree $n\geqslant 1$ PU ; Krasnikov ; Edholm:2016hbt ;
hedao ; Buoninfante:2018mre :

$\gamma_{s}(-k^{2})=\left(\frac{k^{2}}{\mu_{s}^{2}}\right)^{n}\,,\qquad
s=2,0\,.$ (21)

The particular case $n=2$ is sometimes referred to as the Krasnikov form
factor, as it was applied for the first time in gravity in Ref. Krasnikov .

  3. 3.

Kuz’min and Tomboulis form factors. This family of form factors was initially
studied in Refs. Kuzmin ; Tomboulis:1997gg  and more recently in Ref.
Modesto:2011kw  and subsequent works. A general form factor of this type can
be written as

$\gamma_{s}(-k^{2})=\lambda_{s}\left[\Gamma\left(0,P_{s}(-k^{2})\right)+\gamma_{\rm
E}+\ln\left(P_{s}(-k^{2})\right)\right]\,,\qquad s=2,0\,,$ (22)

where $\lambda_{s}$ is a positive constant, $\gamma_{\rm E}$ denotes the
Euler–Mascheroni constant, $\Gamma(0,z)$ is the incomplete gamma function and
$P_{s}(z)$ is a real polynomial such that $\gamma_{s}(0)=0$. The Kuz’min form
factor Kuzmin  consists in the choice $P_{s}(z)=-z/\mu_{s}^{2}$ with an
arbitrary $\lambda_{s}\in\mathbb{N}$, while the Tomboulis form factor
Tomboulis:1997gg  is characterized by $\lambda_{s}=1/2$ and $P_{s}(z)$ of
arbitrary (but even) degree $N_{s}\geqslant 2$. Despite their more complicated
expressions (in comparison to the previous ones), they are useful to formulate
super-renormalizable non-local theories of gravity because, for large
momentum, they behave like a polynomial. In fact (22) yields

$\displaystyle
e^{\gamma_{s}(-k^{2})}\underset{k^{2}\to\infty}{\sim}\,k^{2\lambda_{s}N_{s}},$
(23)

where $N_{s}$ is the degree of the polynomial $P_{s}(z)$.

##  3 Stationary linearised solutions

Most of the solutions in non-local gravity were obtained in the linear
approximation. In this section we describe two types of such solutions,
namely, the Newtonian-limit solution and the weak-field slowly rotating
stationary solution. To this end, let us consider a generic metric with line
element

${\rm d}s^{2}=-(1+2\varphi){\rm d}t^{2}+2\vec{h}\cdot{\rm d}\vec{r}\,{\rm
d}t+(1-2\psi)({\rm d}x^{2}+{\rm d}y^{2}+{\rm d}z^{2})\,,$ (24)

where $\vec{r}=(x,y,z)$ is the radius vector, and $\varphi=-\kappa h_{00}/2,$
$\psi=-\kappa h_{ii}/2$ and $h_{i}=\kappa h_{0i}$ are the potentials sourced
by $T_{\mu\nu}.$

In the non-relativistic limit, the source is pressureless,
$T\equiv\eta^{\mu\nu}T_{\mu\nu}\simeq-T_{00}$, and its only non-vanishing
components are $T_{00}$ and, possibly, $T_{0i}$. The latter is zero for static
point-like sources (so that in this case $h_{i}\equiv 0$) and it is non-zero
for a rotating dust.

The metric potentials $\varphi$ and $\psi$ can be obtained from the
00-component and the trace of the linearised field equations (12), which are
equivalent to the set of coupled differential equations:111In the case of
$T_{0i}\neq 0$, the de Donder gauge $\partial_{\mu}h^{\mu\nu}=0$ is assumed,
or a suitable higher-order generalisation compatible with the metric (24)
(see, e.g., Accioly:2016qeb ; Teyssandier89 ; Buoninfante:2018xif ; BuoGia ).

$\displaystyle\left[a(\Delta)-c(\Delta)\right]\Delta\varphi+2c(\Delta)\Delta\psi=$
$\displaystyle\,\,\kappa^{2}\,T_{00}\,,$ (25a)
$\displaystyle\left[3c(\Delta)-a(\Delta)\right]\left[2\Delta\varphi-\Delta\psi\right]=$
$\displaystyle\,\,\kappa^{2}\,T_{00}\,,$ (25b) $\displaystyle
a(\Delta)\,\Delta h_{i}=$ $\displaystyle\,-2\kappa^{2}\,T_{0i}\,,$ (25c)

where the functions $a$ and $c$ now depend on the spatial Laplacian $\Delta$,
as $\Box\simeq\Delta$ for stationary fields. Notice that for $a=c=1$ (i.e.,
${F}_{1}={F}_{2}=0$) the system (25) reduces to standard Poisson equations,
consistently recovering the Einstein’s GR result.

The first two equations in (25) can be decoupled by using the auxiliary
potentials BreTib1

$\chi_{2}\equiv\frac{\varphi+\psi}{2}\qquad\textrm{and}\qquad\chi_{0}\equiv
2\psi-\varphi\,.$ (26)

In fact, the field equations (25) are equivalent to

$\displaystyle f_{s}(\Delta)\Delta\chi_{s}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{2}T_{00}\,,$ (27) $\displaystyle
f_{2}(\Delta)\Delta h_{i}$ $\displaystyle=$
$\displaystyle-2\kappa^{2}T_{0i}\,,$ (28)

where $s=0,2$ and the functions $f_{s}$ were defined in Eq. (15). Once the
above equations are solved for $\chi_{0,2}$, the original potentials are
obtained through the relations

$\displaystyle\varphi=\frac{4}{3}\chi_{2}-\frac{1}{3}\chi_{0},\qquad\qquad\psi=\frac{2}{3}\chi_{2}+\frac{1}{3}\chi_{0}.$
(29)

The functions $\chi_{s}$ are called spin-$s$ potentials because they only
depend on the spin-$s$ part of the field. In fact, each $\chi_{s}$ only
depends on $f_{s}(\Delta)$, which in its turn is related to the gauge-
invariant spin-$s$ part of the propagator (14). Also, notice that for the
particular case described in (18) we have $a=c=f_{2}=f_{0}$ and all the
Newtonian potentials are equal, namely,

$\displaystyle\varphi=\psi=\chi_{2}=\chi_{0}.$ (30)

In what concerns the off-diagonal components $h_{i}$ of the metric, as Eq.
(28) only depends on $f_{2}$, they are not affected by ${F}_{1}$ in the linear
regime. This is due to the fact that the form factor ${F}_{1}$ modifies only
the scalar part of the propagator, which couples to the trace of the energy-
momentum tensor. Thus, since the components $T_{0i}$ do not enter in the trace
$g^{\mu\nu}T_{\mu\nu}$, they cannot act as sources for $h_{i}$ at this order
in perturbation.

In what follows we show some general results on the solutions for the
Newtonian potentials $\chi_{s}$, and we postpone the consideration of the off-
diagonal components $h_{i}$ to Sec. 3.4.

###  3.1 Solution for the potentials in terms of effective delta sources

It is possible to solve Eq. (27) directly using the Fourier transform method,
or the Laplace transform (see Sec. 3.2 below), but in some cases it is
instructive to consider an intermediate step, which we describe here (see
also, e.g., BreTib2 ; Nos6der ).

An alternative interpretation of Eq. (27) follows from the inversion of the
operator $f_{s}(\Delta)$, so that one obtains a standard Poisson equation with
a modified source,

$\displaystyle\Delta\chi_{s}=4\pi G\rho_{s}.$ (31)

Accordingly, the _effective source_ $\rho_{s}$ satisfies

$\displaystyle\rho=f_{s}(\Delta)\,\rho_{s},$ (32)

where $\rho=T_{00}$ is the mass density.

For a point-like source with mass $m$, $\rho(\vec{r})=m\delta^{(3)}(\vec{r})$
and the _effective delta source_ can be obtained via the Fourier transform
method applied to Eq. (32). The result is

$\displaystyle\rho_{s}(r)=\frac{m}{2\pi^{2}}\int_{0}^{\infty}\mathrm{d}k\,\frac{k\sin(kr)}{rf_{s}(-k^{2})},$
(33)

where $r=|\vec{r}|$.

The solution of Eq. (31) for the potential $\chi_{s}$ with the effective delta
source (33) then reads

$\displaystyle\chi_{s}(r)=G\int_{\infty}^{r}\mathrm{d}x\,\frac{m_{s}(x)}{x^{2}},$
(34)

where $m_{s}(r)$ is the _effective mass function_ ,

$\displaystyle m_{s}(r)=4\pi\int_{0}^{r}\mathrm{d}x\,x^{2}\rho_{s}(x).$ (35)

The formulation of the Newtonian limit of higher-derivative gravity models in
terms of the effective delta source (33) has been fruitfully applied to local
and non-local models (see, e.g., Tseytlin95 ; BreTib2 ; Nos6der ). The main
advantage of the method is the possibility of deriving general results about
the regularity of the potentials that depend only on the behaviour of the form
factor in the UV regime, as we discuss in Sec. 3.3 below.

###  3.2 Heat kernel solution for the Newtonian potentials

In the previous subsection, we showed how the solution for the potential
$\chi_{s}$ can be obtained in the effective source formalism. Here, we present
another integral solution for the potential, using the heat kernel approach
based on the Laplace transform, as carried out in Frolov:Poly ; Frolov:Exp .
This solution is particularly useful in the context of Sec. 4, when dealing
with the dynamic process of mini black hole formation.

Introducing the Green’s function for (27) via

$\displaystyle
f_{s}(\Delta)\Delta\,G_{s}(\vec{x},\vec{x}^{\prime})\,=\,\delta^{(3)}(\vec{x}-\vec{x}^{\prime})\,,$
(36)

the integral solution for the potential is given by

$\displaystyle\chi_{s}(\vec{x})\,=\,4\pi
G\int\mathrm{d}^{3}x^{\prime}\,G_{s}(\vec{x},\vec{x}^{\prime})\,\rho(\vec{x}^{\prime})\,.$
(37)

Let us now assume that the inverse of the differential operator in (36) can be
written as the Laplace transform of a function $h_{s}(s^{\prime})$, that is,

$\displaystyle-\frac{1}{\xi
f_{s}(-\xi)}\,=\,\int_{0}^{\infty}\mathrm{d}s^{\prime}\,h_{s}(s^{\prime})\,e^{-s^{\prime}\xi}\,.$
(38)

Then, the $x$-representation of the Green’s function reduces to

$\displaystyle
G_{s}(\vec{x},\vec{x}^{\prime})\,=\,\int_{0}^{\infty}\mathrm{d}s^{\prime}\,h_{s}(s^{\prime})\,\langle\vec{x}\,|\,e^{s^{\prime}\,\Delta}\,|\,\vec{x}^{\prime}\rangle\,,$
(39)

where

$\displaystyle\langle\vec{x}\,|\,e^{s\,\Delta}\,|\,\vec{x}^{\prime}\rangle\,=\,K(|\vec{x}-\vec{x}^{\prime}|;s)\,=\,\frac{e^{-\frac{|\vec{x}-\vec{x}^{\prime}|^{2}}{4s}}}{(4\pi
s)^{\frac{3}{2}}}\,$ (40)

is the heat kernel of the Laplacian.

For a point-like source $\rho(\vec{x})=m\delta^{(3)}(\vec{x})$ the formula
(37) simplifies to

$\displaystyle\chi_{s}(r)\,=\,4\pi
Gm\int_{0}^{\infty}\mathrm{d}s^{\prime}\,h_{s}(s^{\prime})\,K(r;s^{\prime}),\qquad
s=0,2\,,$ (41)

where $r=|\vec{x}|$. Explicit calculations using this method can be found in
Frolov:Poly ; Frolov:Exp ; BreTib1 ; BFZ-pbranes ; hedao  and in the Sec. 4.4
below.

###  3.3 Resolution of point-like singularities

The resolution of the Newtonian point-like singularity in non-local gravity
models has been studied by several authors Tseytlin95 ; Modesto:2011kw ;
Modesto:2012ys ; Jens ; Edholm:2016hbt ; hedao ; Biswas:2011ar ; Nos6der ;
Nos4der ; BreTib2 ; Buoninfante:2018xiw ; Buoninfante:2018rlq ;
Buoninfante:2018stt ; Ercan18 ; BFZ-pbranes ; Buoninfante:2020ctr ;
Buoninfante:2018lnh ; Abel:2019zou ; delaCruz-Dombriz:2018aal ; Kolar:2020bpo
. In most of the publications, a specific form factor was chosen and then the
potentials (and, sometimes, also some curvature invariants) were evaluated,
either analytically or numerically. Since there is an infinite amount of non-
local form factors and of curvature invariants, it is impossible to proceed
the investigation one by one. This issue was solved in Nos6der , by using the
effective source formalism (see Sec 3.1) and presenting necessary and
sufficient conditions that a form factor must satisfy for the Newtonian limit
to be regular — as defined in terms of gravitational potential, curvature or
curvature-derivative invariants.222We use the term curvature-derivative
invariant to a generic scalar which is polynomial on curvature tensors and
their covariant derivatives.

Regarding the regularity of the potentials, if the function $f_{s}(-k^{2})$
grows at least as fast as fast as $k^{2}$ for large $k$, the smearing of the
original delta source through (33) is enough to yield a potential
$\chi_{s}(r)$ that is finite at $r=0$ BreTib2 . This happens even in the case
of $f_{s}(-k^{2})\sim k^{2}$, for which the effective delta source
$\rho_{s}(r)$ is unbounded (see Nos4der ; Nos6der ). In most cases, however,
the effective delta source is also bounded; in fact, $\rho_{s}(r)$ is finite
if $f_{s}(-k^{2})\sim k^{4}$ or faster BreTib2 . If both $\chi_{0}$ and
$\chi_{2}$ are regular, so are the potentials $\varphi$ and $\psi$ [see Eq.
(29)]. In this spirit, one might say that the improved UV behaviour of the
propagator (14) regularizes the Newtonian singularity of the potentials. We
stress that the resolution of this divergence is _not_ owing to the non-
locality of the theory; on the contrary, it was first noticed in the local
fourth-derivative gravity Stelle77  and later in other more general local
higher-derivative models Newton-MNS ; Giacchini:2016xns ; BreTib1 .

It is possible to achieve an increasing degree of regularity by including more
derivatives and non-localities in the action. This can be better formulated in
terms of the following definition.

######  Definition 1 (Order of regularity of a potential Nos6der )

If there exists a $p\in\mathbb{N}$ such that $\chi_{s}(r)$ is at least
$2p$-times continuously differentiable, $\chi_{s}^{(2p)}(r)$ is bounded and
$\chi_{s}^{(2n+1)}(0)=0$ for all $n\in\\{0,\ldots,p-1\\}$, then $p$ is called
the _order of regularity_ of $\chi_{s}(r)$ (equivalently, it is said that
$\chi_{s}(r)$ is _$p$ -regular_).

With this definition, as the next theorem shows, the considerations about the
usual regularity of the potential can be extended to the regularity of the
curvature invariants associated with the static (i.e., with $h_{i}\equiv 0$)
metric (24).

######  Theorem 3.1 (Regularity of linearised curvature-derivative invariants
Nos6der )

Let the form factors be such that $f_{s}(-k^{2})$ asymptotically grows at
least as fast as $k^{4+2N_{s}}$ for an integer $N_{s}\geqslant 0$ and let
$N\equiv\min\\{N_{0},N_{2}\\}$. Under these conditions, the potential
$\chi_{s}(r)$ is at least $(N_{s}+1)$-regular and all the linearised
curvature-derivative invariants333That is, curvature-derivative invariants
calculated with the metric (24), with $h_{i}\equiv 0$, to the leading order in
the metric perturbation. with at most $2N$ covariant derivatives are bounded.

###### Corollary 1

If $f_{2}(-k^{2})$ and $f_{0}(-k^{2})$ grow faster than any polynomial, then
the potentials $\chi_{0,2}(r)$ are even analytic functions of $r$ and all the
curvature invariants with an arbitrary number of covariant derivatives are
regular.

The proof of the theorem can be found in Nos6der , where the regularity
properties of the potential $\chi_{s}$ were deduced from some basic features
of $f_{s}(-k^{2})$ translated into the effective delta source (33).

If the potentials $\chi_{0,2}$ are only 0-regular (like in the case of the
Kuz’min or Tomboulis form factors (22) that asymptotically tend to $k^{2}$),
then the metric still has curvature singularities. On the other hand, if the
potentials are only 1-regular, all the curvature invariants without covariant
derivatives (such as $R$ and $R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$) are
regular, but there are scalars with two covariant derivatives that diverge,
such as $\Box R$ — this is the case, e.g., of the form factors (22) that tend
to $k^{4}$.

Hence, besides characterizing the models that have a regular Newtonian limit,
the theorem also shows that the regularity of curvature invariants at $r=0$
depends on the absence of odd powers of $r$ in the Taylor expansion of the
potentials $\chi_{s}(r)$. The higher the first odd-order term is, the larger
the set of regular scalars is. A simple example is provided by the family of
scalars $\Box^{n}R$ (for an arbitrary $n$), which in the Newtonian limit reads

$\displaystyle(\Box^{n}R)_{\text{lin}}=2\Delta^{n+1}\chi_{0}=2\left[\chi_{0}^{(2n+2)}+\frac{2(n+1)}{r}\chi_{0}^{(2n+1)}\right].$
(42)

For the above scalar to be regular, it suffices that the potential
$\chi_{0}(r)$ be $(n+1)$-regular — in other words, that $\chi_{0}(r)$ is
$(2n+2)$-times continuously differentiable and the limit $\lim_{r\to
0}\chi_{0}^{(2n+1)}(r)/r$ exists Nos6der . If the former condition is
verified, but the latter is not, then $(\Box^{n}R)_{\text{lin}}$ has a
singularity at $r=0$. Only if the form factor grows faster than any polynomial
[such as (21)], all the local curvature-derivative invariants are regular.

Finally, we remark that the Theorem 3.1 does not distinguish between local and
non-local higher-derivative models, as it only depends on the function
$f_{s}(-k^{2})$. Therefore, the regularization of the Newtonian limit can be
achieved in both types of theories, and it is not a consequence of the
absence/presence of ghosts or non-localities BreTib2 . Nevertheless, the
$\infty$-order regularity can only be achieved in non-local models, but not in
all of them. Namely, only in the ones with form factors that grow faster than
any polynomial [e.g., (21)]; in this case, the effective delta source (33) and
the associated Newtonian potential $\chi_{s}(r)$ are even analytic functions
of $r$ Nos6der .

#### Example 1: Gaussian form factor

As an explicit example, let us consider the form factor (20) with
$f_{0}(-k^{2})=f_{2}(-k^{2})=\exp(k^{2}/\mu^{2})$. The Eq. (30) is valid in
this case, and the potentials can be obtained through the methods described in
the previous sections; the solution reads Tseytlin95 ; Siegel:2003vt ;
Biswas:2005qr ; Modesto:2011kw ; Biswas:2011ar

$\displaystyle\varphi(r)=-\frac{Gm}{r}{\rm erf}\left(\frac{\mu
r}{2}\right)\,,$ (43)

where $\text{erf}(x)$ is the error function. As expected from the above
considerations, the gravitational potential in Eq. (43) is non-singular at
$r=0$ and tends to the finite constant value $\varphi(0)=Gm\mu/\sqrt{\pi}$,
whereas in the large-distance limit ($r\mu\gg 1$) the $1/r$ behaviour of the
Newtonian potential is recovered (see Fig. 1).

Moreover, since the functions $f_{0,2}(-k^{2})$ grow faster than any
polynomial, all the linearised curvature invariants (even those constructed
with covariant derivatives of the curvatures) are bounded. For instance, we
can compute the linearised Kretschmann invariant Buoninfante:2018xiw ,

$\,\,(R_{\mu\nu\alpha\beta}^{2})_{\rm lin}=\frac{G^{2}m^{2}}{3\pi
r^{6}}e^{-\frac{\mu^{2}r^{2}}{2}}\Big{\\{}5\mu^{6}r^{6}+4\Big{[}6\mu
r+\mu^{3}r^{3}-6\sqrt{\pi}e^{\frac{\mu^{2}r^{2}}{4}}{\rm erf}\Big{(}\frac{\mu
r}{2}\Big{)}\Big{]}^{2}\Big{\\}}\,,$ (44)

and explicitly verify that it is non-singular. Indeed,

$\lim\limits_{r\rightarrow 0}(R_{\mu\nu\alpha\beta}^{2})_{\rm
lin}=\frac{5G^{2}m^{2}\mu^{6}}{3\pi}\,,$ (45)

while, in the large-distance limit it consistently recovers the GR expression

$\displaystyle(R_{\mu\nu\alpha\beta}^{2})_{\rm
lin}\sim\frac{48G^{2}m^{2}}{r^{6}}.$ (46)

Figure 1: Behaviour of the metric potential $\varphi(r)$ for the non-local
gravity model in Eq. (43) (red line) and GR (blue line). For convenience, we
have set $\mu=G=m=1$.

From a physical point of view, the regularization that we have just showed can
be interpreted as if the non-local form factor smears out the point-like
source at $r=0$ on a spatial region of size $\sim 1/\mu$. In addition, in
Buoninfante:2018rlq ; Buoninfante:2018xiw  it was shown that the above regular
metric becomes conformally flat in the limit $r\rightarrow 0$, as the Weyl
tensor vanishes at the origin; more generally, this happens provided that the
potential $\chi_{2}(r)$ is at least 1-regular BreTib1 .

Similar qualitative results hold for the form factors in Eq. (21), which also
grow faster than any polynomial. See, e.g., Edholm:2016hbt ; hedao ; Jens ;
BFZ-pbranes  for other explicit examples of this type.

Figure 2: Numerical evaluation of $\chi_{s}(r)$ and its first odd-order
derivatives for the Kuz’min form factor with $\lambda_{s}\in\\{1,2,3,4\\}$ in
(47); we have set $m=G=\mu_{s}=1$.

#### Example 2: Kuz’min form factor

Since the previous example dealt with a form factor that grows faster than any
polynomial, here we discuss the case (presented in Nos6der ) of a form factor
that behaves like a polynomial in UV regime. To this end, let us consider the
Kuz’min form factor (22) Kuzmin , for which

$\displaystyle\gamma_{s}(-k^{2})=\lambda_{s}\left[\gamma_{\rm
E}+\Gamma(0,k^{2}/\mu_{s}^{2})+\ln(k^{2}/\mu_{s}^{2})\right],$ (47)

where $\mu_{s}$ is a mass parameter and $\lambda_{s}\in\mathbb{N}$. For large
momentum it satisfies

$\displaystyle
f_{s}(-k^{2})\,\underset{k^{2}\to\infty}{\approx}\,e^{\lambda_{s}\gamma_{\rm
E}}\,\left(\frac{k^{2}}{\mu_{s}^{2}}\right)^{\lambda_{s}}\,,$ (48)

thus $f_{s}(-k^{2})\sim k^{2\lambda_{s}}$ for $k$ large enough. From the
aforementioned results relating the UV behaviour of the form factor and the
order of regularity of the Newtonian potentials Nos6der  it follows that, if
$\lambda_{s}=1$, the effective delta source is still singular and the
associated potential is only 0-regular. On the other hand, the effective delta
source is bounded for $\lambda_{s}\geqslant 2$, and the potential
$\chi_{s}(r)$ is $(\lambda_{s}-1)$-regular.

These features can be viewed in Fig. 2, which displays the numerical
evaluation of $\chi_{s}(r)$, $\chi_{s}^{\prime}(r)$, $\chi_{s}^{(3)}(r)$ and
$\chi_{s}^{(5)}(r)$ for the parameter $\lambda_{s}\in\\{1,2,3,4\\}$. Notice
that the potential is finite for $\lambda_{s}=1$, but its first derivative
does not vanish at $r=0$, indicating the singularity of the source (and of the
curvature invariants). On the other hand, for $\lambda_{s}\geqslant 2$ the
corresponding potential is at least $(\lambda_{s}-1)$-regular, as discussed
above. More precisely, we see that $\chi_{s}^{(2\lambda_{s}-1)}(0)\neq 0$,
showing that the potential is not $\lambda_{s}$-regular.

Similar results also hold for the form factors considered by Tomboulis
Tomboulis:1997gg  and Modesto Modesto:2011kw , which behave like polynomials
in the UV. See BreTib2  for another example.

###  3.4 Resolution of ring-like singularities

The previous discussion was focused on static solutions; here we return to the
more general metric in the form (24) with non-trivial $h_{i}$ and show that
non-locality is also able to regularize rotating ring-like singularities. We
follow the presentation of Buoninfante:2018xif  and only consider the
particular choice of form factor (20) such that
$f_{0}(-k^{2})=f_{2}(-k^{2})=\exp(k^{2}/\mu^{2})$, like in the example of Sec.
3.3.

In GR, the Kerr metric is plagued by a ring singularity that in Boyer-
Lindquist coordinates is described by the equation $r^{2}+a^{2}{\rm
cos}^{2}\vartheta=0$ or, in Cartesian coordinates, by $z=0$ and
$x^{2}+y^{2}=a^{2},$ where $a$ can be thought of as the radius of the ring. We
can mimic such a ring distribution by modelling the energy-momentum tensor as
a Dirac delta distributed on a ring of radius $a$ rotating with constant
angular velocity $\omega$ Buoninfante:2018xif :

$T_{00}=m\delta(z)\frac{\delta^{(2)}(x^{2}+y^{2}-a^{2})}{\pi},\qquad
T_{0i}=T_{00}v_{i},$ (49)

where $v_{i}$ is the tangential velocity and its magnitude satisfies the
relation $v=\omega\,a.$ By taking the $z$-axis as the direction of the
angular-velocity vector, we can write $v_{x}=-y\,\omega,$ $v_{y}=x\,\omega$
and $v_{z}=0.$ Note that, in the limit $a\rightarrow 0$ we consistently
recover the expression of the point-like source because $v_{i}\rightarrow 0$
and
$\delta^{(2)}(x^{2}+y^{2}-a^{2})\rightarrow\delta^{(2)}(x^{2}+y^{2})=\pi\delta(x)\delta(y).$

In this configuration, the off-diagonal elements of the metric (24) are non-
vanishing, and the set of non-local differential equations (25) reduces to

$\begin{array}[]{rl}\displaystyle
e^{-\Delta/\mu^{2}}\Delta\varphi(\vec{r})=\displaystyle
e^{-\Delta/\mu^{2}}\Delta\psi(\vec{r})=&4Gm\,\delta(z)\delta^{(2)}(x^{2}+y^{2}-a^{2}),\\\\[7.11317pt]
\displaystyle e^{-\Delta/\mu^{2}}\Delta
h_{0x}(\vec{r})=&\displaystyle-16Gm\,\omega\,y\delta(z)\delta^{(2)}(x^{2}+y^{2}-a^{2}),\\\\[7.11317pt]
\displaystyle e^{-\Delta/\mu^{2}}\Delta h_{0y}(\vec{r})=&\displaystyle
16Gm\,\omega\,x\delta(z)\delta^{(2)}(x^{2}+y^{2}-a^{2}).\end{array}$ (50)

We can use the Fourier transform method to solve the modified Poisson
equations. It turns out to be useful to work in cylindrical coordinates, i.e.,
$x=\rho\cos\varphi,$ $y=\rho\sin\varphi,$ $z=z.$ The Fourier transform of the
energy-momentum tensor components read Buoninfante:2018xif

$\displaystyle\mathcal{T}[\delta(z)\delta^{(2)}(x^{2}+y^{2}-a^{2})]$
$\displaystyle=$
$\displaystyle\pi\,I_{0}\left(ia\sqrt{k_{x}^{2}+k_{y}^{2}}\right),$ (51)
$\displaystyle\mathcal{T}[x\delta(z)\delta^{(2)}(x^{2}+y^{2}-a^{2})]$
$\displaystyle=$ $\displaystyle\pi
a\frac{k_{x}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}I_{1}\left(ia\sqrt{k_{x}^{2}+k_{y}^{2}}\right),$
(52) $\displaystyle\mathcal{T}[y\delta(z)\delta(x^{2}+y^{2}-a^{2})]$
$\displaystyle=$ $\displaystyle\pi
a\frac{k_{y}}{\sqrt{k_{x}^{2}+k_{y}^{2}}}I_{1}\left(ia\sqrt{k_{x}^{2}+k_{y}^{2}}\right),$
(53)

where $\mathcal{T}[\cdots]$ stands for the Fourier transform operation,
$I_{0}$ and $I_{1}$ are modified Bessel functions of the first kind.

For simplicity, let us analyse the behaviour of the solution in the $xy$-plane
(i.e. $z=0$), where we expect the singularity to appear in GR, and work with
the cylindrical radial coordinate $\rho=\sqrt{x^{2}+y^{2}}$. Therefore, by
transforming back to coordinate space, the metric potentials are given by
Buoninfante:2018xif

$\varphi(\rho)=\psi(\rho)=-Gm\int_{0}^{\infty}\mathrm{d}\zeta\,I_{0}\left(ia\zeta\right)I_{0}\left(i\zeta\rho\right){\rm
erfc}\left(\frac{\zeta}{\mu}\right)\,,$ (54)
$h_{0x}(x,y)=4Gm\,\omega\,a\frac{y}{\rho}H(\rho),\qquad
h_{0y}(x,y)=-4Gm\,\omega\,a\frac{x}{\rho}H(\rho),$ (55)

where ${\rm erfc}(z)$ is the complementary error function and

$H(\rho)\equiv\int_{0}^{\infty}\mathrm{d}\zeta\,I_{1}(ia\zeta)I_{1}(i\zeta\rho){\rm
erfc}\left(\frac{\zeta}{\mu}\right)\,.$ (56)

The above integrals cannot be solved analytically, but the integrations can be
performed numerically and the solutions are plotted in Fig 3. It shows that in
GR the metric potentials are singular at $\rho=a,$ as expected, whereas in the
non-local gravity model under investigation the would-be ring singularity is
regularized. Also in this case, one can verify that all curvature invariants
are non-singular in the entire spacetime and that the Weyl tensor vanishes at
$r=0$ Buoninfante:2018xif .

Figure 3: Behaviour of the components $-h_{00}=2\varphi$ and $h_{0i}\sim H$ as
functions of the cylindrical radius $\rho,$ for the non-local gravity model
(red line) and GR (blue line). For convenience, we have set $\mu=1,$ $2Gm=1$
and $a=1.$

Let us remark that the linearised regime holds as long as the conditions
$2|\varphi|<1$ and $|h_{0i}|<1$ are satisfied. In particular, they are valid
for any value of $r$ if, neglecting constant factors of order one, $Gm\mu<1$
and $Gm\mu^{2}\omega a^{2}<1,$ respectively Buoninfante:2018xif . From the
last inequality it is evident that the angular velocity $\omega$ cannot be too
large otherwise the linear approximation would break down. Finally, the metric
outside the source was obtained using the multipole expansion also in
Buoninfante:2018xif , and the extension of this procedure for more general
form factors was carried out in BuoGia .

##  4 Mini black hole formation by the collapse of null shells

Up to this point, our considerations were restricted to linearised stationary
configurations without horizons. In this section, we discuss the collapse of
small-mass spherical null shells. By small mass we mean that we continue to
work with the linearised equations for the gravitational field. The interest
in this scenario is the possibility of the emergence of mini black holes.

The gravitational field of a null shell can be built in the linearised limit
using the superposition principle. By boosting the weak-field potential
$\chi_{s}$ of a static point-like mass, the gravitational field of an
ultrarelativistic particle is obtained, and taking a spherical superposition
of such solutions, the linearised metric of a collapsing null shell is
constructed. The formalism we follow was introduced in Frolov:Exp , where it
was applied to non-local gravity with form factor
$f_{s}(-k^{2})=\exp(k^{2}/\mu_{s}^{2})$. It was later generalised to an
arbitrary form factor $f_{s}(-k^{2})$ in Frolov:Poly  and applied to the case
of local higher-derivative polynomial models in Frolov:Poly ; BreTib1 .

One of the conclusions of Refs. Frolov:Exp ; Frolov:Poly ; BreTib1  is the
existence of a mass gap to the formation of mini black holes if
$f_{s}(-k^{2})\sim k^{2}$ for large enough $k$. The presence of a mass gap in
higher-derivative gravity models is known since the 1980s Frolov:Weyl , and it
means that a black hole can only be formed if its mass is larger than a
certain critical value. This contrasts with GR, where any mass can become a
black hole, provided it is concentrated in a sufficiently small region. On top
of that, the curvature invariants remain bounded at $r=0$ for all theories
which $f_{s}(-k^{2})\sim k^{4}$ (or faster) asymptotically, assuming that the
collapsing shell has a finite thickness.

###  4.1 Ultrarelativistic limit

As a first step towards the gravitational field of a collapsing shell, we
consider the field associated with an ultrarelativistic point-like particle,
which can be obtained by the following procedure Frolov:Exp ; Frolov:Poly .
First, we perform a Lorentz transformation of the metric (24) (with
$h_{i}\equiv 0)$,

$\displaystyle t=\gamma\left(t^{\prime}-\beta\,x^{\prime}\right),\qquad
x=\gamma\left(x^{\prime}-\beta\,t^{\prime}\right),\qquad\gamma\equiv\frac{1}{\sqrt{1-\beta^{2}}},$
(57)

which yields the metric of a moving object with velocity $\beta$ in the
$x$-direction. Then, we consider the Penrose limit, i.e., we take $\,\beta\to
1\,$ while keeping the relativistic mass

$\displaystyle M=\lim_{\gamma\to\infty}(\gamma m)$ (58)

of the object fixed.

Therefore, after applying the boost (57), one gets

$\displaystyle\mathrm{d}s^{2}\,=\,\mathrm{d}s_{0}^{2}+\mathrm{d}s_{1}^{2}\,,$
(59)

where

$\displaystyle\mathrm{d}s_{0}^{2}\,=\,-2\mathrm{d}u\mathrm{d}v+\mathrm{d}y^{2}+\mathrm{d}z^{2}$
(60)

is the flat spacetime metric and

$\displaystyle\mathrm{d}s_{1}^{2}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}\left(\varphi+\psi\right)}{2}\left[\left(1-\beta\right)^{2}\mathrm{d}v^{2}+\left(1+\beta\right)^{2}\mathrm{d}u^{2}\right]-\left(\varphi-\psi\right)\,\mathrm{d}u\,\mathrm{d}v$
(61) $\displaystyle-2\psi\,(\mathrm{d}y^{2}+\mathrm{d}z^{2})\,$

is the first-order perturbation. In the above formulas, we introduced the
advanced and retarded null coordinates

$\displaystyle v=t^{\prime}+x^{\prime},\qquad\qquad u=t^{\prime}-x^{\prime}.$
(62)

The form of the flat metric (60) remains unchanged in the limit $\beta\to 1$,
while the perturbation goes to

$\displaystyle\mathrm{d}s_{1}^{2}={\Phi}\,\mathrm{d}u^{2}\,,\qquad\mbox{where}\qquad{\Phi}=-4\,\lim_{\gamma\to\infty}(\gamma^{2}\chi_{2})\,.$
(63)

Hence, the dominant contribution in the ultrarelativistic limit comes from the
spin-$2$ combination of the metric potentials, $\chi_{2}=(\varphi+\psi)/2$.
This happens because, in this regime, the interaction between particles and
the gravitational field is similar to that of photons (see, e.g., BuoGia ;
Giacchini:2018twk ).

The function ${\Phi}$ can be evaluated through (63) by combining the heat
kernel solution (41) for $\chi_{2}$ and Eq. (58). Indeed, taking into account
that after the boost $r^{2}=\gamma^{2}u^{2}+y^{2}+z^{2}$, it follows

$\displaystyle{\Phi}=-4G\lim_{\gamma\to\infty}(\gamma
m)\int_{0}^{\infty}\frac{\mathrm{d}s}{s}\,h_{2}(s)\,e^{-(y^{2}+z^{2})/4s}\lim_{\gamma\to\infty}\frac{\gamma
e^{-\gamma^{2}u^{2}/4s}}{\sqrt{4\pi s}}\,.$ (64)

Recalling that

$\displaystyle\lim_{\gamma\to\infty}\frac{\gamma
e^{-\gamma^{2}u^{2}/4s}}{\sqrt{4\pi s}}\,=\,\delta(u)\,,$ (65)

the Eq. (64) can be written as

$\displaystyle{\Phi}\,=\,-4GM\,H(y^{2}+z^{2})\,\delta(u)\,,$ (66)

where we define

$\displaystyle
H(\zeta)\,\equiv\,\int_{0}^{\eta^{2}}\frac{\mathrm{d}s}{s}\,h_{2}(s)\,e^{-\zeta/4s}\,.$
(67)

Notice that in (67) it was introduced an infrared cutoff $\eta$ for large $s$,
because the integral in (64) typically has an infrared divergence, owing to
the massless nature of the graviton. As discussed in Frolov:Exp , any change
in the cutoff parameter can be absorbed into a redefinition of the
coordinates. Observables, such as the curvature tensors, do not depend on
$\eta$.

The metric (63) generalises, to higher-derivative gravity models, the
Aichelburg–Sexl Aich-Sexl  solution in GR for the gravitational field of an
ultrarelativistic massive particle without angular momentum (non-spinning
gyraton). Furthermore, the extension of (63) to spinning objects can be found
in Boos:2020ccj  (see also Kolar:2021uiu ). Finally, the solution (63) has
been used in Ref. hedao  to study the black hole formation by the head-on
collision of ultrarelativistic particles, with similar conclusions about the
existence of mass gap as the ones derived from the collapse of null shells —
which we mentioned at the beginning of the section and review in Sec. 4.4
below.

###  4.2 Thin null shell collapse

Following Frolov:Exp ; Frolov:Poly , we first consider a shell with vanishing
thickness. At the linearised level, the field associated to a thin null shell,
or $\delta$-shell, can be obtained by the continuous superposition of gyratons
(63) spherically distributed passing through one given point $O$ Frolov:Exp .
This point is the vertex of the null cone representing the shell, such that,
for $t<0$, the shell collapses towards the apex $O$ and, for $t>0$, it
proceeds its expansion after the collapse. The energy-momentum tensor
$T_{\mu\nu}$ associated with the shell is

$\displaystyle T_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}={M\over 4\pi
r^{2}}[\delta(v)\mathrm{d}v^{2}+\delta(u)\mathrm{d}u^{2}].$ (68)

It can be shown that, outside the shell, the averaged metric perturbation
$\langle\mathrm{d}s_{1}^{2}\rangle$ resulting from this distribution of non-
spinning gyratons is given by Frolov:Exp

$\displaystyle\langle\mathrm{d}s_{1}^{2}\rangle=-\dfrac{2GM}{r}H(r^{2}-t^{2})\bigg{[}\bigg{(}\mathrm{d}t-\frac{t}{r}\mathrm{d}r\bigg{)}^{2}+\frac{r^{2}-t^{2}}{2}\mathrm{d}\Omega^{2}\bigg{]}\,,\quad
r\geqslant|t|\,,$ (69)

while in its interior (in the region where $r<|t|$) the spacetime is flat,
$\langle\mathrm{d}s_{1}^{2}\rangle=0$. In (69), $H(z)$ is defined by (67), as
given by the metric (63) associated with a single gyraton.

The linearised Kretschmann scalar calculated with the metric (69) is

$\displaystyle(R_{\mu\nu\alpha\beta}^{2})_{\rm
lin}=\dfrac{48G^{2}M^{2}\zeta^{2}}{r^{6}}Q(\zeta),$ (70) $\displaystyle
Q(\zeta)=2\left[H^{\prime\prime}(\zeta)\right]^{2}\zeta^{2}+2H^{\prime}(\zeta)H^{\prime\prime}(\zeta)\zeta+\left[H^{\prime}(\zeta)\right]^{2}$
, (71)

where $\zeta=r^{2}-t^{2}$. In this case, (70) is singular at $r=0$ in any
higher-derivative model Frolov:Exp ; Frolov:Poly ; BreTib1 . However, this
singularity is a consequence of the non-physical approximation of an
infinitesimally thin shell.

###  4.3 Thick null shell collapse

The metric associated with a thick null shell can be built, in the linear
regime, by superposing a set of $\delta$-shells collapsing to the same spatial
point $O$, which is taken as the origin of the coordinate system. Suppose that
the spherical collapsing null fluid is represented by a pulse with a finite
time duration characterized by a density function $\rho(t)$. The total mass of
the shell is

$\displaystyle M=\int\mathrm{d}t\,\rho(t),$ (72)

and it collapses with the speed of light, shrinking to zero radius at the
moment $t=0$. The corresponding metric perturbation can be obtained by
averaging the metric (69) of the thin null shells with respect to the density
$\rho$ Frolov:Exp ,

$\displaystyle\langle\langle\mathrm{d}s_{1}^{2}\rangle\rangle(t,r)\,=\,\int\mathrm{d}t^{\prime}\,\rho(t^{\prime})\langle\mathrm{d}s_{1}^{2}\rangle(t-t^{\prime},r)\,.$
(73)

Following Frolov:Exp ; Frolov:Poly , we work with the simplest profile,
assuming that the density $\rho(t)$ at $r=0$ remains constant during the
collapse, and that it is null before and after it, namely

$\displaystyle\rho(t)=\left\\{\begin{array}[]{l l}0\,,&\text{if
$\,|t|>\tau/2\,$,}\\\ M/\tau\,,&\text{if $\,-\tau/2<t<\tau/2\,$,}\\\
\end{array}\right.$ (76)

where $\tau$ is the thickness (or duration) of the shell. The distribution
(76) defines specific spacetime domains; see the detailed discussion in
Frolov:Exp . Here we restrict considerations to the $I$-domain (near the point
$t=r=0$), characterized by the intersection of the incoming and the outgoing
fluxes of null fluid. Formally, it is defined as the locus of the spacetime
points where $r+|t|<\tau/2$. In Frolov:Exp ; Frolov:Poly ; BreTib1  it was
proved that the non-singular source (76) suffices to regularizes the
Kretschmann scalar for theories with form factors such that $f_{s}(-k^{2})\sim
k^{4}$ or faster for large $k$.

Taking into account that only the $\delta$-layers which cross $O$ at times
$t^{\prime}\in(t-r,t+r)$ contribute to the field inside the $I$-domain, it is
not difficult to verify that the metric is stationary there and that the Eq.
(73) yields Frolov:Exp

$\displaystyle\langle\langle\mathrm{d}s_{1}^{2}\rangle\rangle=-\frac{2GM}{\tau
r}\left[c_{0}\,\mathrm{d}t^{2}+c_{2}\,\frac{\mathrm{d}r^{2}}{r^{2}}+\frac{1}{2}\left(c_{0}r^{2}-c_{2}\right)\mathrm{d}\Omega^{2}\right],\quad
r+|t|<\frac{\tau}{2}\,,$ (77)

where

$\displaystyle c_{n}(r)\equiv\int_{-r}^{r}\mathrm{d}\xi\,\xi^{n}\
H(r^{2}-\xi^{2})$ (78)

and the function $H(z)$ is defined in (67).

###  4.4 Example: Gaussian form factor

As an explicit example of the procedure outlined in this section, here we
consider the case of the form factor (20), namely,
$f_{s}(-k^{2})=\exp(k^{2}/\mu_{s}^{2})$, which was studied in Frolov:Exp . The
first step is to obtain the inverse Laplace transform $h_{2}(s)$, defined in
(38), and then evaluate the function in (67). It is not difficult to show
that, in this case, $h_{2}(s)=-\theta(s-\mu_{2}^{-2})$ ($\theta(x)$ is the
Heaviside step function) and

$H(\zeta)=\ln(\zeta/\eta^{2})+\gamma_{\rm E}+E_{1}(\mu_{2}^{2}\zeta/4),$ (79)

where $\gamma_{\rm E}$ is the Euler–Mascheroni constant and $E_{1}(x)$ is the
exponential integral function.

Using (79), we get for (70)

$\displaystyle(R_{\mu\nu\alpha\beta}^{2})_{\rm
lin}=\frac{3G^{2}M^{2}\mu_{2}^{4}\zeta^{2}}{r^{6}}+O\left(\zeta^{3}\right),$
(80)

showing that as the collapse of the thin null shell proceeds, the Kretschmann
scalar diverges for $r\to 0$. On the other hand, if the shell has some
thickness, the curvature gets regularized. Indeed, the Eqs. (78) and (79)
imply the small-$r$ behaviour for the components of the metric (77),

$\displaystyle c_{0}=$
$\displaystyle\,\,{\mu_{2}^{2}\,r^{3}(420-21\mu_{2}^{2}r^{2}+\mu_{2}^{4}r^{4})\over
1260}+\ldots\,,$ (81a) $\displaystyle c_{2}=$
$\displaystyle\,\,{\mu_{2}^{2}\,r^{5}(756-27\mu_{2}^{2}r^{2}+\mu_{2}^{4}r^{4})\over
11340}+\ldots\,,$ (81b)

so that we get for the Kretschmann invariant in the $I$-domain,

$\displaystyle\lim_{r\to 0}\,(R_{\mu\nu\alpha\beta}^{2})_{\rm
lin}={32G^{2}M^{2}\mu^{4}_{2}\over 3\tau^{2}}.$ (82)

This shows that the curvature remains finite at $r=0$ for the thick shell
model.

Finally, let us discuss the mini black hole formation, which is related to the
invariant

$\displaystyle(\nabla r)^{2}={1\over
4\varsigma}\,g^{\mu\nu}\nabla_{\mu}\varsigma\nabla_{\nu}\varsigma,\qquad\varsigma=g_{\theta\theta}\,.$
(83)

A curve in the $tr$-plane such that $(\nabla r)^{2}=0$ is an apparent horizon.
Using (77) and (81) one finds that, to the leading order in $M$, this
invariant in the $I$-domain is given by

$\displaystyle(\nabla r)^{2}=1-{2M\mu^{2}_{2}r^{2}\over\tau}.$ (84)

Since in this domain $r<\tau$, we have

$\displaystyle(\nabla r)^{2}>1-{2M\mu^{2}_{2}\tau}\,.$ (85)

This relation means that, for a given $\mu_{2}$ and a fixed duration $\tau$ of
the pulse, the mini black hole does not form if the mass $M$ is small enough
Frolov:Exp .

##  5 Towards the non-linear regime

The presence of higher-derivative terms in the gravitational action makes the
analysis of the complete non-linear equations of motion a highly difficult
task, and non-locality poses an extra obstacle, because of the infinite-order
derivatives. In particular, differently from the linear limit [where any
higher-derivative structure could be reduced to the form of the action (5)],
the space of solutions of the complete field equations depends on the terms
that are actually in the action.

For example, if the Einstein–Hilbert action is enlarged only by the quadratic
structures $R{F}_{1}(\Box)R$ and/or $R_{\mu\nu}{F}_{2}(\Box)R^{\mu\nu}$, the
Ricci-flat solutions of GR are also vacuum solutions in the non-local model;
however, if the term quadratic in the Riemann tensor is included, then those
Ricci-flat spacetimes are not solutions of non-local gravity Li:2015bqa . The
stability of these type of solutions, imported from GR, was studied in
Myung:2017qtc ; Calcagni:2017sov ; Briscese:2019rii , and in Calcagni:2018pro
for theories constructed with form factors that are functions of the
Lichnerowicz operator; thermodynamic aspects of these black holes were also
discussed in Conroy:2015wfa ; Myung:2017axf ; Xiao:2021maa ; Xiao:2021ewv .

It is worth noticing that this process of constructing solutions from the
comparison with GR is not valid in the whole spacetime, namely, it might not
correctly reflect the interaction between gravity and the matter source. For
instance, while the static Schwarzschild solution in GR can be associated with
a Dirac delta source (see, e.g., Balasin:1993fn ), non-locality and higher
derivatives induce a smearing of the delta source at linear level BreTib2 ;
Jens ; Nos6der ; Modesto:2011kw ; Tseytlin95 ; Li:2015bqa ; hedao ;
Koshelev:2018hpt  (see also Sec. 3.3 above), and their effect might be able to
regularize the singularity.

Since exact static black hole solutions in non-local gravity with matter
sources are still an open problem, some insights were obtained from the
linearised limit (as discussed above, in this chapter) and also from a non-
linear approximation of the field equations, which we describe in what
follows. To this end, let us consider the models of the type (19), i.e., with
form factors $F_{2}(\Box)=-2F_{1}(\Box)$, so that
$f_{0}(-k^{2})=f_{2}(-k^{2})=e^{\gamma(\Box)}$. Because of this specific form
of the higher-derivative sector, the form factor gets factored together with
the Einstein tensor in the field equations Modesto:2011kw ; Zhang14 , namely

$\displaystyle e^{\gamma(\Box)}G^{\mu}{}_{\nu}+{O}({R}^{2}_{\ldots})=8\pi
G\,T^{\mu}{}_{\nu}.$ (86)

In the works Modesto:2011kw ; Modesto:2012ys ; Zhang14 ; Bambi:2013gva ;
Bambi:2016uda  the approximated version of this equation was considered, in
the form

$\displaystyle G^{\mu}{}_{\nu}=8\pi G\,\tilde{T}^{\mu}{}_{\nu},$ (87)

where

$\displaystyle\tilde{T}^{\mu}{}_{\nu}=e^{-\gamma(\Box)}{T}^{\mu}{}_{\nu}$ (88)

is an effective energy-momentum tensor.

The Eq. (87) can be regarded as an approximation of (86) by discarding the
terms ${O}({R}^{2}_{\ldots})$ of higher order in curvatures. Furthermore,
taking the d’Alembertian in (88) in its flat-spacetime form, for a point-like
source $T_{00}=m\delta^{(3)}(\vec{r})$, we get

$\displaystyle\tilde{T}_{00}=\rho_{\rm eff},\qquad{\rm where}\qquad\rho_{\rm
eff}(r)=\frac{m}{2\pi^{2}r}\int_{0}^{\infty}\mathrm{d}k\,\frac{k\sin(kr)}{e^{\gamma(-k^{2})}},$
(89)

just like the expression (33). Alternatively, the truncations in the original
equations (86) and (89) might be compensated by imposing the conservation of
the effective energy-momentum tensor, i.e.,
$\nabla_{\mu}\tilde{T}^{\mu}{}_{\nu}=0$ Modesto:2011kw . This leads to the
introduction of the effective radial ($p_{r}$) and tangential ($p_{\theta}$)
pressures in the energy-momentum tensor

$\displaystyle\tilde{T}^{\mu}{}_{\nu}=\text{diag}(-\rho_{\rm
eff},p_{r},p_{\theta},p_{\theta}).$ (90)

In this spirit, generalisations of the Schwarzschild solution were obtained by
using the metric Ansatz

$\displaystyle\mathrm{d}s^{2}=-A(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{A(r)}+r^{2}\mathrm{d}\Omega^{2}.$
(91)

Then, making the redefinition

$\displaystyle A(r)=1-\frac{2G\tilde{m}(r)}{r}$ (92)

and substituting the above expressions into (87), one obtains the equations

$\displaystyle\frac{\mathrm{d}\tilde{m}}{\mathrm{d}r}$ $\displaystyle=$
$\displaystyle 4\pi r^{2}\rho_{\rm eff}\,=\,-4\pi r^{2}p_{r}\,,$ (93)
$\displaystyle\frac{\mathrm{d}p_{r}}{\mathrm{d}r}$ $\displaystyle=$
$\displaystyle\frac{2}{r}\left(p_{\theta}-p_{r}\right)-\frac{\left(p_{r}+\rho_{\rm
eff}\right)}{2A}\frac{\mathrm{d}A}{\mathrm{d}r},$ (94)

which are solved by

$\displaystyle p_{r}=-\rho_{\rm eff},\qquad p_{\theta}=-\rho_{\rm
eff}-\frac{r\rho_{\rm eff}^{\prime}}{2}$ (95)

and

$\displaystyle\tilde{m}(r)=4\pi\int_{0}^{r}\mathrm{d}x\,x^{2}\rho_{\rm
eff}(x).$ (96)

Notice that $\tilde{m}(r)$ is the same mass function as the one defined in
(35).

For the Gaussian form factor (20) the solution is the same as the one obtained
in Modesto:2010uh ; Nicolini:2005vd , namely,

$\displaystyle\mathrm{d}s^{2}=-\left[1-\frac{4Gm}{r\sqrt{\pi}}\gamma\left(\tfrac{3}{2},\tfrac{\mu^{2}r^{2}}{4}\right)\right]\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{1-\frac{4Gm}{r\sqrt{\pi}}\gamma\left(\tfrac{3}{2},\tfrac{\mu^{2}r^{2}}{4}\right)}+r^{2}\mathrm{d}\Omega^{2},$
(97)

where $\gamma\left(a,x\right)$ is the lower incomplete gamma function, or in
terms of the error function,

$\displaystyle\gamma\left(\tfrac{3}{2},\tfrac{\mu^{2}r^{2}}{4}\right)=\frac{1}{2}\left[\sqrt{\pi}\,{\rm
erf}\left(\frac{\mu r}{2}\right)-\mu r\,e^{-\mu^{2}r^{2}/4}\right].$ (98)

Like the case discussed in Sec. 4.4, there exists a mass gap for this solution
to describe a black hole. In fact, if

$\displaystyle m\,<\,1.9\,\frac{M_{\text{P}}^{2}}{\mu},$ (99)

(we denote $1/G=M_{\text{P}}^{2}$) the invariant $(\nabla r)^{2}=A(r)$ is
always positive and the metric does not have any horizon Nicolini:2005vd . On
the other hand, for values of $m$ larger than this critical mass, the solution
displays two horizons. These features are shown in Fig. 4, where we plot the
graph of $A(r)$ for the two scenarios.

Figure 4: Plot of $A(r)$ for $\mu=M_{\rm P}=1$ in two situations. In blue
line, with $m=2.5$ (larger than the critical mass), the metric has two
horizons at $r_{-}=1.87$ and $r_{+}=4.67$. In red line, with $m=1.4$ (smaller
than the critical mass), the metric has no horizon.

In what concerns the regularity of the solution, it is straightforward to
verify that the metric (97) is regular, as well as the related curvature
invariants Nicolini:2005vd ; Modesto:2010uh . This is expected, for the metric
has a de Sitter core,

$\displaystyle A(r)=1+\frac{Gm\mu^{3}r^{2}}{3\sqrt{\pi}}+{O}(r^{4}).$ (100)

Moreover, since the components of (97) are even analytic functions of $r$, it
follows that all its local curvature-derivative invariants are finite
Giacchini:2021pmr ; Nos6der .

Similar solutions were obtained in the cases of more general exponential form
factors (21) of the type $f(-k^{2})=\exp(k^{2}/\mu)^{n}$, and it was shown
that for larger $n$ the solutions can have more than two horizons Zhang14 .
The possibility of having multi-horizon black holes also occurs in the case of
the Kuz’min–Tomboulis form factor (22), as numerically shown in Modesto:2011kw
; Zhang14 .

##  6 Concluding remarks

In this chapter, we reviewed linearised metric solutions in both stationary
and dynamical scenarios. In the former case, we analysed static and rotating
spacetimes and showed that non-locality can regularize point-like and ring-
like singularities. In the latter scenario, we showed that regularized mini
black hole solutions can be found and that the formation of an apparent
horizon can be avoided as long as the mass of the object is smaller than some
critical value set by the scale of non-locality. Furthermore, we discussed an
attempt towards finding full non-linear solutions and showed that regular
solutions can be obtained by working with some simplified field equations.

Understanding the physics of non-locality at the full non-linear level remains
one of the most outstanding open questions in the context of ghost-free non-
local theories of gravity. In fact, it is still not entirely clear whether
singularities can be really avoided in the non-linear regime or whether a
horizon can form. Indeed, it has also been argued that non-locality could
prevent the formation of a horizon, for any value of the mass, in such a way
that black holes could be replaced by ultra-compact horizonless objects in
some non-local gravity models Koshelev:2017bxd ; Buoninfante:2019swn ,
although a rigorous proof of this statement is still lacking.

Providing definite answers to these questions is challenging but at the same
time very important and stimulating. Future investigations and new ideas to
solve infinite-derivative non-linear differential equations are surely needed.
In fact, finding a full non-linear black hole-like solution can place non-
local gravity on firmer ground and lay the foundation for future
phenomenological studies in astrophysics, e.g., in the context of binary
mergers and gravitational waves, and thus offer a new scenario where non-local
extensions of Einstein’s GR can be tested and constrained.

###### Acknowledgements.

Nordita is supported in part by NordForsk.

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