tcb@breakable

#  LYCEN 2023-02
Living on the Edge:
Quantum Black Hole Physics from the Event Horizon

Manuel Del Piano<EMAIL_ADDRESS>Scuola Superiore Meridionale,
Largo S. Marcellino, 10, 80138 Napoli NA, Italy INFN sezione di Napoli, via
Cintia, 80126 Napoli, Italy  Quantum Theory Center QTC & D-IAS, Southern
Denmark Univ., Campusvej 55, 5230 Odense M, Denmark  0000-0003-4515-8787
Stefan Hohenegger<EMAIL_ADDRESS>Institut de Physique des 2
Infinis (IP2I), CNRS/IN2P3, UMR5822, 69622 Villeurbanne, France Université de
Lyon, Université Claude Bernard Lyon 1, 69001 Lyon, France
0000-0001-6564-0795 Francesco Sannino<EMAIL_ADDRESS>Scuola Superiore
Meridionale, Largo S. Marcellino, 10, 80138 Napoli NA, Italy Dept. of Physics
E. Pancini, Università di Napoli Federico II, via Cintia, 80126 Napoli, Italy
INFN sezione di Napoli, via Cintia, 80126 Napoli, Italy  Quantum Theory Center
QTC & D-IAS, Southern Denmark Univ., Campusvej 55, 5230 Odense M, Denmark
0000-0003-2361-5326

###### Abstract

Quantum gravity theories predict deformations of black hole solutions relative
to their classical counterparts. A model-independent approach was advocated in
[1] that uses metric deformations parametrised in terms of physical
quantities, such as the proper distance. While such a description manifestly
preserves the invariance of the space-time under coordinate transformations,
concrete computations are hard to tackle since the distance is defined in
terms of the deformed metric itself. In this work, for spherically symmetric
and static metrics, we provide a self-consistent framework allowing us to
compute the distance function in close vicinity to the event horizon of a
black hole. By assuming a minimal degree of regularity at the horizon, we
provide explicit (series) expansions of the metric. This allows us to compute
important thermodynamical quantities of the black hole, such as the Hawking
temperature and entropy, for which we provide model-independent expressions,
beyond a large mass expansion. Moreover, imposing for example the absence of
curvature singularities at the event horizon leads to non-trivial consistency
conditions for the metric deformations themselves, which we find to be
violated by some models in the literature.

###### Contents

  1. 1 Introduction
  2. 2 Regular geometry close to the black hole horizon
    1. 2.1 Near horizon constraints
    2. 2.2 Impact on the thermodynamics
  3. 3 Series expansions of the distance function
    1. 3.1 Solving the distance function
      1. 3.1.1 Series expansion of the radial coordinate
      2. 3.1.2 Series expansion of the distance and metric function
    2. 3.2 Metric functions and curvature
      1. 3.2.1 Metric functions
      2. 3.2.2 Ricci scalar and Hawking temperature
  4. 4 Examples
    1. 4.1 Example 1: Bonanno-Reuter asymptotically safe black hole
    2. 4.2 Example 2: Minimal metric deformation
      1. 4.2.1 Curvature and temperature
      2. 4.2.2 Extending the metric inside the horizon
  5. 5 Large distance expansion
    1. 5.1 Convergence criteria and derivatives
    2. 5.2 Horizon constraints
    3. 5.3 Thermodynamics
  6. 6 Conclusions
  7. A Regular higher order derivatives
    1. A.1 Distance function
    2. A.2 Conditions for the regularity of higher derivatives $f^{(n)}$
  8. B Towards a fully regular solution
  9. C Series relations
    1. C.1 Series expansion of inverse radial coordinate
    2. C.2 Power of a power series
    3. C.3 Series expansions for $\xi_{1}\neq 0$
    4. C.4 Examples
      1. C.4.1 Schwarzschild distance
      2. C.4.2 Bonanno-Reuter black hole
      3. C.4.3 Minimal example
  10. D Further examples
    1. D.1 Hayward black hole
    2. D.2 Dymnikova black hole
  11. E Conditions for regularity at inner horizons

##  1 Introduction

Black holes are solutions of General Relativity (GR) with an event horizon,
which potentially conceals a central singularity. While quantum corrections
are expected to modify these solutions, our lack of a comprehensive theory of
quantum gravity makes it challenging to provide precise details about these
corrections. Over the years, numerous proposals for black hole deformations
have emerged, drawing from both fundamental theories of gravity and effective
approaches [2, 5, 6, 7, 8, 9, 10, 11, 12, 1, 3, 4]. Notably, certain previous
works [5, 1] have put forward deformations of the Schwarzschild space-time
geometry, incorporating quantum corrections in a manner that allows for the
formulation of universal statements.

Concretely, spherically symmetric and static space-time metrics are
characterized by two functions $f$, $h$ of a radial coordinate [13, 14, 15,
16]. We shall measure the latter in units of the Planck length $\ell_{\rm P}$
and denote it $z$. In general, assuming that $h\neq f$ allows for the
description of a wide range of classical scenarios [17]. This includes the
Tolman-Oppenheimer-Volkov space-time for compact objects and stellar
environments [18, 19, 20, 21]. A horizon in these geometries corresponds to a
zero of the functions $f$ and $h$, which, for simplicity, in this work, we
shall consider to be a simple zero. Asymptotically flat geometries are
characterized by the fact that both $f$ and $h$ tend to $1$ for large values
of $z$.

Quantum corrections to the geometries mentioned above can be incorporated as
deformations of the classical functions $f$ and $h$ and the precise form
evidently depends on a concrete model of quantum gravity. Nevertheless,
inspired by the _renormalization group framework_ [22, 23, 24] it has been
argued in [1, 5, 25] that universal, model-independent statements about
physical quantities in spherically symmetric and static quantum black holes
can be made by demanding independence with respect to spurious scales.
Notably, in order to preserve the invariance of the geometry under coordinate
transformations (similar to those in GR), it has been advocated to write the
deformation functions in terms of a physical quantity. While the concrete
choice of the latter is to some degree ambiguous,111From the perspective of
the renormalization group approach a different choice of this physical
quantity (see _e.g._ [26, 27, 28, 29, 3]) corresponds to a different scheme.
We shall elaborate on this connection in future work [30]. a natural choice in
the context of static, spherically symmetric geometries is the proper distance
from the center of the black hole. However, since the proper distance is
defined in terms of the (deformed) metric function $f$, this prescription
leads to an implicit definition of the quantum geometry. This issue has been
addressed in previous works through different approximations: either by
replacing the proper distance with a simpler function of $z$ (see [5]) or by
assuming a very heavy black hole [1]. In the former case, the approximated
distance generally no longer represents a physical quantity and therefore
constitutes a conceptual departure from the above-mentioned logic. In the
latter case, quantum corrections to physical quantities are suppressed by
inverse powers of the mass.

In this work, we provide a framework that allows us to compute the proper
distance near the event horizon in a self-consistent fashion without the need
for approximations. The framework assumes a certain degree of regularity of
either the metric functions or the proper distance, such that they afford
series expansions, at least up to some order. Furthermore, apart from the
proper distance of the horizon from the center of the black hole $d_{H}$, the
framework only requires information about the black hole exterior, which is
encoded in the deformations of the metric functions. Concretely, within this
setup, we find explicit solutions of the non-linear first-order differential
equation that defines the proper distance in terms of the metric functions.
These solutions completely determine the space-time geometry near the event
horizon, which in turn allows us to compute the thermodynamical properties of
the black hole, namely its Hawking temperature and entropy. Further assuming a
dependence of $d_{H}$ on the mass of the black hole, we calculate mass
expansions of the Hawking temperature, correcting previous results in [1].

Furthermore, using this framework we find non-trivial conditions of the
quantum-deformed black hole geometries. On the one hand, regularity of the
first derivative of the metric functions is required to render the surface
gravity well-defined, which in turn is required for the Hawking temperature to
be well-defined [31]. However, this is not automatic but requires conditions
on the metric deformations. On the other hand, the absence of curvature
singularities at the event horizon also imposes non-trivial constraints: these
can either be found by demanding finiteness of the second derivatives of the
metric functions (which provides sufficient conditions) or by calculating
series expansions of the Ricci and Kretschmann scalar close to (but outside
of) the exterior event horizon. These generically exhibit divergent
contributions which can only be removed if certain conditions for the metric
deformations are met. We formulate these various conditions in the form of
constraints on the original input parameters of the black hole geometry as
mentioned before. Checking these conditions for certain examples in the
literature, we find that they are not always respected. Indeed, the quantum
black hole model proposed in [5] (which we shall refer to as the Bonanno-
Reuter space-time) is based on a deformation of the metric functions that
violate the above-mentioned conditions. This means, treating the Bonanno-
Reuter space-time in a self-consistent fashion leads to an ill-defined Hawking
temperature as well as a divergence of the Ricci scalar at the event horizon.
While approximations to this space-time that were proposed in [5] do not
suffer from such unphysical singularities, they should be interpreted as
metric deformations different from the original ones, which comply with our
consistency conditions. To further show the flexibility of our approach, we
discuss as a different model a minimal solution to our conditions along with
its physical properties.

Finally, in order to make closer contact with our previous work [1] we
consider asymptotic expansions of the metric deformations in inverse powers of
the proper distance. Assuming that the radius of convergence of these
expansions is large enough to be still valid at the event horizon, we use them
as input to the framework explained above. Concretely, we formulate the
consistency conditions for the absence of an unphysical singularity in terms
of the asymptotic expansion coefficients. Solving these conditions, we
furthermore provide self-consistent expressions for the Hawking temperature
and (upon assuming a mass-dependence of $d_{H}$) the entropy.

While our results are derived with a black hole geometry in mind, they can be
generalized in a straightforward manner to spherically symmetric and static
space-times. Therefore, we foresee the further impact of the framework
presented here in the description of quantum effects in gravitational
experiments and cosmology. Furthermore, while in this work we mainly have
deformations of black holes in mind that are due to quantum effects, our
approach is versatile enough to also describe other types of deformations. We,
therefore, expect our work to be useful for studying space-time corrections in
theories of modified gravity.

This paper is organised as follows: in Section 2 we introduce our notation for
deformations of the spherically symmetric and static Schwarzschild geometry.
We derive non-trivial conditions for these deformations by imposing finiteness
of the first and second derivative of the metric functions at the horizon: the
first derivative is a necessary condition for the existence of the surface
gravity and the second provides sufficient conditions for the absence of a
singularity of the Ricci scalar at the horizon. In Section 3, we develop a
more general framework for computing the distance function for deformations of
the Schwarzschild metric. Assuming the existence of a series expansion of the
proper distance, we provide a recursive relation for all expansion
coefficients. The consistency of this approach and the finiteness of the Ricci
scalar at the horizon impose non-trivial conditions on the metric
deformations. In Section 4 we apply these conditions to concrete examples of
black hole solutions. We show that the Bonanno-Reuter space-time does not
abide by the conditions and indeed exhibits a singular behavior at the
horizon. We also provide a novel model based on a minimalistic solution of the
conditions derived previously. In Section 5 we consider asymptotic expansions
of the metric deformations and (assuming that they can be extended all the way
to the black hole horizon) show how to integrate them into the framework
developed in previous Sections. Finally, Section 6 contains our conclusions
and an outlook for further applications. This work is complemented by 5
appendices: Appendix A generalizes the approach of Section 2 by deriving
conditions imposed by assuming that an arbitrarily high order $N$ of
derivatives of the metric function is finite at the horizon. Appendix B
provides a minimal solution for the system of equations established in
Appendix A and shows that this solution for $N\to\infty$ tends to the
Schwarzschild space-time. Appendix C contains several derivations of series
identities that have been deemed too technical for the main body of the paper.
Appendix D discusses further examples of deformed black hole metrics, namely
the Hayward black hole and the Dymnikova space-time. Finally Appendix E gives
a brief outline of how to generalize the conditions derived in Section 2 to
interior black hole horizons.

##  2 Regular geometry close to the black hole horizon

Our starting point is the general form of a spherically symmetric and static
space-time in four dimensions with Lorentzian signature

$\differential{s}^{2}=g_{\mu\nu}\differential x^{\mu}\differential
x^{\nu}=-h(r)\differential{t}^{2}+\frac{\differential{r}^{2}}{f(r)}+r^{2}\differential{\theta}^{2}+r^{2}\sin^{2}\theta\differential{\phi}^{2}\
,$ (1)

where the metric is given by

$g_{\mu\nu}=\mathrm{diag}(-h(r),f(r)^{-1},r^{2},r^{2}\sin^{2}{\theta})\ ,$ (2)

with $h$ and $f$, _a priori_ , general functions of the radial coordinate $r$.
The classical Schwarzschild space-time [32], which is a solution of the
Einstein equations in vacuum, is recovered for
$h(r)=f(r)=(1-r_{\mathrm{S}}/r)$, with $r_{\mathrm{S}}=2G_{\mathrm{N}}M$ the
Schwarzschild radius (and $G_{\mathrm{N}}$ Newton’s constant). For $r\geq
r_{\mathrm{S}}$, this metric describes the space-time outside of a central
body of mass $M$. In the following, we shall be interested in deformations of
this metric, which specifically represent black holes and which are
characterised by (particular) modifications of the metric functions $f$ and
$h$.

To describe these modifications, we first simplify the notation by casting (1)
into a dimensionless form: similar to [1], we write the radial coordinate (and
the mass parameter $M$) in units of the Planck length $\ell_{\rm
P}=1/M_{\mathrm{P}}$ (with $M_{\mathrm{P}}$ the Planck mass), by defining:

$z:=M_{\mathrm{P}}r=\frac{r}{\ell_{\mathrm{P}}}\mbox{\quad
and\quad}\chi:=\frac{M}{M_{\mathrm{P}}}\ .$ (3)

We shall further choose units such that Newton’s constant is equal to $1$,
i.e. $G_{\mathrm{N}}\,M_{\mathrm{P}}^{2}=1$. In this notation, we shall
parametrise deformations of the Schwarzschild geometry by writing the
functions $f$ and $h$ as

$f(z)=1-\frac{2\chi}{z}e^{\widetilde{\Phi}\left(z\right)}\,,\quad
h(z)=1-\frac{2\chi}{z}e^{\widetilde{\Psi}\left(z\right)}\,,$ (4)

where $\widetilde{\Phi}$ and $\widetilde{\Psi}$ encode corrections due to
physical effects beyond GR, either classical or quantum in nature. In order to
describe a black hole, we first require that the geometry is still
asymptotically flat. Concretely, we assume that the geometry approaches the
Schwarzschild metric (with mass parameter $\chi$) for very large distances
from the origin

$\lim_{z\to\infty}\widetilde{\Phi}(z)=0=\lim_{z\to\infty}\widetilde{\Psi}(z)\,.$
(5)

Furthermore, in order for coordinate transformations of the undeformed space-
time to be also realised in the deformed case, we demand that
$\widetilde{\Psi}$ and $\widetilde{\Phi}$ are invariant quantities. This can
be achieved by writing them as functions of a physical quantity, for which in
[5, 1] the proper distance from the origin was proposed. The proper (radial)
distance between two spatial points within the space-time (1) is:

$d(z,z_{0}):=\int_{z_{0}}^{z}\frac{\differential{\tilde{z}}}{\sqrt{\absolutevalue{f(\tilde{z})}}}\,\quad{\rm
and}\quad d(z)=d(z,0)\ .$ (6)

We thus replace in (8)

$\widetilde{\Phi}(z),\,\widetilde{\Psi}(z)\quad\longrightarrow\quad\Phi(1/d(z)),\,\Psi(1/d(z))\
,$ (7)

where $\Phi$ and $\Psi$ are functions of the inverse distance such that

$f(z)=1-\frac{2\chi}{z}e^{\Phi\left(1/d(z)\right)}\,,\quad
h(z)=1-\frac{2\chi}{z}e^{\Psi\left(1/d(z)\right)}\,,$ (8)

Here we choose a dependence on $1/d$, such that asymptotical flatness (5)
amounts to the simple relation

$\displaystyle\Phi(0)=0=\Psi(0)\,.$ (9)

Furthermore, in order to describe a black hole geometry, we require that the
metric (1) has (at least one) horizon, _i.e._ we impose that the functions $f$
and $h$ have a zero at a coordinate $z_{H}\in(0,\infty)$

$\displaystyle f_{H}:=f(z_{H})=0=h(z_{H})=:h_{H}\,.$ (10)

In this paper, we shall assume that (10) are simple zeroes and that

$\displaystyle f(z)>0\,,\mbox{\quad and\quad}h(z)>0\quad\forall z>z_{H}\,,$
(11)

_i.e._ that $z_{H}$ is the location of the outer horizon of the black hole.
Furthermore, we introduce the notation

$\displaystyle d_{H}:=d(z_{H})\,,\mbox{\quad and\quad}d(z)=d_{H}+\rho(z)\,,$
(12)

where $\rho(z)=d(z,z_{H})$ has the interpretation as the proper distance
measured from the horizon of the black hole. With this notation, the condition
(10) can equally be written in the form

$\Phi\left(\frac{1}{d_{H}}\right)=\Psi\left(\frac{1}{d_{H}}\right)=\log\frac{z_{H}}{2\chi}\
.$ (13)

This ensures the vanishing of the norm of time-like Killing vector
$(K^{t})^{\mu}=\delta^{\mu 0}$ required for the existence of an event horizon

$(K^{t})_{\mu}(K^{t})^{\mu}|_{z=z_{H}}=g_{00}(K^{t})^{0}(K^{t})^{0}|_{z=z_{H}}=-h(z_{H})=0\,.$
(14)

We remark that in the remainder of this paper, unless otherwise specified, we
shall consider the space-time outside of the horizon of the black hole, _i.e._
we shall only consider the region $z\geq z_{H}$ (or equivalently $\rho\geq
0$). Furthermore, we consider $d_{H}$ as an additional input into the
(exterior) black hole geometry, which is in fact the only information about
the interior of the black hole that is required in the following.

The surface gravity and the Ricci scalar are fundamental quantities that must
be well-defined at and near the horizon. We shall discuss the surface gravity
in more detail in Section 2.2, while here we give the definition of the Ricci
scalar as a geometric quantity, which directly follows from (1): it is a
scalar quantity which appears in the equation of motion of the gravitational
field (_i.e._ the Einstein equations). Therefore, its regularity (notably at
the horizon) in the deformed metric ensures that in the deformed case no
additional singularities arise beyond the classical ones, which shall be a
central point in the analysis of this paper. The Ricci scalar in terms of $f$
and $h$ is222To save writing factors of $\ell_{\mathrm{P}}$, we have defined a
dimensionless form of $R$ (and $K$), _i.e._ it is measured in units of
$\ell_{\mathrm{P}}$. Furthermore, from here on out we use the notation
$F^{(n)}$ to indicate the $n$-th derivative of a function $F$ with respect to
its argument. For example, we define  $\displaystyle
f^{(n)}(z):=\derivative[n]{f(z)}{z}\,,$ and
$\displaystyle\Phi^{(n)}\left(\frac{1}{x}\right):=\left.\derivative[n]{\Phi(y)}{y}\right|_{y=\frac{1}{x}}\
.$ (15) The subscript H denotes the evaluation of a quantity at the horizon,
which corresponds to taking $z=z_{H}$, $d=d_{H}$, or $\rho=0$, _e.g._
$\displaystyle f^{(n)}_{H}:=f^{(n)}(z=z_{H})\,,$ and
$\displaystyle\Phi^{(n)}_{H}:=\Phi^{(n)}(1/d_{H})\,.$ (16)

$R=-\frac{\left(zf^{(1)}+4f\right)h^{(1)}}{2zh}+\frac{f\,h^{(2)}}{h}-\frac{2\big{(}zf^{(1)}+f-1\big{)}}{z^{2}}+\frac{f(h^{(1)})^{2}}{2h^{2}}\,.$
(17)

We mainly focus on the finiteness of the Ricci scalar at the horizon but, in
some examples, we shall also examine the behavior of the Kretschmann scalar,
given by

$K=\tensor{R}{{}_{\rho}{}_{\sigma}{}_{\mu}{}_{\nu}}\tensor{R}{{}^{\rho}{}^{\sigma}{}^{\mu}{}^{\nu}}\
,$ (18)

where
$\tensor{R}{{}_{\rho}{}_{\sigma}{}_{\mu}{}_{\nu}}=\tensor{g}{{}_{\rho}{}_{\lambda}}\tensor{R}{{}^{\lambda}_{\sigma}{}_{\mu}{}_{\nu}}$
is the fully covariant Riemann tensor.

###  2.1 Near horizon constraints

By examining the expression of the Ricci scalar (17) and the Kretschmann
scalar (18), it is evident that derivatives of the functions $f$ and $h$ up to
the second order (in $(z-z_{H})$) are required. Hence, we start by assuming
that in the vicinity of the black hole horizon, located at $z_{H}$, they can
be expanded using a Taylor series up to the second order, given by

$\displaystyle f(z)$
$\displaystyle=f^{(1)}_{H}(z-z_{H})+\frac{f^{(2)}_{H}}{2}(z-z_{H})^{2}+\mathfrak{o}((z-z_{H})^{2})\,,$
(19) $\displaystyle h(z)$
$\displaystyle=h^{(1)}_{H}(z-z_{H})+\frac{h^{(2)}_{H}}{2}(z-z_{H})^{2}+\mathfrak{o}((z-z_{H})^{2})\,.$
(20)

We remark that similar expansions have been already considered in the
literature, for example in [33, 34]. In the following, we shall assume
$f^{(1)}_{H}>0$ and $h^{(1)}_{H}>0$ such that $z_{H}$ is a simple zero of both
$f$ and $h$, as well as the position of the outer event horizon of the black
hole space-time (_i.e._ $f$ and $h$ have no further zeroes for $z>z_{H}$ and
both change signs at $z=z_{H}$).

The expansions (19) and (20) also afford the following form of the
infinitesimal proper distance (6) from the horizon

$\differential{\rho}=\frac{\differential{z}}{\sqrt{f^{(1)}_{H}(z-z_{H})+\frac{f^{(2)}_{H}}{2}(z-z_{H})^{2}}}+\mathfrak{o}((z-z_{H})^{2})\,,$
(21)

which yields explicitly

$\rho(z)=\frac{2\sqrt{z-z_{H}}}{\sqrt{f^{(1)}_{H}}}-\frac{f^{(2)}_{H}(z-z_{H})^{3/2}}{6(f^{(1)}_{H})^{3/2}}+\mathfrak{o}((z-z_{H})^{3/2})\
.$ (22)

The latter can be locally inverted so that we can write $z(\rho)$, which up to
the fourth order reads:

$z(\rho)=z_{H}+\frac{f^{(1)}_{H}}{4}\rho^{2}+\frac{f^{(1)}_{H}f^{(2)}_{H}}{96}\rho^{4}+\mathfrak{o}(\rho^{4})\,.$
(23)

This allows us to rewrite the Taylor expansions for $f$ and $h$ in terms of
the physical, and therefore coordinate-invariant, distance $\rho$ as
follows333By abuse of notation we shall write in the following $f(\rho)$
instead of $f(z(\rho))$.

$\displaystyle f(\rho)$
$\displaystyle=\frac{(f^{(1)}_{H})^{2}}{4}\rho^{2}+\frac{(f^{(1)}_{H})^{2}f^{(2)}_{H}}{24}\rho^{4}+\mathfrak{o}(\rho^{4})\
,$ (24) $\displaystyle h(\rho)$
$\displaystyle=\frac{f^{(1)}_{H}h^{(1)}_{H}}{4}\rho^{2}+\frac{f^{(1)}_{H}(3f^{(1)}_{H}h^{(2)}_{H}+f^{(2)}_{H}h^{(1)}_{H})}{96}\rho^{4}+\mathfrak{o}(\rho^{4})\
.$ (25)

The derivatives of $f$ with respect to $z$ are consequently computed as (see
(15) for the notation)

$f^{(1)}(z)=\derivative{f}{z}=\derivative{f(\rho(z))}{\rho}\derivative{\rho}{z}=\frac{1}{\sqrt{f(\rho)}}\derivative{f(\rho)}{\rho}\,,$
(26)

and similarly for $h^{(1)}$. In terms of the functions $\Phi$ and $\Psi$ in
(8) we therefore find

$\displaystyle
f^{(1)}(z)=(1-f)\left(\frac{1}{z}+\frac{\Phi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{2}\sqrt{f}}\right)\
,\mbox{\quad
and\quad}h^{(1)}(z)=(1-h)\left(\frac{1}{z}+\frac{\Psi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{2}\sqrt{f}}\right)\
.$ (27)

It is clear from the above that the first derivative shows a divergence at the
horizon due to the second term in the parenthesis $\propto f^{-1/2}$. Using
(23), (24), and (25) we can provide a series expansion around $\rho=0$,
concretely for $f^{(1)}$:

$\displaystyle f^{(1)}$
$\displaystyle=\frac{2\Phi^{(1)}_{H}}{d_{H}^{2}f^{(1)}_{H}}\frac{1}{\rho}+\frac{1}{z_{H}}-\frac{2\left(2d_{H}\Phi^{(1)}_{H}+\Phi^{(2)}_{H}\right)}{d_{H}^{4}f^{(1)}_{H}}+\mathfrak{o}(\rho^{0})\,,$
(28)

and similarly for $h^{(1)}$. Therefore, we conclude that consistency with the
first order in (19) and (20) requires

$\Phi^{(1)}_{H}=0\,,\,\quad{\rm and}\quad\Psi^{(1)}_{H}=0\,,$ (29)

which removes the singularity in (27). With these conditions, we get the
expressions of the first derivatives of the metric functions at the event
horizon444We remark that $f^{(1)}_{H}=\frac{1-\varpi}{2z_{H}}$ is also a
solution compatible with (28). In the following, we shall use the result (30),
which leads to the classical Schwarzschild metric for $\Phi^{(2)}_{H}=0$.

$\displaystyle f^{(1)}_{H}=\frac{1+\varpi}{2z_{H}}\mbox{\quad
and\quad}h^{(1)}_{H}=\frac{1}{z_{H}}-\frac{2\Psi^{(2)}_{H}}{d_{H}^{4}f^{(1)}_{H}}\,,$
(30)

where for later use we have introduced the shorthand notation

$\displaystyle\varpi=\sqrt{1-\frac{8z_{H}^{2}\Phi^{(2)}_{H}}{d_{H}^{4}}}\,.$
(31)

Reality of $f^{(1)}_{H}$ requires that $\varpi\in\mathbb{R}$. Furthermore, in
order for $z_{H}$ to be the position of (a simple) outer event horizon, both
$f^{(1)}_{H}>0$ and $h^{(1)}_{H}>0$. These conditions together impose upper
bounds for the derivatives $\Phi^{(2)}_{H}$ and $\Psi^{(2)}_{H}$ :

$\displaystyle\Phi^{(2)}_{H}\leq\frac{d_{H}^{4}}{8z_{H}^{2}}\,,\mbox{\quad
and\quad}\Psi^{(2)}_{H}<\frac{d_{H}^{4}(1+\varpi)}{4z_{H}^{2}}\,.$ (32)

Moving to the second order of the Taylor polynomials in (19) and (20), we find
for the second derivatives

$\displaystyle f^{(2)}$
$\displaystyle=-f^{(1)}\left(\frac{1}{z}+\frac{\Phi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{2}\sqrt{f}}\right)+$
$\displaystyle-(1-f)\left(\frac{1}{z^{2}}+\frac{2\Phi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{3}f}+\frac{f^{(1)}\Phi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{2(d_{H}+\rho)^{2}f^{3/2}}+\frac{\Phi^{(2)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{4}f^{3/2}}\right),$
$\displaystyle h^{(2)}$
$\displaystyle=-h^{(1)}\left(\frac{1}{z}+\frac{\Psi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{2}\sqrt{f}}\right)+$
$\displaystyle-(1-h)\left(\frac{1}{z^{2}}+\frac{2\Psi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{3}f}+\frac{f^{(1)}\Psi^{(1)}\left(\frac{1}{d_{H}+\rho}\right)}{2(d_{H}+\rho)^{2}f^{3/2}}+\frac{\Psi^{(2)}\left(\frac{1}{d_{H}+\rho}\right)}{(d_{H}+\rho)^{4}f^{3/2}}\right),$
(33)

which we can equally expand around $\rho=0$, similar to eq. (27). By utilising
(29) and the expressions for $f^{(1)}_{H}$ and $h^{(1)}_{H}$ given in equation
(30), we can eliminate the divergent terms by imposing that

$\Phi^{(3)}_{H}=-6d_{H}\Phi^{(2)}_{H}\mbox{\quad
and\quad}\Psi^{(3)}_{H}=-6d_{H}\Psi^{(2)}_{H}\ .$ (34)

As explained in appendix A.2, demanding the finiteness of even higher order
derivatives at the horizon (_i.e._ beyond second order), one can iterate the
above procedure of removing divergent terms in the Taylor expansion, leading
to the general result (174).

Summarising the result for imposing finiteness of the first and second
derivatives of $f$ and $h$ (as in (19) and (20)), as well as the reality and
positiveness of the first derivatives, we have obtained the following general
constraints and upper bounds:

$\displaystyle\Phi_{H}=\Psi_{H}=\log\frac{z_{H}}{2\chi}\
,\quad\Phi^{(1)}_{H}=\Psi^{(1)}_{H}=0\
,\quad\Phi^{(3)}_{H}=-6d_{H}\Phi^{(2)}_{H}\
,\quad\Psi^{(3)}_{H}=-6d_{H}\Psi^{(2)}_{H}\ ,$ $\displaystyle\hskip
113.81102pt\Phi^{(2)}_{H}\leq\frac{d_{H}^{4}}{8z_{H}^{2}}\,,\hskip
42.67912pt\Psi^{(2)}_{H}<\frac{d_{H}^{4}(1+\varpi)}{4z_{H}^{2}}\,.$ (36)

By imposing these conditions, we ensure that there are no curvature
singularities at the horizon: indeed, with (17) (and (18)) it can be verified
that the Ricci and Kretschmann scalars attain finite values, which depend on
$\Phi^{(2)}_{H}$, $\Phi^{(4)}_{H}$, $\Psi^{(2)}_{H}$, and
$\Psi^{(4)}_{H}$.555Due to their complexity, we refrain from presenting the
complete expressions of $R$ and $K$.

However, to provide a more comprehensive and generalized framework, we delve
into a broader approach that encompasses these conditions in Section 3.2.

###  2.2 Impact on the thermodynamics

One of the main applications of the general constraints discussed above is
black hole thermodynamics. To determine the temperature of a black hole we
introduce the _surface gravity_ which is expressed via the time-like Killing
vector $(K^{t})^{\mu}$ introduced earlier (see eq. (14)) and reads [34, 35,
36, 37, 38]

$\kappa^{2}=\left.-\frac{\ell_{\mathrm{P}}^{2}}{2}\nabla_{\mu}(K^{t})_{\nu}\nabla^{\mu}(K^{t})^{\nu}\right|_{z=z_{H}}=\left.\frac{1}{4}\frac{f\cdot(h^{(1)})^{2}}{h}\right|_{z=z_{H}}\,.$
(38)

Using the constraints in (36) the Hawking temperature [37, 39] is then given
by666In order to avoid factors of $\ell_{\mathrm{P}}$ in the following, we
have defined dimensionless versions of both the surface gravity (38) and the
Hawking temperature (39), which are measured in units of $\ell_{\mathrm{P}}$.

$\displaystyle
T_{\mathrm{H}}=\frac{\kappa}{2\pi}=\frac{1}{4\pi}\sqrt{f^{(1)}_{H}h^{(1)}_{H}}=\frac{1}{4\pi}\sqrt{\frac{1+\varpi}{2z_{H}^{2}}-\frac{2\Psi^{(2)}_{H}}{d_{H}^{4}}}\,,$
with
$\displaystyle\varpi=\sqrt{1-\frac{8z_{H}^{2}\Phi^{(2)}_{H}}{d_{H}^{4}}}\,.$
(39)

Notice that due to the upper bound on $\Psi_{H}^{(2)}$, the Hawking
temperature is strictly positive $T_{\text{H}}>0$ for $d_{H}>0$ (and thus
$z_{H}>0$). Indeed, $T_{\text{H}}=0$ would require $f^{(1)}_{H}=0$ and/or
$h_{H}^{(1)}$. With (19) and (20), this would, however, imply that $f$ and/or
$h$ would have a double zero at the horizon, which is thus not compatible with
our initial assumption in this approach.

For the expression of the entropy, we use the first law of thermodynamics

$\differential\chi=T_{\mathrm{H}}\differential S\,.$ (40)

such that the entropy can be written as

$S=\int\frac{\differential\chi}{T_{\mathrm{H}}(\chi)}\,.$ (41)

To perform the integration over the black hole mass $\chi$, one has to provide
the explicit dependence of $d_{H}$, $z_{H}$, $\Psi^{(2)}_{H}$ and
$\Phi^{(2)}_{H}$ on $\chi$ itself.

##  3 Series expansions of the distance function

In the previous Section, we have provided the conditions (36) for the
functions $\Phi,\Psi$ appearing in eq. (8) that guarantee finite first and
second derivatives of the metric functions at the event horizon. These in turn
are sufficient conditions that also the Ricci and Kretschmann scalar are
finite. In this Section, we shall recover more general (but compatible)
results by using a different approach, namely by solving (6) assuming that a
solution in the form of a series expansion exists.

###  3.1 Solving the distance function

####  3.1.1 Series expansion of the radial coordinate

In eq. (22) we have already given a limited series expansion of the distance
to the BH horizon $\rho$ as a function of $z-z_{H}$, the inversion of which is
given in (23). In this Section, we shall provide a general form of these
expansions under the assumption that $z$ can be written as a (convergent)
series in $\rho$ for $\rho\geq 0$. Indeed, we shall start from a general
(integer) series of the form777Other than in the previous Section, we
presently do not make any assumptions on (derivatives of) $f$ or $h$.

$\displaystyle z(\rho)=z_{H}+\sum_{n=1}^{\infty}a_{n}\,\rho^{n}\,$ with
$\displaystyle a_{n}\in\mathbb{R}\hskip 14.22636pt\forall n\in\mathbb{N}\,,$
(42)

which we assume to have an interval of convergence $\rho\in[0,\rho_{A})$ for
some $\rho_{A}>0$. We shall determine the series coefficients $a_{n}$
recursively by solving the differential equation

$\displaystyle\derivative{\rho}{z}=\left(1-\frac{2\chi}{z}\,e^{\Phi\left(\frac{1}{d_{H}+\rho}\right)}\right)^{-1/2}\,,$
$\displaystyle\forall z\geq z_{H}\,.$ (43)

However, we shall consider $z$ as a function of $\rho$, _i.e._ we consider
(43) in the form

$\displaystyle
z\left(1-\left(\derivative{z}{\rho}\right)^{2}\right)=2\chi\,e^{\Phi\left(\frac{1}{d_{H}+\rho}\right)}\,,$
$\displaystyle\forall\rho\geq 0\,.$ (44)

We next assume that $2\chi\,e^{\Phi}$ affords a series expansion in powers of
$\rho$

$\displaystyle
2\chi\,e^{\Phi\left(\frac{1}{d_{H}+\rho}\right)}=\sum_{n=0}^{\infty}\,\xi_{n}\,\rho^{n}\,,$
with $\displaystyle\begin{array}[]{l}\xi_{0}=z_{H}\,,\\\
\xi_{n}\in\mathbb{R}\hskip 14.22636pt\forall n\in\mathbb{N}\end{array}\,,$
(47)

which has an interval of convergence $I\supseteq[0,\rho_{A})$. Concretely, the
coefficients $\xi_{n}$ can be related to the $\Phi^{(n)}_{H}$ as introduced in
(15): starting from the expansion

$\displaystyle\Phi\left(\frac{1}{d_{H}+\rho}\right)=\sum_{n=0}^{\infty}\kappa_{n}\,\rho^{n}\,,$
(48)

with the coefficients

$\displaystyle\kappa_{n}=\frac{(-1)^{n}}{n!}\,\left[\frac{\Phi^{(n)}_{H}}{d_{H}^{2n}}+\sum_{k=1}^{n}\frac{\left(\prod_{i=1}^{k-1}(n-i)\right)\left(\prod_{j=0}^{k}(n-j)\right)}{k!\,d_{H}^{2n-k}}\,\Phi^{(n-k)}_{H}\right]\,,$
(49)

we have verified the following relation up to order $n=8$

$\displaystyle\xi_{n}=2\chi\,e^{\Phi_{H}}\sum_{k=0}^{n}\sum_{\\{u_{1},\ldots,u_{n}\\}}\frac{\kappa_{1}^{n-k}\,\prod_{i=2}^{n}\kappa_{i}^{u_{i}}}{(n-k)!\prod_{j=2}^{n}(u_{j}!)}\,,$
with $\displaystyle u_{j}\in\mathbb{N}\cup\\{0\\}\text{ such that
}\sum_{j=2}^{n}ju_{j}=k\,.$ (50)

For example, we find for the first three coefficients

$\displaystyle\xi_{0}=2\chi\,e^{\Phi_{H}}=z_{H}\,,$
$\displaystyle\xi_{1}=-\frac{2\chi\,e^{\Phi_{H}}}{d_{H}^{2}}\,\Phi^{(1)}_{H}\,,$
$\displaystyle\xi_{2}=\frac{\chi\,e^{\Phi_{H}}}{d_{H}^{4}}\,\left(2d_{H}\,\Phi^{(1)}_{H}+\left(\Phi^{(1)}_{H}\right)^{2}+\Phi^{(2)}_{H}\right)\,.$
(51)

In the following, we shall consider the coefficients $\xi_{n}$ (or
equivalently the derivatives $\Phi^{(n)}_{H}$ of $\Phi$ at the horizon) along
with $d_{H}$ as fixed and as input for the equation (44). We can then expand
the left-hand side of the latter into a series expansion in $\rho$. To this
end, we consider

$\displaystyle\left(\derivative{z}{\rho}\right)^{2}$
$\displaystyle=\sum_{n,m=1}^{\infty}n\,m\,a_{n}\,a_{m}\,\rho^{m+n-2}=a_{1}^{2}+\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n+1}(n-m+2)\,m\,a_{m}\,a_{n-m+2}\right)\,\rho^{n}\,.$

Inserting this expansion into the left-hand side of (44) we find

$\displaystyle
z\left(1-\left(\derivative{z}{\rho}\right)^{2}\right)=z_{H}\,(1-a_{1}^{2})+\sum_{n=1}^{\infty}\left[(1-a_{1}^{2})\,a_{n}-z_{H}\,\sum_{m=1}^{n+1}(n-m+2)\,m\,a_{m}\,a_{n-m+2}\right]\,\rho^{n}$
$\displaystyle\hskip
156.49014pt-\sum_{p=1}^{\infty}\left[\sum_{n=1}^{p-1}\sum_{m=1}^{n+1}(n-m+2)\,m\,a_{p-n}\,a_{m}\,a_{n-m+2}\right]\,\rho^{p}\,,$
(52)

such that equation (44) becomes

$\displaystyle\sum_{p=0}^{\infty}\xi_{p}\rho^{p}=z_{H}\,(1-a_{1}^{2})$
$\displaystyle+\sum_{p=1}^{\infty}\bigg{[}(1-a_{1}^{2})\,a_{p}-z_{H}\,\sum_{m=1}^{p+1}(p-m+2)\,m\,a_{m}\,a_{p-m+2}$
$\displaystyle\hskip
42.67912pt-\sum_{n=1}^{p-1}\sum_{m=1}^{n+1}(n-m+2)\,m\,a_{p-n}\,a_{m}\,a_{n-m+2}\bigg{]}\rho^{p}\,.$
(53)

Identifying the series coefficients order by order for $p\in\\{0,1,2\\}$ leads
to

$\displaystyle\xi_{0}=z_{H}=z_{H}(1-a_{1}^{2})\,,$
$\displaystyle\xi_{1}=(1-a_{1}^{2})a_{1}-4\,z_{H}\,a_{1}\,a_{2}\,,$
$\displaystyle\xi_{2}=(1-a_{1}^{2})\,a_{2}-\left[z_{H}(6\,a_{1}\,a_{3}+4\,a_{2}^{2})+4\,a_{1}^{2}\,a_{2}\right]\,.$
(54)

The first of these equations requires $a_{1}=0$, which imposes the condition
$\xi_{1}=0$.888We remark in passing that a series expansion for $z$ to match
the case $\xi_{1}\neq 0$ requires half-integer powers in eq. (42), as is
discussed in Appendix C.3. We shall consider such a case in more detail in the
context of a specific example in Appendix C.4.2. As we shall see, however,
$\xi_{1}\neq 0$ (and thus $\Phi^{(1)}_{H}\neq 0$) leads in general to a
curvature singularity at the horizon. Using (51), this requires
$\Phi^{(1)}_{H}=0$, which is in fact the second relation in (36). The last
equation in (54) then becomes

$\displaystyle\xi_{2}=a_{2}(1-4\,z_{H}\,a_{2})\,,$ (55)

which has solution

$\displaystyle a_{2}=\frac{1\pm\varpi}{8\,z_{H}}\mbox{\quad
with\quad}\varpi=\sqrt{1-16z_{H}\xi_{2}}\,.$ (56)

For $\xi_{1}=0$ (and thus $\Phi^{(1)}_{H}=0$), $\varpi$ is identical to the
definition in (31), since in this case
$\xi_{2}=\frac{z_{H}\Phi^{(2)}_{H}}{2d_{H}^{4}}$, which follows from (51). The
solution (56) is in agreement with (23) using (30).999Indeed, in (30) the
solution with the $+$ sign has been chosen to recover the classical result in
the limit $\Phi^{(2)}_{H}\to 0$. In the following, we shall make the same
choice. Similar to (32), the requirement for $\varpi$ to be real (such that
$a_{2}\in\mathbb{R}$) imposes an upper bound on $\xi_{2}\leq 1/(16z_{H})$.

Equating the remaining powers of $\rho^{p}$ (for $p>2$) in (53) then becomes a
recursive equation, which allows to express $a_{p}$ in terms of coefficients
$a_{p^{\prime}}$ with $p^{\prime}<p$ (and $\xi_{p}$):

$\displaystyle a_{p}=\frac{1}{1-4\,z_{H}\,p\,a_{2}}\bigg{[}\xi_{p}$
$\displaystyle+z_{H}\,\sum_{n=3}^{p-1}(p-n+2)\,n\,a_{n}\,a_{p-n+2}$
$\displaystyle+\sum_{n=2}^{p-2}\sum_{m=2}^{n}(n-m+2)\,m\,a_{p-n}\,a_{m}\,a_{n-m+2}\bigg{]}\,,\hskip
28.45274pt\forall p\geq 3\,.$ (57)

The equations (57) allow us to compute the coefficients $a_{p}$ explicitly up
to arbitrary order as functions of the $\xi_{n}$.

####  3.1.2 Series expansion of the distance and metric function

In order to expand the metric function $f$ in a series in $z-z_{H}$ (as in eq.
(19)), we first assume an expansion of $\rho$ in (half-integer) powers of
$z-z_{H}$

$\displaystyle\rho=\sum_{n=1}^{\infty}b_{n}\,(z-z_{H})^{n/2}\,,$ (58)

The coefficients $b_{n}$ can be computed from (57) to arbitrarily high order
using series reversion. Indeed, generalizing a result of Whittaker [40] for
the reversion of integer series with $a_{1}\neq 0$ to the current case, we
have verified up to order $n=8$

$\displaystyle b_{1}=\frac{1}{a_{2}^{1/2}}\,,$ $\displaystyle
b_{2}=-\frac{a_{3}}{2a_{2}^{2}}\,,$ $\displaystyle
b_{n}=\frac{(-1)^{n-1}}{2^{n-1}n!\,a_{2}^{n/2}}\,\text{det}(\mathcal{M}_{n})\hskip
14.22636pt\forall n\geq 3\,,$ (59)

where $\mathcal{M}_{n}$ is the following $(n-1)\times(n-1)$ matrix

(80)

Concretely, we find for the first few coefficients with $n\geq 3$

$\displaystyle b_{3}=\frac{5a_{3}^{2}-4a_{2}a_{4}}{8a_{2}^{7/2}}\,,\hskip
28.45274ptb_{4}=\frac{3a_{2}a_{3}a_{4}-2a_{3}^{3}-a_{2}^{2}a_{5}}{2a_{2}^{5}}\,,$
$\displaystyle
b_{5}=\frac{231a_{3}^{4}-504a_{2}a_{3}^{2}a_{4}+224a_{2}^{2}a_{3}a_{5}+112a_{2}^{2}a_{4}^{2}-64a_{2}^{3}a_{6}}{128a_{2}^{13/2}}\,.$
(81)

###  3.2 Metric functions and curvature

In the previous Subsection, we solved the differential equation (42), assuming
an integer series expansion of the radial coordinate as a function of the
distance (from the event horizon) $\rho$. In this Subsection, we use this
result to compute the Ricci scalar.

####  3.2.1 Metric functions

As a first step, similar to (47), we also introduce a series expansion for the
deformation of the metric function $h$

$\displaystyle
2\chi\,e^{\Psi\left(\frac{1}{d_{H}+\rho}\right)}=\sum_{n=0}^{\infty}\,\theta_{n}\,\rho^{n}\,,$
with $\displaystyle\begin{array}[]{l}\theta_{0}=z_{H}\,,\\\
\theta_{n}\in\mathbb{R}\hskip 14.22636pt\forall n\in\mathbb{N}\end{array}\,.$
(84)

The coefficients $\theta_{n}$ can be related to the derivatives
$\Psi^{(n)}_{H}$ in exactly the same way as the coefficients $\xi_{n}$ are
related to the $\Phi^{(n)}_{H}$ in eq. (50). For the moment, we shall keep
$\theta_{1}$ generic and not mimic the constraint $\xi_{1}=0$, which was
required for consistency of the series expansion (42) with (47).

Using (47), (84), as well as the expansion $1/z$ in (188) (with the
coefficients $\mathfrak{p}_{m}$ given recursively in eq. (192)), we can
express the metric functions as power series in $\rho$

$\displaystyle
f=1-\sum_{k=0}^{\infty}\rho^{k}\sum_{n=0}^{k}\,\mathfrak{p}_{n}\,\xi_{k-n}\,,$
and $\displaystyle
h=1-\sum_{k=0}^{\infty}\rho^{k}\sum_{n=0}^{k}\,\mathfrak{p}_{n}\,\theta_{k-n}\,,$
$\displaystyle\forall\rho\geq 0\,,$ (85)

which reads explicitly to leading orders

$\displaystyle f$
$\displaystyle=\frac{a_{2}-\xi_{2}}{z_{H}}\,\rho^{2}+\frac{a_{3}-\xi_{3}}{z_{H}}\,\rho^{3}+\mathcal{O}(\rho^{4})\,,$
(86) $\displaystyle h$
$\displaystyle=-\frac{\theta_{1}}{z_{H}}\,\rho+\frac{a_{2}-\theta_{2}}{z_{H}}\,\rho^{2}+\frac{(a_{3}-\theta_{3})z_{H}+a_{2}\theta_{1}}{z_{H}^{2}}\,\rho^{3}+\mathcal{O}(\rho^{4})\,.$
(87)

We can equivalently write these as expansions in $(z-z_{H})$

$\displaystyle f$
$\displaystyle=\frac{a_{2}-\xi_{2}}{a_{2}\,z_{H}}\,(z-z_{H})+\frac{a_{3}\xi_{2}-a_{2}\xi_{3}}{(a_{2})^{5/2}z_{H}}\,(z-z_{H})^{3/2}+\mathcal{O}((z-z_{H})^{2})\,,$
(88) $\displaystyle h$
$\displaystyle=-\frac{\theta_{1}}{\sqrt{a}_{2}z_{H}}\,(z-z_{H})^{1/2}+\frac{2a_{2}^{2}+a_{3}\theta_{1}-2a_{2}\theta_{2}}{2a_{2}^{2}\,z_{H}}\,(z-z_{H})+\mathcal{O}((z-z_{H})^{3/2})\,,$
(89)

which shows certain similarities with the results of Section 2.1, but also
some differences: on the one hand, comparing the expansion for the function
$f$ with (24), since

$\displaystyle\frac{a_{2}-\xi_{2}}{a_{2}\,z_{H}}=4\,a_{2}=\frac{1+\varpi}{2\,z_{H}}=f^{(1)}_{H}\,,$
(90)

(see (30)) we find agreement with the leading term in (88). On the other hand,
(88) contains a term of order $(z-z_{H})^{3/2}$ (and (86) a term of order
$\rho^{3}$), which is absent in (19) (and (24) respectively). Indeed (88) is
not an integer series expansion and therefore more general than (19).
Similarly, (since we have not imposed $\theta_{1}=0$), the function $h$ even
starts from a term $(z-z_{H})^{1/2}$ in (89) (order $\rho$ in (87)), which is
absent in (20) (and (25) respectively). Our result (85) is therefore more
general than (24) and (25). We remark, however, that for generic values of
$\xi_{3}$ and $\theta_{1}$ we have

$\displaystyle\lim_{z\to z_{H}}f^{(2)}(z)\to\infty\,,$ and
$\displaystyle\lim_{z\to z_{H}}h^{(1)}(z)\to\infty\,.$ (91)

Indeed, in contrast to Section 2.1 finiteness of the derivatives of the metric
functions at the horizon was not our initial assumption and we shall impose in
the next Subsubsection absence of a physical curvature singularity instead as
a more general condition. Before doing so, however, we comment that we can
deduce the conditions for the absence of the singularities (91) from (88) and
(89) in a straightforward manner:

$\displaystyle\theta_{1}=0\,,$ and $\displaystyle
a_{3}\xi_{2}=a_{2}\xi_{3}\,,$ and $\displaystyle
a_{3}\theta_{2}=a_{2}\theta_{3}\,.$ (92)

Using (57), the second of these relations implies

$\displaystyle\frac{1-8z_{H}\xi_{2}+\varpi}{2z_{H}(1+3\varpi)}\,\xi_{3}=0\,,$
(93)

which has as only solution $\xi_{3}=0$, which further implies $\theta_{3}=0$.
Notice that the conditions

$\displaystyle\xi_{1}=\xi_{3}=\theta_{1}=\theta_{3}=0\,,$ (94)

which guarantee existence of the first and second derivative of $f$ and $h$
for $z=z_{H}$ (which are necessary for the expansions (19) and (20)) are
precisely the same as (36). In this case, the positivity of the first
derivative of $h$ leads to an upper bound on
$\theta_{2}<\frac{1+\varpi}{8z_{H}}$, which agrees with the second relation in
(32). The upper bound $\xi_{2}\leq 1/(16z_{H})$ was already obtained
previously to guarantee reality of $\varpi$ in (56).

####  3.2.2 Ricci scalar and Hawking temperature

We next consider the Ricci scalar, however, for simplicity, we shall work out
its series expansions only to leading order. Moreover, we shall start out by
only assuming $\xi_{1}=0$ (which is required for the consistent expansion
(42)), but we shall not assume the remaining conditions in (94). We notably
first consider $\theta_{1}\neq 0$. Inserting (88) and (89) into (17), we
obtain the following series expansion (for $z\geq z_{H}$)

$\displaystyle
R=\frac{a_{2}-\xi_{2}}{8a_{2}z_{H}}\,(z-z_{H})^{-1}+\frac{(2a_{2}^{2}+a_{3}\theta_{1}-2a_{2}\theta_{2})(a_{2}-\xi_{2})}{8a_{2}^{5/2}\theta_{1}z_{H}}\,(z-z_{H})^{-1/2}+\mathcal{O}((z-z_{H})^{0})\,.$
(95)

Using (57) the coefficient of the leading term becomes

$\displaystyle\frac{a_{2}-\xi_{2}}{8a_{2}z_{H}}=\frac{1-8z_{H}\xi_{2}+\varpi}{8z_{H}(1+\varpi)}\,,$
(96)

which is non-vanishing for all values of $\xi_{2}\leq 1/(16z_{H})$ and
therefore signals a curvature singularity at the event horizon. To avoid the
latter, we impose $\theta_{1}=0$, which also leads to a well defined
derivative $h^{(1)}(z_{H})$:

$\displaystyle h^{(1)}(z_{H})=\frac{a_{2}-\theta_{2}}{a_{2}z_{H}}\,,$ (97)

Moreover, the condition $\theta_{1}=0$ also changes the series expansion in
(95)

$\displaystyle
R=\frac{4\sqrt{2}\sqrt{z_{H}(1+\varpi)}\left((1+3\varpi)\theta_{3}+2\xi_{3}\right)}{(1+3\varpi)(1-8z_{H}\theta_{2}+\varpi)}\,(z-z_{H})^{-1/2}+\mathcal{O}((z-z_{H})^{0})\,.$
(98)

We have furthermore verified that the singularity of $R$ at $z=z_{H}$ cannot
be removed if $1-8z_{H}\theta_{2}+\varpi=0$ and we, therefore, require
$\theta_{2}\neq a_{2}$.101010This condition is compatible with
$\theta_{2}<\frac{1+\varpi}{8z_{H}}$ which guarantees that $h^{(1)}(z_{H})>0$
in (97). The latter is necessary such that $h(z)>0$ for $z>z_{H}$ (with a
simple zero at $z=z_{H}$). In this case, the necessary condition for
regularity of $R$ at the horizon is $(1+3\varpi)\theta_{3}+2\xi_{3}=0$. We
have also verified that under the same condition also the Kretschmann scalar
is finite at the horizon. To summarise, the consistency conditions for the
approach outlined in Section 3.1.1, the conditions for the absence of a
singularity of the Ricci scalar, and the bounds for positive metric functions
$f$ and $h$ for $z>z_{H}$ are therefore

$\displaystyle\xi_{1}=0\,,$ $\displaystyle\theta_{1}=0\,,$
$\displaystyle\xi_{3}=-\frac{1}{2}\,(1+3\varpi)\,\theta_{3}\,,$
$\displaystyle\xi_{2}\leq\frac{1}{16z_{H}}\,,$
$\displaystyle\theta_{2}<\frac{1+\varpi}{8z_{H}}\,.$ (99)

As remarked previously (see (94)), the regularity condition (36) found in the
previous Section is compatible with this result and is the particular case
$\xi_{3}=\theta_{3}=0$.

We further remark that the series coefficients (57) and (59) along with the
expansions (85) allow to compute the (finite) value of $R$ at the horizon.
While the general form is rather complicated, here we only give the expression
in the particular case $f=h$ (_i.e._ $\xi_{n}=\theta_{n}$ $\forall n\geq 1$)
with $\xi_{1}=\xi_{3}=0$

$\displaystyle
R\big{|}_{z=z_{H}}=\frac{(1-\varpi^{2})(a_{2}^{2}-a_{4}z_{H})}{8z_{H}^{3}a_{2}^{3}}+\frac{2\xi_{4}}{z_{H}a_{2}^{2}}=\frac{3+(2-5\varpi)\varpi}{2z_{H}^{2}(1+2\varpi)}+\frac{192z_{H}\xi_{4}}{1+3\varpi+2\varpi^{2}}\,.$
(100)

which we shall use in the examples of the following Section.

Before closing this Section, we also provide the expression for the Hawking
temperature

$\displaystyle
T_{\text{H}}=\frac{\sqrt{f^{(1)}_{H}h_{H}^{(1)}}}{4\pi}=\frac{\sqrt{1+\varpi-8z_{H}\theta_{2}}}{4\pi\sqrt{2}\,z_{H}}\,,$
(101)

which agrees with the expression (39) found in the previous Section. The upper
bound on $\theta_{2}$ in (99) guarantees that $T_{\text{H}}>0$. Indeed,
$T_{\text{H}}=0$ would require $f^{(1)}_{H}=0$ and/or $h_{H}^{(1)}=0$, which
translate into $a_{2}=\xi_{2}$ and $a_{2}=\theta_{2}$ respectively. The former
has no real solution, while the latter leads to a singularity of the Ricci
scalar at the horizon. We remark, however, that black hole solutions with
$T_{\text{H}}=0$ are possible upon choosing the solution
$a_{2}=\frac{1-\varpi}{8z_{H}}$ in (56).

##  4 Examples

To illustrate further the approach presented in the previous Sections, in
particular, the conditions (36) and (99) which are sufficient for a finite
Ricci scalar at the horizon, we shall consider two concrete examples: the
first one is the Bonanno-Reuter [5] black hole, while the second one is
specifically constructed to satisfy (36) in a minimal fashion. Further
examples from the literature are discussed in Appendix D.

###  4.1 Example 1: Bonanno-Reuter asymptotically safe black hole

We first consider as an example the black hole metric introduced by Bonanno-
Reuter [5] as a renormalization group improved generalization of the
Schwarzschild space-time. Indeed, in this work, it has been proposed to
replace the (dimensionful) Newton constant $G_{\text{Newton}}$ by a running
Newton constant

$\displaystyle G_{\text{Newton}}\longrightarrow
G(k)=\frac{G(k=0)}{1+\omega\,G(k=0)\,k^{2}}\,,$ (102)

where $\omega\in\mathbb{R}$ is a constant and $k$ a (position dependent) scale
(with reference scale $k=0$). The choice of the latter is ambiguous, but it
has been proposed in [5] to use an inverse physical distance from the center
of the black hole111111Other options discussed in [5] include distances of the
form $\int_{\mathcal{C}}\sqrt{|ds^{2}|}$, for different choices of contours
$\mathcal{C}$ (for example the world-line of a free-falling observer). We
shall discuss in future work (see also [25]) that different such choices
correspond to different schemes from the perspective of the renormalization
group approach.

$\displaystyle k(z)=\xi/\mathfrak{d}(z)\,,$ (103)

with $\xi$ a suitable (dimensionful) constant. Such a distance has physical
meaning, independent of a specific choice of coordinates. Adapting to our
notation, the metric proposed in [5] can be written in the form (1) with

$\displaystyle
h(z)=f(z)=f_{\text{BR}}(z):=1-\frac{2\chi}{z}\frac{1}{1+\widetilde{\omega}/\mathfrak{d}(z)^{2}}\,,$
(104)

where $\widetilde{\omega}=\omega\xi^{2}$ is a dimensionless constant. In [5]
the concrete value $\widetilde{\omega}=\frac{118}{15\pi}$ was given, however,
subsequent works in the literature [41, 42, 43, 44, 45] potentially point
towards different values (and a different sign). In the following, we shall
consider $\widetilde{\omega}$ a generic parameter and discover marked
differences between positive and negative values. Furthermore, we shall
consider (104) to be valid only outside of the event horizon, which is located
at $d_{\text{BH},H}$ (which we take as an input of the model.121212Here we are
allowing for the possibility that the metric inside of the black hole is
different from (104) in which case $d_{\text{BH},H}$ would need to be computed
as a separate input to the model. As we shall see, our conclusions will be
entirely independent of this choice and thus the concrete value of
$d_{\text{BH},H}$.) In order for $f$ to remain well defined at the horizon, we
shall assume $\widetilde{\omega}+d^{2}_{\text{BR},H}=\frac{2\chi
d_{\text{BR},H}^{2}}{z_{H}}\neq 0$.

For concrete computations, a choice for the physical distance needs to be
made. Here we shall discuss three different possibilities that lead to a
geometry with an (outer) event horizon, for which we can verify whether the
conditions (36) are satisfied and whether therefore the Ricci scalar is
finite, namely: _(i)_ the proper distance computed from the metric (1); _(ii)_
the proper distance computed from the Schwarzschild metric; _(iii)_ an
interpolating function. We shall discuss all three possibilities in the
following:

  * _(i)_

choosing $\mathfrak{d}(z)$ as the _proper distance_ $d(z)$ in eq. (6):

$\displaystyle
d_{\text{BR}}(z)=\int_{0}^{z}\frac{1}{\sqrt{|f_{\text{BR}}(z)|}}\,,$
$\displaystyle\forall z\geq 0\,.$ (105)

This is a self-consistent choice in the sense that the proper distance is
compatible with the metric (1). As discussed in [1], this guarantees that the
modified metric exhibits the same diffeomorphism invariance as the (classical)
Schwarzschild black hole. However, explicitly computing the distance becomes
more involved (since (105) is an implicit definition). For negative values of
$\widetilde{\omega}$, a series expansion close to the horizon is developed in
Appendix C.4.2. However, for our purposes, this is not in fact required, since
we can simply verify the regularity conditions developed in the previous
Sections, _i.e._ eq. (36) or (99). For the concrete function (104) with
$\mathfrak{d}(z)=d_{\text{BR}}(z)=d_{\text{BR},H}+\rho(z)$, we obtain

$\displaystyle\Phi^{(1)}_{H}=\Psi^{(1)}_{H}=-\frac{2\widetilde{\omega}}{d_{\text{BR},H}(d_{\text{BR},H}^{2}+\widetilde{\omega})}\neq
0\ ,$
$\displaystyle\xi_{1}=\theta_{1}=\frac{4\chi\widetilde{\omega}d_{\text{BR},H}}{(\widetilde{\omega}+d_{\text{BR},H}^{2})^{2}}\neq
0\,,$ (106)

such that neither (99) nor (36) are satisfied. In fact, (106) implies that
already the first derivative of $f_{\text{BR}}$

$\displaystyle
f_{\text{BR}}^{(1)}(z)=(1-f_{\text{BR}}(z))\left(\frac{1}{z}-\frac{2\widetilde{\omega}}{d_{\text{BR}}(z)(d_{\text{BR}}(z)^{2}+\widetilde{\omega})}\frac{1}{\sqrt{f_{\text{BR}}(z)}}\right)\,,$
$\displaystyle\forall z\geq z_{\text{BR},H}\,,$

diverges at the horizon

$\displaystyle\lim_{\epsilon\to
0^{+}}f_{\text{BR}}^{(1)}(z_{\text{BR},H}+\epsilon)\to\infty\,.$ (107)

This is due to the fact that $f_{\text{BR}}(z_{\text{BR},H})=0$. Following the
discussion of Section 2.2, this poses problems with the interpretation of the
black hole’s thermodynamical properties, notably the Hawking temperature’s
definition. Furthermore, it also leads to a curvature singularity at the
horizon, since for example, the Ricci scalar becomes

$\displaystyle
R=\frac{\widetilde{\omega}(f_{\text{BR}}-1)\left(d_{\text{BR}}(d_{\text{BR}}^{2}+\widetilde{\omega})(1-5f_{\text{BR}})f_{\text{BR}}^{1/2}+6zd_{\text{BR}}^{2}f_{\text{BR}}-2\widetilde{\omega}z\right)}{zd_{\text{BR}}^{2}\left(d_{\text{BR}}^{2}+\widetilde{\omega}\right)^{2}f_{\text{BR}}^{2}}\
,$ $\displaystyle\forall z\geq z_{H}\,.$

This expression diverges at the event horizon due to the factor
$f_{\text{BR}}^{2}$ in the denominator (while the numerator at the horizon
assumes the finite value $2\widetilde{\omega}^{2}z_{H}$).

Finally, we remark since $\xi_{1}\neq 0$, the results of the series expansion
approach developed in Section 3 are not directly applicable. In appendix C.3,
we show how it can be generalised in this case, and the consequences for the
Bonanno-Reuter space-time for $\widetilde{\omega}<0$, are discussed in
appendix C.4.2 (confirming further our above conclusions in this case).

  * _(ii)_

choosing $\mathfrak{d}(z)$ as the _proper distance_ of the (classical)
_Schwarzschild geometry_ , _i.e._

$\displaystyle
d_{\text{S}}(z)=\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\left|1-\frac{2\chi}{z}\right|}}=\left\\{\begin{array}[]{lcl}\pi\chi-\sqrt{z(2\chi-z)}-2\chi\arctan\sqrt{\frac{2\chi}{z}-1}&\text{if}&0<z<2\chi\,,\\\\[2.0pt]
\pi\chi+\sqrt{z(z-2\chi)}+2\chi\text{arctanh}\sqrt{1-\frac{2\chi}{z}}&\text{if}&2\chi<z\,.\end{array}\right.$
(110)

This option was initially advocated in [5] and has the advantage that it can
be computed as a closed expression in terms of $z$. This geometry possesses a
horizon, whose position is corrected by $\widetilde{\omega}$, _e.g._ for large
mass $\chi$ of the black hole

$\displaystyle
z_{\text{BR},H}=2\chi-\frac{2}{\pi^{2}}\,\frac{\widetilde{\omega}}{\chi}+\mathcal{O}\left(|\widetilde{\omega}|^{3/2}/\chi^{2}\right)\,.$
(111)

For $f_{\text{BR}}$ to be well defined at the horizon, we assume that
$\widetilde{\omega}+d_{\text{S}}^{2}(z=z_{\text{BR},H})\neq 0$. The derivative
of $f$ in this case becomes

$\displaystyle
f^{(1)}_{\text{BR}}(z)=\frac{2\chi}{z^{2}\left(1+\frac{\omega}{d_{\text{S}}^{2}}\right)}-\frac{4\chi\omega}{z\,d_{\text{S}}^{3}\left|1-\frac{2\chi}{z}\right|\left(1+\frac{\omega}{d_{\text{S}}^{2}}\right)^{2}}\,,$
(112)

which is finite at the horizon $z_{\text{BR},H}$, but diverges for $z=2\chi$
(which for $\widetilde{\omega}>0$ lies outside the horizon of the black hole).
Because of this, while the Ricci scalar is finite at $z=z_{\text{BR},H}$, the
geometry has a curvature singularity at $z=2\chi$.

  * _(iii)_

Although a closed function of $z$, the Schwarzschild proper distance (110) is
still difficult to work with for concrete computations. Therefore in [5] the
following approximation for (110) was proposed131313We are using a notation
adapted to the current paper.

$d_{\text{S}}(z)\simeq\kappa_{\text{BR}}(z)=\left(\frac{z^{3}}{z+\gamma\,\chi}\right)^{1/2}\,,\mbox{\quad
such that\quad}\begin{array}[]{l}\displaystyle\lim_{z\to
0}\kappa_{\text{BR}}(z)\sim\frac{z^{3/2}}{\sqrt{\gamma\chi}}\,,\\\
\displaystyle\lim_{z\to\infty}\kappa_{\text{BR}}(z)\sim z\,.\end{array}$ (113)

Here $\gamma\in\mathbb{R}$ is a constant, which in order to mimic the same
behaviour as (110) at the origin ($z\to 0$) needs to be chosen as
$\gamma=9/2$. The function $\kappa_{\text{BR}}$ has no inflection points, and
thus does not feature the same behaviour as a proper distance at a horizon for
any value of $z$.

Identifying $\mathfrak{d}$ in (104) with $\kappa_{\text{BR}}$ leads to a zero
of $f_{\text{BR}}$ at

$\displaystyle
z_{\text{BR},H}=2\chi-\frac{2+\gamma}{4}\,\frac{\omega}{\chi}+\mathcal{O}(\omega^{2}/\chi^{3})\,.$
(114)

At this horizon, both the first and second derivatives of $f_{\text{BR}}$ are
finite implying that the Ricci scalar takes a finite value.

To summarise, treating (104) in a _self-consistent fashion_ by identifying
$\mathfrak{d}$ by the proper distance calculated from $f_{\text{BR}}$ itself
leads to a divergent first derivative of the metric function at the horizon,
which in turn leads to significant problems for physical quantities. Notably,
it poses problems for defining a finite Hawking temperature and leads to a
curvature singularity at the (outer) event horizon. This is in line with the
results of Sections 2 and 3, due to the fact that the function
$\frac{2\chi}{1+\widetilde{\omega}/(d_{\text{BR},H}+\rho)^{2}}$ does not
satisfy the conditions (36), independent of the geometry of the black hole
inside the event horizon (_i.e._ independent of the value of
$d_{\text{BR},H}$). The choices _(ii)_ and _(iii)_ constitute a departure from
the original idea presented in [5] (proposed as an approximation in this work)
by replacing $\mathfrak{d}$ by a function of the radial coordinate $z$, which
does _not_ represent a physical distance that has been consistently calculated
from the metric characterised by (104).

While _(ii)_ , depending on the sign of $\widetilde{\omega}$, may have a
curvature singularity outside of the event horizon, choice _(iii)_ is at least
well-behaved from this perspective. However, from the point of view of the
original motivation to deform the Schwarzschild metric by a function of a
(consistently calculated) distance function, the choice _(iii)_ corresponds to
a different deformation function than (104). Using the approach outlined in
Section 3, we can determine this modified deformation function by reverse
engineering the coefficients $\xi_{n}$: indeed, by integrating eq. (43) we
find the coefficients $b_{n}$ in eq. (58)

$\displaystyle
b_{1}=\frac{\sqrt{2}|z_{\text{BR},H}^{3}+\widetilde{\omega}(z_{\text{BR},H}+\gamma\chi)|}{\sqrt{z_{\text{BR},H}\chi(z_{\text{BR},H}^{3}-\widetilde{\omega}(z_{\text{BR},H}+2\gamma\chi))}}\,,\hskip
28.45274ptb_{2}=0\,,$ $\displaystyle
b_{3}=\frac{b_{1}^{3}\chi\left(z_{\text{BR},H}^{6}-3z_{\text{BR},H}^{4}\widetilde{\omega}-7z_{\text{BR},H}^{3}\gamma\chi\widetilde{\omega}+\gamma^{2}\chi^{2}\widetilde{\omega}^{2}\right)}{24\sqrt{2}\left(z_{\text{BR},H}^{3}+\widetilde{\omega}(\gamma\chi+z_{\text{BR},H})\right)^{3}}\,,\hskip
28.45274ptb_{4}=0\,.$ (115)

Therefore with

$\displaystyle a_{1}=0\,,$ $\displaystyle a_{2}=\frac{1}{b_{1}^{2}}\,,$
$\displaystyle a_{3}=0\,,$ $\displaystyle a_{4}=-\frac{2b_{3}}{b_{1}^{5}}\,,$
(116)

we obtain for the leading coefficients of $\xi_{n}$ (which are equal to
$\theta_{n}$)

$\displaystyle\xi_{1}=0\,,$
$\displaystyle\xi_{2}=\frac{2z_{\text{BR},H}^{2}\chi(2z_{\text{BR},H}+3\gamma\chi)\widetilde{\omega}}{b_{1}^{2}(z_{\text{BR},H}^{3}+\widetilde{\omega}(z_{\text{BR},H}+\gamma\chi))^{2}}\,,$
$\displaystyle\xi_{3}=0\,.$ (117)

These indeed satisfy the conditions (99). Therefore, the “approximation” to
use (113) for $\mathfrak{d}$ in (104) instead of the self-consistently
calculated proper distance, corresponds to changing the metric function (104),
in a way characterised by the above expansion coefficients.

###  4.2 Example 2: Minimal metric deformation

As a further (novel) example, we consider the following minimal solution of
the conditions (36)

$\displaystyle\Phi_{H}=\Psi_{H}=\phi_{0}\,,$
$\displaystyle\Phi^{(1)}_{H}=\Psi^{(1)}_{H}=0\,,$
$\displaystyle\Phi^{(2)}_{H}=\Psi^{(2)}_{H}=\phi_{2}\,,$
$\displaystyle\Phi^{(3)}_{H}=\Psi^{(3)}_{H}=-6d_{H}\phi_{2}\,,$ (118)

and $\Phi^{(n)}_{H}=0$ $\forall n\geq 4$. Here
$\phi_{0},\phi_{2}\in\mathbb{R}$ are arbitrary parameters, which, however, are
not independent: indeed, in order for the metric to asymptotically, approach
the Schwarzschild one (with mass parameter $\chi$), we require the asymptotic
limit

$\displaystyle\lim_{\rho\to\infty}e^{\Phi\left(\frac{1}{d_{H}+\rho}\right)}=e^{\phi_{0}+\frac{3\phi_{2}}{2d_{H}^{2}}}\overset{!}{=}1\quad\implies\quad\phi_{0}=-\frac{3\phi_{2}}{2d_{H}^{2}}\
.$ (119)

Here we also consider $d_{H}$ as a parameter of the model, which encodes
information about the interior of the black hole. With this, the solution
(118) of (36) can be written compactly in the form

$\displaystyle\Phi\left(\frac{1}{d_{H}+\rho}\right)=\Psi\left(\frac{1}{d_{H}+\rho}\right)=-\frac{3\phi_{2}}{2d_{H}^{2}}+\frac{\rho^{2}(d_{H}+3\rho)\phi_{2}}{2d_{H}^{2}(d_{H}+\rho)^{3}}\,.$
(120)

Choosing $\Phi^{(n)}_{H}\neq 0$ (or $\Psi^{(n)}_{H}\neq 0$) for $n\geq 4$ in
(118) would yield higher modifications of $e^{\Phi}$ of order
$\mathcal{O}(\rho^{4})$, which are negligible close to the horizon. The
position of the latter is located at

$\displaystyle
z_{H}=2\chi\,e^{\phi_{0}}=2\chi\,\text{exp}\left(-\frac{3\phi_{2}}{2d_{H}^{2}}\right)\,.$
(121)

A schematic plot of $e^{\Phi}$ as a function of $\rho$ is shown in Figure 1.
Here we have plotted $e^{\Phi}$ in the entire space-time (_i.e._ also inside
of the event horizon for $\rho<0$): following our general philosophy, we shall
discuss the metric function (and all associated quantities) only outside of
the horizon (_i.e._ for $\rho\geq 0$) and only make a few brief remarks on the
physics inside of the black hole in Section 4.2.2.

Figure 1: _Function $e^{\Phi}$ in eq. (120) with
$\phi_{0}=-3\phi_{2}/2d_{H}^{2}$ and $d_{H}=1$._

The coefficients $\kappa_{n}$ in a series expansion of $\Phi$ in powers of
$\rho$ (see eq. (48)) are worked out in eq. (245), while the leading
coefficients $\xi_{n}$ (stemming from the series expansion of
$2\chi\,e^{\Phi}$ in eq. (47)) are exhibited in (246). In particular, due to
the choice (118), we find $\xi_{1}=0=\xi_{3}$, which therefore satisfies (36)
and (99). Moreover, since (118) also implies $\Phi=\Psi$ (and thus
$\xi_{n}=\theta_{n}$), the condition (99) is also trivially satisfied, such
that we expect the Ricci scalar of this model to be finite at the event
horizon $z_{H}$.

####  4.2.1 Curvature and temperature

As a first step to calculating physical quantities for the space-time metric
characterised by (118), we compute the derivative (30) of the functions $f$
and $h$ at the horizon $z_{H}=2\chi e^{\phi_{0}}$

$\displaystyle
f^{(1)}_{H}=h^{(1)}_{H}=\frac{e^{-\phi_{0}}}{4\chi}\left(1+\sqrt{1-\frac{32e^{2\phi_{0}}\chi^{2}\phi_{2}}{d_{H}^{4}}}\right)\mbox{\quad
with\quad}\phi_{0}=-\frac{3\phi_{2}}{2d_{H}^{2}}\,,$ (122)

which imposes the condition
$\phi_{2}\leq\frac{d_{H}^{4}}{32\,e^{2\phi_{0}}\,\chi^{2}}$, which with (119)
therefore becomes the non-linear relation for $\phi_{2}$

$\displaystyle\phi_{2}\,e^{3\phi_{2}/d_{H}^{2}}\leq\frac{d_{H}^{4}}{32\chi^{2}}\,.$
(123)

The Ricci scalar at the horizon is finite in this model and takes the value

$\displaystyle
R\big{|}_{z_{H}}=\frac{3+(2-5\varpi)\varpi}{2z_{H}^{2}(1+2\varpi)}+\frac{48e^{-\frac{3\phi_{2}}{2d_{H}^{2}}}z_{H}\chi\phi_{2}(\phi_{2}-12d_{H}^{2})}{d_{H}^{8}(1+\varpi)(1+2\varpi)}\,,$
with $\displaystyle\varpi=\sqrt{1-\frac{16z_{H}\phi_{2}\chi
e^{-\frac{3\phi_{2}}{2d_{H}^{2}}}}{d_{H}^{4}}}\,.$ (124)

Finiteness of the derivatives (122) (at the event horizon) is a necessary
requirement for well-behaved thermodynamical properties of the black hole.
Indeed, the Hawking temperature is given by

$T_{\mathrm{H}}=\frac{f^{(1)}_{H}}{4\pi}=\frac{1}{8\pi
z_{H}}\left(1+\sqrt{1-\frac{8z_{H}^{2}\phi_{2}}{d_{H}^{4}}}\right)\ .$ (125)

Determining the entropy using eq. (41) requires specifying the
$\chi$-dependence of $d_{H}$ and $\phi_{2}$ and thus requires further
refinement of the model. To give a concrete example, we shall consider
$\chi\gg 1$ along with

$\displaystyle\phi_{2}=\mathcal{O}(\chi^{0})\mbox{\quad
and\quad}d_{H}=\pi\chi+\mathcal{O}(\chi^{0})\,,$ (126)

that is, we are assuming only subleading corrections to the classical
Schwarzschild metric. We, therefore, find the horizon position

$\displaystyle
z_{H}=2\chi-\frac{3\phi_{2}}{\pi^{2}\chi}+\mathcal{O}(\chi^{-2})\,,$ (127)

and thus for the Hawking temperature

$\displaystyle
T_{\mathrm{H}}=\frac{1}{8\pi\chi}\left[1+\frac{(3\pi^{2}-16)\phi_{2}}{2\pi^{4}\chi^{2}}+\mathcal{O}(\chi^{-3})\right]\,,$
(128)

and the entropy

$\displaystyle
S=4\pi\chi^{2}\left[1-\frac{\left(3\pi^{2}-16\right)\phi_{2}}{3\pi^{2}\chi^{2}}\log(\chi^{2})+\mathcal{O}\left(\frac{1}{\chi^{4}}\right)\right]+\mathrm{const}\
.$ (129)

This approximation exhibits a logarithmic correction, compatible with previous
results in the literature [46, 49, 50, 51, 52, 53, 54, 55, 56, 47, 48]. We
remind the reader, however, that (129) is based on the assumptions (126),
which are related to the interior of the black hole solution.

####  4.2.2 Extending the metric inside the horizon

Following our general philosophy, so far we have considered the metric
function only outside of the black hole horizon and have used the horizon
distance $d_{H}$ (and $z_{H}$) as the only additional input that is required
for the interior of the black hole. Indeed, due to its definition (6), the
calculation of $d_{H}$ requires knowing the metric in the interior of the
black hole, which we have not specified up to this point. In the current
example, since the function (120) can in fact be extended to the entire space-
time (_i.e._ also for $\rho<0$), as is showcased in Figure 1, one can
contemplate the possibility to use (120) as a model for the entire space-time.
Although this discussion is generally outside of the scope of this paper, here
we shall nevertheless make a few remarks regarding this possibility. For
concreteness, we shall focus on $\phi_{2}>0$ in the entire Subsubsection.

As a first question, we may ask the behavior of the metric at the origin. In
order to get a better understanding of the space-time described by (120) for
$z\ll 1$, we may use a right rectangular approximation for the integral of the
proper distance (6)

$\displaystyle d(z)\simeq\frac{z}{\sqrt{|f(z)|}}\qquad\forall z\ll 1\,,$ (130)

which is an algebraic equation. Numerical analysis suggests that there are no
real positive values for $(z,d(z))$ that satisfy this relation for small $z$
when $f(z)<0$. We have, however, obtained the following solution of (130) for
$z(d)$ in the case $f(z)>0$

$\displaystyle\frac{z}{d}$
$\displaystyle=\frac{\sqrt{d}\left(3^{1/3}d^{2/3}+\left(-9e^{\Phi}\chi+\sqrt{-3d^{2}+81e^{2\Phi}\chi^{2}}\right)\right)^{2/3}}{3^{2/3}\left(-9e^{\Phi}\chi+\sqrt{-3d^{2}+81e^{2\Phi}\chi^{2}}\right)^{1/3}}\,$
$\displaystyle=\cos\left[\frac{1}{3}\,\arcsin\left(\frac{3\sqrt{3}e^{\Phi}\chi}{d}\right)\right]-\frac{1}{\sqrt{3}}\sin\left[\frac{1}{3}\,\arcsin\left(\frac{3\sqrt{3}e^{\Phi}\chi}{d}\right)\right]\,.$
(131)

The function $f$, therefore, approaches $+1$ for $d\to 0$, which hints towards
the absence of a singularity at the origin. To verify this, we realize that
the function $z$ in (131) cannot be expanded in a Taylor series for $d$.
However, noting that (for $\Phi$ given in (120)) for $\phi_{2}>0$ we have the
limit $\displaystyle\lim_{\rho\to-d_{H}}\,e^{\Phi}$ in an exponential fashion,
we can (formally) write $z/d$ as a series expansion in powers of $e^{\Phi}$

$\displaystyle
z/d=1-\sum_{n=1}^{\infty}\frac{2^{n-1}\,\Gamma\left(\frac{3n-1}{2}\right)}{n!\,\Gamma\left(\frac{n+1}{2}\right)}\,\left(\frac{\chi
e^{\Phi}}{d}\right)^{n}\mbox{\quad
for\quad}\frac{\chi\,e^{\Phi}}{d}\in\left[0,\tfrac{1}{3\sqrt{3}}\right)\,.$
(132)

We then find the Ricci scalar close to the origin

$\displaystyle
R=\frac{2\chi\,e^{\Phi}\,\phi_{2}}{d_{H}^{2}d^{9}}\left[d_{H}^{2}d^{3}(7d-6d_{H})+(3d_{H}-4d)^{2}(d-d_{H})^{2}\phi_{2}\right]+\mathcal{O}(e^{2\Phi})\,,$
(133)

which is indeed finite in the limit $d\to 0$, due to the exponential
suppression of $e^{\Phi}$.141414We have also verified that the Kretschmann
scalar is finite at the origin in this model.

While the above result of the absence of a curvature singularity at the origin
is very encouraging conceptually, it also highlights another problem: indeed,
if $f(z)>0$ close to the origin and $f(z)>0$ for $z>z_{H}$, with a simple zero
at $z_{H}$, $f$ necessarily has (at least) one further zero in the interval
$z\in(0,z_{H})$, more concretely for $\rho\leq-d_{H}/3$. In other words, the
space-time described by (120) has at least one more inner horizon. At the
latter, we have to verify again if all necessary conditions are met for the
absence of a curvature singularity. However, since (118) is only a minimal
solution of the conditions (36) (which are tailored to remove unphysical
singularities at $\rho=0$), we cannot guarantee the absence of a curvature
singularity at this inner horizon. This problem can be circumvented by
allowing some of the $\Phi^{(n)}_{H}$ (for $n\geq 4$) to be non-zero. While we
leave a more detailed discussion for the black hole interior to further work,
we show in Appendix E how to derive conditions for the absence of curvature
singularities at an inner horizon.

##  5 Large distance expansion

So far, we have focussed on the consistency conditions arising at and near the
event horizon for generic static and spherically symmetric black hole metrics.
More generally we have investigated the impact of these conditions on an
effective metric expanded in terms of a physical distance from the event
horizon. The range of applicability of this theory is visualized via the green
box in Fig. 2. The generic metric (1) in (8) covers any distance from the
black hole horizon as represented in the blue box.

We now investigate an asymptotic expansion from an infinite distance valid in
the red box. This was considered in [1] and extended in [25]:

$f(z)=1-\frac{2\chi}{z}\left(1+\sum_{n=1}^{\infty}\
\frac{\omega_{n}}{d(z)^{n}}\right)\mbox{\quad
and\quad}h(z)=1-\frac{2\chi}{z}\left(1+\sum_{n=1}^{\infty}\
\frac{\gamma_{n}}{d(z)^{n}}\right)\ ,$ (134)

where $\omega_{n},\gamma_{n}$ are effective coefficients encoding the
deformation from the Schwarzschild solution, and the physical distance $d(z)$
is defined in eq. (6). For $n$ even and $\gamma_{n}=\omega_{n}$ we recover the
metric and results of [1]. By construction, the expansion ensures that the
classical black hole metric is recovered at an infinite distance while its
radius of convergence depends on the $\omega_{n}$ and $\gamma_{n}$
coefficients.

$z,\ d,\ \rho$$\displaystyle
2\chi\left(1+\sum_{n=1}^{\infty}\frac{\omega_{n}}{d^{n}}\right)$$\displaystyle\sum_{n=0}^{\infty}\xi_{n}\rho^{n}$$\displaystyle
2\chi e^{\Phi(1/d)}$$\begin{array}[]{c}z_{H},d_{H}\\\ \rho=0\end{array}$
Figure 2: _Schematic division of the space outside of the black hole in three
main regions. The curved dashed line represents the position of the event
horizon. The blue region is where the _full non-perturbative_ form of the
metric functions, given in eq. (8) is expected to hold (_i.e._ the whole
space-time outside the event horizon). For simplicity, we only give the
expressions for $f$ in the Figure, but a similar form also holds for $h$. The
green region is where the metric is expanded as a convergent series in the
proper distance from the event horizon $\rho$, as given in eqs. (47) and (84).
Finally, the red region refers to the asymptotically large distances from the
event horizon, where the metric can be expanded in inverse powers of the
proper distance, as given in eqs (134)._

###  5.1 Convergence criteria and derivatives

We now investigate the impact of the event horizon constraints (36) and (99)
on the large distance expansion coefficients $\omega_{n}$ and $\gamma_{n}$
under the assumption that (134) is convergent up to the horizon. In other
words, we assume that the red box in Fig. 2 extends all the way to the event
horizon. This imposes certain conditions on the coefficients $\omega_{n}$ and
$\gamma_{n}$, namely that the radius of convergence of the series (134) is
larger than the inverse distance of the horizon

$\limsup_{n\to\infty}|\omega_{n}|^{\frac{1}{n}}\leq d_{H}\mbox{\quad
and\quad}\limsup_{n\to\infty}|\gamma_{n}|^{\frac{1}{n}}\leq d_{H}\ ,$ (135)

where $\limsup$ denotes the limit superior. In the following, we shall find it
useful to re-scale the coefficients $\omega_{n}$ and $\gamma_{n}$ by
$d_{H}^{n}$ 151515For the purpose of a large $\chi$ expansion (see Section
5.3), one could equally rescale by powers of the classical horizon distance
$\pi\chi$.

$\omega_{n}=\bar{\omega}_{n}\,d_{H}^{n}\mbox{\quad
and\quad}\gamma_{n}=\bar{\gamma}_{n}\,d_{H}^{n}\ ,\quad\forall n\in\mathbb{N}\
,$ (136)

where $\bar{\omega}_{n}$ and $\bar{\gamma}_{n}$ are coefficients of a series
with radius of convergence $\geq 1$.

The position of the horizon $z_{H}$ is determined by

$h(z_{H})=1-\frac{2\chi}{z_{H}}\left(1+\sum_{n=1}^{\infty}\bar{\gamma}_{n}\right)=0\mbox{\quad
and\quad}f(z_{H})=1-\frac{2\chi}{z_{H}}\left(1+\sum_{n=1}^{\infty}\bar{\omega}_{n}\right)=0\
,$ (137)

which leads to the following relation for the (convergent) series

$\sum_{n=1}^{\infty}\bar{\omega}_{n}=\sum_{n=1}^{\infty}\bar{\gamma}_{n}\
=\frac{z_{H}}{2\chi}-1\ .$ (138)

We also have

$\Phi_{H}=\log\sum_{n=0}^{\infty}\bar{\omega}_{n}\mbox{\quad
and\quad}\Psi_{H}=\log\sum_{n=0}^{\infty}\bar{\gamma}_{n}\ .$ (139)

The first derivative of $f$ for $z\geq z_{H}$ reads

$\displaystyle
f^{(1)}(z):=\derivative{f}{z}=\frac{1}{z}\left(1-f(z)+\frac{2\chi}{\sqrt{{f(z)}}}\sum_{n=1}^{\infty}\frac{n\omega_{n}}{d(z)^{n+1}}\right)\
.$ (140)

Even assuming (135), such that the sum is convergent for all values of $d(z)$
up to the horizon, this equation is still divergent at $z=z_{H}$ due to the
$\sqrt{f(z)}$ in the denominator. A similar problem occurs for the first
derivative of $h$. Concretely, expanding both derivatives $f^{(1)}$ and
$h^{(1)}$ in terms of $\rho$ (similar to eq. (27)), we obtain

$\displaystyle f^{(1)}$
$\displaystyle=\frac{1}{z_{H}}\left(1+\frac{4\chi}{\rho\,d_{H}f^{(1)}_{H}}\sum_{n=1}^{\infty}n\,\bar{\omega}_{n}-\frac{2}{d_{H}^{2}f^{(1)}_{H}}\sum_{n=1}^{\infty}n(n+1)\,\bar{\omega}_{n}+\mathcal{O}(\rho)\right)\
,$ $\displaystyle h^{(1)}$
$\displaystyle=\frac{1}{z_{H}}\left(1+\frac{4\chi}{\rho\,d_{H}f^{(1)}_{H}}\sum_{n=0}^{\infty}n\,\bar{\gamma}_{n}-\frac{4\chi}{d_{H}^{2}f^{(1)}_{H}}\sum_{n=0}^{\infty}n(n+1)\,\bar{\gamma}_{n}+\mathcal{O}(\rho)\right)\
,$ (141)

which contain terms of order $\rho^{-1}\sim(z-z_{H})^{-1/2}\sim f^{-1/2}$ that
become divergent at the horizon. These can be removed by requiring

$\displaystyle\sum_{n=1}^{\infty}n\,\bar{\omega}_{n}=0\,,$ and
$\displaystyle\sum_{n=0}^{\infty}n\,\bar{\gamma}_{n}=0\,.$ (142)

From the remainder of (141) we now find

$\displaystyle
f^{(1)}_{H}=\frac{1}{2z_{H}}\left(1+\sqrt{1-\frac{16z_{H}\chi}{d_{H}^{2}}\sum_{n=1}^{\infty}n^{2}\,\bar{\omega}_{n}}\right)\,,$
and $\displaystyle
h^{(1)}_{H}=\frac{1}{z_{H}}-\frac{4\chi}{z_{H}d_{H}^{2}f^{(1)}_{H}}\sum_{n=0}^{\infty}n^{2}\,\bar{\gamma}_{n}\
,$ (143)

where we have again chosen a solution for $f^{(1)}_{H}$ which corresponds to
the Schwarzschild geometry for $\omega_{n}\to 0$ $\forall n\geq 1$. Comparing
to the first equation in eq. (30), we have

$\displaystyle\Phi^{(2)}_{H}=\frac{2\chi
d_{H}^{2}}{z_{H}}\sum_{n=1}^{\infty}n^{2}\,\bar{\omega}_{n}\,,$ and
$\displaystyle\Psi^{(2)}_{H}=\frac{2\chi
d_{H}^{2}}{z_{H}}\sum_{n=1}^{\infty}n^{2}\,\bar{\gamma}_{n}\,.$ (144)

###  5.2 Horizon constraints

In order to make contact with the regularity condition (99) in Section 3.1, we
first need to express the coefficients $\xi_{n}$ in (47) in terms of the
$\bar{\omega}_{n}$. For $\rho\in[0,d_{H})$ we can write

$\displaystyle\frac{1}{2\chi}\sum_{p=0}^{\infty}\xi_{p}\,\rho^{p}$
$\displaystyle=1+\sum_{n=1}^{\infty}\frac{\omega_{n}}{(d_{H}+\rho)^{n}}=1+\sum_{n=1}^{\infty}\bar{\omega}_{n}\,\left(\sum_{k=0}^{\infty}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}\,.$
(145)

In order to extract the term of order $\rho^{p}$ on the right-hand side of
this equation, we use relation (196), such that

$\displaystyle\xi_{0}=2\chi\left(1+\sum_{n=1}^{\infty}\bar{\omega}_{n}\right)=z_{H}\,,$
$\displaystyle\xi_{p}=\frac{2\chi}{(-d_{H})^{p}p!}\sum_{n=1}^{\infty}\bar{\omega}_{n}\,\frac{(n+p-1)!}{(n-1)!}\hskip
14.22636pt\forall p\geq 1\,.$ (146)

A similar analysis allows to express the coefficients $\theta_{p}$ in eq. (84)
in terms of the $\bar{\gamma}_{n}$

$\displaystyle\theta_{0}=2\chi\left(1+\sum_{n=1}^{\infty}\bar{\gamma}_{n}\right)=z_{H}\,,$
$\displaystyle\theta_{p}=\frac{2\chi}{(-d_{H})^{p}p!}\sum_{n=1}^{\infty}\bar{\gamma}_{n}\,\frac{(n+p-1)!}{(n-1)!}\hskip
14.22636pt\forall p\geq 1\,.$ (147)

Furthermore, the quantity $\varpi$ in (56) takes the form

$\displaystyle\varpi=\sqrt{1-\frac{16\chi
z_{H}}{d_{H}^{2}}\sum_{n=1}^{\infty}n(n+1)\bar{\omega}_{n}}=\sqrt{1-\frac{16\chi
z_{H}}{d_{H}^{2}}\sum_{n=1}^{\infty}n^{2}\bar{\omega}_{n}}\,,$ (148)

where in the last relation we have used (142). More generally, the conditions
(36) which guarantee the finiteness of the first and second derivatives of the
metric functions at the horizon, translate into

$\displaystyle\sum_{n=1}^{\infty}n\,\bar{\omega}_{n}=0=\sum_{n=1}^{\infty}n^{2}(n+3)\,\bar{\omega}_{n}\,,\hskip
42.67912pt\sum_{n=1}^{\infty}n\,\bar{\gamma}_{n}=0=\sum_{n=1}^{\infty}n^{2}(n+3)\,\bar{\gamma}_{n}\,,$
$\displaystyle\hskip
59.75095pt\sum_{n=1}^{\infty}n^{2}\,\bar{\omega}_{n}\leq\frac{d_{H}^{2}}{16z_{H}\chi}\,,\hskip
45.52458pt\sum_{n=1}^{\infty}n^{2}\,\bar{\gamma}_{n}<\frac{(1+\varpi)d_{H}^{2}}{8\chi
z_{H}}\,.$ (150)

In Appendix B, we delve into the consequences of imposing the condition that
the function $f(z)$ belongs to the class $C^{N}(\Sigma)$, where $\Sigma$ is
the submanifold defined by $z\geq z_{H}$ and $N\geq 2$. We also remark that
the more general conditions (99) can be translated into conditions for the
$\omega_{n}$ and $\gamma_{n}$, using the identifications (146) and (147).
Furthermore, Appendix B provides a minimal solution for the system of
equations obtained by truncating the series in (134) after $N$ terms while
imposing regularity of $N$ derivatives of the metric functions at the horizon.
It also discusses the limit $N\to\infty$.

###  5.3 Thermodynamics

Next we consider the Hawking temperature (39): using expressions (143) and
eliminating $z_{H}$ through the relations (138) we have

$\displaystyle T_{\mathrm{H}}=$
$\displaystyle\frac{1}{8\sqrt{2}\pi\chi\left(1+\sum_{n=1}^{\infty}\bar{\omega}_{n}\right)}\times$
$\displaystyle\times\sqrt{1-\frac{16\chi^{2}}{d_{H}^{2}}\left(1+\sum_{n=1}^{\infty}\bar{\gamma}_{n}\right)\left(\sum_{n=1}^{\infty}n^{2}\bar{\gamma}_{n}\right)+\sqrt{1-\frac{32\chi^{2}}{d_{H}^{2}}\left(1+\sum_{n=1}^{\infty}\bar{\omega}_{n}\right)\left(\sum_{n=1}^{\infty}n^{2}\bar{\omega}_{n}\right)}}\
,$ (152)

which is still complicated to evaluate directly. However, we can gain more
intuition into this temperature by expanding for large mass $\chi\gg 1$. In
this case, we expect that the distance of the horizon compared to the
classical case is only modified by subleading terms, _i.e._
$d_{H}=\pi\chi+\mathfrak{o}(\chi)$, which we assume to also hold true for the
position of the horizon itself: $z_{H}=2\chi+\mathfrak{o}(\chi)$. With
relation (138) this implies

$2\chi\sum_{n=1}^{\infty}\bar{\omega}_{n}=\mathfrak{o}(\chi)=2\chi\sum_{n=1}^{\infty}\bar{\gamma}_{n}\
,\mbox{\quad\emph{i.e.}\quad}\lim_{\chi\to\infty}\sum_{n=1}^{\infty}\bar{\omega}_{n}=0=\lim_{\chi\to\infty}\sum_{n=1}^{\infty}\bar{\gamma}_{n}\
.$ (153)

In the following, we shall furthermore assume that this is due to a genuine
scaling property of the coefficients $\bar{\omega}_{n}$ and $\bar{\gamma}_{n}$
such that

$\lim_{\chi\to\infty}\sum_{n=1}^{\infty}n^{r}\,\bar{\omega}_{n}=0=\lim_{\chi\to\infty}\sum_{n=1}^{\infty}n^{r}\,\bar{\gamma}_{n}\
,\quad\forall r\in\mathbb{N}\ .$ (154)

Under these assumptions, the leading correction to the Hawking temperature
arises from the terms

$T_{\mathrm{H}}=\frac{1}{8\pi\chi}\left[1-\frac{1}{\pi^{2}}\sum_{n=1}^{\infty}\left(4n^{2}(\bar{\omega}_{n}+\bar{\gamma}_{n})+\pi^{2}\bar{\omega}_{n}\right)+\ldots\right]\
.$ (155)

As an example, suppose that the proper distance can be expanded for large
masses as

$\displaystyle
d_{H}=\pi\chi\left(1+\frac{y_{1}}{\sqrt{\chi}}+\mathcal{O}\left(\frac{1}{\chi}\right)\right)\
,$ with $\displaystyle y_{1}\in\mathbb{R}\,.$ (156)

We then find for particular cases

  * •

for $(\omega_{1},\gamma_{1})\neq(0,0)$

$T_{\mathrm{H}}=\frac{1}{8\pi\chi}\left(1-\frac{\pi^{2}\omega_{1}+4(\gamma_{1}+\omega_{1})}{\pi^{3}\chi}+\frac{\left(\pi^{2}\omega_{1}+12(\gamma_{1}+\omega_{1})\right)y_{1}}{\pi^{3}\chi^{3/2}}+\mathcal{O}\left(\frac{1}{\chi^{2}}\right)\right)\
.$ (157)

  * •

for $\omega_{1}=0=\gamma_{1}$ and $(\omega_{2},\gamma_{2})\neq(0,0)$

$T_{\mathrm{H}}=\frac{1}{8\pi\chi}\left(1-\frac{\pi^{2}\omega_{2}+16(\gamma_{2}+\omega_{2})}{\pi^{4}\chi^{2}}+\frac{\left(2\pi^{2}\omega_{2}+64(\gamma_{2}+\omega_{2})\right)y_{1}}{\pi^{4}\chi^{5/2}}+\mathcal{O}\left(\frac{1}{\chi^{3}}\right)\right)\
.$ (158)

We observe that the leading correction to the Hawking temperature is
determined solely by the classical term in the expansion (156) of the proper
distance, with no dependence on higher-order terms.

To compute the entropy, we resort to the first law of thermodynamics (41), and
we consider the two cases examined above

  * •

for $(\omega_{1},\gamma_{1})\neq(0,0)$

$S=4\pi\chi^{2}\left(1+\frac{8\gamma_{1}+2(4+\pi^{2})\omega_{1}}{\pi^{3}\chi}+\right.\\\
\left.+\frac{2(4\gamma_{1}+(4+\pi^{2})\omega_{1})^{2}}{\pi^{6}\chi^{2}}\log(\pi^{3}\chi-4\gamma_{1}-(4+\pi^{2})\omega_{1})+\mathcal{O}\left(\frac{1}{\chi^{3}}\right)\right)\
.$ (159)

  * •

for $\omega_{1}=0=\gamma_{1}$ and $(\omega_{2},\gamma_{2})\neq(0,0)$

$S=4\pi\chi^{2}\left(1+\frac{(16\gamma_{2}+(16+\pi^{2})\omega_{2})^{2}}{\pi^{4}\chi^{2}}\log(\pi^{4}\chi^{2}-16\gamma_{2}-(16+\pi^{2})\omega_{2})+\mathcal{O}\left(\frac{1}{\chi^{3}}\right)\right)\
.$ (160)

We remark that this result corrects the mass expansion of the entropy provided
in the previous paper [1]. There, rather than using a self-consistent approach
as in the current work, an approximation of the distance function (based on
the distance function of the Schwarzschild black hole) was considered. This
leads to a different conclusion for the subleading corrections of the entropy
(as well as the Hawking temperature (158)).

##  6 Conclusions

In this paper, we have provided regularity conditions for generic deformations
of static and spherically symmetric black hole metrics. Following [1, 5] we
have considered deformations of the radially-symmetric and static
Schwarzschild space-time, which are described by corrections of the metric
functions as in eq. (8). In order to remain invariant under the same
coordinate reparametrisations as the classical geometry, it has been proposed
[1, 5] that these deformations are not arbitrary functions of the radial
variable $z$, but only depend on a physical distance. Focusing on the exterior
of the black hole (_i.e._ outside of its event horizon), we have chosen the
latter to be the proper distance $\rho$ measured from the horizon.161616Other
choices shall be considered in upcoming work [57] (see also [29]). Since
$\rho$ is defined through the metric function $f$ itself (concretely through
the differential equation (43)), it needs to be computed in a self-consistent
fashion. In this paper we have solved this problem in a region just outside of
the event horizon (which we assume to be located at $z_{H}$ with distance
$d_{H}$) in two different fashions

  1. 1.

In a first approach (see Section 2.1) we have assumed that the first and
second derivative (with respect to the radial coordinate $z$) of both $f$ and
$h$ at $z_{H}$ are finite such that both functions can be approximated by
their Taylor polynomials (19) and (20) (which also affords a finite expansion
of the distance function in eq.(22)). On the one hand, since the derivative of
$\rho$ with respect to $z$ is divergent at the horizon, these assumptions lead
to non-trivial conditions on the deformations of the metric functions as shown
in eq. (36).171717We have generalised these conditions in Appendix A by
assuming arbitrarily high derivatives of $f$ to remain finite at the event
horizon. On the other hand, these also ensure that important physical
quantities are well behaved: indeed, the existence of the first derivative of
$f$ and $h$ (at the horizon) is required for the finiteness of the Hawking
temperature and the existence of the second derivatives guarantees that the
Ricci-scalar (and Kretschmann scalar) are finite (such that the black hole is
free of curvature singularities) at the horizon.

  2. 2.

The second approach (see Section 3) is schematically summarised in Figure 3:
we assume that (outside of the horizon of the black hole) the deformations of
the metric functions allow for a series expansion in $\rho$, as in eq. (47)
and (84) respectively. Taking the expansion coefficients $\\{\xi_{n}\\}$ and
$\\{\theta_{n}\\}$ respectively, as well as $d_{H}$ as input, we look for a
solution of the non-linear differential equation (44) (which is equivalent to
(43)) in the form of an integer series: more precisely, writing the radial
coordinate $z$ as an integer series in powers of $\rho$ (see eq. (42)) we
determine all coefficients $\\{a_{n}\\}$ recursively in terms of the
coefficients $\\{\xi_{n}\\}$ and $d_{H}$ (see eq. (56) and (57)). Through
series reversion, this allows to write $\rho$ as the series (58) in half-
integer powers of $(z-z_{H})$, whose coefficients $\\{b_{n}\\}$ are given in
(59). These expressions finally allow us to compute the Ricci scalar (as well
as other physical quantities) in the vicinity of the black hole horizon.
Finally, the consistency of this procedure, as well as the absence of a
singularity of the Ricci scalar at the horizon impose the non-trivial
conditions (99) on the expansion coefficients $\\{\xi_{n}\\}$ and
$\\{\theta_{n}\\}$ of the deformation of the metric functions.

$d_{H}$$\\{\xi_{n}\\}$, $\\{\theta_{n}\\}$$\\{\omega_{n}\\}$,
$\\{\gamma_{n}\\}$$e^{\Phi}$, $e^{\Psi}$eq.(47)and (84)eq.(146)and
(147)inputsolvingeq.(44)distance function$\\{a_{n}\\}$eq.(56),
(57)seriesreversion$\\{b_{n}\\}$eq.(59)physical quantitiesCurvature$R$
eq.(17)Hawkingtemperature$T_{\rm H}$ eq.(39)Entropy$S$ eq.(41)conditions
eq.(99)

Figure 3: _Schematic overview of the approach of Section 3 to compute the
distance function $\rho$ and physical quantities of the deformed Schwarzschild
geometry._

Both methods yield compatible results. They are based on assuming regularity
of certain quantities (up to a given order) at the horizon and (apart from
$d_{H}$) only require information about the black hole outside of the event
horizon. They allow, however, to derive non-trivial physical quantities of the
geometry (curvature scalars such as the Ricci and Kretschmann scalar) and the
thermodynamics (notably the Hawking temperature and the entropy of the black
hole). We have tested the conditions (36) in the case of the black hole metric
proposed in [5] and find that they are violated: indeed if considered as a
geometry that is self-consistently defined, it leads to a divergent first
derivative of the metric functions at the horizon, which poses problems for
the thermodynamic interpretation, as well as a curvature singularity at the
horizon. While in [5] “approximations” have been proposed that indeed render
the geometry well defined, these correspond to a modification of the metric
deformations that adhere to the conditions we have found in (99). Along these
lines, other examples can be studied and we have discussed a minimal solution
to these conditions.

Finally, making contact with [1], we have considered a (generic) asymptotic
expansion of the metric deformations in inverse powers of the proper distance
(134). Assuming that the radius of convergence of the latter is sufficiently
large such that these series are still valid at the event horizon, we have
used the previous formalism to convert the conditions (36) on the deformation
functions into non-trivial relations among the asymptotic coefficients (150).
We have furthermore also expressed the Hawking temperature in terms of these
coefficients in a consistent manner in eq. (152), thereby correcting previous
approximations in [1].

In this paper, we have established a framework that is applicable to (quantum)
deformations of the classical Schwarzschild space-time in a model-independent
fashion and moreover allows to extract certain physical quantities. We have
established non-trivial conditions for the deformations themselves, which can
be translated into constraints in the context of concrete quantum gravity
models.

One of the most intriguing future directions is to extend our methodology to
include charged and spinning black holes. Generalising our deformation
approach to the classical Kerr and Reissner-Nordstrøm geometries is a natural
next step, which could reveal new insights into the behavior of quantum-
deformed black holes with other hair parameters [14, 13]. Beyond black holes,
our approach is applicable to various other space-times, such as $AdS$ spaces
and even models of cosmology, for example in the context of cosmic inflation
[58, 59, 60, 61]. Additionally, our work opens up intriguing possibilities for
studying the interior of black holes (a first hint of which is provided in
Appendix E) and the fate of the singularity at the origin within the context
of quantum-deformed geometries [62].

From a broader perspective, our framework offers an exciting opportunity for
quantum gravity phenomenology. By systematically extracting physical
quantities and comparing them with observations, we can test and constrain
concrete quantum gravity models, bridging the gap between theoretical concepts
and experimental verifiability [63, 64]. Finally, it is interesting to explore
the applicability of our approach beyond the realm of (quantum) gravity, for
example in the context of dyons and monopoles in gauge theories.181818We thank
Nikita Nekrasov for suggesting this idea to us during the talk of Francesco
Sannino at the First International Congress of Basic Science at the Yanqi Lake
Beijing Institute of Mathematical Sciences and Applications
2023.07.16-2023.07.28.

## Acknowledgements

We are indebted to Emanuele Binetti for collaboration at an early stage of
this work and for helpful discussions and exchanges. We furthermore thank
Aaron Held for useful conversations and exchanges, and Nikita Nekrasov for
relevant comments and suggestions as well as Hong-Jian He for discussions. FS
wishes to thank Shing-Tung Yau for the hospitality and the organisation of the
First International Congress of Basic Science at the Yanqi Lake Beijing
Institute of Mathematical Sciences and Applications 2023.07.16-2023.07.28
where this work was finalised as well as the hospitality of the CERN
theoretical physics department where this work was initiated. MDP expresses
sincere appreciation to the University of Southern Denmark and D-IAS for their
hospitality during the crucial final stages of the work. MDP thanks also the
Galileo Galilei Institute for Theoretical Physics where part of this work was
carried out.

##  Appendix A Regular higher order derivatives

In Section 2.1 we have derived the conditions (36) by assuming that the first
and second derivatives of the metric functions $f$ and $h$ are finite at the
horizon $z_{H}$. In this appendix, we explore further conditions that stem
from assuming that also higher derivatives (_i.e._ beyond the second) are
finite. For simplicity, we shall focus on the function $f$, while the same
considerations also apply to $h$. Concretely, let $N\in\mathbb{N}$ and let us
assume that all derivatives $f^{(k)}_{H}$ for $k\in\\{1,\ldots,N\\}$ at
$z=z_{H}$ are finite. This allows us to go beyond (19) and write

$\displaystyle
f(z)=\sum_{k=1}^{N}\frac{f^{(k)}_{H}}{k!}(z-z_{H})^{k}+\mathcal{O}((z-z_{H})^{n+1})\,,$
with $\displaystyle
f^{(k)}_{H}:=\left.\derivative[k]{f}{z}\right|_{z=z_{H}}\,.$ (161)

###  A.1 Distance function

Inserting the expansion (161) into the differential equation (43) yields a
series expansion of $\rho$ in powers of $z-z_{H}$

$\displaystyle\rho=\sum_{k=1}^{2N-1}b_{k}\,(z-z_{H})^{k/2}+\mathcal{O}\left((z-z_{H})^{N}\right)\,,$
(162)

which generalises (22). The coefficients $b_{k}$ of this series can be found
as the solutions of the differential equation (43) up to order $N$, which we
re-write in the form

$\displaystyle\left(\derivative{z}{\rho}\right)^{2}=\frac{1}{f(z)}\,.$ (163)

The right-hand side of this equation has a simple pole at $z=z_{H}$ and we can
write the Laurent series expansion

$\displaystyle\frac{1}{f(z)}=\sum_{m=-1}^{N-2}\ell_{m}\,(z-z_{H})^{m}+\mathcal{O}((z-z_{H})^{N-1})\,,$
with $\displaystyle\ell_{m}\in\mathbb{R}\,.$ (164)

Multiplying both sides of this equation by $(z-z_{H})$ and taking the limit
$z\to z_{H}$, we find for the leading coefficient $\ell_{-1}=1/f^{(1)}_{H}$.
To extract the remaining coefficients, we consider the relation

$\displaystyle
f(z)\,\derivative{z}\frac{1}{f(z)}=-\frac{1}{f(z)}\,\derivative{f}{z}$ (165)

and expand both sides in powers of $(z-z_{H})$. Comparing order by order we
then find

$\displaystyle\ell_{0}$
$\displaystyle=-\frac{\ell_{-1}f^{(2)}_{H}}{2f^{(1)}_{H}}=-\frac{f^{(2)}_{H}}{2(f^{(1)}_{H})^{2}}\,,$
$\displaystyle\ell_{p}$
$\displaystyle=\frac{-1}{(p+1)f^{(1)}_{H}}\left[\frac{(p+1)\ell_{-1}\,f^{(p+2)}_{H}}{(p+2)!}+\sum_{k=1}^{p-1}\frac{k\ell_{k}f^{(p-k+1)}_{H}}{(p-k+1)!}+\sum_{k=2}^{p+1}\frac{\ell_{p-k+1}f^{(k)}_{H}}{(k-1)!}\right]\,,\hskip
5.69046pt1\leq p\leq N-2\,,$ (166)

which fixes the coefficients $\ell_{p}$ iteratively (in terms of the
$f^{(n)}_{H}$). Inserting (164) into (163), we find the following recursive
structure for the coefficients $b_{p}$

$\displaystyle b_{1}=2\sqrt{\ell_{-1}}=\frac{2}{\sqrt{f^{(1)}_{H}}}\hskip
56.9055ptb_{2}=0\,,$ $\displaystyle
b_{p}=\left\\{\begin{array}[]{lcl}\frac{2}{(2s+3)b_{1}}\left[\ell_{p}-\sum_{n=2}^{2s+2}\frac{n(2s-n+4)}{4}\,b_{n}\,b_{2s-n+4}\right]&\text{if}&p=2s+3\in\mathbb{N}_{\text{odd}}\,,\\\
-\frac{1}{(s+1)b_{1}}\sum_{n=2}^{2s+1}\frac{n(2s-n+3)}{4}b_{n}b_{2s-n+3}&\text{if}&p=2s+2\in\mathbb{N}_{\text{odd}}\,,\end{array}\right.\hskip
2.84544pt3\leq p\leq 2N-1\,,$ (169)

which allows to fix them iteratively in terms of the $f^{(n)}_{H}$. Since
$b_{2}=0$, the recursive structure in (169) implies that $b_{2s}=0$ for
$s\in\\{1,\ldots,N-2\\}$. For the odd coefficients, we find for the first few
instances (for sufficiently large $N$)

$\displaystyle b_{1}=\frac{2}{(f^{(1)}_{H})^{1/2}}\,,$ $\displaystyle
b_{3}=-\frac{f^{(2)}_{H}}{6(f^{(1)}_{H})^{3/2}}\,,$ $\displaystyle
b_{5}=\frac{9(f^{(2)}_{H})^{2}-8f^{(1)}_{H}f^{(3)}_{H}}{240(f^{(1)}_{H})^{5/2}}\,.$
(170)

These results are indeed compatible with (22) for the case $N=2$.

###  A.2 Conditions for the regularity of higher derivatives $f^{(n)}$

The results of the previous Subsection can be used to derive necessary
conditions for the function $e^{\Phi}$ such that the first $N$ derivatives of
the function $f$ (with respect to $z$) are finite, which is required for
(161). Assuming an expansion of the latter of the form (47), we have with
(162)

$\displaystyle
f(z)=1-\frac{1}{z}\sum_{n=0}^{2N}\xi_{n}\,\left(\sum_{k=1}^{2N-1}b_{k}\,(z-z_{H})^{k/2}\right)^{n}+\mathcal{O}\left((z-z_{H})^{N}\right)\,,$
(171)

which is only a function of $z$. Due to the fact that this expression contains
half-integer powers of $(z-z_{H})$, derivatives of $f$ can contain negative
powers unless certain conditions for the coefficients $\xi_{n}$ are satisfied.
To understand these conditions, we first re-write the summation in (171),
taking into account that $b_{2s}=0$ for $s\in\\{1,\ldots,N-2\\}$

$\displaystyle f(z)=1$
$\displaystyle-\frac{1}{z}\sum_{m=0}^{N}\xi_{2m}(z-z_{H})^{m}\left(\sum_{r=0}^{N-1}b_{2r+1}\,(z-z_{H})^{r}\right)^{2m}$
$\displaystyle-\frac{1}{z}\sum_{m=1}^{N}\xi_{2m-1}\left(\sum_{r=0}^{N-1}b_{2r-1}\,(z-z_{H})^{r+1/2}\right)^{2m-1}+\mathcal{O}\left((z-z_{H})^{N}\right)\,.$
(172)

With the expansion of $1/z$

$\displaystyle\frac{1}{z}=\frac{1}{z_{H}}\sum_{n=0}^{N}\frac{(-1)^{n}}{z_{H}^{n}}\,(z-z_{H})^{n}+\mathcal{O}\left((z-z_{H})^{N+1}\right)\,,$
(173)

it is clear that the terms in the first line of (172) contain no half-integer
powers of $(z-z_{H})$ and thus cannot contribute to singularities of
derivatives of $f$ at $z=z_{H}$. The terms in the second line of (172),
however, contain terms that lead to negative powers of $(z-z_{H})$ for
derivatives of $f$. The conditions to eliminate these singular terms for all
$f^{(k)}_{H}$ for $k\in\\{1,\ldots,N\\}$ are therefore

$\displaystyle\xi_{2n-1}=0\hskip 28.45274pt\forall n\in\\{1,\ldots,N\\}\,.$
(174)

To see this, let $s\in\\{1,\ldots,N\\}$ and assume that $\xi_{2s-1}\neq 0$,
while $\xi_{2m-1}=0$ $\forall m\in\\{1,\ldots,s-1\\}$. In this case,
$f^{(s)}_{H}$ has a singularity of the form $(z-z_{H})^{-1/2}$

$\displaystyle\derivative[s]{z}f(z)$
$\displaystyle=-\frac{\xi_{2s-1}}{z_{H}}\,b_{1}^{2s-1}\,\derivative[s]{z}(z-z_{H})^{s-1/2}+\mathcal{O}((z-z_{H})^{0})$
$\displaystyle=-\frac{(2s-1)!!}{2^{s}z_{H}}\,\xi_{2s-1}\,b_{1}^{2s-1}\,(z-z_{H})^{-1/2}+\mathcal{O}((z-z_{H})^{0})\,.$
(175)

Absence of this singular contribution therefore requires $\xi_{2s-1}=0$.

Using (50), the conditions (174) can also be rewritten as conditions for the
derivatives $\Phi^{(n)}_{H}$. For completeness, we exhibit the first few such
conditions (for sufficiently large $N$)

$\displaystyle\Phi^{(1)}_{H}=0\,,$
$\displaystyle\Phi^{(3)}_{H}=-6\,d_{H}\,\Phi^{(2)}_{H}\,,$
$\displaystyle\Phi^{(5)}_{H}=480d_{H}^{3}\Phi^{(2)}_{H}-20d_{H}\Phi^{(4)}_{H}\,,$
(176)

which are indeed compatible with (36).

##  Appendix B Towards a fully regular solution

In this Appendix, we consider a particular solution for the consistency
conditions found at the end of the previous Appendix. We consider a metric
function $f(z)$ of the form (8), with deformation $e^{\Phi(z)}$, which is
$N$-times differentiable (with $N\in\mathbb{N}$) in the region $z\geq z_{H}$,
_i.e._ outside of the horizon of the black hole. This allows us to define the
Taylor polynomial of $f$ for large $z$

$f_{P}(z)=1-\frac{2\chi}{z}\left(1+\theta\left(\frac{1}{d(z)}\right)\right)\mbox{\quad
with\quad}\theta\left(\frac{1}{d(z)}\right)=\sum_{n=1}^{N}\frac{\omega_{n}}{d(z)^{n}}\
,$ (177)

which we shall assume to be a satisfactory approximation of $f$ for all $z\geq
z_{H}$. In this case, the consistency conditions at the horizon take the form
of the following linear system

$\sum_{n=1}^{N}\bar{\omega}_{n}=\frac{z_{H}(N)-2\chi}{2\chi}\mbox{\quad
and\quad}\sum_{n=1}^{N}\frac{(n+2k)!}{(n-1)!}\,\bar{\omega}_{n}=0\
,\quad\forall k\in\\{0,\ldots,N-2\\}\ .$ (178)

where we have used the rescaled coefficients $\bar{\omega}_{n}$ defined in
(136). Furthermore, $z_{H}(N)$ is the position of the (external) event horizon
computed from the Taylor polynomial (177), which is therefore implicitly a
function of $N$.

For fixed $N$, the system of equations (178) uniquely fixes the coefficients
$\bar{\omega}_{n}$, for $n=1,\ldots,N$, and we have found empirically

$\bar{\omega}_{n}=\frac{z_{H}(N)-2\chi}{2\chi}\,\alpha_{n}(N)\ ,$ (179)

with

$\hskip 28.45274pt\alpha_{n}(N)=\sum_{s=0}^{n-1}(-1)^{s}\begin{pmatrix}n-1\\\
s\end{pmatrix}\frac{\sqrt{\pi}\,\Gamma(N+\tfrac{s}{2})}{\Gamma(N-\tfrac{1}{2})\Gamma(\tfrac{s}{2}+1)}\
,\quad\forall n\in\\{1,\ldots,N\\}\ ,$ (180)

which we have tested up to $N=250$. The individual coefficients $\alpha_{n}$
are plotted for low values of $n$ in the left panel of Fig. 4.

Figure 4: _Left panel: coefficients $\alpha_{n}$ for low values of $n$. The
black dots represent $\text{max}(|\alpha_{1}|,\ldots,|\alpha_{N}|)$, while the
black dashed curve represents the enveloping value according to eq. (183).
Right panel: evaluation of $|\alpha_{n_{\text{max}}}|$ in eq. (182). The
dashed line represents the value $3/\sqrt{5}$._

For $N\to\infty$ individual coefficients are divergent, for example, the
leading contribution behaves as

$\alpha_{n}\sim(-1)^{n-1}\sqrt{\pi}\,\frac{N^{n/2}}{\Gamma\left(\frac{n+1}{2}\right)}\mbox{\quad
for\quad}n\ll N\ .$ (181)

For given $N$, the largest coefficient (in terms of its absolute value)
appears to be

$\text{max}(|\alpha_{1}|,\ldots,|\alpha_{N}|)=\left|\alpha_{n_{\text{max}}}\right|\mbox{\quad
with\quad}n_{\text{max}}=\left\lfloor\frac{N-1}{3}\right\rfloor+1\ .$ (182)

Figure 5: _Function $\theta$ in eq. (177) for different $N$ and for the choice
$\delta_{0}=4/3$._

Numerical evaluations (see right panel of Fig. 4) suggest that this
coefficient behaves as

$\left|\alpha_{n_{\text{max}}}\right|\sim\alpha_{0}\,\left(\frac{4}{3}\right)^{N}\mbox{\quad
with\quad}\alpha_{0}\approx\frac{3}{\sqrt{5}}\ .$ (183)

Therefore, for (178) to make sense also for large $N$, _i.e._ to yield finite
$\bar{\omega}_{n}$, the difference $z_{H}(N)-2\chi$ needs to tend to zero as
well. Indeed, divergent coefficients $\bar{\omega}_{n}$ would indicate the
non-existence of the derivatives of $f$ up to order $N$ and therefore
contradict our initial assumptions. For example, using (183), in order for all
the coefficients $\bar{\omega}_{n}$ (for $n\in\\{1,\ldots,N\\}$) to remain
finite (and not tend to $0$), we may choose the following asymptotic form for
the difference of $z_{H}$ and the classical position of the horizon

$\frac{z_{H}(N)-2\chi}{2\chi}\sim\delta_{0}\,\left(\frac{3}{4}\right)^{N}\ ,$
(184)

for some constant $\delta_{0}\in\mathbb{R}$. The results are shown in Figure
5. Indeed, for large $N$, _i.e._ for a metric function that is infinitely
differentiable for all $d\geq d_{H}$, the solution simply approaches
$\theta\to 0$, _i.e._ simply leads to the Schwarzschild black hole.

Notice, that through (184), the condition of an infinitely differentiable
metric function outside of the horizon, imposes a non-trivial condition for
the position of the horizon. Through

$d_{H}=\int_{0}^{z_{H}}\frac{\differential z}{\sqrt{|f(z)|}}\mbox{\quad
and\quad}f(z_{H})=0\ ,$ (185)

the latter in principle enters also into relations containing the metric
function inside the black hole horizon.

##  Appendix C Series relations

For the reader’s convenience, we compile several series identities in this
Appendix, which are too lengthy to be presented in the main body of the paper.

###  C.1 Series expansion of inverse radial coordinate

In this Appendix, starting from the series (42), we express the inverse radial
coordinate as a series expansion in $\rho$. More precisely, we determine the
coefficients $\mathfrak{p}_{m}$ in

$\displaystyle
P(\rho):=\frac{1}{z(\rho)}=\sum_{m=0}^{\infty}\mathfrak{p}_{m}\,\rho^{m}\,,\,$
with $\displaystyle\begin{array}[]{l}\mathfrak{p}_{m}\in\mathbb{R}\hskip
14.22636pt\forall m\in\mathbb{N}\,,\\\ \mathfrak{p}_{0}=1/z_{H}\,.\end{array}$
(188)

Differentiating both sides with respect to $\rho$ and multiplying by $z$, we
find

$\displaystyle z(\rho)\,P(\rho)^{\prime}$
$\displaystyle=-P(\rho)\,\derivative{z}{\rho}\,,$ (189)

which with (42) becomes the following series identity

$\displaystyle\left(z_{H}+\sum_{n=2}^{\infty}a_{n}\,\rho^{n}\right)\left(\sum_{m=1}^{\infty}m\,\mathfrak{p}_{m}\,\rho^{m-1}\right)=-\left(\sum_{m=0}^{\infty}\mathfrak{p}_{m}\,\rho^{m}\right)\left(\sum_{n=2}^{\infty}n\,a_{n}\,\rho^{n-1}\right)\,.$
(190)

Re-arranging both sides of this equation, we find

$\displaystyle
0=z_{H}\,\mathfrak{p}_{1}+2(z_{H}\,\mathfrak{p}_{2}+a_{2}\,\mathfrak{p}_{0})\,\rho+\sum_{p=2}^{\infty}(p+1)\,\left[z_{H}\,\,\mathfrak{p}_{p+1}+\mathfrak{p}_{0}\,a_{p+1}+\sum_{m=1}^{p-1}\mathfrak{p}_{m}\,a_{p+1-m}\right]\,\rho^{p}\,.$
(191)

Order by order we therefore obtain the relations

$\displaystyle\mathfrak{p}_{1}=0\,,$
$\displaystyle\mathfrak{p}_{2}=-\frac{a_{2}}{z_{H}^{2}}\,,$
$\displaystyle\mathfrak{p}_{p}=-\frac{a_{p}}{z_{H}^{2}}-\frac{1}{z_{H}}\sum_{m=1}^{p-2}\,\mathfrak{p}_{m}\,a_{p-m}\hskip
14.22636pt\forall p\geq 3\,.$ (192)

###  C.2 Power of a power series

For $\rho\in[0,d_{H})$ and $n\in\mathbb{N}$, we consider the following power
series

$\displaystyle\left(\sum_{k=0}^{\infty}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}=\sum_{p=0}^{\infty}\mathfrak{c}_{p}\,\rho^{p}\,.$
(193)

For the coefficients $\mathfrak{c}_{n}$ we find the following explicit
expression

$\displaystyle\mathfrak{c}_{p}:=\left(\sum_{k=0}^{\infty}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}\bigg{|}_{\rho^{p}}=\left(-\frac{1}{d_{H}}\right)^{p}\,\left(\begin{array}[]{c}n+p-1\\\
p\end{array}\right)=\left(-\frac{1}{d_{H}}\right)^{p}\frac{(n+p-1)!}{p!(n-1)!}\,.$
(196)

For $p=0$, we indeed have $\mathfrak{c}_{0}=1$. In order to show (196) for
$p\in\mathbb{N}$, we begin by demonstrating

$\displaystyle\left(\sum_{k=0}^{p-\ell}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{\ell}\bigg{|}_{\rho^{p-\ell}}=\left(-\frac{1}{d_{H}}\right)^{p-\ell}\,\frac{(p-1)!}{(p-\ell)!(\ell-1)!}\,,$
$\displaystyle\begin{array}[]{l}\forall p\in\mathbb{N}\,,\\\
\forall\ell\in\\{0,\ldots,p\\}\,.\end{array}$ (199)

For $p\geq 1$ it can be verified for all (finitely) many values of
$\ell\in\\{0,\ldots,p\\}$, concretely:

  * •

for $\ell=0$ both sides are vanishing, since we use the convention
$1/((-1)!)\to 0$.

  * •

for $\ell=1$ we find directly

$\displaystyle\sum_{k=0}^{p}\left(-\frac{\rho}{d_{H}}\right)^{k}\bigg{|}_{\rho^{p-1}}=\left(-\frac{1}{d_{H}}\right)^{p-1}\,,$
(200)

which indeed agrees with (199).

  * •

for general $2\leq\ell\leq p$ we have

$\displaystyle\left(\sum_{k=0}^{p-\ell}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{\ell}\bigg{|}_{\rho^{p-\ell}}=\sum_{u_{2,\ldots,\ell}=0}^{p-\ell}\left(\sum_{k_{1}=0}^{p-\ell-
u_{2}-\ldots-
u_{\ell}}\left(-\frac{\rho}{d_{H}}\right)^{k_{1}}\bigg{|}_{\rho^{p-\ell-
u_{2}-\ldots-u_{\ell}}}\right)$ $\displaystyle\hskip
5.69046pt\times\left(\sum_{k_{2}=0}^{u_{2}}\left(-\frac{\rho}{d_{H}}\right)^{k_{2}}\bigg{|}_{\rho^{u_{2}}}\right)\times\ldots\times\left(\sum_{k_{\ell}=0}^{u_{\ell}}\left(-\frac{\rho}{d_{H}}\right)^{k_{\ell}}\bigg{|}_{\rho^{u_{\ell}}}\right)=\left(-\frac{1}{d_{H}}\right)^{p-\ell}\,\sum_{{u_{2,\ldots,\ell}=0}\atop{u_{2}+\ldots+u_{\ell}\leq
p-\ell}}^{p-\ell}1\,.$

The last summation can be re-written in the form

$\displaystyle\sum_{{u_{2,\ldots,\ell}=0}\atop{u_{2}+\ldots+u_{\ell}\leq
p-\ell}}^{p-\ell}1=\sum_{k_{1}=0}^{p-\ell}\sum_{{u_{3,\ldots,\ell}=0}\atop{u_{3}+\ldots+u_{\ell}\leq
k_{1}}}1=\sum_{k_{1}=0}^{p-\ell}\sum_{k_{2}=0}^{k_{1}}\ldots\sum_{k_{\ell-1}=0}^{k_{\ell-2}}1=\sum_{k_{1}=0}^{p-\ell}\sum_{k_{2}=0}^{k_{1}}\ldots\sum_{k_{\ell-2}=0}^{k_{\ell-3}}\left(\begin{array}[]{c}k_{\ell-2}+1\\\
k_{\ell-2}\end{array}\right)\,,$ (203)

which, upon using the identity

$\displaystyle\sum_{u=0}^{k}\left(\begin{array}[]{c}u+s\\\
u\end{array}\right)=\left(\begin{array}[]{c}k+s+1\\\ k\end{array}\right)$
$\displaystyle\forall s\in\mathbb{N}\,,$ (208)

leads to

$\displaystyle\sum_{{u_{2,\ldots,\ell}=0}\atop{u_{2}+\ldots+u_{\ell}\leq
p-\ell}}^{p-\ell}1=\left(\begin{array}[]{c}p-1\\\
p-\ell\end{array}\right)=\frac{(p-1)!}{(p-\ell)!(\ell-1)!}\,.$ (211)

This result therefore indeed demonstrates (199).

With the result (199) we can prove (196) (for $p\geq 1$). To this end, we
consider

$\displaystyle\left(\sum_{k=0}^{\infty}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}\bigg{|}_{\rho^{p}}=\left(\sum_{k=0}^{p}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}\bigg{|}_{\rho^{p}}=\sum_{\ell=0}^{n}\left(\begin{array}[]{c}n\\\
\ell\end{array}\right)\left(\sum_{k=1}^{p}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{\ell}\bigg{|}_{\rho^{p}}$
(214) $\displaystyle\hskip
56.9055pt=\sum_{\ell=0}^{n}\left(-\frac{\rho}{d_{H}}\right)^{\ell}\left(\begin{array}[]{c}n\\\
\ell\end{array}\right)\left(\sum_{k=0}^{p-1}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{\ell}\bigg{|}_{\rho^{p}}$
(217) $\displaystyle\hskip
56.9055pt=\sum_{\ell=1}^{n}\left(-\frac{\rho}{d_{H}}\right)^{\ell}\left(\begin{array}[]{c}n\\\
\ell\end{array}\right)\left[\left(\sum_{k=0}^{p-\ell}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{\ell}\bigg{|}_{\rho^{p-\ell}}\right]\,.$
(220)

With the relation (199) we therefore obtain

$\displaystyle\left(\sum_{k=0}^{\infty}\left(-\frac{\rho}{d_{H}}\right)^{k}\right)^{n}\bigg{|}_{\rho^{p}}$
$\displaystyle=\left(-\frac{1}{d_{H}}\right)^{p}\sum_{\ell=1}^{\text{min}(n,p)}\left(\begin{array}[]{c}n\\\
\ell\end{array}\right)\,\frac{(p-1)!}{(p-\ell)!(\ell-1)!}$ (223)
$\displaystyle=\left(-\frac{1}{d_{H}}\right)^{p}\frac{1}{p}\sum_{\ell=1}^{\text{min}(n,p)}\ell\,\left(\begin{array}[]{c}n\\\
\ell\end{array}\right)\left(\begin{array}[]{c}p\\\
\ell\end{array}\right)=\left(-\frac{1}{d_{H}}\right)^{p}\,\left(\begin{array}[]{c}n+p-1\\\
p\end{array}\right)\,,$ (230)

which indeed demonstrates (196).

###  C.3 Series expansions for $\xi_{1}\neq 0$

For completeness, we shall generalise the approach of Section 3.1 also to
accommodate functions $e^{\Phi}$ such that the coefficient $\xi_{1}\neq 0$ in
eq. (47) is non-zero. Indeed, assuming that $z\sim
z_{H}+\mathcal{O}(\rho^{p})$ for some $p\in\mathbb{R}_{+}$, for the left-hand-
side of the equation (44) to have a term of order $\mathcal{O}(\rho)$ requires
either $p=1$ or $p=3/2$. Since we have seen in Section 3.1 that $p=1$ still
requires $\xi_{1}=0$, we shall here explore the case $p=3/2$, _i.e._ instead
of (42) we shall consider the expansion

$\displaystyle
z=z_{H}+\sum_{n=1}^{\infty}\widehat{a}_{n}\,\rho^{\frac{n+2}{2}}\,,$ with
$\displaystyle\widehat{a}_{n}\in\mathbb{R}\hskip 5.69046pt\forall
n\in\mathbb{N}\,,$ (231)

which as before we assume to have an interval of convergence
$\rho\in[0,\rho_{A})$ (with some $\rho_{A}>0$). We then obtain

$\displaystyle z\left(1-\left(\derivative{z}{\rho}\right)^{2}\right)$
$\displaystyle=z_{H}-\frac{9}{4}\,\rho\,z_{H}\,\widehat{a}_{1}^{2}+\rho^{3/2}\left(\widehat{a}_{1}-6\,z_{H}\,\widehat{a}_{1}\,\widehat{a}_{2}\right)+\rho^{2}\,\left[\widehat{a}_{2}-z_{H}\left(4\,\widehat{a}_{2}^{2}+\frac{15}{2}\,\widehat{a}_{1}\,\widehat{a}_{3}\right)\right]$
$\displaystyle\hskip
14.22636pt+\sum_{p=5}^{\infty}\rho^{p/2}\bigg{[}\widehat{a}_{p-2}-z_{H}\sum_{m=1}^{p-1}\frac{(m+2)(p-m+2)}{4}\,\widehat{a}_{m}\,\widehat{a}_{p-m}$
$\displaystyle\hskip
73.97733pt-\sum_{k=1}^{p-4}\widehat{a}_{k}\sum_{m=1}^{p-k-3}\frac{(m+2)(p-k-m)}{4}\,\widehat{a}_{m}\,\widehat{a}_{p-k-m-2}\bigg{]}\,.$
(232)

Comparing the coefficients of $\rho^{0}$, $\rho^{1}$, $\rho^{3/2}$ and
$\rho^{2}$ to (47) yields the relations

$\displaystyle z_{H}=\xi_{0}\,,$
$\displaystyle\xi_{1}=-\frac{9}{4}\,z_{H}\,\widehat{a}_{1}^{2}\,,$
$\displaystyle
0=\widehat{a}_{1}-6\,z_{H}\,\widehat{a}_{1}\,\widehat{a}_{2}\,,$
$\displaystyle\xi_{2}=\widehat{a}_{2}-z_{H}\left(4\,\widehat{a}_{2}^{2}+\frac{15}{2}\,\widehat{a}_{1}\,\widehat{a}_{3}\right)\,,$
(233)

which has solution

$\displaystyle\widehat{a}_{1}=\pm\frac{2}{3}\sqrt{-\frac{\xi_{1}}{z_{H}}}\,,$
$\displaystyle\widehat{a}_{2}=\frac{1}{6z_{H}}\,,$
$\displaystyle\widehat{a}_{3}=\frac{1-18\,z_{H}\,\xi_{2}}{135\,\widehat{a}_{1}\,z_{H}^{2}}\,.$
(234)

The coefficient $\widehat{a}_{1}$ is real only for $\xi_{1}<0$. In the case
$\xi_{1}>0$, there exists no real solution of (44), which is of the form
(231). In the following, we shall assume $\xi_{1}<0$ and furthermore pick the
positive sign for $\widehat{a}_{1}$ in (234): indeed, for the negative sign,
the function $z(\rho)$ would not be monotonically growing for $\rho>0$.

Comparing the remaining terms in (232) order by order, we can express
$\widehat{a}_{p-1}$ in terms of $\widehat{a}_{k}$ with $k<p-1$

$\displaystyle\widehat{a}_{p-1}$
$\displaystyle=\frac{2}{3\,\widehat{a}_{1}\,z_{H}\,(p+1)}\bigg{[}\widehat{a}_{p-2}-z_{H}\sum_{m=2}^{p-2}\frac{(m+2)(p-m+2)}{4}\,\widehat{a}_{m}\,\widehat{a}_{p-m}$
$\displaystyle\hskip
14.22636pt-\sum_{k=1}^{p-4}\widehat{a}_{k}\sum_{m=1}^{p-k-3}\frac{(m+2)(p-k-m)}{4}\,\widehat{a}_{m}\,\widehat{a}_{p-k-m-2}\bigg{]}-\left\\{\begin{array}[]{lcl}0&\text{if}&p\in\mathbb{N}_{\text{odd}}\\\
\frac{2\xi_{p/2}}{3\widehat{a}_{1}\,z_{H}\,(p+1)}&\text{if}&p\in\mathbb{N}_{\text{even}}\end{array}\right.$
(237)

###  C.4 Examples

To showcase the approach developed in Section 3.1, we consider three simple
examples, corresponding to different choices of the coefficients $\xi_{n}$.

####  C.4.1 Schwarzschild distance

The simplest choice is to set $\xi_{0}=2\chi$ and $\xi_{n}=0$ $\forall n>0$,
which corresponds to the Schwarzschild black hole (_i.e._ $e^{\Phi}=1$). In
this case, we also choose $z_{H}=2\chi$ and $d_{H}=\pi\chi$. Using (56) and
(57), the first few coefficients $a_{2n}$ (and their reversions $b_{n}$) can
be tabulated as follows

| $n=1$ | $n=2$ | $n=3$ | $n=4$ | $n=5$ | $n=6$
---|---|---|---|---|---|---
$a_{2n}$ | $\tfrac{1}{8\chi}$ | $-\tfrac{1}{384\chi^{3}}$ | $\tfrac{11}{92160\chi^{5}}$ | $-\tfrac{73}{10321920\chi^{7}}$ | $\tfrac{887}{1857945600\chi^{9}}$ | $-\tfrac{136883}{3923981107200\chi^{11}}$
$b_{2n-1}$ | $2\sqrt{2\chi}$ | $\tfrac{1}{3\sqrt{2\chi}}$ | $-\tfrac{1}{40\chi\sqrt{2\chi}}$ | $\tfrac{1}{224\chi^{2}\sqrt{2\chi}}$ | $-\tfrac{5}{4608\chi^{3}\sqrt{2\chi}}$ | $\tfrac{7}{22528\chi^{4}\sqrt{2\chi}}$

which allows to compute a series expansion of the distance function, as in eq.
(58). These coefficients have previously been obtained in [65, 66] (see also
[67]) in a different context, namely solutions for free-falling bodies in
Newtonian gravity. We have verified up to $(z-2\chi)^{500-1/2}$ that these
coefficients follow the pattern

$\displaystyle
b_{2n-1}=\frac{(-1)^{n}}{\sqrt{2\chi}}\,\frac{(2n-5)!!}{4^{n-2}\chi^{n-2}(2n-1)(n-1)!}\,,$
(238)

such that the series expansion of $\rho$ in terms of $(z-z_{H})^{1/2}$ has an
interval of convergence of $z\in[0,4\chi]$. A graphical example (for $\chi=5$)
is shown in Figure 6, with an expansion up to order $(z-z_{H})^{500-1/2}$. We
have furthermore verified that the coefficients $b_{n}$ agree with a series
expansion of $\rho(z)$: indeed, in the case of the Schwarzschild geometry, the
distance $\rho$ can in fact be computed in closed form as a function of $z$
(see eq. (110))

$\displaystyle\rho(z)=\sqrt{z(z-2\chi)}+2\chi\,\text{arctanh}\sqrt{1-\frac{2\chi}{z}}\mbox{\quad
for\quad}z>2\chi\,,$ (239)

allowing us to verify whether (58) is indeed a good representation of the
distance function (see Figure 6).

Figure 6: _Comparison of the series expansion ( 58) up to order
$\mathcal{O}((z-z_{H})^{500-1/2})$ (dashed black line) to the analytic result
(239) (red curve) for $\chi=5$. The blue line $4\chi=20$ denotes the boundary
of the interval of convergence for the series expansion._

####  C.4.2 Bonanno-Reuter black hole

For the choice (104) of the metric deformation (along with $f=h$), the
coefficients $\xi_{n}$ in the expansion (47) are given by

$\displaystyle\xi_{0}$
$\displaystyle=\frac{2\chi\,d_{H}^{2}}{\widetilde{\omega}+d_{H}^{2}}\,,$
$\displaystyle\xi_{n}$
$\displaystyle=\frac{(-1)^{n}\,2\chi}{(\widetilde{\omega}+d_{H}^{2})^{n+1}}\,\sum_{k=1}^{\lfloor\frac{n}{2}+1\rfloor}(-1)^{k}\,\left(\begin{array}[]{c}n+1\\\
2k-1\end{array}\right)\,d_{H}^{n+2-2k}\,\widetilde{\omega}^{k}\,,$
$\displaystyle\forall n\geq 1\,.$ (242)

Here the interval of convergence (for $\widetilde{\omega}<0$) is given by
$\rho\in\left[0,\frac{|d_{H}^{2}+\widetilde{\omega}|}{d_{H}+\sqrt{-\widetilde{\omega}}}\right)$.
Moreover, since
$\xi_{1}=\frac{4\chi\widetilde{\omega}\,d_{H}}{(\widetilde{\omega}+d_{H}^{2})^{2}}\neq
0$, the results of Section 3.1 cannot be directly applied. However, as
explained in Appendix C.3, for $\widetilde{\omega}<0$ (such that $\xi_{1}<0$),
this approach can be adapted, leading to a series expansion (231) with
coefficients $\widehat{a}_{n}$ in (234) and (237). This expansion provides a
solution of the differential equation (44) (albeit with a divergent first
derivative $f^{(1)}_{H}$ of $f$ at the horizon). Inversion of the series (231)
leads to an expansion of the proper distance to the horizon of the form

$\displaystyle\rho_{\text{BR}}(z)=\sum_{n=0}^{\infty}\,\widehat{b}_{n}\,(z-z_{H})^{\frac{2+n}{3}}\,,$
(243)

where for concreteness, we provide explicitly the first few coefficients

$\displaystyle\widehat{b}_{0}=\frac{1}{\widehat{a}_{1}^{2/3}}=\frac{3^{2/3}z_{H}^{1/3}(d_{H}+\widetilde{\omega})^{2/3}}{2^{4/3}(-d_{H}\chi\widetilde{\omega})^{1/3}}\,,$
$\displaystyle\widehat{b}_{1}=\frac{2\widehat{a}_{2}}{3\widehat{a}_{1}^{2}}=\frac{(d_{H}+\widetilde{\omega})^{2}}{16d_{H}\chi\widetilde{\omega}}\,,$
$\displaystyle\widehat{b}_{2}=\frac{7\widehat{a}_{2}^{2}-6\widehat{a}_{1}\widehat{a}_{3}}{9\widehat{a}_{1}^{10/3}}\,.$
(244)

A numerical plot of the three different distance functions used in Section 4.1
for the Bonanno-Reuter black hole is shown in Figure 7.

Figure 7: _Comparison of three different distance functions for the Bonanno-
Reuter space-time:_(i)_ $\rho_{\text{BR}}$ (represented by the expansion
(243)), _(ii)_ the Schwarzschild proper distance
$\rho_{\text{S}}=d_{\text{S}}-d_{H}$ and _(iii)_ the approximating function
$\kappa_{\text{BR}}-d_{H}$ in eq. (113). Here we have chosen
$\widetilde{\omega}=-1$, $\chi=5$ and $z_{H}=10.08$ and $d_{H}=12.96$._

####  C.4.3 Minimal example

As a last example, we consider the metric function characterised by the
minimal choice (118), leading to the function $\Phi$ in eq. (120). In order to
make contact with the approach in Section 3, we first remark that the
coefficients $\kappa_{n}$ in the series expansion of (120) (see (48)) are
explicitly given by

$\displaystyle\kappa_{n}=-\frac{(-1)^{n}\,(n-1)(n-3)}{2d_{H}^{n+2}}\,\phi_{2}\,,$
$\displaystyle\forall\,n\geq 0\,.$ (245)

Therefore, the series expansion (48) of (120) has interval of convergence
$\rho\in[0,d_{H})$. Using (50), the coefficients $\kappa_{n}$ allow to
calculate the expansion coefficients $\xi_{n}$ of $e^{\Phi}$. The first few
$\xi_{n}$ read explicitly

$\displaystyle\xi_{0}=2\chi\,e^{-\frac{3\phi_{2}}{2d_{H}^{2}}}\,,$
$\displaystyle\xi_{1}=0\,,$
$\displaystyle\xi_{2}=\frac{2\chi}{d_{H}^{4}}\,\phi_{2}\,e^{-\frac{3\phi_{2}}{2d_{H}^{2}}}\,,$
$\displaystyle\xi_{3}=0\,,$ (246)

which therefore satisfy the conditions (94), as expected. The leading
coefficients $a_{n}$ in the expansion of $z(\rho)$ in (42) therefore become

$\displaystyle a_{2}=\frac{1+\varpi}{8z_{H}}\,,$ $\displaystyle
a_{3}=-\frac{z_{H}\phi_{2}}{d_{H}^{4}(1+3\varpi)}\,,$ $\displaystyle
a_{4}=-\frac{1}{1+2\varpi}\left(\frac{(1+\varpi)^{3}}{256z_{H}^{3}}+\frac{9z_{H}^{3}\phi_{2}^{2}}{d_{H}^{8}(1+3\varpi)^{2}}\right)\,,$
(247)

where we have again used the shorthand notation
$\varpi=\sqrt{1-16z_{H}\xi_{2}}$. The series inversion yields the following
coefficients for the distance function (58)

$\displaystyle b_{1}=\frac{2\sqrt{2z_{H}}}{\sqrt{1+\varpi}}\,,$ $\displaystyle
b_{2}=\frac{32z_{H}^{3}\phi_{2}}{d_{H}^{4}(1+\varpi)^{2}(1+3\varpi)}\,,$
$\displaystyle
b_{3}=\frac{d_{H}^{8}(1+\varpi)^{4}(1+3\varpi)^{3}+256z_{H}^{6}(19+29\varpi)\phi_{2}^{2}}{2\sqrt{2z_{H}}d_{H}^{8}(1+\varpi)^{7/2}(1+2\varpi)(1+3\varpi)^{2}}\,.$
(248)

##  Appendix D Further examples

In this Appendix, we discuss two further examples from the literature that
describe non-singular, static, and spherically symmetric black holes: the
first is the Hayward black hole [10] and the second one is the Dymnikova
space-time [68]. Although we are aware that these examples do not exhibit any
divergent physical quantities, it is still interesting to demonstrate how our
approach can be applied to deformation functions that explicitly depend on
coordinates other than the proper distance.

###  D.1 Hayward black hole

We begin by examining the Hayward black hole, which was introduced in [10] as
the first model to describe a non-singular black hole (notably at the origin)
without committing to any specific modification of General Relativity. The
metric function for the Hayward space-time can be written as

$\displaystyle f(z)=h(z)=f_{\text{Hay}}(z):=1-\frac{2\chi
z^{2}}{z^{3}+2\chi\gamma}\,,$ $\displaystyle\forall z\in[0,\infty)\,,$ (249)

where $\gamma$ is a free parameter that determines the scale at which the
departure from the classical Schwarzschild solution becomes significant [69].
Here we assume $f_{\text{Hay}}$ to hold in the entire space-time. From the
metric element, we can directly deduce the form of the deformation function,
which explicitly depends on the coordinate $z$ rather than the proper distance
$d$

$e^{\Phi\left(\frac{1}{d(z)}\right)}:=\frac{z^{3}}{z^{3}+2\chi\gamma}\ .$
(250)

The Hayward space-time exhibits two event horizons, indicated by the existence
of two zeroes of the function $f_{\mathrm{Hay}}$. The position of the outer
horizon is:

$\displaystyle
z_{H,+}=\frac{2\chi}{3}\,\left(1+2\cos\left[\frac{1}{3}\,\arccos\left(1-\frac{27\gamma}{8\chi^{2}}\right)\right]\right)\,.$
(251)

Furthermore, the proper distance can be computed using the integral expression
(6):

$d_{\mathrm{Hay}}(z)=\int_{0}^{z}\sqrt{\absolutevalue{\frac{\tilde{z}^{3}+2\chi\gamma}{\tilde{z}^{3}-2\chi(\tilde{z}^{2}+\gamma)}}}\differential\tilde{z}\
.$ (252)

Figure 8: _Comparison between the proper distance in the Hayward space-time
(blue, calculated using equation ( 252)) and the proper distance in the
Schwarzschild space-time $d_{\mathrm{S}}$ (green) for specific parameter
values: $\chi=10$ and $\gamma=3$. The dashed orange line corresponds to the
Taylor series expansion of the proper distance from the event horizon,
obtained from equation (256)._

Numerical evaluation of this integral suggests the following form of the
distance of the horizon for large values of $\chi$

$\displaystyle
d_{H,+}=\pi\chi+c_{1}\,\gamma^{c_{2}}\,\ln\chi+\mathcal{O}(\chi^{0})\,,$ with
$\displaystyle\begin{array}[]{l}c_{1}=0.3334\pm 0.0002\,,\\\ c_{2}=0.4999\pm
0.0002\,.\end{array}$ (255)

More importantly, the distance (for a generic point outside of the horizon)
can be expanded in powers of $(z-z_{H,+})$, which takes the explicit form

$\displaystyle d_{\text{Hay,ex}}$
$\displaystyle=d_{H,+}+\frac{\sqrt{2}(z_{H,+}^{3}+2\gamma\chi)\sqrt{z-z_{H,+}}}{\sqrt{z_{H,+}\chi(z_{H,+}^{3}-4\gamma\chi)}}+\frac{\chi(z_{H,+}^{6}-14z_{H,+}^{3}\gamma\chi+4\gamma^{2}\chi^{2})}{3\sqrt{2}(z_{H,+}\chi(z_{H,+}^{3}-4\gamma\chi))^{3/2}}\,(z-z_{H,+})^{3/2}$
$\displaystyle\hskip 38.41139pt+\mathcal{O}((z-z_{H,+})^{5/2})\,.$ (256)

A comparison between the numerical solution for the distance function (252)
and (256) is shown in Figure 8. Using expansion (58) we can read off the
coefficients $b_{n}$ from eq. (256) and express them in terms of the $a_{n}$’s
using the recursive relation listed in (59)

$a_{2}^{-1/2}=\frac{\sqrt{2}(z_{H,+}^{3}+2\gamma\chi)}{\sqrt{z_{H,+}\chi(z_{H,+}^{3}-4\gamma\chi)}}\,,\,-\frac{a_{3}}{2a_{2}^{2}}=0\,,\,\frac{5a_{3}^{2}-4a_{2}a_{4}}{8a_{2}^{7/2}}=\frac{\chi(z_{H,+}^{6}-14z_{H,+}^{3}\gamma\chi+4\gamma^{2}\chi^{2})}{3\sqrt{2}(z_{H,+}\chi(z_{H,+}^{3}-4\gamma\chi))^{3/2}}\,.$
(257)

Solving this system yields the solutions for $a_{2}$, $a_{3}$ and $a_{4}$

$\displaystyle a_{2}$
$\displaystyle=\frac{z_{H,+}\chi(z_{H,+}^{3}-4\gamma\chi)}{2(z_{H,+}^{3}+2\gamma\chi)^{2}}\,,\qquad
a_{3}=0\,,$ (258) $\displaystyle a_{4}$
$\displaystyle=\frac{-\chi^{2}z_{H,+}^{10}+18\gamma\chi^{3}z_{H,+}^{7}-60\gamma^{2}\chi^{4}z_{H,+}^{4}+16\gamma^{3}\chi^{5}z_{H,+}}{384\gamma^{5}\chi^{5}+12z_{H,+}^{15}+120\gamma\chi
z_{H,+}^{12}+480\gamma^{2}\chi^{2}z_{H,+}^{9}+960\gamma^{3}\chi^{3}z_{H,+}^{6}+960\gamma^{4}\chi^{4}z_{H,+}^{3}}\
.$ (259)

Plugging these coefficients into the series (58) allows us to write $z$ as a
power series in $\rho$ and then expand around $\rho=0$

$\frac{2\chi
z(\rho)^{3}}{z(\rho)^{3}+2\chi\gamma}=z_{H,+}+\frac{6\gamma\chi^{3}z_{H,+}^{3}\left(z_{H,+}^{3}-4\gamma\chi\right)}{\left(2\gamma\chi+z_{H,+}^{3}\right)^{4}}\rho^{2}+\\\
+\frac{\gamma\chi^{4}\left(-7z_{H,+}^{12}+72\gamma\chi
z_{H,+}^{9}-204\gamma^{2}\chi^{2}z_{H,+}^{6}+112\gamma^{3}\chi^{3}z_{H,+}^{3}\right)}{\left(2\gamma\chi+z_{H,+}^{3}\right)^{7}}\rho^{4}+\mathcal{O}(\rho^{5})\
,$ (260)

from which we can read the coefficients $\xi_{n}$. In particular, we obtain

$\xi_{1}=0\,,\quad\xi_{2}=\frac{6\gamma\chi^{3}z_{H,+}^{3}\left(z_{H,+}^{3}-4\gamma\chi\right)}{\left(2\gamma\chi+z_{H,+}^{3}\right)^{4}}\mbox{\quad
and\quad}\xi_{3}=0\ .$ (261)

Alternatively, we can use the last relation in equation (54) and the recursive
relation in equation (57) for $p=3$ to determine the values of the
coefficients $\xi_{2}$ and $\xi_{3}$. By doing so, we can establish that the
Hayward space-time satisfies the condition (94). Consequently, it is not
surprising that the Ricci scalar and the Hawking temperature in this space-
time are well-defined and free from singularities at the event horizon
$z_{H,+}$.

###  D.2 Dymnikova black hole

We now turn our attention to the Dymnikova space-time, which was proposed in
[68, 70]. This space-time describes a static, spherically symmetric non-
singular black hole embedded in an effective energy-momentum tensor. The
metric function in the Dymnikova space-time is given by

$\displaystyle
f(z)=h(z)=f_{\text{Dymn}}(z):=1-\frac{2\chi}{z}\,\left(1-e^{-z^{3}/z_{*}^{3}}\right)\mbox{\quad
with\quad}\begin{array}[]{l}z_{\ast}^{3}=2\chi z_{0}^{2}\,,\\\ \forall z\geq
0\,,\end{array}$ (264)

which (as for the Hayward black hole) we take to hold for the entire space-
time. Also, similar to the Hayward black hole, the parameter $z_{0}$ ensures
the regularity of the solution near the origin. Additionally, the density
profile of the effective energy-momentum tensor is chosen such that the metric
possesses a de Sitter core. Furthermore, the Dymnikova space-time possesses
two horizons, which for large masses are located at

$z_{H,+}=2\chi\left(1-\mathcal{O}\left(e^{-4\chi^{2}/z_{0}^{2}}\right)\right)\mbox{\quad
and\quad}z_{H,-}=2\chi\left(1-\mathcal{O}\left(\frac{z_{0}}{8\chi}\right)\right)\
.$ (265)

Figure 9: _Comparison of the proper distance in the Dymnikova space-time,
computed using the definition ( 6) (blue), with the proper distance in the
Schwarzschild space-time $d_{\mathrm{S}}$ (green) for $\chi=10$ and $z_{0}=3$.
The dashed orange line represents the Taylor series expansion of the proper
distance from the event horizon, as given by equation (266)._

We find from (22) that the power series expansion of the proper distance has
the following form

$d_{\mathrm{Dymn,ex}}=d_{H,+}+\frac{2z_{H,+}\sqrt{z-z_{H,+}}}{\sqrt{2\chi-e^{-z_{H,+}^{3}/2\chi
z_{0}^{2}}\left(2\chi+3z_{H,+}^{3}/z_{0}^{2}\right)}}+\\\
+\frac{e^{-z_{H,+}^{3}/2\chi
z_{0}^{2}}\left(8\chi^{2}z_{0}^{4}\left(e^{z_{H,+}^{3}/2\chi
z_{0}^{2}}-1\right)-9z_{H,+}^{6}\right)}{24\chi
z_{0}^{4}\left(2\chi+e^{-z_{H,+}^{3}/2\chi
z_{0}^{2}}\left(-2\chi-{3z_{H,+}^{3}}/{z_{0}^{2}}\right)\right)^{3/2}}(z-z_{H,+})^{3/2}+\mathcal{O}((z-z_{H,+})^{2})\
.$ (266)

In Figure 9 we provide a numerical approximation of the proper distance using
the definition (6) and compare this expansion for $z>z_{H,+}$. Furthermore, we
can recover the coefficients $a_{2}$, $a_{3}$ and $a_{4}$

$\displaystyle a_{2}=\frac{e^{-z_{H,+}^{3}/2\chi z_{0}^{2}}\left(2\chi
z_{0}^{2}\left(e^{z_{H,+}^{3}/2\chi
z_{0}^{2}}-1\right)-3z_{H,+}^{3}\right)}{4z_{0}^{2}z_{H,+}^{2}}\,,\qquad
a_{3}=0\,,$ (267) $\displaystyle a_{4}=\frac{e^{-z_{H,+}^{3}/2\chi
z_{0}^{2}}\left(2\chi z_{0}^{2}\left(e^{z_{H,+}^{3}/2\chi