LAPTH-1178/07
Hawking radiation of linear dilaton black holes
G. Clémenta∗, J.C. Fabrisb†and G.T. Marquesa,b‡
aLaboratoire de Physique Théorique LAPTH (CNRS),
B.P.110, F-74941 Annecy-le-Vieux cedex, France
b Departamento de F́ısica, Universidade Federal do Esṕırito Santo,
Vitória, 29060-900, Esṕırito Santo, Brazil
April 3, 2007
Abstract
We compute exactly the semi-classical radiation spectrum for a
class of non-asymptotically flat charged dilaton black holes, the so-
called linear dilaton black holes. In the high frequency regime, the
temperature for these black holes generically agrees with the surface
gravity result. In the special case where the black hole is massless,
we show that, although the surface gravity remains finite, there is
no radiation, in agreement with the fact that massless objects cannot
radiate.
e-mail: gclement@lapp.in2p3.fr
e-mail: fabris@cce.ufes.br
e-mail:gtadaiesky@cce.ufes.br
http://arxiv.org/abs/0704.0399v1
Quantum field theory in curved spacetime predicts new phenomena such
as particle emission by a black hole [1]. This is due to the fact that the vac-
uum for a quantum field near the horizon is different from the observer’s
vacuum at spatial infinity. A distant observer thus receives from a black
hole a steady flux of particles exhibiting, in the high frequency regime, a
black body spectrum with a temperature proportional to the surface grav-
ity [2]. Although Hawking’s original derivation of this black hole evaporation
dealt with realistic collapsing black holes, Unruh [3] showed that the same
results are obtained when the collapse is replaced by appropriate boundary
conditions on the horizon of an eternal black hole. In the semi-classical ap-
proximation, the black hole radiation spectrum may be evaluated by com-
puting the Bogoliubov coefficients relating the two vacua. An equivalent
procedure is to compute the reflection and absorption coefficients of a wave
by the black hole. Usually, the wave equation cannot be solved exactly,
and one must resort to match solutions in an overlap region between the
near-horizon and asymptotic regions [4, 5]. In the special case of the (2+1)-
dimensional BTZ black hole [6], an exact solution of the wave equation is
available, which allows for an exact computation of the radiation spectrum,
leading to the Hawking temperature [7, 8, 9].
In this Letter, we discuss another case of black holes also allowing for
an exact semi-classical computation of their radiation spectrum, that of lin-
ear dilaton black hole solutions to Einstein-Maxwell dilaton (EMD) theory
in four dimensions. Linear dilaton black holes are a special case of the
more general class of non-asymptotically flat black hole solutions to EMD
[10, 11], which we first briefly present. We discuss the evaporation of these
non-asymptotically flat black holes and show that they either collapse to a
naked singularity in a finite time, or evaporate in an infinite time. We then
specialize to linear dilaton black holes, and outline the analytical computa-
tion of their radiation spectrum. For massive black holes, this computation
leads, in the high frequency regime, to the same temperature which is ob-
tained from the surface gravity. However in the case of massless extreme
black holes, we find that, although the surface gravity remains finite, there
is no radiation, in agreement with the fact that a massless object cannot
radiate.
EMD is defined by the following action
R− 2∂µφ∂µφ− e−2αφFµνFµν
, (1)
where Fµν is the electromagnetic field, and φ is the dilatonic field, with cou-
pling constant α. This theory admits static spherically symmetric solutions
representing black holes. Among these black hole solutions there are asymp-
totically flat ones [12, 13] as well as non-asymptotically flat configurations
[10, 11]. In the present work, we are interested in the non-asymptotically
flat black hole solutions
ds2 =
rγ(r − b)
dt2 −
rγ(r − b)
dr2 + r(r − b)dΩ2
, (2)
1 + γ
dr ∧ dt , e2αφ = ν2
. (3)
1− α2
1 + α2
. (4)
The constants b and r0 are related to the mass and to the electric charge of
the black hole through
M = (1− γ)b/4 , Q =
1 + γ
. (5)
The solutions (2),(3) interpolate between the Schwarzschild solution for
γ = −1 (α2 → ∞) and the Bertotti-Robinson solution for γ = +1 (α2 = 0).
For b > 0 the horizon at r = b hides the singularity at r = 0, while in the
extreme black hole case b = 0 the horizon coincides with the singularity.
This is a curious case, with vanishing mass but a finite electric charge. For
−1 < γ < 0 (α2 > 1) the central singularity is timelike and clearly naked
[11]. On the other hand, for 0 ≤ γ < 1 (0 < α2 ≤ 1), the central singularity
is null and marginally trapped [14], so that signals coming from the centre
never reach external observers. Thus in this case, extreme black holes can
be still considered as black holes indeed.
The statistical Hawking temperature of the black holes (2), computed as
usual by dividing the surface gravity by 2π is given by
. (6)
It is finite for all γ if b 6= 0. For b = 0 and −1 < γ < 0 (naked singularity).
the temperature is infinite, while for b = 0 and 0 < γ < 1 (extreme black
hole), the temperature vanishes.
The case b = γ = 0 is intriguing. Although this an extreme black
hole, the situation is different from that of asymptotically flat extreme black
holes. The near-horizon Euclidean extreme Reissner-Nordström geometry
is cylindrical, rather than conical, so that its statistical temperature is ar-
bitrary, contrary to the zero value derived from surface gravity [15]. In the
present case the two-dimensional Euclidean continuation of the metric (2)
with γ = 0 clearly has a conical singularity at r = b for all values of b,
including b = 0, leading for this particular extreme black hole to the finite
temperature TH = 1/4πr0, in agreement with the value (6). However this
result is questionable. A black hole with pointlike horizon and zero mass
clearly cannot radiate, so one should rather expect its temperature to be
zero. We will return to this question presently.
As black holes (2) radiate, they loose mass according to Stefan’s law
= −σAhT 4H , (7)
where σ is Stefan’s constant, and Ah = 4πr
1−γ is the horizon area.
Assuming that only electrically neutral quanta are radiated, (7) implies that
the horizon area decreases according to
(4π)3(1− γ)
−3(1+γ)
1+3γ , (8)
which is solved by
b(t) = r0
t− t0
)−1/3γ
(γ 6= 0) ,
b(t) = r0 exp
t− t0
(γ = 0) , (9)
where c = 3σ/16π3, and t0 is an integration constant. The outcome de-
pends on the sign of γ. For γ < 0, the Hawking temperature increases with
decreasing mass and the black hole collapses to a naked singularity (or evap-
orates away altogether in the Schwarzschild case γ = −1) in a finite time
according to b ∼ (t0 − t)1/3|γ|. On the other hand, for γ ≥ 0, the Hawking
temperature decreases (or is constant for γ = 0) with decreasing mass, and
the black hole evaporates in an infinite time, reaching the extreme black
hole state b = 0 only asymptotically.
We now proceed to a more precise evaluation of the temperature of non-
asymptotically flat black holes from the study of wave scattering in these
spacetimes. The wave equation
∇2φ = 0 (10)
does not generically allow for an exact solution in the spacetimes (2). How-
ever, it can be solved analytically [16] in the case of linear dilaton black
holes with γ = 0 and b 6= 0, with the metric
ds2 =
r − b
dt2 −
r − b
dr2 + r(r − b)dΩ2
, (11)
Considering the harmonic eigenmodes
φ(x) = ψ(r, t)Ylm(θ, ϕ) , ψ(r, t) = R(r)e
−iωt , (12)
we obtain the following radial equation:
r(r − b)∂rR
r − b
− l(l + 1)
R = 0 (13)
(ω̄2 ≡ ω2r20). Putting
, R = yiω̄f , (14)
reduces (13) to the equation
y(1−y)∂2yf+
1+2iω̄−2(1+ iω̄)y
ω̄2− iω̄− λ̄2−1/4
f = 0 , (15)
λ̄2 = ω̄2 − (l + 1/2)2 . (16)
This is a hypergeometric equation
y(1− y)∂2yf +
c− (a+ b+ 1)y
∂yf − abf = 0 , (17)
+ i(ω̄ + λ̄) , b =
+ i(ω̄ − λ̄) , c = 1 + 2iω̄ . (18)
It follows that the general solution of equation (13) is
R = C1
r − b
+ i(ω̄ + λ̄),
+ i(ω̄ − λ̄), 1 + 2iω̄; b− r
r − b
)−iω̄
− i(ω̄ + λ̄), 1
− i(ω̄ − λ̄), 1 − 2iω̄; b− r
.(19)
Putting
r − b
= ex/r0 , (20)
the partial wave near the horizon (x→ −∞) is thus
ψ ≃ C1eiω(x−t) +C2e−iω(x+t) . (21)
To obtain the behavior of the partial wave near spatial infinity, we must
expand the solutions of (15) in hypergeometric functions of argument 1/y.
The relevant transformation is
F (a, b, c; y) =
Γ(c)Γ(b− a)
Γ(b)Γ(c− a)
(−y)−aF (a, a+ 1− c, a+ 1− b; 1/y)
Γ(c)Γ(a − b)
Γ(a)Γ(c − b)
(−y)−bF (b, b+ 1− c, b+ 1− a; 1/y) . (22)
This leads to the asymptotic behavior
)−1/2(
i(λx−ωt) +B2e
−i(λx+ωt)
(λ = λ̄/r0), where the amplitudes of the asymptotic outgoing and ingoing
waves B1 and B2 are related to the amplitudes of the near-horizon outgoing
and ingoing waves C1 and C2 by
B1 = Γ(2iλ̄)
Γ(1 + 2iω̄)
Γ(1/2 + i(ω̄ + λ̄))2
Γ(1− 2iω̄)
Γ(1/2 − i(ω̄ − λ̄))2
B2 = Γ(−2iλ̄)
Γ(1 + 2iω̄)
Γ(1/2 + i(ω̄ − λ̄))2
Γ(1− 2iω̄)
Γ(1/2− i(ω̄ + λ̄))2
. (24)
Hawking radiation can be considered as the inverse process of scattering
by the black hole, with the asymptotic boundary condition B1 = 0 (the
outgoing mode is absent). The coefficient for reflection by the black hole is
then given by
|C1|2
|C2|2
|Γ(1/2 + i(ω̄ + λ̄))2|2
|Γ(1/2 + i(ω̄ − λ̄))2|2
cosh2 π(ω̄ − λ̄)
cosh2 π(ω̄ + λ̄)
. (25)
The resulting radiation spectrum is
= (eω/TH − 1)−1 . (26)
For high frequencies, λ̄ ≃ ω̄ = ω/r0, and we recover from (25) the Hawking
temperature as computed from the surface gravity,
. (27)
The above computation fails in the linear dilaton vacuum case b = 0.
The question of assigning a temperature to such massless black holes might
be evacuated by arguing that they cannot be formed, either through cen-
tral collapse of matter, or (as we have seen above) through evaporation of
massive black holes. Nevertheless, as a matter of principle one should con-
sider the possibility of primordial massless black holes. From the general
temperature law (6) these should have a finite temperature. On the other
hand, being massless they cannot radiate energy away, so their temperature
should vanish.
The question can be settled by solving the massless Klein-Gordon equa-
tion in the metric (11) with b = 0,
ds2 =
dt2 − r0
dr2 − r0rdΩ2 . (28)
This metric can be rewritten as
ds2 = Σ2
dτ2 − dx2 − dΩ2
, (29)
x = ln(r/r0) , τ = t/r0 , Σ = r0e
x/2 , (30)
showing that the linear dilaton vacuum metric is conformal to the product
M2 × S2 of a two-dimensional Minkowski spacetime with the two-sphere.
Performing also the redefinition
φ = Σ−1ψ , (31)
the Klein-Gordon equation (10) is reduced to
∇2φ = Σ−3
∂ττ − ∂xx −∇2Ω +
ψ = 0 , (32)
where ∇2Ω is the Laplacian operator on the two-sphere.
For a given spherical harmonic with orbital quantum number l, the re-
duced Klein-Gordon equation is thus
∇22ψl + (l + 1/2)2ψl = 0 , (33)
with ∇22 the Dalembertian operator on M2. Also, for a given spherical
harmonic the four-dimensional Klein-Gordon norm reduces to theM2 norm:
‖φ‖2 = 1
|g|g0µφ∗
∂µ φ =
dxψ∗l
∂τ ψl . (34)
Thus, the problem of wave propagation in the linear dilaton vacuum reduces
to the propagation of eigenmodes of a free Klein-Gordon field in two dimen-
sions, with effective mass µ = l+1/2. Clearly there is no reflection, so that
the linear dilaton vacuum does not radiate and hence its Hawking temper-
ature vanishes, contrary to the naive surface gravity value (6). A similar
reasoning holds in 2+1 dimensions for the BTZ vacuum [6] (M = L = 0),
which is conformal to M2 × S1.
We have shown that a complete analytical computation of the radia-
tion spectrum is possible for linear dilaton black hole solutions of EMD. For
massive black holes, this leads in the high frequency regime to a Planckian
distribution with a temperature independent of the black hole mass, in ac-
cordance with the surface gravity value. On the other hand, we find that
extreme, massless black holes do not radiate, thereby solving the paradox
presented by apparently hot (if the surface gravity temperature is taken
seriously) yet massless black holes.
Acknowledgements: J.C.F. thanks the LAPTH for the warm hospitality
during the elaboration of this work. He also thanks CNPq (Brazil) for
partial support. J.C.F. and G.T.M. thank the French-Brazilian scientific
cooperation CAPES/COFECUB for partial financial support.
References
[1] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space,
Cambridge University Press, Cambridge (1982).
[2] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.
[3] W.G. Unruh, Phys. Rev. D14 (1976) 870.
[4] D. Page, Phys. Rev. D13 (1976) 198.
[5] W.G. Unruh, Phys. Rev. D14 (1976) 3251.
[6] M. Bañados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992)
1849.
[7] K. Ghoroku and A.L. Larsen, Phys. Lett. B328 (1994) 28.
[8] M. Natsuume, N. Sakai and M. Sato, Mod. Phys. Lett. A11 (1996)
1467.
[9] D. Birmingham, I. Sachs and S. Sen, Phys. Lett. B413 (1997) 281.
[10] K.C.K. Chan, J.H. Horne and R.B. Mann, Nucl. Phys. B447 (1995)
[11] G. Clément and C. Leygnac, Phys. Rev. D70 (2004) 084018.
[12] G.W. Gibbons and K. Maeda, Nucl. Phys. B298 (1988) 741.
[13] D. Garfinkle, G.T. Horowitz and A. Strominger, Phys. Rev. D43 (1991)
3140.
[14] S.A. Hayward, Class. Quantum Grav. 17 (2000) 4021.
[15] S.W. Hawking, G.T. Horowitz and S.F. Ross, Phys. Rev. D51 (1995)
4302.
[16] G. Clément, D. Gal’tsov and C. Leygnac, Phys. Rev. D67 (2003)
024012.

  We compute exactly the semi-classical radiation spectrum for a class of
non-asymptotically flat charged dilaton black holes, the so-called linear
dilaton black holes. In the high frequency regime, the temperature for these
black holes generically agrees with the surface gravity result. In the special
case where the black hole is massless, we show that, although the surface
gravity remains finite, there is no radiation, in agreement with the fact that
massless objects cannot radiate.

LAPTH-1178/07
Hawking radiation of linear dilaton black holes
G. Clémenta∗, J.C. Fabrisb†and G.T. Marquesa,b‡
aLaboratoire de Physique Théorique LAPTH (CNRS),
B.P.110, F-74941 Annecy-le-Vieux cedex, France
b Departamento de F́ısica, Universidade Federal do Esṕırito Santo,
Vitória, 29060-900, Esṕırito Santo, Brazil
April 3, 2007
Abstract
We compute exactly the semi-classical radiation spectrum for a
class of non-asymptotically flat charged dilaton black holes, the so-
called linear dilaton black holes. In the high frequency regime, the
temperature for these black holes generically agrees with the surface
gravity result. In the special case where the black hole is massless,
we show that, although the surface gravity remains finite, there is
no radiation, in agreement with the fact that massless objects cannot
radiate.
e-mail: gclement@lapp.in2p3.fr
e-mail: fabris@cce.ufes.br
e-mail:gtadaiesky@cce.ufes.br
http://arxiv.org/abs/0704.0399v1
Quantum field theory in curved spacetime predicts new phenomena such
as particle emission by a black hole [1]. This is due to the fact that the vac-
uum for a quantum field near the horizon is different from the observer’s
vacuum at spatial infinity. A distant observer thus receives from a black
hole a steady flux of particles exhibiting, in the high frequency regime, a
black body spectrum with a temperature proportional to the surface grav-
ity [2]. Although Hawking’s original derivation of this black hole evaporation
dealt with realistic collapsing black holes, Unruh [3] showed that the same
results are obtained when the collapse is replaced by appropriate boundary
conditions on the horizon of an eternal black hole. In the semi-classical ap-
proximation, the black hole radiation spectrum may be evaluated by com-
puting the Bogoliubov coefficients relating the two vacua. An equivalent
procedure is to compute the reflection and absorption coefficients of a wave
by the black hole. Usually, the wave equation cannot be solved exactly,
and one must resort to match solutions in an overlap region between the
near-horizon and asymptotic regions [4, 5]. In the special case of the (2+1)-
dimensional BTZ black hole [6], an exact solution of the wave equation is
available, which allows for an exact computation of the radiation spectrum,
leading to the Hawking temperature [7, 8, 9].
In this Letter, we discuss another case of black holes also allowing for
an exact semi-classical computation of their radiation spectrum, that of lin-
ear dilaton black hole solutions to Einstein-Maxwell dilaton (EMD) theory
in four dimensions. Linear dilaton black holes are a special case of the
more general class of non-asymptotically flat black hole solutions to EMD
[10, 11], which we first briefly present. We discuss the evaporation of these
non-asymptotically flat black holes and show that they either collapse to a
naked singularity in a finite time, or evaporate in an infinite time. We then
specialize to linear dilaton black holes, and outline the analytical computa-
tion of their radiation spectrum. For massive black holes, this computation
leads, in the high frequency regime, to the same temperature which is ob-
tained from the surface gravity. However in the case of massless extreme
black holes, we find that, although the surface gravity remains finite, there
is no radiation, in agreement with the fact that a massless object cannot
radiate.
EMD is defined by the following action
R− 2∂µφ∂µφ− e−2αφFµνFµν
, (1)
where Fµν is the electromagnetic field, and φ is the dilatonic field, with cou-
pling constant α. This theory admits static spherically symmetric solutions
representing black holes. Among these black hole solutions there are asymp-
totically flat ones [12, 13] as well as non-asymptotically flat configurations
[10, 11]. In the present work, we are interested in the non-asymptotically
flat black hole solutions
ds2 =
rγ(r − b)
dt2 −
rγ(r − b)
dr2 + r(r − b)dΩ2
, (2)
1 + γ
dr ∧ dt , e2αφ = ν2
. (3)
1− α2
1 + α2
. (4)
The constants b and r0 are related to the mass and to the electric charge of
the black hole through
M = (1− γ)b/4 , Q =
1 + γ
. (5)
The solutions (2),(3) interpolate between the Schwarzschild solution for
γ = −1 (α2 → ∞) and the Bertotti-Robinson solution for γ = +1 (α2 = 0).
For b > 0 the horizon at r = b hides the singularity at r = 0, while in the
extreme black hole case b = 0 the horizon coincides with the singularity.
This is a curious case, with vanishing mass but a finite electric charge. For
−1 < γ < 0 (α2 > 1) the central singularity is timelike and clearly naked
[11]. On the other hand, for 0 ≤ γ < 1 (0 < α2 ≤ 1), the central singularity
is null and marginally trapped [14], so that signals coming from the centre
never reach external observers. Thus in this case, extreme black holes can
be still considered as black holes indeed.
The statistical Hawking temperature of the black holes (2), computed as
usual by dividing the surface gravity by 2π is given by
. (6)
It is finite for all γ if b 6= 0. For b = 0 and −1 < γ < 0 (naked singularity).
the temperature is infinite, while for b = 0 and 0 < γ < 1 (extreme black
hole), the temperature vanishes.
The case b = γ = 0 is intriguing. Although this an extreme black
hole, the situation is different from that of asymptotically flat extreme black
holes. The near-horizon Euclidean extreme Reissner-Nordström geometry
is cylindrical, rather than conical, so that its statistical temperature is ar-
bitrary, contrary to the zero value derived from surface gravity [15]. In the
present case the two-dimensional Euclidean continuation of the metric (2)
with γ = 0 clearly has a conical singularity at r = b for all values of b,
including b = 0, leading for this particular extreme black hole to the finite
temperature TH = 1/4πr0, in agreement with the value (6). However this
result is questionable. A black hole with pointlike horizon and zero mass
clearly cannot radiate, so one should rather expect its temperature to be
zero. We will return to this question presently.
As black holes (2) radiate, they loose mass according to Stefan’s law
= −σAhT 4H , (7)
where σ is Stefan’s constant, and Ah = 4πr
1−γ is the horizon area.
Assuming that only electrically neutral quanta are radiated, (7) implies that
the horizon area decreases according to
(4π)3(1− γ)
−3(1+γ)
1+3γ , (8)
which is solved by
b(t) = r0
t− t0
)−1/3γ
(γ 6= 0) ,
b(t) = r0 exp
t− t0
(γ = 0) , (9)
where c = 3σ/16π3, and t0 is an integration constant. The outcome de-
pends on the sign of γ. For γ < 0, the Hawking temperature increases with
decreasing mass and the black hole collapses to a naked singularity (or evap-
orates away altogether in the Schwarzschild case γ = −1) in a finite time
according to b ∼ (t0 − t)1/3|γ|. On the other hand, for γ ≥ 0, the Hawking
temperature decreases (or is constant for γ = 0) with decreasing mass, and
the black hole evaporates in an infinite time, reaching the extreme black
hole state b = 0 only asymptotically.
We now proceed to a more precise evaluation of the temperature of non-
asymptotically flat black holes from the study of wave scattering in these
spacetimes. The wave equation
∇2φ = 0 (10)
does not generically allow for an exact solution in the spacetimes (2). How-
ever, it can be solved analytically [16] in the case of linear dilaton black
holes with γ = 0 and b 6= 0, with the metric
ds2 =
r − b
dt2 −
r − b
dr2 + r(r − b)dΩ2
, (11)
Considering the harmonic eigenmodes
φ(x) = ψ(r, t)Ylm(θ, ϕ) , ψ(r, t) = R(r)e
−iωt , (12)
we obtain the following radial equation:
r(r − b)∂rR
r − b
− l(l + 1)
R = 0 (13)
(ω̄2 ≡ ω2r20). Putting
, R = yiω̄f , (14)
reduces (13) to the equation
y(1−y)∂2yf+
1+2iω̄−2(1+ iω̄)y
ω̄2− iω̄− λ̄2−1/4
f = 0 , (15)
λ̄2 = ω̄2 − (l + 1/2)2 . (16)
This is a hypergeometric equation
y(1− y)∂2yf +
c− (a+ b+ 1)y
∂yf − abf = 0 , (17)
+ i(ω̄ + λ̄) , b =
+ i(ω̄ − λ̄) , c = 1 + 2iω̄ . (18)
It follows that the general solution of equation (13) is
R = C1
r − b
+ i(ω̄ + λ̄),
+ i(ω̄ − λ̄), 1 + 2iω̄; b− r
r − b
)−iω̄
− i(ω̄ + λ̄), 1
− i(ω̄ − λ̄), 1 − 2iω̄; b− r
.(19)
Putting
r − b
= ex/r0 , (20)
the partial wave near the horizon (x→ −∞) is thus
ψ ≃ C1eiω(x−t) +C2e−iω(x+t) . (21)
To obtain the behavior of the partial wave near spatial infinity, we must
expand the solutions of (15) in hypergeometric functions of argument 1/y.
The relevant transformation is
F (a, b, c; y) =
Γ(c)Γ(b− a)
Γ(b)Γ(c− a)
(−y)−aF (a, a+ 1− c, a+ 1− b; 1/y)
Γ(c)Γ(a − b)
Γ(a)Γ(c − b)
(−y)−bF (b, b+ 1− c, b+ 1− a; 1/y) . (22)
This leads to the asymptotic behavior
)−1/2(
i(λx−ωt) +B2e
−i(λx+ωt)
(λ = λ̄/r0), where the amplitudes of the asymptotic outgoing and ingoing
waves B1 and B2 are related to the amplitudes of the near-horizon outgoing
and ingoing waves C1 and C2 by
B1 = Γ(2iλ̄)
Γ(1 + 2iω̄)
Γ(1/2 + i(ω̄ + λ̄))2
Γ(1− 2iω̄)
Γ(1/2 − i(ω̄ − λ̄))2
B2 = Γ(−2iλ̄)
Γ(1 + 2iω̄)
Γ(1/2 + i(ω̄ − λ̄))2
Γ(1− 2iω̄)
Γ(1/2− i(ω̄ + λ̄))2
. (24)
Hawking radiation can be considered as the inverse process of scattering
by the black hole, with the asymptotic boundary condition B1 = 0 (the
outgoing mode is absent). The coefficient for reflection by the black hole is
then given by
|C1|2
|C2|2
|Γ(1/2 + i(ω̄ + λ̄))2|2
|Γ(1/2 + i(ω̄ − λ̄))2|2
cosh2 π(ω̄ − λ̄)
cosh2 π(ω̄ + λ̄)
. (25)
The resulting radiation spectrum is
= (eω/TH − 1)−1 . (26)
For high frequencies, λ̄ ≃ ω̄ = ω/r0, and we recover from (25) the Hawking
temperature as computed from the surface gravity,
. (27)
The above computation fails in the linear dilaton vacuum case b = 0.
The question of assigning a temperature to such massless black holes might
be evacuated by arguing that they cannot be formed, either through cen-
tral collapse of matter, or (as we have seen above) through evaporation of
massive black holes. Nevertheless, as a matter of principle one should con-
sider the possibility of primordial massless black holes. From the general
temperature law (6) these should have a finite temperature. On the other
hand, being massless they cannot radiate energy away, so their temperature
should vanish.
The question can be settled by solving the massless Klein-Gordon equa-
tion in the metric (11) with b = 0,
ds2 =
dt2 − r0
dr2 − r0rdΩ2 . (28)
This metric can be rewritten as
ds2 = Σ2
dτ2 − dx2 − dΩ2
, (29)
x = ln(r/r0) , τ = t/r0 , Σ = r0e
x/2 , (30)
showing that the linear dilaton vacuum metric is conformal to the product
M2 × S2 of a two-dimensional Minkowski spacetime with the two-sphere.
Performing also the redefinition
φ = Σ−1ψ , (31)
the Klein-Gordon equation (10) is reduced to
∇2φ = Σ−3
∂ττ − ∂xx −∇2Ω +
ψ = 0 , (32)
where ∇2Ω is the Laplacian operator on the two-sphere.
For a given spherical harmonic with orbital quantum number l, the re-
duced Klein-Gordon equation is thus
∇22ψl + (l + 1/2)2ψl = 0 , (33)
with ∇22 the Dalembertian operator on M2. Also, for a given spherical
harmonic the four-dimensional Klein-Gordon norm reduces to theM2 norm:
‖φ‖2 = 1
|g|g0µφ∗
∂µ φ =
dxψ∗l
∂τ ψl . (34)
Thus, the problem of wave propagation in the linear dilaton vacuum reduces
to the propagation of eigenmodes of a free Klein-Gordon field in two dimen-
sions, with effective mass µ = l+1/2. Clearly there is no reflection, so that
the linear dilaton vacuum does not radiate and hence its Hawking temper-
ature vanishes, contrary to the naive surface gravity value (6). A similar
reasoning holds in 2+1 dimensions for the BTZ vacuum [6] (M = L = 0),
which is conformal to M2 × S1.
We have shown that a complete analytical computation of the radia-
tion spectrum is possible for linear dilaton black hole solutions of EMD. For
massive black holes, this leads in the high frequency regime to a Planckian
distribution with a temperature independent of the black hole mass, in ac-
cordance with the surface gravity value. On the other hand, we find that
extreme, massless black holes do not radiate, thereby solving the paradox
presented by apparently hot (if the surface gravity temperature is taken
seriously) yet massless black holes.
Acknowledgements: J.C.F. thanks the LAPTH for the warm hospitality
during the elaboration of this work. He also thanks CNPq (Brazil) for
partial support. J.C.F. and G.T.M. thank the French-Brazilian scientific
cooperation CAPES/COFECUB for partial financial support.
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