August 2023

HU-EP-23/46-RTG

MIT-CTP/5595

On black hole singularity resolution in $D=2$
via duality-invariant $\alpha^{\prime}$ corrections

Tomas Codina†, Olaf Hohm† and Barton Zwiebach∗

† Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

∗ Center for Theoretical Physics,
Massachusetts Institute of Technology,
Cambridge MA 02139, USA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

> Abstract
>
> Starting with the two-derivative limit of $D=2$ string theory, we explore
> the space of T-duality invariant $\alpha^{\prime}$ corrections, a space that
> contains a point representing the fully $\alpha^{\prime}$-corrected
> classical string theory. Using a parametrization introduced by Gasperini and
> Veneziano we obtain black hole solutions in this theory space. We prove that
> the dual of a solution with a regular horizon must have a curvature
> singularity. We find regions in the theory space where the black hole is
> deformed while preserving the horizon and the singularity, and regions where
> no black hole appears to exist. Furthermore, we find subregions in this
> theory space, probably not containing string theory, in which the black hole
> geometry exhibits a horizon leading to an interior that, having no
> singularity in the metric, curvature, or dilaton, is a regular cosmology.

###### Contents

  1. 1 Introduction
  2. 2 Black hole in 2D string theory
  3. 3 The Gasperini-Veneziano parametrization
  4. 4 $\alpha^{\prime}$ corrected black hole solutions
  5. 5 Horizons and singularities
  6. 6 Regular black holes
  7. 7 Concluding remarks

##  1 Introduction

Black holes are perhaps the most mysterious objects predicted by general
relativity. They are characterized by an event horizon that forms, for
instance, when the size of a collapsing star falls below its Schwarzschild
radius. It follows from the singularity theorems of Penrose that in general
relativity the gravitational collapse inevitably leads to a singularity at
which the spacetime curvature becomes infinite [1]. Since singularities cannot
exist in nature we learn that general relativity breaks down close to the
black hole singularity. In contrast, general relativity should be applicable
at the event horizon, since the curvature there might be relatively small. In
addition, we have excellent reasons to believe that black holes with an event
horizon exist in nature. This then leads to the problem of resolving the black
hole singularity: finding a new theory that replaces general relativity in the
appropriate regime so that there are solutions with an event horizon but no
spacetime singularity in the interior [2, 3, 4, 5, 6].

In this paper we ask if the black hole (BH) singularity can be resolved in
string theory by means of the higher-derivative $\alpha^{\prime}$ corrections
that are already a feature of classical string theory. We consider non-
critical string theory in two dimensions (2D). Apart from its simplicity this
case is particularly promising in that there is an exact worldsheet CFT whose
target space interpretation is that of a 2D BH [7, 8, 9]. Thus there should be
an exact BH solution to all orders in $\alpha^{\prime}$. We were motivated by
the recent result of Gasperini and Veneziano [10] who showed that, remarkably,
the big-bang singularity may be resolved in the $\alpha^{\prime}$-complete
theory. Since the BH interior may be visualized as a cosmology, one could
envision that a resolution of the BH singularity is feasible. Recent
approaches trying to remove the BH singularity using the setup developed in
[16, 17] for cosmological backgrounds were studied in [18, 19, 20, 21]. An old
result in this direction is the $D=2$ black-hole geometry of [11], which was
suggested to be $\alpha^{\prime}$-exact. The maximally-extended space-time
geometry and regularity of such ansatz was studied in [12, 13, 14] and an
action describing this background and its (singular) dual was built in [15].

This paper is based on our recent results in [22], in which we classify the
possible higher-derivative corrections that are compatible with an $O(1,1)$
duality symmetry that string theory is known to possess for such backgrounds
to all orders in $\alpha^{\prime}$ [23, 24, 25, 26, 27]. In this we
generalized the earlier work in [16, 17] on cosmology. Specifically, we
consider the class of 2D string backgrounds with metrics
$ds^{2}=-m^{2}(x)dt^{2}+n^{2}(x)dx^{2}$ that admit a time-like isometry since
the metric components are independent of time $t$. The outside region of the
2D BH is of this form. For any local solution of this form, in string theory
there is also the T-dual solution obtained by sending
$m\rightarrow\frac{1}{m}$, while $n$ and the dilaton $\Phi$, defined in terms
of the standard dilaton via $\sqrt{-g}e^{-2\phi}=ne^{-\Phi}$, are invariant.
By exploiting all possible perturbative field redefinitions that are
consistent with duality we showed, under certain assumptions on the fall-off
behavior of the coefficients, that the most general duality invariant action
is of the form

$I=\int dx\,n\,e^{-\Phi}\Bigl{[}1+(D\Phi)^{2}+F(M)\Bigr{]}\,,$ (1.1)

where $D:=\frac{1}{n}\partial_{x}$ is a covariant derivative, $M:=D\log m$,
and

$F(M)\equiv\sum_{i=0}^{\infty}\epsilon_{i}M^{2i+2}=-M^{2}+\epsilon_{1}M^{4}+\dots\,,\quad\epsilon_{0}\equiv-1\,.$
(1.2)

The $\epsilon_{i}$ are the free coefficients that are not fixed by duality and
that must be determined by other methods from string theory (see for instance
[28, 29, 30]). We currently do not have the techniques to determine them and
hence the function $F(M)$ to all orders. Nevertheless, here we explore the
question whether there are points in the ‘theory space’ of all choices of
$F(M)$ for which there are BH solutions that carry an event horizon but are
regular everywhere.

In order to deal efficiently with the equations following from the
$\alpha^{\prime}$-complete action (1.1) we will use a powerful representation
recently introduced by Gasperini and Veneziano [10], to study the resolution
of the big-bang singularity (for earlier work in this direction see [31, 32]).
In the present black-hole context, instead of seeking a solution given by
explicit functions $m(x)$ and $\Phi(x)$ (say fixing the gauge $n=1$) one seeks
a solution of the form $m(f)$, $\Phi(f)$ and $x(f)$, with $f$ a new parameter.
Remarkably, such a parametrized form can be obtained by inverting the function
$f(M):=F^{\prime}(M)$ to obtain $M(f)$, which parametrizes the metric variable
$M$ in terms of $f$. For the two-derivative theory $f(M)=-2M$ and
$M(f)=-\tfrac{1}{2}f$. While perturbatively $\alpha^{\prime}$ corrections
build up the function $f(M)$ as a power series, the inverse function $M(f)$
affords a more direct solution of the equations of motion. We are thus led to
think of the ‘theory space’ of duality-invariant derivative corrections as one
described by the possible choices of the function $M(f)$. This leads to subtle
questions on how to encode a complete solution by such a Gasperini-Veneziano
parametrization, which in general requires the use of several contours in
‘$f$-space.’ Each contour represents a patch in coordinate space, and together
they represent the whole space. In fact, the equations of motion imply the
existence of branch points and branch cuts in the $f$ plane, and the
underlying structure of the solution is determined by these features. The
relation $x(f)$ is given in terms of an integral $x=\int^{f}\cdots$. We divide
a complete solution into exterior and interior regions, a division that has a
clear meaning in the context of black hole solutions. We will use $f$ to
parameterize the exterior and $\tilde{f}$ for the interior, to make clear they
represent different patches.

For the two-derivative BH the exterior geometry is obtained with an $f$ plane
contour going from zero to infinity on the real line (Fig. 1(a)). Here $f=0$
is the asymptotically flat (AF) region and $f=\infty$ is the horizon. For the
interior, we find that the $\tilde{f}$ plane has a branch point at
$\tilde{f}=2$ and a cut running from this point to infinity. The contour that
gives the interior solution begins under the cut at infinity goes down to the
branch point and then returns to infinity at the top of the cut (Fig. 1(b)).
The horizon and the singularity are at $\tilde{f}=\infty$, below and above the
cut, respectively. The $x$-space picture has the exterior going from
$x=-\infty$ (AF) to $x=0$ (the horizon). The interior, drawn on the same
picture goes from the horizon at $x=0$ to the singularity at $x=\pi$ ((Fig.
1(c)). It must be emphasized that, in general, different $x$-space coordinate
patches are used to cover the BH solution. For instance, in the coordinate
systems we mainly use for the standard BH, the $x$ coordinate ranges from
$-\infty$ to $0$ and then from $0$ to $\pi$, which are two different patches,
despite being displayed in the same graph.

It is worth noting that since the roles of space and time are interchanged
when passing the event horizon, the isometry becomes space-like, and hence the
interior of a black hole is actually a cosmology [12]. For this cosmology the
time coordinate is $x$, with time $x=0$ corresponding to the event horizon.
Since the curvature is finite at the event horizon, the cosmology so obtained
does not have a big-bang singularity. In contrast, after a finite time the
scale factor becomes infinite, with infinite curvature, leading to a ‘big
rip’.

Figure 1: The two-derivative black hole. (a) The exterior is produced by a
contour $f\in(0,\infty)$ joining the asymtotically flat (AF) region to the
horizon (H). (b) The interior requires a contour going under and then over the
cut, with branch point $\tilde{f}=2$ and the horizon and singularity at
$\tilde{f}=\infty$, under and over the cut, respectively. (c) The $x$-space
representation of the black hole. The exterior is in $(-\infty,0)$ and the
interior in $(0,\pi)$.

In this paper we show that $T$ duality implies that the dual of a solution
with a regular horizon must have a curvature singularity. This result, valid
for any solution in the full theory space, is a formalization of the early
observation of Giveon [9] noting this fact in the context of the two-
derivative version of the string theory black hole. It has some implications
in our work; in particular, two options seem available. First, solutions may
be self-dual in the sense that the full BH (extended) geometry is invariant
under T-duality. In such a case, the BH must have a curvature singularity
somewhere if it has a horizon. This is the case for the two-derivative black
hole solution. Second, a complete solution with a horizon and no curvature or
dilaton singularity would not be self-dual, but it would still be physically
equivalent (in the sense of string theory) to its $T$-dual, which necessarily
would exhibit a curvature singularity. It should be stressed that throughout
this paper we employ the standard general relativity notion of singularities,
which is probed by point particles. In string theory, however, spacetime is
probed by strings, which requires further investigation.

We show that for a large class of functions $M(f)$ there are black hole
solutions that, just like that of the two-derivative theory, are not regular
and necessarily carry a singularity. More precisely, parametrizing the space
of functions as $M(f)=-\frac{f}{2}\left(1+h(f^{2})\right)$, we find a number
of constraints that the function $h$ has to obey in order to obtain solutions
with a horizon. These conditions thus pose rather stringent constraints on the
$\alpha^{\prime}$ corrections for string theory to permit BH solutions with a
horizon. Conversely, if one could show that the functions $h$ arising in
string theory do not obey these constraints one could prove that string theory
does not permit BH solutions.111The work of S. Ying in [19] appears to imply
that a cubic $M(f)$ would yield a regular BH. Our analysis of cubic
deformations in this paper, however, finds no such regular solution. In fact,
[20] indicates that this regular solution requires changing the value of the
string theory constant term in the action.

The singularity in this class of theories can be understood using the above
horizon/singularity duality. For these black holes, the interior is
represented by a contour in $\tilde{f}$ space that goes both under and above a
branch cut extending to infinity and ending at a branch point at some finite
$\tilde{f}=f_{0}$ (Fig. 2(a)). The interior solution is self-dual here, with
the above and below the branch contours mapped into each other. As a result,
the singularity is present (at infinity of the top contour) because the
horizon is present (at infinity on the bottom contour).

Figure 2: (a) The interior of a deformed black hole with a full contour
containing a horizon H and a singularity S. (b) The interior of the
regularized black hole, without a singularity. The bottom contour containing
the horizon ends at $f_{0}$ and the top contour (dotted) is not needed for
completeness. (c) The $x$ space representation of the regularized black hole.

Our strategy to construct a BH solution with a horizon but without a
singularity consists in modifying the nature of the branch point $f_{0}$ in
such a way that an $\tilde{f}$ contour going only under the cut suffices to
describe a complete geometry (Fig. 2(b)). We identify a smaller class of
functions $M(f)$ that can be engineered so that the branch point $f_{0}$ has
the desired properties and the singularity is avoided. The solution is
complete, but not self-dual. This black hole singularity resolution, which is
non-perturbative in $\alpha^{\prime}$ and closely analogous to the mechanism
introduced in [16] to obtain de Sitter vacua in string cosmology, has the
following general features. The interior is again a cosmology with no big-bang
singularity at $x=0$. In contrast to the standard black hole, however, it
describes an expanding universe that approaches flat Minkowski space in the
infinite future (Fig. 2(c)). Moreover, the dilaton goes to a fixed value in
that infinite future. Such an asymptotic background, however, defines a $c=2$
matter conformal field theory (CFT), which is not a critical string theory
background.222We thank Ashoke Sen for a question that led to this conclusion,
and discussions about it. It thus appears that these regular black holes are
solutions of duality-invariant theories, but apparently not of string theory.

In exploring the geometry of the regular black hole we emphasize that in a
subtle but important way, the $f$ parameterization extends the definition of
the $\alpha^{\prime}$ corrections beyond what the series expansion $f(M)$ can
tell us. Indeed, we are postulating an $M(f)$ that is a well defined function,
and for the case of the regular black hole, this function is not invertible
($M(f)$ is also not invertible for the cases considered in [10]). Thus, $f(M)$
would have a finite radius of convergence. The physics of the resolved
singularity involves values of $f$ that are not achieved within its
convergence region. This allows the late time cosmology to have a fixed
dilaton instead of a rolling one. In our re-examination of the derivation of
the equations of motion in the $f$ parameterization (section 3.1) we discuss
in detail how the issue of non-invertibility affects the analysis.

This paper is organized as follows. In section 2 we review the general
framework for the 2D black hole and the structure of higher-derivative
duality-invariant corrections, considering the equations of motion relevant
for the exterior and interior geometries. Section 3 introduces the new
Gasperini-Veneziano parametrization by $f$ and shows how the equations of
motion are solved in this framework. We use the example of the two-derivative
BH to understand the basic structure of branch points and cuts in $f$ space.
The next step is displaying a large class of solutions that can be viewed as a
small general deformation of the two-derivative black hole function
$M(f)=-f/2$. We do this in section 4, where we also examine cubic deformations
of the function $M(f)$, which happen not to yield black holes. In section 5 we
give the proof that the T-dual of a geometry with a regular horizon must have
a curvature singularity. This result is used to explain why the back-and-forth
contour in $\tilde{f}$ space used for the interior solution necessarily leads
to an interior with a singularity. Finally, in section 6 we find functions
$M(f)$ for which the exterior solution is largely preserved but the interior
solution has no singularity. This is done by adjusting $M(f)$ in such a way
that $x$ as a function of $\tilde{f}$ diverges at the branch point. We discuss
our results and some possible extensions in section 7.

##  2 Black hole in 2D string theory

Here we begin by reviewing our notation for the two-dimensional geometry,
giving the general formulae for curvature and the action of T-duality on the
field variables (for more details see [22]). We reconsider the two-derivative
black hole and discuss the equations of motion on the exterior and in the
interior, the relevant equations being related by a number of sign changes.
Finally, we consider general $\alpha^{\prime}$ corrections and, again,
formulate the exterior and interior versions of the equations of motion.

###  2.1 Metric, curvature, and duality

We consider the target space theory of bosonic strings in two dimensions (2D)
of Lorentzian signature, with the field content restricted to the universal
massless sector. Since the antisymmetric $b$-field is trivial in two
dimensions, we thus include the metric $g_{\mu\nu}$ and the dilaton $\phi$.
Denoting coordinates by $x^{\mu}=(t,x)$, we further impose that the fields,
and hence the gauge parameters, do not depend on time $t$. This truncated
theory posses a global $O(1,1;\mathbb{R})$ duality symmetry together with one-
dimensional diffeomorphism invariance: reparametrizations of $x$. We can
assume a diagonal ansatz for the metric

$g_{\mu\nu}=\begin{pmatrix}-m^{2}(x)&0\\\ 0&n^{2}(x)\end{pmatrix}\,,\qquad
ds^{2}=\,-m^{2}(x)dt^{2}\,+n^{2}(x)dx^{2}\,.$ (2.1)

Here, $m(x)$ and the lapse function $n(x)$ are real. Under coordinate
transformations of $x$, the filed $m(x)$ is a scalar while $n(x)$ is a scalar
density. The metric $m(x)$ enters the theory through

$M\equiv\frac{Dm}{m}\,,$ (2.2)

in terms of the covariant derivative

$D\equiv\frac{1}{n}\frac{d}{dx}\,.$ (2.3)

The scalar dilaton $\phi(x)$ is related to the duality invariant dilaton
$\Phi(x)$ via

$\phi(x)=\tfrac{1}{2}\left(\Phi(x)+\log|m(x)|\,\right)\,,$ (2.4)

which makes the measure of the effective theory to be of the standard form
$\sqrt{-g}e^{-2\phi}=ne^{-\Phi}$. The scalar curvature $R$ of the two
dimensional metric is given by

$R=-2(M^{2}+DM)=-2\,\frac{D^{2}m}{m}\,.$ (2.5)

The curvature $R$ encodes the full Riemann tensor and hence allows one to
detect geometric singularities.

In these 2D backgrounds with a time-like isometry, the duality group acts on
the fields as

$m(x)\to\hat{m}(x)=\frac{1}{m(x)}\,,\quad
n(x)\to\hat{n}(x)=n(x)\,,\quad\Phi(x)\to\hat{\Phi}(x)=\Phi(x)\,.$ (2.6)

We will use hatted fields for the fields after duality. Note that the
covariant derivative $D$ is unchanged by duality. As a consequence of the
transformation of $m$ one also has

$M(x)\to\hat{M}(x)=-M(x)\,,$ (2.7)

The dual scalar curvature, namely, the curvature of the dual metric, is then
given by

$R\to\hat{R}=-2(\hat{M}^{2}+D\hat{M})=-2(M^{2}-DM)\,,$ (2.8)

so we have

$\hat{R}=-2(M^{2}-DM)=-R-4M^{2}=R+4DM\,.$ (2.9)

A useful gauge choice: We can pick a gauge for one-dimensional diffeomorphism
invariance by fixing the lapse function $n(x)$. A familiar gauge choice takes

$n(x)=1\,.$ (2.10)

In this gauge the field $M(x)$, if known, is easily integrated to find $m(x)$:

$M(x)={\partial_{x}m\over m}=\partial_{x}\ln m\quad\Rightarrow\quad
m(x)=\exp\Bigl{(}\int_{c}^{x}M(x^{\prime})dx^{\prime}\Bigr{)}\,,$ (2.11)

and consequently

${m(x_{2})\over
m(x_{1})}=\exp\Bigl{(}\int_{x_{1}}^{x_{2}}M(x^{\prime})dx^{\prime}\Bigr{)}\,.$
(2.12)

The scalar curvature (2.5) now reads

$R=-2(M^{2}+\partial_{x}M)=-2\,{\partial^{2}_{x}m\over m}\,.$ (2.13)

It is worth emphasizing that all relations above hold for arbitrary 2D metrics
of the form (2.1), independent of any equations of motion, and are hence
applicable to generic higher-derivative theories.

###  2.2 The two-derivative, two-dimensional black hole

Before considering $\alpha^{\prime}$ corrections, in the following we recall
the 2D string equations with up to two derivatives and review the black hole
(BH) solution, both for the exterior region and the interior region. We will
set up the solutions, both in the $n=1$ gauge in such a way that the exterior
solution is in the coordinate interval $(-\infty,0)$ and the interior solution
uses the coordinate interval $(0,\pi)$. While in both cases we use the label
$x$ to denote the coordinate, these two solutions do not form together a
single solution over the full real line; the two $x$’s are really different.

The target space action for the above string backgrounds reads to leading
order

$I=\int dx\,n\,e^{-\Phi}\left[1+(D\Phi)^{2}-M^{2}\right]\,,$ (2.14)

where we use a unit-free notation that absorbs factors of $\alpha^{\prime}$,
denoted by a bar in [22] that we drop in this paper in order not to clutter
the equations. Sometimes we refer to the zero-derivative term as the
cosmological term, although it is not a cosmological constant in the standard
sense. The equations of motion for $m,\Phi$ and $n$ are equivalent to

$\displaystyle\frac{d}{dx}\left(e^{-\Phi}M\right)$ $\displaystyle=0\,,$
(2.15a) $\displaystyle\left(\frac{d\Phi}{dx}\right)^{2}-M^{2}-1$
$\displaystyle=0\,,$ (2.15b) $\displaystyle\frac{d^{2}\Phi}{dx^{2}}-M^{2}$
$\displaystyle=0\,,$ (2.15c)

where we picked $n=1$ gauge after performing the variation.

Exterior BH solution: The above equations admit a unique solution describing
the outside region of a 2D BH [7, 8, 9]. We use the region $x\in(-\infty,0)$,
and the metric and duality-invariant dilaton take the form

$m(x)=-\tanh\tfrac{x}{2}\,,\qquad\Phi(x)=-\log|\sinh x|+\Phi_{0}\,,\ \ \ \
x<0\,,$ (2.16)

with $\Phi_{0}$ an integration constant, and the sign of $m(x)$ is fixed to
get the familiar asymptotic behavior. The scalar curvature (2.13) is given by

$R(x)=\frac{1}{\cosh^{2}\frac{x}{2}}\,.$ (2.17)

Here $x=0$ is the position of the horizon, $x\rightarrow-\infty$ is the
asymptotically flat (AF) region with $m$ approaching one.333The solution can
also be used for $x\in(0,\infty)$, but our later work is simplified by this
choice of range of $x$. Indeed, at $x=0$ the metric vanishes and the curvature
is finite

$m(0)=0\,,\qquad R(0)=1\,,$ (2.18)

while in the AF region we have

$\lim\limits_{x\rightarrow-\infty}m(x)=1\,,\qquad\lim\limits_{x\rightarrow-\infty}R(x)=0\,.$
(2.19)

On the other hand, the duality-invariant dilaton $\Phi(x)$ becomes infinite at
both extremes

$\lim\limits_{x\rightarrow
0^{-}}\Phi(x)=\infty\,,\qquad\lim\limits_{x\rightarrow-\infty}\Phi(x)=-\infty\,.$
(2.20)

This does not necessarily indicate a pathology since $\Phi$ is not the scalar
dilaton $\phi$ that determines the string coupling as $g_{s}=e^{\phi}$. Using
(2.4), we get

$\phi=-\tfrac{1}{2}\log\cosh^{2}\tfrac{x}{2}+\phi_{0}\quad\Rightarrow\quad\phi(0)=\phi_{0}=\tfrac{1}{2}\left(\Phi_{0}-\log
2\right)\,,$ (2.21)

and so the dilaton and hence the string coupling are finite at the horizon. We
also infer $\lim_{x\rightarrow\infty}\phi(x)=-\infty$, so the string coupling
$g_{s}$ goes to zero in the AF region.

Interior BH solution: The solution (2.16) describes only the exterior region
of the BH. Therefore, it has no information on the BH singularity, which lies
in the interior. This region, together with others, can be obtained by
analytic extension of the exterior solution. Moreover, for the 2D BH, it is
possible to obtain the maximally extended solution using a single patch, by
the so-called Kruskal-Szekeres coordinates. This coordinate system covers not
only the exterior and interior region of the BH, but also a white hole region
and a disconnected asymptotically flat space. Since we are mainly interested
in the BH singularity, for now it is sufficient to describe the interior
region only.

The most straightforward approach to get the interior solution is to look for
a transformation that preserves the measure and changes the metric signature
or, more precisely, that maps a metric with time-like isometry to a metric
with space-like isometry as follows:

$\displaystyle ne^{-\Phi}\ $ $\displaystyle\to\ \tilde{n}e^{-\tilde{\Phi}}\,,$
(2.22) $\displaystyle ds^{2}=-m^{2}(x)dt^{2}+n^{2}(x)dx^{2}\quad$
$\displaystyle\to\quad
ds^{2}=\tilde{m}^{2}(x)dt^{2}-\tilde{n}^{2}(x)dx^{2}\,.$ (2.23)

This can be achieved by setting

$m=i\tilde{m}\,,\quad n=-i\tilde{n}\,,\quad\Phi=\tilde{\Phi}-\log i\,,$ (2.24)

which preserves the measure and the product $mn$. Therefore, from (2.2) and
(2.3) we have

$D=i\tilde{D}\,,\ \ \ M=i\tilde{M}\,,$ (2.25)

with $\tilde{D}={1\over\tilde{n}}{d\over dx}$ and
$\tilde{M}={1\over\tilde{m}\tilde{n}}\partial_{x}\tilde{m}$. Applying these
substitutions to the curvature (2.5) we find $R\to\tilde{R}$ with

$\tilde{R}=2(\tilde{M}^{2}+\tilde{D}\tilde{M})=\,2\,\frac{\tilde{D}^{2}\tilde{m}}{\tilde{m}}\,.$
(2.26)

Fixing the gauge $\tilde{n}(x)=1$,

$\tilde{R}=2(\tilde{M}^{2}+\partial_{x}\tilde{M})=\,2\,\frac{\partial_{x}^{2}\tilde{m}}{\tilde{m}}\,.$
(2.27)

The scalar dilaton in (2.4) is mapped to

$\tilde{\phi}(x)=\tfrac{1}{2}\left(\tilde{\Phi}(x)+\log|\tilde{m}|\right)\,.$
(2.28)

From the original action (2.14), we obtain,

$\tilde{I}=\int
dx\,\tilde{n}e^{-\tilde{\Phi}}\left[1-(\tilde{D}\tilde{\Phi})^{2}+\tilde{M}^{2}\right]\,.$
(2.29)

Varying this action and fixing the gauge $\tilde{n}=1$, the equations of
motion read

$\displaystyle\frac{d}{dx}\left(e^{-\tilde{\Phi}}\tilde{M}\right)$
$\displaystyle=0\,,$ (2.30a)
$\displaystyle\Bigl{(}\frac{d\tilde{\Phi}}{dx}\Bigr{)}^{2}-\tilde{M}^{2}+1$
$\displaystyle=0\,,$ (2.30b)
$\displaystyle\frac{d^{2}\tilde{\Phi}}{dx^{2}}-\tilde{M}^{2}$
$\displaystyle=0\,.$ (2.30c)

The solution in this case is given by trigonometric functions

$\tilde{m}(x)=\tan\tfrac{x}{2}\,,\quad\tilde{\Phi}(x)=-\log\sin
x+\Phi_{0}\quad\Rightarrow\quad\tilde{\phi}(x)=-\tfrac{1}{2}\log\cos^{2}\tfrac{x}{2}+\phi_{0}\,,$
(2.31)

where we pick the same integration constant $\Phi_{0}$ as for the exterior
solution in (2.16), and hence we got the same $\phi_{0}$ as in (2.21). The
curvature is obtained by inserting this solution into the right-hand side of
(2.27):

$\tilde{R}(x)=\frac{1}{\cos^{2}\frac{x}{2}}\,.$ (2.32)

This solution is now valid for the finite range $x\in(0,\pi)$. For $x=0$, the
metric, scalar dilaton and curvature take the values

$\tilde{m}(0)=0\,,\qquad\tilde{\phi}(0)={\phi}_{0}\,,\qquad\tilde{R}(0)=1\,,$
(2.33)

which is consistent with a horizon interpretation. More importantly, the
curvature and scalar dilaton on both sides match. At the other end $x=\pi$,
all fields and curvature diverge

$\tilde{m}(\pi)=\infty\,,\qquad\tilde{\phi}(\pi)=\infty\,,\qquad\tilde{R}(\pi)=\infty\,,$
(2.34)

signaling that this is the position of the BH singularity.

Dual BH solutions: As shown in [9], under T-duality the exterior region of the
2D BH is mapped to a region beyond the singularity, while the interior region
is mapped into itself. Here we revisit briefly these phenomena.

Starting from the exterior solution (2.16) and performing the duality
transformation (2.6), one obtains the new solution, also valid for
$x\in(-\infty,0):$

$\hat{m}(x)=-\coth\tfrac{x}{2}\,,\quad\hat{\Phi}(x)=-\log|\sinh
x|+\Phi_{0}\,,\quad\hat{\phi}(x)=-\tfrac{1}{2}\log\sinh^{2}\tfrac{x}{2}+\phi_{0}\,,$
(2.35)

where the dual metric is the inverse of the original one and the duality-
invariant dilaton remains the same. Inserting the dual metric into the dual
curvature (2.27) we get

$\hat{R}(x)=-\frac{1}{\sinh^{2}\tfrac{x}{2}}\,.$ (2.36)

In this dual solution, the former horizon $x=0^{-}$ is mapped to a curvature
singularity

$\lim_{x\rightarrow 0^{-}}\hat{m}(x)=\infty\,,\quad\lim_{x\rightarrow
0^{-}}\hat{\phi}(x)=\infty\,,\quad\lim_{x\rightarrow
0^{-}}\hat{R}(x)=-\infty\,,$ (2.37)

while the asymptotic region is again a flat space-time which lies beyond the
BH singularity.

The dual solution to the interior solution (2.31) is given by

$\hat{\tilde{m}}(x)=\text{cot}\tfrac{x}{2}\,,\quad\hat{\tilde{\Phi}}(x)=-\log\sin
x+\Phi_{0}\,,\quad\hat{\tilde{\phi}}(x)=-\tfrac{1}{2}\log\sin^{2}\tfrac{x}{2}+\phi_{0}\,,$
(2.38)

together with the curvature

$\hat{\tilde{R}}(x)=\frac{1}{\sin^{2}\frac{x}{2}}\,.$ (2.39)

This geometry still corresponds to the interior region, with the horizon and
the singularity exchanged. This can be made clear by noting that444We thank M.
Gasperini and G. Veneziano for pointing this out. duality plus the ‘time
reversal’ transformation $x\to\pi-x$ leaves the interior solution invariant:
$\hat{\tilde{m}}(\pi-x)=\tilde{m}(x)$.

###  2.3 General $\alpha^{\prime}$ corrections

In [22] we considered higher-derivative modifications to (2.14) and pointed
out that, due to the presence of the cosmological term, contributions with
more derivatives are generally not subleading relative to terms with less
derivatives. A naive perturbative expansion is therefore not meaningful unless
one assumes that the numerical coefficients of the higher-derivative terms
obey certain fall-off conditions so that terms with more derivatives are
subleading after all. Under this assumption we classified in [16] the higher-
derivative terms modulo field redefinitions that preserve the fall-off
behavior. This leads to the minimal all-order action

$I=\int dx\,n\,e^{-\Phi}\biggl{[}1+(D\Phi)^{2}+F(M)\biggr{]}\,,$ (2.40)

where

$F(M)\equiv\sum_{i=0}^{\infty}\epsilon_{i}M^{2i+2}=-M^{2}+\epsilon_{1}M^{4}+\dots\,,\quad\epsilon_{0}\equiv-1\,.$
(2.41)

The fall-off conditions we assumed amount to the following constraints for the
coefficients $\epsilon_{i}$

$\epsilon\equiv\epsilon_{1}\ll 1\,,\quad\epsilon_{i}\sim(\epsilon)^{i}\,,\
i\geq 1\,,\quad\epsilon_{p}\epsilon_{k}\sim\epsilon_{p+k}\,,$ (2.42)

where $\sim$ indicates equality up to numerical constants of order one.

Taking the variation of (2.40) with respect to $m,\Phi$ and $n$, combining the
dilaton and lapse equations and fixing $n=1$, we arrive at the all-order
extension of (2.15)

$\displaystyle\frac{d}{dx}\left(e^{-\Phi}f(M)\right)$ $\displaystyle=0\,,$
(2.43a) $\displaystyle\left(\frac{d\Phi}{dx}\right)^{2}+\check{g}(M)-1$
$\displaystyle=0\,,$ (2.43b)
$\displaystyle\frac{d^{2}\Phi}{dx^{2}}+\tfrac{1}{2}Mf(M)$ $\displaystyle=0\,.$
(2.43c)

Here we introduced555In our previous work [22], we used $g(M)$ for what is now
written as $\check{g}(M)$. This change of notation is useful to deal with the
issue of non-invertibility of $M(f)$.

$\begin{split}f(M)&\equiv\
F^{\prime}(M)=\sum_{i=0}^{\infty}(2i+2)\epsilon_{i}M^{2i+1}=-2M+4\epsilon
M^{3}+\mathcal{O}(\epsilon^{2})\,,\\\
\check{g}(M)&\equiv\sum_{i=0}^{\infty}(2i+1)\epsilon_{i}M^{2i+2}=-M^{2}+3\epsilon
M^{4}+\mathcal{O}(\epsilon^{2})\,,\end{split}$ (2.44)

which satisfy the following relation

$\check{g}^{\prime}(M)=Mf^{\prime}(M)\,,$ (2.45)

where ′ denotes the derivative with respect to $M$.

For the two-derivative theory it was straightforward to solve (2.15) to obtain
(2.16), the exterior region of the BH. While it is much harder to find
solutions of (2.43) for generic $f(M)$ and $g(M)$, assuming that we have such
a solution describing the exterior of a BH, we can obtain the solution
describing the interior by performing the ‘signature change’ discussed above.
To this end we apply (2.24) to (2.40) to get

$\tilde{I}=\int
dx\,\tilde{n}e^{-\tilde{\Phi}}\left[1-(\tilde{D}\tilde{\Phi})^{2}+\tilde{F}(\tilde{M})\right]\,,$
(2.46)

where we have defined

$\tilde{F}(\tilde{M})\equiv
F(i\tilde{M})=\sum_{i=0}^{\infty}(-1)^{i+1}\epsilon_{i}\tilde{M}^{2i+2}=\tilde{M}^{2}+\epsilon\tilde{M}^{4}+\dots$
(2.47)

The equations of motion in the $\tilde{n}=1$ gauge are now given by

$\displaystyle\frac{d}{dx}\left(e^{-\tilde{\Phi}}\tilde{f}(\tilde{M})\right)$
$\displaystyle=0\,,$ (2.48a)
$\displaystyle\Bigl{(}\frac{d\tilde{\Phi}}{dx}\Bigr{)}^{2}-\tilde{\check{g}}(\tilde{M})+1$
$\displaystyle=0\,,$ (2.48b)
$\displaystyle\frac{d^{2}\tilde{\Phi}}{dx^{2}}+\tfrac{1}{2}\tilde{M}\tilde{f}(\tilde{M})$
$\displaystyle=0\,.$ (2.48c)

The new function $\tilde{f}(\tilde{M})$ is defined analogously to (2.44):

$\tilde{f}(\tilde{M})\equiv\tilde{F}^{\prime}(\tilde{M})\,,$ (2.49)

while $\tilde{\check{g}}(\tilde{M})$ can be defined via the identity

$\tilde{\check{g}}^{\prime}(\tilde{M})=\tilde{M}\tilde{f}^{\prime}(\tilde{M})\,.$
(2.50)

One can check that these functions are related to the ones in (2.44) via

$\tilde{f}(\tilde{M})=if(i\tilde{M})\,,\quad\tilde{\check{g}}(\tilde{M})={\check{g}}(i\tilde{M})\,.$
(2.51)

Equations (2.43) and (2.48) govern the exterior and interior of higher-
derivative BH solutions, respectively.

For later purposes we need to find out how the inverse functions $f^{-1}$ and
$\tilde{f}^{-1}$ are related: if $f^{-1}$ is known, what is $\tilde{f}^{-1}$?
We claim that acting on a variable $y$ we have

$\tilde{f}^{-1}(y)=-if^{-1}(-iy)\,.$ (2.52)

This can be checked using the first relation in (2.51) as follows:

$\tilde{f}^{-1}(\tilde{f}(\tilde{M}))=-if^{-1}(-i\tilde{f}(\tilde{M}))=-if^{-1}(-i\cdot
if(i\tilde{M}))=-if^{-1}(f(i\tilde{M}))=-i\cdot i\tilde{M}=\tilde{M}\,.$
(2.53)

Noting that we have $f(M)$ and $\tilde{f}(\tilde{M})$ functions, the inverse
$f^{-1}$ is the function $M(f)$ and the inverse $\tilde{f}^{-1}$ is the
function $\tilde{M}(\tilde{f})$. With this notation, (2.52) reads

$\tilde{M}(\tilde{f})=-iM(-i\tilde{f})\,.$ (2.54)

##  3 The Gasperini-Veneziano parametrization

In order to study solutions of (2.43) and (2.48) we will use a useful
parameterization inspired by recent work by Gasperini and Veneziano in the
context of pre-big bang string cosmology [10]. Instead of describing the
fields as functions of the coordinate $x$, they are described as functions of
a new parameter, which arises as follows. One considers the function $f(M)$ in
(2.44), a typically infinite series in the metric variable $M$, and inverts it
to find $M(f)$. In $M(f)$ the metric variable is parameterized by $f$, the
Gasperini-Veneziano parameter. One can then find $x(f)$ by performing an
integral analytically, or numerically if needed. We will distinguish the
parameter for each region, using $f$ to describe the exterior and $\tilde{f}$
for the interior.

There is a shift in perspective here. While one originally would view the
theory space as described by the set of possible functions $f(M)$, we now
consider the theory space as described by the set of functions $M(f)$. The
invertibility issues are actually central to the development and we will argue
that the $f$ parameterization leads to a natural extension of the original
equations of motion, beyond their original domain of validity. Clarifying this
will be our first task in this section.

In this section we show how to solve the exterior and interior equations of
motion using this parameterization. We then do this explicitly for the two-
derivative BH. Here the nature of the contours required in $f$ space becomes
evident.

###  3.1 Multivalued functions and equations of motion

As mentioned above and reviewed in section 2.3, the conventional definition of
$\alpha^{\prime}$ corrections begins by stating that the Lagrangian contains a
function $F(M)$ given as an infinite power series expansion in (even) powers
of $M$, each successive term containing an additional power of
$\alpha^{\prime}$. It is quite possible, perhaps even likely, that this series
only has a finite radius of convergence, making the definition of the action
incomplete. From $F(M)$ one defines $f(M)\equiv F^{\prime}(M)$ and one usually
works with $f(M)$. The viewpoint adopted here is that $\alpha^{\prime}$
corrections are described by the function $M(f)$ which we assume to be well
defined (single valued), and that $f(M)$ is just its inverse. Thus we have

$M(f)\ \hbox{ is a well defined function (single valued).}$ (3.1)

It follows that the inverse $f(M)$ is usually multivalued. Indeed, assume
$M(f)$ grows monotonically from zero at $f=0$ to some maximum $M_{*}$ at some
value $f_{*}>0$, and then falls down. Then the inverse $f(M)$ is not single
valued. In fact if one expands $M(f)$ as a series in $f$ and perturbatively
inverts it, $f(M)$ will only converge up to $M=M_{*}$, where it has a branch
point. In general, let $M_{*}$ denote the maximum value of $M$ for which the
series $f(M)$ converges. In summary, we have two important additional facts:

> The inverse $f(M)$ of $M(f)$ is not single valued. The series definition of
> $f(M)$ converges for $M\in[0,M_{*}]$ and $f(0)=0$.

The convergent series is a precise definition of $f(M)$ in this range.

The equations of motion also feature a quantity $\check{g}(M)$ defined by two
relations

${d\check{g}\over dM}=M{df\over dM}\ \ \ \hbox{and}\ \ \ \check{g}(0)=0\,.$
(3.2)

Both of these are implemented by the integral definition

$\check{g}(M)\equiv\int_{u=0}^{u=M}u{df(u)\over du}du\,,\ \ \
M\in[0,M_{*}]\,.$ (3.3)

The condition on the range of validity of the definition is needed because
$\check{g}$ is a multivalued function. Indeed, $df/du$ appearing inside the
integral, is multivalued, a property it inherits from $f(u)$. But now, in this
range we can change the variable of integration from $u$ to $f$. In doing so
$u$ must be thought as $u(f)$, where this is the inverse function of $f(u)$.
The integral then becomes

$\check{g}(M)\equiv\int_{f(u=0)}^{f(u=M)}u(f)df=\int_{0}^{f(M)}u(f^{\prime})df^{\prime}\,,$
(3.4)

so that, recalling that we call $M$ the inverse function of $f$,

$\check{g}(M)\equiv\int_{0}^{f(M)}M(f^{\prime})df^{\prime}\,,\ \ \
M\in[0,M_{*}]\,.$ (3.5)

The above definition suggests the following construction of a $g(f)$ well
defined for all $f$:

$g(f)\equiv\int_{0}^{f}M(f^{\prime})df^{\prime}\,.$ (3.6)

It follows immediately from the last two equations that

$\check{g}(M)=g(f(M))\,,\ \ \ \,M\in[0,M_{*}]\,.$ (3.7)

This states that in the domain of definition, $\hat{g}(M)$ is given by the
well-defined $g(f)$.

Let us now consider the field equations, following the steps considered in
[10], but in the light of the above observations. In the two-dimensional
theory we have (2.43b) that reads

$\Bigl{(}{d\Phi\over dx}\Bigr{)}^{2}=1-\check{g}(M)\,,\ \ M\in[0,M_{*}]\,.$
(3.8)

Using the above definition of $\check{g}$ we have

$\Bigl{(}{d\Phi\over dx}\Bigr{)}^{2}=1-g(f(M))\,,\ \ M\in[0,M_{*}]\,.$ (3.9)

Here the dilaton derivative is effectively set equal to a function of $f(M)$.
But then we may as well forget $M$, declaring that this equation sets the
dilaton derivative equal to a function of $f$:

$\Bigl{(}{d\Phi\over
dx}\Bigr{)}^{2}=1-g(f)=1-\int_{0}^{f}M(\tilde{f})d\tilde{f}\,,\ \ \ \ \
\forall\,f\,.$ (3.10)

This equation is now valid for all $f$, thus going beyond the limitations of
the power series definition of the $\alpha^{\prime}$ corrections. The equation
indeed yields a parameterization of the dilaton derivative in terms of $f$:

${d\Phi\over dx}(f)=\pm\sqrt{P(f)}\,,\ \ \hbox{with}\ \ P(f)\equiv
1-{\int_{0}^{f}}M(f^{\prime})df^{\prime}\,.$ (3.11)

The plus/minus signs correspond to using different branches of the square
root. The equations of motion also imply that $f$ ends up being a coordinate,
which can be related to the $x$ coordinate. Indeed, consider (2.43a) and
distribute the $x$ derivative

${d\Phi\over dx}f(M)={df\over dx}\quad\Rightarrow\quad dx={df\over
f}{1\over{d\Phi\over dx}(f)}\,.$ (3.12)

Using (3.11), the second equality leads to an equation for the coordinate $x$
as a function of $f$

$dx=\pm{df\over f\sqrt{P(f)}}\,.$ (3.13)

Finally, consider again (2.43a), which states that the combination
$e^{-\Phi}f(M)$ is a constant relative to the $x$ coordinate. It is therefore
a constant relative to $f$, and therefore,

$\Phi(f)=\log|f|+\Phi_{1}\,,$ (3.14)

with $\Phi_{1}\equiv\Phi(f=1)$ an integration constant that absorbs the sign
of $f$ and encodes the mass of the black hole.

The above equations represent an extension of the original equations, and we
will assume these provide the non-perturbative definition of the theory with
$\alpha^{\prime}$ corrections.

###  3.2 Parametrization in terms of $f$

In this section we consider the equations of motion in the $f$
parameterization, both for the exterior and for the interior. Note that we
have passed from using $\check{g}(M)$ to using $g(f)$, in the exterior, and
from using $\tilde{\check{g}}(\tilde{M})$ to using $\tilde{g}(\tilde{f})$ in
the interior. The analog of (3.10) for the interior follows from (2.48c) and
becomes

$\Bigl{(}{d\tilde{\Phi}\over dx}\Bigr{)}^{2}=-1+\tilde{g}(\tilde{f})\,.$
(3.15)

Exterior solution: Much of the work was already done above, so we just make
some comments, consider a few extra equations, and collect results. Consider
again (3.11):

${d\Phi\over dx}(f)=\pm\sqrt{P(f)}\,,\ \ \hbox{with}\ \ P(f)\equiv
1-{\int_{0}^{f}}M(f^{\prime})df^{\prime}\,.$ (3.16)

The allowed range of $f$ is determined by the condition $P(f)\geq 0$. We see
that $P(f=0)=1>0$, so by continuity there is at least an interval around $f=0$
where the solution exists. Moreover, if the integral appearing above is
negative for all values of $f$ (as will be the case for the standard BH), then
the solution is valid for the whole real line $f\in\mathbb{R}$.

Using (3.13) in (2.12) we obtain an expression for the metric components

${m(x_{2})\over m(x_{1})}={m(f_{2})\over
m(f_{1})}=\exp\biggl{(}\pm\int_{f_{1}}^{f_{2}}{df\over
f}{M(f)\over\sqrt{P(f)}}~{}\biggr{)}\,.$ (3.17)

With $m(f)$, we can relate $\Phi(f)$ to the scalar dilaton $\phi(f)$ via
(2.4), where now all quantities are parameterized by $f$. For the curvature,
we have from (2.13),

$R_{\pm}(f)=-2\Bigl{(}M^{2}+{dM\over df}{df\over
dx}\Bigr{)}=-2\Bigl{(}M^{2}\pm f\,{dM\over df}\sqrt{P(f)}\ \ \Bigr{)}\,.$
(3.18)

Collecting the results, for ease of reference, we have

$\displaystyle{d\Phi\over dx}(f)$ $\displaystyle=\
\pm\sqrt{P(f)}\,,\quad\quad\Phi(f)=\log|f|+\Phi_{1}\,,$ (3.19a) $\displaystyle
dx$ $\displaystyle=\ \pm{df\over f\sqrt{P(f)}}\,,$ (3.19b)
$\displaystyle{m(x_{2})\over m(x_{1})}$ $\displaystyle={m(f_{2})\over
m(f_{1})}=\exp\biggl{(}\pm\int_{f_{1}}^{f_{2}}{df\over
f}{M(f)\over\sqrt{P(f)}}~{}\biggr{)}\,,$ (3.19c) $\displaystyle R_{\pm}(f)$
$\displaystyle=-2\Bigl{(}M^{2}+{dM\over df}{df\over
dx}\Bigr{)}=-2\Bigl{(}M^{2}\pm f\,{dM\over df}\sqrt{P(f)}\ \ \Bigr{)}\,.$
(3.19d)

Here the argument $P(f)$ of the square roots is

$P(f)\equiv 1-\int_{0}^{f}M(f^{\prime})df^{\prime}\,.$ (3.20)

Equations (3.19) encode the general solution to (2.43) in terms of $f$ as a
parameter. The solution is completely determined once an ansatz for $M(f)$ is
given, and the physics depends on such choice. To make this point more
emphatically, we can write the metric $ds^{2}=-m^{2}(x)dt^{2}+dx^{2}$ by
passing from the $x$ coordinate to the $f$ coordinate using (3.19b), and
writing $m(x)$ using (3.19c):

$ds^{2}=-\exp\Biggl{(}\pm\,2\int^{f}{du\over
u}{M(u)\over\sqrt{P(u)}}\Biggr{)}\,(dt)^{2}\ +\ {(df)^{2}\over f^{2}P(f)}\,,\
\ \ \Phi(f)=\log|f|+\Phi_{1}\,,$ (3.21)

where we also included the dilaton. It may be that eventually $x$ can be
traded for $f$, but we will not explore this here. Still, the above makes
clear that we have a solution if we know $M(f)$.

Without $M(f)$ we cannot say much about the specifics of each solution, but we
can still analyze some global aspects of them: the first thing to notice is
that, from our starting point in (2.44), since $f(M)$ has a power series
expansion in $M$, its inverse $M(f)$, at least perturbatively, is expected to
have the following expansion

$M(f)=-\tfrac{1}{2}f\left[1+\frac{\epsilon}{2}f^{2}+\mathcal{O}(\epsilon^{2}f^{4})\right]=-\tfrac{1}{2}\,f\left[1+h(f^{2})\right]\,,$
(3.22)

where we introduced the function $h$ as follows

$h(\xi)\equiv\frac{\epsilon}{2}\,\xi+\mathcal{O}(\epsilon^{2}\xi^{2})\,.$
(3.23)

Here $M(f)=-\frac{f}{2}$, obtained for $h(\xi)=0$, corresponds to the standard
two-derivative case, for which we recover the exterior BH solution of (2.16),
as we will show in section 3.3. The expansion (3.22) implies the following two
properties for generic $M(f)$

$M(f=0)=0\,,\quad M(-f)=-M(f)\,,$ (3.24)

and

$h(\xi=0)=0\,,\quad h^{\prime}(\xi=0)=\text{finite}\,.$ (3.25)

These observations have the important consequence that $f=0$ corresponds to
the faraway region in all solutions. To see this we study the behavior of
(3.19) and (3.20) near $f=0$ for generic $M(f)$. Using (3.22), (3.24) and
$P(f=0)=1$, we get

$\Phi(0)=-\infty\,,\quad dx\simeq\pm\frac{df}{f}\,,\quad m(f)\simeq
m(0)e^{\mp\frac{1}{2}f}\,,\quad R(0)=0\,.$ (3.26)

From the second relation we infer $x\simeq\pm\ln f+{\rm const}$ and hence that
$f=0$ corresponds to an asymptotic region with infinite $x$. From the third
relation we see that we can choose the integration constant such that
$m(0)=1$. Using this together with $\Phi(0)=-\infty$ in (2.4) we obtain
$\phi(0)=-\infty$. All these results are compatible with the interpretation as
a faraway region.

This makes $f=0$ the end of the space-time, which implies that solutions with
$f<0$ and with $f>0$ should be treated separately, not as two regions of the
same exterior solution. On top of this distinction, we have the two branches
of the square root of (3.11) corresponding to the $\pm$ choices. While this
seems to suggest we have four different solutions, all of them should be
physically equivalent in string theory. More precisely, the negative and
positive regions of $f$ are related by T-duality since $M\to-M$ and (3.22)
imply

$f\to\hat{f}=-f\,.$ (3.27)

Moreover, solutions with different signs of the square root (3.11) are
connected via a trivial sign flip of the $x$ coordinate. More precisely, the
solution with minus sign and $f<0$ is identical to the solution with plus sign
and $f>0$, upon changing $x\to-x$. Their T-dual solutions can be obtained by
(3.27).

Interior solution:

In order to get the interior solutions in the Gasperini-Veneziano
parameterization, we repeat the procedure we just performed for the exterior
case, but this time using the equations (2.48). Without going into details,
the solutions read

$\displaystyle{d\tilde{\Phi}\over dx}(\tilde{f})$ $\displaystyle=\
\pm\sqrt{\tilde{P}(\tilde{f})}\,,\quad\quad\tilde{\Phi}(\tilde{f})=\log|\tilde{f}|+\Phi_{1}\,,$
(3.28a) $\displaystyle dx$ $\displaystyle=\
\pm{d\tilde{f}\over\tilde{f}\sqrt{\tilde{P}(\tilde{f})}}\,,$ (3.28b)
$\displaystyle{\tilde{m}(x_{2})\over\tilde{m}(x_{1})}$
$\displaystyle={\tilde{m}_{\pm}(\tilde{f}_{2})\over\tilde{m}_{\pm}(\tilde{f}_{1})}=\exp\biggl{(}\pm\int_{\tilde{f}_{1}}^{\tilde{f}_{2}}{df\over
f}{\tilde{M}(f)\over\sqrt{\tilde{P}(f)}}~{}\biggr{)}\,,$ (3.28c)
$\displaystyle\tilde{R}_{\pm}(\tilde{f})$
$\displaystyle=2\Bigl{(}\tilde{M}^{2}+{d\tilde{M}\over
d\tilde{f}}{d\tilde{f}\over
dx}\Bigr{)}=2\Bigl{(}\tilde{M}^{2}\pm\tilde{f}\,{d\tilde{M}\over
d\tilde{f}}\sqrt{\tilde{P}(\tilde{f})}\ \ \Bigr{)}\,.$ (3.28d)

Here the argument of the square roots, called $\tilde{P}(\tilde{f})$, is
defined as follows

$\tilde{P}(\tilde{f})\equiv-1+{\int_{0}^{{\tilde{f}}}}\tilde{M}(\tilde{f}^{\prime})d\tilde{f}^{\prime}\,.$
(3.29)

Since $\tilde{P}(\tilde{f}=0)=-1<0$, this time the solution excludes the point
$\tilde{f}=0$. Moreover, in general it is not even guaranteed that the
interior solution exist at all. There would be no interior solution if
$\tilde{P}(\tilde{f})$ is negative for all $\tilde{f}$.

Using (2.54) we can determine the function $\tilde{M}(\tilde{f})$ for the
interior region from the function $M(f)$ for the exterior region given in
(3.22):

$\tilde{M}(\tilde{f})=\tfrac{1}{2}\tilde{f}\left[1-\tfrac{\epsilon}{2}\,\tilde{f}^{2}+\mathcal{O}(\epsilon^{2}\tilde{f}^{4})\right]=\tfrac{1}{2}\tilde{f}\left[1+h(-\tilde{f}^{2})\right]\,,$
(3.30)

with the same $h(\xi)$ defined in (3.23).

The range of validity of the interior solution is very different from the one
of the exterior solution. From now on we assume there is always at least one
positive branch point $f_{0}>0$ such that $\tilde{P}(f_{0})=0$ and
$\tilde{P}(\tilde{f})>0$ for some interval $\tilde{f}\in(f_{0},f_{1})$, where
$f_{1}>f_{0}$ can be another branch point or the point at infinity. If this is
the case, due to the even parity of $\tilde{P}(\tilde{f})$, the interior
solution exists at least for an interval
$\tilde{f}\in(-f_{1},-f_{0})\cup(f_{0},f_{1})$. However, since for the
interior solution T-duality also acts as

$\tilde{f}\to\hat{\tilde{f}}=-\tilde{f}\,,$ (3.31)

both negative and positive regions are related via a duality transformation.
We are imposing to have at least one branch point for positive $f$. With
multiple such branch points it is possible to have multiple separated domains
within $\tilde{f}>0$ such that $\tilde{P}(\tilde{f})$ is positive in each of
those disconnected regions.

###  3.3 The standard black hole

Here we work out the presentation of the standard two-derivative BH solution
of section 2.1 in the Gasperini-Veneziano parametrization. The experience
gained here will prove essential for the later generalizations.

Exterior solution: The $M(f)$ ansatz for the exterior region of the two-
derivative theory corresponds to the limit $\epsilon\to 0$ in (3.22):

$M(f)=-\tfrac{1}{2}f\,.$ (3.32)

Associated to $M$ we have $P(f)$ given in (3.20):

$P(f)=1+\tfrac{1}{4}\,f^{2}\ >\ 0\ \quad\forall f\in\mathbb{R}\,,$ (3.33)

which implies that all $f$ are in principle allowed. The duality-invariant
dilaton does not depend on the specific ansatz for $M(f)$, so it is still
given by (3.14). Using (3.33) in (3.11), its $x$-derivative is given by

${d\Phi\over dx}=\sqrt{1+\tfrac{1}{4}f^{2}}\,,$ (3.34)

where we choose the plus sign to fit conventions where $x=0^{-}$ is the
horizon and the exterior region is $x<0$, with the AF region at $x\to-\infty$.
Moreover, to stick to this convention, we need to choose $f\geq 0$. The
relation between $x$ and $f$ can be obtained from (3.13), which for this
ansatz takes the form

$dx={df\over f}{1\over\sqrt{1+\tfrac{1}{4}f^{2}}}=\
-d\,\hbox{ArcSinh}\,\tfrac{2}{f}\,.$ (3.35)

Integrating,

$x(f)=\ -\hbox{ArcSinh}\tfrac{2}{f}\ \ \to\ \
\tfrac{f}{2}=-\,\tfrac{1}{\sinh\,x}\,,$ (3.36)

where we choose the integration constant to be zero. The $x=0^{-}$ horizon
corresponds to $f=\infty$ and the AF region $x=-\infty$ corresponds to
$f=0^{+}$:

$\hbox{AF region:}\ x=-\infty\,,\ \ f=0^{+}\,,\quad\ \hbox{Horizon:}\ x=0,\ \
f=+\infty\,.$ (3.37)

Increasing $x$ corresponds to increasing $f$.

Figure 3: Black hole exterior. The solid line is the plot of the dilaton
derivative $d\Phi/dx$ and $M$ as a function of $f\in[0,\infty)$. The point on
the horizontal axis is $f=0$ and represents the AF region. As $f\to\infty$ we
reach the horizon and the curve asymptotes to the dashed line
$|M|=|{d\Phi\over dx}|$.

In order to determine the metric we take the plus sign in (3.19c) and use
(3.32) and (3.33):

${m(f_{2})\over
m(f_{1})}=\exp\Biggl{(}-\frac{1}{2}\int_{f_{1}}^{f_{2}}{df\over\sqrt{1+{f^{2}\over
4}}}\Biggr{)}\,.$ (3.38)

We can integrate this analytically, but before doing so it is instructive to
consider the ratio of the scale factor at the horizon $f=\infty$ divided by
the value at the AF region about $f=0$:

${m(f=\infty)\over
m(f=0)}=\exp\Biggl{(}-\frac{1}{2}\int_{0}^{\infty}{df\over\sqrt{1+{f^{2}\over
4}}}\Biggr{)}=0\,,$ (3.39)

due to the logarithmic divergence at the top limit of integration, and
confirming the vanishing of the metric at the horizon: $m(f=\infty)=0$. Since
the scale of $m$ is not detected by the equations of motion, we can set
$m(f=0)=1$. Setting $f_{2}=f$ and $f_{1}=0$ in (3.38) we get

$m(f)=\exp\Biggl{(}-\frac{1}{2}\int_{0}^{f}{df^{\prime}\over\sqrt{1+{f^{\prime
2}\over
4}}}\Biggr{)}=\exp\Bigl{(}-\hbox{ArcSinh}\tfrac{f}{2}\Bigr{)}=-\tfrac{f}{2}+\sqrt{1+\tfrac{f^{2}}{4}}\,.$
(3.40)

This indeed gives $m(0)=1$, and $m(f)\simeq 1/f$ for large $f$, so that
$m(\infty)=0$. We now compute the curvature from (3.18), using the top sign
and finding

$\displaystyle R(f)=\ f\left(-{f\over 2}+\sqrt{1+{f^{2}\over
4}}\right)=fm(f)\,.$ (3.41)

Note that $R(0)=0$, as befits the AF region, and $R(f)\simeq 1$ for
$f\to\infty$, which is the value of the curvature at the horizon.

At this point we have $M(f),\Phi(f),m(f)$ and $R(f)$, and so the full
$f$-parameterized solution is determined. As a consistency check one may use
the relation between $f$ and $x$ to recover the BH solution in the familiar
form reviewed in section 2.2. For the metric, for example, we have, using
(3.36), and recalling that $x$ is negative,

$m(x)=\tfrac{1}{\sinh x}+\sqrt{1+\tfrac{1}{\sinh^{2}x}}=\tfrac{1}{\sinh
x}\bigl{(}1-\sqrt{1+\sinh^{2}x}\bigr{)}=-\tanh\tfrac{x}{2}\,.$ (3.42)

For the dilaton we have, since $f>0$,

$\Phi=\log f+c=\log\bigl{(}\tfrac{-2}{\sinh x}\bigr{)}+c=-\log|\sinh
x|+c^{\prime}\,.$ (3.43)

Interior solution:

The BH interior solution of (2.31) can also be recovered in the $\tilde{f}$
parameterization, using (3.28). The ansatz for $\tilde{M}(\tilde{f})$ is
obtained by using the formula (2.54) on the ansatz for the exterior (3.32):

$\tilde{M}(\tilde{f})=\tfrac{1}{2}\tilde{f}\,.$ (3.44)

The argument of the square root this time takes the form

$\tilde{P}(\tilde{f})=-1+\tfrac{1}{4}\tilde{f}^{2}\,,$ (3.45)

and so the square root has branch cuts on the real line with $|\tilde{f}|>2$,
with branch points at $\tilde{f}=\pm 2$. Therefore, the interior solution is
valid for the interval $\tilde{f}\in(-\infty,-2)\cup(2,\infty)$, where
negative and positive regions are related via T-duality (3.31). From now on we
choose $\tilde{f}\geq 2$.

In order to describe all of the internal region in the
$\tilde{f}$-parameterization we need to cover the range $\tilde{f}\geq 2$
twice. We begin from $\tilde{f}=\infty$ to $\tilde{f}=2$ traveling under the
branch cut, and then we return from $\tilde{f}=2$ to $\tilde{f}=\infty$ over
the cut. The difference between paths is given by the choice of $\pm$ sign in
front of the square root in the solution (3.28). The square root is assumed to
take negative values below the cut ($-$ sign) and positive values above the
cut ($+$ sign).

When going under the cut, the dilaton derivative is given by

$\frac{d\tilde{\Phi}}{dx}(\tilde{f})=-\sqrt{-1+\tfrac{1}{4}\tilde{f}^{2}}\,.$
(3.46)

The relation between $x$ and $\tilde{f}$ in (3.28b) takes the form

$dx=-\frac{d\tilde{f}}{\tilde{f}\sqrt{-1+\tfrac{1}{4}\tilde{f}^{2}}}=-\frac{d\tilde{f}}{|\tilde{f}|\sqrt{-1+\tfrac{1}{4}\tilde{f}^{2}}}=d\,\hbox{arccsc}\,\tfrac{\tilde{f}}{2}\,,\quad\tilde{f}\geq
2\,.$ (3.47)

Integrating this equation and setting the integration constant to zero we get

$x(\tilde{f})=\hbox{arccsc}\,\tfrac{\tilde{f}}{2}\quad\to\quad\tfrac{\tilde{f}(x)}{2}=\csc
x=\frac{1}{\sin x}\,.$ (3.48)

In this parameterization as $\tilde{f}$ decreases $x$ increases. Indeed, we
have $x(\tilde{f}=\infty)=0$ and $x(\tilde{f}=2)=\tfrac{\pi}{2}$:

$\tilde{f}\in[\infty,2]_{-}\ \to\ x\in[0,\tfrac{\pi}{2}]\,,$ (3.49)

with the minus subscript indicating that we are traveling under the cut. As we
will see, $\tilde{f}=\infty$ ($x=0$) corresponds to the position of the
horizon, while $\tilde{f}=2$ ($x=\tfrac{\pi}{2}$) is just an intermediate
point in the interior solution.

For the metric we need (3.28c) with the minus sign. Choosing the lower
boundary to be $\tilde{f}_{1}=2$ and leaving the upper boundary arbitrary we
get

$\tilde{m}_{-}(\tilde{f})=\tilde{m}_{-}(2)\exp\Biggl{(}-\tfrac{1}{2}\int_{2}^{\tilde{f}}\frac{df}{\sqrt{-1+\tfrac{1}{4}f^{2}}}\Biggr{)}=\tilde{m}_{-}(2)\exp\left(-\hbox{arccosh}\tfrac{\tilde{f}}{2}\right)\,,$
(3.50)

and since $\hbox{arccosh}(1)=0$ we can take $\tilde{m}_{-}(2)=1$. Since
$\hbox{arccosh}\,y=\log\left(y+\sqrt{-1+y^{2}}\right)$, we obtain

$\tilde{m}_{-}(\tilde{f})=\tfrac{\tilde{f}}{2}-\sqrt{-1+\tfrac{\tilde{f}^{2}}{4}}=\tfrac{\tilde{f}}{2}\left(1-\sqrt{-\tfrac{4}{\tilde{f}^{2}}+1}\right)\,,$
(3.51)

where for the second equality we used that $\tilde{f}$ is positive. For the
curvature one can show with (3.28d) that

$\tilde{R}_{-}(\tilde{f})=\tilde{f}\left(\tfrac{\tilde{f}}{2}-\sqrt{-1+\tfrac{\tilde{f}^{2}}{4}}\right)=\tilde{f}\tilde{m}_{-}(\tilde{f})\,.$
(3.52)

As anticipated, $\tilde{m}_{-}(\tilde{f})$ and $\tilde{R}_{-}(\tilde{f})$ are
consistent with $\tilde{f}=\infty$ being the location of the horizon since

$\lim_{\tilde{f}\to\infty}\tilde{m}_{-}(\tilde{f})=0\,,\quad\lim_{\tilde{f}\to\infty}\tilde{R}_{-}(\tilde{f})=1\,,$
(3.53)

as can be checked with the expansion around $\tilde{f}\to\infty$ of the second
equality in (3.51). On the other hand, $\tilde{f}=2$ is just an intermediate
point in the interior solution such that

$\tilde{m}_{-}(2)=1\,,\quad\tilde{R}_{-}(2)=2\,.$ (3.54)

The other half of the solution is recovered by going over the cut. In this
situation, the dilaton derivative (3.28a) and the relation between $x$ and
$\tilde{f}$ from (3.28b) have the opposite sign as in (3.46) and (3.47). For
the latter we have

$dx=-d\,\hbox{arccsc}\,\tfrac{\tilde{f}}{2}=d\,\hbox{arccos}\,\tfrac{2}{\tilde{f}}\quad\Rightarrow\quad
x(\tilde{f})=\hbox{arccos}\,\tfrac{2}{\tilde{f}}+c_{0}\,,$ (3.55)

where after integration we picked a non-trivial integration constant $c_{0}$.
This is necessary since at $\tilde{f}=2$ we already had $x=\tfrac{\pi}{2}$
from under the cut, so by continuity we must have

$\quad\tfrac{\pi}{2}=\hbox{arccos}\,1+c_{0}=c_{0}\quad\to\quad
c_{0}=\tfrac{\pi}{2}\,.$ (3.56)

Therefore,

$x(\tilde{f})-\tfrac{\pi}{2}=\hbox{arccos}\,\tfrac{2}{\tilde{f}}\quad\to\quad\tfrac{2}{\tilde{f}(x)}=\cos(x-\tfrac{\pi}{2})=\sin
x\,.$ (3.57)

Indeed we have $x(\tilde{f}=2)=\tfrac{\pi}{2}$ and $x(\tilde{f}=\infty)=\pi$,
since $x$ grows with $\tilde{f}$. So we have

$\tilde{f}\in[2,\infty]_{\tiny+}\ \to\ x\in[\tfrac{\pi}{2},\pi]\,,$ (3.58)

where the subscript $+$ indicates going on top of the cut. It is interesting
to note that the relation $\tilde{f}(x)$ is the same for under the cut, c.f.
(3.48), and over the cut, c.f. (3.57). The former covers
$x\in(0,\tfrac{\pi}{2})$ and the latter the remaining half of the interior
solution $x\in(\tfrac{\pi}{2},\pi)$.

Figure 4: Black hole interior. The solid line plots the relation between the
dilaton spatial derivative and $\tilde{M}$. The horizon H is on the upper
left, the singularity S on the upper right (both at $\tilde{f}=\infty$, below
and above the cut). The branch point at $\tilde{f}=2$ corresponds to the point
with minimal $\tilde{M}=1$. The $\tilde{f}$ contour is that shown in Fig.1
(b).

This time, taking the plus sign choice in (3.28c) and (3.28d), we obtain the
expressions for the metric and curvature over the cut:

$\tilde{m}_{+}(\tilde{f})=\tfrac{\tilde{f}}{2}+\sqrt{-1+\tfrac{\tilde{f}^{2}}{4}}\,,\quad\tilde{R}_{+}(\tilde{f})=\tilde{f}\tilde{m}_{+}(\tilde{f})\,.$
(3.59)

The intermediate point $\tilde{f}=2$ ($x=\tfrac{\pi}{2}$) gives again (3.54),
but this time the boundary $\tilde{f}=\infty$ ($x=\pi$) corresponds to the
singularity, where

$\lim_{\tilde{f}\to\infty}\tilde{m}_{+}(\tilde{f})=\infty\,,\quad\lim_{\tilde{f}\to\infty}\tilde{R}_{+}(\tilde{f})=\infty\,.$
(3.60)

As a consistency check one may again verify that the solution in the form
(2.31) follows from the relation between $\tilde{f}$ and $x$. A plot of the
dilaton derivative $d\tilde{\Phi}/dx$ and the metric derivative $\tilde{M}$,
both as a function of $\tilde{f}$ helps visualize the solution (Fig. 4). As we
go from the horizon to the singularity the dilaton derivative first decreases
and then increases. The metric $\tilde{m}$ goes from zero to infinity, while
$\tilde{M}$ first decreases and then increases.

##  4 $\alpha^{\prime}$ corrected black hole solutions

In the first part of this section we present a class of functions $M(f)$ that
lead to BH solutions in the conventional sense. These solutions have an
exterior and an interior region. The exterior includes in $f$-space the point
zero, which can be identified with the AF region, and the point at infinity,
which can be identified with a horizon. The interior region begins at a
horizon and ends in a singularity, that is not removed in this class of
solutions. Both regions, we believe, form a single solution because the
curvature on both regions attain the same value at the horizon. We make no
claims that this is the most general class of $M$’s that lead to conventional
BH solutions.

We then consider a cubic ansatz for $M(f)$, the simplest deformation of the
expression $M(f)=-\tfrac{1}{2}f$ valid in the two-derivative theory. We do not
find black hole solutions here. Instead we find situations where the exterior
and interior solutions do not seem to form a single solution, situations with
naked singularities, and situations where only the exterior solution exists.

###  4.1 A family of BH solutions

This family of solutions is parameterized by a function $h(\xi)$ via the
relation

$M(f)=-\tfrac{f}{2}\left(1+h(f^{2})\right)\,,$ (4.1)

where we take

$h(0)=0\,,\ \ \ h^{\prime}(0)<\infty\,,$ (4.2)

conditions that allow making $f=0$ the far away region of the black hole. We
also require that the correction to $M=-f/2$ implied by $h$ is ‘small’. For
this we impose

$|h(\xi)|\leq 1\,,\quad\ \forall\,\xi\in\mathbb{R}\,.$ (4.3)

This implies, in particular, that $M(f)\leq 0$ for all $f>0$. To get a horizon
we need conditions on the behavior of $h$ for large argument:

$\lim_{\xi\rightarrow\infty}h(\xi)=0\,,\ \ \
\int_{0}^{\infty}h(\xi)d\xi=\alpha<\infty\,,\ \ \ \
\lim_{\xi\rightarrow\infty}\xi h^{\prime}(\xi)=0\,.$ (4.4)

The first condition implies that $M(f)\simeq-f/2$ for very large $f$. The
second condition strengthens the bound $|h(\xi)|\leq 1$ by requiring that the
full integral over positive arguments be finite. The third condition follows
from the first for a regular function at infinity.

For the interior of the black hole, we need conditions on $h(\xi)$ for large
negative $\xi$. At the cost of some generality, we simply demand that $h$ be
an odd function of its argument:

$h(-\xi)=-h(\xi)\,.$ (4.5)

The interior solution demands one further constraint. Let $f_{0}>0$ be the the
value of $f$ where the function $\tilde{P}(f)$, defined in (3.29), first goes
from negative to positive, so that $\tilde{P}(f_{0})=0$. We will demand that

$h(f_{0}^{2})<1\,,$ (4.6)

in particular, $h(f^{2})$ is not equal to one at this point.

Exterior solution: For the exterior solution, we have equations (3.19).
Picking the plus sign and inserting the function $M(f)$ as given in (4.1), the
equations for the metric and curvature reduce to

$\displaystyle{m(f_{2})\over m(f_{1})}$
$\displaystyle=\exp\biggl{(}\int_{f_{1}}^{f_{2}}dfI_{e}(f)~{}\biggr{)}\,,\quad
I_{e}(f)\equiv\,-\ \frac{1+h(f^{2})}{2\sqrt{P(f)}}\,,$ (4.7a) $\displaystyle
R(f)$
$\displaystyle=-\frac{f}{2}\left[f(1+h(f^{2}))^{2}-2\left(1+h(f^{2})+2f^{2}\frac{dh(f^{2})}{df^{2}}\right)\sqrt{P(f)}\
\right]\,,$ (4.7b)

with the subscript $e$ for ‘exterior’ and

$P(f)=1+\tfrac{1}{4}f^{2}+\tfrac{1}{4}\int_{0}^{f^{2}}d\xi\,h(\xi)\,,$ (4.8)

which determines the location of branches. From the bound (4.3) on $h(\xi)$,
we infer that

$-f^{2}\leq\int_{0}^{f^{2}}d\xi\,h(\xi)\leq f^{2}\,,$ (4.9)

and we conclude that

$P(f)\geq 1\quad\forall\,f\in\mathbb{R}\,,$ (4.10)

showing that there are no branch cuts in the $f$ plane and we can work with

$f\in(0,\infty)\,.$ (4.11)

We now consider the behavior of the integrand $I_{e}$. Using the vanishing of
$h(\xi=0)$ and the first two conditions in (4.4), we conclude

$\lim_{f\rightarrow
0}I_{e}(f)=-\tfrac{1}{2}\,,\quad\lim_{f\rightarrow\infty}I_{e}(f)\simeq-\tfrac{1}{f}\quad\Rightarrow\quad\int_{0}^{f}df^{\prime}I_{e}(f^{\prime})\simeq-\log
f\,,\ \hbox{as}\ f\to\infty\,.$ (4.12)

For the ratio of metric values, these limits imply

${m(\infty)\over
m(0)}=\exp\biggl{(}\int_{0}^{\infty}dfI_{e}(f)~{}\biggr{)}=\exp(-\infty)=0\,,$
(4.13)

which allows us to pick the integration constant such that

$m(0)=1\,,\quad m(\infty)=0\,,$ (4.14)

the former consistent with $f=0$ being the faraway region and the latter
consistent with $f=\infty$ being the horizon.

Additionally, as required, the curvature goes to zero for $f\rightarrow 0$ and
is finite for $f\rightarrow\infty$. While the former is a general feature of
solutions coming from the classification, the behavior at the horizon is a
consequence of (4.4), which imply

$\lim_{f\rightarrow\infty}R(f)\simeq-\frac{f^{2}}{2}\left[1-\sqrt{1+\frac{4+\alpha}{f^{2}}}\right]\simeq
1+\frac{\alpha}{4}\,.$ (4.15)

Let us now show that the behavior of the dilaton is as in the standard two-
derivative black hole: the duality invariant dilaton diverges at the horizon
but the scalar dilaton is finite. For this recall that

$\Phi=\log f+\Phi_{1}\,,$ (4.16)

where $\Phi_{1}$ is a constant, and we can take the logarithm without concern
as $f\in(0,\infty)$. The asymptotically flat region is $f\sim 0$ and
corresponds to $\Phi\to-\infty$ which is weak coupling. Using (2.4) and that
$m\simeq 1$ in this region, the scalar dilaton also indicates weak coupling.
The horizon is obtained as $f\to\infty$ and this means $\Phi\to\infty$ so we
need to consider the metric contribution. We can obtain $\log m(f)$ by taking
the log of (4.7a), and picking the upper bound of the integration to be
$f_{2}=f$:

$\log m(f)=\log m(f_{1})+\int_{f_{1}}^{f}df^{\prime}I_{e}(f^{\prime})\,.$
(4.17)

For large $f$, using the last relation in (4.12) we find that

$\log m\simeq-\log f+c_{2}\,,\ \ \ \hbox{as}\ \ f\to\infty\,.$ (4.18)

where $c_{2}$ is a constant. Therefore, at the horizon $x_{h}$, which
corresponds to $f\to\infty$, from (2.4) we have

$\phi(x_{h})\simeq\tfrac{1}{2}(\log f+\Phi_{1}-\log f+c_{2})\simeq c\,,$
(4.19)

a constant, as we wanted to confirm.

Interior solution: For the interior solution, we use (2.54) to find that the
ansatz (4.1) gives

$\tilde{M}(\tilde{f})=-iM(-i\tilde{f})=\tfrac{\tilde{f}}{2}\left(1+h(-\tilde{f}^{2})\right)=\tfrac{\tilde{f}}{2}\left(1-h(\tilde{f}^{2})\right)\,,$
(4.20)

where the last equality follows because $h$ is odd. The expressions for the
metric ratios in (3.28c) and the curvature in (3.28d), become

$\displaystyle{\tilde{m}_{\pm}(\tilde{f}_{2})\over\tilde{m}_{\pm}(\tilde{f}_{1})}$
$\displaystyle=\exp\biggl{(}\pm\int_{\tilde{f}_{1}}^{\tilde{f}_{2}}dfI_{i}(\tilde{f})~{}\biggr{)}\,,\quad
I_{i}(\tilde{f})\equiv\frac{1-h(\tilde{f}^{2})}{2\sqrt{\tilde{P}(\tilde{f})}}\,,$
(4.21a) $\displaystyle\tilde{R}_{\pm}(\tilde{f})$
$\displaystyle=\frac{\tilde{f}}{2}\left[\tilde{f}(1-h(\tilde{f}^{2}))^{2}\pm
2\left(1-h(\tilde{f}^{2})-2\tilde{f}^{2}\frac{dh(\tilde{f}^{2})}{d\tilde{f}^{2}}\right)\sqrt{\tilde{P}(\tilde{f})}\
\right]\,,$ (4.21b)

with $i$ for ‘interior’. Regarding branch cuts, this time the range of
$\tilde{f}$ depends on the roots of the function

$\tilde{P}(\tilde{f})=-1+\tfrac{1}{4}\tilde{f}^{2}-\tfrac{1}{4}\int_{0}^{\tilde{f}^{2}}d\xi\,h(\xi)\,.$
(4.22)

This function picks the following values at the boundaries

$\tilde{P}(0)=-1<0\,,\quad\lim_{\tilde{f}\rightarrow\pm\infty}\tilde{P}(\tilde{f})=\lim_{\tilde{f}\rightarrow\pm\infty}\bigl{(}-1-\tfrac{1}{4}\alpha+\tfrac{1}{4}\tilde{f}^{2}\bigr{)}=\infty\,,$
(4.23)

where we used the second equation of (4.4). This shows that
$\tilde{P}(\tilde{f})$ must change sign as $\tilde{f}$ grows from zero. Let us
call $f_{0}$ the value where $\tilde{P}(f_{0})=0$; a branch point. Notice also
that

${d\tilde{P}\over
d\tilde{f}}=\tfrac{1}{2}\tilde{f}\,\bigl{(}1-h(\tilde{f}^{2})\bigr{)}\geq
0\,,$ (4.24)

since $|h|\leq 1$. Note, however, that $h(f_{0}^{2})<1$ by assumption so that

${d\tilde{P}\over d\tilde{f}}\Bigl{|}_{f_{0}}>0\,.$ (4.25)

Once $\tilde{P}(\tilde{f})$ becomes positive it will remain positive all the
way since its derivative is never negative. Therefore, the range of validity
of the solution is taken to be

$\tilde{f}\in(f_{0},\infty)\,.$ (4.26)

Using (4.9) one can show that $f_{0}>\sqrt{2}$. For the standard BH we had
$f_{0}=2$.

We now investigate the condition for the horizon. We have from (4.21a)

${\tilde{m}_{\pm}(\infty)\over\tilde{m}_{\pm}(f_{0})}=\exp\biggl{(}\pm\int_{f_{0}}^{\infty}d\tilde{f}I_{i}(\tilde{f})~{}\biggr{)}\,.$
(4.27)

We need the right hand side to be infinite for the plus sign; this makes
$\tilde{m}(\infty)$ over the cut infinite, assuming $\tilde{m}(f_{0})$ is
finite. This is as desired. For the minus sign the right hand side should be
zero, which makes $\tilde{m}(\infty)$ below the cut equal to zero — this is as
desired as it corresponds to the horizon. To show this, we consider the
integral of $I_{i}$ on the right-hand side. At the branch point $f_{0}$ there
is an integrable square-root singularity: $\tilde{P}(f_{0})=0$ and the
derivative $\tilde{P}^{\prime}(f_{0})$ is nonzero, see (4.24). We thus need
only focus on the behavior of $I_{i}$ for large $\tilde{f}$. For the
$\tilde{f}\rightarrow\infty$ limit, using (4.4) we find

$I_{i}(\tilde{f})\simeq\frac{1}{\tilde{f}}\ \hbox{as}\ \
\tilde{f}\to\infty\,\quad\Rightarrow\
\quad\int_{f_{0}}^{\tilde{f}}d\tilde{f}^{\prime}I_{i}(\tilde{f}^{\prime})\simeq\ln\tilde{f}\,,\
\ \ f\to\infty\,.$ (4.28)

Finally, inserting this into (4.27) we get
${\tilde{m}_{\pm}(\infty)\over\tilde{m}_{\pm}(f_{0})}=e^{\pm\infty}$, from
where we can choose as integration constant a finite value of $\tilde{m}$ at
the branch point $\tilde{m}_{+}(f_{0})=\tilde{m}_{-}(f_{0})=\tilde{m}_{0}$, so
that we have, as desired

$\tilde{m}_{-}(\infty)=0\,,\ \ \ \hbox{and}\ \ \
\tilde{m}_{+}(\infty)=\infty\,.$ (4.29)

The curvature at the branch point $f_{0}$ is easily evaluated since
$\tilde{P}(f_{0})$ vanishes:

$\tilde{R}_{-}(f_{0})=\tilde{R}_{+}(f_{0})=2\tilde{M}^{2}(f_{0})=\tfrac{1}{2}f_{0}^{2}\left(1-h(f_{0}^{2})\right)^{2}\,.$
(4.30)

For the infinite limit we have, from (4.21b)

$\tilde{R}_{\pm}(\tilde{f})\simeq\tfrac{1}{2}{\tilde{f}^{2}}\Bigl{[}1\pm\sqrt{1-\tfrac{4+\alpha}{\tilde{f}^{2}}}\
\Bigr{]}\,,\ \ \ \tilde{f}\to\infty\,.$ (4.31)

As a result we find

$\lim_{\tilde{f}\rightarrow\infty}\tilde{R}_{-}(\tilde{f})=1+\tfrac{\alpha}{4}\,,\quad\lim_{\tilde{f}\rightarrow\infty}\tilde{R}_{+}(\tilde{f})=\infty\,.$
(4.32)

The first is the curvature at the horizon as computed from the interior. As
required, it coincides with the curvature at the horizon computed from the
exterior in (4.15). The second result is the infinite curvature at the
singularity.

For the duality-invariant dilaton in the interior region we have

$\tilde{\Phi}=\log\tilde{f}+\Phi_{1}\,,$ (4.33)

which is well defined for $\tilde{f}\in(f_{0},\infty)$. $\tilde{\Phi}$ does
not differentiate between lower or upper branches. It diverges at
$\tilde{f}\to\infty$ but it is finite for $\tilde{f}=f_{0}$. For the scalar
dilaton we use

$\tilde{\phi}_{\pm}(\tilde{f})=\tfrac{1}{2}\left(\tilde{\Phi}(\tilde{f})+\log\tilde{m}_{\pm}(\tilde{f})\right)\,,$
(4.34)

which does distinguish branches through $\tilde{m}_{\pm}$. While it is clear
that $\tilde{\phi}_{+}(f_{0})=\tilde{\phi}_{-}(f_{0})=\text{finite}$, for
$\tilde{f}\to\infty$ under and over the cut we need $\log\tilde{m}_{\pm}$. To
this end, as we did for the exterior region, we integrate the metric ratio in
(4.21a)

$\log\tilde{m}_{\pm}(f)=\log\tilde{m}_{\pm}(f_{1})\pm\int_{f_{1}}^{\tilde{f}}d\tilde{f}^{\prime}I_{i}(\tilde{f}^{\prime})\,.$
(4.35)

Using large $\tilde{f}$ value of the integral from (4.28), we find

$\log\tilde{m}_{\pm}(f)\simeq\pm\log\tilde{f}+c_{\pm}\,,\ \ \ \
\tilde{f}\to\infty\,,$ (4.36)

where $c_{\pm}$ are constants. Plugging this expansion back into (4.34),
together with (4.33) we find that for $\tilde{f}\to\infty$

$\tilde{\phi}_{-}(\tilde{f})\simeq\tilde{c}\,,\quad\ \
\tilde{\phi}_{+}(\tilde{f})\simeq\log\tilde{f}\,.$ (4.37)

The first one is consistent with the horizon interpretation, where we need
$\tilde{c}=c$ with $c$ in (4.19), in order to match exterior and interior
regions. The behavior of $\tilde{\phi}_{+}$ close to infinity signals, once
more, the presence of a singularity there.

To sum up, the general ansatz (4.1) parameterized with a function $h(\xi)$
satisfying (4.3), (4.4) and (4.5) lead to BH solutions. There are infinitely
many functions satisfying these conditions. For example

$h(\xi)=\xi\exp(-\xi^{2})\,,$ (4.38)

which satisfies $\int_{0}^{\infty}h(\xi)d\xi=\tfrac{1}{2}$ and
$|h(\xi)|\leq{1\over\sqrt{2e}}<1$.

The way the more general solutions are parameterized in terms of $f$ mimics
the standard BH. The exterior corresponds to $f\in(0,\infty)$ with $f=0$ the
faraway region and $f=\infty$ the horizon. The interior region is covered by
two patches: $\tilde{f}\in(f_{0},\infty)_{-}$ and
$\tilde{f}\in(f_{0},\infty)_{+}$, where the first goes under the branch cut
(picking the minus sign in (4.21)) and the second over it (with the plus sign
in (4.21)). We begin under the branch at $\tilde{f}=\infty$ (the horizon)
until $\tilde{f}=f_{0}$, the branch point, where all quantities are finite.
From there, we return to $\tilde{f}=\infty$ over the branch cut. This time,
the metric and curvature diverge at this point, which is identified as the
singularity.

###  4.2 Cubic ansatz for $M(f)$

The class of solutions introduced above assume a function $h(\xi)$ that
vanishes at zero and at infinity, among other conditions. Here we assume that
$h(\xi)$ is just linear in $\xi$, leading to a cubic ansatz for $M(f)$:

$M(f)=-\tfrac{1}{2}f+c\,f^{3}\,,\quad h(\xi)=-2c\,\xi\,,$ (4.39)

with $c\neq 0$, otherwise we are back to the two-derivative BH. This $h(\xi)$
does not satisfy the conditions in (4.3) and (4.4), and so the physics is very
different. We will find that the solutions do not seem to have a BH
interpretation due to the presence of naked singularities and disconnected
regions. This shows that arbitrary $\alpha^{\prime}$ corrections can easily
eliminate BH solutions. We were also motivated to consider this case because
some of the pre-big bang solutions of [10] were constructed with a cubic
ansatz for $M(f)$.

Without going into the calculations which are straightforward, we proceed to
enumerate the main characteristics of these solutions. To this end, we need
the branch structures of exterior and interior solutions, which can be
inferred respectively from

$P(f)=1+\tfrac{1}{4}f^{2}-\tfrac{c}{4}f^{4}\,,\quad\hbox{and}\quad\tilde{P}(\tilde{f})=-1+\tfrac{1}{4}\tilde{f}^{2}+\tfrac{c}{4}\tilde{f}^{4}\,.$
(4.40)

The analysis is different depending on the sign of $c$. For simplicity we give
the results for $f>0$, but similar results hold for the negative region, which
is connected to the positive one via T-duality.

  * •

$c>0$:

The exterior region is parameterized by $f\in(0,f_{0})$, where

$f_{0}^{2}=\frac{\sqrt{1+16c}+1}{2c}\simeq\frac{1}{c}+\mathcal{O}(\sqrt{c})\,,\
\ c\to 0\,.$ (4.41)

For positive $f$, the function $P(f)$ only vanishes at $f_{0}$. Since
$f_{0}\sim 1/\sqrt{c}$, the appearance of a branch cut for the exterior
solution is non-perturbative in $c$. As it should be, $f=0$ still corresponds
to the far away region. This time, the boundary at $f_{0}$ is a point with
non-vanishing metric and finite curvature.

The interior region exists for $|\tilde{f}|\geq\tilde{f}_{0}$, with
$\tilde{f}_{0}^{2}=\tfrac{\sqrt{1+16c}-1}{2c}$. As in the standard BH and the
class of deformations studied above, we cover the solution by the two patches
$\tilde{f}\in(\tilde{f}_{0},\infty)_{-}\cup(\tilde{f}_{0},\infty)_{+}$,
corresponding to going under the cut and then coming back over the cut. At one
end, $\tilde{m}_{-}=0$ but $\tilde{R}_{-}=\infty$. At the other end
$\tilde{m}_{+}=\infty$ and $\tilde{R}_{+}=\infty$.

The absence of a horizon in the exterior and the presence of singularities in
the interior makes this solution incompatible with a black hole. Moreover, the
two geometries do not connect. Each one is a separate solution.

  * •

$c<0$:

In this situation, $P(f)$ is always positive and therefore the exterior
solution is valid for $f\in(0,\infty)$ as in the two-derivative case. As
usual, $f\sim 0$ is the AF region, but $f\to\infty$ is not a horizon but a
point with vanishing metric with infinite curvature. This region then
corresponds to a naked singularity.

The interior region has now two branch points, $\tilde{f}_{0}$ and
$\tilde{f}_{1}$. The point $\tilde{f}_{0}$ is the same as for the $c>0$ case,
and $\tilde{f}_{1}$ is given from
$\tilde{f}_{1}^{2}=\tfrac{-\sqrt{1+16c}-1}{2c}$. The interior region only
exists for $c>-\tfrac{1}{16}$ and $\tilde{f}\in(\tilde{f}_{0},\tilde{f}_{1})$.
Since this $\tilde{f}$ interval is finite, is easy to see that
$\tilde{m}_{\pm}$ and $\tilde{R}_{\pm}$ are regular everywhere but the metric
never vanishes. For $c<-\tfrac{1}{16}$, there is no interior solution, just an
exterior one.

When both exist, the exterior and the interior solutions do not combine. They
are separate solutions, and we have no black hole.666The construction of [19]
was in this class of negative $c$ theories, so while there seems to be a
regular interior solution, the exterior solution does not work out.

##  5 Horizons and singularities

Having gained experience on the constraints of black hole solutions in the
previous section, we now want to discuss some general lessons. We show that
for dilaton-gravity theories in two-dimensional time-independent backgrounds
T-duality implies a constraint on the construction of regular black holes
(geometries with a horizon but no curvature singularity). We will prove that T
duality maps a geometry with a regular horizon to one with a curvature
singularity. This shows that the observation of Giveon [9] for the two-
derivative black hole is general. For a solution that is expected to be self-
dual, this means one cannot have a horizon and no singularity. Moreover, we
explain, as a corollary, how the usual branch structure of the interior
solution implies self-duality and thus a singularity. In the next section we
will give black hole solutions that have an unusual branch structure and that
are not self-dual so that the singularity can be avoided.

To prove that a horizon implies a singularity in the dual geometry we begin
with a configuration where, without loss of generality, we can take $x=0$ as
the position of the horizon, the point where the metric $m(x)$ vanishes. A
regular configuration is one where the scalar curvature is finite everywhere.
Since the conclusions should be independent of any gauge choice, we pick $n=1$
for simplicity.

Demanding $R=-2m^{\prime\prime}/m$ ((2.13)) to be finite everywhere implies,
in particular, ${m^{\prime\prime}(x)}/{m(x)}$ must be finite at the horizon.
Then, from $\hat{R}=-R-4M^{2}$, ((2.9)), we see that regularity of the dual
curvature $\hat{R}$ additionally forces $M=m^{\prime}/m$ to be finite
everywhere, in particular at the horizon. Together with the horizon condition
we obtain

$\lim_{x\to 0}m(x)=0\,,\ \ \ \lim_{x\to 0}{m^{\prime\prime}(x)\over
m(x)}=\hbox{finite}\,,\ \ \ \ \ \ \lim_{x\to 0}{m^{\prime}(x)\over
m(x)}=\hbox{finite}\,.$ (5.1)

To gain intuition, we now show that (5.1) cannot be satisfied for analytic
metric components. If $m(x)$ is analytic, it admits a regular Taylor expansion
around $x=0$, the location of the horizon,

$m=a_{1}x+\tfrac{1}{2}a_{2}x^{2}+\tfrac{1}{3}a_{3}x^{3}+\cdots$ (5.2)

where the first condition in (5.1) rules out the presence of a constant term.
Plugging (5.2) into the second condition of (5.1) we find

$\lim_{x\to 0}{m^{\prime\prime}(x)\over
m(x)}\simeq\frac{a_{2}}{a_{1}}\frac{1}{x}+\dots$ (5.3)

which is finite only if $a_{2}=0$. Finally, with $a_{2}=0$ the third condition
on (5.1) now reads

$\lim_{x\to 0}{m^{\prime}(x)\over m(x)}\simeq\frac{1}{x}+\dots$ (5.4)

which diverges regardless the coefficients of the Taylor expansion. This
logarithmic derivative diverges at the position of the horizon. As claimed the
conditions cannot be satisfied.

A proof that a horizon implies a curvature singularity

Having considered the case when $m(x)$ is analytic we now build a general
proof. We define the neighborhood $N_{\epsilon}=(0,\epsilon)$, and the
neigborhood $\bar{N}_{\epsilon}=[0,\epsilon)$ where we include the point $0$.
We assume that there is an $\epsilon$ sufficiently small such that the
following hold:

  1. 1.

The function $m(x)$ is continuous in $\bar{N}_{\epsilon}$ and vanishing at
$x=0$. This is an isolated zero: the neighborhood $\bar{N}_{\epsilon}$ does
not contain another zero of $m$.

  2. 2.

The curvature $R(x)$ in $\bar{N}_{\epsilon}$ is finite.

These are the natural conditions satisfied by a conventional horizon.

Claim: The dual curvature $\hat{R}$ cannot be finite in $\bar{N}_{\epsilon}$.

Proof. Since the curvature $R$ and the dual curvature $\hat{R}$ are related as
follows

$\hat{R}=-R-4\Bigl{(}{m^{\prime}\over m}\Bigr{)}^{2}\,,$ (5.5)

and $R$ is assumed finite in $\bar{N}_{\epsilon}$ this means that the
finiteness condition

${m^{\prime}(x)\over m(x)}<\infty\,,\ \ \ \ \ \forall
x\in\bar{N}_{\epsilon}\,,$ (5.6)

cannot hold. Let us assume it holds and derive a contradiction.

Since the sign of $m(x)$ is immaterial to the definition of the metric
component $g_{00}$, we can change it at will so that we can require $m(x)>0$
for $x\in N_{\epsilon}$. This follows from the continuity of $m$ and the
isolated zero. Given the positivity of $m(x)$ in $\bar{N}_{\epsilon}$ we can
write $m(x)$ in terms of its logarithm $g(x)$:

$m(x)=e^{g(x)}\,,\ \ \hbox{with}\ \ m(0)=0\quad\to\quad g(0)=-\infty\,.$ (5.7)

Since $m(x)\not=0$ in $N_{\epsilon}$, the function $g(x)$ is finite in
$N_{\epsilon}$. Taking a derivative of $m$ we have

${m^{\prime}(x)\over m(x)}=g^{\prime}(x)\,.\ $ (5.8)

Given the hypothesis (5.6) we conclude that $g^{\prime}(x)<\infty$ in
$\bar{N}_{\epsilon}$. Now consider the relation777We thank Daniel Harlow for a
discussion where he suggested this line of reasoning.

$g(0)=g(x)-\int_{0}^{x}g^{\prime}(u)du\,,\ \ \ \forall\ x\in N_{\epsilon}\,,$
(5.9)

and examine the right-hand side above. For $x\in N_{\epsilon}$, the function
$g(x)$ is finite. Moreover, since $g^{\prime}$ is finite in
$\bar{N}_{\epsilon}$ and $x$ is finite, the integral is also finite. But with
both terms on the right-hand side finite, we cannot have $g(0)=-\infty$ as
required. This contradiction means that (5.6) does not hold; we need
$g^{\prime}$ to diverge in order for the integral above to give an infinite
$g(0)$. This is what we wanted to show. The proof does not tell us where is
$\hat{R}$ infinite, but the intuition of the analytic case indicates that it
is expected to be at $x=0$. $\square$

The result above is reasonable: In one dimension, T-duality maps
$m(x)\rightarrow{1}/{m(x)}$, which makes a vanishing metric unavoidably dual
to a divergent one. All we had to prove was that such dual metric has
divergent curvature.

This result actually gives a simple proof that the interior solution based on
the contour extending below and above the branch cut in $f$ space must lead to
a singularity. The main observation here is simple. Recall that $\tilde{R}$ is
given in (2.26) and its dual $\hat{\tilde{R}}$ is obtained by letting
$\tilde{M}\to-\tilde{M}$,

$\begin{split}\tilde{R}\ &=\ 2\tilde{M}^{2}+2D\tilde{M}\,,\\\ \hat{\tilde{R}}\
&=\ 2\tilde{M}^{2}-2D\tilde{M}\,.\end{split}$ (5.10)

Compare with equation (3.28d) for the curvatures above ($+$) and below ($-$)
the cut:

$\tilde{R}_{\pm}(\tilde{f})=2\tilde{M}^{2}\pm 2\tilde{f}\,{d\tilde{M}\over
d\tilde{f}}\sqrt{\tilde{P}(\tilde{f})}\ \,.$ (5.11)

Since $\tilde{M}(\tilde{f})$ and $\tilde{P}(\tilde{f})$ are functions without
branches, each term in the above curvature formula takes the same value above
or below the cut; the only difference being that they enter with different
sign combinations for the curvature and for its dual. It thus follows that we
can identify $\tilde{R}=\tilde{R}_{+}$ and $\hat{\tilde{R}}=\tilde{R}_{-}$.
The curvatures above and below the cut are related by T-duality! Since the
$\tilde{f}$ contour defining the $x$ domain extends both above and below the
cut, this means that having a horizon below the cut (as we conventionally set
it up) will imply a curvature singularity above the cut. The interior solution
is self-dual: T-duality, supplemented with the time reparameterization that
exchanges the values of $x(\tilde{f})$ for points immediately above and below
the cut, leaves the fields invariant.

##  6 Regular black holes

In this section we modify the analysis of section 4.1 where we found black
hole solutions with conventional properties. We will show that with a couple
of extra constraints on the function $M(f)$ we can obtain regular black holes:
solutions where the exterior, as usual, has an asymptotically flat region and
a horizon, and the interior has no singularity in the metric, the curvature,
or the dilaton. As in section 4.1 we parameterize $M$ in terms of a function
$h$

$M(f)=-\tfrac{1}{2}f\left[1+h(f^{2})\right]\,,$ (6.1)

with $h(\xi)$ satisfying the same properties as before: $h(0)=0$,
$h^{\prime}(0)$ is finite, $|h(\xi)|\leq 1$ for all $\xi$, $h(\infty)=0$, $\xi
h^{\prime}(\xi)|_{\xi=\infty}=0$, the parity condition $h(-\xi)=-h(\xi)$, and
$\int_{0}^{\infty}h(\xi)d\xi=\alpha<\infty$. If this the case, the interior
solution is parameterized by

$\tilde{M}(\tilde{f})=\tfrac{1}{2}\tilde{f}\left[1-h(\tilde{f}^{2})\right]\,.$
(6.2)

There was one additional condition on $h(\xi)$ (recall that $\xi=f^{2}$): it
is not supposed to reach the value of one at the point $f_{0}>0$ where the
function $\tilde{P}(f)$ first goes from negative to positive. We will
explicitly relax this condition; this will be essential for the interior
solution. Letting $\xi_{0}=f_{0}^{2}$, we demand that

$h(\xi_{0})=1\,,\ \ \ h^{\prime}(\xi_{0})=0\,,\quad
h^{\prime\prime}(\xi_{0})<0\,,\quad\hbox{and}\quad\int_{0}^{\xi_{0}}d\xi\,h(\xi)=\xi_{0}-4\,,$
(6.3)

We will confirm that with these conditions $\tilde{P}(f_{0})=0$. This fact and
the motivation for the above set of conditions will be explained below.

If we call ${\cal T}$ the space of all possible $\alpha^{\prime}$ corrections
(the space of all $M(f)$’s), and ${\cal S}$ the subspace of section 4.1
leading to conventional 2D black holes, the regular black holes arise from the
boundary of ${\cal S}$. This is clear because the inequality $h(f_{0}^{2})<1$,
needed for conventional black holes, is saturated for regular black holes.
They do not cover the full boundary, because we require additional conditions.

We will now examine the equations of motion in the $f$-parameterization, both
in the exterior and the interior, finding solutions. We will see that in the
interior the non-invertibility of $\tilde{M}(\tilde{f})$ plays a central role.
We will also explain that the solution for the regular black hole has an
interior that is a cosmology that at late times is asymptotic to Minkowski
space with a constant dilaton. This behavior suggests strongly that this is
not a solution in string theory.

###  6.1 Regular BH solutions

Exterior solution: The additional conditions (6.3) preserve all good features
of the exterior BH solution. In particular, for the exterior region $f_{0}$
has no special meaning, it is just a finite intermediate point in the
solution’s domain. The object that determines the latter is the argument of
the square root:

$P(f)=1+\tfrac{1}{4}{f^{2}}+\tfrac{1}{4}\int_{0}^{f^{2}}d\xi\,h(\xi)\,,$ (6.4)

and the domain $f\in(0,\infty)$ that we use remains unchanged since, as a
consequence of the bound $|h(\xi)|\leq 1$, we have $P(f)>0$ for such domain.
The analysis of the exterior solution in section 4.1 remains unchanged, and
the results for the metric, curvature, and dilaton derived there hold here as
well.

Interior solution: For the interior solution, the role of $f_{0}$ is crucial
to get a regular solution. The idea is that the conditions (6.3) render the
position $x(f_{0})$ infinitely faraway. Therefore, after going from the
horizon ($\tilde{f}=\infty$) to $\tilde{f}=f_{0}$ under the cut, we do not
need to go back to $\tilde{f}=\infty$ on the top of the cut, where the
singularity always was. Now, we can just stop at $f_{0}$ because in terms of
$x$ this is the end of space-time.

We begin with the function $\tilde{P}(\tilde{f})$ appearing in the interior
formulae under a square root:

$\tilde{P}(\tilde{f})=-1+\int_{0}^{\tilde{f}}\tilde{M}(\tilde{f}^{\prime})d\tilde{f}^{\prime}=-1+\tfrac{1}{4}\tilde{f}^{2}-\tfrac{1}{4}\int_{0}^{\tilde{f}^{2}}d\xi\,h(\xi)\,.$
(6.5)

We then have

$\tilde{P}(\tilde{f}=0)=-1\,,\quad\tilde{P}(f_{0})=0\,,\quad\tilde{P}(\tilde{f})\simeq\tfrac{1}{4}{\tilde{f}^{2}}\
\ {\rm for}\ \ \tilde{f}\to\infty\,.$ (6.6)

The first equality is manifest, the second equality follows from the last
condition in (6.3), and the last one from the convergence of
$\int_{0}^{\infty}h(\xi)d\xi$. To understand the nature of the interior
solution we must consider derivatives of $\tilde{P}(\tilde{f})$. The first
derivative coincides with $\tilde{M}(\tilde{f})$:

$\tilde{P}^{\prime}(\tilde{f})=\tilde{M}(\tilde{f})=\tfrac{\tilde{f}}{2}\left[1-h(\tilde{f}^{2})\right]\geq
0\quad\forall\,\tilde{f}\geq 0\,,$ (6.7)

where the inequality follows from $|h|\leq 1$. We then have the following
values

$\tilde{P}^{\prime}(0)=\tilde{M}(0)=0\,,\quad\tilde{P}^{\prime}(f_{0})=\tilde{M}(f_{0})=0\,,\quad\tilde{P}^{\prime}(\tilde{f})=\tilde{M}(\tilde{f})\simeq\tfrac{1}{2}{\tilde{f}}\
\ {\rm for}\ \ \tilde{f}\to\infty\,.$ (6.8)

The first equality is manifest, the second follows from $h(\xi_{0})=1$, the
third from the vanishing of $h$ for large argument. The second derivative is
given by

$\tilde{P}^{\prime\prime}(\tilde{f})=\tilde{M}^{\prime}(\tilde{f})=\tfrac{1}{2}\bigl{(}1-h(\xi)\bigr{)}-\xi\,h^{\prime}(\xi)\,,$
(6.9)

using $\xi=\tilde{f}^{2}$. We obtain

$\tilde{P}^{\prime\prime}(\tilde{f}=0)=\tilde{M}^{\prime}(\tilde{f}=0)=\tfrac{1}{2}\,,\quad\tilde{P}^{\prime\prime}(f_{0})=\tilde{M}^{\prime}(f_{0})=0\,.$
(6.10)

The second equality holds because $h(\xi_{0})=1$ and $h^{\prime}(\xi_{0})=0$.
For the third derivative we have

$\tilde{P}^{\prime\prime\prime}(\tilde{f})=\tilde{M}^{\prime\prime}(\tilde{f})=-3\tilde{f}h^{\prime}(\xi)-2\tilde{f}^{3}h^{\prime\prime}(\xi)\,.$
(6.11)

The special values here are

$\tilde{P}^{\prime\prime\prime}(\tilde{f}=0)=\tilde{M}^{\prime\prime}(\tilde{f}=0)=0\,,\quad\tilde{P}^{\prime\prime\prime}(f_{0})=\tilde{M}^{\prime\prime}(f_{0})=-2f_{0}^{3}\,h^{\prime\prime}(\xi_{0})>0\,,$
(6.12)

the second relation following from the assumption
$h^{\prime\prime}(\xi_{0})<0$.

All these results can be represented collectively in the two plots in Fig.5,
where we show $\tilde{M}(\tilde{f})$ and $\tilde{P}(\tilde{f})$.

Figure 5: Left: Sketch of $\tilde{M}(\tilde{f})$. The shaded region must have
unit area, the point $\tilde{f}_{*}$ indicates the position of the first local
maximum $\tilde{M}(\tilde{f}_{*})=\tilde{M}_{*}$, the point $f_{0}$ is a
minimum, and for large $\tilde{f}$ the curve approaches the line
$\tilde{f}/2$. Right: Sketch of $\tilde{P}(\tilde{f})$. This function changes
sign at $f_{0}$, which is an inflection point.

The plot of $\tilde{M}$ (left) shows a function that begins at zero with
positive slope and remains strictly positive until $f_{0}$ where
$\tilde{M}(f_{0})=\tilde{M}^{\prime}(f_{0})=0$ and
$M^{\prime\prime}(f_{0})>0$, which makes $f_{0}$ a minimum. The area under the
curve in the interval $(0,f_{0})$ is exactly one because
$\tilde{P}(f_{0})=-1+\int_{0}^{f_{0}}d\tilde{f}\tilde{M}(\tilde{f})=0.$ For
$\tilde{f}>f_{0}$ the function remains always positive (see (6.7)) and
approaches infinity exactly as the standard BH.

Let us now turn to the graph of $\tilde{P}(\tilde{f})$. This function equals
$-1$ for $\tilde{f}=0$, the minimum of $\tilde{P}(\tilde{f})$. The function
then increases with $\tilde{f}$ until $f_{0}$, where it vanishes together with
its first and second derivative. Since the third derivative is nonzero, in
fact, $\tilde{P}^{\prime\prime\prime}(f_{0})>0$, we see that $f_{0}$ is an
inflection point. Afterwards, $\tilde{P}(\tilde{f})$ keeps increasing
monotonically, as can be seen from (6.7), and at infinity it behaves as
${\tilde{f}^{2}}/{4}$.

Note now that with $h(\xi)$ as discussed, the domain for the interior solution
is $\tilde{f}\in(f_{0},\infty)$. With the function $\tilde{P}(\tilde{f})$, its
first derivative, and its second derivative all vanishing at $f_{0}$, we see
that $\tilde{P}$ near $f_{0}$ is given by

$\tilde{P}(\tilde{f})\simeq\
\tfrac{1}{6}\tilde{P}^{\prime\prime\prime}(f_{0})(\tilde{f}-f_{0})^{3}\ =\
-\tfrac{1}{3}f_{0}^{3}h^{\prime\prime}(f_{0}^{2})(\tilde{f}-f_{0})^{3}>0\quad\
\ {\rm for}\ \ \tilde{f}\sim f_{0}\,.$ (6.13)

This behavior will allow us to regularize the BH’s interior.

We first confirm the presence of the horizon in this interior solution (below
the cut). Consider the third equation in (3.28), picking the minus-sign and
integrating over the whole range of $\tilde{f}$:

$\frac{\tilde{m}(\infty)}{\tilde{m}(f_{0})}=\exp\Biggl{(}-\int_{f_{0}}^{\infty}d\tilde{f}\
{1-h(\tilde{f}^{2})\over 2\sqrt{\tilde{P}(\tilde{f})}}\Biggr{)}\,.$ (6.14)

The integral is convergent at the lower limit since the integrand in fact
vanishes there (the numerator vanishes like $(\tilde{f}-f_{0})^{2}$ and the
denominator vanishes like $(\tilde{f}-f_{0})^{3/2}$. At the upper limit we get
a logarithmic divergence since $\sqrt{\tilde{P}}\sim\tilde{f}$ (see (6.6)) and
the numerator approaches a constant. Thus the integral is infinite and the
right-hand side of (6.14) becomes zero. Since $\tilde{m}(f_{0})$ is finite we
can choose

$\tilde{m}(f_{0})=1\,,\quad\tilde{m}(\infty)=0\,.$ (6.15)

This is the horizon at $\tilde{f}=\infty$.

For the curvature, we can use (5.11) with the lower sign, as we are working
below the cut,

$\tilde{R}_{-}(\tilde{f})=2\Bigl{[}(\tilde{P}^{\prime}(\tilde{f}))^{2}-\tilde{f}\tilde{P}^{\prime\prime}(\tilde{f})\sqrt{\tilde{P}(\tilde{f})}\
\Bigr{]}\,,$ (6.16)

where we rewrote the dependence on $\tilde{M}$ in terms of derivatives of
$\tilde{P}$. The curvature vanishes at $\tilde{f}=f_{0}$, because
$\tilde{P},\tilde{P}^{\prime},$ and $\tilde{P}^{\prime\prime}$ all vanish at
$f_{0}$. For $\tilde{f}\rightarrow\infty$ the behavior of $\tilde{P}$ is the
same as we had in section 4.1, and therefore the curvature computation we did
there applies here as well. The relevant result is given by the first
expression in (4.32). In summary, we have

$\tilde{R}_{-}(f_{0})=0\,,\quad\tilde{R}_{-}(\infty)=1+\tfrac{\alpha}{4}\,.$
(6.17)

For the dilaton we have, as usual,
$\tilde{\Phi}(\tilde{f})=\log\tilde{f}+\Phi_{1}$, which is finite for $f_{0}$
and diverges for $\tilde{f}\to\infty$. The scalar dilaton is given by
$\tilde{\phi}=\tfrac{1}{2}(\tilde{\Phi}+\log\tilde{m})$. For $\tilde{f}=f_{0}$
we get $\tilde{\phi}(f_{0})=\tfrac{1}{2}\tilde{\Phi}(f_{0})$ since
$\tilde{m}(f_{0})=1$. The scalar dilaton, as opposed to the duality invariant
dilaton $\tilde{\Phi}$, has no divergence as $\tilde{f}\to\infty$. The
argument follows identically as the one used in section 4.1 to arrive at
(4.37), which in this case reads

$\tilde{\phi}(\tilde{f})\simeq\tilde{c}\ \ {\rm for}\ \ \tilde{f}\to\infty\,.$

Finally, the relation between $x$ and $\tilde{f}$ is very different from that
in the family of singular black holes: the point half-way from the horizon to
the singularity is moved to infinite distance. This is the branch point at
$f_{0}$. Using the behavior (6.13) of $\tilde{P}(\tilde{f})$ close to $f_{0}$,
the differential equation (3.13) relating $x$ and $\tilde{f}$ takes the form

$dx=-d\tilde{f}\frac{1}{\tilde{f}\sqrt{\tilde{P}(\tilde{f})}}\simeq-d\tilde{f}\sqrt{\frac{3}{f_{0}^{5}|h^{\prime\prime}(\xi_{0})|}}\
(\tilde{f}-f_{0})^{-\tfrac{3}{2}}\ \ {\rm for}\ \ \tilde{f}\sim f_{0}\,.$
(6.18)

This integral diverges as $\tilde{f}\to f_{0}$. On the other hand, for
$\tilde{f}\rightarrow\infty$ we get a finite integral: here
$\tilde{P}(\tilde{f})\simeq\tfrac{1}{2}\tilde{f}$ and therefore
$dx\simeq-2{d\tilde{f}}/\tilde{f}^{2}$, which is integrable as
$\tilde{f}\to\infty$. This allows us to define $x(\tilde{f})$ with the
following values at the boundaries

$x(f_{0})=\infty\,,\quad x(\infty)=0\,.$ (6.19)

These results make the interior region to be a regular cosmology, identifying
$x$ with a time coordinate $\tau$ and $t$ with a spatial coordinate $w$, as
required by the signature of the metric

$ds^{2}=\tilde{m}^{2}(x)dt^{2}-dx^{2}=-d\tau^{2}+\tilde{m}^{2}(\tau)dw^{2}\,.$
(6.20)

This cosmology begins at time $\tau=0$ with finite curvature
$\tilde{R}(\tau=0)=1+\tfrac{\alpha}{4}$ and evolves as $\tau\rightarrow\infty$
to a geometry with zero curvature and hence to flat space. In this cosmology
there is no big-bang, and time begins at the position of the horizon.

We now perform a local analysis to see how the solution approaches the flat
region to compare it with standard cosmology. For clarity we continue to use
$x$ (rather than $\tau$). We begin by integrating (6.18) to get the relation
between $x$ and $\tilde{f}$ close to $f_{0}$

$x(\tilde{f})\simeq\sqrt{\frac{12}{f_{0}^{5}|h^{\prime\prime}(\xi_{0})|}}\
{1\over\sqrt{\tilde{f}-f_{0}}}\ \ {\rm for}\ \ \tilde{f}\sim f_{0}\,,$ (6.21)

where we dropped an integration constant. Inverting this expression we get

$\tilde{f}(x)\simeq
f_{0}+\frac{12}{f_{0}^{5}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{2}}\ \ {\rm
for}\ \ x\to\infty\,.$ (6.22)

With this relation we can express all quantities locally in terms of $x$. For
$\tilde{M}(x)$ we use (6.2) to expand $\tilde{M}(\tilde{f})$ around
$\tilde{f}\sim f_{0}$:

$\tilde{M}(\tilde{f})=\tilde{M}(\tilde{f}_{0})+\tilde{M}^{\prime}(f_{0})(\tilde{f}-f_{0})+\tfrac{1}{2}\tilde{M}^{\prime\prime}(f_{0})(\tilde{f}-f_{0})^{2}+{\cal
O}((\tilde{f}-f_{0})^{3})\,.$ (6.23)

Recalling that $\tilde{M}(f_{0})=\tilde{M}^{\prime}(f_{0})=0$ and using the
value for the second derivative from (6.12) we get

$\tilde{M}(\tilde{f})\simeq\ f_{0}^{3}|h^{\prime\prime}(\xi_{0})|\
(\tilde{f}-f_{0})^{2}\ \ {\rm for}\ \ \tilde{f}\sim f_{0}\,,$ (6.24)

which in terms of $x$ reads

$\tilde{M}(x)\simeq\frac{144}{f_{0}^{7}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{4}}\
\ {\rm for}\ \ x\to\infty\,.$ (6.25)

Now $\tilde{m}(x)$ is given by

$\tilde{m}(x)=C\exp\left(\int^{x}dx^{\prime}\tilde{M}(x^{\prime})\right)\simeq
C\exp\left(-\frac{48}{f_{0}^{7}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{3}}\right)\simeq
C\Bigl{(}1-\frac{48}{f_{0}^{7}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{3}}\Bigr{)}\,,$
(6.26)

with $C$ a constant of integration. For the curvature we use (2.27) to get

$\tilde{R}(x)=2\left(\tilde{M}^{2}(x)+\tilde{M}^{\prime}(x)\right)\simeq
2\tilde{M}^{\prime}(x)\simeq-\frac{1152}{f_{0}^{7}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{5}}\,,$
(6.27)

where we noted that $\tilde{M}^{2}\sim x^{-8}$ and thus can be ignored to
lowest order. Finally, the duality invariant dilaton is given by

$\tilde{\Phi}=\log\tilde{f}+\Phi_{1}\simeq\tfrac{1}{f_{0}}(\tilde{f}-f_{0})+\log
f_{0}+\Phi_{1}\ \ {\rm for}\ \ \tilde{f}\sim f_{0}\,,$ (6.28)

which in terms of $x$ reads

$\tilde{\Phi}(x)\simeq\
\frac{12}{f_{0}^{6}|h^{\prime\prime}(\xi_{0})|}\,\frac{1}{x^{2}}+\log
f_{0}+\Phi_{1}\,.$ (6.29)

The scalar dilaton is given by
$\tilde{\phi}(x)\simeq\tfrac{1}{2}\tilde{\Phi}(x)$ since the contribution from
the logarithm of $\tilde{m}$ is of order $x^{-3}$ and thus subleading.

Remark: It is important to note that the interior solution, for infinite time
$x$, becomes flat Minkowski space with a constant dilaton. This may seem
surprising, given the well-known fact that the ‘cosmological’ term in the 2D
action requires a linear dilaton when the spacetime becomes flat. Indeed, the
black hole exterior in the far away region operates in this way, both in the
two-derivative theory and in the case above: the metric approaches the flat
metric and the dilaton profile approaches a linear function. To see what is
happening in the interior consider the exterior and interior relations (3.10),
(3.11), and (3.15), (3.28a):

$\begin{split}\Bigl{(}{d\Phi\over dx}\Bigr{)}^{2}=&\ \ \ \ 1-g(f)\ =P(f)\,,\\\
\Bigl{(}{d\tilde{\Phi}\over dx}\Bigr{)}^{2}=&\
-1+\tilde{g}(\tilde{f})=\tilde{P}(\tilde{f})\,.\end{split}$ (6.30)

Moreover, just as we have (3.7) in the exterior, there is a similar equation
in the interior

$\begin{split}\check{g}(M)\ &=\ g(f(M))\,,\qquad M\in[0,M_{*}]\,,\\\
\tilde{\check{g}}(\tilde{M})\ &=\
\tilde{g}(\tilde{f}(\tilde{M}))\,,\qquad\tilde{M}\in[0,\tilde{M}_{*}]\,,\end{split}\
$ (6.31)

where $\tilde{M}_{*}$ is the value beyond which the inverse
$\tilde{f}(\tilde{M})$ of $\tilde{M}(\tilde{f})$ is not expected to have a
convergent series. Looking at Fig. 5, $\tilde{M}_{*}$ is the maximum of
$\tilde{M}(\tilde{f})$ attained for $\tilde{f}_{*}\in(0,f_{0})$. The non-
invertibility issue can be better appreciated in Fig. 6.

Figure 6: Sketch of $\tilde{f}(\tilde{M})$, the inverse of
$\tilde{M}(\tilde{f})$ in Fig. 5, making its multi-valuedness evident. The
bold curve is $\tilde{f}(\tilde{M})$ on the domain of expected convergence of
its series representation.

Consider first the exterior, in particular the AF region $f\sim 0$ and $M\sim
0$. Here $P(f=0)=1$ (see (6.4)), and therefore $g(0)=0$ ((6.30)). This is
consistent with the top equation in (6.31): perturbatively
$\check{g}(M)=-M^{2}+\cdots$ which gives (left-hand side) $\check{g}(0)=0$
consistent with (right-hand side) $g(f(0))=g(0)=0$. Indeed, this requires a
rolling dilaton ((6.30)).

Consider now the interior, in particular the AF region $\tilde{f}\sim f_{0}$
and $\tilde{M}\sim 0$. Here $\tilde{P}(f_{0})=0$ ((6.6)), therefore
$\tilde{g}(f_{0})=1$ ((6.30)), and there is no need for a rolling dilaton
((6.30)). The value $\tilde{g}(f_{0})=1$ may sound surprising given that
$\tilde{M}(f_{0})=0$ and $\tilde{\check{g}}(\tilde{M}=0)=0$. There is no
contradiction, however, with the bottom equation in (6.31):
$\tilde{\check{g}}(\tilde{M})=\tilde{M}^{2}+\cdots$ gives (left-hand side)
$\tilde{\check{g}}(\tilde{M}=0)=0$ consistent with (right-hand side)
$\tilde{g}(\tilde{f}(0))=\tilde{g}(0)=0$. Since both $\tilde{M}(0)$ and
$\tilde{M}(f_{0})$ are zero, the inverse function $\tilde{f}(M)$ is
necessarily multivalued. In the domain of definition of the perturbative
expansion $\tilde{f}(0)=0$, so this expansion cannot see what is happening for
$\tilde{f}\sim f_{0}$.

The above discussion shows how this regular BH convincingly emerges from a
suitable choice of $M(f)$. Still, the interior solution is approaching a
background defined by two-dimensional Minkowski space and a fixed dilaton.
Such a background is associated with a $c=2$ conformal field theory, not a
$c=26$ one. Since classical backgrounds of (bosonic) string field theory are
$c=26$ conformal field theories, one does not expect the regular BH solution
to be a string theory solution. More likely, the choice of $M(f)$ leading to
it defines $\alpha^{\prime}$ corrections that do not occur in string theory.

###  6.2 A particular example

As a proof of the existence of a function with the properties mentioned above,
we built a concrete example:

$h(\xi)={16\,\xi_{0}^{3}\,\xi\over(\xi^{2}+3\xi_{0}^{2})^{2}}\,.$ (6.32)

In here $\xi_{0}$ is a parameter that will be adjusted to make $h$ satisfy all
the requisite conditions. The derivatives of $h$ are

$h^{\prime}(\xi)=\ {48\,\xi_{0}^{3}\
(\xi_{0}^{2}-\xi^{2})\over(\xi^{2}+3\xi_{0}^{2})^{2}}\,,\quad
h^{\prime\prime}(\xi)={192\,\xi_{0}^{3}\
\xi(\xi^{2}-3\xi_{0}^{2})\over(\xi^{2}+3\xi_{0}^{2})^{4}}\,.$ (6.33)

We see that consistent with conditions (4.2) we have $h(0)=0$ and finite
$h^{\prime}(0)$. It is simple to check that, as required by (4.3), $h(\xi)\leq
1$ for all $\xi$, with $|h(\pm\xi_{0})|=1$. The integral of $h$ is easily
calculated

$\int_{0}^{\xi}d\xi^{\prime}\,h(\xi^{\prime})={8\xi_{0}\over
3}{\xi^{2}\over\xi^{2}+3\xi_{0}^{2}}\,.$ (6.34)

This implies that

$\int_{0}^{\infty}d\xi^{\prime}\,h(\xi^{\prime})={8\xi_{0}\over
3}=\alpha<\infty\,,$ (6.35)

so that the finiteness condition expressed in the second relation in (4.4) is
obeyed. The first and third condition in (4.4) (vanishing of $h$ and $\xi
h^{\prime}$ as $\xi\rightarrow\infty$) are also satisfied. Note also that $h$
is an odd function, as required.

Figure 7: Plot of
$h(\xi)={16\,\xi_{0}^{3}\,\xi\over(\xi^{2}+3\xi_{0}^{2})^{2}}$ with
$\xi_{0}=12$. The dotted lines indicate the position of the maximum. Figure 8:
Parametric plot for ${d\tilde{\Phi}\over dx}$ and $\tilde{M}$ both as
functions of $\tilde{f}\in(12,\infty)$ for the interior of the black hole with
resolved singularity (continuous line). Shown dashed is the analogous plot for
the two-derivative black hole. The faint lines are asymptotes at $45^{\circ}$
and $135^{\circ}$. The horizon is far on the $135^{\circ}$ asymptote, and the
late-time asymptotically Minkowski part of the cosmology is around the origin.

All of the above conditions were already imposed in the previous section, but
here we have the new conditions concerning the point $\xi_{0}$, where $h$
reaches the value of one. We indeed have, as required by the conditions listed
in (6.3),

$h(\xi_{0})=1\,,\ \ \ h^{\prime}(\xi_{0})=0\,,\quad
h^{\prime\prime}(\xi_{0})\,=\,-\tfrac{3}{2}\tfrac{1}{\xi_{0}^{2}}<0\,.$ (6.36)

The value of $\xi_{0}$ is determined by the integral constraint in (6.3)

$\int_{0}^{\xi_{0}}d\xi^{\prime}\,h(\xi^{\prime})=\tfrac{2\xi_{0}}{3}=\xi_{0}-4\quad\to\quad\xi_{0}=12\,.$
(6.37)

We see that for this value of $\xi_{0}$ we have $\alpha=32$. The function $h$
has a maximum at $\xi_{0}$, and being odd, a minimum at $-\xi_{0}$. It
vanishes both for $\xi=0$ and as $\xi\to\infty$. A plot of this function can
be seen in Fig. 7. Fig. 8 shows the plot of $d\tilde{\Phi}/dx$ and $\tilde{M}$
as a function of $\tilde{f}$ for this regular black hole (continuous line).
Superposed in the figure we see the similar plot (Fig. 4) for the two-
derivative black hole (dashed lines). The coordinate $x$ for the interior as a
function of $\tilde{f}$ is given by the integral

$x\,=\,\int_{\tilde{f}}^{\infty}{df\over f\sqrt{\tilde{P}(f)}}\,,$ (6.38)

which correctly gives the horizon $x=0$ for $\tilde{f}=\infty$ and a
$x\to\infty$ for $\tilde{f}\to f_{0}$. The integral above is best done
numerically, and a plot of it is shown in Fig. 9.

Figure 9: The coordinate $x$ as a function of the parameter
$\tilde{f}\in(f_{0},\infty)$ for the resolved BH interior. Here
$f_{0}=\sqrt{12}\sim 3.464$.

##  7 Concluding remarks

In this paper we have considered the theory space of duality-invariant
derivative corrections to the lowest order action that gives a solution
identified as the string theory two-dimensional black hole. We describe the
theory space by the choice of function $M(f)$, whose inverse is related to the
usual perturbatively defined $\alpha^{\prime}$ corrections. Using $f$ as a
parameter, and with square roots of nontrivial functions appearing in the
equations, $f$ space naturally becomes a space with branch points and branch
cuts. In this $f$ plane, the underlying structure of the solution is
determined by the type of branch points which determine the lines or contours
that parameterize the solution. We have argued that the $f$ parameterization
provides an extension of the original $\alpha^{\prime}$ corrected theory that
applies for situations where the series defining the corrections in the action
does not converge. We have used a notation where $f$ and $\tilde{f}$ are the
parameters for the exterior and interior solutions, respectively.

In fact, we saw that the black hole interior leads to a branch point at some
real, positive $f_{0}$ with a cut going all the way to $\tilde{f}=\infty$. For
the black hole interior with a singularity, the $x$ coordinate reaches a
finite value at the branch point $f_{0}$, a point that can be reached in
finite proper time. This means that the space cannot end there, and one must
indeed return to $\tilde{f}=\infty$ over the cut. To avoid the singularity,
while preserving a horizon, we altered the nature of the branch point $f_{0}$.
Now $x$ reaches an infinite value as we approach $f_{0}$, and it takes
infinite proper time to get there. Thus we get a complete space without having
to go over the branch. This yields a black hole with a horizon but a regular
interior, a regular black hole. The main caveat is that such solution does not
appear to be a string theory one, as discussed in section 6.1. It could be
that while general T-duality invariant $\alpha^{\prime}$ corrections allow for
regular black holes, such regular two-dimensional black holes do not arise in
string theory.

Our work can be viewed as an exploration of the space ${\cal T}$ of all
possible $\alpha^{\prime}$ corrections in two dimensions; the space of all
possible $M(f)$. If we call ${\cal S}$ the subspace we showed leads to
conventional singular black holes, the regular black holes we identified
appear on some subspace ${\cal R}\subset\partial{\cal S}$ of the boundary of
${\cal S}$. Note that we have not delineated the maximal space $\hat{\cal
S}\supset{\cal S}$ that leads to conventional black holes. Of course, we would
very much like to know where $D=2$ string theory sits in ${\cal T}$. If the
string theory black hole has a conventional singularity, then string theory
would lie in $\hat{\cal S}$.

We have also discussed duality in the black hole solutions. The maximally
extended geometry of the black hole of the two-derivative theory describes a
self-dual solution where, as noted by Giveon, duality maps the horizon to the
curvature singularity. We showed this is actually a general result. In
particular, complete solutions that have horizons but no singularities are not
self-dual. Their T-duals give an alternative description of the space.

A few issues could be investigated next:

  1. 1.

The present work makes some progress but does not completely describe the type
of situations that can arise throughout the theory space ${\cal T}$. We have
seen some unusual solutions, and it would be of interest to know all that can
happen. We identified a subspace ${\cal R}\subset{\cal T}$ leading to regular
black holes, but there may be a larger subspace $\hat{\cal R}$ giving non-
singular black holes, perhaps containing some that could be expected to be
string theory solutions. Since 2D string theory is strongly believed to have a
black hole, the regions of ${\cal T}$ for which no black hole solution exists
could be thought of as the ‘swampland’ in the theory space.

  2. 2.

To better understand solutions, instead of using the $n=1$ gauge, which
results in disconnected $f$ contours for the exterior and interior solutions,
one may try for a better, more geometric understanding in the $nm=1$ gauge,
where maximal extensions may be more easily constructed.

  3. 3.

It would indeed be important to determine the maximal extensions of the
regular black holes obtained here and to display their Penrose diagrams. The
challenge here is doing the needed work analytically.

  4. 4.

To get hints if the 2D string theory black hole is regular, it may be of
interest to do a more careful study of the Dijkgraaf, Verlinde, Verlinde
ansatz [11] to understand its geometry and the behavior of the dilaton. In
[13], the geometry is first extended to include a region with Euclidean
signature, which is later removed, with completeness restored by gluing
infinite number of copies with such excision. Of course, we still lack insight
on how this configuration could be a solution of the $\alpha^{\prime}$
corrected string equations of motion. In [22] we showed that the ansatz, in
its given form, does not fit the framework of our analysis.

  5. 5.

Our work could help constrain the $\alpha^{\prime}$ corrections of string
theory. We have seen that if the string theory BH remains singular then the
$\alpha^{\prime}$ corrections are strongly constrained. Indeed for generic
classes of corrections the BH presumably does not exist. At present the class
of $M(f)$’s for which the BH has no singularity seems strongly restricted, but
further investigation could change this.

  6. 6.

As emphasized in the introduction, our search for a regular BH geometry is
based on standard general relativity notions, where a singularity is probed by
point particles and geodesics. Since in string theory spacetime should be
probed by strings, the ultimate picture might be quite different. As shown
here, the T-dual of a regular BH geometry necessarily features a curvature
singularity, but it might be that when probed by winding modes this
singularity is harmless. This could be explored with a genuine double field
theory [33, 34] on 2D black hole backgrounds.

### Acknowledgements

We thank Netta Engelhardt, Maurizio Gasperini, Daniel Grumiller, Daniel
Harlow, Matthew Kleban, Massimo Porrati, Gabriele Veneziano, and Ashoke Sen,
for comments and discussions.

This work is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (grant agreement
No 771862). T. C. is supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - Projektnummer 417533893/GRK2575 “Rethinking
Quantum Field Theory”.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics of U.S. Department of Energy
under grant Contract Number DE-SC0012567.(High Energy Theory research).

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