Stability analysis and absorption cross-section in three-dimensional black string

18 Pages·0.17 MB·English

by  H. W. Lee

INJE–TP–98–1, hep–th/9801152
Stability analysis and absorption cross–section in
three–dimensional black string
H.W. Lee1, N.J. Kim1, Y. S. Myung1 and Jin Young Kim2
1Department of Physics, Inje University, Kimhae 621-749, Korea
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9 2Division of Basic Science, Dongseo University, Pusan 616-010, Korea
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J

0 Abstract
3

2
v It is shown that all string fields except dilaton are non–propagating in the
2
5
(2+1)–dimensional black string. One finds that the perturbation around the
1
1
0 black string reveals a mixing between the dilaton and other fields. Under
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9 the new gauge(dilaton gauge), we disentangle this mixing and obtain one
/
h
t decoupled dilaton equation. It turns out that this black string is stable.
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e From thescattering of dilaton off theneutralblack string(N = 0), we findthe
h
:
v absorption cross–section. Further the absorption cross–section for minimally
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X
coupled scalar is obtained and we compared it with that of dilaton.
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Compiled at February 1, 2008.
1
I. INTRODUCTION
Black stringsplayanimportant roleinunderstanding themicroscopic originfortheblack
hole entropy. The statistical interpretation of the Bekenstein-Hawking entropy was made
first in the extremal or near-extremal five-dimensional (5D) black holes [1]. The microstates
of 5D extremal black hole arise mainly from the fields moving around a circle in the internal
dimension. In order to understand this situation, it is useful to take this internal direction
as a spacetime direction explicitly. This is the six-dimensional (6D) black string [2]. The
entropy calculation of 6D black string gives us the same result as in the 5D black hole.
On the other hand, to calculate the entropy of Schwarzschild black hole, we need a
nonperturbative formulation of string theory, M–theory. This is because the Schwarzschild
black hole belongs to the non–extremal one. Recently, there have been many attempts
along this direction [3]. The idea is to map the Schwarzschild black hole to extremal or
near-extremal configurations, where the entropy counting might be possible [4]. It is shown
that black strings give rise to Schwarzschild black hole with the compactification radius (R )
1
or charged dilaton black hole with the radius (R ), when compactified to lower dimensions
2
before or after boosting(α) along uncompactified direction. One finds R = R coshα and
1 2
the Newton constant GN = GN/coshα. Thus the entropy of two holes are the same.
1 2
Further a 7D Schwarzschild black hole is obtained as a compactification of a three brane
in 11D supergravity [5]. And we relate it to a charged black hole with the same statistical
entropy. The charged black hole will be found from subjecting the three brane to a boost
in uncompactified space–time, followed by Kaluza-Klein compactification. A near–extremal
chargedholeisdefinedsuch thattheSchwarzschild radiusremainsarbitrarilylargeatinfinite
boost(α ). This case is used to derive the entropy of neutral Schwarzschild hole from
→ ∞
the charged ones.
In this paper, we study the propagation of string fields in the three–dimensional(3D)
black string background with the new(dilaton) gauge. The 3D black string is used as a
toy model for investigating higher–dimensional black strings [6]. Especially, the s–mode
2
approximation of higer-dimensional black string is exactly the same as in the 3D black
string [7]. Here we are interested in the stability of 3D balck string. The transition between
Schwarzschild black holes to black branes in M–theory corresponds to the balck hole–black
string transition in a boosted frame [3,4]. The latter is related to the instability of a black
string [7]. In this sense, it is very important to analyze the stability of a black string.
Further we investigate the dynamical behavior(absorption cross–section=greybody factor)
of 3D black string rather than the static behavior(entropy) [8,9]. Apart from counting the
microstates of black holes, the dynamical behavior is also an important issue. This is so
because the greybody factor for the black hole arises as a consequence of the gravitational
potential barrier surrounding the horizon. That is, this is an effect of space–time curvature.
Together with the Bekenstein–Hawking entropy, this seems to be the strong hint of a deep
and mysterious connection between curvature and statistical mechanics.
The organization of this paper is as follows. In Sec.II we set up the equations of motion
for the 3D black string and linearize these equations around the background solution. And
we introduce both the transverse gauge and the new (dilaton) gauge. In Sec.III, we study
the propagation of string fields with the gauge conditions. It turns out that a physically
propagating mode is just the dilaton. Further we show that the 3D black string is stable
against the external perturbation. Section IV is concerned with the dynamical behavior of
the neutral (N=0) black string. We derive the absorption cross section from the scattering
process of dilaton off the 3D black string. Finally we discuss our results in Sec.V.
II. FORMALISM
Let us start with the σ-model action of string theory [10]
1 1
S = − d2ξ √γγab∂ Xµ∂ Xνg +ǫab∂ Xµ∂ XνB α′√γR(2)Φ , (1)
σ a b µν a b µν
4πα′ − 2
Z (cid:18) (cid:19)
where R(2) is the Ricci curvature of the world sheet, B two–form field, and Φ dilaton. The
µν
conformal invariance requires the β-function equations
3
1
R Φ H Hρσ = 0, (2)
µν −∇µ∇ν − 4 µρσ ν
8 1
2Φ+( Φ)2 H2 = 0, (3)
∇ ∇ − k − 6
Hµνρ +( Φ)Hµνρ = 0, (4)
µ µ
∇ ∇
where H = 3∂ B isthe Kalb–Ramondfield corresponding toB . The above equations
µνρ [µ νρ] νρ
are also derived from the requirement that the fields must be an extremum of the low-energy
string action in string frame,
8 1
S = d3x√ geΦ R+( Φ)2 + H2 . (5)
l−e
− { ∇ k − 12 }
Z
The static black string solution to (2)–(4) is found to be
Q 1 k
H¯ = , Φ¯ = lnr ln( ),
rtx
r2 − 2 2
(1 M) 0 0
− − r
 
g¯µν = 0 (1 N) 0 , (6)
 − r 
 
 
 0 0 k (1 M)−1(1 N)−1 
 8r2 − r − r 
 
 
with N Q2/M(M > N). The Christoffel symbols are given by
≡
1 1 1
Γt = ,
tr 2 r M − r
(cid:16) − (cid:17)
1 1 1
Γx = ,
xr 2 r N − r
(cid:16) − (cid:17)
4 M N
Γr = M 1 1 ,
tt k − r − r
(cid:16) (cid:17)(cid:16) (cid:17)
4 M N
Γr = N 1 1 ,
xx −k − r − r
1 (cid:16)1 (cid:17)(cid:16) 1 (cid:17)
Γr = + . (7)
rr −2 r M r N
(cid:16) − − (cid:17)
A simple extension of Witten’s construction for a gauged WZW model also yields the 3D
charged black string [11]. The solution (6) is characterized by three parameters: M (mass),
Q (axion charge per unit length), and k (cosmological constant). For 0 < Q < M, black
| |
string is similar to the 4D Reissner-Nordstr¨om solution. In addition to the event (outer)
horizon (r = M), there exist an inner horizon (r = N). When Q = M, this becomes
EH IH
| |
4
the extremal black string. Finally, when Q > M, the spacetime has neither a horizon nor
| |
a curvature singularity and this case is not relevant to us.
To study the propagation of fields specifically, we introduce the small perturbation fields
around the background solution as [12,13]
¯
H = H + ,
rtx rtx rtx
H
Φ = Φ¯ +φ,
g = g¯ +h . (8)
µν µν µν
In order to obtain the equations governing the perturbations, we introduce the notation
1
ˆ
h = h g¯ h. (9)
µν µν µν
− 2
And then one needs to linearize (2)–(4) to obtain
1 1
δR (h) ¯ ¯ φ+δΓρ (h)¯ Φ¯ H¯ ρσ + H¯ H¯ σhρα = 0, (10)
µν −∇µ∇ν µν ∇ρ − 2 µρσHν 2 µρσ να
hµν ¯ ¯ Φ¯ +hµν∂ Φ¯∂ Φ¯ + ¯ hˆµρ∂ Φ¯ ¯2φ 2g¯µν∂ Φ¯∂ φ
µ ν µ ν µ ρ µ ν
∇ ∇ ∇ −∇ −
1
+ 2H¯ µρσ 3H¯ H¯αρσhµ = 0, (11)
6 µρσH − µρσ α
n o
(¯ +∂ Φ¯) µνρ (¯ h ν)H¯µβρ +(¯ h ρ)H¯µβν (¯ hˆµ )H¯ανρ +(∂ φ)H¯µνρ = 0, (12)
∇µ µ H − ∇µ β ∇µ β − ∇µ α µ
where
1 1 1 1
δR (h) = ¯2h ¯ ¯ hρ + ¯ρ¯ h + ¯ρ¯ h , (13)
µν −2∇ µν − 2∇µ∇ν ρ 2∇ ∇ν ρµ 2∇ ∇µ νρ
1
δΓρ (h) = g¯ρσ(¯ h + ¯ h ¯ h ). (14)
µν 2 ∇ν µσ ∇µ νσ −∇σ µν
Here δR can be transformed to the Lichnerowicz operator [7]
µν
1
δR = ¯2h +R¯ hσ R¯ hσρ + ¯ ¯ hˆρ . (15)
µν −2∇ µν σ(ν µ) − σµρν ∇(ν∇|ρ| µ)
These are the bare perturbation equations. We have to examine whether there exist
any choice of gauge which can simplify (10)–(12). Conventionally, we choose the har-
monic(transverse) gauge(¯ hˆµρ = 0) to describe the propagation of gravitons. Further
µ
∇
5
one requires the traceless condition(h = 0), which corresponds to de Donder gauge. How-
ever this does not eliminate all of the gauge degrees of freedom for the metric. There still
exist the residual gauge transformations,
δ h = ¯ ξ + ¯ ξ (16)
ξ µν µ ν ν µ
∇ ∇
with ¯ ξµ = 0 and ¯2ξµ = 0. The counting of gravitational degrees of freedom is as
µ
∇ ∇
follows. A symmetric traceless tensor has D(D+1)/2 1 in D–dimensions. D of them are
−
eliminated by the harmonic gauge condition. Also D 1 are eliminated from our freedom
−
to take further residual gauge transformations. Thus gravitational degrees of freedom are
D(D+1)/2 1 D (D 1) = D(D 3)/2. We have no gravitational degrees of freedom
− − − − −
in D=3. Without loss of generality, the Kalb–Ramond perturbation may be taken to
rtx
H
be zero in the black string background [7]. Hence the physical degree of freedom in the 3D
balck string turns out to be the dilaton field. As we will see later, however, the conventional
(harmonic)gauge is not appropriate for studying the propagation of dilaton.
Here we introduce a new gauge (dilaton gauge) to investigate the propagation of dilaton.
The harmonic gauge is originated from the harmonic coordinates which satisfy
gµνΓλ = 0. (17)
µν
Linearization of (17) leads to
hµνΓλ g¯µνδΓλ = 0 (18)
µν − µν
with g¯µνδΓλ = ¯ hˆµλ. For the Minkowski background (g¯µν = ηµν), (18) becomes the
µν ∇µ
harmonic gauge (¯ hˆµλ = 0) [14]. But we may choose a new gauge hµνΓλ = ¯ hˆµλ when
∇µ µν ∇µ
g¯µν = ηµν, instead of the harmonic gauge. In our coordinate system, one finds
6
8
g¯µνΓt = 0, g¯µνΓx = 0, g¯µνΓr = (r2 MN), (19)
µν µν µν −kr −
which means that (t,x,r) do not belong to the harmonic coordinates. Requiring that (19)
remainunchangedatthelinearizedlevel, onealwayschoosethenewgauge(hµνΓλ = ¯ hˆµλ).
µν ∇µ
This gauge is suitable for studying the propagation of dilaton.
6
III. PROPAGATION OF STRING FIELDS
First we consider the symmetries of the background space–time in (6). In our case, the
t and x–translational symmetries means that we can decompose h into frequency modes
µν
in these variables [7]. Hence we take the form for h ,
µν
H (r) H (r) H (r)
tt tx tr
 
hµν(t,x,r) = eiωteiµxHxt(r) Hxx(r) Hxr(r) . (20)
 
 
 H (r) H (r) H (r) 
 rt rx rr 
 
 
Similarly, one chooses the perturbations for Kalb–Ramond field and dilaton as
(t,x,r)= H¯ (r) (t,x,r) = H¯ (r)eiωteiµx ˜(r), (21)
rtx rtx rtx
H H H
φ(t,x,r)= eiωteiµxφ˜(r). (22)
Since we start with full degrees of freedom (20), we should choose a gauge to study the
propagation of string fields.
A. Harmonic Gauge
With the harmonic gauge condition( ¯ hˆµρ = 0), the equations (10)–(12) become
µ
∇
¯2h R¯ h σ R¯ hσ +2R¯ hσρ +2¯ ¯ φ
∇ µν − σν µ − σµ ν σµρν ∇µ∇ν
2δΓρ (h)¯ Φ¯ +H¯ ρσ H¯ H¯ σhρα = 0, (23)
− µν ∇ρ µρσHν − µρσ να
hµν ¯ ¯ Φ¯ +hµν Φ¯ Φ ¯2φ 2gµν ¯ Φ¯ φ
µ ν µ ν µ ν
∇ ∇ ∇ ∇ −∇ − ∇ ∇
1
+ 2H¯ µρσ 3H¯ H¯αρσhµ = 0, (24)
6 µρσH − µρσ α
n o
(¯ +∂ Φ¯) µνρ (¯ h ν)H¯µβρ +(¯ h ρ)H¯µβν +(∂ φ)H¯µνρ = 0, (25)
∇µ µ H − ∇µ β ∇µ β µ
Hereafter we are interested only in the dilaton propagation. From (24) one finds
1 1 1 kr ω2 µ2
φ′′ + + + φ′ + φ
(cid:18)r r −M r −N(cid:19) 8(r −M)(r −N) r −M − r −N!
1
M(r +N)ht +N(r +M)hx
− 2r2(r M)(r N) t x
− − h
+ 2r2 (M +N)r +2MN hr 4MN = 0, (26)
− r − H
n o i
7
where the prime(′) means the derivative with respect to r. We note that the analysis of dila-
ton fluctuation around the black string reveals a surprising mixing between the dilaton and
other fields. We have to disentangle this mixing and obtain one decoupled dilaton equation
by using the harmonic gauge(¯ hˆµρ = 0) and Kalb–Ramond equation (25). However we fail
µ
∇
to obtain a decoupled dilaton equation.
B. Dilaton Gauge
We recognize that it is not easy to decouple the dilaton equation with the harmonic
gauge condition. But if we choose the dilaton gauge( hµνΓρ = ¯ hˆµρ), the dilaton equation
µν ∇µ
can be diagonalized easily. Under this gauge the equations (10)–(12) are given by
¯2h R¯ h σ R¯ hσ +2R¯ hσρ +2¯ ¯ φ g¯ ¯ (hαβΓσ ) g¯ ¯ (hαβΓσ )
∇ µν − σν µ − σµ ν σµρν ∇µ∇ν − µσ∇ν αβ − νσ∇µ αβ
2δΓρ (h)¯ Φ¯ +H¯ ρσ H¯ H¯ σhρα = 0, (27)
− µν ∇ρ µρσHν − µρσ να
hµν ¯ ¯ Φ¯ +hµν ¯ Φ¯ Φ+g¯µνδΓρ ∂ Φ¯ 2φ 2g¯µν ¯ Φ¯ φ
∇µ∇ν ∇µ ∇ν µν ρ −∇ − ∇µ ∇ν
1
+ 2H¯ µρσ 3H¯ H¯αρσhµ = 0, (28)
6 µρσH − µρσ α
n o
(¯ +∂ Φ¯) µνρ ¯ h νH¯µβρ + ¯ h ρH¯µβν +∂ φH¯µνρ hδηΓα H¯ νρ = 0. (29)
∇µ µ H −∇µ β ∇µ β µ − δη α
Thanks to the dilaton gauge, the first three terms in (28) cancel out. Then the dilaton
equation (28) leads to
1 1 1
φ′′ + + + φ′
r r M r N
(cid:18) − − (cid:19)
kr ω2 µ2 MN
+ φ (h 2 ) = 0. (30)
8(r M)(r N) r M − r N! − r2(r M)(r N) − H
− − − − − −
Now we attempt to disentangle the final term in (30) by using the dilaton gauge (18) and
Kalb–Ramond equation (29). Each component of dilaton gauge condition gives rise to
1
t : (∂ )htr +iωhtt +iµhtx = 0, (31)
r
− r
1
x : (∂ )hxr +iωhxt +iµhxx = 0, (32)
r
− r
1
r : (∂ )hrr +iωhrt +iµhrx = 0. (33)
r
− r
8
And the Kalb–Ramond equation (29) leads to
8
tx : r(φ′ + ′ ht ′ hx ′)+ (r2 MN)h +iωht +iµhx = 0, (34)
H − t − x k − rr r r
8
tr : iµ(φ+ ht hr ) (r M)h +iωht +hr ′ = 0, (35)
H− t − r − k − xr x x
8
xr : iω(φ+ hx hr ) (r N)h +iµhx +hr ′ = 0. (36)
H− x − r − k − tr t t
Solving six equations (31)–(36), one finds an important equation
∂ (φ+ h) = 0. (37)
µ
H−
The solution to this is given by
h = φ+ . (38)
H
This means that the trace of h (h) is a redundant field. On the other hand, the Einstein
µν
equations from (27) are given by
8 8(2r2 (3M +N)r +2MN) ω2 µ2
tt : (r M)(r N)h′′ + − h′ +r h
k − − tt k r tt r M − r N! tt
− −
8M2(r N) 8MN(r M) 64M(r M)(r N)(r2 MN)
+ − h − h − − − h
tt xx rr
k r2(r M) − k r2(r N) − k kr3
− −
16 (r M)(r N)
iω − − h
tr
− k r
8M(r M)(r N) MN(r M)
− − φ′ 2ω2φ+16 − = 0, (39)
− k r2 − kr3 H
8 8(2r2 (M +3N)r +2MN) ω2 µ2
xx : (r M)(r N)h′′ + − h′ +r h
k − − xx k r xx r M − r N! xx
− −
8MN(r N) 8N2(r M) 64N(r M)(r N)(r2 MN)
− h + − h + − − − h
tt xx rr
− k r2(r M) k r2(r N) k kr3
− −
16 (r M)(r N)
iµ − − h
xr
− k r
8N(r M)(r N) MN(r N)
+ − − φ′ 2µ2φ 16 − = 0, (40)
k r2 − − kr3 H
8 16(3r2 (M +N)r MN) ω2 µ2
rr : (r M)(r N)h′′ + − − h′ +r h
k − − rr k r rr r M − r N! rr
− −
M N M(M N) N(M N)
h′ + h′ − h − h
− (r M)2 tt (r N)2 xx − (r M)3(r N) tt − (r M)(r N)3 xx
− − − − − −
8(6r4 5(M +N)r3 +4MNr2 MN(M +N)r +2M2N2)
+ − − h
rr
k r2(r M)(r N)
− −
9
M N
+2iω h 2iµ h
tr xr
(r M)2 − (r N)2
− −
1 1 2MN
+2φ′′ + + φ′ = 0, (41)
r M r N − r2(r M)(r N)H
(cid:18) − − (cid:19) − −
8 16(r M)(r N) ω2 µ2
tx : (r M)(r N)h′′ + − − h′ +r h
k − − tx k r tx r M − r N! tx
− −
8 (r2 2Nr +MN) 8 (r2 2Mr +MN)
iµ − h iω − h 2ωµφ = 0, (42)
tr xr
− k r − k r −
8 8(r M)(3r N) ω2 µ2
tr : (r M)(r N)h′′ + − − h′ +r h
k − − tr k r tr r M − r N! tr
− −
8(r M) N
+ − h iµ h
tr tx
k r − (r N)2
−
iω M iω N 4iω (3M +N)r 4MN
+ h + h + { − }h
tt xx rr
2 (r M)2 2 (r N)2 k r
− −
M
+2iωφ′ iω φ = 0, (43)
− r(r M)
−
8 8(3r M)(r N) ω2 µ2
xr : (r M)(r N)h′′ + − − h′ +r h
k − − xr k r xr r M − r N! xr
− −
8(r N) M
+ − h +iω h
xr tx
k r (r M)2
−
iµ M iµ N 4iµ (M +3N)r 4MN
h h + { − }h
tt xx rr
− 2 (r M)2 − 2 (r N)2 k r
− −
N
+2iµφ′ iµ φ = 0. (44)
− r(r N)
−
From (34)–(36) and (39)–(41) we may set = 0 without loss of generality. Also this is
H
confirmed from Ref. [7]. With = 0, (30) becomes a decoupled dilaton equation
H
1 1 1
φ˜′′ + + + φ˜′
r r M r N
(cid:18) − − (cid:19)
kr ω2 µ2 MN
˜ ˜
+ φ φ = 0. (45)
8(r M)(r N) r M − r N! − r2(r M)(r N)
− − − − − −
C. Stability Analysis
Since the physical field in the 3D black string is dilaton, we will consider the stability of
dilaton. We remind the reader that the stability analysis should be based on the physical
fields. In order to check whether there exists an exponentially growing mode, we replace ω
10