††institutetext: International Centre for Theoretical Physics (ICTP),
Strada Costiera 11, Trieste 34151, Italy

# Self-Similar Gravitational Dynamics, Singularities and Criticality in 2D

Upamanyu Moitra<EMAIL_ADDRESS>
###### Abstract

We initiate a systematic study of continuously self-similar (CSS)
gravitational dynamics in two dimensions, motivated by critical phenomena
observed in higher dimensional gravitational theories. We consider CSS
spacetimes admitting a homothetic Killing vector (HKV) field. For a general
two-dimensional gravitational theory coupled to a dilaton field and Maxwell
field, we find that the assumption of continuous self-similarity determines
the form of the dilaton coupling to the curvature. Certain limits produce two
important classes of models, one of which is closely related to two-
dimensional target space string theory and the other being Liouville gravity.
The gauge field is shown to produce a shift in the dilaton potential strength.
We consider static black hole solutions and find spacetimes with uncommon
asymptotic behaviour. We show the vacuum self-similar spacetimes to be special
limits of the static solutions. We add matter fields consistent with self-
similarity (including a certain model of semi-classical gravity) and write
down the autonomous ordinary differential equations governing the
gravitational dynamics. Based on the phenomenon of finite-time blow-up in
ODEs, we argue that spacetime singularities are generic in our models. We
present qualitatively diverse results from analytical and numerical
investigations regarding matter field collapse and singularities. We find
interesting hints of a Choptuik-like scaling law.

##  1 Introduction

The general theory of relativity keeps springing surprises well past a century
after its discovery. Studies of different aspects of gravity continue to be
rewarding. The formation of black holes through gravitational collapse is one
of the most striking predictions of general relativity. For a better
understanding of actual physical systems observed in Nature and of subtle
conceptual issues, the study of the _dynamics_ of gravitational systems is
very important. As is true of most things important, such studies are also
very difficult. Even with assumptions that simplify a problem, studying
gravitational dynamics is a formidable task. This fact has not prevented
progress. One can use several techniques, including numerical methods, to
glean important information about gravitational dynamics.

One of the most surprising features in gravitational collapse was unravelled
by numerical investigations by Choptuik Choptuik:1992jv . Choptuik considered
a family of initial data for the spherically symmetric collapse (in $3+1$
dimensional flat spacetime ) of a massless scalar field, characterised by a
parameter $p$ (which, roughly speaking, can be regarded as the “strength” of
the initial data). It turns out there exists a critical value $p^{*}$ (which
is not universal) which governs whether or not the end state will contain a
black hole. If $p<p^{*}$, then the scalar field leaves behind an
asymptotically flat spacetime with no black holes. On the other hand, if the
strength $p$ exceeds $p^{*}$, then a black hole is formed, whose mass near
this threshold is governed by a scaling relation,

$\displaystyle M\sim(p-p^{*})^{\alpha},$ (1)

now commonly known as the _Choptuik scaling law_. While $p^{*}$ is non-
universal, the scaling exponent $\alpha$ is universal, which was determined
numerically. It was further observed that around criticality, the spacetime
exhibits self-similarity. It was also suggested that the critical solution
itself corresponded to a naked singularity. While the self-similarity observed
in the original investigation was discrete in nature, it was soon realised
Evans:1994pj  that this similarity can be either discrete or continuous.
Behaviours of this kind (i.e., scaling, universality) are ubiquitous in
statistical mechanics. Since the work of Choptuik, there has been a
considerable body of work in the exploration of gravitational critical
phenomena. An exposition of various aspects of critical behaviour in
gravitational collapse can be found in the review Gundlach:2007gc .

Since the solutions to the gravitational equations of motion show self-
similarity near criticality, a good place to start is to look for solutions
with self-similarity as an _input_. The assumption of self-similarity is not
overly restrictive, since it is observed in physical systems in a wide class
of situations. For a discussion on the importance of self-similarity in
general relativity, see the review Carr:1998at , which also points out its
relevance in astrophysics and cosmology. From a practical point of view, the
assumption of continuous self-similarity (CSS) accords an enormous
simplification. In this article, we would concern ourselves with spacetimes
admitting a homothetic Killing vector (HKV) field $\xi$, defined by,

$\displaystyle\pounds_{\xi}g_{\mu\nu}=2g_{\mu\nu},$ (2)

where $\pounds_{\xi}$ denotes the Lie derivative.111The factor of $2$ on the
right-hand-side of eq. 2 is conventional and corresponds to a normalisation of
the vector field $\xi$. The equations of motion become much simpler under the
assumption of spacetimes admitting such a symmetry — in our case (and in
previous studies involving spherically symmetric spacetimes in higher
dimensions), the equations of motion turn into _ordinary_ differential
equations (ODE) in a suitable self-similar variable, which we call $x$.

Some important insights about critical behaviour in gravitational collapse
have come from studying ODEs of this kind. Under a judicious set of
transformations of the variables, the ODEs in question become autonomous
differential equations (see Christodoulou:1994hg  and Brady:1994aq  for early
works in this regard). There exist methods and powerful theorems associated
with solutions of autonomous ODEs under certain conditions — this fact makes
it easier to study the problem in question. From the point of view of
autonomous ODEs, each integral curve in $x$ associated with a different
initial condition corresponds to a different spacetime. The aforementioned
findings can be conveniently put in this language — the final solutions of
scalar collapse with or without black holes represent two basins of attraction
to which the integral curves flow depending on the initial condition $p$; on
the other hand, the critical solution is a fixed point (saddle) of the ODEs
which corresponds to a finely tuned initial condition $p=p^{*}$. Indeed, as is
common in a renormalisation group (RG) analysis, one can read off various
critical exponents from the eigenvalues obtained from a linear stability
analysis around the fixed point Koike:1995jm . The reader can refer to the
review Gundlach:2007gc  and the references therein for more details;
relatively recent applications of the assumption of homothety in different
contexts include Zhang:2015rsa ; Rocha:2018lmv .

The focus of the present article is on two-dimensional spacetimes with an HKV.
Before we describe our motivations for studying this problem in detail, let us
address an immediate concern some might have. A homothetic Killing vector
field defined by eq. (2) is a special case of a conformal Killing vector (CKV)
field — with the conformal pre-factor being a constant instead of an arbitrary
function of spacetime coordinates. As is well known, _every_ two-dimensional
metric admits infinitely many CKVs. Given this fact, for any metric it is
almost certain that there would exist an HKV. However, our interests are
broader than the metric — we consider two-dimensional dilaton gravity (coupled
to matter) in which we have additional fields. Since we want to find solutions
exhibiting CSS, we insist that the equations of motion transform appropriately
under the action of the HKV — we would thus need the transformation laws of
the other fields under its action as well. Therefore, the task of finding
spacetimes with this symmetry is a non-trivial one and we will see that this
symmetry is quite restrictive.

We now discuss the motivations for studying the problem in two dimensions.
First of all, virtually all discussions of critical phenomena in gravity in
four dimensions take place in the context of spherical symmetry. The essential
dynamics therefore takes place in a two-dimensional subspace. Under a
spherically symmetric reduction of higher-dimensional gravity, we end up
precisely with dilaton gravity, which offers a convenient setting to study
such questions. However, two-dimensional dilaton gravity theories need not
descend from higher dimensions — they are quite interesting in themselves. In
fact, we will consider dilaton gravity coupled to a Maxwell field. When
uplifted, such a model can describe a wide variety of scenarios: rotating
black holes, charged black holes in arbitrary dimensions with asymptotically
flat or (A)dS spacetimes Moitra:2019bub , black holes with toroidal geometry.
This work is motivated partly by the possibility of understanding critical
behaviour in such scenarios.

Another motivation for this work is the prospect of an improved understanding
of gravitational collapse and dynamical singularity from a controlled
microscopic perspective. In two dimensions, certain questions related to
quantum gravity and string theory can be dealt with in a relatively tractable
manner — matrix models are an important tool in this regard. The resurgence of
interest in two-dimensional gravity has led to several promising avenues in
the recent years. There is a considerable body of work on spacetime
singularities in string theory, especially in the context of matrix models.
See for example, the review Craps:2006yb  and the relatively recent work
Das:2019cgl  and references therein. (See also Moitra:2022glw .) The question
of black hole (non-)formation is discussed in Karczmarek:2004bw .

Let us now describe our organisation of and the results obtained in this
article. In Section 2, we describe our basic set-up of the “pure” or vacuum
sector222In the sense that there are no bulk perturbative degrees of freedom.
This is in contrast with the situation when there are matter fields.: metric,
dilaton and a gauge field. We consider a general dilaton gravity action, which
can be reduced to a convenient form by a certain redefinition of the dilaton
field. With this redefinition, there is a function $A(\phi)$ of the dilaton
field $\phi$ which couples to the Ricci scalar. We assume a very general
transformation law for the dilaton field. Demanding that some components of
the equations of motion transform to themselves under the HKV allows us to fix
the form of the function $A(\phi)$. For the most general form of this function
$A(\phi)$, it appears difficult to make further progress — it turns out that
under two different limits, the form of $A(\phi)$ makes the analysis simple.
These two limits produce very well-known models. The first model is closely
related to two-dimensional target space string theory Elitzur:1990ubs ;
Mandal:1991tz ; Witten:1991yr , we loosely refer to this class of models as
the “stringy” models in the remainder of the paper. The other model is nothing
but Liouville gravity. It is easy to fix the form of the remaining dilaton
potential in both these models. What about the gauge field? Owing to the fact
that we are in two dimensions, the homothety properties of the gauge field
strength is completely determined by the remaining fields. We find the curious
result that the only effect of the gauge field is to renormalise the strength
of the dilaton potential. This is a considerable simplification, which allows
us to eschew the gauge field altogether and work with an effective dilaton
potential even when there is uncharged matter present.

In Section 3, we first find the coordinate system which makes the study of CSS
solutions very convenient — with the particular choice we make, all the
relevant equations of motion become manifestly autonomous, obviating the need
for any further transformation of variables. We first solve the vacuum
equations of motion under the CSS ansatz and find, rather unsurprisingly, that
the vacuum CSS solutions are some particular static solutions in disguise and
also provide the explicit coordinate transformation between the static and CSS
solutions. We now add scalar matter to the theory. We find some simple (yet
conceptually relevant) matter models consistent with the CSS ansatz. In
particular, for the Liouville model, we find that a certain semi-classical
model is consistent with CSS — which makes it an interesting object of study.
We write down the autonomous differential equations governing the dynamics for
all the systems. Here, we observe a significant departure from related studies
— the previously mentioned linear stability analysis is no longer useful in
our case, forcing us to look at the problem from a different angle.

In Section 4, we use this fact (lack of utility of linearisation) to our
advantage to argue that in our models, spacetime singularities are always
formed on account of the phenomenon of _finite-time blow-up_ , a well-known
feature of certain dynamical systems. We illustrate our arguments with some
simple dynamical systems.

In Section 5, we present our results regarding the singularity and its nature
in self-similar collapse with matter. In certain simple cases, we are able to
make statements analytically. In most cases, however, we have to take recourse
to high-precision numerics to make proper statements about the solutions. (We
emphasise that, within the models, the solutions discussed are fully back-
reacted and no approximation like the probe limit has been used anywhere.) We
find very interesting results about the singularity in all these models. In
particular, we find numerical evidence of a behaviour analogous to Choptuik
scaling. Virtually all the singularities in the dynamical situation turn out
to be null or spacelike, but not timelike. Even when a timelike singularity is
present initially, it is rendered unstable by the dynamics. The results are
thus pleasing vis-à-vis the weak cosmic censorship conjecture. (See
Moitra:2020ojo  and Moitra:2021eom  for discussions on the _strong_ cosmic
censorship conjecture in two dimensions.)

We conclude this article with a discussion on many exciting future directions
in Section 6.

In Appendix A,  we write down the static solutions corresponding to the
actions obtained in Section 2. We make a detailed study of the black hole
solutions for different values of the parameters in the theory. Although these
solutions are not self-similar in general, the numerical analysis in Section 5
is informed by the results of this appendix. We find, rather surprisingly, a
class of spacetimes which have a similar asymptotic causal structure as AdS
spacetimes even though the curvature is vanishingly small asymptotically. Such
spacetimes of higher dimensionality have been observed previously Chan:1995fr
. To the best of my knowledge, this is the first time explicit examples of
such unusual asymptotics have been constructed in two dimensions. Since this
appendix is quite useful in building the intuition for the analysis in the
main body of this article, some readers might want to study this appendix
before starting Section 3.

An existing definition of quasi-local mass, which is useful in the main text,
is discussed in Appendix B.

Since some of our two-dimensional solutions asymptote to spacetimes which look
similar to AdS spacetimes, this also raises the exciting possibility that we
may gain insight about self-similar spacetimes using some holographic model. A
constant non-zero value of the scalar curvature $R$ is inconsistent with the
existence of an HKV (2). The authors of Pretorius:2000yu  found continuously
self-similar behaviour near criticality, which was _not_ associated with an
HKV. Diverse investigations in the context of higher dimensional AdS
spacetimes include Birmingham:1999yt ; Birmingham:2001hc ;
Bhattacharyya:2009uu ; Chesler:2019ozd ; Emparan:2021ewh . As an example of
the interesting possibilities of self-similar gravitational collapse involving
holography, such a collapse was related to high-energy scattering in QCD in
Alvarez-Gaume:2008emf . It would be interesting indeed if the bulk dynamics
explored in this article could be phrased entirely in terms of some boundary
degrees of freedom.

Before we proceed further, let us mention some other relevant references. See
Wald:1997wa  for an early review of weak cosmic censorship in the context of
gravitational collapse (see also Joshi:2008zz ). Questions related to Choptuik
scaling and critical collapse in two spacetime dimensions have been considered
in a few articles Strominger:1993tt ; Peleg:1996ce ; Frolov:2006is ;
Dhar:2018pii . Such investigations differ significantly from the one presented
here — the assumption of continuous self-similarity plays the central role in
our discussions. In the previous works, self-similarity, if present, was
discrete in nature. Indeed, in a specific model, there were arguments
_against_ continuous self-similarity Chiba:1997ex . Instead of starting with
some specific models, we begin systematically by asking the question: what
models are allowed at all? Furthermore, many such specific models are subsumed
as special cases of the broad class of models described in this article. We
hope the work presented in this article will be useful in further
investigations of critical collapse.

##  2 Self-Similarity in Two Dimensions

We start our discussion with the “pure” or vacuum sector, described by a
general two-derivative action with the metric, dilaton and Maxwell field333The
gauge potential is of course $A_{\mu}$ and the field strength is
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Though we have used
the same letter for the gauge potential $A_{\mu}$ and the function $A(\phi)$,
the difference should be obvious from the context. Actually, we have not used
the gauge potential explicitly anywhere in the article. ,

$\displaystyle S=\frac{1}{16\pi
G_{2}}\int\differential[2]{x}\,\sqrt{-g}A(\phi)\bqty{R+\gamma(\nabla\phi)^{2}+V(\phi)-G(\phi)F_{\mu\nu}F^{\mu\nu}},$
(3)

and examine what restrictions are imposed by the criterion of self-similarity.
The reader might have noticed that we have chosen the same function of the
dilaton field $A(\phi)$ before the Ricci scalar and the dilaton kinetic term.
In the most general possible action, the coefficient of the Ricci scalar term
would be different from the one of the kinetic term. However, we can redefine
the field $\phi$ to make the two terms equal.

We could also eliminate the kinetic term for the dilaton by making a dilaton-
dependent Weyl transformation of the metric. Since we wish to study the
singularity properties and also connect the two-dimensional physics to higher
dimensions, we work in a fixed Weyl frame throughout this article, which makes
the connection more transparent. We shall work with some fixed $A(\phi)$ — we
keep the coefficient $\gamma$ as an independent parameter as different values
of $\gamma$ illustrate qualitatively different aspects of the physics, as will
be apparent below. See also Lechtenfeld:1992rt  for related comments.

The equation of motion obtained by varying the scalar $\phi$ is given by,

$\displaystyle R-\gamma(\nabla\phi)^{2}-\frac{2\gamma
A(\phi)}{A^{\prime}(\phi)}\nabla^{2}\phi+\pqty{V(\phi)+\frac{V^{\prime}(\phi)A(\phi)}{A^{\prime}(\phi)}}-\pqty{G(\phi)+\frac{G^{\prime}(\phi)A(\phi)}{A^{\prime}(\phi)}}F^{2}=0.$
(4)

The equation of motion obtained by varying the (inverse) metric is given by,

$\displaystyle
g_{\mu\nu}A^{\prime}(\phi)\nabla^{2}\phi+g_{\mu\nu}A^{\prime\prime}(\phi)(\nabla\phi)^{2}-A^{\prime\prime}(\phi)\nabla_{\mu}\phi\nabla_{\nu}\phi-A^{\prime}(\phi)\nabla_{\mu}\nabla_{\nu}\phi$
(5) $\displaystyle\quad+\gamma
A(\phi)\pqty{\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}(\nabla\phi)^{2}g_{\mu\nu}}-\frac{1}{2}g_{\mu\nu}A(\phi)V(\phi)$
$\displaystyle\quad-2A(\phi)G(\phi)\pqty{F_{\mu\rho}F_{\nu}{}^{\rho}-\frac{1}{4}g_{\mu\nu}F^{2}}=0.$

Its trace gives,

$\displaystyle
A^{\prime}(\phi)\nabla^{2}\phi+A^{\prime\prime}(\phi)(\nabla\phi)^{2}-A(\phi)V(\phi)-A(\phi)G(\phi)F^{2}=0.$
(6)

###  2.1 Most General Curvature Coupling

Since we are looking for self-similar solutions, we demand that under the
action of the HKV, the equations of motion (4) and (6) should transform to
themselves, up to some overall multiplicative factor.

Under the homothety transformation (2), the Ricci scalar transforms with
weight $(-2)$,

$\pounds_{\xi}R=-2R.$ (7)

Under our assumptions, we will see that we will be able to constrain the form
of $A(\phi)$ by simply considering the transformation properties of
$(\nabla\phi)^{2}$ and $\nabla^{2}\phi$.

Let us note that for any HKV $\xi$,

$\displaystyle\pounds_{\xi}\nabla_{\mu}\phi=\nabla_{\mu}\pounds_{\xi}\phi,$
(8)

and,

$\displaystyle\pounds_{\xi}\nabla^{2}\phi=\nabla^{2}\pounds_{\xi}\phi-2\nabla^{2}\phi.$
(9)

Let us make the reasonable assumption under the Lie action of the HKV, the
scalar field transforms to a function of itself (which is not a function of
the metric or the gauge field)444One could possibly consider here derivatives
of $\phi$ — but we would necessarily have to include the metric to soak up the
indices. The ensuing analysis would be rather complicated.

$\displaystyle\pounds_{\xi}\phi=f(\phi).$ (10)

To complete the discussion we would also need to consider the homothety
transformation properties of the gauge field. However, since our immediate
goal is to determine the form of the function $A(\phi)$, we can ignore the
gauge field term for the moment. It will turn out that the gauge field
homothetic properties will be uniquely determined by those for the metric and
the dilaton.

Under the assumed conditions, only the first term of the LHS of eq. (4)
generates the Ricci scalar under the HKV action. By the property (7), we can
thus say that for self-similar solutions, the homothety action transforms the
LHS of (4) to $(-2)$ times itself.

We see for the terms involving derivatives of $\phi$, using eqs. (8), (9) and
(10),

$\displaystyle\pounds_{\xi}\pqty{-\gamma(\nabla\phi)^{2}}$
$\displaystyle=(-2)(-\gamma(\nabla\phi)^{2})-2\gamma
f^{\prime}(\phi)(\nabla\phi)^{2},$ (11)
$\displaystyle\pounds_{\xi}\bqty{-\frac{2\gamma
A(\phi)}{A^{\prime}(\phi)}\nabla^{2}\phi}$
$\displaystyle=(-2)\bqty{-\frac{2\gamma
A(\phi)}{A^{\prime}(\phi)}\nabla^{2}\phi}-2\gamma\frac{A(\phi)}{A^{\prime}(\phi)}f^{\prime\prime}(\phi)(\nabla\phi)^{2}$
$\displaystyle\quad-2\gamma\derivative{\phi}(\frac{A(\phi)f(\phi)}{A^{\prime}(\phi)})\nabla^{2}\phi.$

Since no other terms can generate derivatives of $\phi$ under the homothety
transformation rules, for eq. 4 to be covariant under a homothety
transformation, we obtain the following condition on the derivatives of
$\phi$,

$\displaystyle-2\gamma\pqty{f^{\prime}(\phi)+\frac{A(\phi)}{A^{\prime}(\phi)}f^{\prime\prime}(\phi)}(\nabla\phi)^{2}-2\gamma\derivative{\phi}(\frac{A(\phi)f(\phi)}{A^{\prime}(\phi)})\nabla^{2}\phi=0.$
(12)

The coefficients of these terms should vanish independently and we immediately
obtain,

$\displaystyle f(\phi)=a_{1}\frac{A^{\prime}(\phi)}{A(\phi)},$ (13)

and,

$\displaystyle f^{\prime}(\phi)=\frac{a_{2}}{A(\phi)},$ (14)

where $a_{1,2}$ are constants.

Let us see if homothety action on the derivative terms of (6) produces more
constraints. Since the coefficients of $(\nabla\phi)^{2}$ and $\nabla^{2}\phi$
are $A^{\prime\prime}(\phi)$ and $A^{\prime}(\phi)$ respectively, we insist
that under a homothety transformation, ratio of the coefficient of
$(\nabla\phi)^{2}$ to that of $(\nabla^{2}\phi)$ must be the same as
$A^{\prime\prime}(\phi)/A^{\prime}(\phi)$.

We see,

$\displaystyle\pounds_{\xi}\bqty{A^{\prime\prime}(\phi)(\nabla\phi)^{2}}=\bqty{-2A^{\prime\prime}(\phi)+2A^{\prime\prime}(\phi)f^{\prime}(\phi)+A^{\prime\prime\prime}(\phi)f(\phi)}(\nabla\phi)^{2},$
(15)

and,

$\displaystyle\pounds_{\xi}\bqty{A^{\prime}(\phi)\nabla^{2}\phi}=\bqty{-2A^{\prime}(\phi)+A^{\prime\prime}(\phi)f(\phi)+A^{\prime}(\phi)f^{\prime}(\phi)}\nabla^{2}\phi+A^{\prime}(\phi)f^{\prime\prime}(\phi)(\nabla\phi)^{2}.$
(16)

Therefore, on acting with $\pounds_{\xi}$ on equation (6), the resulting
coefficient of $(\nabla\phi)^{2}$ is given by,

$\displaystyle
A^{\prime\prime\prime}f+2A^{\prime\prime}f^{\prime}+A^{\prime}f^{\prime\prime}-2A^{\prime\prime}=(A^{\prime}f-2A)^{\prime\prime},$
(17)

and that of $\nabla^{2}\phi$ is given by,

$\displaystyle
A^{\prime\prime}f+A^{\prime}f^{\prime}-2A^{\prime}=(A^{\prime}f-2A)^{\prime}.$
(18)

Therefore, we must have,

$\displaystyle\frac{(A^{\prime}f-2A)^{\prime\prime}}{(A^{\prime}f-2A)^{\prime}}=\frac{A^{\prime\prime}}{A^{\prime}}.$
(19)

This determines the form of $f(\phi)$ to be,

$\displaystyle f(\phi)=\frac{a_{3}A(\phi)+a_{4}}{A^{\prime}(\phi)},$ (20)

with new constants $a_{3,4}$. Substituting eq. (13) in eq. (20) and solving
the resulting differential equation, we obtain the general solution for
$A(\phi)$,

$\displaystyle
A(\phi)=\frac{a_{4}}{a_{3}}\sinh^{2}\pqty{\frac{1}{2}\sqrt{\frac{a_{3}}{a_{1}}}(\phi-\phi_{0})},$
(21)

where $\phi_{0}$ is an arbitrary constant of integration. Recall from eq. (3)
that $A(\phi)$ couples to the Ricci scalar and hence must be positive. We
could also insist on this coupling being monotonic in $\phi$. These conditions
would imply that $a_{1}$, $a_{3}$ and $a_{4}$ are of the same sign.

As a check, we find that the form of $f(\phi)$ determined from (13) is
consistent with the behaviour from (14), with the constant $a_{2}$ being
determined to be $-a_{4}/2$.

Although we have obtained the form of $A(\phi)$ consistent with the assumed
homothety (2) and the assumption (10), it seems difficult to proceed further
with the general form (21). We will instead focus on a subset of the most
general solution in the theory space, which are still quite generic, and we
find that these models are rather well-known.

###  2.2 Models Related to 2D Target Space String Theory

One of the simplest results emerges when we set $a_{4}=0$ in eq. (20). The
$a_{4}\to 0$ limit of eq. (21) is trivial so we examine this case separately.
When $a_{4}=0$, eqs. (20) and (13) together imply that
$A^{\prime}(\phi)/A(\phi)$ must be a constant. Since $A(\phi)$ is real, we
reach the conclusion that $A(\phi)$ is an exponential function in $\phi$. We
can use our freedom to rescale the Newton constant and the coefficient
$\gamma$ to set

$\displaystyle A(\phi)=\exp(-2\phi).$ (22)

This form (involving the Ricci scalar and the dilaton kinetic term) is exactly
the same as two-dimensional target space string theory. For string theory, the
value of the coupling $\gamma$ is 4.

Once we have fixed the scale and origin of $\phi$, we can now fix $f(\phi)$ to
be some constant. Following the conventions in the literature, we take the Lie
derivative of the scalar field to be,

$\displaystyle\pounds_{\xi}\phi=-\kappa.$ (23)

We would now like to determine the forms of the potential $V(\phi)$ and the
“gauge coupling” $G(\phi)$. The Maxwell equations are given by,

$\displaystyle\nabla_{\mu}(A(\phi)G(\phi)F^{\mu\nu})=0.$ (24)

Since the discussion concerns a two-dimensional spacetime, we must have,

$\displaystyle A(\phi)G(\phi)F_{\mu\nu}=Q\sqrt{-g}\epsilon_{\mu\nu},$ (25)

where $\frac{1}{2}\sqrt{-g}\,\epsilon_{\mu\nu}\,\differential
x^{\mu}\wedge\differential x^{\nu}$ is the volume form and $Q$ is a constant,
which is the charge carried by the system. Using eq. 25 and the properties of
the volume form, we readily obtain,

$\displaystyle F_{\mu\rho}F_{\nu}{}^{\rho}$
$\displaystyle=-\frac{Q^{2}}{A^{2}G^{2}}g_{\mu\nu},$ (26) $\displaystyle
F^{2}$ $\displaystyle=-\frac{2Q^{2}}{A^{2}G^{2}}.$

Inserting eq. 26 in eq. 6 and using eq. 22 we obtain,

$\displaystyle-\nabla^{2}\phi+2(\nabla\phi)^{2}-\frac{1}{2}V(\phi)+\frac{Q^{2}e^{4\phi}}{G(\phi)}=0.$
(27)

If we demand that the homothety properties be independent of the some specific
solutions to the equation of motion, we could analyse the properties of the
two functions appearing in eq. 27 independently. Since both $\nabla^{2}\phi$
and $(\nabla\phi)^{2}$ transform under homothety with weight $-2$, we must
demand that,

$\displaystyle\pounds_{\xi}V(\phi)=-2V(\phi).$ (28)

This implies that,

$\displaystyle-\kappa V^{\prime}(\phi)=-2V(\phi),$ (29)

which constrains the potential to be of the form,

$\displaystyle V(\phi)=V_{0}\exp(\frac{2\phi}{\kappa}).$ (30)

By a similar argument, we must have $e^{4\phi}/G(\phi)=e^{2\phi/\kappa}$ and
thus, we obtain the required functional form for $G(\phi)$,

$\displaystyle G(\phi)=\exp(4\phi-\frac{2\phi}{\kappa}).$ (31)

Before proceeding further, let us determine the homothety property of the
field strength tensor $F_{\mu\nu}$ and check for consistency. We have from eq.
25,

$\displaystyle\exp(2\phi-\frac{2\phi}{\kappa})F_{\mu\nu}=Q\sqrt{-g}\epsilon_{\mu\nu}.$
(32)

We note that under the homothety transformation $\sqrt{-g}\epsilon_{\mu\nu}$
transforms to itself with weight 2,

$\displaystyle\pounds_{\xi}\sqrt{-g}\epsilon_{\mu\nu}=2\sqrt{-g}\epsilon_{\mu\nu}.$
(33)

Therefore, the left hand side must transform with weight 2 as well. Since the
quantity $\exp(2\phi(1-1/\kappa))$ transforms with weight $2(1-\kappa)$ (by
(23)), the weight of $F_{\mu\nu}$ is uniquely determined to be $2\kappa$:

$\displaystyle\pounds_{\xi}F_{\mu\nu}=2\kappa F_{\mu\nu}.$ (34)

Therefore, from (6), we see that $G(\phi)F^{2}$ must transform with weight
$(-2)$. Noting that $F^{2}$ has two factors of the inverse metric and two
factors of the field strength $F_{\mu\nu}$, we find that the weight of this
quantity is,

$\displaystyle\pqty{4-\frac{2}{\kappa}}(-\kappa)+2(-2)+2(2\kappa)=-2.$ (35)

Thus the homothety properties are consistent, as they must be.

Let us write down the final equations of motion. The $\phi$ equation of motion
is given by,

$\displaystyle
R-\gamma(\nabla\phi)^{2}+\gamma\nabla^{2}\phi+(V_{0}-2Q^{2})\pqty{1-\frac{1}{\kappa}}\exp(\frac{2\phi}{\kappa})=0.$
(36)

On the other hand,

$\displaystyle
2\nabla_{\mu}\nabla_{\nu}\phi-2g_{\mu\nu}\nabla^{2}\phi+\frac{8-\gamma}{2}g_{\mu\nu}(\nabla\phi)^{2}+(\gamma-4)\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}e^{\frac{2\phi}{\kappa}}(V_{0}-2Q^{2})=0.$
(37)

It is thus easy to see that the effect of the gauge field is to additively
shift the value of potential strength and therefore, as far the source-free
equations are concerned, we could work with only an effective potential
strength

$\displaystyle V_{\mathrm{eff}}\equiv V_{0}-2Q^{2},$ (38)

and no gauge field.

###  2.3 Liouville Gravity

We can examine another case, which is given by $a_{3}=0$. In this case, the
solution for $A(\phi)$ is given by,

$\displaystyle A(\phi)=\frac{a_{4}}{4a_{1}}(\phi-\phi_{0})^{2}.$ (39)

Rescaling and shifting $\phi$, we can take

$\displaystyle A(\phi)=\phi^{2}.$ (40)

This is actually nothing but Liouville gravity in disguise. The simple field
redefinition,

$\displaystyle\phi^{2}=\sigma_{0}+\sigma,$ (41)

yields the action for Liouville gravity,

$\displaystyle S=\frac{1}{16\pi
G_{2}}\int\differential[2]{x}\sqrt{-g}\pqty{(\sigma_{0}+\sigma)R-(\nabla\sigma)^{2}+V(\sigma)-G(\sigma)F^{2}}.$
(42)

Here, $\sigma_{0}$ is a constant which we have separated by hand from the
dynamical field $\sigma$ — this artificial splitting has been done for later
convenience when we discuss the black hole solutions and collapse.

Note that in this case, we can absorb $\gamma$ before the kinetic term by a
redefinition of the $\sigma$ field and the Newton constant $G_{2}$. Note that
we have chosen a particular sign for the kinetic term for clearly establishing
the connection between static black holes (see Appendix A) in this model and
those of the stringy model.

The equations of motion are given by,

$\displaystyle R+2\nabla^{2}\sigma+V^{\prime}(\sigma)-G^{\prime}(\sigma)F^{2}$
$\displaystyle=0,$ (43)

and

$\displaystyle g_{\mu\nu}\nabla^{2}\sigma-\nabla_{\mu}\nabla_{\nu}\sigma$
$\displaystyle-\pqty{\nabla_{\mu}\sigma\nabla_{\nu}\sigma-\frac{1}{2}(\nabla\sigma)^{2}g_{\mu\nu}}$
(44)
$\displaystyle-\frac{1}{2}g_{\mu\nu}V-2G\pqty{F_{\mu\rho}F_{\nu}{}^{\rho}-\frac{1}{4}g_{\mu\nu}F^{2}}=0.$

We now have,

$\displaystyle\pounds_{\xi}\sigma=-\kappa,$ (45)

and similar considerations as in the previous subsection fix $V(\sigma)$ and
$G(\sigma)$ to take the forms,

$\displaystyle V(\sigma)$ $\displaystyle=V_{0}\exp(\frac{2\sigma}{\kappa}),$
(46) $\displaystyle G(\sigma)$ $\displaystyle=\exp(-\frac{2\sigma}{\kappa}).$

As before, if we insert the solution to the Maxwell equations, the effective
potential strength is shifted by the charge and we obtain the same relation as
eq. (38), i.e.,

$V_{\mathrm{eff}}=V_{0}-2Q^{2}.$

In the remainder of this article, we work with an effective potential strength
and no gauge field.

##  3 Self-Similar Solutions: Setting the Stage

###  3.1 The Coordinate System

We need to work with a coordinate system well-adapted to the problem at hand —
namely, describing the self-similar behaviour of gravitating systems. In
higher dimensions, typically the retarded or advanced Bondi gauge is
particularly useful. In studies in higher dimensions (see, for example,
Brady:1994aq ) an ansatz like the following is used,

$\displaystyle\differential
s^{2}=-g\quantity(\frac{r}{v})\bar{g}\quantity(\frac{r}{v})\differential{v}^{2}+2\bar{g}\quantity(\frac{r}{v})\differential{v}\differential{r}+r^{2}\,\differential\Omega_{2}^{2}.$
(47)

Here $v$ is an advanced null coordinate, and one has to consider a patch of
spacetime, say $v>0$. This metric transforms appropriately under the action of
the homothetic Killing vector field,

$\displaystyle\xi=v\partialderivative{v}+r\partialderivative{r}.$ (48)

In other words, under the transformation $v\to\lambda v\,\,r\to\lambda r$, the
line element transforms as $\differential s^{2}\to\lambda^{2}\,\differential
s^{2}$. Since we are interested in a two-dimensional spacetime, we could start
with the line element by simply omitting the transverse part of the metric
(47). We thus have the initial metric ansatz,

$\displaystyle\differential
s^{2}=\bar{g}\quantity(\frac{r}{v})\bqty{-g\quantity(\frac{r}{v})\differential
v^{2}+2\,\differential v\,\differential r}.$ (49)

It turns out that we can simplify this ansatz considerably by using
appropriate coordinate transformations and get rid of the Weyl factor
$\bar{g}$ altogether. This will aid the later calculations considerably. Let
us first define,

$\displaystyle r=yv.$ (50)

In the $(y,v)$-coordinate system the metric takes the form,

$\displaystyle\differential
s^{2}=\bqty{-\bar{g}(y)g(y)+2y\bar{g}(y)}\differential
v^{2}+2v\bar{g}(y)\differential{y}\differential{v}.$ (51)

Let us define a new coordinate $x$,

$\displaystyle x=\int^{y}\differential{y^{\prime}}\bar{g}(y^{\prime}),$ (52)

which can be inverted to write $y$ as a function of $x$,

$\displaystyle y=y(x).$ (53)

In terms of this new coordinate, the line element can be written as,

$\displaystyle\differential
s^{2}=\bqty{-\bar{g}(y(x))g(y(x))+2y(x)\bar{g}(y(x))}\differential
v^{2}+2v\differential{x}\differential{v}.$ (54)

We retain the old $v$-coordinate, and define a new “radial” coordinate $\rho$,

$\displaystyle x=\frac{\rho}{v}.$ (55)

The line element then reads,

$\displaystyle\differential
s^{2}=-\bqty{\bar{g}(y(x))g(y(x))-2y(x)\bar{g}(y(x))+2x}\differential
v^{2}+2\differential{\rho}\differential{v}.$ (56)

Defining $F(x)\equiv\bar{g}(y(x))g(y(x))-2y(x)\bar{g}(y(x))$, we can write the
simplified self-similar ansatz as,

$\displaystyle\differential
s^{2}=-(F(x)+2x)\differential{v}^{2}+2\differential{v}\differential{\rho},$
(57)

which we shall use in the rest of the article. We emphasise at this point that
$(\rho,v)$ coordinates used here are different from the static coordinates
either in the Schwarzschild gauge or in the Eddington-Finkelstein gauge used
below.

Note that we have done away with one function here in deriving the final form
starting from (49). This can be done in the $(r,v)$ subspace in higher
dimensions as well, where the “extra” undetermined function gets transferred
to the transverse part of the metric. However, in two dimensions there is no
transverse part of the metric. When the dilaton gravity arises from
dimensional reduction of higher-dimensional gravity, it is the dilaton that
plays role of the transverse sphere. Note that the ansatz above has the same
homothety property as before,

$\displaystyle\xi=v\partialderivative{v}+\rho\partialderivative{\rho}.$ (58)

This coordinate chart has the advantage of being well-adapted to describe
gravitational collapse.

We note that the Ricci scalar assumes a particularly simple form in this
coordinate chart,

$\displaystyle R=-\frac{F^{\prime\prime}(x)}{v^{2}}.$ (59)

We will often use the $x$ variable in our investigations to describe our
solutions. In trying to find the self-similar solutions, we will adopt exactly
the same strategy as for the static solutions discussed in Appendix A. We use
the ansatz (57) and insert it in the equations of motion to solve for it. We
would also need an ansatz for the dilaton (and other matter fields).

Recalling that in all cases of interest, the action of the vector field (58)
on the scalar $\phi$ (or the Liouville field $\sigma$) is the constant
$(-\kappa)$, we immediately solve for $\phi$,

$\displaystyle\phi(\rho,v)=\Phi\pqty{x}-\kappa\log v.,$ (60)

where $\Phi(x)$ is a function to be determined from the equations of motion.

Let us mention briefly here that we could have also worked with retarded null
coordinate $u$, which could be obtained by the substitution $v\to-u$ in the
formulæ above. In this case, one describes the patch $u<0$. The form of all
the equations of motion remain unchanged under this transformation — therefore
the solutions we obtain later can also be adapted to this coordinate. This
fact can be used for patching together different regions of spacetime. The
advanced null coordinate seems to be suitable for the initial value problem we
are interested in.

###  3.2 Why Matter Matters: The Vacuum Self-Similar Solutions

The most obvious thing to do at first would be to consider the vacuum
equations and solve them using the using the self-similar ansätze (57) and
(60). The procedure is roughly the same as that of finding the static
solutions; see Appendix A. We do not go through the steps in detail at this
point — some explicit details are given later when we deal with the system in
the presence of additional matter. The results in this subsection can be
treated as a limiting case for the more general scenarios discussed later.

Let us then write down the solutions for the metric function $F(x)$ and the
dilaton function $\Phi(x)$. For the stringy system with $\gamma<4$, the
solutions are given by,

$\displaystyle F(x)=\frac{(4-\gamma)^{2}\kappa
V_{\mathrm{eff}}}{4((8-\gamma)\kappa-4)}x^{2-\frac{4}{(4-\gamma)\kappa}}-(4-\gamma)\kappa
x,$ (61)

and,

$\displaystyle\Phi(x)=-\frac{2}{4-\gamma}\log x.$ (62)

Before we proceed further, it is important to address one important question –
whether the solution we just found represents a genuinely new solution or it
is a familiar solution in a different guise. It should be no surprise that the
latter turns out to be correct. To see this, let us write down the static
solution (140) with $r_{h}\to 0$ in the ingoing Eddington-Finkelstein
coordinate chart (we use $\bar{v}$ as the null coordinate to distinguish it
from the $v$ used in the self-similar ansatz). The line element reads,

$\displaystyle\differential s^{2}=-\frac{(4-\gamma)^{2}\kappa
V_{\mathrm{eff}}}{4((8-\gamma)\kappa-4)}r^{2-\frac{4}{(4-\gamma)\kappa}}\,\differential\bar{v}^{2}+2\,\differential\bar{v}\,\differential
r.$ (63)

The coordinate transformation that brings this line element and dilaton into
the aforementioned forms is,

$\displaystyle r$ $\displaystyle=\rho v^{\frac{(4-\gamma)\kappa}{2}-1},$ (64)
$\displaystyle\bar{v}$
$\displaystyle=\frac{1}{2-\frac{1}{2}(4-\gamma)\kappa}v^{2-\frac{1}{2}(4-\gamma)\kappa}.$

Therefore, the self-similar vacuum solution is nothing but the zero-mass
static solutions discussed in Appendix A. The nature of the singularity at
$r=0$ depends on the value of the parameters $\gamma$ and $\kappa$.

The static $(r,t)$ coordinates are best suited for determining the causal
structure of the spacetimes. Let us bear in mind the discussion on the
restrictions on these parameters from Appendix A — we will continue to impose
these restrictions for the remainder of this article as they are physically
well motivated. If $V_{\mathrm{eff}}\neq 0$ and
$4/(8-\gamma)<\kappa<2/(4-\gamma)$, then there is a timelike singularity at
$r=0$. Demanding only null singularities, one has
$\kappa>2/(4-\gamma)$.555Actually, the value $\kappa=2/(4-\gamma)$ with is
$V_{\mathrm{eff}}\neq 0$ is a special case, for which the curvature is zero
everywhere — the strong coupling surface (where $\phi\to\infty$) is timelike —
it is analogous to the origin of the spherically symmetric coordinate system.
in higher dimensions. We will study this special case later.

(a)

(b)

(c)

Figure 1: Possibilities for the Carter-Penrose diagrams for the self-similar
spacetimes in this section: (a) and (b) correspond to an asymptotic causal
structure similar to Minkowski, while (c) corresponds to an AdS-like
asymptotic causal structure. The wavy lines represent curvature singularities
or regions of strong coupling, in case the curvature is zero everywhere.

It is worth pointing out that for $\kappa>\ifrac{2}{(4-\gamma)}$, $v=0$
corresponds to a curvature singularity and hence, it is not possible to
continue the $v$ coordinate to the region $v<0$. Furthermore, the map (64)
between the static coordinates and the self-similar coordinates obscures an
easy intuition about the self-similar coordinates, which is rather
straightforward in the static coordinate system. In spite of this issue, we
note a useful fact about the $(\rho,v)$ coordinates for all the systems:
$x\to\infty$ with finite non-zero $v$ describes the asymptotic region
$r\to\infty$, where the curvature is vanishingly small for our choice of range
of parameters.

For the stringy coupling, $\gamma=4$, we have,

$\displaystyle F(x)=\frac{\kappa
V_{\mathrm{eff}}}{4(\kappa-1)}\exp(-\frac{2x}{\kappa})-2\kappa,$ (65)

and,

$\displaystyle\Phi(x)=-x.$ (66)

This is equivalent to the $r_{h}\to-\infty$ limit of the metric function
(149), i.e., the line element,

$\displaystyle\differential s^{2}=\frac{\kappa
V_{\mathrm{eff}}}{4(\kappa-1)}\exp(-\frac{2r}{\kappa})\differential\bar{v}^{2}+2\,\differential{v}\differential{r}.$
(67)

In this case, for $V_{\mathrm{eff}}\neq 0$, the curvature grows unboundedly as
$r\to-\infty$, which describes a timelike “singularity” (see Figure 1(c)).
However, as mentioned in Appendix A, this is not a true singularity
Wald:1984rg , since it takes an infinite amount of affine time for a null
geodesic to reach this singularity. For $V_{\mathrm{eff}}=0$, the curvature is
zero everywhere. The coordinate transformation that brings this metric to the
form described by (65) is given by,

$\displaystyle r$ $\displaystyle=\frac{\rho}{v}+\kappa\log v,$ (68)
$\displaystyle\bar{v}$ $\displaystyle=\frac{1}{2}v^{2}.$

Finally, for the Liouville system, the self-similar solution is given by,

$\displaystyle\Sigma(x)=\log x,$ (69)

(here, we use notation analogous to (60), i.e,
$\sigma(r,v)=\Sigma(x)-\kappa\log v$) and,

$\displaystyle
F(x)=\frac{V_{\mathrm{eff}}\kappa}{2+\kappa}x^{2+\frac{2}{\kappa}}+2\kappa x.$
(70)

The coordinate transformation from the $r_{h}\to 0$ static metric (see eq.
(152)) is given by,

$\displaystyle r$ $\displaystyle=\frac{\rho}{v^{\kappa+1}},$ (71)
$\displaystyle\bar{v}$ $\displaystyle=\frac{1}{\kappa+2}v^{\kappa+2}.$

Note that we have taken $\kappa<-2$. See the discussion in the paragraph
before eq. (65) and Appendix A.

Let us also emphasise here that we need a large value of $\sigma_{0}$ (the
coefficient of the topological term in (42)). Since the self-similar solution
is the $r_{h}\to 0$ limit of the static solution, the singularity of the
geometry, at $r=0$, is null in nature. However, strictly speaking if we want
$\sigma_{0}+\sigma$ to be positive at finite $\sigma_{0}$, $r=0$ is not part
of the spacetime and the spacetime terminates on the timelike surface
$r=e^{-\sigma_{0}}$. To ensure that the curvature singularity coincides with
this surface, we always take the limit $\sigma_{0}\to\infty$. Since
$\sigma_{0}$ does not affect the dynamics, we will only need to determine the
curvature singularity. In the remainder of this article, we shall be using
this limit. See also Appendix A.

We therefore see why additional matter truly matters — all the vacuum self-
similar solutions are just zero mass static solutions. To make things more
interesting we add to our systems matter contents — mainly scalar fields,
consistent with the assumption of homothety. We will see that addition of
matter fields makes the dynamics quite rich and interesting.

At the risk of repetition, it is worth emphasising here that the scenario
being considered in this subsection is different from gravity coupled to
matter in higher dimensions — the gravity-dilaton system here is more similar
to pure gravity in higher dimensions. What is truly interesting here is the
fact that a certain non-trivial static solution is also a non-trivial self-
similar solution. This feature is special to two-dimensional dilaton gravity
and can happen only trivially in higher dimensions — i.e., the Minkowski
spacetime.

Let us also point out that under the assumption of continuous self-similarity,
the only possible static solution that could have been self-similar is the
zero mass solution. If there were a mass present, it would have implied the
presence of a length scale in the geometry, which is inconsistent with CSS.

###  3.3 Autonomous Equations for the Stringy System with Matter

Let us consider adding matter to the stringy system first. We consider a
single scalar field $\psi$. If we assume that this scalar has the same sort of
homothety property (23) as the dilaton, i.e.,

$\displaystyle\pounds_{\xi}\psi=-\lambda,$ (72)

then the first obvious candidate — a minimally coupled scalar with zero mass —
is actually not consistent with homothety. This can be readily fixed by
considering a non-minimal dilaton coupling to the scalar — we simply add the
common pre-factor $\exp(-2\phi)$. For good measure, we could also add a
potential term for the scalar so that the combined action reads,

$\displaystyle S=\frac{1}{16\pi
G_{2}}\int\differential[2]{x}\,\sqrt{-g}e^{-2\phi}\bqty{R+\gamma(\nabla\phi)^{2}+V_{\mathrm{eff}}e^{2\phi/\kappa}-(\nabla\psi)^{2}-V_{\psi}e^{2\psi/\lambda}}.$
(73)

One could think of trying to incorporate scalar matter without the dilaton
coupling — it appears to be quite complicated. One would not have a simple
relation of the form (72) and the homothety properties of the scalar might be
dependent on other fields — which is not a very attractive scenario.

From the point of view of gravity in higher spacetime dimensions, however,
this particular form of dilaton coupling with matter is the appropriate one.
If one starts with a $(d+2)$-dimensional gravitational theory with a minimally
coupled scalar $\psi$, (with the potential term as well), then after
compactifying down to two dimensions, the scalar kinetic term and the Ricci
scalar will have the same pre-factor. This is what happens here.

The $\phi$, $g^{\mu\nu}$ and $\psi$ equations of motion are given by,

$\displaystyle
R-\gamma(\nabla\phi)^{2}+\gamma\nabla^{2}\phi+V_{\mathrm{eff}}\pqty{1-\frac{1}{\kappa}}\exp(\frac{2\phi}{\kappa})-(\nabla\psi)^{2}-V_{\psi}e^{2\psi/\lambda}=0,$
(74) $\displaystyle 2\nabla_{\mu}\nabla_{\nu}\phi-2g_{\mu\nu}\nabla^{2}\phi$
$\displaystyle+\frac{8-\gamma}{2}g_{\mu\nu}(\nabla\phi)^{2}+(\gamma-4)\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}e^{2\phi/\kappa}V_{\mathrm{eff}}$
(75)
$\displaystyle-\pqty{\nabla_{\mu}\psi\nabla_{\nu}\psi-\frac{1}{2}g_{\mu\nu}(\nabla\psi)^{2}}+\frac{1}{2}V_{\psi}e^{2\psi/\lambda}g_{\mu\nu}=0,$

and,

$\displaystyle
4\nabla_{\mu}\phi\nabla^{\mu}\psi-2\nabla^{2}\psi+\frac{2}{\lambda}V_{\psi}e^{2\psi/\lambda}=0,$
(76)

respectively.

We use the ansätze (57), (60) and the one that follows from (72), i.e,

$\displaystyle\psi(\rho,v)=\Psi(x)-\lambda\log v.$ (77)

Inserting these in the equations of motion yields ordinary differential
equations in the independent variable $x$.

The set of equations of motion are not all independent. For example, (74)
follows from the (76) and the $(rr)$ and $(rv)$ components of (75). The
assumption of self-similarity imposes a stronger constraint than the static
case, and the metric function $F(x)$ is _uniquely_ (i.e., not even up to a
constant) determined by $\Phi(x),\Psi(x)$ and their first derivatives. The
only dynamical equations to solve are given by a pair of autonomous second
order ordinary differential equations,

$\displaystyle\Phi^{\prime\prime}(x)=\frac{1}{2}\left((4-\gamma)\Phi^{\prime}(x)^{2}+\Psi^{\prime}(x)^{2}\right),$
(78)

and,

$\displaystyle\Psi^{\prime\prime}(x)=\frac{\mathcal{N}}{\lambda\left(\kappa\left(Ve^{\frac{2\Phi(x)}{\kappa}}-V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}\right)+2\left(\lambda^{2}-\kappa((\gamma-8)\kappa+4)\right)\Phi^{\prime}(x)\right)},$
(79)

where the numerator $\mathcal{N}$ is given by,

$\displaystyle\mathcal{N}$
$\displaystyle=\Psi^{\prime}(x)^{2}\left(\lambda^{2}V_{\mathrm{eff}}e^{\frac{2\Phi(x)}{\kappa}}-V_{\psi}\left(\kappa+\lambda^{2}\right)e^{\frac{2\Psi(x)}{\lambda}}+\lambda\left(\lambda^{2}-\kappa((\gamma-6)\kappa+4)\right)\Psi^{\prime}(x)\right)$
(80)
$\displaystyle\quad+\lambda\Phi^{\prime}(x)\Psi^{\prime}(x)\left(-V_{\mathrm{eff}}((\gamma-6)\kappa+2)e^{\frac{2\Phi(x)}{\kappa}}+V_{\psi}((\gamma-6)\kappa+4)e^{\frac{2\Psi(x)}{\lambda}}+6\kappa\lambda\Psi^{\prime}(x)\right)$
$\displaystyle\quad+\Phi^{\prime}(x)^{2}\left(\lambda\left((\gamma-6)\kappa((\gamma-8)\kappa+4)-\gamma\lambda^{2}\right)\Psi^{\prime}(x)-V_{\psi}((\gamma-8)\kappa+4)e^{\frac{2\Psi(x)}{\lambda}}\right)$
$\displaystyle\quad+2\lambda^{2}((\gamma-8)\kappa+4)\Phi^{\prime}(x)^{3}.$

We wrote the above pair of equations to illustrate the autonomous nature of
the differential equations. It would perhaps be helpful to write down the
equations in a simpler form. We will also be able to explain easily how we
derived these equations.

The equation (79) is actually nothing but the $(\rho\rho)$ component of the
metric equation (75). Inserting (75) in the $(v\rho)$ component of the metric
equation, we can obtain a simple expression for $F^{\prime}(x)$ in terms of
$\Phi(x)$, $\Psi(x)$, their derivatives and $F(x)$:

$\displaystyle
F^{\prime}(x)=\frac{V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}-V_{\text{eff}}e^{\frac{2\Phi(x)}{\kappa}}+\gamma
F(x)\Phi^{\prime}(x)^{2}-F(x)\Psi^{\prime}(x)^{2}-8\kappa\Phi^{\prime}(x)}{2\Phi^{\prime}(x)}.$
(81)

We can now insert (81) and (78) in the last remaining metric equation, i.e.,
the $(vv)$ component. Solving the resulting equation for $F(x)$ yields $F(x)$
of the promised form:

$\displaystyle
F(x)=\frac{\kappa\left(V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}-V_{\text{eff}}e^{\frac{2\Phi(x)}{\kappa}}\right)+2\left(\kappa((\gamma-8)\kappa+4)-\lambda^{2}\right)\Phi^{\prime}(x)}{((\gamma-8)\kappa+4)\Phi^{\prime}(x)^{2}+\kappa\Psi^{\prime}(x)^{2}-2\lambda\Phi^{\prime}(x)\Psi^{\prime}(x)}.$
(82)

If we now look at the $\psi$ equation of motion (76) and solve for
$\Psi^{\prime\prime}(x)$, we obtain,

$\displaystyle\Psi^{\prime\prime}(x)=\frac{V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}-\lambda\Psi^{\prime}(x)\left(F^{\prime}(x)-2F(x)\Phi^{\prime}(x)+2\kappa\right)-2\lambda^{2}\Phi^{\prime}(x)}{\lambda
F(x)}.$ (83)

Inserting (81) and (82) in (83) yields the rather complicated expression that
is (79).

Complicated as the equation (79) may be, we can make some non-trivial
statements by virtue of the simplicity of the equation (78). Since the
equations (78) and (79) are a pair of second-order autonomous equations, our
first instinct would be to convert each second order equation into a pair of
first order equations by defining,

$\displaystyle\Phi^{\prime}(x)$ $\displaystyle=P_{\Phi}(x),$ (84)
$\displaystyle\quad\Psi^{\prime}(x)$ $\displaystyle=P_{\Psi}(x).$

The two equations (78) and (79) can now be written as
$P^{\prime}_{\Phi}(x)=\cdots$ and $P^{\prime}_{\Psi}(x)=\cdots$, where the
same expressions as the right-hand-sides of (78) and (79) appear in the
ellipses, with the replacement (84).

Given an autonomous system such as the above, we would like to find the fixed
points of the system and perform a linear stability analysis around the fixed
point. In case of the system in question, such a fixed point represents an
entire spacetime.

Looking at eqs. (78), (79) and (84), we see immediately that the fixed points
are characterised by $P_{\Phi}=0=P_{\Psi}$. Since the phase space of the
autonomous system is four-dimensional, we actually have a co-dimension two
surface, i.e. a two-dimensional plane of non-isolated fixed points. If we set
$V_{\psi}=0$, then only $\Psi^{\prime}(x)$ appears in the differential
equations (78), (79) and not $\Psi(x)$. Therefore, there is an effective
reduction of phase space and the phase space now becomes three-dimensional —
i.e., the second of (84) is just a definition and no longer an equation
governing autonomous evolution. In this case, too, the fixed points are co-
dimension two, (i.e., a line of fixed points) given by $P_{\Phi}=0=P_{\Psi}$.

Let us now come to the issue of linear stability of these fixed points. For
the four-dimensional phase space (when $V_{\psi}\neq 0$), the non-isolated
fixed points lie on a plane. We therefore immediately see that two of the
eigenvalues about each fixed point would be zero. These correspond to constant
values $\Phi(x)=\Phi_{0}$ and $\Psi(x)=\Psi_{0}$. What would be interesting is
the behaviour along the remaining two dimensional subspace even after we fix
$\Phi,\Psi$ to some constant values. It is quite remarkable that the remaining
eigenvalues are zero as well. To see this, note that all the terms on the RHS
of (78) and the numerator of (79) are at least of quadratic order in
$P_{\Phi}$ or $P_{\Psi}$. Therefore, taking a derivative with respect to
either $P_{\Phi}$ or $P_{\Psi}$ and then setting $P_{\Phi}=0=P_{\Psi}$ yields
zero. Therefore, the eigenvalues are zero.

The results in the previous paragraph are in sharp contrast with self-similar
collapse models studied in higher dimensional spacetimes. In such studies,
there were typically more than one fixed point of the autonomous system of
differential equations. One of the fixed points was associated with the flat
spacetime. One of the fixed points usually corresponded to a black hole
spacetime. Previous studies also found a saddle point which governed the
critical behaviour and was responsible for Choptuik-like scaling. In our case,
however, linearisation yields no significant insight into the dynamics —
clearly, we should explore different options to understand the solutions to
the differential equations (78) and (79).

Taken as a statement on the differential equations (78) and (79) the comments
made above the linearisation are true. However, when we think a little more
carefully and take the physics into account, we will conclude that the
particular linearisation we discussed is itself not a valid step. To see this,
we can go back to the equations of motion and ask whether the constant
solutions, viz., $\Phi(x)=\Phi_{0}$ and $\Psi(x)=\Psi_{0}$ meet the equations
of motion. It turns out that the constants are valid solutions only if the
constraint

$\displaystyle
V_{\psi}\exp(\frac{2\Psi_{0}}{\lambda})-V_{\text{eff}}\exp(\frac{2\Phi_{0}}{\kappa})=0$
(85)

is met. When we impose this constraint in conjunction with
$P_{\Phi}=0=P_{\Psi}$, we find that the right hand side of (79) becomes of the
form $0/0$. We will therefore adopt new strategies to attack the problem in
the following sections. Before we do that, however, let us obtain the
requisite differential equations for the Liouville system coupled to two
different classes of matter.

###  3.4 Liouville System and a Minimally Coupled Scalar

In the previous subsection we discussed the difficulty of introducing a
minimally coupled scalar field into the picture while respecting the homothety
symmetry. We now consider the Liouville system earlier discussed in §2.3.
Thanks to the nature of the couplings in this case, it becomes possible to
introduce a minimally coupled scalar field $\psi$ (with perhaps an additional
appropriate potential) that satisfies the homothety property (72). The action
is given by,

$\displaystyle S=\frac{1}{16\pi
G_{2}}\int\differential[2]{x}\sqrt{-g}\pqty{(\sigma_{0}+\sigma)R-(\nabla\sigma)^{2}+V_{\mathrm{eff}}e^{2\sigma/\kappa}-(\nabla\psi)^{2}-V_{\psi}e^{2\psi/\lambda}}.$
(86)

The equations of motion obtained by varying $\sigma$, $g^{\mu\nu}$ and $\psi$
are

$\displaystyle
R+2\nabla^{2}\sigma+\frac{2}{\kappa}V_{\mathrm{eff}}e^{2\sigma/\kappa}$
$\displaystyle=0,$ (87) $\displaystyle
g_{\mu\nu}\nabla^{2}\sigma-\nabla_{\mu}\nabla_{\nu}\sigma-\pqty{\nabla_{\mu}\sigma\nabla_{\nu}\sigma-\frac{1}{2}g_{\mu\nu}(\nabla\sigma)^{2}}-\frac{1}{2}g_{\mu\nu}V_{\mathrm{eff}}e^{2\sigma/\kappa}$
(88)
$\displaystyle-\pqty{\nabla_{\mu}\psi\nabla_{\nu}\psi-\frac{1}{2}g_{\mu\nu}(\nabla\psi)^{2}}+\frac{1}{2}V_{\psi}e^{2\psi/\lambda}g_{\mu\nu}=0,$

and,

$\displaystyle\nabla^{2}\psi-\frac{1}{\lambda}V_{\psi}e^{2\psi/\lambda}$
$\displaystyle=0,$ (89)

respectively.

We use the ansätze (57) and (77) for the metric and the scalar $\psi$ and use
the analogous ansatz for $\sigma$,

$\displaystyle\sigma(\rho,v)=\Sigma(x)-\kappa\log v.$ (90)

Since we have described the process in some detail in the previous subsection,
we will be brief here. As before, we obtain second order autonomous
differential equations for $\Sigma(x)$ and $\Psi(x)$, which are given by,

$\displaystyle\Sigma^{\prime\prime}(x)=-\Sigma^{\prime}(x)^{2}-\Psi^{\prime}(x)^{2},$
(91)

and

$\displaystyle\Psi^{\prime\prime}(x)=-\frac{\lambda
F^{\prime}(x)\Psi^{\prime}(x)-V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}}{\lambda
F(x)},$ (92)

respectively, where,

$\displaystyle
F^{\prime}(x)=\frac{V_{\text{eff}}e^{\frac{2\Sigma(x)}{\kappa}}+F(x)\Sigma^{\prime}(x)^{2}+F(x)\Psi^{\prime}(x)^{2}-V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}}{\Sigma^{\prime}(x)},$
(93)

and,

$\displaystyle F(x)=\frac{\kappa
V_{\text{eff}}e^{\frac{2\Sigma(x)}{\kappa}}-\kappa
V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}+2\left(\kappa(\kappa+2)+\lambda^{2}\right)\Sigma^{\prime}(x)}{(\kappa+2)\Sigma^{\prime}(x)^{2}-\kappa\Psi^{\prime}(x)^{2}+2\lambda\Sigma^{\prime}(x)\Psi^{\prime}(x)}.$
(94)

Similar comments as those after (83) apply in this case too. These appear to
be quite generic features (e.g., limited utility of the linear stability
analysis around a fixed point) of the systems we have considered so far.

Let us point out an interesting fact here. The Ricci scalar (59) is determined
in this case only by the potential functions,

$\displaystyle
F^{\prime\prime}(x)=\frac{2(\kappa+1)V_{\text{eff}}e^{\frac{2\Sigma(x)}{\kappa}}}{\kappa}-2V_{\psi}e^{\frac{2\Psi(x)}{\lambda}}.$
(95)

It can be checked that such is not the case for the stringy system —
derivatives of $\Phi,\Psi$ terms do contribute to the scalar curvature.

###  3.5 Liouville System and a Semi-classical Model

As mentioned in the introduction, two-dimensional gravity is a rich playground
for dealing with tractable aspects of quantum gravity. Models in which quantum
corrections are under calculational control typically involve some sort of
semi-classical approximation. One realisation of such a scenario is given by
adding $N$ free massless scalar fields,

$\displaystyle
S_{\mathrm{M}}=-\frac{1}{2}\int\differential[2]{x}\sqrt{-g}\sum_{i}(\nabla\psi_{i})^{2}.$
(96)

and studying the back-reaction of these scalars in the large $N$ limit. More
precisely, one demands that $N\to\infty$ and the two-dimensional Newton
constant $G_{2}\to 0$ in such a way that $G_{2}N=$ finite. Such a semi-
classical set-up would remind the reader of the Callan-Giddings-Harvey-
Strominger (CGHS) model Callan:1992rs  in 2D gravity. The scenario here is
slightly different. Had we added this matter content to the stringy model (the
pure-gravity sector of (73)) with $\gamma=4$ and $\kappa\to\infty$, this would
have been completely equivalent to the CGHS model. As mentioned previously,
with minimally coupled matter in the stringy model, the homothety
transformation property would be very complicated, if it exists.

A particular aspect of the quantum-corrected system, namely the trace anomaly,
is captured by a simple classical action involving a scalar $\chi$ that also
couples to the Ricci scalar Almheiri:2014cka  (see also Moitra:2019xoj ),

$\displaystyle
S_{\mathrm{M}}=-\frac{N}{24\pi}\int\differential[2]{x}\sqrt{-g}(\chi
R+(\nabla\chi)^{2}).$ (97)

Addition of this matter action to the Liouville system (i.e., eq. (86) without
the $\psi$) is consistent with the homothety properties discussed so far. The
$\sigma$ equation of motion is the same as before, i.e., eq. (87). The
$g^{\mu\nu}$ and the $\chi$ equations of motion are given by,

$\displaystyle{}g_{\mu\nu}\nabla^{2}(\sigma-\zeta\chi)-\nabla_{\mu}\nabla_{\nu}(\sigma-\zeta\chi)$
$\displaystyle-\pqty{\nabla_{\mu}\sigma\nabla_{\nu}\sigma-\frac{1}{2}g_{\mu\nu}(\nabla\sigma)^{2}}$
(98) $\displaystyle-\frac{1}{2}g_{\mu\nu}V_{\mathrm{eff}}e^{2\sigma/\kappa}$
$\displaystyle-\zeta\pqty{\nabla_{\mu}\chi\nabla_{\nu}\chi-\frac{1}{2}g_{\mu\nu}(\nabla\chi)^{2}}=0,$

and,

$\displaystyle R-2\nabla^{2}\chi=0,$ (99)

respectively. Note that we have defined the quantity,

$\displaystyle\zeta=\frac{2}{3}GN,$ (100)

which is the effective quantity controlling the semi-classical dynamics.

Taking, as before, $\pounds_{\xi}\chi=-\lambda$, we get,

$\displaystyle\chi(\rho,v)=X(x)-\lambda\log v,$ (101)

using which we can analogously obtain the two second order differential
equations in $\Sigma$ and $X$.

$\displaystyle\Sigma^{\prime\prime}(x)-\zeta X^{\prime\prime}(x)=-\zeta
X^{\prime}(x)^{2}-\Sigma^{\prime}(x)^{2},$ (102) $\displaystyle
X^{\prime\prime}(x)=\frac{-F^{\prime\prime}(x)-2F^{\prime}(x)X^{\prime}(x)}{2F(x)},$
(103)

where the metric function $F(x)$ and its derivative are given by,

$\displaystyle
F(x)=\frac{V_{\text{eff}}(\kappa-\zeta\lambda)e^{\frac{2\Sigma(x)}{\kappa}}+2(\zeta(\lambda-2)\lambda+\kappa(\kappa+2))\left(\Sigma^{\prime}(x)-\zeta
X^{\prime}(x)\right)}{-2\zeta(\kappa-\lambda+2)\Sigma^{\prime}(x)X^{\prime}(x)+(\zeta\lambda+\kappa+2)\Sigma^{\prime}(x)^{2}-\zeta(\zeta(\lambda-2)+\kappa)X^{\prime}(x)^{2}},$
(104)

and

$\displaystyle
F^{\prime}(x)=\frac{V_{\text{eff}}e^{\frac{2\Sigma(x)}{\kappa}}+F(x)\left(\zeta
X^{\prime}(x)^{2}+\Sigma^{\prime}(x)^{2}\right)}{\Sigma^{\prime}(x)-\zeta
X^{\prime}(x)}$ (105)

respectively. The second derivative of $F(x)$, which determines the Ricci
scalar, is determined only by the field $\Sigma(x)$, like in the previous
model,

$\displaystyle
F^{\prime\prime}(x)=\frac{2(\kappa+1)V_{\text{eff}}e^{\frac{2\Sigma(x)}{\kappa}}}{(\zeta+1)\kappa}.$
(106)

It is worth contrasting the result here with a previous work, Chiba:1997ex
which had ruled out the possibility of continuously self-similar solutions in
a certain model of two-dimensional semi-classical gravity. The scenario in
Chiba:1997ex  is rather different from the one explored in this subsection:
the authors examined the semi-classical equation of motion in a particular
model and concluded that the two sides of the equation did not transform
covariantly under the homothety transformation. Our semi-classical model is
different from theirs and, as is obvious above, consistent with continuous
self-similarity.

##  4 Interlude: Some Simple Dynamical Systems

The results of the previous section suggest that linear stability analysis is
perhaps not the best approach in dealing with the systems under consideration.
Quite interestingly, we will be able to say something about singularity
formation in these models precisely because of the reason linear stability
analysis fails.

Let us consider first the one-dimensional dynamical system governed by the
equation,

$\displaystyle\dot{X}=X^{2},$ (107)

where the overdot ($\dot{}$) indicates a time-derivative. The equation is
suitably non-dimensionalised. The phase plot for the system is shown in Figure
2(a).

This system has a fixed point at $X=0$. However, linear stability analysis
does not yield any useful information about the dynamics as
$\evaluated{\derivative*{(X^{2})}{X}}_{X=0}=0$. This fixed point is clearly
half-stable. This particular dynamical system has one remarkable feature,
though, which goes by the name of _finite-time blow-up_ strogatz2000nonlinear
. In quantum field theory, an analogous phenomenon is a divergent coupling
constant at a finite energy scale Weinberg:1996kr . As we argue now, this
feature is intimately related to spacetime singularities in the two-
dimensional theory of gravity.

(a)

(b)

(c)

Figure 2: The phase plots for the dynamical systems described by eqs. (107),
(109) and (110) respectively

If we have $X=0$ at any time, then clearly, we shall have $X=0$ for all times,
as follows from (107). We ask what happens when we consider slight
perturbations from the equilibrium solution. Let us solve the equation with
the initial condition $X(0)=\epsilon$ with $|\epsilon|\ll 1$. The solution is,
of course,

$\displaystyle X(t)=\frac{1}{\frac{1}{\epsilon}-t}.$ (108)

If $\epsilon>0$, then $X(t)\to\infty$ as $t\to\epsilon^{-1}$ — i.e., an
infinite value of $X$ is reached in a _finite_ time. If we had assumed
$\epsilon<0$, then we would have concluded that there was a blow-up
$X\to-\infty$ at a finite time in the _past_.

The flow from one linear fixed point to another usually takes an infinite
amount of time. However, if the dynamics of the system is dominated by non-
linear terms, then the occurrence of finite-time blow-up is quite generic. Let
us consider the system (see Figure 2(b)) ,

$\displaystyle\dot{X}=-X+X^{2},$ (109)

In this case, there are linear fixed points at $X=0$ and $X=1$. The fixed
points at $X=0$ and $X=1$ are stable and unstable respectively. If at $t=0$,
$0<X<1$, then $X(t\to-\infty)=0$ and $X(t\to-\infty)=1$. However, even though
both are linear fixed points, this system shows finite blow-up in the future
if the initial condition satisfies $X(0)>1$, since the dynamics is now
dominated by the $X^{2}$ term in (109). If $X(0)<0$, then the system shows
finite-time blow-up in the past due to the eventual dominance of $X^{2}$.

We can understand the phenomenon of finite-time blow-up more physically. We
can imagine a particle whose dynamics is governed by an equation of the form
$\dot{X}=X^{\alpha}$. For $\alpha>0$, a particle released from $X=\epsilon$
continues to travel with increasing speed to the right. For $0<\alpha<1$, the
increase in speed is slow, at most polynomial in time. When $\alpha=1$, the
behaviour is exponential in time. When $\alpha>1$, the process is even faster
— it shows a runaway behaviour and the particle reaches any point in finite
time. In most physical systems, such runaway behaviour is typically opposed by
higher order effects.

In the context of the present article, the relevance of the interlude is
hopefully clear. We encounter precisely the same form of equation as (107) for
the self-similar systems in the absence of matter, see e.g., eqs. (78) and
(91). In this section, the time variable $t$ plays the role of the self-
similar variable $x$ and $X$ is equivalent to the derivative of the dilaton
field, i.e., $\Phi^{\prime}(x)$ or $\Sigma^{\prime}(x)$. What the foregoing
discussion implies is that we will always find that $\Phi^{\prime}(x)$ or
$\Sigma^{\prime}(x)$ would diverge for a finite value of $x$. Clearly, so
would $\Phi$ or $\Sigma$. We have chosen our parameters in such a way that it
is ensured that effective Newton constant becomes large. The curvature
singularity, if any, is present precisely on this surface.

Does the inclusion of matter improve things? In the previous examples (78) and
(91), we see that the matter contributes to the dynamical equation with the
_same sign_ as the extant quadratic term. This implies that the finite-time
blow-up occurs even more “quickly” (i.e., at a value $x>0$) in the presence of
matter.

The coupled differential equation for the matter-dilaton system is of course
rather hard to solve in general. Let us show the reduction of the duration
using a simpler system. We add a small perturbation $\Delta^{2}$ to the
equation (107) (see Figure 2(c))

$\displaystyle\dot{X}=X^{2}+\Delta^{2}.$ (110)

Let us solve the equation with the same initial condition as before,
$X(0)=\epsilon$. Let us assume $\Delta^{2}\ll\epsilon^{2}$ so that the
perturbation is really small. The solution to the equation with this initial
condition is,

$\displaystyle X(t)=\Delta\tan\left(\Delta
t+\tan^{-1}\left(\frac{\epsilon}{\Delta}\right)\right),$ (111)

which blows up at,

$\displaystyle t$
$\displaystyle=\frac{1}{\Delta}\pqty{\pm\frac{\pi}{2}-\tan^{-1}\quantity(\frac{\epsilon}{\Delta})}$
(112)
$\displaystyle=\frac{1}{\epsilon}\pqty{1-\frac{\Delta^{2}}{3\epsilon^{2}}+\mathcal{O}(\Delta^{4}/\epsilon^{4})},$

where the upper (lower) sign is to be chosen for a positive (negative)
$\epsilon$. Therefore, this system shows finite-time blow-up both in the past
and the future. The duration of blow-up is, however, much shorter.

##  5 Analytical and Numerical Results

Now that we know that we will encounter singularities generically, we want to
learn about the gravitational dynamics of the systems at a more detailed
level. We shall be using a combination of analytical and numerical techniques.
The goal of this section is to look at certain important qualitative aspects
of collapse. The very first questions involving gravitational collapse one has
in mind concern the nature of the curvature singularity and formation of
(apparent) horizons when the singularity is spacelike. Besides analytical
results, we will present a sampling of numerical results to illustrate the
variety of possibilities. There are several parameters in the problem, which
can give rise to a rather diverse set of results. One need look no further
than Appendix A to see an example.

In discussing the self-similar coordinate system, we had assumed that $v>0$.
Since we saw that $v=0$ corresponds to a curvature singularity for some
geometries, we will always consider the finite interval $\varepsilon\leq
v<\Lambda$, where $\varepsilon>0$ is a fixed number (recall that we have set
the characteristic length scale to unity) and $\Lambda$ is a suitable large
number (to avoid any complication from $v\to\infty$ limits). We will be
turning on matter only at $v=\varepsilon$. This choice of coordinates
describes only a part of the spacetime. We attach the geometry with matter in
$v>\varepsilon$ to the vacuum geometry in the region $v<\varepsilon$ and match
the two geometries along the null surface $v=\varepsilon$. Since the scalar
field is turned on suddenly at $v=\varepsilon$, there will be some
discontinuity across $v=\varepsilon$, which can be interpreted as a shock
wave.

We would essentially be solving a pair of coupled differential equations
(involving the dilaton field and the matter field). We will need some initial
conditions in $x$ to solve the differential equations. Physical considerations
readily tell us at which value of $x$ we should impose the conditions. When
the asymptotics are Minkowski-like, the initial conditions should be imposed
on the past null infinity $\mathscr{J}^{-}$, where the effective gravitational
coupling is very weak. It is easy to see from previous discussions that this
happens as $x\to\infty$, for fixed and finite $v$ (this feature is common to
all models). Since we have taken $v\geq\epsilon$, the $x\to\infty$ limit is
not associated with $v\to 0$ and finite $\rho$. For numerical analysis, it is
not possible to impose initial conditions at $x=\infty$. We will impose them
at $x=x_{L}$, where $x_{L}$ is some large number and integrate “back in time”.

Since the boundary of the spacetime is _defined_ by the dilaton fields, we
will use the same boundary conditions for these fields and their derivatives
as the vacuum solution, namely,

$\displaystyle\Phi(x_{L})$ $\displaystyle=-\frac{2}{4-\gamma}\log x_{L},$
$\displaystyle\quad\Phi(x_{L})$
$\displaystyle=-\frac{2}{4-\gamma}\frac{1}{x_{L}},$ (113)
$\displaystyle\Sigma(x_{L})$ $\displaystyle=\log x_{L},$
$\displaystyle\quad\Sigma^{\prime}(x_{L})$ $\displaystyle=\frac{1}{x_{L}}.$

For the stringy model with $\gamma=4$, we will do essentially the same thing,
with a slight adaptation to suit the numerics. In this section, we have set
$V_{\psi}=0$ throughout, which corresponds to a free massless scalar in higher
dimensions (for the stringy system) or a massless free two-dimensional scalar
in two dimensions (for the Liouville system). For the stringy models, it turns
out that even with

$\displaystyle\Psi(x_{L})=0=\Psi^{\prime}(x_{L}),$ (114)

non-trivial dynamics is generated due to a non-zero $\lambda$. Therefore, we
use the condition (114) for the stringy systems and vary $\lambda$, which is a
very natural boundary condition on the null infinity. For the Liouville
systems, we have to have a non-zero value of $\Psi^{\prime}(x_{L})$ to start
the numerics and we use initial values,

$\displaystyle\Psi^{\prime}(x_{L})=\epsilon,\quad\Psi(x_{L})=0,$ (115)

to start the numerical integration, where $\epsilon$ is some small non-zero
number.

Due to the relative complexity of the $(\rho,v)$ coordinate system, we will
refrain from making comments that depend on this coordinate system and instead
make statements that are coordinate-invariant. The characterisation of the
singularity is not difficult. If the singular hypersurface is given by
$x=x_{0}$, then the Ricci scalar generally blows up in the vicinity of
$x=x_{0}$. Since we approach this point from the right, we shall find666Since
we have restricted $v\geq\epsilon$, there is no divergence associated with the
$v^{2}$ in the denominator of the definition of the Ricci scalar (59). ,

$\displaystyle\quantity|F^{\prime\prime}(x_{0}^{+})|\to\infty.$ (116)

On the other hand the causal nature of the singularity is given simply by the
sign of $F(x)$ in the vicinity of $x=x_{0}$:

$\displaystyle F(x_{0}^{+})\begin{cases}&>0\Rightarrow\text{timelike
singularity},\\\ &=0\Rightarrow\text{null singularity},\\\
&<0\Rightarrow\text{spacelike singularity}.\end{cases}$ (117)

The condition above is easily derived from considering the hypersurface where
the singularity occurs,

$\displaystyle\frac{\rho}{v}-x_{0}=0.$ (118)

The causal nature of this surface is given simply by the sign of the norm of
the vector field normal to this surface. The norm in question is given by
$F(x)/v^{2}$, i.e, in terms of the function that appears in the metric ansatz
(57). it goes without saying that this result is robust even if the
hypersurface were defined by the relation,

$\displaystyle H(\rho,v)(x-x_{0})=0,$ (119)

where $H(\rho,v)$ is some arbitrary function (at least $C^{1}$) of $\rho,v$.
In that case, there do exist additional contributions to the norm but these
vanish as $x\to x_{0}$. The norm near the hypersurface (119) is then given by
$F(x_{0}^{+})H(x_{0}v,v)^{2}/v^{2}$, which means that the characterisation
(117) holds true no matter what the function $H(\rho,v)$ is. Based on our
results, we have qualitatively depicted some possible causal structures in
Figure 3. We see that the dynamics of the singularity is sometimes reminiscent
of the Dray–’t Hooft geometries Dray:1984ha . In such geometries, a high
energy shock-wave would be sent towards a black hole and one would then match
the geometries across the shock-wave. The black hole horizon would be shifted
relative to its initial position — we observe a similar phenomenon here vis-à-
vis the singularities.

(a)

(b)

(c)

(d)

Figure 3: Some qualitatively different possible causal structures under
dynamical evolution. The static spacetime is matched to a dynamical spacetime
(grey) along $v=\epsilon$. The region surrounded by the thin lines does not
exist. For asymptotically Minkowski-like spacetimes with an initially null
singularity, the matter pulse can result in a spacelike (a) or a shifted null
singularity (b). When the initial singularity (or the strong coupling region)
is timelike, one finds a spacelike singularity (c) under dynamical evolution.
For AdS-like asymptotics with a null singularity, one finds a shifted null
singularity (d).

We will also attempt to learn whether there exists an apparent horizon in the
spacetime, when there is a spacelike singularity present. In two-dimensional
gravity, in analogy with gravity in higher spacetime dimensions, the locus of
the apparent horizon is given by a zero value of the norm of the gradient of
the pre-factor of the Ricci scalar in the action (see Moitra:2019xoj ). For
the stringy system, for example, the relevant equation is given by

$\displaystyle\quantity(\nabla\phi)^{2}=\frac{\Phi^{\prime}(x)\left(F(x)\Phi^{\prime}(x)-2\kappa\right)}{v^{2}}=0.$
(120)

The generalisation to the Liouville system is straightforward. By our boundary
conditions, we have $\Phi^{\prime}(x\to\infty)\to 0$. Therefore, if this
condition describes an apparent horizon, it is a trivial one, at asymptotic
infinity. We call the function,

$\displaystyle G(x)=F(x)\Phi^{\prime}(x)-2\kappa,$ (121)

a testing function for the non-trivial apparent horizon.

When there is a spacelike singularity and a non-trivial apparent horizon, an
interesting quantity to study would be the quasi-local mass function, which is
well-suited for application in dynamical situations such as this. In previous
works such as Brady:1994xfa ; Aniceto:2019mco , it was found that the quasi-
local mass function evaluated on the apparent horizon exhibits a Choptuik-like
scaling law. We will try to see if such a scaling law exists for the non-
trivial apparent horizon.

Before we describe the results, let us make a comment on the numerical methods
used. We argued that the existence of singularity is generally guaranteed —
this fact also makes the numerics a very difficult exercise. One has to be
extremely precise in the vicinity of the singularity of the solution, see
antia2012numerical . In particular, one needs to have some form of adaptive
step size control to get reliable results around the singularity, especially
when $F(x)$ goes to zero at the singularity. We have taken this into account
and used different methods to compare the results around the singularity to
get reliable results.

###  5.1 Zero Potential: 2D Phase Space

One of the simplest possibilities in the parameter space is when we consider
zero values of the potential for both the dilaton and matter fields
$V_{\mathrm{eff}}=0=V_{\psi}$. Since the variables $\Phi(x)$ (or $\Sigma(x)$)
and $\Psi(x)$ (or $X(x)$) do not appear explicitly in the equations of motion,
the phase space is effectively two-dimensional.

For the Liouville systems with the scalar $\psi$, the solution can be put in a
closed form. To see this, we write down the ODEs for $\Sigma,\Psi$. They are
given by,

$\displaystyle\Sigma^{\prime\prime}(x)$
$\displaystyle=-\Sigma^{\prime}(x)^{2}-\Psi^{\prime}(x)^{2},$ (122)
$\displaystyle\Psi^{\prime\prime}(x)$
$\displaystyle=-\frac{\Psi^{\prime}(x)\left(\Sigma^{\prime}(x)^{2}+\Psi^{\prime}(x)^{2}\right)}{\Sigma^{\prime}(x)}.$

Imposing the initial conditions (113) and (115), we easily find the solution,

$\displaystyle\Sigma^{\prime}(x)$
$\displaystyle=\frac{1-x_{0}/x_{L}}{x-x_{0}},$ (123)
$\displaystyle\Psi^{\prime}(x)$
$\displaystyle=x_{L}\epsilon\frac{1-x_{0}/x_{L}}{x-x_{0}},$

where,

$\displaystyle x_{0}=x_{L}\frac{(x_{L}\epsilon)^{2}}{1+(x_{L}\epsilon)^{2}}.$
(124)

These derivatives diverge as $x\to x_{0}$, the fields $\Sigma,\Psi$ themselves
diverge logarithmically around this point. Note that if we take
$x_{L}\to\infty$ and $\epsilon\to 0$ in such a way that $x_{L}\epsilon^{2}=$
finite, then $x_{L}-x_{0}>0$ and finite. Therefore, under this scaling, the
singularity is reached at a shorter distance in $x$.

On account of (95), we find the Ricci scalar is zero everywhere. The function
$F(x)$ vanishes linearly in $(x-x_{0})$. Therefore the hypersurface of strong
coupling is null in nature. Clearly, there is also no non-trivial apparent
horizon in the spacetime. The analogue of the function $G(x)$ defined
previously, eq. (121), evaluates to a constant function on the solution.

We get completely analogous results for the semi-classical model, so we do not
repeat the analysis here. In this case, the with the boundary condition
$X^{\prime}(x_{L})=\epsilon$, the blow-up occurs at,

$\displaystyle x_{0}=\zeta
x_{L}\frac{x_{L}\epsilon(1+x_{L}\epsilon)}{1+\zeta(x_{L}\epsilon)^{2}},$ (125)

which tells us that the value of $x_{0}$ becomes independent of $\zeta$ at
sufficiently large values of $\zeta$. This is a rather curious result — adding
more effective degrees of freedom does not shift the “singularity” around!
However, we have to exercise caution with taking a large $\zeta$ limit, since
taking an arbitrarily large $\zeta$ might lead to inconsistencies
Moitra:2019xoj . Interestingly, we see analytically that for $\epsilon<0$, the
“singularity” moves inwards. For $V_{\mathrm{eff}}=0$, we get an
asymptotically flat spacetime and not the AdS-like asymptotics we discussed
for the Liouville system, since that analysis relied on a non-zero value of
$V_{\mathrm{eff}}$.

Let us finally discuss the stringy system. In contrast with the Liouville
systems discussed above, there is no closed form solution, but we can make
some comments analytically. Looking at the expression for $F(x)$ in eq. (82),
one might be tempted to consider the special surface,

$\displaystyle\kappa((8-\gamma)\kappa-4)+\lambda^{2}=0,$ (126)

in the parameter space and conclude that $F(x)=0$ for parameters satisfying
this constraint. This constraint corresponds to a singular behaviour777Not the
sort we are interested in. in the differential equations. The RHS of eq. (79)
blows up when this constraint is met. Interestingly, if the inequality (143)
is met, the we can never have (126).

We next examine the possibility of formation of a non-trivial apparent
horizon, the locus of which is given by,

$\displaystyle-\frac{2\left(\lambda\Phi^{\prime}(x)-\kappa\Psi^{\prime}(x)\right)^{2}}{((\gamma-8)\kappa+4)\Phi^{\prime}(x)^{2}+\kappa\Psi^{\prime}(x)^{2}-2\lambda\Phi^{\prime}(x)\Psi^{\prime}(x)}=0.$
(127)

This is zero along the surface
$\lambda\Phi^{\prime}(x)-\kappa\Psi^{\prime}(x)=0$. However, one can show
using the differential equations (78) and (79) that the relation,
$\lambda\Phi^{\prime}(x)-\kappa\Psi^{\prime}(x)=0$, if true at one value of
$x$, will be true for all values of $x$. In other words, this condition is
preserved along the vector field $(\Phi^{\prime\prime},\Psi^{\prime\prime})$
in the $(\Phi^{\prime},\Psi^{\prime})$-plane. A phase portrait is shown in
Figure 4. Therefore, there is no formation of an apparent horizon in this
particular system for any set of values of the parameters.

Figure 4: The phase plot in the relevant region on the
$\Phi^{\prime}-\Psi^{\prime}$ plane. Each curve corresponds to a different
solution. The red curve is the locus of $G(x)=0$.

Let us now describe what information we can glean by a numerical analysis. Let
us note that in the initial geometry, there is no curvature singularity
anywhere and the strong coupling surfaces are null in nature.

For $\gamma=2$ and $\kappa=1$, a variation of $\lambda$ gives rise to the
following behaviour: For very small $\lambda$, say $\lambda=10^{-6}$, the
solution looks very much like the unperturbed solution for a long extent, as
one would expect. It turns out, however, that besides $\Phi(x)\to+\infty$ at a
small value $x=x_{0}$, there is a spacelike curvature singularity at
$x=x_{0}$. The curvature singularity was previously absent. As we dial up
$\lambda$, the singularity becomes much stronger and “comes outward”, i.e.,
$x_{0}$ increases. Therefore, for this choice of parameters, turning on a
profile for the matter field turns the null surface of strong coupling into a
genuine spacelike curvature singularity with increasing strength. Since there
is no non-trivial apparent horizon, we conclude that the singularity stretches
to the end of $\mathscr{J}^{-}$. See Figure 3(a) for a possible causal
structure.

When we try doing the same exercise for larger values of $\kappa$, i.e.,
$\kappa>1$ with fixed $\gamma$. It turns out that from very small values of
$\lambda$ up until very large values of $\lambda$, we get a curvature
singularity at $x=x_{0}$ which is always _null_. The result indicates that the
null singularity after the pulse has moved out. The causal structure is given
by Figure 3(b).

One of the questions not addressed by the analysis in §4 is whether a finite-
time blow-up in case of $\gamma=4$ is guaranteed. Indeed, in the absence of
matter, one simply has the equation,

$\displaystyle\Phi^{\prime\prime}(x)=0,$ (128)

and there is no finite-time blow-up even if we consider the solution
$\Phi^{\prime}(x)=-a$. ($|\Phi(x)|$ diverges only at $x=\pm\infty$). When one
includes matter, the equations are changed and on numerically solving the
equations, we find that there is indeed a finite-time blow-up occurring here
as well and a spacelike singularity is formed here — this suggests that the
singularity is now a “true singularity”, reachable in a finite affine time.
(Similar numerical results with $V_{\mathrm{eff}}\neq 0$ are shown in Figure
5(c).)

It is worth pointing out one very interesting feature from the numerical
analysis here. When $V_{\mathrm{eff}}=0$, the strong coupling surface is a
null surface in the vacuum for all values of $\kappa$. When
$V_{\mathrm{eff}}\neq 0$, there is a singularity which is timelike or null
depending on the sign of $\kappa-\ifrac{2}{(4-\gamma)}$. It is a very
interesting that there is a transition in the nature of singularity with
$V_{\mathrm{eff}}=0$ at precisely this value of $\kappa$!

Thus, even in this rather simple setting where we have set all the potentials
to zero, we see rather interesting qualitative behaviour.

###  5.2 Numerical Results for Non-Zero Effective Potential

We first look at the stringy system. We now consider the case where
$V_{\mathrm{eff}}>0$ and try to answer the same question as before (however,
we continue to have $V_{\psi}=0$). One could observe several important
departures from the previous subsection. For one, the function $G(x)$ can now
smoothly cross zero, which could signify the existence of an apparent horizon.
Turning on $V_{\mathrm{eff}}$ springs some surprises and we get rather
interesting results.

At first, we take a set of parameters for which the singularity in the vacuum
solution is null. To be concrete, we study the cases where the asymptotic
structure is AdS-like, i.e., $\gamma=2$, $\kappa>2$ (the numerical plots for
the case $\kappa=6$ case is shown in Figure 5(a)). We choose the initial value
at $x_{L}=10^{6}$. We find that analogous to the $V_{\mathrm{eff}}=0$
situation, there is always a null singularity for all the values of $\lambda$
that we have used. This structure persists for reasonably large values of
$\kappa$ with a fixed $\gamma$. The causal structure looks like Fig 3(d). For
$\gamma=2$ and the range $1<\kappa\leq 2$, the asymptotic structure is
Minkowski-like. However, we see that the singularity continues to be null in
character. The function $F(x)$ for these cases decays to zero near the
singularity.

(a)

(b)

(c)

(d)

Figure 5: A sample of the numerical results. In the first three plots, various
quantities are plotted against $x$ in the vicinity of the singularity at
$x=x_{0}$ (the origin). For all the plots $V_{\mathrm{eff}}=1$ and
$x_{L}=10^{6}$. For (a), $\gamma=2$, $\kappa=6$, $\lambda=10^{-4}$; for (b),
$\gamma=2$, $\kappa=1$, $\lambda=10^{-6}$, and for (c) $\gamma=4$,
$\kappa=10$, $\lambda=10^{-10}$, $a=10^{-4}$. Many quantities diverge near
$x=x_{0}$. In (d), a log-log plot of $M(x)$ (at the apparent horizon) vs
$(\lambda-\lambda_{c})$ is shown, with two piecewise linear fits. The
parameters for (d) are $\gamma=2$ and $\kappa=1$.

We next turn our discussion to the special values of parameters $\gamma=2$,
$\kappa=1$, the values of parameters related to the Schwarzschild-like black
hole (144). The initial geometry has zero curvature everywhere. For small
enough values of $\lambda$, there appears to be a null singularity. As we dial
up $\lambda$, above a certain critical value of $\lambda=\lambda_{c}$, we find
that the singularity is spacelike (see Figure 3(c)). Furthermore, we find that
the function $G(x)$ does cross zero. In other words, there seems to be a non-
trivial apparent horizon in the geometry when we find the spacelike
singularity. The quasi-local mass function $M(x)$ (suitably multiplied with
some power of $v$) defined in Appendix B is useful now. One naturally wonders
whether there is some Choptuik-like scaling phenomenon taking place here. As
mentioned before, a possible form of scaling Brady:1994xfa ; Aniceto:2019mco
is the scaling of the mass function $M(x)$ at the location of the apparent
horizon (i.e., the zero of $G(x)$) with $\lambda$.

Plotting $\log M$ against $\log(\lambda-\lambda_{c})$, we see a rather
interesting behaviour, as shown in Figure 5(d). A straight line fits the
numerical data beautifully in the vicinity of $\lambda=\lambda_{c}$. As one
moves further away from this value of $\lambda$, there is a new linear slope
governing the mass. This suggests that $M(x)$ is governed by different scaling
exponents:

$\displaystyle M\sim
c_{1}(\lambda-\lambda_{c})^{\alpha}+c_{2}(\lambda-\lambda_{c})^{\beta}.$ (129)

Numerically we find $\alpha\approx 0.8$ and $\beta\approx 1.9$. There is some
transition taking place around the point where these two exponents exchange
dominance. One naturally wonders whether the exponents etc. are dependent on
the regulator $x_{L}$. We find that although $\lambda_{c}$ does depend on
$x_{L}$, there does not seem to be a strong dependence of the exponents on
$x_{L}$. In fact, $\lambda_{c}$ seems to scale inversely with $x_{L}$ — thus,
when we take $x_{L}\to\infty$, $\lambda_{c}\to 0$. Therefore, turning on even
a small $\lambda$ would lead to a spacelike singularity with a black hole-like
geometry. The results above are strongly reminiscent of Choptuik-like
criticality, with the critical parameter being $\lambda$.

There are more surprises. The results above suggest that as we increase
$\lambda$ further, $M$ will also increase. However, there comes a point for
sufficiently large $\lambda$ at which the non-trivial apparent horizon is not
formed. In this case, only the trivial apparent horizon at infinity survives.
In fact, the non-trivial apparent horizon appears to be an artefact of finite
$x_{L}$. When we increase $x_{L}$ by some factor $a$, the location of the
apparent horizon increases by a much bigger factor — this suggests that this
non-trivial apparent horizon might not survive in the $x_{L}\to\infty$ limit.
However, even on increasing $x_{L}$ by four orders of magnitude, we find
roughly the same exponents $\alpha$ and $\beta$, which indicates these
exponents might actually be universal.

Some of these results also seem to hold true when we vary $\kappa$ slightly so
the initial geometry has an timelike singularity — this timelike singularity
turns out to be unstable and collapses to a spacelike singularity, with the
causal structure given by Figure 3(c). Analogous non-trivial apparent horizons
are seen in this case as well.

In all the cases discussed so far, the initial condition $\Psi^{\prime}(x)=0$
is consistent with a positive mass function at $x=x_{L}$. In fact, the mass
function behaves as

$\displaystyle M(x_{L})\sim x_{L}^{4/(4-\gamma)}\lambda^{2}.$ (130)

Therefore, as we take the limit $x_{L}\to\infty$, keeping $\lambda$ fixed, the
mass function diverges. This fact is not too surprising in the context of
self-similar collapse. In situations such as this, one has to normalise
various quantities appropriately to extract physically meaningful quantities.
Put this way, we can perhaps interpret the above as a Choptuik-like scaling in
$\lambda$ with exponent being given by the integer 2! In fact, $\beta$ is
quite close to 2. It would be interesting to examine the scaling behaviour
further.

However, the issue of defining quasi-local mass in general relativity is
extremely subtle — more so when there is additional matter present. We have
used one proposed definition from the literature. It is conceivable that other
(inequivalent) proposals of mass function could be of more direct utility for
the problem at hand. We leave such a possibility for a future study.

We finally study the case of stringy coupling, $\gamma=4$. In this case, for
the numerics to be better controlled, we choose the following initial
conditions, $\Phi(x_{L})=-ax_{L}$ and $\Phi^{\prime}(x_{L})=-a$, where $a$ is
a small number. For the data presented in Figure 5(c), we have chosen
$a=10^{-4}$, $\lambda=10^{-10}$ and $\kappa=10$. In this case, we find a
strong spacelike singularity accompanied by an apparent horizon. This is very
similar to the $V_{\mathrm{eff}}=0$ case. It is worth mentioning that if we
take $a$ to be very small keeping all the other parameters fixed, we find a
very strong _timelike_ singularity. If this transition between the nature of
the singularities is not some finite cut-off or numerical artefact and has
some genuine physical origin, it would be interesting to examine it. We hope
to return to this question in a future work. It is also worth mentioning that
for $V_{\mathrm{eff}}=0$, the singularity always seems to be spacelike, no
matter how small $a$ is.

We end this section with remarks on the numerical results for the Liouville
system with $V_{\mathrm{eff}}\neq 0=V_{\psi}$. Using allowed values of the
parameters $\kappa$, we found that for a wide sampling of values of
$\Psi^{\prime}(x_{L})$ and $\lambda$, we get a null singularity, which has
moved “outward”. Therefore, the causal diagram is that of Figure 3(d). What
happens when we study the semi-classical $\chi$ system? Again, the results are
not very different from what is observed in the Liouville system with the
minimal scalar. We always form a null singularity.

In summary, we find very interesting behaviour in all corners of the parameter
space for the models we considered.

##  6 Discussion and Future Directions

The investigation reported in this article clarified some aspects of self-
similar gravitational collapse and nature of singularities in two dimensions.
One can think of several ways to ask physically relevant question in related
contexts.

Our starting point was the existence of a homothetic Killing vector. We found
a rather wide class of theories consistent with this symmetry. It would be
interesting to see what happens when one relaxes this assumption — we might
see even richer behaviour than has been found in this article. It is worth
noting that the original Choptuik scaling result was associated with a
discrete self-similarity (DSS). While the evolution equations were ODEs with
the assumption of CSS ansatz, it would be interesting to see whether a
solution to the partial differential equations in these systems shows some
behaviour pertaining to DSS. It is conceivable that other forms of the dilaton
potential might be useful in this regard. Furthermore, we imposed some
constraints on the parameters of the theory seeking some nice properties of
the static spacetimes. Relaxing these constraints might prove to be useful,
especially when one considers microscopic realisations to model self-similar
collapse. It would also be interesting to see if variations of the matter
content lead to results of similar nature.

We worked in a particularly simple coordinate system, in which the ODEs were
autonomous — which made the resulting analysis easy. It would be interesting
to see if these features can be extended to some other coordinate systems
which are better suited to answer some specific questions. As mentioned in the
introduction, it would be interesting to use the results in this article to
understand similar questions in higher dimensional scenarios. Theories like
the Jackiw-Teitelboim model Teitelboim:1983ux ; Jackiw:1984je  arise naturally
in the near-horizon region of near-extremal black holes. In the set-ups of our
consideration, there were singularities instead of black holes. With
appropriate ingredients, one could construct analogous singularities in higher
dimensional set-ups. It would be interesting to explore whether, in the
relevant limit, aspects of the dynamics of such singularities are captured by
the set-up explored in this article or deformations thereof.

Only a selected sample of extensive numerical investigations has been
presented in Section 5 of this article. Many of the results gleaned from the
numerical analysis are quite intriguing. For example, we found that a timelike
singularity if present initially, quickly disappears and becomes a spacelike
singularity stretching to null infinity. This is a very pleasing result from
the point of view of the weak cosmic censorship conjecture, and it would be
nice to understand this result from a microscopic perspective. We got a
tantalising hint of Choptuik-like criticality in our numerical results — it
would be interesting to explore this aspect in a wider region of the parameter
space. Another very interesting result was the fact that whenever the
asymptotic causal structure resembled AdS asymptotics, the singularity in the
dynamical situation was null.

Besides addressing some of the questions raised above, we also hope to present
some more detailed aspects of the numerical analysis in future works.

###### Acknowledgements.

I thank the ICTP for support. I also thank the following organisations for
their hospitality during the progress of this work: the Max Planck Institute
for the Physics of Complex Systems, the Simons Center for Geometry and
Physics, Stony Brook University, (during the 2022 Simons Summer Workshop), the
University of Chicago, the University of Wisconsin–Madison and the University
of Michigan, Ann Arbor.

##  Appendix A Static Black Hole Solutions

In Section 3, we set-up the stage for discussing self-similar solutions in a
detailed manner. In this Appendix, we explore something simpler which arises
from the actions derived in Section 2 – static solutions without additional
matter. We will construct black hole solutions in these models with a
curvature singularity behind the event horizon. These static solutions are not
self-similar in general. We explore these solutions to build more intuition
about the spacetimes under consideration. We will examine the nature of the
asymptotic structure of these spacetimes and also constrain the parameters
that appear in the Lagrangian. Some related solutions are well studied in the
literature — see the early works Lechtenfeld:1992rt ; Mann:1993rf  where the
focus was on finding asymptotically flat black holes.

To find out the static solutions, we will consider a coordinate system in
which the metric is put in the Schwarzschild gauge,

$\displaystyle\differential s^{2}=-f(r)\,\differential
t^{2}+\frac{\differential r^{2}}{f(r)},$ (131)

and the dilaton too depends only on the spatial coordinate $r$,

$\displaystyle\phi=\phi(r).$ (132)

(When we consider the Liouville model, we will similarly assume that
$\sigma=\sigma(r)$). The vector field $\partial/\partial t$ is a Killing
vector of the gravity-dilaton system. In this coordinate system the expression
for the Ricci scalar is particularly simple,

$\displaystyle R=-f^{\prime\prime}(r).$ (133)

We therefore carry out the usual steps of inserting these ansätze in the
equations of motion and solving for the functions $f(r)$ and $\phi(r)$ (or
$\sigma(r)$).

###  A.1 Stringy System

We find that the off-diagonal $(tr)$ component of the metric equations of
motion (37) is identically zero. A certain combination of the $rr$ and $tt$
components yields the simple differential equation for the dilaton,

$\displaystyle 2\phi^{\prime\prime}(r)+(\gamma-4)\phi^{\prime}(r)^{2}=0,$
(134)

which immediately yields the solution (for $\gamma\neq 4$),

$\displaystyle\phi(r)=\phi_{0}-\frac{2}{4-\gamma}\log(r-r_{0}).$ (135)

We fix the two constants of integration $\phi_{0}$ and $r_{0}$ as follows:
$\phi_{0}$ can be set to zero by a redefinition of the effective potential
strength $V_{\mathrm{eff}}$ (at the level of the equations of motion), while
on the other hand, $r_{0}$ simply corresponds to the “origin” of the $r$
coordinate and can thus be set to zero. In other words, we take the $r$
coordinate to lie in the range $0<r<\infty$. A characteristic length scale
appears within the logarithm to make its argument dimensionless – we have
conveniently set this length scale to unity. Finally, the dilaton takes the
form,

$\displaystyle\phi(r)=-\frac{2}{4-\gamma}\log r.$ (136)

The dilaton can thus assume any real value. When $\gamma<4$, the dilaton
$\phi$ monotonically decreases from $+\infty$ to $-\infty$ as a function of
$r$. Note that the Ricci scalar couples to,

$\displaystyle\exp(-2\phi)=r^{\frac{4}{4-\gamma}}.$ (137)

Therefore, for identification of $r$ with the conventional radial coordinate
(where the asymptotic region is located for large values of $r$), we shall
take,

$\displaystyle\gamma<4,$ (138)

(we will treat the “stringy coupling” $\gamma=4$ separately). We will also
impose the constraint $\gamma\geq 0$, which is relevant for generic
dimensional reductions.

Let us now solve for $f$. We obtain from the $rr$ component of (37) the
following first order equation in $f$,

$\displaystyle 2f^{\prime}(r)\phi^{\prime}(r)-\gamma
f(r)\phi^{\prime}(r)^{2}+V_{\mathrm{eff}}e^{\frac{2\phi(r)}{\kappa}}=0.$ (139)

The solution is given by,

$f(r)=\frac{(4-\gamma)^{2}\kappa
V_{\mathrm{eff}}}{4((8-\gamma)\kappa-4)}r^{2-\frac{4}{(4-\gamma)\kappa}}\bqty{1-\pqty{\frac{r}{r_{h}}}^{\frac{\gamma\kappa-8\kappa+4}{(4-\gamma)\kappa}}},$
(140)

where $r_{h}$ is a constant of integration, carefully chosen so that the event
horizon of the black hole lies at $r=r_{h}>0$.

We must have $f(r)>0$ outside the black hole, i.e., when $r>r_{h}$. This
requirement implies, in conjunction with the condition (138) that,

$\displaystyle V_{\mathrm{eff}}>0.$ (141)

We can find the form of Ricci scalar from eq. (133),

$\displaystyle
R=(\cdots)r^{-\frac{4}{(4-\gamma)\kappa}}+(\cdots)r^{-\frac{8-\gamma}{4-\gamma}}.$
(142)

Since we want our asymptotic region to lie near $r\to\infty$ and the
singularity at $r=0$, we require $\kappa>0$. Note that with this constraint,
$R\to 0$ as $r\to\infty$ and $|R|\to\infty$ as $r\to 0$. Therefore, the
asymptotic region $r\to\infty$ has zero curvature and in the region near
$r=0$, the curvature becomes arbitrarily large. Whenever $r_{h}>0$, this
(spacelike) singularity is shielded by the event horizon – there is a null or
timelike singularity otherwise. The parameter $r_{h}$ can be related to the
mass carried by the black hole (see Appendix B). It so happens that we
actually have a stronger constraint on $\kappa$,

$\kappa>\frac{4}{8-\gamma}.$ (143)

We can derive this constraint as follows. We want the function $f(r)$, eq.
(140), to be positive everywhere outside of the black hole $r>r_{h}$. For all
positive values of $\kappa$ (except for the singular point
$\kappa=4/(8-\gamma)$), the function $f(r)$ is positive. However, since we
shall identify some positive power of $r_{h}$ with the (quasi-local) mass of
the black hole, if we want the $r_{h}\to 0$ limit to be smooth, we easily see
from (140) that we need the constraint (143). (See Appendix B)

At large values of $r$, the function $f(r)$ may either decay to zero, or grow
unboundedly or assume a constant value depending on the leading exponent of
$r$ in $f(r)$, i.e., $2-4/(4-\gamma)\kappa$. Although the curvature is
vanishingly small in the asymptotic region, the asymptotic causal structure
depends on the value of $\kappa$.

When $\kappa<\ifrac{2}{(4-\gamma)}$, the leading power of $r$ is negative and
the function $f(r)$ decays to zero at large $r$. The asymptotic causal
structure for this range of $\kappa$ is similar to that of Minkowski
spacetime. A more careful analysis reveals that this causal structure persists
till the value $\kappa\leq\ifrac{4}{(4-\gamma)}$. For the regime
$\kappa>\ifrac{4}{(4-\gamma)}$, the asymptotic causal structure is similar to
that of the AdS spacetime, even though the curvature vanishes in this region.
As mentioned in the Introduction, spacetimes with such exotic asymptotics have
been noted previously in case of higher dimensional dilatonic black holes
Chan:1995fr . In two dimensions, the dilaton and “gravity” are inseparable
entities — it is somewhat of a surprise that such 2D spacetimes did not find
much (if any) emphasis in the literature.

Let us clarify the aforementioned discussion with some familiar examples for
some special values of $\kappa$. For $\kappa=\ifrac{2}{(4-\gamma)}$, for
instance, the function $f(r)$ is given by,

$\displaystyle
f(r)=\frac{(4-\gamma)^{2}}{4\gamma}V_{\mathrm{eff}}\bqty{1-\pqty{\frac{r_{h}}{r}}^{\frac{\gamma}{4-\gamma}}},$
(144)

which shows that for this value of $\kappa$, the metric in this coordinate
system assumes the form similar to the Schwarzschild black hole. For example,
for a $d$-dimensional Schwarzschild black hole, $\gamma$ would correspond to
$\gamma=\ifrac{4(d-3)}{(d-2)}$. To see how this value of $\gamma$ comes about,
one can consider a general dimensional reduction for a spherically symmetric
black hole. See, for example, Moitra:2021eom , eq. (2.255), which yields this
value of $\gamma$ after appropriate field redefinitions. Note that this value
of $\gamma$ is always less than 4 (the string theory coupling) and reaches
this value only as $d\to\infty$.

Another well-known example is the borderline case
$\kappa=\ifrac{4}{(4-\gamma)}$,

$\displaystyle
f(r)=\frac{(4-\gamma)^{2}}{16}V_{\mathrm{eff}}\,r\,\bqty{1-\pqty{\frac{r_{h}}{r}}^{\frac{4}{4-\gamma}}}.$
(145)

For large $r$, $f(r)$ grows linearly in $r$, which is just the Rindler metric
in disguise. Therefore, the asymptotic causal structure is that of Minkowski
spacetime.

On the other hand, the limit $\kappa\to\infty$ illustrates the rather
interesting nature of the asymptotic behaviour for sufficiently large values
of $\kappa$. In this case, the function $f(r)$ grows quadratically in $r$ for
large $r$, as in asymptotically AdS spacetimes,

$\displaystyle
f(r)\approx\frac{(4-\gamma)^{2}V_{\mathrm{eff}}}{4(8-\gamma)}\,r^{2}\bqty{1-\pqty{\frac{r_{h}}{r}}^{\frac{8-\gamma}{4-\gamma}}}.$
(146)

For a general $\gamma$, the spacetime described by this function still has a
curvature singularity at $r=0$. However, at the value $\gamma=0$, the
spacetime becomes exactly $\mathrm{AdS}_{2}$, as it must, because the modified
action with $\gamma=0$ and $\kappa=\infty$ is nothing but the Jackiw-
Teitelboim model Teitelboim:1983ux ; Jackiw:1984je .

As mentioned previously, a non-zero constant value of the scalar curvature,
$R=\Lambda$ is inconsistent with the homothety transformation property of the
Ricci scalar, (7). Therefore, for studying self-similar collapse in
asymptotically AdS-like spacetimes with an HKV, our models (also see the
following Liouville models) seem to be very convenient. Such spacetimes are
interesting because they are often helpful in realising the holographic
correspondence — a one-dimensional quantum mechanical theory lives on the
time-like conformal boundary of the two-dimensional spacetime. It would thus
be interesting to apply holographic ideas in this scenario, to understand the
dynamics from a quantum-mechanical point of view.

Before we end this subsection, there is an obvious yet important comment to
make. It is immediately obvious that a constant dilaton $\phi(r)=\phi_{0}$ is
a solution of the equation (134). This fact is, however, obscured in the
solution (135) — we cannot recover the constant solution by taking any smooth
limit of the integration constants in (135). This constant solution when
inserted in (139) immediately tells us that such a solution is possible only
if the potential vanishes: $V_{\mathrm{eff}}=0$. Under this condition, we
immediately obtain $R=0$ everywhere and therefore the solution describes
simply flat spacetime.

###  A.2 Stringy Coupling $\gamma=4$

We solve the system of equations as before, and we obtain, after fixing an
additive constant to zero and setting the constant associated with the
characteristic length scale to unity,

$\displaystyle\phi(r)=-r.$ (147)

In this case, however, in contrast to the case for $\gamma<4$, the range of
the coordinate $r$ is given by,

$\displaystyle-\infty<r<\infty.$ (148)

The solution is given by,

$\displaystyle f(r)=\frac{\kappa
V_{\mathrm{eff}}}{4(\kappa-1)}\exp(-\frac{2r}{\kappa})\bqty{1-\exp(-\frac{2(\kappa-1)(r-r_{h})}{\kappa})},$
(149)

where $r_{h}$ as before denotes the location of the event horizon.

The positivity of $f(r)$ outside the event horizon leads to the constraints,

$\displaystyle V_{\mathrm{eff}}>0,\quad\kappa>1.$ (150)

The curvature grows unbounded as one takes $r\to-\infty$. However, in contrast
with the usual singularities in general relativity, it takes an infinite
affine time to reach the “singularity” at $r=-\infty$, so this is not really a
singularity Wald:1984rg .

In this case, in contrast with the scenario for $\gamma<4$, the asymptotic
causal structure is the same as that of Minkowski spacetime for all values of
$\kappa$. In particular, the $\kappa\to\infty$ limit is the black hole
discussed in Mandal:1991tz ; Elitzur:1990ubs ; Witten:1991yr  (which is also
the “pure gravity sector” of the CGHS model Callan:1992rs ).

###  A.3 Liouville System

In this case, the equations of motion give us,

$\displaystyle\sigma(r)=\log r.$ (151)

Since we want the coupling to the Ricci scalar to be non-negative, we restrict
$r$ to lie in the range $[e^{-\sigma_{0}},\infty)$. We take $\sigma_{0}$ (the
coefficient of the topological term) to be so large that for all practical
purposes (especially while performing numerical analysis), we take the lower
limit of the $r$ coordinate to be zero. Here, $r=0$ corresponds to the
curvature singularity (the Ricci scalar is exponentially large) and $r=\infty$
is the asymptotic region. The function $f(r)$ is given by,

$\displaystyle
f(r)=\frac{V_{\mathrm{eff}}\kappa}{2+\kappa}r^{2+\frac{2}{\kappa}}\bqty{1-\pqty{\frac{r_{h}}{r}}^{1+\frac{2}{\kappa}}}.$
(152)

In this case, the previously mentioned physical constraints necessitate that,

$\displaystyle V_{\mathrm{eff}}>0,$ (153)

and

$\displaystyle\kappa<-2.$ (154)

The Ricci scalar for this geometry is given by,

$\displaystyle R=-2\quantity(1+\frac{1}{\kappa})V_{\mathrm{eff}}r^{2/\kappa}.$
(155)

Clearly, for finite $\kappa<-2$, the curvature is exponentially large near
$r=0$ and vanishes at $r=\infty$.

The asymptotic causal structure is similar to that of AdS spacetime. Indeed in
the limit $\kappa\to-\infty$, the metric is that of $\mathrm{AdS}_{2}$ (which
of course has no curvature singularity) with the $\mathrm{AdS}_{2}$ length
$V_{\mathrm{eff}}^{-1/2}$.

##  Appendix B Quasi-Local Mass

We consider the question of quasi-local mass in the two-dimensional theories
of gravity discussed in this article. The authors of Cai:2016rcv  proposed a
definition of quasi-local mass in two-dimensional dilaton gravity in analogy
with the Misner-Sharp mass in higher dimensional theories. The proposal of
Cai:2016rcv  can be summarised as follows.

If the action describing the gravity dilaton system with matter takes the
form, (setting $16\pi G_{2}=1$)

$\displaystyle S=\int\differential[2]{x}\sqrt{-g}\pqty{\Pi
R+U(\Pi)(\nabla\Pi)^{2}+V(\Pi)}+S_{\mathrm{matter}},$ (156)

the proposed definition of quasi-local mass is given by,

$\displaystyle E=\frac{1}{2}\pqty{w-e^{Q}(\nabla\Pi)^{2}},$ (157)

where the functions $Q,w$ are defined by the relations,
$U(\Pi)=-Q^{\prime}(\Pi)$ and $V(\Pi)=w^{\prime}(\Pi)\exp[-Q(\Pi)]$. In
certain cases, we have to modify (157) by an overall constant of
proportionality, but for the problem we are interested in, this is not an
issue.

Comparing with the stringy system, we find,

$\displaystyle\Pi=e^{-2\phi},\quad U(\Pi)=\frac{\gamma}{4\Pi},\quad
Q=-\frac{\gamma}{4}\log\Pi,\quad
W(\Pi)=V_{\mathrm{eff}}\frac{\Pi^{2-\frac{1}{\kappa}-\frac{\gamma}{4}}}{2-\frac{1}{\kappa}-\frac{\gamma}{4}}$
(158)

which gives us the form of the mass function in terms of $\phi$

$\displaystyle
E=\exp(\frac{\gamma-8}{2}\phi)\pqty{V_{\mathrm{eff}}\frac{e^{2\phi/\kappa}}{2-\frac{1}{\kappa}-\frac{\gamma}{4}}-4(\nabla\phi)^{2}}.$
(159)

For the Liouville system, we would have,

$\displaystyle\Pi=\sigma,\quad U(\Pi)=-1,\quad Q(\Pi)=\Pi,\quad w(\Pi)=\kappa
V_{\mathrm{eff}}\frac{\exp(\frac{2+\kappa}{\kappa}\Pi)}{2+\kappa},$ (160)

which yields,

$\displaystyle E=\frac{\kappa
V_{\mathrm{eff}}}{2+\kappa}\exp(\frac{2+\kappa}{\kappa}\sigma)-\exp(\sigma)(\nabla\sigma)^{2}.$
(161)

This can be easily generalised to the semi-classical model as well. It can be
seen that for the static solutions in Appendix A, these are constant and give
the correct dependence on $r_{h}$ (which can be found out from the first law
of black hole mechanics) namely,

$\displaystyle
E\propto\begin{cases}r_{h}^{\frac{(8-\gamma)\kappa-4}{(4-\gamma)\kappa}},\quad&\text{(stringy
model with $\gamma<4$)},\\\
e^{\frac{2(\kappa-1)r_{h}}{\kappa}}\quad&\text{(stringy model with
$\gamma=4$)},\\\ r_{h}^{1+2/\kappa},\quad&\text{(Liouville
model)}.\end{cases}$ (162)

In the main text, for the self-similar system, we have studied the mass
function $M(x)$ defined by taking out an overall power of $v$ in $E$,

$\displaystyle E=v^{-\frac{1}{2}(\gamma-8)\kappa-2}M(x),$ (163)

where,

$\displaystyle M(x)=-\frac{4e^{\frac{1}{2}(\gamma-8)\Phi(x)}\left(\kappa
V_{\text{eff}}e^{\frac{2\Phi(x)}{\kappa}}-((\gamma-8)\kappa+4)\Phi^{\prime}(x)\left(2\kappa-F(x)\Phi^{\prime}(x)\right)\right)}{(\gamma-8)\kappa+4}.$
(164)

Analogous relations can be derived for the other models as well.

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