0704.0955
Charges from Attractors
Nemani V. Suryanarayana 1 and Matthias C. Wapler 2
1 Theoretical Physics Group and
Institute for Mathematical Sciences
Imperial College London, UK
E-mail: v.nemani@imperial.ac.uk
2 Perimeter Institute for Theoretical Physics,
Waterloo, ON, N2L 2Y5, Canada
Department for Physics and Astronomy,
University of Waterloo, Waterloo, ON, N2L 3G1, Canada
Kavli Institute for Theoretical Physics,
University of California, Santa Barbara, CA, 93106, USA
E-mail: mwapler@perimeterinstitute.ca
September 26, 2018
Abstract
We describe how to recover the quantum numbers of extremal black holes from their near
horizon geometries. This is achieved by constructing the gravitational Noether-Wald charges
which can be used for non-extremal black holes as well. These charges are shown to be equiva-
lent to the U(1) charges of appropriately dimensionally reduced solutions. Explicit derivations
are provided for 10 dimensional type IIB supergravity and 5 dimensional minimal gauged su-
pergravity, with illustrative examples for various black hole solutions. We also discuss how to
derive the thermodynamic quantities and their relations explicitly in the extremal limit, from
the point of view of the near-horizon geometry. We relate our results to the entropy function
formalism.
http://arxiv.org/abs/0704.0955v2
1 Introduction
Studies of extremal black holes in string theory have regained importance with the advent of
the attractor mechanism. In its simplest form the attractor mechanism states that the near
horizon geometry of an extremal black hole is fixed in terms of its charges. Further, it has been
realized that there is a single function, called the entropy function, which determines the near
horizon geometry of extremal black holes [1] (see also [2]). Even though the entropy function
provides the non-zero charges such as the electric, magnetic charges and angular momenta, for
many extremal black holes, it does not always give the correct charges. For instance, there are
apparent discrepancies when there are Chern-Simons terms for the gauge fields present in the
Lagrangian. This is the case, for instance, in 5d minimal (and minimally gauged) supergravities.
On the other hand it has been believed [4] that the near horizon geometry of an extremal
rotating black hole of 5d supergravities knows about only part of the the full black hole angular
momentum, called the horizon angular momentum. In [4] this has been argued to be the case
for the BMPV black hole [16].
Given that finding the near horizon geometries of the yet to be discovered extremal black
hole solutions might be easier than finding the full black hole solutions, it will be useful to
have a prescription to extract the quantum numbers of the full black hole from its near horizon
geometry. In this note we show, by careful analysis of the near horizon geometries of these black
holes, that one can find the full set of asymptotic charges and angular momenta of extremal
rotating black holes that satisfy certain assumptions.
For this, we first construct gravitational Noether charges following Wald [5] for several su-
pergravity theories. These charges can be defined for Killing vectors of any given solution of the
theory of interest. We mainly focus on type IIB in 10d, minimal and gauged supergravities in 5d.
We present closed form expressions for the Nother-Wald charges of these theories as integrals
over compact submanifolds of co-dimension 2 of any given solution.
The 5d minimal gauged supergravity can be obtained by a consistent truncation of type IIB
reduced on S5 [22] (see also [23]). We show that the charges of the 5d theory can be obtained by
the same dimensional reduction of the corresponding 10d charges. We further reduce the theory
down to 3 dimensions and show that the Nother-Wald charges corresponding to Killing vectors
that generate translations along compact directions are the same as the usual Noether charges
for the corresponding Kaluza-Klein gauge fields in the dimensionally reduced theory. We use
the understanding of the charges in the reduced theory to show how the entropy function may
be modified to reproduce the charges of the 5d black holes.
We will argue that these Noether-Wald charges can be used to extract the charges of extremal
black holes from their near horizon geometries under certain assumptions which will be discussed
later on. Thus the formulae presented in this paper should prove useful in extracting the con-
served charges of an extremal black hole from only its near-horizon geometry without having to
know the full black hole solution. We exhibit the successes and limitations of our formulae by
considering the examples of Gutowski-Reall black holes [12] and their generalizations [17] and
BMPV [16, 4] black holes, black rings [18] and the 10d lift of Gutowski-Reall black holes [13].
The analysis of the conserved charges in this paper can be applied to many geometries other
than the extremal black holes considered here and in particular to non-extremal black holes too.
In addition to the charges of a black hole, one is typically interested in the entropy, the mass,
as well as the laws of black hole thermodynamics. Up to now, the entropy has been defined in
terms of a Noether charge only for non-extremal black holes [5]. To find these thermodynamic
quantities and the laws of thermodynamics on the “extremal shell”, it was necessary to take
the extremal limit of the relations defined for the non-extremal black holes (see for instance [1]).
Furthermore, computations of quantities such as the mass, the euclidean action and relations like
the first law and the Smarr formula relied on computing quantities in the asymptotic geometry.
Hence, it would be desirable to derive appropriate relations intrinsically for extremal black holes,
and with only minimal reference to the existence of an asymptotic geometry.
With this motivation, in the second part of the paper, we propose a definition of the entropy
for extremal black holes in the near horizon geometry that does not require taking the extremal
limit of Wald’s entropy, but agrees with it. With a similar approach, we also derive the extremal
limit of the first law from the extremal geometry, assuming only that the near-horizon geometry
be connected to some asymptotic geometry. This definition of the entropy further allows us
to derive a statistical version of the first law [6]. We also show that this gives us the entropy
function directly from a study of the appropriate Noether charge in the near-horizon geometry
of extremal black holes. We will comment on the interpretation of the mass as well, from the
point of view of the near horizon solution.
The rest of the paper is organized as follows. In section 2, we review Wald’s construction of
gravitational Noether charges and use it to derive the charges for type IIB supergravity (with
the metric and the five-form fields) and for the 5d minimal and gauged supergravity theories and
show that they are related by dimensional reduction. In section 3, we show that the Noether-
Wald charges are identical to the standard Noether charges for the Kaluza-Klein U(1) gauge
fields of the corresponding compact Killing vectors. We also discuss various assumptions under
which these charges, when evaluated anywhere in the interior of the geometry, match with the
standard Komar integrals evaluated in the asymptotes. Some issues of gauge (in)dependence
of our charges are also address there. In section 4, we demonstrate how our formulae work on
several examples of interest. The readers who are only interested in the formalism may skip this
section. In section 5, we turn to modifying the entropy function formalism to include the Chern-
Simons terms. In section 6, we discuss thermodynamics of the extremal black holes and define
various physical quantities like the entropy, chemical potentials for the charges and the mass.
We end with conclusions in section 7. The example for black rings is given in the appendix.
2 Charges from Noether-Wald construction
Here we derive expressions for the gravitational Noether charges corresponding to Killing isome-
tries of the gravitational actions we are interested in following Wald [5, 7]. We review first
the general formalism and point out some relevant subtleties. Then we construct these charges
for 10d type IIB supergravity and for minimally gauged supergravity and Einstein-Maxwell-CS
theory in 5d. Finally, we show how the 10d and 5d expressions can be related by dimensional
reduction.
2.1 Review of Noether construction
Let us first review the construction of the charges and discuss some of the relevant properties. In
[7], Lee and Wald described how to construct the Noether charges for diffeomorphism symmetries
of a Lagrangian L(φi = gµν , Aµ, · · · ), a d-form in d spacetime dimensions. For this, one first
writes the variation of L under arbitrary field variations δφi as
δ L = Ei(φ) δφ
i + dΘ(δφ) (1)
where Ei(φ) = 0 are the equations of motion and Θ is a (d − 1)-form. Secondly, one finds the
variation of the Lagrangian under a diffeomorphism
δξL = d(iξ L), (2)
where ξa is the (infinitesimal) generator of a diffeomorphism. Then one defines the (d− 1)-form
current Jξ
Jξ = Θ(δξφ)− iξ L (3)
where δξφ
i are the variations of the fields under the particular diffeomorphism. Then Jξ are
conserved, i.e. dJξ = 0, for any configuration satisfying the equations of motion. Since Jξ is
closed, one can write (for trivial cohomology)
Jξ = dQξ (4)
for some (d − 2)-form charge Qξ. Now consider ξ to be a Killing vector and suppose that the
field configurations on the given solution respect the symmetry generated by it, Lξφi = 0. Since
Θ(δξφ
i) is linear in Lξφi we have Θ(δξφi) = 0 and so Jξ = −iξL. Next, let us illustrate that the
charge defined as the integral
Qξ over a compact (d-2)-surface Σr is conserved when (i) ξ is
a Killing vector generating a periodic isometry or (ii) when the current Jξ = 0 (as for Killing
vectors in theories with L = 0 on the solutions). Consider a (d − 1)-hypersurface M12 which is
foliated by compact (d−2)-hypersurfaces Σr over some interval R12 ⊂ R. Using Gauss’ theorem
one has
Jξ (5)
for ∂M12 = {Σ1,Σ2}. If Jξ = 0, it follows that the charge
Qξ does not depend on Σr and
therefore is conserved along the direction r. Next, let us assume that ξ generates translations
along a periodic direction of Σr. In general,
Jξ receives contributions from terms in Jξ that
contain the one-form ξ̂ dual to the Killing vector field ξ and terms that do not. The terms not
involving ξ̂ vanish by the periodicity of ξ. Since Jξ = −iξL, there are no terms involving ξ̂.
Therefore
Qξ is again independent of Σr.
We will now discuss two important ambiguities in the above prescription. The first one is
that the charge density defined by the equation Jξ = dQξ is ambiguous as Qξ → Qξ + dΛξ does
not change Jξ for some (d-3)-form Λξ. The extra term does not contribute to the integrated
charge only if Λξ is a globally defined (d-3)-form on Σr, that is, it is periodic in the coordinates
of Σr and non-singular. While this is the case for most of our examples, there may be situations
in which, for instance, some gauge potentials that go into Qξ are only locally defined. Similarly,
conservation of Qξ is not guaranteed if any component of Qξ ∈ Ωd−1
is not globally defined.
To illustrate this, consider the Jξ = dQξ = 0 case and let n be a normal to Σr, such that dn = 0.
Qξ = (ind)
indQξ +
d (inQξ) =
d (inQξ) , (6)
which is only forced to vanish if inQξ is globally defined on Σr. The second, and a more impor-
tant, ambiguity comes from possible boundary terms in the Lagrangian L. For the boundary
terms Sbdy. =
Lbdy. =
dLbdy., the variation that gives the equations of motion is done on
the boundary,
δξSbdy. =
δLbdy.
δLbdy.
δξLbdy. =
d(δξLbdy.). (7)
Since δξLbdy. = iξ(dLbdy.) + d(iξLbdy.), the current is just given by
Jξ = −iξ(dLbdy.) + iξ(dLbdy.) + d(iξLbdy.) (8)
and hence the charge is Qξ = iξLbdy.. This implies that boundary terms contribute only to
conserved charges
Qξ of (Killing) vectors that do not lie in Σr.
2.2 The Noether-Wald charges for type IIB supergravity
Now we would like to find the Noether-Wald charges in 10d type IIB supergravity for config-
urations with just the metric and the 5-form turned on. As is standard, we work with the
action
LIIB =
16πG10
−g [R−
4 · 5!
F 2(5)] (9)
neglecting the self-duality of the 5-form and impose it only at the level of the equations of motion.
We follow the procedure outlined in section 2.1 to find the Noether-Wald currents. Using the
variations
−g R) =
−g [Rµν − 12Rgµν ] δg
−g gµν [∇σδ̄Γσµν −∇ν δ̄Γσµσ] and
−g F 2(5)) =
−g [5F (5)
µκσωλ
F (5) κσωλν − 12 gµν F
(5)] δg
−2 · 5![δC(4)
−g Fµνσωλ
)− ∂µ(δC
µνσωλ
−g)], (10)
where δ̄Γλµν =
gλσ [∇µδgσν +∇νδgµσ −∇σδgµν ], one can find the equations of motion
Rµν −
µκσωλ
F (5)ν
= 0 and ∂µ(
−g Fµνσωλ
) = 0. (11)
These are supplemented by the self-duality condition ⋆(10)F
(5) = F (5). The self-duality constraint
F (5) = ⋆F (5) implies that F 2
= 0, and then the metric equation of motion in (11) implies R = 0
for any solution. Hence the Lagrangian vanishes on the solutions and therefore the Noether-Wald
current in (3) is given entirely by the 9-form Θ (or equivalently by its dual vector field). This
can be found from the total derivative terms in δL by substituting δξg
µν = ∇µξν +∇νξµ and
and δξC
= 4 ∂[ν|(ξ
θ|σωλ]) + ξ
θνσωλ
. This gives us the current
J α = −2
−g gασ [Rσλ −
λνθωλ
F (5) νθωλσ ]ξ
+∂µ[−
−g gµνgασ(∇νξσ −∇σξν) +
2 · 3!
−g ξθC(4)
αµσωλ
] , (12)
where the first term vanishes by the equations of motion and the second term gives us the charge
density
16πG(10)
∇αξµ −∇µξα +
αµσωλ
. (13)
Noting that the self-duality constraint
−g Fµ0···µ4
ǫµ0···µ9F
µ5···µ9 implies
αµσωλ
= ξνC
3! 5!
ǫαµσωλµ5···µ9F
µ5···µ9 , (14)
the Noether-Wald charge density (13) can be equivalently written as the 8-form
= − 1
16πG10
⋆ dξ̂ − 1
(4) ∧ F (5)
where ξ̂ is the dual 1-form of the vector field ξµ. This can be integrated over a compact 8d
submanifold to get the corresponding conserved charge. A quick calculation verifies that the
current for this charge vanishes identically as expected because of the vanishing Lagrangian.
Hence, all charges that are computed from it are conserved as discussed in section 2.1. If we
further assume that LξC(4) = 0, we have iξF (5) = −d(iξC(4)). This can be used to rewrite (15)
16πG(10)
⋆ dξ̂ +
C(4) ∧ iξF (5)
up to an additional term proportional to d(C(4) ∧ iξC(4)). This extra term does not contribute
when integrated over a compact 8-manifold provided that C(4) ∧ iξC(4) is a globally well defined
7-form as we discussed in section 2.1. In such cases (16) can be used instead of (15).
In section 4, we will demonstrate that this formula reproduces conserved charges [12] of
Gutowski-Reall black holes of type IIB in 10 dimensions successfully. We hope this expression
may be useful in obtaining the charges of the yet to be discovered black holes from their near
horizon geometries alone.
2.3 The Noether-Wald charges for 5d Einstein-Maxwell-CS
The action for 5d Einstein-Maxwell-Chern-Simons gravity is
16πG5
−g (R− FµνFµν)−
ǫmnpqrAmFnpFqr
which is the same as the action for the 5d minimal gauged supergravity up to the cosmological
constant, which turns out not to contribute to the Noether charge. After a straight forward but
slightly lengthy calculation it is easy to show that the Noether current for this action is
16πG5
(Rαλ − 1
gαλR)− 2 (F λµFαµ − 14g
λαF 2)
+ 4 (ξ · A)
−gFαµ) + 2√
ǫανσωλFνσFωλ
−ggµνgαλ (∇νξλ −∇λξν)− 4
−g(ξ · A)Fαµ − 8
(ξ · A) ǫαµσωλAσFωλ
.(18)
The first two lines are simply proportional to the equations of motion and vanish on-shell and
hence the Noether-Wald charges for this theory are
16πG5
−g (∇αξµ −∇µξα) + 4(ξ ·A)(
−g Fαµ +
ǫαµσωλAσFωλ)
. (19)
These expressions have also appeared recently in [8]. An alternative derivation of (19) in terms
of KK charges will be presented in section 3.3. The charge density (19) can equivalently be
written as the 3-form
16πG5
⋆dξ̂ + 4 (iξA)
⋆ F − 4
A ∧ F
. (20)
As before the charges can be obtained by integrating Qξ over a 3d compact sub-manifold. Note
that if we set the gauge fields to zero we recover the standard Komar integral for the angular
momentum.
2.4 Reduction from 10 dimensions
Now, we will find the dimensional reduction of the 10d formula of conserved charges to the 5d
formula to show that they are indeed identical, so let us first review the reduction formulae to
obtain the equations of motion of 5d minimal gauged supergravity from 10d type IIB supergravity
with only the metric and the self-dual 5-form F (5) turned on [13, 14].
As usual, we express the metric in terms of the frame fields e0, . . . , e9 and do the dimensional
reduction along the compact 5-manifold Σc that is spanned by the 5-form e
5∧e6∧e7∧e8∧e9 =:
e56789. Then, the lift formula is [22] (see also [23])
ds210 = ds
5 + l
(dµi)
2 + µ2i
dξi +
F (5) = (1 + ∗(10))
vol(5) +
d(µ2i ) ∧ dξi ∧ ∗(5)F
, (21)
where µ1 = sinα, µ2 = cosα sin β, µ3 = cosα cosβ with 0 ≤ α ≤ π/2, 0 ≤ β ≤ π/2, 0 ≤ ξi ≤ 2π
and together they parametrise S5. Note that we define the Hodge star of a p-form ω in n-
dimensions as ∗(n)ωi1...in−p = 1p!ǫi1...in−p
j1...jpωj1...jp , with ǫ0123456789 = 1 and ǫ01234 = 1 in an
orthonormal frame. The 10d geometry is specified by {e0, · · · e4}, an orthonormal frame for the
5d metric ds25, together with
e5 = l dα, e6 = l cosαdβ, e7 = l sinα cosα [dξ1 − sin2β dξ2 − cos2β dξ3], (22)
e8 = l cosα sinβ cosβ[dξ2 − dξ3], e9 = −2√3A− l sin
2α dξ1 − l cos2α(sin2β dξ2 + cos2β dξ3).
and the five form [22, 23, 13]
F (5)=
e0···4 + e5···9
(e57 + e68) ∧ (∗(5)F − e9 ∧ F ) (23)
One can write the 5-form RR field strength as F (5) = dC(4) where
C(4) = Ω4 + cotα e
678 ∧ (e9 + 2√
A ∧ (e57 + e68) ∧ (e9 + 2√
(e9 + 2√
A) ∧ (⋆F + 2√
A ∧ F )
. (24)
where Ω4 is a 4-form such that e
01234 = dΩ4. Now we are ready to do the reduction of the 10d
charge
Qχ := −
16π G10
⋆ dχ̂− 1
(4) ∧ F (5)
where Σ8 is a compact 8d submanifold that is composed of a spacelike 3-surface Σ in 5d and
Σc. Hence, only e
5...9 will contribute to the integral. Let us consider χ to be a Killing vector of
the 10d geometry which also reduces to a Killing vector of the 5d geometry and χ̂ be its dual
1-form. Then we find from the expression for the frame fields (21, 22):
χ̂ = χ̂5 + (iχe
9) e9 = χ̂5 − 2√3(iχA) e
9 , so
⋆dχ̂ = ⋆dχ̂5 − 2√
(iχA) ⋆ d e
9 + . . . = ⋆dχ̂5 +
(iχA) ⋆ F + . . . (26)
where “. . .” denotes terms that do not contribute to Qξ. Next, let us find the relevant terms in
C(4) and F (5) (23,24). Noting that iχ
e9 + 2√
= 0, they are:
(4) = iχΩ4 − 2√
(iχA)
e57 + e68
e9 + 2√
e9 + 2√
iχ ⋆ F +
iχ(A ∧ F )
+ . . . (27)
F (5) = −4
e56789 + 2√
⋆ F − F ∧ e9
e57 + e68
+ . . . (28)
(4) ∧ F (5) = −2
iχΩ4 +
(iχA) +A ∧ iχ
⋆ F + 2√
A ∧ F
e56789 + . . . . (29)
After some algebra, the charge reads
Qχ = −
16π G5
⋆dχ̂5 + 4 (iχA) ⋆ F +
(iχA)A ∧ F +
iχΩ4 −
iχ(A ∧ ⋆F )
. (30)
We see immediately that for vectors in the directions of Σ it just reproduces the 5d Noether
charge (19). For vectors orthogonal to Σ, it is different, as is not unexpected, since typically in
dimensional reduction the actions agree only up to boundary terms.
3 Charges from dimensional reduction
In this section we will rederive the Noether-Wald charges for 5d supergravity of section (2.3)
using further dimensional reduction. In particular, we will demonstrate that the 5d Noether-
Wald charges can alternatively be obtained from Kaluza-Klein U(1) charges. For this, we will
first dimensionally reduce the 5d theory along the relevant Killing vectors and then find the
Noether charges of the resulting gauge theory.1 Then we will lift the results back to 5d and
show that they agree with the corresponding 5d Noether-Wald charges. Finally, we will discuss
in which cases the charges obtained by our methods in the interior of the solution agree with
the asymptotic ones.
3.1 Dimensional reduction
In 5 dimensions one can have two independent angular momenta, so we consider dimensional
reduction over both compact Killing vector directions which generate translations along which we
have the independent angular momenta. We will again assume that all fields obey the isometries
and hence only need to consider zero-modes in the compact directions.
We take lower case greek letters α, β, . . . ∈ {t, r, θ, φ, ψ} to be the 5d indices, upper case latin
A,B, . . . ∈ {t, r, θ} to be the 3d indices and lower case latin a, b, . . . , i, j, l,m, . . . ∈ {θ, φ} to be
the indices for the compactified directions in 5d or scalar fields in 3d. The appropriate reduction
ansatz is:
Gµν =
gMN + hijB
, Am =: Am and AM =: A
M + AaB
M , (31)
such that we get
Fµν =
FMN + (dAa ∧Ba)MN An,M
−Am,N 0
, (32)
in terms of the 3d gauge fields Ha = dBa and F 3d = dA3d, and we defined for simplicity F =
F 3d + AaH
a. The definition of A3d in (31) is needed to have the appropriate transformations
of the KK and Maxwell U(1) symmetries and arises naturally from the reduction using frame
fields (see, for instance, [9] for details). Now, we find
µν = FMNF
MN − 2habA,MA,M and
ǫαµνρσAαFµνFρσ = 4ǫ
LMNǫab
Aa,LFMNAb − A3dLAa,MAb,N
, (33)
such that the 5d Lagrangian (17) can be rewritten as :
G5 × L3d =
R3d −
HaMNH
b MN − FMNFMN + 2habAa,MA ,Mb
1This dimensional reduction has been used recently in [10, 11] for defining the entropy functions for such theories.
ǫLMNǫab
Aa,LFMNAb − A3dLAa,MAb,N
, (34)
where VT 2 is the “volume” of the compact coordinates. One can now construct conserved currents
using the Noether procedure for the gauge symmetries of the two U(1) gauge fields Baµ and A
We find the corresponding Noether charges for Baµ to be
Ja = −
16πG5
a rt + 4AaF
ǫLrtǫmnAm,LAn
. (35)
which we identify as the two independent angular momenta. The Noether charge for A3dµ works
out to be
Q = − VT 2
hF rt +
ǫLrtǫmnAm,LAn
which we identify with the 5d electric charge. Alternatively, these charges can be read off by
writing the left hand side of the equations of motion for the Lagrangian (34)
a MN + 4AFMN
+ 16Aa
ǫLMNǫmnAm,LAn
= 0 (37)
−g hFMN + 4
ǫLMNǫabAa,LAb
ǫLMNǫmnAm,LAn,M , (38)
as a total derivative and interpreting the resulting total conserved quantities as the charges.
For geometries with just one independent angular momentum, one can apply the above
formulae in a straight forward way, or do a reduction only down to 4d as in such cases only one
U(1) isometry is expected in the geometry. The computations for the latter are identical to the
ones here, so we just state the expressions for the angular momentum along ∂ξ and the charge:
J = − VT1
16πG5
e2σHrt + 4A F rt
ǫrtAB
A FAB − 2A,AA4dB
, (39)
Q = − VT1
−geσF rt + 1
ǫrtAB
3A FAB + A F
AB − 4A,AA4dB
, (40)
where e2σ = gψψ , VT 1 is the periodicity of ψ, and the conservation follows by the equations of
motion
e2σHMN + 4A FMN
ǫABMN
A FAB − 2A,AA4dB
= 0, (41)
−geσFMN + 2
ǫABMN
A FAB − 2A,AA4dB
ǫABMNFABA,M . (42)
3.2 Oxidation of the angular momentum
Now we would like to demonstrate that the lower dimensional Noether charges above, when
lifted back to 5d, give the Noether-Wald charges for the compactified Killing vectors. For
simplicity, we look at the expression with only one independent angular momentum and only
one dimension (along ψ) reduced. Our results will hold in general though, as the gauge theory
corresponding to the angular momentum is abelian, so we can examine different Killing vectors
independently. First, we note that the dimensional reduction ansatz can be obtained with the
following triangular form of the frame fields [9]:
V Iµ =
viM e
and the inverse V MI =
0 e−σ
, (43)
with (bold latin) tangent space indices A,B, . . . ∈ {0, . . . , 4} and a,b, . . . ∈ {0, . . . , 3} such that
we can write the 4d fields in terms of the 5d fields (but still in 4d coordinates):
BM = e
−σV 4M , HMN = e
−σ(dV 4
− 2e−σ
deσ) ∧B
σ = ξµV Iµ and A = ξ
µAµ . (44)
Now the conservation equation (41) for the angular momentum Jψ reads in flat indices
ηacηbd
ξµV Kµ ηKLdV
L− 2eσ(deσ) ∧B
+ 4ξµAµ(F− 2(dA ) ∧B)cd
8ξµAµ
ǫcdij
F− 2(dA ) ∧B
− 2(dA )cAd + (dA 2)cBd
= 0 . (45)
Extending the summations to A,B, .. and using the form of the frame fields and the indepen-
dence from ψ yields:
ηACηBD
d(ξµV IµηIJV
+ 4ξµAµFCD
8ξνAν
ǫCDABEA
(dξ̂)µN + 4ξ ·AFµN
8ξ · A
ǫµNαρσAαFρσ
= 0 . (46)
The conserved charge extracted from this equation exactly reproduces the charge in (19).
3.3 Generalization and Limitations
3.3.1 Relation to the Asymptotes
Let us now discuss in which situations the charges computed in the spacetime interior give
the charges as defined on the asymptotic boundary. We see most easily from (20) that when
evaluated on a hypersurface on which iξA = 0, such as a suitable asymptotic boundary, our
formulae match with the appropriate Komar integral.
We can compute a (possibly zero) KK or Noether-Wald charge, that corresponds in a specific
geometry to the angular momentum, for every U(1) isometry. However, the asymptotic hyper-
surface on which the angular momentum of a black hole is defined is an Sd−2. When in such
a geometry angular momenta are turned on, its SO(d − 1) isometry breaks (generically) down
to its U(1) subgroups whose charges give the angular momenta, so only the local U(1) factors
that correspond to the asymptotic U(1) subgroups will be related to the angular momentum.
Furthermore, the normalization of the period generated by the Killing vector also has to be
taken into account.
We saw in sections 2.1 and 3.1 how the charges of compact Killing vectors are conserved
whenever the source-free equations of motion hold. That is, they are independent of the position
of the surface on which they are computed, QΣr2 −QΣr1 =
M dMM ∂NQ
MN = 0 where Σr1 and
Σr2 are the boundaries of the volume M - provided that the U(1) theory is defined throughout
the bulk volume and we can consistently compactify the manifold (at least outside the horizon).
Hence, the black hole charge and angular momentum as defined on a spacelike d-2 hyper-
surface Σ∞ at the asymptotes are given by the corresponding KK or Noether-Wald charge,
computed over any spacelike d-2 hypersurface Σr0 in the spacetime for any (not necessarily ex-
tremal) black hole (or in general any spacetime with a suitable asymptotic boundary). That
is, provided there exists a spacelike d-1 hypersurface M with ∂M = {Σr,Σ∞} on which the
following sufficient conditions are satisfied:
1. The relevant compact Killing vector is a restriction to Σr of a Killing vector field that is
globally defined on M and generates a constant periodicity.
2. There are no sources, i.e. the vacuum equations of motion for the gauge fields are satisfied.
3. There exists a smooth fibration of surfaces
π→ [r0,∞[
= M such that π−1r0 = Σr0
limr→∞ π
−1r = Σ∞.
An example where these conditions are satisfied is the region outside the (outer) horizon
of a stationary black hole solution with an Sd−2 horizon topology, embedded in a geodesically
complete spacetime with an asymptotic Sd−2 boundary. One example where these conditions
are violated is that of black rings [18] which will be considered separately in an appendix.
3.3.2 Gauge Issues
The contributions of the CS term in the conserved quantities in (3.1) depend explicitly on the
gauge potentials. This does not however make them gauge dependent. To see this in 5d, let
us consider the electric charge computed by the Noether procedure which is given in [4] as
⋆ F + 2√
A ∧ F
. We notice that the charges get contributions of the form
A ∧ F ,
that change under a transformation δA = dΛ as
dΛ∧F =
d(ΛF ) = 0 because Σ is compact.
From the 3d point of view the KK scalars A may depend on a 5d gauge transformation. However
Λ must be periodic in the angular coordinates so that the contributions from dΛ vanish after
integration. This is also the reason why the term containing ξ ·A in eq. (19) is gauge independent
for compact Killing vectors. On the other hand, the Noether charge for a non-compact Killing
vector is gauge-dependent and hence is only physically relevant when measured with respect to
some boundary condition or as a difference of charges.
4 Examples
So far we have derived Noether charges for various supergravity theories that may be used to
calculate the electric charges and angular momenta of the solutions. In particular, they can be
used on the near horizon geometries to calculate the conserved charges of the corresponding black
holes. In this section we will demonstrate with several examples how our charges successfully
reproduce the known black hole charges in different dimensions, for equal or unequal angular
momenta and independent of the asymptotic geometries. We will start with a 10d example and
then cover 5d examples, first with one angular momentum in AdS and flat asymptotics, and
then with unequal angular momenta in asymptotic AdS.
4.1 The 10d Gutowski-Reall black hole
In [12], Gutowski and Reall found the first example of a supersymmetric black hole which
asymptotes to AdS5 as a solution to minimal gauged supergravity in 5d (see also [34, 17, 35, 36]).
Their solution was lifted to a solution to 10d type IIB supergravity in [13] and shown to admit
two supersymmetries. In [14] (see also [15]), the near horizon geometry of this 10d black hole was
studied. Here we use the formulae found in section 2.2 to calculate the Noether-Wald charges in
the near horizon geometry and show that they agree with the charges of the black hole measured
from the asymptotes. The 10d metric of this near horizon geometry is ds210 = ηabe
aeb with the
orthonormal frame
σL3 , e
, e2 =
σL1 , e
σL2 , e
λ σL3 , (47)
and the five-form is
F (5) =
(e0···4+e5···9)−
(e57+e68)∧ [−3e023+e014−
e234+e9∧ (3e14−e23−
e01)] (48)
where e5 . . . e9 are given in (22) and
dt+ ω
σL3 ) =
(e0 + 2ω
e4), λ =
l2 + 3ω2 and
σL1 = sinφdθ − sin θ cosφdψ, σL2 = cosφdθ + sin θ sinφdψ, σL3 = dφ+ cos θ dψ. (49)
The potential C(4) for the above field strength was given in section 2.4 with Ω4 =
e0234 [14].
Here we concentrate on the compact Killing vectors ∂φ and ∂ξ1 + ∂ξ2 + ∂ξ3 of this geometry and
calculate the corresponding conserved charges. For χ = ∂φ which has a period 4π, we have
χ̂ = 3ω
e0 + ωλ
e4 − ω2
e9 and
dχ̂ = −2ωλ
e01 + 3ω
e14 − (1 + ω2
)e23 + ω
(e57 + e68) (50)
and hence the relevant terms in ⋆dχ̂ are ω
(4l2 + 3ω2) e2···9. Similarly, we find
C(4) ∧ iχF (5) = ω
(2 l2 + ω2) 1
σL1 ∧ σL2 ∧ σL3 ∧ e56789 . (51)
After noting that the integral over 1
σ123 ∧ e56789 gives a factor of 2π5l5, we find
Q∂φ = −
16π4 l5G5
S3∧S5
[⋆dχ̂+
C(4) ∧ iχF (5)] = −
8 l G5
) , (52)
which agrees with the angular momentum, up to a minus sign, that comes from the definition
of the angular momentum as minus the Noether charge [12]. For χ = ∂ξ1 + ∂ξ2 + ∂ξ3 , we have
9 = −l. One can calculate the 10d current and find that
⋆ dχ̂+
C(4) ∧ iχF (5) =
(⋆5F +
A ∧ F ) ∧ e5678 ∧ (e9 +
A) + · · · . (53)
Therefore the corresponding charge is
Q∂ξ1+∂ξ2+∂ξ3
π l ω2
) . (54)
This differs from the answer Q(GR) =
3π ω2
(1 + ω
) [12] by a factor of −l/
12. The minus
sign is because of a difference in our conventions from those of [12] and the factor of l is there
to make the charge Q(GR) dimensionless. The killing vector ∂ξ1 + ∂ξ2 + ∂ξ3 has a period of 6π
and to normalise it to have a period of 2π we have to multiply it by a factor of 3. If we take this
into account the extra factor reduces to
3/2. This is precisely the factor required to define the
5d gauge field in the conventions of dimensional reduction from 10d to 5d [22]. Thus we find
complete agreement between our 10d computation of charges from the NHG and the asymptotic
black hole charges of [12].
4.2 5d Black Holes
Now we turn to black hole solutions in 5d Einstein-Maxwell-CS and minimal gauged supergravity.
4.2.1 Equal Angular Momenta: BMPV and GR
Let us consider two examples that are similar in the near-horizon geometry, with a squashed
S3 horizon, but differ by their asymptotic behaviour; the BMPV black hole [4, 16] with asymp-
totically flat geometry and the Gutowski-Reall (GR) black hole [12] with asymptotically AdS5
geometry.
Their near-horizon solutions can be put in to the form
ds2 = v1
− r2dt2 + dr
σ21 + σ
2 + η(σ3 − αr dt)2
, A = −e r dt+ p(σ3 − αr dt) (55)
which, when dimensionally reduced along the ψ-direction, gives ds24 = v1
− r2dt2 + dr2
dθ2 + sin2θ dφ2
. This has AdS2 × S2 symmetry as expected. The fields take the form
B = −rαdt+ cos θ dφ, e2σ = v2η, A = p and A4d = −e r dt. For the BMPV case, we find:
v1 = v2 =
, η = 1−
, α =
µ3 − j2
, e = −
µ3 − j2
and p =
. (56)
Evaluating the 4d quantities and noting that ǫtrφθ = 1 and VT 1 = 4π, (39, 40) gives us J = πj4G5
which is equal in magnitude to the angular momentum in [4] up to a factor of 2, which arises
from the canonical normalization of the Killing vector ξ = 2∂ψ , and Q =
For the GR case, we have:
, v2 =
, η = 1 + 3
, α = − 3ωl
4l2 + 3ω2
, e =
α, p =
. (57)
Note that we have defined A with an overall factor of −1 compared to [14] to account for a
different convention for the CS term. This gives the results J = −3πω2
(1 + 2ω
) and Q =
(1 + ω
) as expected. Note that [12] do not use the canonical normalization for ∂ψ of [4].
4.2.2 Non-equal Angular Momenta: Supersymmetric Black Holes
Here, we present as the most simple example the N=2 supersymmetric black holes with non-
equal angular momenta of [17], which are asymptotically AdS5, just as the GR case. We start
off with the metric in the form [17]
gtt =
(ρ2ΞaΞb)
ρ2ΞaΞb(1 + r
2) − ∆t(2mρ2 − q2 + 2abrρ2)
, grr =
, gθθ =
gtφ =
−∆t sin2θ
ρ4Ξ2aΞb
a(2mρ2 − q2) + bqρ2(1 + a2)
, gtψ = gtφ(a↔ b, sin θ ↔ cos θ)
gφφ =
sin2θ
ρ2Ξ2a
(r2 + a2)ρ4Ξa + a sin
a(2mρ2 − q2) + 2bqρ2
gψψ = gφφ(a↔ b, sin θ ↔ cos θ), gφψ = sin
2θ cos2θ
ρ4ΞaΞb
ab(2mρ2 − q2) + (a2 + b2)qρ2
with the the gauge field
∆tΞaΞbdt −
a sin2θ
dφ − b cos
where
ρ2 = r2 + a2 cos2θ + b2 sin2θ, ∆t = 1− a2 cos2θb2 sin2θ,
(r2+a2)(r2+b2)(1+r2)+q2+2abq
r2−2m , Ξa = 1− a
2 and Ξb = 1− b2 . (60)
We consider the case with saturated BPS-limit and no CTC’s, which requires:
1 + a+ b
, m = (a+ b)(1 + a)(1 + b)(1 + a+ b). (61)
Now we can find the near horizon geometry with explicit AdS2 symmetry as in [18], by re-defining
t̃ = ǫt, r̃ =
4(1+3a+a2+3b+b2+3ab)
(1+a)(1+b)(a+b)
a+b+ab
, d̃φ = dt+ dφ, d̃ψ = dt+ dψ, (62)
then taking the limit of ǫ→ 0 and applying a gauge transformation to get rid of a constant term
in At. We can read off the 3d scalar fields hmn and A and find
BmN = h
maGaN , gMN = GMN − BaMhabBbN and A3dM = AM − AaBaM . (63)
Noting that VT 2 = 4π2, eqns. (35) give us the angular momenta Jφ̃ = π
a2+2b2+3ab+a2b+ab2
4G5(1−a)(1−b)2 and
= π b
2+2a2+3ab+a2b+ab2
4G5(1−b)(1−a)2 . These agree precisely with the corresponding asymptotic angular
momenta of [18].
5 Charges from the entropy function
The original incarnation of the entropy function formalism [3, 1] was not only a useful tool for
finding near-horizon solutions, but also for extracting the conserved charges from a given solution.
However, in the presence of Chern-Simons terms, the entropy function formalism captures only
part of the conserved charges. We demonstrate here two equivalent ways to cure this problem.
Let us first recall the entropy function formalism [3, 1]:
One considers a general theory of gravity described by the Lagrangian density L with abelian
gauge fields F i(x) and scalar fields Φj(x). Then one writes down the most general ansatz for the
near horizon geometry assuming the isometries of AdS2 × S1 (for simplicity, we consider here
d=4 as in [3, 1]):
ds2 = v1(θ)
− r2dt2 +
dθ2 + v2(θ)
dφ2 − α r dt
F i =
ei − αbi(θ)
dr ∧ dt + ∂θbi(θ)dθ ∧ (dφ − α r dt) and Φj = uj(θ) , (64)
in terms of the parameters {α, ei, β} and θ-dependent scalars {vi(θ), bi(θ), ui(θ)}. Then, one de-
fines the “reduced action” f(α, ~e, β, ~v(θ), ~b(θ), ~u(θ)) =
dθdφL - a functional that generates
the equations of motion
δbi(θ)
δvi(θ)
δui(θ)
= 0, where the functional derivatives
can be understood in terms of the Fourier coefficients in the expansion along θ, and
= qi ,
= j , (65)
where qi and j are supposed to give the charges of the black hole. Then the entropy function is
defined to be the Legendre-transform of the reduced action
E(j, qi, β, ~v(θ), ~b(θ), ~u(θ)) = 2π(eiqi + αj − f) . (66)
Finally, the entropy of the black hole is S = E , evaluated on the solution.
5.1 Completing the equations of motion
In section 3.1, we learned how to find the conserved charges in the presence of Chern-Simons
by writing the KK gauge field equations of motion in a conserved form. Since we now know the
right reduction ansatz, we just need to find a mechanism to parametrize both the variation with
respect to At and Bt and the integration of the right hand side of the equations of motion to
obtain the closed form. One such mechanism is a modification of the ansatz with the pure gauge
terms {ǫi,ℵa} to do the variations δL
and δL
; and with a dummy function c(r), that introduces
an artificial and unphysical r-dependence into fields that are constant by the symmetries. c(r)
then allows to keep track of their, otherwise vanishing, derivatives and to do their integration
on the right hand side of the equations of motion. Hence, we write
Ai = −(ǫi + ei r)dt + c(r) pia(θ)
dφa − (ℵa + αar) dt
, (67)
ds2 = v(θ)
− r2dt2 + dr
dθ2 + ηab(θ)(dφ
a − (ℵa + αar)dt)(dφb − (ℵb + αbr)dt)
and we also wrap all scalar fields that appear in the Chern-Simons terms with a factor of c(r),
ui(θ, r) = c(r)Φi(θ). The solution corresponds to setting c(r) = 1 and c′(r) = 0, which we can
either implement by furnishing c(r) with a control parameter, or by choosing c(r), s.t. c(r0) = 1
and c′(r0) = 0 for some r0, but c
′(r0) 6= 0 for r 6= r0. The equations of motion for the gauge
fields are then ∂r
and ∂r
∂ℵa and give rise to the conserved charges
and Ja =
, (68)
evaluated on the solution. A simple variation of this is c(r) = 1 + 1
r, n being the number of
3d scalar fields in the CS term, which automatically takes care of the integration of the second
term and ensures that all remnant dummy terms will disappear in the first term at r = 0.
The other computations follow just as in the original form of the Entropy function, using
c = 1, c′ = 0 throughout. Note that the entropy function is still computed as originally defined,
E = 2π
αa + ∂L
ei − f
, i.e. not using the conserved charges.
One can easily see that this gives the equations of motion, and it also gives the correct value
for the entropy as the original derivation [3, 1] is independent of what the conserved charges are.
This can also be seen by repeating the derivation in section 6.4 with the original action (34).
As a simple example we have already written the 4d ansatz (55) in section 4.2.1 in a suggestive
form, such that the coefficients can be read off from (56) and (57) with β2 = v2. We note that
the ℵa parameters do not appear here in the action. A simple computation reveals that this
gives indeed the results in section 4.2.1.
5.2 Gauge invariance from boundary terms
In section 3.3, we found that the charges are gauge invariant. However, it would be desirable if
we could impose gauge invariance at the level of the Lagrangian of the 3d action (34). The result
can, in principle, be oxidized back to 5d, but we will stick for simplicity to 3d. The only term of
concern is the A3d ∧ dA[a ∧ dAb] in the CS term in (34), which varies under A3d → A3d + dΛ as
dΛ ∧ dA[a ∧ dAb]. This variation is a total derivative d(ΛdA[a ∧ dAb]) which, after integration,
gives a boundary term ΛdA[a ∧ dAb]. This can be re-expressed as d(ΛA[adAb])− A[adΛ ∧ dAb],
where the first term vanishes if we consider a stationary boundary. The second term is suitably
cancelled by adding a boundary term Abdy. [aA
∧ dAbdy. b], which is identical to a bulk term
d(A[aA
3d ∧ dAb]). Expressed in index notation, and furnished with appropriate factors, the
boundary term that we need to add corresponds to the bulk term is
δL 3d = −
16πG5
ǫLMNǫab
Aa,LF
MNAb + 2 A
LAa,MAb,N
, (69)
which brings the Lagrangian to
G5 × L3d =
R3d −
Ha MNH
b MN − FMNFMN + 2habAa,MA ,Mb
ǫLMNǫab
2Aa,LFMNAb + Aa,LF
, (70)
eliminating the gauge dependent term. A quick calculation shows that this does not affect the
value of the charges (35, 36). Effectively, what we have done is to differentiate the components
of the 5d gauge field in the CS term whose gauge transformations do not vanish automatically
by periodicity constraints, and remove the derivative from other components by an integration
by parts. Hence, the right hand side of each of the 3d gauge field equations of motion does
vanish, and the charges are just the conjugate momenta of the gauge fields B and A3d:
Q = −
δF 3dµν
ǫρµνdx
ρ and Ja = −
δHaµν
ǫρµνdx
ρ , (71)
as in the absence of CS terms. It is easy to verify that the value of the charges remains unchanged.
This means that, if we compute the reduced action from the gauge independent action, the
original formalism will give us the right charges. The entropy function, now computed with the
full charges, does not depend on the extra boundary term and hence also gives us the correct
value of the entropy as we shall derive directly from the Poincaré time Noether charge in section
6 Thermodynamic Charges
Having computed the charges of the Sd−2 isometries, we now turn to the charges of the AdS2
isometries. In particular, we will concentrate on the charge of ∂t, as this will be related to the
thermodynamic quantities entropy S and mass M . First we will compute the Poincar’e time
Noether charge from the Hamiltonian in the NHG and propose a new definition of the black
hole entropy for extremal black holes in the NHG in terms of this charge - similar to Wald’s
definition for non-extremal black holes. Then we (i) justify this definition by showing that it
gives the right extremal limit of the first law, (ii) derive from the Noether charge a statistical
version of the first law suitable for extremal black holes and (iii) re-derive the entropy function
directly from the definition of the entropy. Finally, we discuss the notion of mass as seen from
the NHG by deriving a Smarr-like formula.
6.1 Poincaré Time Hamiltonian
For the Poincaré time Killing vector ∂t, one expects the Noether charge to be related to the
Hamiltonian, which we will explore now.
Since the theory is generally diffeomorphism invariant, we expect the bulk contribution to
vanish. So we concentrate on boundary terms Sbdy. =
B Lbdy., that are necessary to cancel total
derivatives dΘ in the variation of the bulk action δS =
(Eiδφ
i + dΘ(δφ)). In our example,
we have to consider both the variations of the metric and of the 3d gauge fields. For the gauge
fields, the term that we ignored in the derivation of the equations of motion was
µ = ∂µ
δ Aν,µ
δAν +
δ Baν,µ
. (72)
For a complete spacetime, the textbook answer is to place the usual restriction δA|bdy. =
δB|bdy. = 0. Then, the only boundary term that one needs to add in order to make the vari-
ational principle consistent is a Gibbons-Hawking-like term, that compensates for a variation
proportional to the normal derivative of δg at the boundary. For the Einstein-Hilbert action,
that is the usual Gibbons-Hawking term
SGH =
LGH =
hK = − VT 2
16πG5
hγMNn
M ;N , (73)
where γ is the boundary metric and K is the surface gravity of the boundary B, which, in our
geometry, is just an S1 fibred over time. Note that we took n = −∂r to be inward-pointing
in order to define the bi-normal NMN :=
(∂t)[MnN]
|∂t||n| of Σbdy. with a positive signature. Now,
we can read off the Hamiltonian of the NHG if it were an isolated solution. By definition,
Lξgµν = 0, such that the canonical Hamiltonian is just HI = −
i∂tLGH with the time
slice of B being Σbdy = S1. Since ∂t is a Killing vector, a quick calculation shows |∂t|
−γK =√
−gNMN (d ∂̂t)MN , and hence the Hamiltonian is just
HI = −
Σbdy.
i∂tLGH =
16πG5
hNMN (d ∂̂t)
MN . (74)
Now, if we consider the near-horizon geometry being embedded in the full black hole solution,
we cannot put δA|bdy. = δB|bdy. = 0, but we need to satisfy the variational principle by adding
a Hawking-Ross-like boundary term as in [28]:
LHR = nM
δ AN,M
=: −nN
Q̃MNAN + J
and impose the condition to keep the charges fixed under variations of the boundary fields. Now,
the boundary action varies as:
δSHR = −
d2σ nM
δQ̃MN
δJMNa
d2σ nM
Q̃MNδAN + J
where the second term cancels the total derivative in the variation of the bulk action (note the
inward-pointing n), and the first term vanishes as the charges are fixed. A little caveat occurs if
we use the gauge-dependent form of the action (34), when Q̃ 6= Q, however the missing bit does
not depend on the 3d gauge fields, but only on the scalar fields, and hence it is invariant under
variations of the gauge fields. If we consider the gauge-independent form of the action (70), then
Q̃ = Q. Again, by definition we have LξBi = 0, and we will choose a gauge such that LξA=0,
and the canonical Hamiltonian is just
H = −
i∂t(LHR + LGH) . (77)
Because of the AdS2 symmetries, we have
Σbdy.
i∂t (Q∧A) =
Σbdy.
Q(i∂tA) and similar for Ji∧Bi.
This puts the Hawking-Ross contribution to the boundary Hamiltonian to−
Σbdy.
dθ NMN
Q̃MN (i∂tA)+
JMNa (i∂tB
. This gives for the action (34)
H = − VT 2
16πG5
dθNMN
(d ∂̂t)
MN + HaMNhab(i∂tB
b)+4FMN i∂t
ǫPMNǫabAa,PAb i∂t
We now compare (78) with the Noether charge obtained by dimensional reduction of the 5d
expression (20). For this, we work out how the individual terms look like in 3d with the notation
of section 3.1. We consider only the components QMN
in the non-compact directions, and only
zero modes of the fields in the compact directions. Hence we get from the reduction formulae
(31 - 34):
(dξ̂)MN =
dξ̂3d
ξ3d ·Bjhji + χihij
H iMN , FMN = FMN ,
ǫMNαβγ = 2ǫMNLǫijAi,LAj and ξ ·A = ξ3d ·A3d + ξ3d ·BiAi + χiAi . (79)
Now, we can write down the charges of ξ3d, the non-compact components of ξ, and χ, its compact
components, separately:
QMNξ3d = −
16πG5
dξ̂3d
+ ξ3d ·Bj
j MN + 4AiF
+ 4ξ3d ·A3dFMN
ξ3d · A3d + ξ3d ·BiAi
ǫMNLǫijAi,LAj
QMNχ = −
16πG5
iMN + 4AiF
ǫMNLǫkjAk,LAj
, (81)
where we have implicitly done an integration over the compact coordinates. Thus we see that
(78) is just the Noether charge Q∂t in 3d (80) as expected, and we have yet another confirmation
of the KK charge (35), as it matches with (80).
6.2 Entropy
The entropy S of non-extremal black holes was shown by Wald [5] to be given by the Noether
charge κS = 2π
Qξ of the timelike Killing vector ξ that generates the horizon, evaluated on
the bifurcate d-2 surface B of the horizon, and κ is the surface gravity of the horizon. Jacobsen,
Myers and Kang [19] later showed that the charge can be evaluated anywhere on the horizon,
provided all fields are regular at the bifurcation surface. After a coordinate transformation, one
sees that this requires all gauge fields to vanish on the horizon, such that the gauge is fixed to
ξ · A = 0 at the horizon, and hence eliminates the ambiguity of the gauge-dependence of the
Noether charge.
For extremal black holes, κ = 0 on the horizon (r = 0), so Wald does not give a suitable
definition of S, and furthermore there is no bifurcation surface - putting in doubt the gauge fixing.
In the AdS NHG, there should be no special point where to compute physical quantities. Using
the concept that the entropy is intrinsic to the horizon, and hence does not require embedding
the NHG into an asymptotic geometry, those problems are cured by defining the entropy as
κ(rbdy.)
HI(rbdy.) , (82)
in the dimensionally reduced theory with the boundary placed at any radius rbdy. 6= 0. The fact
that the 3d theory is static allows us to use
κ = −
gtt,r
−gttgrr
[9] that is well-defined and physically motivated as the acceleration of a probe at any radius r
with respect to an asymptotic observer and hence related to the temperature of Unruh radiation.
It also ensures that the entropy is independent of rbdy. with well-defined limits rbdy. → 0 and
rbdy. → ∞. Now, in terms of the Noether charge (80), the entropy is just as expected
Q∂t(r) (84)
in the gauge ξ · A(r) = ξ · B(r) = 0; but evaluated at r 6= 0, rather than r = 0 that one would
näıvely expect. We will see in the following three subsections that this definition of the entropy
naturally arises from black hole thermodynamics.
6.3 First Law
Since we have now an expression for the entropy intrinsic to the extremal limit, let us see whether
we can also find an expression for its variation as derived for non-extremal black holes by Wald
in [5]. First let us write the the Noether charge for the gauge-invariant action (70) in 3d for
ξ3d = ∂t as
Qξ3d(r) =
S − ξ3d · A(r)Qel. − ξ3d · Ba(r)Ja . (85)
Then, we consider variations of the dynamical fields δφi that keep the solution on-shell and use
the identity δdQξ3d = d
ξ3d · Θ
[5], with Θ defined in section 2, such that we can relate the
variation of the charge evaluated over two boundaries Σ1 and Σ2 of a spacelike d-1 surface:
δQξ3d − ξ3d ·Θ
δQξ3d − ξ3d ·Θ
. (86)
Now, let us move the boundaries into the near-horizon geometry (→ ΣH) and into some asymp-
totic limit (→ Σ∞). On ΣH , we have
ξ3d ·Θ =
L dθM ǫLMN
gOP δ̄ΓNOP + g
ON δ̄ΓPOP
δAO,N
δAO +
δκ − Qelδ(ξ3d ·A) − Jiδ(ξ3d ·Bi) , (87)
where we used for the second equality the AdS2 isometries, and assumed an Einstein-Hilbert
term for the gravitational action, and any gauge field term that can be written with only first
derivatives of A, such as (70). The right hand side of (86) can be interpreted by following Wald,
and defining the canonical energy, i.e. the Hamiltonian measured by an asymptotic observer at
Σ∞, E =
(Qξ3d − ξ3d·V ) with some d-1 form V: δ
ξ3d ·V =
ξ3d ·Θ. This corresponds,
for the asymptotic boundary conditions A = B = 0 and suitable normalization of ξ3d, to the
mass. Altogether, (87) gives us now an expression similar to the first law
δS + Φ(r) δQel. + Ω
i(r) δJi = δE (88)
at some r 6= 0, where Φ(r) = −ξ3d · A(r) and Ωi(r) = −ξ3d · Bi(r) measure the co-rotating
electric potential and angular frequency2 at r in the NHG with respect to the definition of E .
This, however is not yet a relation for the full black hole, but captures only physics outside Σr.
The extremal limit of the non-extremal first law of the full black hole solution is reproduced by
taking the limit r → 0:
ΦH δQel. + Ω
H δJi = δE , (89)
where ΦH = −ξ3d ·A(0) and ΩH = −ξ3d ·B(0) are the horizon co-rotating electric potential and
angular frequency. It is interesting to observe though, that (88) and corresponding expressions
for the Smarr formula resemble the first law of a finite temperature black hole, even though its
physical significance is limited, as Σrfor r 6= 0 is not a horizon.
An interesting observation and lesson is that when embedding the near horizon solution into
an asymptotic solution, but computing Noether charges in the NHG, we need to use the gauge
invariant action (70) and the full Noether charge, because there is no boundary of the NHG on
which we were allowed to fix the gauge fields and its gauge variations.
2To illustrate that this definition of Ω corresponds to the one in [5], consider a vector ξ = ∂t − Ω∂φ in static
coordinates with a diagonal metric g, and ξ = ∂t′ in co-rotating coordinates with a non-diagonal metric g
′. Then
ξ̂ = gttdt− Ωgφφdφ = gt′t′dt′ + Bφt′gφφdφ. A similar argument follows from requiring constant normalization of ξ and
considering gtt + gφφ = gt′t′ in the explicit coordinate transformation.
We see that our version of the first law also holds also for perturbations away from extremality,
which connects it smoothly (in a thermodynamic sense) to the near-extremal limit of the non-
extremal black hole, again supporting our definition of the entropy.
6.4 Entropy Function and the Euclidean Action
Now, let us continue following Wald [5] and relate the (integrated) mass (or energy E) to the
entropy. Starting with (85), we apply Gauss’ law to find
S − ξ3d ·A(r)Qel. − ξ3d · Ba(r)Ja = E −
Jξ3d +
ξ3d · V =: E −
I(r) , (90)
where the euclidean action3 I is now, in principle, a function of the radial position of ΣH , since
∂M = {ΣH ,Σ∞}. Even though I is defined only for κ 6= 0 as the integral of the analytically
continued Lagrangian, with τ = it having period 2π
, one would like to find a well-defined limit
as κ→ 0, i.e. r → 0, representing the full extremal black hole solution. This requires
ΦHQel. + Ω
HJa = E . (91)
This relation can be taken as a (gauge-dependent) definition of the mass of the black hole in
the near-horizon geometry. We note that since the action is gauge-invariant, (91) is gauge-
independent in the sense that a gauge transformation that changes ΦH and ΩH on Σ0 changes E
at Σ∞ accordingly. In the appropriate gauge in which E =M , it should agree with the BPS (or
extremality) condition - as we verified for BMPV and GR - and with an applicable Smarr-like
formula, supposed one has a full solution at hand. Now, let us study the remaining terms of (90).
Again, we make use of the AdS2 geometry to find that ξ3d ·
A(r)−A(0)
�κ(r) = F 3drt =: −EH
is the constant co-rotating electric field-strength in the NHG, as is ξ3d ·
Bi(r)−Bi(0)
�κ(r) =
Hrt =: −HH the field strength of the KK gauge field. Now, (90) reads
S = −2π
EHQel. + H
− I , (92)
with all terms, including I, being independent of the position r 6= 0 of ΣH in the NHG. (92)
holds also in the limit as r → 0. A similar expression was proposed and discussed in a statistical
context by Silva in [6], where it was motivated by taking the extremal limit of non-extremal black
holes, assuming an appropriate expansion of ΦH and ΩH in terms of the inverse temperature.
This is identical to (92), provided one identifies the NHG field strengths with the appropriate
expansion coefficients in [6]. Note that this relation is particular for extremal black holes and
profoundly different from the relation of the entropy to the euclidean action for non-extremal
black holes [29, 30].
Let us now show how this relates to the entropy function formalism. Given I = −2π
M iξ3dL +
iξ3dV
[5], we use the fact that the spacetime in the NHG can be trivially
foliated with spheres to re-write this as
I = −
iξ3dL +
iξ3dV −
iξ3dL
=: I0 +
L , (93)
where ∂M0 = {Σr=0,Σ∞}. Since
L is supposed to be invariant under the AdS2 isometries,
it is proportional to the volume form on AdS2 and (
L)�κ(r) = ⋆
L = const. Now,
the fact that I = const. implies that I0 = 0 and we are left with
S = −2π
EHQel. + H
HJi + ⋆
. (94)
3I equals the euclidean action only for stationary spacetimes, see [5].
This is just the entropy function for the gauge invariant action (70). The same derivation can
be applied to the original action (34) to give its corresponding entropy function. In that case E
in (91) will have a different value, because of the boundary terms in the action, stressing again
the need to work with (70) when relating the NHG to the asymptotic geometry.
6.5 Mass
Even though the mass of extremal black holes is fixed by the extremality (or BPS) relation 91,
let us now study its physical interpretation from the point of view of the NHG by deriving a
Smarr-like formula for the 5d Einstein-Maxwell-CS case.
Let us suppose there is some asymptotic geometry attached to the near horizon geometry in a
way that the conditions in section 3.3 are satisfied, and follow closely the derivation by Gauntlett,
Myers and Townsend in [4] for a few steps. The mass, E in a gauge in which A = B = 0 at Σ∞,
can be re-written using Gauss’s law in 5d as
M = −
16πG5
⋆dk̂ =
16πG5
⋆dk̂ +
, (95)
for some ∂M = {Σ,Σ∞} and k being the asymptotic unit norm timelike Killing vector. Assuming
we work in a gauge in which LξA = 0, and using the relations ✷kµ = −Rµνkν , LkΩ = ik(dΩ) +
d(ikΩ) for any form Ω and the equations of motion for g and A, the result is
16πG5
⋆dk̂ + 4(k · A) ⋆ F −
k̂ ∧ (Â · F )
(k · A)A ∧ F
, (96)
plus a term at Σ∞ that vanishes as A→ 0. In dimensions other than d = 5, there will be an extra
term that cannot be expressed as a surface integral at ΣH . For details see [4]. Now, we see that
the first, second and last terms combine to give the Noether charge (19). Decomposing k into
its compact and non-compact components, k = ∂t + Ω
iχi, and choosing Σ to be an r = const.
surface in the NHG, we find from the 3d expressions (80,81) that this gives us
S + ΩiJi
+Φ(r)Qel.−
(∂t · A) ⋆ F −
(∂̂t +Ω
iχ̂i) ∧ (Â · F )
In (∂̂t + Ω
iχ̂i) ∧ (Â · F ), we find that in terms of frame fields the relevant components are
(∂̂t+Ω̂
iχi)0, A0 and F01, since the AdS2 symmetries restrict non-vanishing FM1 to M = 0. This
makes the last term vanishing, such that we get in the limit r → 0 the Smarr formula
ΩiHJi + ΦHQel. , (98)
that agrees with the near-horizon limit of the non-extremal one. From the point of view of
the near-horizon solution, we find that the mass is now a gauge-dependent expression, with the
gauge given by the embedding of the near-horizon solution in the asymptotic solution. We find
that (98) looks different from (91), however they are in agreement since ΩH vanishes for BMPV
black holes [4].
7 Conclusions
In this paper we presented expressions for conserved currents and charges of 10d type IIB
supergravity (with the metric and five-form) and minimal (gauged) supergravity theories in 5
dimensions. These have been obtained following Wald’s construction of gravitational Noether
charges. Those of the 5d gauged supergravity can also be obtained by dimensional reduction of
the 10d formulae. We further showed that the Noether charges of the higher dimensional theories,
after dimensional reduction, match precisely with the Noether charges of gauge fields obtained
by Kaluza-Klein reduction over the compact Killing vector directions of interest. Our expressions
for the charges should be valid generally for both extremal and non-extremal geometries. We
then turned to their applications to extremal black holes and demonstrated that, when evaluated
in the near horizon geometries, our charges reproduce the conserved charges of the corresponding
extremal black holes under certain assumptions. In particular, we exhibited that our methods
give the correct electric charges and angular momenta for the BMPV and Gutowski-Reall black
holes.
A host of new solutions to supergravity theories with AdS2 isometries have been found
recently [20] and many more such solutions are expected to be found in the future. These
solutions may be interpreted as the near horizon geometries of some yet to be found black holes.
In such cases, our results should be useful in extracting the black hole charges without having to
know the full black hole solutions but just the near horizon geometries. On the other hand, the
holographic duals of string theories in the NHG are expected to be supersymmetric conformal
quantum mechanics. Our conserved charges should be part of the characterising data of these
conformal quantum mechanics.
We argued that the black holes with AdS3 near horizons do not satisfy our assumptions when
embedded in black hole asymptotes with Sd−2 isometries (rather than black string asymptotes).
Supersymmetric black rings are the main examples for which our formulae do not seem to apply.
More generally for black holes with AdS3 one has to find the correct way to extract the conserved
charges separately which we would like to return to in future.
We then presented a new entropy function valid for rotating black holes in 5d with CS terms
which gives the correct electric charges as well as the entropy. This is an improvement over [21].
We used appropriate boundary terms, that make the action fully gauge-independent which turns
out to be relevant to obtain the thermodynamics in the second part of the paper.
In the second part of the paper we exhibited a new definition of the entropy as a Noether
charge, and a derivation of the first law, which are applicable for extremal black holes directly.
We used this definition to produce the statistical version of the first law and moved on to re-
derive the entropy function from a more physical perspective. Finally, we commented on the
physical interpretation of the mass in the near-horizon solution. The relevant calculations were
done in the near-horizon geometry, only assuming an embedding into some asymptotic solution
for the purpose of formally defining the Mass. We did not, however, produce a conserved charge
corresponding to the the level number. In terms of the 5d fields, the expression in [27] is just
proportional to
⋆F , which is conserved in the NHG by the symmetries, but not by the
equations of motion in a general geometry. Various potentially interesting candidates, such as
the R-charge and global AdS2 time Noether-Wald charge did not produce an interesting result.
We find that the gauge-independent thermodynamic quantities can be evaluated everywhere
in the near-horizon geometry, as they are a statement about the near-horizon geometry. In
particular, they are the entropy, euclidean action and charges and their chemical potentials,
as well as the statistical version of the first law (92). Relations and quantities related to the
asymptotic geometry and to thermodynamics of non-extremal black holes (the mass, horizon
electric potential and angular frequency, as well as the first law and Smarr formula) however are
gauge-dependent from the point of view of the near-horizon geometry. They need to be evaluated
on a specific hypersurface, r = 0, as they come from position-dependent statements in the near-
horizon geometry. This means that the former ones may be more relevant for characterising
attractors.
Acknowledgements
We thank Rob Myers for helpful discussions and suggestions and helpful comments on the
manuscript. MW was supported by funds from the CIAR and from an NSERC Discovery grant.
Research at the KITP is supported in part by the National Science Foundation under Grant No.
PHY05-51164 and research at the Perimeter Institute in part by funds from NSERC of Canada
and MEDT of Ontario.
A Black Rings
The non-equal angular momentum generalization of the BMPV case is the supersymmetric black
ring [18]. It is an excellent counter-example in which the conditions in section 3.3 are not satisfied.
To demonstrate this, we sketch out the derivation of the asymptotic and near horizon limits as
given in [18]. The general form of the solution is given by:
ds1 = −f2(dt + ωφdφ + ωψdψ)2 +
f−1R2
(x−y)2
( dy2
+ (1−x2)dφ2 + (y2−1)dψ2
f(dt+ ω) − q
(1 + x)dφ + (1 + y)dψ
, (99)
where y ∈]−∞,−1] , x ∈ [−1, 1] , φ, ψ ∈ R�2πZ and f−1 = 1 + Q−q
(x − y) − q
(x2 − y2),
ωφ = − q8R2 (1− x
3Q− q2(3 + x+ y)
and ωψ =
(1 + y) +
(1− y2)
3Q− q2(3 + x+ y)
The asymptotic limit is given by (x + 1) → +0 and (y + 1) → −0, and its geometry of a
squashed sphere with broken isometry SO(4) → U(1)2 can be made manifest by combining
(x, y) into a radial coordinate ρ ∈ R+ and an angular coordinate Θ ∈ [−π2 ,
ρ sinΘ =
x−y and ρ cosΘ =
x−y (100)
The near horizon limit, on the other hand, is given by y → −∞, such that appropriate radial
and angular coordinates are r = −R
and cos θ = x. A first observation is that the two limits
are just points in the “opposite” coordinates, (ρ,Θ) → (R, π
) and (r, θ) → (R,π). To obtain
the near horizon geometry in a suitable form, we define χ = φ − ψ, take the limit r = ǫr̃R−1,
t = ǫ−1t̃, ǫ→ 0 and get:
ds2 =
q2dr̃2
dt̃dψ +
(q2 −Q)2 − 4q2R2
dψ2 +
dθ2 + sin2θdχ2
A = −
(q2 +Q)dψ + q2(1 + cos θ)dχ
. (101)
Now, we also see that the topology of the horizon is S1×S2 with U(1)×SO(3) ∋ U(1)2 isometry
and whose subgroup U(1)2 is not guaranteed to agree with the U(1)2 of the asymptotic geometry.
The AdS2 geometry is more apparent after dimensional reduction, when gtt ∝ r̃2 is restored,
and after suitably rescaling t̃. [18] show furthermore that the AdS2 and S
1 combine into a local
AdS3. The conserved charges are now Jψ =
(q2 −Q)2 − 12q2R2
, Jχ = − π8G5 q(q
2 +Q)
and Qel. =
(q2+Q), or in the old coordinates Jψ =
(q2−Q)2+2q2(q2−2Q−6R2
qQ . They compare to the asymptotic quantities computed in [18] Jψ =
q(3Q− q2),
q(6R2 + 3Q− q2) and Qel. =
The distinguishing feature here is that black rings have an AdS3×S2 near-horizon geometry.
Thus the S1 × S2 of the horizon and the S3 of the asymptotic hypersurface are topologically
distinct, such that there is no continuous fibration of hypersurfaces over r between them. In
particular, The coordinates that describe the asymptotic S3 shrink the horizon and the area
bounded by the black ring to a point in 3d (or an S1 × S1 in 5d), and are missing part of the
boundary of the full solution because of the difference in topology. This missing part shrinks
into the coordinate singularity that also contains the horizon, so flux that passes though that
part of the boundary will not be seen from the asymptotic geometry.
It is not inconceivable that if we consider the black rings on Taub-Nut spaces like in [31, 32, 33]
and obtain a 4d black hole which satisfies our criteria one may yet be able to recover the charges
of such black rings.
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	Introduction
	Charges from Noether-Wald construction
	Review of Noether construction
	The Noether-Wald charges for type IIB supergravity
	The Noether-Wald charges for 5d Einstein-Maxwell-CS
	Reduction from 10 dimensions
	Charges from dimensional reduction
	Dimensional reduction
	Oxidation of the angular momentum
	Generalization and Limitations
	Relation to the Asymptotes
	Gauge Issues
	Examples
	The 10d Gutowski-Reall black hole
	5d Black Holes
	Equal Angular Momenta: BMPV and GR
	Non-equal Angular Momenta: Supersymmetric Black Holes
	Charges from the entropy function
	Completing the equations of motion
	Gauge invariance from boundary terms
	Thermodynamic Charges
	Poincaré Time Hamiltonian
	Entropy
	First Law
	Entropy Function and the Euclidean Action
	Mass
	Conclusions
	Black Rings

  We describe how to recover the quantum numbers of extremal black holes from
their near horizon geometries. This is achieved by constructing the
gravitational Noether-Wald charges which can be used for non-extremal black
holes as well. These charges are shown to be equivalent to the U(1) charges of
appropriately dimensionally reduced solutions. Explicit derivations are
provided for 10 dimensional type IIB supergravity and 5 dimensional minimal
gauged supergravity, with illustrative examples for various black hole
solutions. We also discuss how to derive the thermodynamic quantities and their
relations explicitly in the extremal limit, from the point of view of the
near-horizon geometry. We relate our results to the entropy function formalism.

Introduction
Studies of extremal black holes in string theory have regained importance with the advent of
the attractor mechanism. In its simplest form the attractor mechanism states that the near
horizon geometry of an extremal black hole is fixed in terms of its charges. Further, it has been
realized that there is a single function, called the entropy function, which determines the near
horizon geometry of extremal black holes [1] (see also [2]). Even though the entropy function
provides the non-zero charges such as the electric, magnetic charges and angular momenta, for
many extremal black holes, it does not always give the correct charges. For instance, there are
apparent discrepancies when there are Chern-Simons terms for the gauge fields present in the
Lagrangian. This is the case, for instance, in 5d minimal (and minimally gauged) supergravities.
On the other hand it has been believed [4] that the near horizon geometry of an extremal
rotating black hole of 5d supergravities knows about only part of the the full black hole angular
momentum, called the horizon angular momentum. In [4] this has been argued to be the case
for the BMPV black hole [16].
Given that finding the near horizon geometries of the yet to be discovered extremal black
hole solutions might be easier than finding the full black hole solutions, it will be useful to
have a prescription to extract the quantum numbers of the full black hole from its near horizon
geometry. In this note we show, by careful analysis of the near horizon geometries of these black
holes, that one can find the full set of asymptotic charges and angular momenta of extremal
rotating black holes that satisfy certain assumptions.
For this, we first construct gravitational Noether charges following Wald [5] for several su-
pergravity theories. These charges can be defined for Killing vectors of any given solution of the
theory of interest. We mainly focus on type IIB in 10d, minimal and gauged supergravities in 5d.
We present closed form expressions for the Nother-Wald charges of these theories as integrals
over compact submanifolds of co-dimension 2 of any given solution.
The 5d minimal gauged supergravity can be obtained by a consistent truncation of type IIB
reduced on S5 [22] (see also [23]). We show that the charges of the 5d theory can be obtained by
the same dimensional reduction of the corresponding 10d charges. We further reduce the theory
down to 3 dimensions and show that the Nother-Wald charges corresponding to Killing vectors
that generate translations along compact directions are the same as the usual Noether charges
for the corresponding Kaluza-Klein gauge fields in the dimensionally reduced theory. We use
the understanding of the charges in the reduced theory to show how the entropy function may
be modified to reproduce the charges of the 5d black holes.
We will argue that these Noether-Wald charges can be used to extract the charges of extremal
black holes from their near horizon geometries under certain assumptions which will be discussed
later on. Thus the formulae presented in this paper should prove useful in extracting the con-
served charges of an extremal black hole from only its near-horizon geometry without having to
know the full black hole solution. We exhibit the successes and limitations of our formulae by
considering the examples of Gutowski-Reall black holes [12] and their generalizations [17] and
BMPV [16, 4] black holes, black rings [18] and the 10d lift of Gutowski-Reall black holes [13].
The analysis of the conserved charges in this paper can be applied to many geometries other
than the extremal black holes considered here and in particular to non-extremal black holes too.
In addition to the charges of a black hole, one is typically interested in the entropy, the mass,
as well as the laws of black hole thermodynamics. Up to now, the entropy has been defined in
terms of a Noether charge only for non-extremal black holes [5]. To find these thermodynamic
quantities and the laws of thermodynamics on the “extremal shell”, it was necessary to take
the extremal limit of the relations defined for the non-extremal black holes (see for instance [1]).
Furthermore, computations of quantities such as the mass, the euclidean action and relations like
the first law and the Smarr formula relied on computing quantities in the asymptotic geometry.
Hence, it would be desirable to derive appropriate relations intrinsically for extremal black holes,
and with only minimal reference to the existence of an asymptotic geometry.
With this motivation, in the second part of the paper, we propose a definition of the entropy
for extremal black holes in the near horizon geometry that does not require taking the extremal
limit of Wald’s entropy, but agrees with it. With a similar approach, we also derive the extremal
limit of the first law from the extremal geometry, assuming only that the near-horizon geometry
be connected to some asymptotic geometry. This definition of the entropy further allows us
to derive a statistical version of the first law [6]. We also show that this gives us the entropy
function directly from a study of the appropriate Noether charge in the near-horizon geometry
of extremal black holes. We will comment on the interpretation of the mass as well, from the
point of view of the near horizon solution.
The rest of the paper is organized as follows. In section 2, we review Wald’s construction of
gravitational Noether charges and use it to derive the charges for type IIB supergravity (with
the metric and the five-form fields) and for the 5d minimal and gauged supergravity theories and
show that they are related by dimensional reduction. In section 3, we show that the Noether-
Wald charges are identical to the standard Noether charges for the Kaluza-Klein U(1) gauge
fields of the corresponding compact Killing vectors. We also discuss various assumptions under
which these charges, when evaluated anywhere in the interior of the geometry, match with the
standard Komar integrals evaluated in the asymptotes. Some issues of gauge (in)dependence
of our charges are also address there. In section 4, we demonstrate how our formulae work on
several examples of interest. The readers who are only interested in the formalism may skip this
section. In section 5, we turn to modifying the entropy function formalism to include the Chern-
Simons terms. In section 6, we discuss thermodynamics of the extremal black holes and define
various physical quantities like the entropy, chemical potentials for the charges and the mass.
We end with conclusions in section 7. The example for black rings is given in the appendix.
2 Charges from Noether-Wald construction
Here we derive expressions for the gravitational Noether charges corresponding to Killing isome-
tries of the gravitational actions we are interested in following Wald [5, 7]. We review first
the general formalism and point out some relevant subtleties. Then we construct these charges
for 10d type IIB supergravity and for minimally gauged supergravity and Einstein-Maxwell-CS
theory in 5d. Finally, we show how the 10d and 5d expressions can be related by dimensional
reduction.
2.1 Review of Noether construction
Let us first review the construction of the charges and discuss some of the relevant properties. In
[7], Lee and Wald described how to construct the Noether charges for diffeomorphism symmetries
of a Lagrangian L(φi = gµν , Aµ, · · · ), a d-form in d spacetime dimensions. For this, one first
writes the variation of L under arbitrary field variations δφi as
δ L = Ei(φ) δφ
i + dΘ(δφ) (1)
where Ei(φ) = 0 are the equations of motion and Θ is a (d − 1)-form. Secondly, one finds the
variation of the Lagrangian under a diffeomorphism
δξL = d(iξ L), (2)
where ξa is the (infinitesimal) generator of a diffeomorphism. Then one defines the (d− 1)-form
current Jξ
Jξ = Θ(δξφ)− iξ L (3)
where δξφ
i are the variations of the fields under the particular diffeomorphism. Then Jξ are
conserved, i.e. dJξ = 0, for any configuration satisfying the equations of motion. Since Jξ is
closed, one can write (for trivial cohomology)
Jξ = dQξ (4)
for some (d − 2)-form charge Qξ. Now consider ξ to be a Killing vector and suppose that the
field configurations on the given solution respect the symmetry generated by it, Lξφi = 0. Since
Θ(δξφ
i) is linear in Lξφi we have Θ(δξφi) = 0 and so Jξ = −iξL. Next, let us illustrate that the
charge defined as the integral
Qξ over a compact (d-2)-surface Σr is conserved when (i) ξ is
a Killing vector generating a periodic isometry or (ii) when the current Jξ = 0 (as for Killing
vectors in theories with L = 0 on the solutions). Consider a (d − 1)-hypersurface M12 which is
foliated by compact (d−2)-hypersurfaces Σr over some interval R12 ⊂ R. Using Gauss’ theorem
one has
Jξ (5)
for ∂M12 = {Σ1,Σ2}. If Jξ = 0, it follows that the charge
Qξ does not depend on Σr and
therefore is conserved along the direction r. Next, let us assume that ξ generates translations
along a periodic direction of Σr. In general,
Jξ receives contributions from terms in Jξ that
contain the one-form ξ̂ dual to the Killing vector field ξ and terms that do not. The terms not
involving ξ̂ vanish by the periodicity of ξ. Since Jξ = −iξL, there are no terms involving ξ̂.
Therefore
Qξ is again independent of Σr.
We will now discuss two important ambiguities in the above prescription. The first one is
that the charge density defined by the equation Jξ = dQξ is ambiguous as Qξ → Qξ + dΛξ does
not change Jξ for some (d-3)-form Λξ. The extra term does not contribute to the integrated
charge only if Λξ is a globally defined (d-3)-form on Σr, that is, it is periodic in the coordinates
of Σr and non-singular. While this is the case for most of our examples, there may be situations
in which, for instance, some gauge potentials that go into Qξ are only locally defined. Similarly,
conservation of Qξ is not guaranteed if any component of Qξ ∈ Ωd−1
is not globally defined.
To illustrate this, consider the Jξ = dQξ = 0 case and let n be a normal to Σr, such that dn = 0.
Qξ = (ind)
indQξ +
d (inQξ) =
d (inQξ) , (6)
which is only forced to vanish if inQξ is globally defined on Σr. The second, and a more impor-
tant, ambiguity comes from possible boundary terms in the Lagrangian L. For the boundary
terms Sbdy. =
Lbdy. =
dLbdy., the variation that gives the equations of motion is done on
the boundary,
δξSbdy. =
δLbdy.
δLbdy.
δξLbdy. =
d(δξLbdy.). (7)
Since δξLbdy. = iξ(dLbdy.) + d(iξLbdy.), the current is just given by
Jξ = −iξ(dLbdy.) + iξ(dLbdy.) + d(iξLbdy.) (8)
and hence the charge is Qξ = iξLbdy.. This implies that boundary terms contribute only to
conserved charges
Qξ of (Killing) vectors that do not lie in Σr.
2.2 The Noether-Wald charges for type IIB supergravity
Now we would like to find the Noether-Wald charges in 10d type IIB supergravity for config-
urations with just the metric and the 5-form turned on. As is standard, we work with the
action
LIIB =
16πG10
−g [R−
4 · 5!
F 2(5)] (9)
neglecting the self-duality of the 5-form and impose it only at the level of the equations of motion.
We follow the procedure outlined in section 2.1 to find the Noether-Wald currents. Using the
variations
−g R) =
−g [Rµν − 12Rgµν ] δg
−g gµν [∇σδ̄Γσµν −∇ν δ̄Γσµσ] and
−g F 2(5)) =
−g [5F (5)
µκσωλ
F (5) κσωλν − 12 gµν F
(5)] δg
−2 · 5![δC(4)
−g Fµνσωλ
)− ∂µ(δC
µνσωλ
−g)], (10)
where δ̄Γλµν =
gλσ [∇µδgσν +∇νδgµσ −∇σδgµν ], one can find the equations of motion
Rµν −
µκσωλ
F (5)ν
= 0 and ∂µ(
−g Fµνσωλ
) = 0. (11)
These are supplemented by the self-duality condition ⋆(10)F
(5) = F (5). The self-duality constraint
F (5) = ⋆F (5) implies that F 2
= 0, and then the metric equation of motion in (11) implies R = 0
for any solution. Hence the Lagrangian vanishes on the solutions and therefore the Noether-Wald
current in (3) is given entirely by the 9-form Θ (or equivalently by its dual vector field). This
can be found from the total derivative terms in δL by substituting δξg
µν = ∇µξν +∇νξµ and
and δξC
= 4 ∂[ν|(ξ
θ|σωλ]) + ξ
θνσωλ
. This gives us the current
J α = −2
−g gασ [Rσλ −
λνθωλ
F (5) νθωλσ ]ξ
+∂µ[−
−g gµνgασ(∇νξσ −∇σξν) +
2 · 3!
−g ξθC(4)
αµσωλ
] , (12)
where the first term vanishes by the equations of motion and the second term gives us the charge
density
16πG(10)
∇αξµ −∇µξα +
αµσωλ
. (13)
Noting that the self-duality constraint
−g Fµ0···µ4
ǫµ0···µ9F
µ5···µ9 implies
αµσωλ
= ξνC
3! 5!
ǫαµσωλµ5···µ9F
µ5···µ9 , (14)
the Noether-Wald charge density (13) can be equivalently written as the 8-form
= − 1
16πG10
⋆ dξ̂ − 1
(4) ∧ F (5)
where ξ̂ is the dual 1-form of the vector field ξµ. This can be integrated over a compact 8d
submanifold to get the corresponding conserved charge. A quick calculation verifies that the
current for this charge vanishes identically as expected because of the vanishing Lagrangian.
Hence, all charges that are computed from it are conserved as discussed in section 2.1. If we
further assume that LξC(4) = 0, we have iξF (5) = −d(iξC(4)). This can be used to rewrite (15)
16πG(10)
⋆ dξ̂ +
C(4) ∧ iξF (5)
up to an additional term proportional to d(C(4) ∧ iξC(4)). This extra term does not contribute
when integrated over a compact 8-manifold provided that C(4) ∧ iξC(4) is a globally well defined
7-form as we discussed in section 2.1. In such cases (16) can be used instead of (15).
In section 4, we will demonstrate that this formula reproduces conserved charges [12] of
Gutowski-Reall black holes of type IIB in 10 dimensions successfully. We hope this expression
may be useful in obtaining the charges of the yet to be discovered black holes from their near
horizon geometries alone.
2.3 The Noether-Wald charges for 5d Einstein-Maxwell-CS
The action for 5d Einstein-Maxwell-Chern-Simons gravity is
16πG5
−g (R− FµνFµν)−
ǫmnpqrAmFnpFqr
which is the same as the action for the 5d minimal gauged supergravity up to the cosmological
constant, which turns out not to contribute to the Noether charge. After a straight forward but
slightly lengthy calculation it is easy to show that the Noether current for this action is
16πG5
(Rαλ − 1
gαλR)− 2 (F λµFαµ − 14g
λαF 2)
+ 4 (ξ · A)
−gFαµ) + 2√
ǫανσωλFνσFωλ
−ggµνgαλ (∇νξλ −∇λξν)− 4
−g(ξ · A)Fαµ − 8
(ξ · A) ǫαµσωλAσFωλ
.(18)
The first two lines are simply proportional to the equations of motion and vanish on-shell and
hence the Noether-Wald charges for this theory are
16πG5
−g (∇αξµ −∇µξα) + 4(ξ ·A)(
−g Fαµ +
ǫαµσωλAσFωλ)
. (19)
These expressions have also appeared recently in [8]. An alternative derivation of (19) in terms
of KK charges will be presented in section 3.3. The charge density (19) can equivalently be
written as the 3-form
16πG5
⋆dξ̂ + 4 (iξA)
⋆ F − 4
A ∧ F
. (20)
As before the charges can be obtained by integrating Qξ over a 3d compact sub-manifold. Note
that if we set the gauge fields to zero we recover the standard Komar integral for the angular
momentum.
2.4 Reduction from 10 dimensions
Now, we will find the dimensional reduction of the 10d formula of conserved charges to the 5d
formula to show that they are indeed identical, so let us first review the reduction formulae to
obtain the equations of motion of 5d minimal gauged supergravity from 10d type IIB supergravity
with only the metric and the self-dual 5-form F (5) turned on [13, 14].
As usual, we express the metric in terms of the frame fields e0, . . . , e9 and do the dimensional
reduction along the compact 5-manifold Σc that is spanned by the 5-form e
5∧e6∧e7∧e8∧e9 =:
e56789. Then, the lift formula is [22] (see also [23])
ds210 = ds
5 + l
(dµi)
2 + µ2i
dξi +
F (5) = (1 + ∗(10))
vol(5) +
d(µ2i ) ∧ dξi ∧ ∗(5)F
, (21)
where µ1 = sinα, µ2 = cosα sin β, µ3 = cosα cosβ with 0 ≤ α ≤ π/2, 0 ≤ β ≤ π/2, 0 ≤ ξi ≤ 2π
and together they parametrise S5. Note that we define the Hodge star of a p-form ω in n-
dimensions as ∗(n)ωi1...in−p = 1p!ǫi1...in−p
j1...jpωj1...jp , with ǫ0123456789 = 1 and ǫ01234 = 1 in an
orthonormal frame. The 10d geometry is specified by {e0, · · · e4}, an orthonormal frame for the
5d metric ds25, together with
e5 = l dα, e6 = l cosαdβ, e7 = l sinα cosα [dξ1 − sin2β dξ2 − cos2β dξ3], (22)
e8 = l cosα sinβ cosβ[dξ2 − dξ3], e9 = −2√3A− l sin
2α dξ1 − l cos2α(sin2β dξ2 + cos2β dξ3).
and the five form [22, 23, 13]
F (5)=
e0···4 + e5···9
(e57 + e68) ∧ (∗(5)F − e9 ∧ F ) (23)
One can write the 5-form RR field strength as F (5) = dC(4) where
C(4) = Ω4 + cotα e
678 ∧ (e9 + 2√
A ∧ (e57 + e68) ∧ (e9 + 2√
(e9 + 2√
A) ∧ (⋆F + 2√
A ∧ F )
. (24)
where Ω4 is a 4-form such that e
01234 = dΩ4. Now we are ready to do the reduction of the 10d
charge
Qχ := −
16π G10
⋆ dχ̂− 1
(4) ∧ F (5)
where Σ8 is a compact 8d submanifold that is composed of a spacelike 3-surface Σ in 5d and
Σc. Hence, only e
5...9 will contribute to the integral. Let us consider χ to be a Killing vector of
the 10d geometry which also reduces to a Killing vector of the 5d geometry and χ̂ be its dual
1-form. Then we find from the expression for the frame fields (21, 22):
χ̂ = χ̂5 + (iχe
9) e9 = χ̂5 − 2√3(iχA) e
9 , so
⋆dχ̂ = ⋆dχ̂5 − 2√
(iχA) ⋆ d e
9 + . . . = ⋆dχ̂5 +
(iχA) ⋆ F + . . . (26)
where “. . .” denotes terms that do not contribute to Qξ. Next, let us find the relevant terms in
C(4) and F (5) (23,24). Noting that iχ
e9 + 2√
= 0, they are:
(4) = iχΩ4 − 2√
(iχA)
e57 + e68
e9 + 2√
e9 + 2√
iχ ⋆ F +
iχ(A ∧ F )
+ . . . (27)
F (5) = −4
e56789 + 2√
⋆ F − F ∧ e9
e57 + e68
+ . . . (28)
(4) ∧ F (5) = −2
iχΩ4 +
(iχA) +A ∧ iχ
⋆ F + 2√
A ∧ F
e56789 + . . . . (29)
After some algebra, the charge reads
Qχ = −
16π G5
⋆dχ̂5 + 4 (iχA) ⋆ F +
(iχA)A ∧ F +
iχΩ4 −
iχ(A ∧ ⋆F )
. (30)
We see immediately that for vectors in the directions of Σ it just reproduces the 5d Noether
charge (19). For vectors orthogonal to Σ, it is different, as is not unexpected, since typically in
dimensional reduction the actions agree only up to boundary terms.
3 Charges from dimensional reduction
In this section we will rederive the Noether-Wald charges for 5d supergravity of section (2.3)
using further dimensional reduction. In particular, we will demonstrate that the 5d Noether-
Wald charges can alternatively be obtained from Kaluza-Klein U(1) charges. For this, we will
first dimensionally reduce the 5d theory along the relevant Killing vectors and then find the
Noether charges of the resulting gauge theory.1 Then we will lift the results back to 5d and
show that they agree with the corresponding 5d Noether-Wald charges. Finally, we will discuss
in which cases the charges obtained by our methods in the interior of the solution agree with
the asymptotic ones.
3.1 Dimensional reduction
In 5 dimensions one can have two independent angular momenta, so we consider dimensional
reduction over both compact Killing vector directions which generate translations along which we
have the independent angular momenta. We will again assume that all fields obey the isometries
and hence only need to consider zero-modes in the compact directions.
We take lower case greek letters α, β, . . . ∈ {t, r, θ, φ, ψ} to be the 5d indices, upper case latin
A,B, . . . ∈ {t, r, θ} to be the 3d indices and lower case latin a, b, . . . , i, j, l,m, . . . ∈ {θ, φ} to be
the indices for the compactified directions in 5d or scalar fields in 3d. The appropriate reduction
ansatz is:
Gµν =
gMN + hijB
, Am =: Am and AM =: A
M + AaB
M , (31)
such that we get
Fµν =
FMN + (dAa ∧Ba)MN An,M
−Am,N 0
, (32)
in terms of the 3d gauge fields Ha = dBa and F 3d = dA3d, and we defined for simplicity F =
F 3d + AaH
a. The definition of A3d in (31) is needed to have the appropriate transformations
of the KK and Maxwell U(1) symmetries and arises naturally from the reduction using frame
fields (see, for instance, [9] for details). Now, we find
µν = FMNF
MN − 2habA,MA,M and
ǫαµνρσAαFµνFρσ = 4ǫ
LMNǫab
Aa,LFMNAb − A3dLAa,MAb,N
, (33)
such that the 5d Lagrangian (17) can be rewritten as :
G5 × L3d =
R3d −
HaMNH
b MN − FMNFMN + 2habAa,MA ,Mb
1This dimensional reduction has been used recently in [10, 11] for defining the entropy functions for such theories.
ǫLMNǫab
Aa,LFMNAb − A3dLAa,MAb,N
, (34)
where VT 2 is the “volume” of the compact coordinates. One can now construct conserved currents
using the Noether procedure for the gauge symmetries of the two U(1) gauge fields Baµ and A
We find the corresponding Noether charges for Baµ to be
Ja = −
16πG5
a rt + 4AaF
ǫLrtǫmnAm,LAn
. (35)
which we identify as the two independent angular momenta. The Noether charge for A3dµ works
out to be
Q = − VT 2
hF rt +
ǫLrtǫmnAm,LAn
which we identify with the 5d electric charge. Alternatively, these charges can be read off by
writing the left hand side of the equations of motion for the Lagrangian (34)
a MN + 4AFMN
+ 16Aa
ǫLMNǫmnAm,LAn
= 0 (37)
−g hFMN + 4
ǫLMNǫabAa,LAb
ǫLMNǫmnAm,LAn,M , (38)
as a total derivative and interpreting the resulting total conserved quantities as the charges.
For geometries with just one independent angular momentum, one can apply the above
formulae in a straight forward way, or do a reduction only down to 4d as in such cases only one
U(1) isometry is expected in the geometry. The computations for the latter are identical to the
ones here, so we just state the expressions for the angular momentum along ∂ξ and the charge:
J = − VT1
16πG5
e2σHrt + 4A F rt
ǫrtAB
A FAB − 2A,AA4dB
, (39)
Q = − VT1
−geσF rt + 1
ǫrtAB
3A FAB + A F
AB − 4A,AA4dB
, (40)
where e2σ = gψψ , VT 1 is the periodicity of ψ, and the conservation follows by the equations of
motion
e2σHMN + 4A FMN
ǫABMN
A FAB − 2A,AA4dB
= 0, (41)
−geσFMN + 2
ǫABMN
A FAB − 2A,AA4dB
ǫABMNFABA,M . (42)
3.2 Oxidation of the angular momentum
Now we would like to demonstrate that the lower dimensional Noether charges above, when
lifted back to 5d, give the Noether-Wald charges for the compactified Killing vectors. For
simplicity, we look at the expression with only one independent angular momentum and only
one dimension (along ψ) reduced. Our results will hold in general though, as the gauge theory
corresponding to the angular momentum is abelian, so we can examine different Killing vectors
independently. First, we note that the dimensional reduction ansatz can be obtained with the
following triangular form of the frame fields [9]:
V Iµ =
viM e
and the inverse V MI =
0 e−σ
, (43)
with (bold latin) tangent space indices A,B, . . . ∈ {0, . . . , 4} and a,b, . . . ∈ {0, . . . , 3} such that
we can write the 4d fields in terms of the 5d fields (but still in 4d coordinates):
BM = e
−σV 4M , HMN = e
−σ(dV 4
− 2e−σ
deσ) ∧B
σ = ξµV Iµ and A = ξ
µAµ . (44)
Now the conservation equation (41) for the angular momentum Jψ reads in flat indices
ηacηbd
ξµV Kµ ηKLdV
L− 2eσ(deσ) ∧B
+ 4ξµAµ(F− 2(dA ) ∧B)cd
8ξµAµ
ǫcdij
F− 2(dA ) ∧B
− 2(dA )cAd + (dA 2)cBd
= 0 . (45)
Extending the summations to A,B, .. and using the form of the frame fields and the indepen-
dence from ψ yields:
ηACηBD
d(ξµV IµηIJV
+ 4ξµAµFCD
8ξνAν
ǫCDABEA
(dξ̂)µN + 4ξ ·AFµN
8ξ · A
ǫµNαρσAαFρσ
= 0 . (46)
The conserved charge extracted from this equation exactly reproduces the charge in (19).
3.3 Generalization and Limitations
3.3.1 Relation to the Asymptotes
Let us now discuss in which situations the charges computed in the spacetime interior give
the charges as defined on the asymptotic boundary. We see most easily from (20) that when
evaluated on a hypersurface on which iξA = 0, such as a suitable asymptotic boundary, our
formulae match with the appropriate Komar integral.
We can compute a (possibly zero) KK or Noether-Wald charge, that corresponds in a specific
geometry to the angular momentum, for every U(1) isometry. However, the asymptotic hyper-
surface on which the angular momentum of a black hole is defined is an Sd−2. When in such
a geometry angular momenta are turned on, its SO(d − 1) isometry breaks (generically) down
to its U(1) subgroups whose charges give the angular momenta, so only the local U(1) factors
that correspond to the asymptotic U(1) subgroups will be related to the angular momentum.
Furthermore, the normalization of the period generated by the Killing vector also has to be
taken into account.
We saw in sections 2.1 and 3.1 how the charges of compact Killing vectors are conserved
whenever the source-free equations of motion hold. That is, they are independent of the position
of the surface on which they are computed, QΣr2 −QΣr1 =
M dMM ∂NQ
MN = 0 where Σr1 and
Σr2 are the boundaries of the volume M - provided that the U(1) theory is defined throughout
the bulk volume and we can consistently compactify the manifold (at least outside the horizon).
Hence, the black hole charge and angular momentum as defined on a spacelike d-2 hyper-
surface Σ∞ at the asymptotes are given by the corresponding KK or Noether-Wald charge,
computed over any spacelike d-2 hypersurface Σr0 in the spacetime for any (not necessarily ex-
tremal) black hole (or in general any spacetime with a suitable asymptotic boundary). That
is, provided there exists a spacelike d-1 hypersurface M with ∂M = {Σr,Σ∞} on which the
following sufficient conditions are satisfied:
1. The relevant compact Killing vector is a restriction to Σr of a Killing vector field that is
globally defined on M and generates a constant periodicity.
2. There are no sources, i.e. the vacuum equations of motion for the gauge fields are satisfied.
3. There exists a smooth fibration of surfaces
π→ [r0,∞[
= M such that π−1r0 = Σr0
limr→∞ π
−1r = Σ∞.
An example where these conditions are satisfied is the region outside the (outer) horizon
of a stationary black hole solution with an Sd−2 horizon topology, embedded in a geodesically
complete spacetime with an asymptotic Sd−2 boundary. One example where these conditions
are violated is that of black rings [18] which will be considered separately in an appendix.
3.3.2 Gauge Issues
The contributions of the CS term in the conserved quantities in (3.1) depend explicitly on the
gauge potentials. This does not however make them gauge dependent. To see this in 5d, let
us consider the electric charge computed by the Noether procedure which is given in [4] as
⋆ F + 2√
A ∧ F
. We notice that the charges get contributions of the form
A ∧ F ,
that change under a transformation δA = dΛ as
dΛ∧F =
d(ΛF ) = 0 because Σ is compact.
From the 3d point of view the KK scalars A may depend on a 5d gauge transformation. However
Λ must be periodic in the angular coordinates so that the contributions from dΛ vanish after
integration. This is also the reason why the term containing ξ ·A in eq. (19) is gauge independent
for compact Killing vectors. On the other hand, the Noether charge for a non-compact Killing
vector is gauge-dependent and hence is only physically relevant when measured with respect to
some boundary condition or as a difference of charges.
4 Examples
So far we have derived Noether charges for various supergravity theories that may be used to
calculate the electric charges and angular momenta of the solutions. In particular, they can be
used on the near horizon geometries to calculate the conserved charges of the corresponding black
holes. In this section we will demonstrate with several examples how our charges successfully
reproduce the known black hole charges in different dimensions, for equal or unequal angular
momenta and independent of the asymptotic geometries. We will start with a 10d example and
then cover 5d examples, first with one angular momentum in AdS and flat asymptotics, and
then with unequal angular momenta in asymptotic AdS.
4.1 The 10d Gutowski-Reall black hole
In [12], Gutowski and Reall found the first example of a supersymmetric black hole which
asymptotes to AdS5 as a solution to minimal gauged supergravity in 5d (see also [34, 17, 35, 36]).
Their solution was lifted to a solution to 10d type IIB supergravity in [13] and shown to admit
two supersymmetries. In [14] (see also [15]), the near horizon geometry of this 10d black hole was
studied. Here we use the formulae found in section 2.2 to calculate the Noether-Wald charges in
the near horizon geometry and show that they agree with the charges of the black hole measured
from the asymptotes. The 10d metric of this near horizon geometry is ds210 = ηabe
aeb with the
orthonormal frame
σL3 , e
, e2 =
σL1 , e
σL2 , e
λ σL3 , (47)
and the five-form is
F (5) =
(e0···4+e5···9)−
(e57+e68)∧ [−3e023+e014−
e234+e9∧ (3e14−e23−
e01)] (48)
where e5 . . . e9 are given in (22) and
dt+ ω
σL3 ) =
(e0 + 2ω
e4), λ =
l2 + 3ω2 and
σL1 = sinφdθ − sin θ cosφdψ, σL2 = cosφdθ + sin θ sinφdψ, σL3 = dφ+ cos θ dψ. (49)
The potential C(4) for the above field strength was given in section 2.4 with Ω4 =
e0234 [14].
Here we concentrate on the compact Killing vectors ∂φ and ∂ξ1 + ∂ξ2 + ∂ξ3 of this geometry and
calculate the corresponding conserved charges. For χ = ∂φ which has a period 4π, we have
χ̂ = 3ω
e0 + ωλ
e4 − ω2
e9 and
dχ̂ = −2ωλ
e01 + 3ω
e14 − (1 + ω2
)e23 + ω
(e57 + e68) (50)
and hence the relevant terms in ⋆dχ̂ are ω
(4l2 + 3ω2) e2···9. Similarly, we find
C(4) ∧ iχF (5) = ω
(2 l2 + ω2) 1
σL1 ∧ σL2 ∧ σL3 ∧ e56789 . (51)
After noting that the integral over 1
σ123 ∧ e56789 gives a factor of 2π5l5, we find
Q∂φ = −
16π4 l5G5
S3∧S5
[⋆dχ̂+
C(4) ∧ iχF (5)] = −
8 l G5
) , (52)
which agrees with the angular momentum, up to a minus sign, that comes from the definition
of the angular momentum as minus the Noether charge [12]. For χ = ∂ξ1 + ∂ξ2 + ∂ξ3 , we have
9 = −l. One can calculate the 10d current and find that
⋆ dχ̂+
C(4) ∧ iχF (5) =
(⋆5F +
A ∧ F ) ∧ e5678 ∧ (e9 +
A) + · · · . (53)
Therefore the corresponding charge is
Q∂ξ1+∂ξ2+∂ξ3
π l ω2
) . (54)
This differs from the answer Q(GR) =
3π ω2
(1 + ω
) [12] by a factor of −l/
12. The minus
sign is because of a difference in our conventions from those of [12] and the factor of l is there
to make the charge Q(GR) dimensionless. The killing vector ∂ξ1 + ∂ξ2 + ∂ξ3 has a period of 6π
and to normalise it to have a period of 2π we have to multiply it by a factor of 3. If we take this
into account the extra factor reduces to
3/2. This is precisely the factor required to define the
5d gauge field in the conventions of dimensional reduction from 10d to 5d [22]. Thus we find
complete agreement between our 10d computation of charges from the NHG and the asymptotic
black hole charges of [12].
4.2 5d Black Holes
Now we turn to black hole solutions in 5d Einstein-Maxwell-CS and minimal gauged supergravity.
4.2.1 Equal Angular Momenta: BMPV and GR
Let us consider two examples that are similar in the near-horizon geometry, with a squashed
S3 horizon, but differ by their asymptotic behaviour; the BMPV black hole [4, 16] with asymp-
totically flat geometry and the Gutowski-Reall (GR) black hole [12] with asymptotically AdS5
geometry.
Their near-horizon solutions can be put in to the form
ds2 = v1
− r2dt2 + dr
σ21 + σ
2 + η(σ3 − αr dt)2
, A = −e r dt+ p(σ3 − αr dt) (55)
which, when dimensionally reduced along the ψ-direction, gives ds24 = v1
− r2dt2 + dr2
dθ2 + sin2θ dφ2
. This has AdS2 × S2 symmetry as expected. The fields take the form
B = −rαdt+ cos θ dφ, e2σ = v2η, A = p and A4d = −e r dt. For the BMPV case, we find:
v1 = v2 =
, η = 1−
, α =
µ3 − j2
, e = −
µ3 − j2
and p =
. (56)
Evaluating the 4d quantities and noting that ǫtrφθ = 1 and VT 1 = 4π, (39, 40) gives us J = πj4G5
which is equal in magnitude to the angular momentum in [4] up to a factor of 2, which arises
from the canonical normalization of the Killing vector ξ = 2∂ψ , and Q =
For the GR case, we have:
, v2 =
, η = 1 + 3
, α = − 3ωl
4l2 + 3ω2
, e =
α, p =
. (57)
Note that we have defined A with an overall factor of −1 compared to [14] to account for a
different convention for the CS term. This gives the results J = −3πω2
(1 + 2ω
) and Q =
(1 + ω
) as expected. Note that [12] do not use the canonical normalization for ∂ψ of [4].
4.2.2 Non-equal Angular Momenta: Supersymmetric Black Holes
Here, we present as the most simple example the N=2 supersymmetric black holes with non-
equal angular momenta of [17], which are asymptotically AdS5, just as the GR case. We start
off with the metric in the form [17]
gtt =
(ρ2ΞaΞb)
ρ2ΞaΞb(1 + r
2) − ∆t(2mρ2 − q2 + 2abrρ2)
, grr =
, gθθ =
gtφ =
−∆t sin2θ
ρ4Ξ2aΞb
a(2mρ2 − q2) + bqρ2(1 + a2)
, gtψ = gtφ(a↔ b, sin θ ↔ cos θ)
gφφ =
sin2θ
ρ2Ξ2a
(r2 + a2)ρ4Ξa + a sin
a(2mρ2 − q2) + 2bqρ2
gψψ = gφφ(a↔ b, sin θ ↔ cos θ), gφψ = sin
2θ cos2θ
ρ4ΞaΞb
ab(2mρ2 − q2) + (a2 + b2)qρ2
with the the gauge field
∆tΞaΞbdt −
a sin2θ
dφ − b cos
where
ρ2 = r2 + a2 cos2θ + b2 sin2θ, ∆t = 1− a2 cos2θb2 sin2θ,
(r2+a2)(r2+b2)(1+r2)+q2+2abq
r2−2m , Ξa = 1− a
2 and Ξb = 1− b2 . (60)
We consider the case with saturated BPS-limit and no CTC’s, which requires:
1 + a+ b
, m = (a+ b)(1 + a)(1 + b)(1 + a+ b). (61)
Now we can find the near horizon geometry with explicit AdS2 symmetry as in [18], by re-defining
t̃ = ǫt, r̃ =
4(1+3a+a2+3b+b2+3ab)
(1+a)(1+b)(a+b)
a+b+ab
, d̃φ = dt+ dφ, d̃ψ = dt+ dψ, (62)
then taking the limit of ǫ→ 0 and applying a gauge transformation to get rid of a constant term
in At. We can read off the 3d scalar fields hmn and A and find
BmN = h
maGaN , gMN = GMN − BaMhabBbN and A3dM = AM − AaBaM . (63)
Noting that VT 2 = 4π2, eqns. (35) give us the angular momenta Jφ̃ = π
a2+2b2+3ab+a2b+ab2
4G5(1−a)(1−b)2 and
= π b
2+2a2+3ab+a2b+ab2
4G5(1−b)(1−a)2 . These agree precisely with the corresponding asymptotic angular
momenta of [18].
5 Charges from the entropy function
The original incarnation of the entropy function formalism [3, 1] was not only a useful tool for
finding near-horizon solutions, but also for extracting the conserved charges from a given solution.
However, in the presence of Chern-Simons terms, the entropy function formalism captures only
part of the conserved charges. We demonstrate here two equivalent ways to cure this problem.
Let us first recall the entropy function formalism [3, 1]:
One considers a general theory of gravity described by the Lagrangian density L with abelian
gauge fields F i(x) and scalar fields Φj(x). Then one writes down the most general ansatz for the
near horizon geometry assuming the isometries of AdS2 × S1 (for simplicity, we consider here
d=4 as in [3, 1]):
ds2 = v1(θ)
− r2dt2 +
dθ2 + v2(θ)
dφ2 − α r dt
F i =
ei − αbi(θ)
dr ∧ dt + ∂θbi(θ)dθ ∧ (dφ − α r dt) and Φj = uj(θ) , (64)
in terms of the parameters {α, ei, β} and θ-dependent scalars {vi(θ), bi(θ), ui(θ)}. Then, one de-
fines the “reduced action” f(α, ~e, β, ~v(θ), ~b(θ), ~u(θ)) =
dθdφL - a functional that generates
the equations of motion
δbi(θ)
δvi(θ)
δui(θ)
= 0, where the functional derivatives
can be understood in terms of the Fourier coefficients in the expansion along θ, and
= qi ,
= j , (65)
where qi and j are supposed to give the charges of the black hole. Then the entropy function is
defined to be the Legendre-transform of the reduced action
E(j, qi, β, ~v(θ), ~b(θ), ~u(θ)) = 2π(eiqi + αj − f) . (66)
Finally, the entropy of the black hole is S = E , evaluated on the solution.
5.1 Completing the equations of motion
In section 3.1, we learned how to find the conserved charges in the presence of Chern-Simons
by writing the KK gauge field equations of motion in a conserved form. Since we now know the
right reduction ansatz, we just need to find a mechanism to parametrize both the variation with
respect to At and Bt and the integration of the right hand side of the equations of motion to
obtain the closed form. One such mechanism is a modification of the ansatz with the pure gauge
terms {ǫi,ℵa} to do the variations δL
and δL
; and with a dummy function c(r), that introduces
an artificial and unphysical r-dependence into fields that are constant by the symmetries. c(r)
then allows to keep track of their, otherwise vanishing, derivatives and to do their integration
on the right hand side of the equations of motion. Hence, we write
Ai = −(ǫi + ei r)dt + c(r) pia(θ)
dφa − (ℵa + αar) dt
, (67)
ds2 = v(θ)
− r2dt2 + dr
dθ2 + ηab(θ)(dφ
a − (ℵa + αar)dt)(dφb − (ℵb + αbr)dt)
and we also wrap all scalar fields that appear in the Chern-Simons terms with a factor of c(r),
ui(θ, r) = c(r)Φi(θ). The solution corresponds to setting c(r) = 1 and c′(r) = 0, which we can
either implement by furnishing c(r) with a control parameter, or by choosing c(r), s.t. c(r0) = 1
and c′(r0) = 0 for some r0, but c
′(r0) 6= 0 for r 6= r0. The equations of motion for the gauge
fields are then ∂r
and ∂r
∂ℵa and give rise to the conserved charges
and Ja =
, (68)
evaluated on the solution. A simple variation of this is c(r) = 1 + 1
r, n being the number of
3d scalar fields in the CS term, which automatically takes care of the integration of the second
term and ensures that all remnant dummy terms will disappear in the first term at r = 0.
The other computations follow just as in the original form of the Entropy function, using
c = 1, c′ = 0 throughout. Note that the entropy function is still computed as originally defined,
E = 2π
αa + ∂L
ei − f
, i.e. not using the conserved charges.
One can easily see that this gives the equations of motion, and it also gives the correct value
for the entropy as the original derivation [3, 1] is independent of what the conserved charges are.
This can also be seen by repeating the derivation in section 6.4 with the original action (34).
As a simple example we have already written the 4d ansatz (55) in section 4.2.1 in a suggestive
form, such that the coefficients can be read off from (56) and (57) with β2 = v2. We note that
the ℵa parameters do not appear here in the action. A simple computation reveals that this
gives indeed the results in section 4.2.1.
5.2 Gauge invariance from boundary terms
In section 3.3, we found that the charges are gauge invariant. However, it would be desirable if
we could impose gauge invariance at the level of the Lagrangian of the 3d action (34). The result
can, in principle, be oxidized back to 5d, but we will stick for simplicity to 3d. The only term of
concern is the A3d ∧ dA[a ∧ dAb] in the CS term in (34), which varies under A3d → A3d + dΛ as
dΛ ∧ dA[a ∧ dAb]. This variation is a total derivative d(ΛdA[a ∧ dAb]) which, after integration,
gives a boundary term ΛdA[a ∧ dAb]. This can be re-expressed as d(ΛA[adAb])− A[adΛ ∧ dAb],
where the first term vanishes if we consider a stationary boundary. The second term is suitably
cancelled by adding a boundary term Abdy. [aA
∧ dAbdy. b], which is identical to a bulk term
d(A[aA
3d ∧ dAb]). Expressed in index notation, and furnished with appropriate factors, the
boundary term that we need to add corresponds to the bulk term is
δL 3d = −
16πG5
ǫLMNǫab
Aa,LF
MNAb + 2 A
LAa,MAb,N
, (69)
which brings the Lagrangian to
G5 × L3d =
R3d −
Ha MNH
b MN − FMNFMN + 2habAa,MA ,Mb
ǫLMNǫab
2Aa,LFMNAb + Aa,LF
, (70)
eliminating the gauge dependent term. A quick calculation shows that this does not affect the
value of the charges (35, 36). Effectively, what we have done is to differentiate the components
of the 5d gauge field in the CS term whose gauge transformations do not vanish automatically
by periodicity constraints, and remove the derivative from other components by an integration
by parts. Hence, the right hand side of each of the 3d gauge field equations of motion does
vanish, and the charges are just the conjugate momenta of the gauge fields B and A3d:
Q = −
δF 3dµν
ǫρµνdx
ρ and Ja = −
δHaµν
ǫρµνdx
ρ , (71)
as in the absence of CS terms. It is easy to verify that the value of the charges remains unchanged.
This means that, if we compute the reduced action from the gauge independent action, the
original formalism will give us the right charges. The entropy function, now computed with the
full charges, does not depend on the extra boundary term and hence also gives us the correct
value of the entropy as we shall derive directly from the Poincaré time Noether charge in section
6 Thermodynamic Charges
Having computed the charges of the Sd−2 isometries, we now turn to the charges of the AdS2
isometries. In particular, we will concentrate on the charge of ∂t, as this will be related to the
thermodynamic quantities entropy S and mass M . First we will compute the Poincar’e time
Noether charge from the Hamiltonian in the NHG and propose a new definition of the black
hole entropy for extremal black holes in the NHG in terms of this charge - similar to Wald’s
definition for non-extremal black holes. Then we (i) justify this definition by showing that it
gives the right extremal limit of the first law, (ii) derive from the Noether charge a statistical
version of the first law suitable for extremal black holes and (iii) re-derive the entropy function
directly from the definition of the entropy. Finally, we discuss the notion of mass as seen from
the NHG by deriving a Smarr-like formula.
6.1 Poincaré Time Hamiltonian
For the Poincaré time Killing vector ∂t, one expects the Noether charge to be related to the
Hamiltonian, which we will explore now.
Since the theory is generally diffeomorphism invariant, we expect the bulk contribution to
vanish. So we concentrate on boundary terms Sbdy. =
B Lbdy., that are necessary to cancel total
derivatives dΘ in the variation of the bulk action δS =
(Eiδφ
i + dΘ(δφ)). In our example,
we have to consider both the variations of the metric and of the 3d gauge fields. For the gauge
fields, the term that we ignored in the derivation of the equations of motion was
µ = ∂µ
δ Aν,µ
δAν +
δ Baν,µ
. (72)
For a complete spacetime, the textbook answer is to place the usual restriction δA|bdy. =
δB|bdy. = 0. Then, the only boundary term that one needs to add in order to make the vari-
ational principle consistent is a Gibbons-Hawking-like term, that compensates for a variation
proportional to the normal derivative of δg at the boundary. For the Einstein-Hilbert action,
that is the usual Gibbons-Hawking term
SGH =
LGH =
hK = − VT 2
16πG5
hγMNn
M ;N , (73)
where γ is the boundary metric and K is the surface gravity of the boundary B, which, in our
geometry, is just an S1 fibred over time. Note that we took n = −∂r to be inward-pointing
in order to define the bi-normal NMN :=
(∂t)[MnN]
|∂t||n| of Σbdy. with a positive signature. Now,
we can read off the Hamiltonian of the NHG if it were an isolated solution. By definition,
Lξgµν = 0, such that the canonical Hamiltonian is just HI = −
i∂tLGH with the time
slice of B being Σbdy = S1. Since ∂t is a Killing vector, a quick calculation shows |∂t|
−γK =√
−gNMN (d ∂̂t)MN , and hence the Hamiltonian is just
HI = −
Σbdy.
i∂tLGH =
16πG5
hNMN (d ∂̂t)
MN . (74)
Now, if we consider the near-horizon geometry being embedded in the full black hole solution,
we cannot put δA|bdy. = δB|bdy. = 0, but we need to satisfy the variational principle by adding
a Hawking-Ross-like boundary term as in [28]:
LHR = nM
δ AN,M
=: −nN
Q̃MNAN + J
and impose the condition to keep the charges fixed under variations of the boundary fields. Now,
the boundary action varies as:
δSHR = −
d2σ nM
δQ̃MN
δJMNa
d2σ nM
Q̃MNδAN + J
where the second term cancels the total derivative in the variation of the bulk action (note the
inward-pointing n), and the first term vanishes as the charges are fixed. A little caveat occurs if
we use the gauge-dependent form of the action (34), when Q̃ 6= Q, however the missing bit does
not depend on the 3d gauge fields, but only on the scalar fields, and hence it is invariant under
variations of the gauge fields. If we consider the gauge-independent form of the action (70), then
Q̃ = Q. Again, by definition we have LξBi = 0, and we will choose a gauge such that LξA=0,
and the canonical Hamiltonian is just
H = −
i∂t(LHR + LGH) . (77)
Because of the AdS2 symmetries, we have
Σbdy.
i∂t (Q∧A) =
Σbdy.
Q(i∂tA) and similar for Ji∧Bi.
This puts the Hawking-Ross contribution to the boundary Hamiltonian to−
Σbdy.
dθ NMN
Q̃MN (i∂tA)+
JMNa (i∂tB
. This gives for the action (34)
H = − VT 2
16πG5
dθNMN
(d ∂̂t)
MN + HaMNhab(i∂tB
b)+4FMN i∂t
ǫPMNǫabAa,PAb i∂t
We now compare (78) with the Noether charge obtained by dimensional reduction of the 5d
expression (20). For this, we work out how the individual terms look like in 3d with the notation
of section 3.1. We consider only the components QMN
in the non-compact directions, and only
zero modes of the fields in the compact directions. Hence we get from the reduction formulae
(31 - 34):
(dξ̂)MN =
dξ̂3d
ξ3d ·Bjhji + χihij
H iMN , FMN = FMN ,
ǫMNαβγ = 2ǫMNLǫijAi,LAj and ξ ·A = ξ3d ·A3d + ξ3d ·BiAi + χiAi . (79)
Now, we can write down the charges of ξ3d, the non-compact components of ξ, and χ, its compact
components, separately:
QMNξ3d = −
16πG5
dξ̂3d
+ ξ3d ·Bj
j MN + 4AiF
+ 4ξ3d ·A3dFMN
ξ3d · A3d + ξ3d ·BiAi
ǫMNLǫijAi,LAj
QMNχ = −
16πG5
iMN + 4AiF
ǫMNLǫkjAk,LAj
, (81)
where we have implicitly done an integration over the compact coordinates. Thus we see that
(78) is just the Noether charge Q∂t in 3d (80) as expected, and we have yet another confirmation
of the KK charge (35), as it matches with (80).
6.2 Entropy
The entropy S of non-extremal black holes was shown by Wald [5] to be given by the Noether
charge κS = 2π
Qξ of the timelike Killing vector ξ that generates the horizon, evaluated on
the bifurcate d-2 surface B of the horizon, and κ is the surface gravity of the horizon. Jacobsen,
Myers and Kang [19] later showed that the charge can be evaluated anywhere on the horizon,
provided all fields are regular at the bifurcation surface. After a coordinate transformation, one
sees that this requires all gauge fields to vanish on the horizon, such that the gauge is fixed to
ξ · A = 0 at the horizon, and hence eliminates the ambiguity of the gauge-dependence of the
Noether charge.
For extremal black holes, κ = 0 on the horizon (r = 0), so Wald does not give a suitable
definition of S, and furthermore there is no bifurcation surface - putting in doubt the gauge fixing.
In the AdS NHG, there should be no special point where to compute physical quantities. Using
the concept that the entropy is intrinsic to the horizon, and hence does not require embedding
the NHG into an asymptotic geometry, those problems are cured by defining the entropy as
κ(rbdy.)
HI(rbdy.) , (82)
in the dimensionally reduced theory with the boundary placed at any radius rbdy. 6= 0. The fact
that the 3d theory is static allows us to use
κ = −
gtt,r
−gttgrr
[9] that is well-defined and physically motivated as the acceleration of a probe at any radius r
with respect to an asymptotic observer and hence related to the temperature of Unruh radiation.
It also ensures that the entropy is independent of rbdy. with well-defined limits rbdy. → 0 and
rbdy. → ∞. Now, in terms of the Noether charge (80), the entropy is just as expected
Q∂t(r) (84)
in the gauge ξ · A(r) = ξ · B(r) = 0; but evaluated at r 6= 0, rather than r = 0 that one would
näıvely expect. We will see in the following three subsections that this definition of the entropy
naturally arises from black hole thermodynamics.
6.3 First Law
Since we have now an expression for the entropy intrinsic to the extremal limit, let us see whether
we can also find an expression for its variation as derived for non-extremal black holes by Wald
in [5]. First let us write the the Noether charge for the gauge-invariant action (70) in 3d for
ξ3d = ∂t as
Qξ3d(r) =
S − ξ3d · A(r)Qel. − ξ3d · Ba(r)Ja . (85)
Then, we consider variations of the dynamical fields δφi that keep the solution on-shell and use
the identity δdQξ3d = d
ξ3d · Θ
[5], with Θ defined in section 2, such that we can relate the
variation of the charge evaluated over two boundaries Σ1 and Σ2 of a spacelike d-1 surface:
δQξ3d − ξ3d ·Θ
δQξ3d − ξ3d ·Θ
. (86)
Now, let us move the boundaries into the near-horizon geometry (→ ΣH) and into some asymp-
totic limit (→ Σ∞). On ΣH , we have
ξ3d ·Θ =
L dθM ǫLMN
gOP δ̄ΓNOP + g
ON δ̄ΓPOP
δAO,N
δAO +
δκ − Qelδ(ξ3d ·A) − Jiδ(ξ3d ·Bi) , (87)
where we used for the second equality the AdS2 isometries, and assumed an Einstein-Hilbert
term for the gravitational action, and any gauge field term that can be written with only first
derivatives of A, such as (70). The right hand side of (86) can be interpreted by following Wald,
and defining the canonical energy, i.e. the Hamiltonian measured by an asymptotic observer at
Σ∞, E =
(Qξ3d − ξ3d·V ) with some d-1 form V: δ
ξ3d ·V =
ξ3d ·Θ. This corresponds,
for the asymptotic boundary conditions A = B = 0 and suitable normalization of ξ3d, to the
mass. Altogether, (87) gives us now an expression similar to the first law
δS + Φ(r) δQel. + Ω
i(r) δJi = δE (88)
at some r 6= 0, where Φ(r) = −ξ3d · A(r) and Ωi(r) = −ξ3d · Bi(r) measure the co-rotating
electric potential and angular frequency2 at r in the NHG with respect to the definition of E .
This, however is not yet a relation for the full black hole, but captures only physics outside Σr.
The extremal limit of the non-extremal first law of the full black hole solution is reproduced by
taking the limit r → 0:
ΦH δQel. + Ω
H δJi = δE , (89)
where ΦH = −ξ3d ·A(0) and ΩH = −ξ3d ·B(0) are the horizon co-rotating electric potential and
angular frequency. It is interesting to observe though, that (88) and corresponding expressions
for the Smarr formula resemble the first law of a finite temperature black hole, even though its
physical significance is limited, as Σrfor r 6= 0 is not a horizon.
An interesting observation and lesson is that when embedding the near horizon solution into
an asymptotic solution, but computing Noether charges in the NHG, we need to use the gauge
invariant action (70) and the full Noether charge, because there is no boundary of the NHG on
which we were allowed to fix the gauge fields and its gauge variations.
2To illustrate that this definition of Ω corresponds to the one in [5], consider a vector ξ = ∂t − Ω∂φ in static
coordinates with a diagonal metric g, and ξ = ∂t′ in co-rotating coordinates with a non-diagonal metric g
′. Then
ξ̂ = gttdt− Ωgφφdφ = gt′t′dt′ + Bφt′gφφdφ. A similar argument follows from requiring constant normalization of ξ and
considering gtt + gφφ = gt′t′ in the explicit coordinate transformation.
We see that our version of the first law also holds also for perturbations away from extremality,
which connects it smoothly (in a thermodynamic sense) to the near-extremal limit of the non-
extremal black hole, again supporting our definition of the entropy.
6.4 Entropy Function and the Euclidean Action
Now, let us continue following Wald [5] and relate the (integrated) mass (or energy E) to the
entropy. Starting with (85), we apply Gauss’ law to find
S − ξ3d ·A(r)Qel. − ξ3d · Ba(r)Ja = E −
Jξ3d +
ξ3d · V =: E −
I(r) , (90)
where the euclidean action3 I is now, in principle, a function of the radial position of ΣH , since
∂M = {ΣH ,Σ∞}. Even though I is defined only for κ 6= 0 as the integral of the analytically
continued Lagrangian, with τ = it having period 2π
, one would like to find a well-defined limit
as κ→ 0, i.e. r → 0, representing the full extremal black hole solution. This requires
ΦHQel. + Ω
HJa = E . (91)
This relation can be taken as a (gauge-dependent) definition of the mass of the black hole in
the near-horizon geometry. We note that since the action is gauge-invariant, (91) is gauge-
independent in the sense that a gauge transformation that changes ΦH and ΩH on Σ0 changes E
at Σ∞ accordingly. In the appropriate gauge in which E =M , it should agree with the BPS (or
extremality) condition - as we verified for BMPV and GR - and with an applicable Smarr-like
formula, supposed one has a full solution at hand. Now, let us study the remaining terms of (90).
Again, we make use of the AdS2 geometry to find that ξ3d ·
A(r)−A(0)
�κ(r) = F 3drt =: −EH
is the constant co-rotating electric field-strength in the NHG, as is ξ3d ·
Bi(r)−Bi(0)
�κ(r) =
Hrt =: −HH the field strength of the KK gauge field. Now, (90) reads
S = −2π
EHQel. + H
− I , (92)
with all terms, including I, being independent of the position r 6= 0 of ΣH in the NHG. (92)
holds also in the limit as r → 0. A similar expression was proposed and discussed in a statistical
context by Silva in [6], where it was motivated by taking the extremal limit of non-extremal black
holes, assuming an appropriate expansion of ΦH and ΩH in terms of the inverse temperature.
This is identical to (92), provided one identifies the NHG field strengths with the appropriate
expansion coefficients in [6]. Note that this relation is particular for extremal black holes and
profoundly different from the relation of the entropy to the euclidean action for non-extremal
black holes [29, 30].
Let us now show how this relates to the entropy function formalism. Given I = −2π
M iξ3dL +
iξ3dV
[5], we use the fact that the spacetime in the NHG can be trivially
foliated with spheres to re-write this as
I = −
iξ3dL +
iξ3dV −
iξ3dL
=: I0 +
L , (93)
where ∂M0 = {Σr=0,Σ∞}. Since
L is supposed to be invariant under the AdS2 isometries,
it is proportional to the volume form on AdS2 and (
L)�κ(r) = ⋆
L = const. Now,
the fact that I = const. implies that I0 = 0 and we are left with
S = −2π
EHQel. + H
HJi + ⋆
. (94)
3I equals the euclidean action only for stationary spacetimes, see [5].
This is just the entropy function for the gauge invariant action (70). The same derivation can
be applied to the original action (34) to give its corresponding entropy function. In that case E
in (91) will have a different value, because of the boundary terms in the action, stressing again
the need to work with (70) when relating the NHG to the asymptotic geometry.
6.5 Mass
Even though the mass of extremal black holes is fixed by the extremality (or BPS) relation 91,
let us now study its physical interpretation from the point of view of the NHG by deriving a
Smarr-like formula for the 5d Einstein-Maxwell-CS case.
Let us suppose there is some asymptotic geometry attached to the near horizon geometry in a
way that the conditions in section 3.3 are satisfied, and follow closely the derivation by Gauntlett,
Myers and Townsend in [4] for a few steps. The mass, E in a gauge in which A = B = 0 at Σ∞,
can be re-written using Gauss’s law in 5d as
M = −
16πG5
⋆dk̂ =
16πG5
⋆dk̂ +
, (95)
for some ∂M = {Σ,Σ∞} and k being the asymptotic unit norm timelike Killing vector. Assuming
we work in a gauge in which LξA = 0, and using the relations ✷kµ = −Rµνkν , LkΩ = ik(dΩ) +
d(ikΩ) for any form Ω and the equations of motion for g and A, the result is
16πG5
⋆dk̂ + 4(k · A) ⋆ F −
k̂ ∧ (Â · F )
(k · A)A ∧ F
, (96)
plus a term at Σ∞ that vanishes as A→ 0. In dimensions other than d = 5, there will be an extra
term that cannot be expressed as a surface integral at ΣH . For details see [4]. Now, we see that
the first, second and last terms combine to give the Noether charge (19). Decomposing k into
its compact and non-compact components, k = ∂t + Ω
iχi, and choosing Σ to be an r = const.
surface in the NHG, we find from the 3d expressions (80,81) that this gives us
S + ΩiJi
+Φ(r)Qel.−
(∂t · A) ⋆ F −
(∂̂t +Ω
iχ̂i) ∧ (Â · F )
In (∂̂t + Ω
iχ̂i) ∧ (Â · F ), we find that in terms of frame fields the relevant components are
(∂̂t+Ω̂
iχi)0, A0 and F01, since the AdS2 symmetries restrict non-vanishing FM1 to M = 0. This
makes the last term vanishing, such that we get in the limit r → 0 the Smarr formula
ΩiHJi + ΦHQel. , (98)
that agrees with the near-horizon limit of the non-extremal one. From the point of view of
the near-horizon solution, we find that the mass is now a gauge-dependent expression, with the
gauge given by the embedding of the near-horizon solution in the asymptotic solution. We find
that (98) looks different from (91), however they are in agreement since ΩH vanishes for BMPV
black holes [4].
7 Conclusions
In this paper we presented expressions for conserved currents and charges of 10d type IIB
supergravity (with the metric and five-form) and minimal (gauged) supergravity theories in 5
dimensions. These have been obtained following Wald’s construction of gravitational Noether
charges. Those of the 5d gauged supergravity can also be obtained by dimensional reduction of
the 10d formulae. We further showed that the Noether charges of the higher dimensional theories,
after dimensional reduction, match precisely with the Noether charges of gauge fields obtained
by Kaluza-Klein reduction over the compact Killing vector directions of interest. Our expressions
for the charges should be valid generally for both extremal and non-extremal geometries. We
then turned to their applications to extremal black holes and demonstrated that, when evaluated
in the near horizon geometries, our charges reproduce the conserved charges of the corresponding
extremal black holes under certain assumptions. In particular, we exhibited that our methods
give the correct electric charges and angular momenta for the BMPV and Gutowski-Reall black
holes.
A host of new solutions to supergravity theories with AdS2 isometries have been found
recently [20] and many more such solutions are expected to be found in the future. These
solutions may be interpreted as the near horizon geometries of some yet to be found black holes.
In such cases, our results should be useful in extracting the black hole charges without having to
know the full black hole solutions but just the near horizon geometries. On the other hand, the
holographic duals of string theories in the NHG are expected to be supersymmetric conformal
quantum mechanics. Our conserved charges should be part of the characterising data of these
conformal quantum mechanics.
We argued that the black holes with AdS3 near horizons do not satisfy our assumptions when
embedded in black hole asymptotes with Sd−2 isometries (rather than black string asymptotes).
Supersymmetric black rings are the main examples for which our formulae do not seem to apply.
More generally for black holes with AdS3 one has to find the correct way to extract the conserved
charges separately which we would like to return to in future.
We then presented a new entropy function valid for rotating black holes in 5d with CS terms
which gives the correct electric charges as well as the entropy. This is an improvement over [21].
We used appropriate boundary terms, that make the action fully gauge-independent which turns
out to be relevant to obtain the thermodynamics in the second part of the paper.
In the second part of the paper we exhibited a new definition of the entropy as a Noether
charge, and a derivation of the first law, which are applicable for extremal black holes directly.
We used this definition to produce the statistical version of the first law and moved on to re-
derive the entropy function from a more physical perspective. Finally, we commented on the
physical interpretation of the mass in the near-horizon solution. The relevant calculations were
done in the near-horizon geometry, only assuming an embedding into some asymptotic solution
for the purpose of formally defining the Mass. We did not, however, produce a conserved charge
corresponding to the the level number. In terms of the 5d fields, the expression in [27] is just
proportional to
⋆F , which is conserved in the NHG by the symmetries, but not by the
equations of motion in a general geometry. Various potentially interesting candidates, such as
the R-charge and global AdS2 time Noether-Wald charge did not produce an interesting result.
We find that the gauge-independent thermodynamic quantities can be evaluated everywhere
in the near-horizon geometry, as they are a statement about the near-horizon geometry. In
particular, they are the entropy, euclidean action and charges and their chemical potentials,
as well as the statistical version of the first law (92). Relations and quantities related to the
asymptotic geometry and to thermodynamics of non-extremal black holes (the mass, horizon
electric potential and angular frequency, as well as the first law and Smarr formula) however are
gauge-dependent from the point of view of the near-horizon geometry. They need to be evaluated
on a specific hypersurface, r = 0, as they come from position-dependent statements in the near-
horizon geometry. This means that the former ones may be more relevant for characterising
attractors.
Acknowledgements
We thank Rob Myers for helpful discussions and suggestions and helpful comments on the
manuscript. MW was supported by funds from the CIAR and from an NSERC Discovery grant.
Research at the KITP is supported in part by the National Science Foundation under Grant No.
PHY05-51164 and research at the Perimeter Institute in part by funds from NSERC of Canada
and MEDT of Ontario.
A Black Rings
The non-equal angular momentum generalization of the BMPV case is the supersymmetric black
ring [18]. It is an excellent counter-example in which the conditions in section 3.3 are not satisfied.
To demonstrate this, we sketch out the derivation of the asymptotic and near horizon limits as
given in [18]. The general form of the solution is given by:
ds1 = −f2(dt + ωφdφ + ωψdψ)2 +
f−1R2
(x−y)2
( dy2
+ (1−x2)dφ2 + (y2−1)dψ2
f(dt+ ω) − q
(1 + x)dφ + (1 + y)dψ
, (99)
where y ∈]−∞,−1] , x ∈ [−1, 1] , φ, ψ ∈ R�2πZ and f−1 = 1 + Q−q
(x − y) − q
(x2 − y2),
ωφ = − q8R2 (1− x
3Q− q2(3 + x+ y)
and ωψ =
(1 + y) +
(1− y2)
3Q− q2(3 + x+ y)
The asymptotic limit is given by (x + 1) → +0 and (y + 1) → −0, and its geometry of a
squashed sphere with broken isometry SO(4) → U(1)2 can be made manifest by combining
(x, y) into a radial coordinate ρ ∈ R+ and an angular coordinate Θ ∈ [−π2 ,
ρ sinΘ =
x−y and ρ cosΘ =
x−y (100)
The near horizon limit, on the other hand, is given by y → −∞, such that appropriate radial
and angular coordinates are r = −R
and cos θ = x. A first observation is that the two limits
are just points in the “opposite” coordinates, (ρ,Θ) → (R, π
) and (r, θ) → (R,π). To obtain
the near horizon geometry in a suitable form, we define χ = φ − ψ, take the limit r = ǫr̃R−1,
t = ǫ−1t̃, ǫ→ 0 and get:
ds2 =
q2dr̃2
dt̃dψ +
(q2 −Q)2 − 4q2R2
dψ2 +
dθ2 + sin2θdχ2
A = −
(q2 +Q)dψ + q2(1 + cos θ)dχ
. (101)
Now, we also see that the topology of the horizon is S1×S2 with U(1)×SO(3) ∋ U(1)2 isometry
and whose subgroup U(1)2 is not guaranteed to agree with the U(1)2 of the asymptotic geometry.
The AdS2 geometry is more apparent after dimensional reduction, when gtt ∝ r̃2 is restored,
and after suitably rescaling t̃. [18] show furthermore that the AdS2 and S
1 combine into a local
AdS3. The conserved charges are now Jψ =
(q2 −Q)2 − 12q2R2
, Jχ = − π8G5 q(q
2 +Q)
and Qel. =
(q2+Q), or in the old coordinates Jψ =
(q2−Q)2+2q2(q2−2Q−6R2
qQ . They compare to the asymptotic quantities computed in [18] Jψ =
q(3Q− q2),
q(6R2 + 3Q− q2) and Qel. =
The distinguishing feature here is that black rings have an AdS3×S2 near-horizon geometry.
Thus the S1 × S2 of the horizon and the S3 of the asymptotic hypersurface are topologically
distinct, such that there is no continuous fibration of hypersurfaces over r between them. In
particular, The coordinates that describe the asymptotic S3 shrink the horizon and the area
bounded by the black ring to a point in 3d (or an S1 × S1 in 5d), and are missing part of the
boundary of the full solution because of the difference in topology. This missing part shrinks
into the coordinate singularity that also contains the horizon, so flux that passes though that
part of the boundary will not be seen from the asymptotic geometry.
It is not inconceivable that if we consider the black rings on Taub-Nut spaces like in [31, 32, 33]
and obtain a 4d black hole which satisfies our criteria one may yet be able to recover the charges
of such black rings.
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	Introduction
	Charges from Noether-Wald construction
	Review of Noether construction
	The Noether-Wald charges for type IIB supergravity
	The Noether-Wald charges for 5d Einstein-Maxwell-CS
	Reduction from 10 dimensions
	Charges from dimensional reduction
	Dimensional reduction
	Oxidation of the angular momentum
	Generalization and Limitations
	Relation to the Asymptotes
	Gauge Issues
	Examples
	The 10d Gutowski-Reall black hole
	5d Black Holes
	Equal Angular Momenta: BMPV and GR
	Non-equal Angular Momenta: Supersymmetric Black Holes
	Charges from the entropy function
	Completing the equations of motion
	Gauge invariance from boundary terms
	Thermodynamic Charges
	Poincaré Time Hamiltonian
	Entropy
	First Law
	Entropy Function and the Euclidean Action
	Mass
	Conclusions
	Black Rings
