#### Exact solutions and critical chaos in dilaton gravity with a boundary

Received: February
Exact solutions and critical chaos in dilaton gravity with a boundary
Maxim Fitkevich 0 1 3 5 6 7 8
Dmitry Levkov 0 1 3 6 7 8
Yegor Zenkevich 0 1 2 3 4 7 8
Open Access, c The Authors.
0 Piazza della Scienza 3 , I-20126 Milano , Italy
1 Institutskii per. 9, Dolgoprudny 141700, Moscow Region , Russia
2 INFN , sezione di Milano-Bicocca
3 60th October Anniversary Prospect 7a , Moscow 117312 , Russia
4 National Research Nuclear University MEPhI
5 Moscow Institute of Physics and Technology
6 Institute for Nuclear Research of the Russian Academy of Sciences
7 Moscow 115409 , Russia
8 I-20126 Milano , Italy
We consider (1 + 1)-dimensional dilaton gravity with a re ecting dynamical boundary. The boundary cuts o the region of strong coupling and makes our model causally similar to the spherically-symmetric sector of multidimensional gravity. demonstrate that this model is exactly solvable at the classical level and possesses an on-shell SL(2; R) symmetry. After introducing general classical solution of the model, we study a large subset of soliton solutions. The latter describe re ection of matter waves o the boundary at low energies and formation of black holes at energies above critical. They can be related to the eigenstates of the auxiliary integrable system, the Gaudin spin chain. We argue that despite being exactly solvable, the model in the critical regime, i.e. at the verge of black hole formation, displays dynamical instabilities speci c to chaotic systems. We believe that this model will be useful for studying black holes and gravitational scattering.
with; a; boundary; cDipartimento di Fisica; Universita di Milano-Bicocca
1 Introduction 2 The model
3 Integrable sector 4 5 Critical chaos
Adding the boundary
Solution in the bulk and re ection laws
Simple equation for the boundary
On-shell conformal symmetry
General solution
Soliton solutions with power-law singularities
Simplifying the coe cient equations
SL(2; C) symmetry
Relation to the Gaudin model
Positivity condition
Perturbative expansion in the critical regime
Shock-wave instability
A Field equations and boundary conditions
A.1 Derivation
A.2 Solution in the conformal gauge
B Bethe Ansatz for the Gaudin model
Introduction
The models of two-dimensional dilaton gravity were popular for decades [1{3]. Some of
them describe spherically-symmetric sectors of multidimensional gravities with dilaton
related to the sizes of the extra spheres.1 Some others are exactly solvable at
the semiclassical [4, 5] or quantum [3] levels which makes them valuable for studying black
holes and gravitational scattering [6{8].
These models become particularly important in the context of information
paradox [9, 10] confronting an apparent loss of quantum coherence during black hole evaporation
1In particular, gravitational sector of the CGHS model [4] can be obtained by spherical reduction of
D-dimensional gravity at D ! +1 [3].
J −
J −
J −
with a boundary. The dashed lines are light rays extending from J
with the principles of quantum theory. Since unitarity of quantum gravity is strongly
supported by the AdS/CFT correspondence [11, 12], modern AMPS argument [13, 14] suggests
dramatic violation of the equivalence principle (\ rewalls") in the vicinity of old black hole
horizons, see [15, 16] for earlier works. This feature, if exists, may leave \echoes" in the
gravitational wave signal [17, 18] to be detected by LIGO [19, 20], cf. [21, 22]. From the
theoretical viewpoint, further progress can be achieved by understanding unitary evolution
of black holes outside of the explicit AdS/CFT framework. This brings us to the arena of
two-dimensional models which may, in addition, clarify relation of black holes to quantum
chaos [23{28], cf. [29].
Unfortunately, solvable models of two-dimensional dilaton gravity essentially di er
from their multidimensional cousins. Consider e.g. the celebrated
Callan-Giddings-HarveyStrominger (CGHS) model [4], see [1, 2] for reviews. Its two-dimensional Minkowski vacuum
in gure 1a, unlike the multidimensional vacua, has disconnected sets of \left" and \right"
in nities J
L and JR , and transitions between those are expected [30] to be important
for the information loss problem. Besides, the CGHS model is strongly coupled [31] near
the \left" in nities which puts its semiclassical results on shaky ground. It was recently
suggested [32] that due to the above peculiarities evaporation of the CGHS black holes
leads to remnants rather than rewalls.
We consider the modi ed CGHS model proposed2 in [33, 34], see also [31, 39{42]. The
region of strong coupling in this model is cut o by the re ective dynamical boundary placed
0 of the dilaton
eld, see gure 1b. Parameter e2 0
1 plays the
role of a small coupling constant. We explicitly obtain reparametrization-invariant action
of the model by restricting CGHS action to the space-time region
0 and adding
appropriate boundary terms. Note that the original CGHS model is formally restored in
the limit 0 ! +1 which shifts the regulating boundary in
gure 1b all the way the left.
We do not consider this limit avoiding potential problems with strong coupling, cf. [43{45].
2Similar models appeared recently in the context of near AdS2 / near CFT1 holography [35{38].
(a) low-energy
(c) high-energy
values of parameters.
Finite-range light-cone coordinates (u; v) are used.
The centers of the
incoming and re ected matter wave packets are marked by the dashed lines.
As an additional bonus, the above model with a boundary is causally similar to
spherically-symmetric multidimensional gravity, cf. gure 1b. The price to pay, however, is
nonlinear equation of motion for the boundary which, if non-integrable, may damage major
attractive property of the CGHS model | its solvability. Note that the previous studies
of this or similar models were relying on numerical [40{42, 46] or shock-wave [33, 34, 39]
In this paper we demonstrate that the CGHS model with a boundary is exactly
solvable at the classical level. We obtain general solution of the classical eld equations and
construct an in nite number of particular soliton solutions. The latter describe re ection
of matter waves o the boundary at low energies and formation of black holes at energies
above some critical values, see gures 2a and 2c. Each solution is characterized by N
integers or half-integers s1; : : : ; sN and the same number of real parameters. The parameters
of the solitons satisfy inequalities ensuring positivity of energy.
We establish one-to-one correspondence between the above solitons and the eigenstates
of the auxiliary integrable system | the rational Gaudin model [47{49]. This allows us to
classify these solutions and study their properties.
We nd that equation of motion for the boundary is invariant under conformal
transformations v ! w(v), where v is the light-cone coordinate, w(v) is an arbitrary function.
These transformations relate physically distinct solutions, and one should not confuse them,
say, with the residual reparametrization symmetry in [39, 40]. In particular, the
transformations from the global SL(2; R) subgroup change massless matter eld(s) f of the model
as the standard zero-weight elds. They also map the solitons into solitons. The
transformations with nonzero Schwarzian derivative act non-linearly on f , and we do not consider
them in detail.
Finally, we study dynamics of the model in the critical regime, i.e. at the verge of
black hole formation, cf. gure 2b. We demonstrate that in this limit scattering of matter
waves o the boundary displays instabilities speci c to chaotic systems: the nal state of
the process becomes extremely sensitive to the initial Cauchy data. This feature is in tune
with the near-horizon chaos suggested in [25]. We argue that it impedes global integrability
of the model, i.e. prevents one from choosing a complete set of smooth conserved quantities
in the entire phase space.
In section 2 we introduce dilaton gravity with a boundary and study its properties.
We construct exact solutions in section 3. Critical chaos is considered in section 4. In
section 5 we discuss possible applications of our results.
The model
Adding the boundary
We consider two-dimensional model with classical action
S =
(rf )2=2
= 0
where3 the integrand in the rst line is the CGHS Lagrangian [4] describing interaction
of the metric g
and dilaton
with massless scalar f ; the dimensionful parameter
the energy scale of the model. In eq. (2.1) we modi ed the CGHS action by restricting
integration to the submanifold
0 and adding the boundary terms4 at
unit outer normal n
In fact, the choice of the boundary action in eq. (2.1) is limited. First, the
GibbonsHawking term with extrinsic curvature ensures consistency of the gravitational action.
Without this term the boundary conditions following from eq. (2.1) would be incompatible
with the Dirichlet condition
0, see [51] and cf. appendix A.1. Second, we assume
no direct interaction of the matter
eld f with the boundary. Then the only natural
generalization of our model would include an arbitrary constant in the last term of eq. (2.1).
However, this parameter needs to be ne-tuned in order to retain Minkowski solution (see
below). Thus, the action (2.1) describing interaction of the boundary with the gravitational
sector of the CGHS model is xed [33].
The quantity e2 0 is a coupling constant controlling loop expansion in the model (2.1).
Indeed, change of variables ~ =
the model is classical at e2 0
It is clear that the bulk equations in the model (2.1) are the same as in the original
CGHS model [1, 2, 4]. However, extremizing the action with respect to the boundary values
3We use ( ; +) signature and Greek indices ; ;
= 0; 1. We denote covariant derivatives by r and
Ricci scalar by R.
4Similar boundary terms appear in the path integral formulation of dilaton gravity [50].
and f , one also obtains the boundary conditions
n r f = 0
= 0 ;
see appendix A.1 for details. Note that the constant
in the right-hand side of the rst
equation comes from the last term in eq. (2.1). Besides, the second equation guarantees
zero energy
ux through the boundary.
Let us now recall [4] that linear dilaton vacuum
f = 0 ;
= .
satis es the CGHS equations, cf. appendix A.1. In this case the boundary
= 0 is static,
xboundary =
0= , and the
rst of eqs. (2.2) is automatically satis ed. Note that the
Minkowski vacuum (2.3) is a solution in our model due to exact matching between the
bulk and boundary terms with
in the action (2.1).
Solution in the bulk and re ection laws
The CGHS equations in the bulk are exactly solvable [1, 2] in the light-cone frame (u; v),
ds2 =
Let us review their general solution leaving technical details to appendix A.2. In what
follows we x the remaining gauge freedom in eq. (2.4) with the on-shell \Kruskal"
condiof incoming and outgoing parts,
The respective energy uxes are
f = fin(v) + fout(u)
Tvv(v) = (@vfin)2
Tuu(u) = (@ufout)2 :
This speci es the Cauchy problem in our model: one prepares fin or Tvv at the past null
in nity and calculates fout or Tuu at J +, see gure 1b.
The solution for the scale factor
and dilaton eld
e 2 = e 2 =
2vu + g(v) + h(u) ;
1 Z
g(v) =
dv00 Tvv(v00) ;
h(u) =
du00 Tuu(u00) :
We xed the integration constants in these expressions by requiring, rst, that the
spacetime is at in the in nite past, i.e. no white hole preexists the scattering process. Second,
we chose the coordinates in such a way that the quadrant u 2 (
1; 0), v 2 (0; +1) covers
and v ! +1 at u < 0 lead to J
and J +, respectively, see gure 3.
all space-time accessible to the distant observer. In particular, the limits u !
Now, consider the boundary
\Kruskal" coordinates. Substituting the bulk solution (2.5), (2.7) into the boundary
conditions (2.2), one obtains equation for U (v) and re ection law for the matter eld f ,
fout(U (v)) = fin(v) ;
energy conservation.
Tuu = 0,
see appendix A.2 for the derivation of these equations and proof that they are compatible
with the de nition
(U (v); v) =
0 of the boundary. Note that the second of eqs. (2.9)
relates the incoming and outgoing waves by conformal transformation v ! U (v). The rst
appropriate terms, it coincides5 with the boundary equation obtained in [33, 34, 39] using
One easily
U (v) =
e 2 0 =( 2v) ;
e 2 = e 2 =
f = 0 ;
where the integration constant in the rst expression was chosen to make U (v) smooth
and invertible in the interval 0 < v < +1. Solution (2.10) is the linear dilaton vacuum:6
coordinate transformation
v = e (t+x) ;
u =
brings it to the standard form (2.3). In what follows we impose at asymptotics (2.10) in
the in nite past v ! 0, u !
Note that the space-time (2.7) is always at far away from the boundary, i.e. at large juj
and v. Below we transform to the asymptotic Minkowski coordinates (t; x) using eq. (2.11).
5It does not conform, however, with the boundary conditions introduced at one-loop level in [43{45]: in
the classical model the latter conditions imply that the boundary is space-like.
6Recall that we excluded solutions with eternal black holes in eq. (2.7).
We have got a receipt for solving the Cauchy problem in the CGHS model with a
boundary. In this case the initial Cauchy data are represented by the incoming wave fin(v)
or its energy
ux Tvv(v). One solves eqs. (2.9) with the initial condition (2.10) at v ! 0
and nds U (v), fout(u). The scale factor of the metric, dilaton and matter elds are then
given by eqs. (2.7) and (2.5).
Simple equation for the boundary
One notices that eq. (2.9) for U (v) is, in fact, a Riccati equation. The standard substitution
brings it to the form of a Schrodinger equation for the new unknown (v),
U = @vg
v2 (v) =
Tvv(v) (v) :
Note that
(v) is de ned up to a multiplicative constant. Now, one can solve for
given the initial data Tvv(v). After that the entire solution is determined by eq. (2.12) and
expressions from the previous section. For example, the outgoing energy ux equals
Tuu(u) =
e 0 =@v
Tvv v=V (u)
re ection law (2.9) into the de nition (2.6) of the ux and then expressing the derivative
of U (v) from the rst of eqs. (2.9) and eq. (2.12).
Importantly, eq. (2.13) is well-known in mathematical physics. Similar equation
appears in Liouville theory at classical and semiclassical levels [52]. Besides, the eigenstates
of the Gaudin model [47] can be related to the solutions of eq. (2.13) with monodromies
1 and rational Tvv(v) [48]. In what follows we exploit these similarities for studying exact
solutions in dilaton gravity.
The function
(v) in eq. (2.12) has simple geometric meaning. First, the value of
related to the proper time
along the boundary,
d 2 = e2 0 dU (v) dv = (@v =
(v) =
where we used eqs. (2.4), (2.9), (2.12) and introduced the arbitrary constant 0 related to
the origin of . Function (v) is illustrated in gure 3. Second, recall that v is the exponent
of the at light-cone coordinate (t + u) far away from the boundary, eq. (2.11). Thus, (v)
maps the a ne coordinate at J
to . Equation (2.13) relates this coordinate-independent
function to the asymptotic Cauchy data Tvv(v).
Consider general properties of classical solutions in the model with a boundary.
Expression (2.15) implies that (v) vanishes in the in nite past,
(0) = 0 :
= 0
purposes. Dashed line in this graph is the space-like \singularity"
Indeed, behavior
! c0v as v ! 0 corresponds to the linear dilaton vacuum (2.10) in
well-localized Tvv(v) and therefore linear asymptotics
(v) ! Cv + D
of the solution to eq. (2.13). If Tvv is small, one has C
1. The respective \low-energy" solutions describe re ection of matter waves o the time-like boundary, see gures 4a,b.
As Tvv grows, the function
2
(v) becomes more concave and C decreases because @v
this case the boundary is null in the asymptotic future because its proper time (v) in
eq. (2.15) remains
nite as v ! +1. The respective \critical" solution in
gures 4 is at
the brink of black hole formation: we will see that the asymptotically null boundary sits
precisely at the horizon of would-be black hole.
At su ciently high energies we get C < 0 and therefore (v) has a maximum (point
Tvv(vA) h
4d2=3
u)2=3
d2=3(v
where eqs. (2.9), (2.12), and (2.7) were solved to the leading order in u
vA. Thus,
A is a singularity of
in coordinates (u; v).
Besides, one discovers that the condition
0 de nes two intersecting curves u
vA)3 near A, and only one of those is the time-like boundary considered
7Recall that (v) is de ned up to a multiplicative constant.
so far. The second curve is space-like, it is shown by the dashed line in
gure 4b. The
boundary conditions (2.9) are not met at this line. We obtained the analog of the black
hole singularity in the model with a boundary. Indeed, our model is formulated at
i.e. in the space-time region to the right of both solid and dashed graphs in gure 4b. The
space-like \edge"
Except for the point A itself, the solution is smooth at the space-like \singularity"
. This fact was not appreciated in the previous studies. The mass of the formed
black hole, by energy conservation, is related to the value of the dilaton eld at the future
Mbh =
jujdu Tuu = 2 [g(+1) + h(0)] = 2
where we subtracted the
nal matter energy from the initial one in the rst equality
(cf. eq. (2.11)), integrated by parts and used eqs. (2.8) in the second equality, and then
expressed the result in terms of , eq. (2.7). Since
< 0, this implies that all black hole
masses are larger than
Mcr = 2 e 2 0 ;
precisely at the horizon. They are formed in the critical solutions.
The solutions in
gure 4b, when replotted in the
nite-range coordinates (u; v) =
(arctan u; arctan v), look like Penrose diagrams, see gure 2. From now on, we will exploit u
and v for visualizing the solutions. We will also mark the (smooth) space-like \singularities"
= 0 by zigzag lines, see the one in gure 2c.
On-shell conformal symmetry
We nd that the boundary equation (2.13) is invariant under conformal transformations
~(w) =
Tvv 7! T~vv(w) =
which change
(v) as an h =
24 e 2 0 [53]. In eq. (2.19) we introduced
the Schwarzian derivative fv; wg
mations (2.19), (2.20) relate physically distinct solutions8 with di erent energy uxes Tvv.
Acting with them on the vacuum
= v, Tvv = 0 one can obtain any solution.
v000=v0
3(v00)2=2(v0)2 with v0
dv=dw. The
transfor8Unlike the transformations in [39, 40], they do not represent residual gauge symmetry. The latter was
xed, see discussion after eq. (2.4).
Note that the symmetry (2.19), (2.20) does not make our model a CFT in a
conventional sense.9 First, the full energy-momentum tensor T
( ) of the model includes
contribution of the dilaton
eld and vanishes by Einstein equations, cf. eq. (A.1).
Second, eq. (2.20) is not a conformal transformation fin ! fin(v(w)) of the massless scalar
eld f far away from the boundary: the latter changes classical Tvv
2 without
the Schwarzian derivative. At the quantum level, healthy conformal matter has positive
central charge c > 0 [53], and transformations of its energy-momentum tensor Tvv do not
match eq. (2.20) as well.
Transformations from the SL(2; R) subgroup of (2.19), (2.20),
v ! w(v) =
= 1 ;
re ection t + x !
of our model.
have vanishing Schwarzian derivative and therefore change f in the standard way
fin ! fin(v(w)). Besides trivial translations of v they include v-dilatations due to shifts
(t + x). These transformations constitute the global symmetry group
As a side remark, let us argue that (2.19), (2.20) is a symmetry of the gravitational
u2 (u) =
Tuu(u) (u) ;
cf. eqs. (2.15) and (2.13); now, (u) is the boundary proper time parametrized with u. It
is convenient to combine (v) and (u) into a single free eld
transforming in a simple Liouville-like manner under eq. (2.19). To describe the
gravitational degrees of freedom with
, we extract its energy-momentum tensor T
Einstein equations T
= 0,
Tvv = 8e 2 0 (@v )
v2 =2 ;
where eq. (2.13) was used in the left equality; similar expression for Tu(u) can be obtained
using eq. (2.22). One observes that Tv(v ) transforms under eq. (2.19) as an energy-momentum
Now, the entire scattering problem can be reformulated in terms of . One sends the
incoming energy
All these equations and boundary conditions can be summarized in the at-space action
= 0. The ux
9Thus, one may still hope that our model is unitary at the quantum level despite negative primary
dimension in eq. (2.19) and negative central charge in eq. (2.20).
In this setup (2.19), (2.20) is an apparent conformal symmetry of
far away from the
boundary, whereas the symmetry of the matter sector is hidden in the re ection laws.
Integrable sector
General solution
Indeed, introducing
One can use eq. (2.13) to express the entire solution in terms of one arbitrary function.
@v =
= e2 0 @vg
= eR dv0 W (v0) ;
Tvv = W 2 + @vW :
Then U , Tuu, , and f are given by eqs. (3.1), (2.14), (2.7), and (2.6). We obtained general
classical solution in the model with a boundary.
By itself, this solution is of little practical use because the function
(v) has a zero at
Tvv(v) in eq. (3.2) is singular at these points. Indeed, eq. (3.1) gives
W (v) = R(v) + 1=v + 1=(v
where R(v) is regular at v
R(v~1) =
1=v~1 on parameters of R(v).
Choosing multiparametric R(v) and solving the constraints, one nds an arbitrary
number of smooth solutions. The physical ones satisfy
In what follows we will concentrate on a large class of soliton solutions with power-law
singularities. We will argue that some of them satisfy eq. (3.3).
Soliton solutions with power-law singularities
Let us follow the Painleve test [54] and guess the form of Tvv(v) which guarantees that
the general solution
(v) of eq. (2.13) has power-law singularities
complex v-plane. One introduces Laurent series at v
Tvv =
where the expansion of Tvv starts from (v
v0) 2 due to eq. (2.13). Substituting eqs. (3.4)
into eq. (2.13), we obtain an in nite algebraic system for k s,
1) k s = T 2 k s + T 1 k s 1 +
T1 = T0T 1
T2 = 23 T1T 1
158 T0(T 1)2 + 14 (T0)2 + 316 (T 1)
1 in terms of arbitrary
s and fTmg. Expression (3.4) is a general solution of
the second-order equation (2.13) if precisely two of its parameters,
s and some
remain arbitrary. Thus, (k0
One concludes that s is integer or half-integer.
Note that the two equations from the system (3.5) which do not determine the coe
T 2 = 3=4 ;
T0 = (T 1)2 ;
where we expressed all k 1=2 via fTkg and
and higher-order equations listed in table 1.
We arrived at the practical method for obtaining the soliton solutions in our model.
One speci es N singularities of
(v): selects their integer or half-integer powers sn and
This analytic structure gives expressions,
1=2. For larger s, one obtains T 2 = s(s + 1)
Tvv =
sn(sn + 1)
= C
m=1(v
n=1(v
where we required Tvv ! 0 as v ! +1 and introduced a polynomial in the nominator
of (v) with M zeroes v~m and a normalization constant C. Next, one solves equations
in table 1 at each singularity and determines T n1. After that
(v) is obtained by
substituting eqs. (3.7) into eqs. (2.13) or (3.5). Two parameters | say, C and v~M |
remain arbitrary because eq. (3.7) is a general solution of the second-order equation. One
f (v); Tvv(v)g characterized by N complex parameters vn and the same number of integers
or half-integers sn.
We consider solutions with nite total energy of incoming matter,
Ein =
vdv Tvv(v) ;
see eq. (2.11). Convergence of this integral implies Tvv
eq. (3.7), linear relations
o(v 2) as v ! +1 or, given
X T n1 = 0 ;
sn(sn + 1) + vnT n1 = 0 :
(a) a =
(b) a = 0
(c) a = 0:2
1 can be restored in
the classical solution, see discussion in section 2.1.
Moreover, asymptotic (2.17) of (v) suggests fallo Tvv
O(v 4) at large v and additional
2vnsn(sn + 1) + vn2T n1 = 0 ;
which should hold for noncritical solutions. Equations (3.8) and (3.9) are useful for
obtaining the lowest solitons.
conditions (3.8), one obtains T 11 =
T 21 = 3=[2(v2
v1)]. It is straightforward to check
ib. Then eqs. (3.7) give,
Tvv =
a)2 + b2]2
(v) =
a)2 + b2]1=2 ;
where (v) was obtained by substituting eqs. (3.7) into eq. (2.13). One observes that the
ux (3.10) peaks near v
av as v ! +1, the solution (3.10) describes re ection of matter waves o
the boundary and formation of black holes at a < 0 and a > 0, respectively, see gure 4a.
This fact is clearly seen in
nite-range coordinates (u; v). In
gure 5c we also plotted the space-like \singularity"
The simplest exact solution in eq. (3.10) describes the incoming matter ux with a
single peak. Solutions with multiple peaks can be obtained by adding singularities at
v = an
ibn, see gure 6. Unfortunately, it is hard to nd these solutions explicitly at
10Note that Tvv(v) with one singularity does not satisfy eqs. (3.8).
large N . Besides, it is not clear whether they will satisfy the positivity condition (3.3). We
will clarify these issues in the subsequent sections.
Simplifying the coe cient equations
Instead of solving the equations in table 1, one can extract Tvv(v) from the general solution.
Namely, substituting the solitonic (v) into the rst of eqs. (3.2), we nd,
Then the second of eqs. (3.2) gives the incoming ux. However, in this case Tvv(v) receives
at these poles, we obtain equations for fv~mg,
W (v) =
n=1 v~m
m06=m
T n1 =
n06=n vn
which are, in fact, equivalent to the ones in table 1. Indeed, after solving eqs. (3.12) one
obtains Tvv(v) of the form (3.7) with
In practice one
nds v~m numerically from eqs. (3.12), then computes Tvv and
eqs. (3.13) and (3.7).
Unlike in section 3.1, we impose eqs. (3.12) at all v~m, not just the ones at the real
positive axis. The goal is to obtain solutions with transparent properties, see the forthcoming
discussion in section 3.5.
SL(2; C) symmetry
The global SL(2; C) transformations (2.21) are invertible and therefore preserve the
singularity structure of the solitons. One obtains,
Tvv ! T~vv(w) =
~(w) = (
This symmetry relates solitons with di erent parameters. Real solutions at v
under SL(2; R).
= to in nity. If the original solution
was regular at this point, its image receives asymptotics ~ ! Cw + D and T~vv ! O(w 4
as w ! +1. In eq. (2.17) we obtained the same asymptotics from physical considerations.
Solutions with other asymptotics, i.e. those violating the nite-energy conditions (3.8) or
eq. (3.9), have singularities \sitting" at in nity.
transformation11 (3.14) and get,
One can use the above property to construct new solutions. Consider e.g. the
trivial solution
= v s
vs+1, Tvv =
Tvv =
2e 2 0 s(s + 1)(v2
where the constant in front of (v) was ignored. This is the soliton with two singularities
of power s. Taking v1 = v
2 = a + ib, one obtains Tvv(v)
0 at real v. Note that the
incoming ux in eq. (3.15) is the same as in eq. (3.10) albeit with di erent multiplicative
factor. The behaviors of the boundaries are also similar, as one can see by comparing the
Relation to the Gaudin model
In this section we establish one-to-one correspondence between the solitons (3.7) and
eigenstates of the auxiliary integrable system, the Gaudin model [47{49]. This will allow us to
count the number of solitons and explain some of their properties.
model [47] describes a chain
three-dimensional spins
model is equipped with N commuting Hamiltonians
T^n =
l6=n
where vn are complex parameters and (s^n; s^l)
P s^n s^l is the scalar product. The
where Tn are complex eigenvalues.
It is convenient to pack all spins and Hamiltonians into the operator-valued functions
[s^(v)]2 =
11With parameters
= (1=v2
1=v1) 1=2,
= =v2,
= =v1.
s^n. The
Tvv =
T n1 = 2Tn
positions of singularities
parameters of the Hamiltonians
powers of singularities
zeros of (v)
incoming energy ux
coe cients of Tvv
Eigenstates of the Gaudin model
representations of s^n
parameters of eigenstates
eigenvalue of T^ (v)
eigenvalues of T^n
A complete set of eigenvectors and eigenvalues in the model (3.16) is provided by the
algebraic Bethe Ansatz [47{49]. We review this method in appendix B and list its main
results below.
half-integers. The simplest eigenstate j0i of the Gaudin model has all spins down,
s^n j0i = 0 ;
s^3nj0i =
for all n ;
where s^n
is^2n are the lowering operators. The other eigenstates are obtained by
acting on j0i with rising operators s^+(v)
s^1(v) + is^2(v),
jv~1; : : : ; v~M i = s^+(v~1)s^+(v~2) : : : s^+(v~M )j0i
at certain points v~m which satisfy the Bethe equations,
The eigenvalue of T^ (v) corresponding to the state (3.19) is
T (v) = W 2 + @vW ;
n=1 v~m
= 0 :
m06=m
W (v) =
eigenvalues of T^ (v).
To sum up, one solves eqs. (3.20) for every M and nds all Qn(2sn + 1) eigenvectors and
have zero total spin13
Importantly, the Bethe equations (3.20) coincide with the algebraic equations (3.12)
for the parameters v~m of the solitons in dilaton gravity. This establishes one-to-one
correspondence between our exact solutions and the eigenstates (3.19) of the Gaudin model.
The singularities fsn; vng and zeros fv~mg of
(v) are related to the parameters of the
Gaudin Hamiltonians (3.16) and Bethe states (3.19), respectively. Besides, the
incomeqs. (3.2), (3.11) and (3.21). The related quantities of the two models are listed in table 2.
One can use the Gaudin model to study solitons in dilaton gravity. We are interested
13Note that S^ commutes with all Gaudin Hamiltonians.
S^ =
number of solitons with correct asymptotics by adding up spins. For example, there are
representations are marked with their highest weights.
solitons coalesce. In this limit the spin operator (3.17),
Besides, now we can explain what happens at v1 ! v2 when two singularities of the
powers js1 s2j; js1 s2j+1; : : : ; (s1+s2) in accordance with the irreducible representations
The second-order equations (3.6) at these singularities have four solutions corresponding
Finally, one can obtain more general solutions with in nite number of singularities
using the thermodynamic Bethe Ansatz for the Gaudin model [55].
conjugate pairs v1;2 = a1
ib1, v3;4 = a2
ib2. Solving eqs. (3.8), (3.9), (3.6), we obtain,
as expected above, two solutions
2, where the spin
( ) =
a2)2 + b21 + b22
= (a1
+ 12b21b22 > 0. In the limit a1 ! a2, b1 ! b2 the pairs of
singularities in the upper and lower parts of the complex v-plane coalesce, and one obtains
again in accordance with the above expectations.
Note that Tv(v+)(v) is not positive-de nite at real positive v and therefore unphysical.
The function Tv(v )(v) describes incoming matter ux with two peaks at v
a1 and a2, see
Positivity condition
Physical solutions have real (v) at real v. Thus, their singularities vn and zeros v~m are
either real or organized in complex conjugate pairs like in gure 6. Besides, the singularities
vn may not be placed at the physical part v
0 of the real axis.
14One can explicitly demonstrate this by solving eqs. (3.6) to the leading order in v1 v2 ! 0.
0:1, b1 = 1,
dashed lines. For this choice of parameters, the second peak forms the black hole.
The remaining nontrivial condition is Tvv(v)
0, eq. (3.3). This
inequality is not satis ed automatically. For example, our solutions with two singularities (3.15)
ib, respectively. In fact,
any solution with all singularities placed at v < 0 is unphysical. In this case the
operator s^(v) at real v is Hermitean, and therefore T^ (v) in eq. (3.17) has positive-de nite
eigenvalues T (v) /
In the opposite case when all singularities are organized in complex conjugate pairs
v2k 1; v2k = ak
ibk with s2k 1 = s2k, one expects to
nd at least one physical
solution. Indeed, consider the state j 1i (not an eigenstate) of the Gaudin model satisfying
v. On the other hand, the variational principle implies that for any N real points wn there
exists an eigenstate j i minimizing all h jT^ (wn)j i
. The respective eigenvalue T (v) is
T (v) is positive at the entire real axis.
Let us explicitly select the above physical solution at bk ! 0. In this case Tvv(v) falls
into a collection of peaks at v
ak near the singularities v2k 1, v2k. At jv
and yet, far away from other singularities, the operator (3.17) takes the form T^ (v)
(s^2k 1 + s^2k)2=(v
ak)2. Its eigenvalue T (v) /
Tvv(v) is positive-de nite unless the
coincides with the state j 1i introduced above. The respective energy ux Tvv(v) is the
sum of two-spin terms (3.15),
k=1 (v
s2k(s2k + 1)b2k
at small bk :
One expects that this solution remains physical at nite bk.
de nite Tvv(v) at v
hole formation.
0 (gray region). The upper right corner of this region corresponds to black
Example. In general case the positivity condition bounds parameters of the solutions.
Solving eqs. (3.8), (3.9), one obtains,
Tvv =
a)2 + b2)2
The second (negative) term in this expression represents contribution of the singularity
v3 < 0. It can be compensated by the rst term if the singularities v1 and v2 are close
(a2 + b2)=(a
bp3), see the gray region in
enough to v3. Namely, the function (3.24) is positive-de nite at v
gure 8b. The solutions with these parameters
involve one peak of the incoming ux, just like the solutions (3.15).
Critical chaos
Perturbative expansion in the critical regime
In section 2.3 we argued that the critical solutions at the verge of black hole formation
have constant (v) and null boundary U (v) at large v, see gure 4. One can say that they
describe formation of the minimal-mass black holes with the boundary placed precisely at
the horizon [46, 56], cf. [57, 58].
At energies somewhat below critical the boundary has long almost null part
(\plateau"), see
gure 9a. The energy
ux re ected from this part is strongly
amplied by the Lorentz factor of the boundary and forms a high and narrow peak in Tuu(u),
see gure 9b. We will argue below that in the critical limit the peak tends to a -function
(shock-wave) with energy equal to the minimal black hole mass Mcr. In the overcritical
solutions the shock-wave is swallowed by the black hole. Besides, we will see in the next
10 3, where acr =
1=p3 and we use units with
= 1. In this case C
1. Figure (a) shows the boundary u = U (v) in the asymptotically at light-cone coordinates t+x = log( v)= , t x =
u)= , see eq. (2.11). In
gures (b), (c) we plot the outgoing and incoming energy
uxes u2Tuu and v2Tvv as functions of t
x and t + x, respectively.
section that the structure of the peak is highly sensitive to the initial data. This feature
impedes global integrability of the model.
Let us nd the boundary U (v) in the \plateau" region where v is large and Tvv(v) is
small. In this case eq. (2.13) can be solved perturbatively by representing
= 1 +
(2) + : : : , where
1 in the r.h.s. of eq. (2.13), we obtain,
(1)(v) = Cv + e2 0 [g(v)
where the function g(v) is introduced in eq. (2.8) and g
1 is its value at v ! +1. Note
that the linear asymptotics Cv
1 of the solution appears at rst order of expansion in
eq. (4.1) because in the near-critical regime @v
C is small at large v. In what follows
we will regard C as a parameter of the expansion. Using
(1) in the r.h.s. of
eq. (2.13), we get
@v (2)(v) = e2 0 (g
The higher-order corrections (n) are obtained in similar way.
Now, we compute the re ected energy ux Tuu(u) and the boundary function U (v)
using eqs. (2.14) and (2.12),
Tuu(U (v))
e 2 0 C + e 2 0 C2v + 2C(g
with gure 9b.
We kept one and two orders of the expansion in eqs. (4.2) and (4.3), respectively. Note that
the leading ( rst) term in U (v) is constant; this behavior corresponds to the \plateau" in
gure 9a. At the same time, the re ected
ux (4.2) has a peak at large v corresponding
Using the soliton asymptotics Tvv / v 4 and @vg / v 3, one nds that the peak in
value of U (v) is approximately given by the rst term in eq. (4.3), while the peak width
of Tuu(u) is in nitely high and narrow.
v is of the same order. The respective
Calculating the total energy within the shock-wave at C ! 0, we obtain,
Epeak =
jujdu Tuu(u) !
2 C
dv @v2g(v)
where eqs. (4.2), (4.3) were used. The value of Epeak coincides with the minimal black hole
mass Mcr implying that the peak of Tuu(u) tends to a -function in the critical limit.
Shock-wave instability
Since our model is equipped with the general solution, one may think that it is integrable,
i.e. has a complete set of conserved quantities fIkg smoothly foliating the phase space. In
the in-sector these quantities are arbitrary functionals Ik[fin] of conserved fin(v), cf. [59].
Then, Ik can be computed at arbitrary space-like line: to this end one evolves the
classical elds from this line to J , extracts the incoming wave15 fin(v), and calculates Ik[fin].
The quantities fIkg obtained in this way are conserved by de nition. For example, in the
out-sector one gets Ik[fout]
Ik[fin] if fout(u) and fin(v) are related by classical
evolution, eq. (2.9).
Let us argue, however, that fIkg cannot be smoothly de ned in the near-critical regime
because the map fin ! fout in this case is essentially singular. To simplify the argument,
we consider solutions with the modulated ux at large v,
Tvv = (@vfin)2 ;
A v 2 cos(! ln ( v))
v & C 1=3 ;
where C is the small parameter of the near-critical expansion. If ! is small as well, the
asymptotics of Tvv is almost power-law, like in the ordinary solitons. However, the
shockwave part of the re ected ux represents squeezed and ampli ed tail of Tvv at v
C 1=3,
15Recall that all our solutions start from at space-time in the in nite past.
solution (4.4) with A2 = 12e 2 0 = 2.
I3 > 0 (white) and
I3 < 0 (blue) in the (C; !) plane. We use the
see gure 9. It should be essentially modulated. For simplicity, let us characterize the
outgoing wave packet with a single quantity
I3 =
x) (@t xfout)3 =
I3(C; !) + const ;
C2(@vfin)3
where we used the
shock-wave part
at coordinates (2.11) in the de nition of I3, then separated the
I3 of the integral at t
& log C from the (C;
!)independent contribution at smaller t
x. In the second line we substituted the
shockwave pro le (4.2), (4.3) and extended the integration range to v
0. Now, one
substitutes the asymptotics (4.4) into eq. (4.5) and
I3(e6 n=!C; !) = e 2 n=!
I3(C; !). Thus,
I3 = C 1=3 J (! log C), where J (x) is
I3(C; !) is quasi-periodic.
6 -periodic.
We see that
1; 0, or +1, see gure 10.
Thus, any value of
I3 can be obtained by adjusting the limiting path.
The above property ascertains dynamical chaos in the critical limit of our model.
Indeed, in nitesimally small changes (4.4) of the initial data at small C produce outgoing
uxes with essentially di erent values of I3. This prevents one from characterizing the
critical evolution with a set of smooth conserved quantities Ik. Indeed, all functionals
Ik[fin], being smooth in the in-sector, are not sensitive to ! at small values of latter.
Thus, they fail to describe essentially di erent out-states fout(u) at di erent !. From a
more general perspective, one can introduce the integrals which are smooth either in the
in-sector or in the out-sector, but not in both.
see eqs. (2.8).
In this paper we considered two-dimensional CGHS model with a regulating dynamical
boundary [33, 34]. This model is weakly coupled and causally similar to the
sphericallysymmetric gravity in many dimensions. We demonstrated that classical eld equations in
this model are exactly solvable. We constructed their general solution and studied in detail
a large subset of soliton solutions with transparent properties. We illustrated the results
with many explicit examples hoping that this model will serve as a practical playground
for black hole physics.
In the critical regime i.e. at the verge of black hole formation, our model displays
dynamical instabilities speci c to chaotic systems. This property is similar to the
nearhorizon chaos suggested recently in the context of AdS/CFT correspondence [23{28]. We
argued that it hinders global integrability of the model.
We see several applications of our results. First, exact solvability may extend to
oneloop semiclassical level if one adds a re ective boundary to the RST model [5]. This
approach, if successful, will produce analytic solutions describing black hole formation and
evaporation. The singularities of such solutions should be either covered by the boundary
or hidden behind the space-like line
gure 4b. Then a complete Penrose
diagram for the evaporation process may be obtained, cf. [39, 41, 42, 46].
Second, in the alternative approach one directly adds one-loop corrections to the
classical equations of our model with a boundary and integrates the resulting system
numerically, cf. [60, 61]. By the same reasons as above, the respective solutions should completely
describe the process of black hole evaporation.
Third and nally, the model of this paper is ideal for applying the semiclassical method
of [62, 63] which relates calculation of the exponentially suppressed S-matrix elements to
certain complex classical solutions. The results of such calculations may be used to test
unitarity of the gravitational S-matrix [63].
Acknowledgments
We thank S. Sibiryakov for participating at early stages of this project. We are grateful
to Ecole Polytechnique Federale de Lausanne for hospitality during our visits. This work
was supported by the grant RSCF 14-22-00161.
Field equations and boundary conditions
Field equations in the bulk are obtained by varying the action (2.1) with respect to g ,
, and f , and ignoring the boundary terms,
2 = r f r f
2 = R=4 ;
position of the boundary
its outer normal, n
variations. One obtains,
S =
= 0
The rst line here relates the energy-momentum tensors of
second line implies, in addition, that the rescaled metric e 2 g
nd the boundary conditions at the line
0, we keep the boundary terms in
the variation of the action. For a start, let us consider variations preserving the coordinate
. We take
= 0 along this line and
x the direction of
/ n . The integration domains in eq. (2.1) are unchanged by such
e 2 = e 2 =
v0) + g(v) + h(u) :
In this expression M , u0, and v0 are integration constants; functions g(v) and h(u) were
the mass of white hole in the in nite past [1, 2]. Indeed, the past time in nity i in
17Namely, the transformations u ! u~(u), v ! v~(v) preserving the metric (2.4).
are integrated into
) + f n r f = 0 ;
where we canceled the bulk terms using eqs. (A.1){(A.3) and introduced the induced metric
n n . The variation (A.4) gives the boundary conditions (2.2). Note that
0 using the rst of
eqs. (2.2), eq. (A.2) and the trace of eq. (A.1).
Now, let us consider general variations shifting the position of the boundary. They are
combinations of the general coordinate transformations and position-preserving variations
considered above. The action is unchanged by these variations: it is covariant and already
extremized at xed coordinate position of the boundary.
Solution in the conformal gauge
Let us review the general solution [4] of the bulk equations (A.1){(A.3), see [1, 2] for details.
) = 0
where the residual coordinate freedom17 was
xed in the last equation.
After that
eqs. (A.1), namely,
@u@ve 2 =
gure 1b is reached at u !
curvature remains nonzero in this limit,
R = 4e2 (@ue 2 )(@ve 2 )
0 as v ! +1 (i+), see gure 3.
at space-time. Thus, M
It is worth noting that the patch u 2 (
1; 0) and v 2 (0; +1) covers all space-time
accessible to the outside observer. Indeed, we already mentioned that the time in nities i
and i+ are reached in the limits u !
1 and v ! +1 at nite values of the dilaton eld
. By eq. (2.7), the product uv remains nite in these limits implying v ! +0 as u !
0 =
e 2 0 = U 0 @uh
u = U (v) ;
where U 0
from the boundary conditions (2.2). Introducing the unit outer normal
nu = e 0 U 0 ;
nv =
e 0 =pU 0
and using eq. (A.6), we rewrite eqs. (2.2) in the form (2.9).
At this point, we have three equations, eqs. (A.6) and (2.9), for the two unknown
functions fout(u) and U (v). Note, however, that eq. (A.6) follows from the other two
equations. Indeed,
chosen correctly.
2U ) 1 =
(@vfin)2=2 + 2U 0
where we expressed U 0 and g via eqs. (2.9) and (2.8) in the rst and second equalities, then
turned fin ! fout by the second of eqs. (2.9) and used the equation for U 0, again. One
concludes that eq. (A.6) is automatically satis ed once the initial conditions for U (v) are
Bethe Ansatz for the Gaudin model
In this appendix we review Bethe Ansatz for the Gaudin model (3.16), see [47{49] for
is^2(v) for the
positiondependent spin (3.17). The commutation rules of these operators are
[s^ (v); s^+(w)] = 2
[s^3(v); s^ (w)] =
The Hamiltonian T^ (v) in eq. (3.17) takes the form
T^ (v) =
s^ (v)s^+(v) + (s^3(v))2 :
Now, it is straightforward to check that the spin-down state (3.18) is an eigenstate:
T^ (v)j0i = (W0)2 + @vW0 j0i ;
W0(v) =
One explicitly acts with T^ (v), eq. (B.1), on the state (3.19) and obtains,
T^ (v)jv~1; : : : ; v~M i = T (v)jv~1; : : : ; v~M i
v~m jv~1; : : : ; v~m 7! v; : : : ; v~M i ;
where T (v) is given by eq. (3.21), Lm is the left-hand side of eq. (3.20), and arrow denotes
substitution. Note that the relations
[T^ (v); s^+(w)] =
s^+(w)s^3(v)
s^+(v)s^3(w) ;
s^3(v)jv~1; : : : ; v~M i = W (v)jv~1; : : : ; v~M i
v~m jv~1; : : : ; v~m ! v; : : : ; v~M i ;
where W (v) is de ned in eq. (3.21), are helpful for deriving eq. (B.2).
We conclude that eq. (B.2) coincides with the eigenproblem for T^ (v) if the Bethe
eigenstates of the Gaudin Hamiltonians (3.16). Moreover, one can prove [47{49] that the
basis (3.19) is complete.
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