On the Entropy Function and the Attractor Mechanism for Spherically
  Symmetric Extremal Black Holes
On the Entropy Function and the Attractor Mechanism for Spherically
Symmetric Extremal Black Holes
Rong-Gen Cai∗
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
Li-Ming Cao†
Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China
Graduate School of the Chinese Academy of Sciences, Beijing 100039, China
In this paper we elaborate on the relation between the entropy formula of Wald and the
“entropy function” method proposed by A. Sen. For spherically symmetric extremal black
holes, it is shown that the expression of extremal black hole entropy given by A. Sen can
be derived from the general entropy definition of Wald, without help of the treatment of
rescaling the AdS2 part of near horizon geometry of extremal black holes. In our procedure,
we only require that the surface gravity approaches to zero, and it is easy to understand the
Legendre transformation of f , the integration of Lagrangian density on the horizon, with
respect to the electric charges. Since the Noether charge form can be defined in an “off-shell”
form, we define a corresponding entropy function, with which one can discuss the attractor
mechanism for extremal black holes with scalar fields.
e-mail address: cairg@itp.ac.cn
e-mail address: caolm@itp.ac.cn
http://arxiv.org/abs/0704.1239v4
I. INTRODUCTION
The attractor mechanism for extremal black holes has been studied extensively in the past few
years in supergravity theory and superstring theory. It was initiated in the context supersymmetric
BPS black holes [1, 2, 3, 4, 5, 6] and generalized to more general cases, such as supersymmetric
black holes with higher order corrections [7, 8, 9, 10] and non-supersymmetric attractors [11, 12,
13, 14, 15].
Recently, A. Sen has proposed a so-called “entropy function” method for calculating the entropy
of n-dimensional extremal black holes, where the extremal black holes are defined to be the space-
times which have the near horizon geometry AdS2 × Sn−2 and corresponding isometry [16, 17, 18,
19]. It states that the entropy of such kind of extremal black holes can be obtained by extremizing
the “entropy function” with respect to some moduli on the horizon, where the entropy function
is defined as 2π times the Legendre transformation ( with respect to the electric charges ) of the
integration of the Lagrangian over the spherical coordinates on the horizon in the near horizon
field configurations. This method does not depend upon supersymmetry and has been applied
or generalized to many solutions in supergravity theory, such as extremal black objects in higher
dimensions, rotating extremal black holes, various non-supersymmetric extremal black objects and
even near-extremal black holes [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41].
In general, for spherically symmetric extremal black holes in a theory with Lagrangian L =
L(gab, Rabcd,Φs, AIa), the near horizon geometry of these black holes has the form AdS2 × Sn−2
[17, 18]. Due to SO(1, 2) × SO(n − 1) isometry of this geometry, the field configuration have the
form as follows: The metric can be written down as
ds2 = gabdx
adxb = v1
−ρ2dτ2 + 1
+ v2dΩ
n−2 , (1.1)
where v1, v2 are constants which stand for the sizes of AdS2 and S
n−2. Some other dynamical fields
such as the scalar fields and U(1) gauge fields are also taken to be constant: Φs = us and F
ρτ = eI .
The magnetic-type fields are also fixed with magnetic-charges pi. Then, for this configuration,
defining
f(v1, v2, us, eI ; pi) =
dx2 ∧ · · · ∧ dxn−1
−gL , (1.2)
where the integration is taken on the horizon, and {x2 · · · xn−1} are angle coordinates of Sn−2,
those constant moduli can be fixed via the equations of motion
= qI , (1.3)
where qI are electrical-like charges for U(1) gauge fields A
a. To relate the entropy of the black
holes to these definitions, one defines fλ as (1.2) with the Riemann tensor part in L multiplied by a
factor λ, and then one finds a relation between fλ and the Wald formula for spherically symmetric
black holes [44]: SBH = −2π∂fλ/∂λ|λ=1. Consider the structure of the Lagrangian, one can find
− fλ = 0 . (1.4)
When the equations of motion are satisfied, the entropy of black holes turns out to be SBH =
2π(eIqI − f).
Therefore, one can introduce the “entropy function” for the extremal black holes
E(v1, v2, us, eI ; pi) = 2π (eIqI − f(v1, v2, us, eI ; pi)) , (1.5)
which is obtained by carrying an integral of the Lagrangian density over Sn−2 and then taking
the Legendre transformation with respect to the electric fields eI . For fixed electric changes qI
and magnetic charges pi, these fields us and v1 and v2 are determined by extremizing the entropy
function with respect to the variables us and v1 and v2. And then the entropy of the extremal black
holes is given by the extremum of the entropy function by substituting the values of v1, v2 and
us back into the entropy function. In addition, let us notice that if the moduli fields us are only
dependent of the charges qI and pi, the attractor mechanism is then manifested, and the entropy
is a topological quantity.
This is a very simple and powerful method for calculating the entropy of such kind of extremal
black holes. In particular, one can easily find the corrections to the entropy due to the higher
derivative terms in the effective action. However, we notice that this method is established in a
fixed coordinate system (1.1). If one uses another set of coordinates for the AdS2 part, instead
of the coordinates {ρ, τ}, it seems that one can not define an entropy function as (1.5) because
the function f is not invariant under the coordinate transformation. In addition, the reason that
to get the entropy of black holes, one should do the Legendre transformation with respect to the
electric charges, but not include magnetic charges seems unclear in this procedure. Some authors
have pointed out that the entropy function E resulting from this Legendre transformation of the
function f with respect to electric charges transforms as a function under the electric-magnetic dual,
while the function f does not [37]. But it is not easy to understand the Legendre transformation
with respect to the angular-momentum J in the rotating attractor cases [32]. There might be a
more general formalism for the entropy function, and the Legendre transformation can be naturally
understood in this frame. In this paper, we will elaborate these issues in the “entropy function”
method and show that a general formalism of the “entropy function” method can be extracted
from the black hole entropy definition due to Wald et al. [44, 45, 46]. In this procedure, we only
require that the surface gravity of the black hole approaches to zero. Our entropy expression will
reduce to the expression of A. Sen if we choose the same coordinates as in [17, 18].
The extremal black holes are different objects from the non-extremal ones due to different
topological structures in Euclidean sector [49, 50, 51]. The extremal black hole has vanishing
surface gravity and has no bifurcation surface, so the Noether charge method of Wald can not be
directly used [44]. Thus, in this paper we regard the extremal black holes as the extremal limit of
non-extremal black holes as in [17, 18, 42]. That is, we will first consider non-extremal black holes
and then take the extremal limit. In this sense, the definitions of Wald are applicable.
The paper is organized as follows. In section II, we make a brief review on the entropy definition
of Wald and give the required formulas. In section III, we give the near horizon analysis for the
extremal black holes and derive the general form of the entropy. In section IV, we define the entropy
function and discuss the attractor mechanism for the black holes with various moduli fields. The
conclusion and discussion are given in section V.
II. THE DEFINITION OF WALD
In differential covariant theories of gravity, Wald showed that the entropy of a black hole is a
kind of Noether charge [44, 45]. In this paper, we will use the Wald’s method to define the entropy
functions for spherically symmetric black holes. Assume the differential covariant Lagrangian of
n-dimensional space-times (M,gab) is
L = L(gab, Rabcd,Φs, AIa) ǫ, (2.1)
where we have put the Lagrangian in the form of differential form and ǫ is the volume element.
Rabcd is Riemann tensor (since we are mainly concerning with extremal black holes, therefore
we need not consider the covariant derivative of the Riemann tensor). {Φs, s = 0, 1, · · · } are
scalar fields, {AIa, I = 1, · · · } are U(1) gauge potentials, and the corresponding gauge fields are
= ∂aA
− ∂bAIa. We will not consider the Chern-Simons term as [18].
The variation of the Lagrange density L can be written as
δL = Eψδψ + dΘ, (2.2)
where Θ = Θ(ψ, δψ) is an (n− 1)-form, which is called symplectic potential form, and it is a local
linear function of field variation (we have denoted the dynamical fields as ψ = {gab,Φs, AIa}). Eψ
corresponds to the equations of motion for the metric and other fields. Let ξ be any smooth vector
field on the space-time manifold, then one can define a Noether current form as
J[ξ] = Θ(ψ,Lξψ)− ξ · L , (2.3)
where “ · ” means the inner product of a vector field with a differential form, while Lξ denotes the
Lie derivative for the dynamical fields. A standard calculation gives
dJ[ξ] = −ELξψ . (2.4)
It implies that J[ξ] is closed when the equations of motion are satisfied. This indicates that there
is a locally constructed (n − 2)-form Q[ξ] such that, whenever ψ satisfy the equations of motion,
we have
J[ξ] = dQ[ξ] . (2.5)
In fact, the Noether charge form Q[ξ] can be defined in the so-called “off shell” form so that the
Noether current (n− 1)-form can be written as [46]
J[ξ] = dQ[ξ] + ξaCa , (2.6)
where Ca is locally constructed out of the dynamical fields in a covariant manner. When the
equations of motion hold, Ca vanishes. For general stationary black holes, Wald has shown that
the entropy of the black holes is a Noether charge [44], and may be expressed as
SBH = 2π
Q[ξ] , (2.7)
here ξ be the Killing field which vanishes on the bifurcation surface of the black hole. It should be
noted that the Killing vector field has been normalized here so that the surface gravity equals to
“1”. Furthermore, it was shown in [45] that the entropy can also be put into a form
SBH = −2π
EabcdR ǫabǫcd, (2.8)
where ǫab is the binormal to the bifurcation surface H, while EabcdR is the functional derivative of
the Lagrangian with respect to the Riemann tensor with metric held fixed. This formula is purely
geometric and does not include the surface gravity term. In this paper, since we will treat a limit
procedure with surface gravity approaching to zero, we will not normalize the Killing vector such
that the surface gravity equal to one. So we use the formula (2.8) to define the entropy of black
holes as in [17, 18, 42]. For an asymptotically flat, static spherically symmetric black hole, one can
simply choose ξ = ∂t =
For the Lagrangian as (2.1), we have
δL = Eabδgab +E
sδΦs + dΘ , (2.9)
where
I = −2ǫ∇b
, (2.10)
s = ǫ
∂∇aΦs
, (2.11)
ab = ǫ
gabL+ ∂L
∂Rcdea
b + 2∇c∇d
∂Rcabd
(2.12)
are the equations of motion for the U(1) gauge fields, the scalar fields and the metric gab, respec-
tively. The symplectic potential form has the form
Θa1···an−1 =
∂∇aΦs
δΦs + 2
∂Rabcd
∇dδgbc − 2∇d
∂Rdbca
ǫaa1···an−1 . (2.13)
Let ξ be an arbitrary vector field on the space-time, The Lie derivative of ξ on the fields are
LξΦs = ξa∇aΦs, Lξgab = ∇aξb +∇bξa , LξAIa = ∇a(ξbAIb) + ξbF Iba . (2.14)
Substituting these Lie derivatives into the symplectic potential form, we find
Θa1···an−1 =
∂∇aΦs
ξb∇bΦs + 2
∇b(ξcAIc) + 2
ξcF Icb
∂Rabcd
∇d(∇bξc +∇cξb)− 2∇d
∂Rdbca
(∇bξc +∇cξb)
ǫaa1···an−1
∂∇aΦs
ξb∇bΦs + 2∇b
ξcAIc
− 2∇b
ξcAIc + 2
ξcF Icb
∂Rabcd
∇d(∇bξc +∇cξb)− 2∇d
∂Rdbca
(∇bξc +∇cξb)
ǫaa1···an−1 . (2.15)
Then, we have
Θa1···an−1 =
ξcAIc
∂Rabcd
∇[cξd]
ǫaa1···an−1 + · · ·
∂∇aΦs
ξb∇bΦs + 2
ξcF Icb + · · · · · ·
ǫaa1···an−1
− 2∇b
ξcAIcǫaa1···an−1 . (2.16)
The first line in the above equation will give the Noether charge form, while the second line together
with the terms in ξ · L in Eq. (2.3) will give the constraint which corresponds to the equations
of motion for the metric. For example, the first term in the second line combined with scalar
fields terms in ξ · L will give the energy-momentum tensor for scalar fields. Similarly the second
term in the second line will enter the energy-momentum tensor for the U(1) gauge fields in the
equations of motion for the metric. The last line in the above equation will give the constraint
which corresponds to the equations of motion for the U(1) gauge fields. Thus, we find
J[ξ] = dQ[ξ] + ξaCa , (2.17)
where
Q = QF +Qg + · · · (2.18)
a1···an−2
ξcAIcǫaba1···an−2 , (2.19)
a1···an−2
= − ∂L
∂Rabcd
∇[cξd]ǫaba1···an−2 . (2.20)
The “ · · · ” terms are not important for our following discussion, so we brutally drop them at first.
We will give a discussion at the end of the next section for these additional terms. Especially, the
constraint for the U(1) gauge fields is simply
ca1···an−1
= −2∇b
AIcǫaa1···an−1 . (2.21)
The term QF in the Q was not discussed explicitly in the earlier works of Wald et al. [44, 45, 46].
This is because that the killing vector vanishes on the bifurcation surface and the dynamical fields
are assumed to be smooth on the bifurcation surface. However, in general, the U(1) gauge fields are
singular on the bifurcation surface, so one have to do a gauge transformation, A→ A′ = A−A|H,
such that the ξaA′a are vanished on this surface, and then Q
F . This gauge transformation will
modify the data of gauge potential at infinity and an additional potential-charge term ΦδQ into the
dynamics of the charged black holes from infinity, where Φ = ξcAc|H is the electrostatic potential
on the horizon of the charged black hole and Q is the electric charge [47]. Another treatment is: We
only require the smoothness of the gauge potential projecting on the bifurcation surface, i.e., ξaAa
instead of the gauge potential itself, so QF will generally not vanish on the bifurcation surface, and
then Φ = ξcAc|H is introduced into the law of black hole without help of gauge transformation [48].
Similarly, in the next sections of this paper we only require that the projection of the gauge potential
on the bifurcation surface is smooth. Since our final result will not depend on the gauge potential,
the gauge transformation mentioned above will not effect our discussion. One can do such gauge
transformation if necessary. In this paper, however, we will merely use the explicit form of the
Noether charge (n − 2)-form and we will not discuss the first law. Certainly, it is interesting to
give a general discussion on the thermodynamics of these black holes. The relevant discussion can
be found in a recent paper [43].
III. ENTROPY OF EXTREMAL BLACK HOLES
In this section, we will use the formulas above to give the general entropy function for static
spherically symmetric extremal black holes. Assume that the metric for these black holes is of the
ds2 = −N(r)dt2 +
dr2 + γ(r)dΩ2n−2 , (3.1)
where N, γ are functions of radial coordinate r, and dΩ2n−2 is the line element for the (n − 2)-
dimensional sphere. The horizon r = rH corresponds to N(rH) = 0. If the equations of motion
are satisfied, the constraint Ca = 0, and we have
J[ξ] = dQ[ξ] .
Consider a near horizon region ranged from rH to rH +∆r, we have
rH+∆r
Q[ξ]−
Q[ξ] =
Θ− ξ · L . (3.2)
If ξ is a Killing vector, then Θ = 0, and
rH+∆r
Q[ξ]−
Q[ξ] = −
ξ · L . (3.3)
Thus we arrive at
rH+∆r
g[ξ]−
rH+∆r
F [ξ] +
F [ξ]−
ξ · L . (3.4)
Taking ξ = ∂t, (since we consider the asymptotically flat space-time, N(r) has the property
limr→∞N(r) = 1, such that ∂t has a unit norm at infinity.), we have ∇[aξb] = 12N
ǫab, and
rH+∆r
g[∂t]−
g[∂t]
N ′(rH +∆r)B(rH +∆r)−N ′(rH)B(rH)
N ′′(rH)B(rH) +N
′(rH)B
′(rH)
+O(∆r2) , (3.5)
where we have defined a function B(r)
B(r) ≡ −
(n− 2)!
∂Rabcd
ǫcdǫaba1···an−2dx
a1 ∧ · · · ∧ dxan−2 . (3.6)
Note that the QF terms in the right hand side of Eq. (3.4) can be written as
rH+∆r
F [∂t] +
F [∂t]
= AIt (rH +∆r)qI −AIt (rH)qI
= qIA
t (rH)∆r +O(△r2)
= qIF
rt(rH)∆r +O(△r2) = qI ẽI∆r +O(△r2) , (3.7)
where AIt = (∂t)
aAIa, ẽI ≡ F Irt(rH), and the U(1) electrical-like charges are defined to be
qI = −
(n− 2)!
ǫaba1···an−2dx
a1 ∧ · · · ∧ dxan−2 . (3.8)
They do not change with the radii r. This is ensured by the Gaussian law. Note that there is an
integration on the sphere part in (3.8), therefore the only F Irt in F
is relevant, so that we can
simply write F I
(rH) as −ẽIǫab. Considering −2ẽI2 = ẽIǫabẽIǫab we have
= − ∂L
∂(ẽIǫab)
ab . (3.9)
Substituting this result into the definition of the electric charges, we find
qI = −
2(n− 2)!
ǫaba1···an−2dx
a1 ∧ · · · ∧ dxan−2 = ∂f̃(rH)
. (3.10)
Here f̃(rH) will be defined below in Eq. (3.12). The last term in the right hand side of Eq. (3.4)
can be written as
∂t · L =
∫ rH+∆r
dx2 ∧ · · · ∧ dxn−1
−gL =
∫ rH+∆r
drf̃(r) , (3.11)
where
f̃(r) =
dx2 ∧ · · · ∧ dxn−1
−gL . (3.12)
Thus we arrive at
∂t · L = ∆rf̃(rH) +O(△r2) , (3.13)
up to the leading order of △r. Substituting Eqs. (3.5), (3.7) and (3.13) into Eq. (3.4), we get
N ′′(rH)B(rH) +N
′(rH)B
′(rH)
+O(∆r2)
= ∆rqI ẽI −∆rf̃(rH) . (3.14)
Considering the limit ∆r → 0, we find
N ′′(rH)B(rH) +N
′(rH)B
′(rH)
= qI ẽI − f̃(rH) . (3.15)
So far, we have not specialized to extremal black holes; therefore, the above results hold for general
non-extremal black holes. For the extremal black holes limit with N ′(rH) → 0, while N ′′(rH) 6= 0,
from (3.15) we have
B(rH) =
N ′′(rH)
qI ẽI − f̃(rH)
. (3.16)
Since we view the extremal black holes as the extremal limit of non-extremal black holes, the
entropy formula of Wald is applicable for the extremal black holes. Note that B(rH) is nothing
but the integration in Eq. (2.8) without the 2π factor. Thus, the entropy of the extremal black
holes can be expressed as
SBH = 2πB(rH) =
N ′′(rH)
qI ẽI − f̃(rH)
. (3.17)
This is one of main results in this paper. It is easy to see that this entropy form is very similar to
the one in the “entropy function” method of A. Sen. But some remarks are in order:
(i). We have not stressed that the extremal black holes have the near horizon geometry
AdS2 ×Sn−2 as in [17, 18] although the vanishing surface gravity and the metric assumption (3.1)
may coincide with the definition through the near horizon geometry. However, let us notice that
some extremal black holes have near horizon geometries of the form AdS3 products some compact
manifold X. In our procedure, the near horizon geometry is not necessary to be AdS2 × Sn−2 and
the only requirement is to have vanishing surface gravity. Therefore our procedure can be used to
discuss that kind of extremal black holes whose near horizon geometry is of the form AdS3 ×X by
simply modifying the metric assumption in Eq.(3.1).
(ii). Our result is explicitly invariant under coordinate transformation, and this can be easily
seen from the above process. We have not used the treatment method Eq.(1.4) employed by A.
(iii). The Legendre transformation with respect to the electric charges appears naturally in
this procedure, while the Legendre transformation with respect to the magnetic charges does not
appear.
(iv). If we choose a set of coordinates as the one in [17, 18], our expression for the entropy
is exactly same as the one given by A. Sen. This can be seen as follows. In the extremal limit
N ′(rH) = 0, we can rewrite the metric near the horizon as
ds2 = −
N ′′(rH)(r − rH)2dt2 +
N ′′(rH)(r − rH)2
dr2 + γ(rH)dΩ
n−2 . (3.18)
Redefine the coordinates as
ρ = r − rH , τ =
N ′′(rH)t . (3.19)
Then, the near horizon metric can be further rewritten as
ds2 =
N ′′(rH)
−ρ2dτ2 +
+ γ(rH)dΩ
n−2 . (3.20)
The components of gauge fields F Irt and f̃ are dependent of coordinates, in this new set of coordi-
nates they are
ẽI =
N ′′(rH)eI , (3.21)
f̃(rH) =
N ′′(rH)f . (3.22)
where
eI = F
ρτ (rH), f =
dx2 ∧ · · · ∧ dxn−1
−g′L . (3.23)
Since the entropy is invariant under the coordinate transformation, we find in these coordinates
like {τ, ρ, · · · },
SBH = 2π (qIeI − f) . (3.24)
This is nothing but the entropy formula given by A. Sen for extremal black holes. Since the factor
2/N ′′(rH) in (3.17) disappears in this new set of coordinates, the entropy formula becomes more
simple and good look. This is an advantage of this set of coordinates. But we would like to
stress that the entropy expression with the factor “2/N ′′(rH)” makes it invariant under coordinate
transformation.
(v). Finally the function f̃(rH) is evaluated for the solution of the equations of motion, i.e. all
the fields: {gab,Φs, F Iab} are on shell. For example, if the near horizon geometry has the form
ds2 = v1(−ρ2dτ2 +
dρ2) + v2dΩ
n−2 , (3.25)
and the equations of motion are satisfied, then we can express the entropy in the form (3.24).
There v1 and v2 should equal to 2/N
′′(rH) and γ(rH). N , γ, and other fields, should satisfy the
equations of motion.
One may worry about that the conserved charge form Q in Eq.(2.18) is not complete: For
example, we will have an additional term ǫaba1···an−2ξ
a∇bD(φ) if the action has a dilaton coupling
termD(φ)R. In general, the conserved charge form can be written asQ = QF+Qg+ξaWa+Y+dZ,
where Wa, Y and Z are smooth functions of fields and their derivatives, and Y = Y(ψ,Lξψ) is
linear for the field variation [45, 46]. Obviously, Y and dZ will not give contributions to the
near horizon integration (3.2) if ξ is a killing vector. It seems that ξaWa will give an additional
contribution to this integration. For the extremal case, this contribution will vanish due to the
smoothness of Wa and the vanishing surface gravity. For example, the term corresponding to the
dilaton coupling mentioned above will vanish in the near horizon integration. So the final form
of the entropy (3.17) will not change. For the non-extremal case, this term essentially appear in
the near horizon integration if we add the ξaWa into Q. However, if necessary, we can always
change the Lagrangian L to be L+dµ and put the conserved charge form Q into the form of (2.18)
without the “ · · · ” terms, where µ is a (n − 1)-form. This change of Lagrangian will not affect
the equations of motion and the entropy of the black holes [45, 46]. Then, the formulas (3.4) and
therefore (3.15) are still formally correct for the non-extremal case after considering that ambiguity
of the Lagrangian and therefore f̃(rH). But this ambiguity has no contribution to Eq. (3.17) which
describes the entropy of the black hole in the extremal case.
IV. ENTROPY FUNCTION AND ATTRACTOR MECHANISM
In this section we show further that one can define an entropy function with the help of the
entropy definition of Wald. The Noether current can always be written as J[ξ] = dQ[ξ] + ξaCa
where Ca corresponds to constraint. The constraint for the U(1) gauge fields is (2.21). If the
equations of motion for the U(1) gauge fields hold, this constraint vanishes. In this section, we
will assume the equations of motion for the U(1) gauge fields are always satisfied, but not for the
metric and scalar fields. In other word, we will not consider the constraint for the gauge fields.
Assuming that the metric of the extremal black holes has the form
ds2 = −N(r)dt2 +
dr2 + γ(r)dΩ2n−2 ,
on the horizon r = rH of an extremal black hole, one has N(rH) = 0, N
′(rH) = 0, but N
′′(rH) 6= 0.
Thus the near horizon geometry will be fixed if N ′′(rH) and γ(rH) are specified. This means the
“off-shell” of the near horizon geometry corresponds to the arbitrariness of the parameter N ′′(rH)
and γ(rH). In the near horizon region ranged from rH to rH +∆r, we have
rH+∆r
Q[ξ]−
Q[ξ] +
J[ξ] =
Θ− ξ · L . (4.1)
If ξ is a Killing vector for the field configuration space for our discussion (the solution space is a
subset of this space), then Θ = 0, and we have
rH+∆r
Q[ξ]−
Q[ξ] +
ξaCa = −
ξ · L . (4.2)
With this, we obtain
rH+∆r
g[ξ]−
g[ξ] +
rH+∆r
F [ξ] +
F [ξ]−
ξ · L . (4.3)
Define our “entropy function” as
E = lim
N ′′(rH)∆r
rH+∆r
g[∂t]−
g[∂t] +
. (4.4)
If the equations of motion are satisfied, obviously, this E will reduce to the entropy of extremal
black holes given in the previous section. Therefore this definition is meaningful. Further, from
Eq. (4.3), we have
E = lim
N ′′(rH)∆r
rH+∆r
F [∂t] +
F [∂t]−
∂t · L
. (4.5)
Recalling that the equations of motion for the U(1) gauge fields have been assumed to hold always,
and following the calculations in the previous section, we have
N ′′(rH)
ẽIqI − f̃(rH)
. (4.6)
This expression looks the same as the one given in the previous section. However, a crucial
difference from the one in the previous section is that here the fields need not be the solutions of
the equations of motion. To give the entropy of the extremal black holes, we have to solve the
equations of motion or extremize the entropy function with respect to the undetermined values of
fields on the horizon. It is easy to find that entropy function has the form
E = E(N ′′, γ, us, ẽI ; pi) =
ẽIqI − f̃H(N ′′, γ, us, ẽI ; pi)
, (4.7)
where, for simplicity, we have denoted the N ′′(rH) and γ(rH) by N
′′ and γ, respectively. The
terms u′s will not appear because those kinetic terms of scalar fields in the action always have a
vanishing factor N(rH) = 0 on the horizon. Similarly, γ
′(rH), γ
′′(rH) will not appear because
that the components of the Riemann tensor which include these terms have to contract with the
vanished factors N(rH) or N
′(rH). Certainly, this point can be directly understood from the near
horizon geometry in Eq. (3.20). So, extremizing the entropy function becomes
∂N ′′
= 0 . (4.8)
The electric charges are determined by
= 0 or qI =
∂f̃(rH)
. (4.9)
The entropy of the black hole can be obtained by solving these algebraic equations, and substituting
the solutions for N ′′, γ, us back into the entropy function. If the values of moduli fields on the
horizon are determined by charges of black holes, then the attractor mechanism is manifest. Then
the entropy has the form
SBH = SBH(qI ; pi) = E|extremum piont , (4.10)
a topological quantity which is fully determined by charges [17, 18]. These definitions will become
more simple if one chooses the coordinates {τ, ρ, · · · } so that one can define
N ′′(rH)
, v2 = γ(rH) , (4.11)
then, the entropy function can be written as
E = E(v1, v2, us, eI ; pi) = 2π (eIqI − f(v1, v2, us, eI ; pi)) , (4.12)
where eI are gauge fields on the horizon in this set of coordinates, and qI =
are electric charges
which are not changed with the coordinate transformation. So, in this set of coordinates, our
entropy function form reduces to the entropy function defined by A. Sen [17, 18].
V. CONCLUSION AND DISCUSSION
In this paper, we have shown that the “entropy function” method proposed by A. Sen can be
extracted from the general black hole entropy definition of Wald [44]. For a spherically symmetry
extremal black hole as described by metric (3.1), we find that the entropy of the black hole can be
put into a form
SBH =
N ′′(rH)
ẽIqI − f̃(rH)
which is similar to the one given in Ref. [17, 18]. To get this entropy form, we have regarded the
extremal black hole as the extremal limit of an non-extremal black hole, i.e., we have required (and
only required) that the surface gravity approaches to zero. In a special set of coordinates, i.e.,
{τ, ρ · · · }, this entropy is exactly of the same form as the one given by A. Sen. We have obtained
a corresponding entropy function (4.7). After extremizing this entropy function with respect to
N ′′, γ and other scalar fields, one gets the entropy of the extremal black holes. Similarly, in
the coordinates {τ, ρ · · · }, our entropy function reduces to the form of A. Sen. Note that in our
procedure, we have neither used the treatment of rescaling AdS2 part of the near horizon geometry
of extremal black holes, nor especially employed the form of the metric in the coordinates {τ, ρ, · · · }
as Eq.(1.1). In this procedure, it can be clearly seen why the electric charge terms eIqI appear,
but not the magnetic charges terms in the entropy function.
Recently it was shown that for some near-extremal black holes with BTZ black holes being a
part of the near horizon geometry, that the “entropy function” method works as well [40]. A similar
discussion for non-extremal D3,M2 and M5 branes has also been given in [41]. Therefore it is
interesting to see whether the procedure developed in this paper works or not for near-extremal
black holes. In this case, N ′(rH) is an infinitesimal one instead of vanishing. Eq. (3.15) then gives
SBH = 2πB(rH) = S0
N ′(rH)
N ′′(rH)
, (5.1)
where
N ′′(rH)
(ẽIqI − f̃(rH)) , (5.2)
and r∗ = B(rH)/B
′(rH) approximately equals to “
· radius of the black hole” if the higher
derivative corrections in the effective action are small. Thus, after considering that ambiguity in
f̃(rH) becomes very small and for large r∗ (sometimes, this corresponds to large charges), the
entropy function method gives us an approximate entropy for near-extremal black holes, but the
attractor mechanism will be destroyed [15]. In addition, it is also interesting to discuss the extremal
rotating black holes with the procedure developed in this paper. Certainly, in this case, the Killing
vector which generates the horizon should be of the form χ = ∂t+ΩH∂φ instead of ξ = ∂t. A term
including angular-momentum J will naturally appear in the associated entropy function [32]. This
issue is under investigation.
Acknowledgements
L.M.Cao thanks Hua Bai, Hui Li, Da-Wei Pang, Ding Ma, Yi Zhang and Ya-Wen Sun for useful
discussions and kind help. This work is supported by grants from NSFC, China (No. 10325525
and No. 90403029), and a grant from the Chinese Academy of Sciences.
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	Introduction
	The Definition of Wald
	Entropy of extremal black holes
	Entropy function and attractor mechanism
	Conclusion and discussion
	Acknowledgements
	References
