Classical and Semiclassical Properties of Extremal Black Holes with Dilaton and Modulus Fields

28 Pages·0.19 MB·English

by  M. Cadoni

December 1993 INFN-CA-20-93
4
9 CLASSICAL AND SEMICLASSICAL PROPERTIES OF EXTREMAL
9 BLACK HOLES WITH DILATON AND MODULUS FIELDS
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7 M. Cadoni
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2 S. Mignemi
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9 Dipartimento di Scienze Fisiche, Universit`a di Cagliari
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h and Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari
t Via Ada Negri 18, I-09127 Cagliari, Italy
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ABSTRACT
We discuss both classical and semiclassical properties of extremal black holes in the-
ories where the dilaton and a modulus field are present. We find that the corresponding
2-dim geometry is asymptotically anti-de Sitter rather then asymptotically flat as in the
purely dilatonic case. This fact has many important consequences, which we analyze at
length, both for the classical behaviour and for the thermodynamical properties of the
black hole. We also study the Hawking evaporation process in the semiclassical approxi-
mation. The calculations strongly indicates the emergence of a stable ground state as the
end point of the process. Some comments are made about the relevance of our results for
the problem of information loss in black hole physics.
1
1. Introduction
String inspired black hole models have been extensively studied in recent years. A
first motivation for this interest is the fact that exact charged solutions of the 4-dim lowest
order effective action of string theory have been found in a variety of cases [1-3].
These solutions exhibit peculiar thermodynamical properties, which are different from
those of the ordinary Einstein theory black holes [4]. In particular, the extremal limit,
where the values of the mass and of the charge are such that the black hole is on the edge
of becoming a naked singularity, presents a behaviour which resembles that of elementary
particles [5]. Even if some of these features are spoiled by going to higher order in pertur-
bation theory [6], these models still present a relevant theoretical interest. The extremal
limit of a black hole of given charge represents, in fact, the ground state for its Hawking
evaporation process. The study of the scattering of low-energy particles by an extremal
black hole should therefore shed some light on the problem of black hole evaporation and
on the fate of the information after the evaporation process has taken place [7]. In par-
ticular, one hopes to clarify the issue whether the black hole completely disappears or a
massive remnant is left. In this context, a very useful property of extremal string black
hole solutions is the fact that in proximity of the horizon they split into a direct product
of a 2-dim solution with a 2-sphere of constant radius [8]. It is therefore possible to study
the properties of the black hole by means of a reduced 2-dimensional model, which makes
the problem much easier to treat [9]. Many papers have in fact been devoted to the in-
vestigation of these 2-dim models [10,11]. If the back-reaction of the gravitational field
is neglected, exact solution of the field equations including one-loop quantum corrections
can be found, which describe the formation of a black hole and the early stages of its
evaporation [9]. Moreover, if the back-reaction is also taken into account, the behaviour
of the black hole during the final stages of the evaporation can be qualitatively described
[10].
The peculiar properties of string black holes are essentially due to the non-minimal
coupling of gravity and gauge fields to a massless scalar field which is contained in the
low-energy spectrum of string, namely the dilaton. The non-minimality of the coupling
permits to circumvent the no-hair theorems which render essentially unique the black hole
solutions of Einstein theory. It is therefore interesting to consider the other non-minimally
coupled massless scalars which appear in the low-energy effective action of the string [3,12].
It turns out, in fact, that modulus fields coming from the compactification of the string
to 4-dimensions can acquire non minimal couplings to the gauge fields of the 4-dim action
owing to string one-loop effects [13]. In a recent paper [3], we have studied a model where
these effects are taken into account and found a class of exact 4-dim solutions, whose
properties are slightly different from those where modulus fields are decoupled. In this
paper we extend the investigation of such solutions to their extremal limit and study the
black hole evaporation by means of a 2-dim effective model.
The main result of our investigation is that, contrary to the pure dilatonic case, the
solutions of the 2-dim effective theory are not asymptotically flat, but have as asymptotic
a space of constant negative curvature. This implies that in this approximation the 4-
dim extremal black hole is screened by an infinite potential barrier, which prevents it to
irradiate to the asymptotically flat region. This explains why the temperature vanishes in
2
the 4-dim theory. We also study the quantum evaporation of the 2-dim black hole . We
find that for a wide range of the parameters which characterize the model, the Hawking
radiationrateisnot asymptoticallymass independent as inthe case of pure dilatonic2-dim
gravity, but goes to zero with the mass of the hole. This strongly suggests the emergence
of a stable state at the end point of the evaporation process.
The paper is organized as follows: in section 2 we review the 4-dim solutions found
in [3] and study their extremal limit. In section 3 we examine the propagation of a per-
turbation in the throat region of the solution and show that it is exponentially damped at
infinity. In section 4 we study the 2-dimensional effective theory and discuss its solutions
in different gauges. We also review 2-dim spacetimes of constant curvature. In section 5
we study the 2-dimensional theory in the conformal gauge and in section 6 we analyse the
2-dim black hole evaporation process using various methods. We state our conclusions in
section 7.
2. The solution in 4 dimensions
a) The general solution
In a recent paper [3], we have studied the magnetic charged black hole solution of a
4-dim gravitational action obtained by low-energy effective string theory when moduli of
the compactified manifold are taken into account.
In terms of the metric to which the string couples, which we shall use in this paper,
the action had the form:
2
S = √g d4x e−2φ R+4( φ)2 ( ψ)2 F2 e2(φ−q3ψ)F2 , (2.1)
∇ − 3 ∇ − −
Z (cid:20) (cid:21)
where φ is the dilaton field, ψ the compacton and F the Maxwell field strength, q being
µν
a coupling constant.
Exactchargedblackholesolutionstothisactionwerefoundinthecaseψ = 3φ+const,
q
which have the form:
r r k r 1 r 1
ds2 = 1 + 1 − dt2 + 1 + − 1 − − dr2 +r2dΩ2,
− − r − r − r − r
(cid:16) (cid:17)(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
r (k 1)/2 Q
e2(φ−φ0) = 1− r− − , Fµν = rM2 ǫµν, (2.2)
(cid:16) (cid:17)
with
3 2q2
k = − , 1 k 1. (2.3)
3+2q2 − ≤ ≤
The two parameters r and r are related to the charge Q and to the mass M of the
+
−
black hole by the relations:
k +1 1 k
2M = r + r , Q2 = − r r . (2.4)
+ 2 − M 4 + −
In the limit k = 1 (i.e. q ), the solution reduces to that found in in ref. [1] for the
− → ∞
case where the compacton field is not taken into account.
3
The spatial part of the metric (2.2) is actually identical to that of ref. [1]: it describes
an asymptotically flat region attached to a long tube (the ”throat”), terminated by a
horizon. In the extremal limit r = r , the tube becomes infinitely long.
+
−
In our case, however, the full spacetime has a slightly different structure with respect
to ref. [1]: as we shall see, the infinitely long tube in the extremal limit is replaced by the
direct product of a 2-dim negative curvature spacetime with a 2-sphere of constant radius.
This fact will have important consequences on the physical properties of the model.
Also of interest are the thermodynamical properties of the black hole described by
(2.2). The temperature is given by
1+k
1 r 2
T = 1 − . (2.5)
4πr − r
+ (cid:18) +(cid:19)
which goes to zero in the extremal limit r = r , while the entropy is
+
−
S = πr2. (2.6a)
+
One could therefore think of the extremal limit as a non-radiating ground state. This point
of view will be confirmed in the following.
It should be noticed, however, that for the ”canonical” metric g˜ = e 2φg, while the
−
expression for the temperature remains the same, the entropy is given by
1−k
r 2
S = πr2 1 − , (2.6b)
+ − r
(cid:18) +(cid:19)
whichalsovanishesforr r . Thediscrepance isofcoursedue tothedifferent definition
+
→ −
of the volume element in the two metrics.
b) The extremal limit
In the rest of this paper we shall study the extremal limit r = r = Q of the
+
solution (2.2), where Q = 2 Q , and the ensuing 2-dim effective the−ory, along the
√1 k M
lines of ref. [8]. For this purp−ose, it is useful to define a new coordinate σ, such that
σ = arcsh (r−r+) .
(r+−r−)
In terqms of σ, the metric and the dilaton field (2.2) take the form
∆k+1sinh2σcosh2kσ
ds2 = 4Q2 dt2 +(r +∆sinh2σ)2(4dσ2 +dΩ2),
− (r +∆sinh2σ)k+1 +
+
1−k
r +∆sinh2σ 2
e2(φ−φ0) = + , (2.7)
∆cosh2σ
(cid:20) (cid:21)
where ∆ = r r .
+
− −
There are several regimes under which the extremal limit can be approached, which
correspond to different solutions of the action (2.1):
4
1) σ 1: this limit is reached by taking the asymptotically flat region and the throat
≫
fixed and allowing the horizon to move off to infinity as ∆ 0. The solution is then:
→
(k+1) 2
Q − Q
ds2 = 4Q2 1+ dt2 + 1+ (dy2 +y2dΩ2),
− y y
(cid:18) (cid:19) (cid:18) (cid:19)
1−k
Q 2 √1 k Q
e2(φ−φ0) = 1+ y , Fµν = 2− (y +Q)2ǫµν, (2.8)
(cid:18) (cid:19)
with y 0. The metric is everywhere regular and describes the transition between an
≥
asymptotically flat spacetime for y and one with topology H2 S2 for y 0,
→ ∞ × →
H2 being 2-dim anti-de Sitter spacetime . This solution can therefore be considered a
generalization of the solitonic solutions of string theory described in [14].
2) 1 σ ln(Q/∆): this limit corresponds to the infinite throat with linear dilaton
≫ ≫
and is reached by sending to infinity both the horizon and the asymptotically flat region:
ds2 = 4Q2e2(k+1)σdt2 +Q2(4dσ2 +dΩ2),
−
k 1
φ = − σ. (2.9)
2
At variance with the GHS case, the metric in this limit does not describe an infinite
cylinder, but rather the direct product of 2-dim anti-de Sitter spacetime with a 2-sphere
of radius Q.
3) σ 1: this limit is obtained by keeping the horizon and ∆1−2ke−2φ0 fixed, and
≪
letting the asymptotically flat region go to infinity when ∆ 0 and describes the horizon
→
plus the infinite throat. The solution is given by:
ds2 = 4Q2sinh2σcosh2kσ dt2 +Q2(4dσ2 +dΩ2),
−
1−k
Q 2
e2(φ−φ0) = , (2.10)
cosh2σ
(cid:18) (cid:19)
and again has the form of a direct product of a 2-dim solutions with an horizon at σ = 0
and a 2-sphere. It will therefore be useful in the discussion of black hole solutions of the
2-dim effective action. Its 2-dim sections will be discussed at length in the following.
3. Perturbations on the throat
Near the extremal limit, the two essential features of the geometry are the asymptot-
ically flat region and the attached throat. It is therefore crucial to study the propagation
of the fields along the throat. From the anti-de Sitter form of the metric can be expected
that the fields are in some way confined into the throat, due to the infinitely increasing
gravitational potential for σ .
→ ∞
Given a perturbation χ on the throat, one can easily check that the kinetic term in
the action at the linearized level is
S = √g¯ d4x e 2bφ( χ)2, (3.1)
χ −
− ∇
Z
5
where g¯ is the flat metric and b is a constant depending on the mode considered.
The effect of the linear dilaton background can be seen by defining the new field
χ˜ = e bφχ. The linearized action becomes therefore, modulo a total derivative,
−
S = √g¯ d4x ( χ˜)2 +χ˜2(b2( φ)2 b 2φ) =
χ
− ∇ ∇ − ∇
Z
(cid:2) (cid:3)
= √g¯ d4x ( χ˜)2 +M2(φ)χ˜2 , (3.2)
− ∇
Z
(cid:2) (cid:3)
where
1+k
M2(φ) = const exp 4 φ ,
× − 1 k
(cid:18) − (cid:19)
isa space dependent mass termfor χ˜, which becomes infinite in theregion ofweak coupling
e2φ 0, where the excitations are thus suppressed by an infinite mass gap. They are
→
therefore not allowed to escape to infinity, which is in our case the asymptotically flat
region.
Similar results have been obtained in [4] for the ”canonical” metric g˜ = e 2φg. In
−
particular, these results can help to interpret the vanishing of the temperature (2.5) in the
extremal limit : a potential barrier hinders the interaction between the black hole and the
external fields, creating a mass gap. In our treatment, the geometrical origin of the barrier
is however more evident.
4. The two-dimensional effective theory
a) Dimensional reduction
In order to investigate the quantum properties of the extremal black hole , it is useful
to study the 2-dim effective model obtained by retaining only the radial modes of the 4-dim
theory. This approximation is justified by the fact that the background solution, as we
have seen, reduces to the direct product of two 2-dim metrics near the horizon. The 2-dim
theory is renormalizable and is a generalization of the model considered in [9].
We then start with a discussion of the classical aspects of the 2-dim effective theory.
†
The action (2.1) can be dimensionally reduced by taking the angular coordinates to span
a 2-sphere of constant radius Q: the resulting action is
1 2
S = 2π √g d2x e−2φ R+4(∇φ)2 − 3(∇ψ)2 +λ21 −λ22 e2(φ−q3ψ) , (4.1)
Z (cid:20) (cid:21)
where
2(q2 +3) 1 3+k 2q2 1 1 k
λ2 = = , λ2 = = − .
1 2q2 +3 Q2 2Q2 2 2q2 +3Q2 2Q2
The ensuing equations of motion are:
2 1 λ2 λ2
2 µ νφ µψ νψ = gµν 2 2φ 2( φ)2 ( ψ)2 + 1 2e2(φ−q3ψ) ,
∇ ∇ − 3∇ ∇ ∇ − ∇ − 3 ∇ 2 − 2
(cid:20) (cid:21)
These properties will be discussed in more detail in a forthcoming paper [15]. Related
†
2-dim models are discussed also in [16, 11].
6
2
4 2φ 4( φ)2 ( ψ)2 +λ2 +R = 0, (4.2)
∇ − ∇ − 3 ∇ 1
q
∇2ψ −2(∇φ)(∇ψ)+ 2λ22e2(φ−3qψ) = 0.
With the ansatz
q2
e23qψ = e2φ, (4.3)
3
the 2-dim action admits the exact solution:
ds2 = sinh2(κσ)cosh2k(κσ)dt2 +dσ2,
−
e2(φ−φ0) = coshk−1(κσ), (4.4)
where κ = λ2 , with k defined by (2.3).
√2(1 k)
−
These solutions are everywhere regular for any value of k, and have a horizon at σ = 0.
They of course coincide with the 2-dim section of the solutions (2.10). For k = 1 the
6 −
solutions (4.4) behave as anti-de Sitter for σ . For k = 1 they are asymptotically
→ ∞ −
flat and reduce to those obtained in [17].
Another class of solutions to (4.2,3) is given by the anti-de Sitter background with
linear dilaton :
ds2 = e2(k+1)κσdt2 +dσ2,
−
k 1
φ φ = − κσ, (4.5)
0
− 2
which we shall refer as ADS linear dilaton vacuum. These solutions are asymptotic of (4.4)
for σ and correspond to the dimensional reduction of (2.9).
→ ∞
We also notice that, substituting the ansatz (4.3) directly into the action, one has:
1 8k
S = √g d2x e 2φ R+ ( φ)2 λ2 . (4.6)
2π − k 1 ∇ − 2
Z (cid:20) − (cid:21)
For k = 0 the action reduces to that of the Jackiw-Teitelboim theory [18].
b) The Schwarzschild gauge
In the following, it will be useful to write the metric in Schwarzschild coordinates. In
such coordinates it is in fact possible to continue the metrics besides the horizon at σ = 0
and to get a more immediate insight of their physical properties. The new coordinates
are defined so that the metric takes the form ds2 = Υ(r)dt2 +Υ 1(r)dr2. The general
−
−
solutions in this gauge are:
ds2 = (b2r2 a2r12+kk)dt2 +(b2r2 a2r12+kk)−1dr2,
− −
e2(φ−φ0) = (br)k1+−k1, (4.7)
7
where b = (1+k)λ2 and a is a free parameter which, from a 4-dim point of view, can
√2(1 k)
−
be interpreted as parametrizing the departure of the solution from extremality. These
coordinates can be related to the previous ones by the transformation br = cosh1+k(κσ),
with a = b.
The metric asymptotes anti-de Sitter spacetime for r , and has a horizon at
→ ∞
r = a 1+k, which shields a singularity at r = 0, except in the special cases k = 0,1. This
0 b
is easily seen by considering the curvature:
(cid:0) (cid:1)
k(1 k)
R = 2 b2 + − a2r−1+2k . (4.8)
− (1+k)2
(cid:18) (cid:19)
If a = 0, the metric reduces to that of 2-dim anti-de Sitter spacetime with curvature 2b2:
−
ds2 = (br)2dt2 +(br)−2dr2, e2(φ−φ0) = (br)k1+−k1, (4.9)
−
which corresponds to the solution (4.5).
The mass of the solutions can easily be obtained by the ADM procedure and is given
by:
M = 1−k e−2φ0a2b11−+kk, (4.10a)
2(1+k)
or, in terms of the value φ of φ at the horizon,
H
λ 1 k 1+k 4
2
M = − exp 2 φ φ . (4.10b)
0 H
2 2 1 k − 1 k
r (cid:18) − − (cid:19)
Analogously, using standard procedures, one can calculate the thermodynamical pa-
rameters:
1+k 1−k
1 2M 2 b 2
T = e(1+k)φ0 , (4.11)
2π 1 k 1+k
(cid:18) − (cid:19) (cid:18) (cid:19)
1−k
1+k 2M 2
S = 2πe(1+k)φ0 , (4.12)
1 k b
(cid:18) − (cid:19)
where the entropy S is computed with respect to the asymptotic anti-de Sitter solution.
The temperature increases monotonically with the mass and goes to zero for M = 0,
which should therefore be considered as a stable ground state for the quantum black hole .
The entropy displays essentially the same behaviour as T. Also is remarkable the relation
ST = 2M , valid for φ = 0.
1 k 0
Fro−m the previous discussion is evident that for a = 0 all the metrics (4.7) describe
6
black holes with asymptotically anti-de Sitter behaviour. There are however three special
cases, which should be considered separately:
1) k = 1:
−
ds2 = (1 a2e−br)dt2 +(1 a2e−br)−1dr2, e2(φ−φ0) = ebr. (4.13)
− − −
8
This is the asymptotically flat solution thoroughly discussed in [17]. Therefore we do
not discuss it here.
2) k = 0:
ds2 = (b2r2 a2)dt2 +(b2r2 a2)−1dr2, e2(φ−φ0) = (br)−1. (4.14)
− − −
For a = 0 the metric, while having a constant negative curvature has different prop-
6
erties from anti-de Sitter spacetime . In particular, it possesses a horizon at r = a/b, but
no singularities. We consider it in more detail below.
3) k = 1:
ds2 = (b2r2 a2r)dt2 +(b2r2 a2r) 1dr2, φ = const. (4.15)
−
− − −
In this case, the metric can be put in the same form as for the k = 0 case, by simply
shifting r. Now, however, the dilaton is everywhere constant.
c) 2-dim spacetimes of constant negative curvature.
In order to discuss the properties of the k = 0 solution and for future reference, it is
useful to summarize the properties of the 2-dim spacetimes of constant negative curvature.
They can be easily constructed by considering parabolic hyperboloids embedded in 3-dim
Minkowski spacetime , with metric ds2 = dz2 dx2 dy2. The standard anti-de Sitter
− −
spacetime is then represented by a 1-sheet hyperboloid of equation [19]:
†
x2 +y2 z2 = 1. (4.16)
−
The surface can be parameterized by the coordinates :
x = coshσsint, y = coshσcost, z = sinhσ, (4.17)
with < σ < , 0 t 2π, giving rise to the metric
−∞ ∞ ≤ ≤
ds2 = cosh2σdt2 +dσ2, (4.18a)
−
or, in the Schwarzschild gauge,
ds2 = (r2 +1)dt2 +(r2 +1) 1dr2. (4.18b)
−
−
Another parametrization, which however does not cover the whole hyperboloid, is
given by:
1 1
x = eσt, y = coshσ eσt2, z = sinhσ + eσt2, (4.19)
− 2 2
For simplicity we put Λ = 1 in the following.
†
9
with < σ < , < t < . In these coordinates ,
−∞ ∞ −∞ ∞
ds2 = e2σdt2 +dσ2, (4.20a)
−
or
ds2 = r2dt2 +r 2dr2. (4.20b)
−
−
Contrary to the 4-dim case, in 2 dimensions there is another spacetime of constant
negative curvature. This is represented by a 2-sheet hyperboloid of equation
x2 +y2 z2 = 1, (4.21)
− −
embedded as before in 3-dim flat spacetime . With the choice of coordinates :
x = sinhσsint, y = sinhσcost, z = coshσ, (4.22)
0 σ < , 0 t 2π, the metric becomes :
†
≤ ∞ ≤ ≤
ds2 = sinh2σdt2 +dσ2, (4.23a)
−
or, in Schwarzschild coordinates ,
ds2 = (r2 1)dt2 +(r2 1) 1dr2. (4.23b)
−
− − −
To our knowledge, this metric has not been previously discussed in the literature,
presumably because it cannot be obtained by a dimensional reduction of 4-dim anti-de
Sitter spacetime (see however [20]).
From its construction, it is evident that the solution is everywhere regular. Its most
striking difference from anti-de Sitter spacetime is the presence of a horizon at r = 1 (the
vertex of the hyperboloid). The asymptotic properties instead, are identical in both cases.
Thus the metric describes a non-singular black hole . Its horizon has temperature T = 1
2π
at variance with the anti-de Sitter metric (4.18), whose temperature of course vanishes.
We shall discuss in extent this feature in the following sections.
Toconcludewerecallsomeproblemsintheinterpretationofanti-deSitter-likemetrics
which emerge when a quantum theory is constructed on such background [22]. First of
all, from the geometry of the spacetime and the definitions (4.17), (4.22) is evident that
the time coordinate is periodic, 0 t 2π. The consequent presence of closed timelike
≤ ≤
paths can however be avoided by going to the universal covering space, < t < .
−∞ ∞
The second problem is the lack of global hyperbolicity of anti-de Sitter spacetime , due
to the fact that spacelike infinity is at finite coordinate distance in conformal coordinates.
A great care is therefore needed in the choice of the boundary conditions [22]. We shall
discuss at length these problems in section 6.
In this form, the metric has been independently discussed in [20], in the context of
†
the Jackiw-Teitelboim theory [18]
10