The First Law for Boosted Kaluza-Klein
Black Holes
David Kastor, Sourya Ray and Jennie Traschen
Department of Physics
University of Massachusetts
Amherst, MA 01003
Abstract
We study the thermodynamics of Kaluza-Klein black holes with
momentum along the compact dimension, but vanishing angular
momentum. These black holes are stationary, but non-rotating.
We derive the first law for these spacetimes and find that the
parameter conjugate to variations in the length of the compact
direction is an effective tension, which generally differs from the
ADM tension. For the boosted black string, this effective ten-
sion is always positive, while the ADM tension is negative for
large boost parameter. We also derive two Smarr formulas, one
that follows from time translation invariance, and a second one
that holds only in the case of exact translation symmetry in the
compact dimension. Finally, we show that the ‘tension first law’
derived by Traschen and Fox in the static case has the form of a
thermodynamic Gibbs-Duhem relation and give its extension in
the stationary, non-rotating case.
http://arxiv.org/abs/0704.0729v2
1 Introduction
The physics of Kaluza-Klein black holes, i.e. black hole spacetimes asymp-
totic at infinity to M × S1, has proved to be a surprisingly rich subject, in-
cluding such phenomena as the Gregory-Laflamme instability, non-uniform
static black strings and the black hole/black string phase transition (see e.g.
the reviews [1, 2]). Research to date has focused primarilly on the static
case. However, it is also of interest to explore the properties of stationary
solutions. Accordingly, in this paper we will study the thermodynamics of
stationary Kaluza-Klein black holes1.
Static Kaluza-Klein black holes are characterized at infinity by the mass
M, tension T and the length L of the compact direction. The physical
meaning of the tension follows from its role in the first law for static S1
Kaluza-Klein black holes [6][7][8]
dM = κ
dA + T dL. (1)
We see that the tension determines the variation of the mass with varying
length of the compact direction, under the constraint that the horizon area is
held fixed. Within the thermodynamic analogy, it appears to be an intensive
parameter of the system, like temperature or pressure.
Stationary Kaluza-Klein black holes can carry linear momentum in the
compact direction, as well as angular momentum. In this paper, we will be
interested in this linear momentum, which we denote by P, and will assume
that the angular momentum vanishes. The simplest solutions with P 6= 0
are boosted black strings. These are obtained by starting from the infinite
uniform black string, boosting in the z direction and then identifying the new
z coordinate with period L. The boosted black string is then locally, but not
globally, the same as the static uniform black string. Further stationary, but
not z-translationally invariant, solutions may be obtained by giving localized
black holes or non-uniform black strings velocity in the compact direction.
In subsequent sections, we present the following results. We use the
Hamiltonian methods of [9][10][8] to establish the first law for stationary, non-
rotating Kaluza-Klein black holes. We also derive two Smarr formulas for
these spacetimes. These are exact relations between the geometric quantities
1Aspects of stationary Kaluza-Klein black holes have been studied in [3][4]. The ther-
modynamics of asymptotically AdS, boosted domain walls have been investigated in ref-
erence [5].
M, κA, vHP and T L, where the quantity VH is defined below. The first of
these formulas holds for the entire class of spacetimes under consideration.
The second Smarr formula holds under the additional assumption of exact
translation invariance in the compact direction. A linear combination of these
two formulas gives the relation between mass and tension for the boosted
black string. We derive each of these Smarr formulas in two ways, first using
scaling arguments (as in e.g. reference [11]) and second using Komar integral
relations (as in reference [12]). Finally, we present a Gibbs-Duhem formula
that relates variations in the tension to variations in the other intensive
parameters. This result generalizes the ‘tension first law’ of reference [13].
Our result for the first law resolves a small puzzle related to the boosted
black string, which formed part of the motivation for this work. It was found
in reference [3] that the tension of the boosted black string becomes negative
for values of the boost parameter in excess of a certain critical value, which
depends only on the spacetime dimension. If the physical interpretation of
the tension based on equation (1) were to continue to hold in the stationary
case, then the energy of the system would decrease with increasing L, which
seems counter-intuitive. This puzzle is resolved by showing that the coeffi-
cient of the dL term in the first law for black holes is an effective tension T̂ .
The effective tension T̂ is equal to the ADM tension in the static case, but
includes a contribution from the momentum in the stationary case. For the
boosted black string T̂ is always positive, and is in fact given by the tension
of the unboosted black string with the same horizon radius.
2 Stationary, non-rotating Kaluza-Klein black
holes
We consider stationary D-dimensional vacuum black hole spacetimes that
are asymptotic to MD−1 × S1, and assume that the black hole horizon is a
bifurcate Killing horizon. In accordance with our focus on linear momentum
around the S1, we take the ADM angular momentum to vanish. We denote
the horizon generating Killing field by la, and assume that at infinity it has
the form
la = T a + vHZ
a (2)
where T a = (∂/∂t)a and Za = (∂/∂z)a, with z being the coordinate around
the S1. The surface gravity κ of the black hole horizon is defined, as usual,
via the relation on the horizon
∇a(lblb) = −2κla. (3)
The form (2) of the horizon generating Killing field at infinity resem-
bles the decomposition of the horizon generating Killing field for a rotating,
asymptotically flat black hole. In that case, i.e. upon replacing vHZ
a, with φ an azimuthal coordinate, one can show that T a and φa are
themselves Killing vectors [14, 15, 16]. The quantity ΩH can then be in-
terpreted as the angular velocity of the horizon, and further shown to be
constant on the horizon [14].
Returning to the case of Kaluza-Klein black holes, the situation is quite
different. Already in the static case, solutions exist which are non-uniform in
the z direction. In the stationary case then, it will not generally be the case
that T a and Za are Killing vectors. For localized black holes or non-uniform
black strings with velocity around the S1, only the linear combination la is
a Killing vector.
2.1 Two commuting Killing fields
It is nonetheless useful to separately consider the case in which T a = (∂/∂t)a
and Za = (∂/∂z)a are two commuting Killing fields, and that the relation
(2) holds throughout the spacetime. The boosted black string falls into this
class of spacetimes. If both Za and T a are Killing fields, then the quantity
vH in equation (2) may be considered to be the velocity of the black hole
horizon. This identification follows in a similar way to that of ΩH as the
angular velocity in the rotating case (see e.g. the article by Carter in [17]).
It follows from equation (3), together with our assumption that T a and Za
are commuting Killing vectors, that in addition to lala = 0 on the horizon,
one also has there the orthogonality relations
laTa = 0, l
aZa = 0. (4)
Given these, one can then show that the metric components on the horizon
satisfy the two relations
(T aZa)
2 = (T aTa) Z
bZb, vH = −
T aZa
(ZbZb)
The second of these leads to the interpretation of vH as the velocity of the
horizon in the following manner. For rotating black holes, one considers ‘zero
angular momentum observers’ or ZAMO’s. The angular velocity ΩH of the
horizon is the limit of a ZAMO’s angular velocity as it approaches the horizon
radius. For a boosted black string, we may analogously consider observers
with zero linear momentum along the string, which we might be justified
in calling ZELMO’s. Let pa = m dxa/dτ be the momentum of a particle
following a geodesic. It’s energy E = −T apa and the z−component of its
momentum P = Zapa are both constants of motion. The condition P = 0
of vanishing linear momentum is then dz/dt = −gtz/gzz, and we see that on
the horizon the coordinate velocity of a ZELMO is equal to vH .
3 ADM mass, tension and momentum
We review the formulas for the ADM mass, tension and momentum. Let us
write the spacetime metric near infinity as gab = ηab + γab, where ηab is the
D-dimensional Minkowski metric. The components of γab are assumed to fall-
off sufficiently rapidly that the integral expressions for the mass, tension and
momentum are well-defined. In the asymptotic region, write the spacetime
coordinates as xa = (t, z, xi), where i = 1, . . . , D − 2 and the coordinate z
running around the S1 is identified with period L. Let Σ be a spatial slice
and ∂Σ∞ its boundary at spatial infinity. The ADM mass and momentum
in the z direction are then given in asymptotically Cartesian coordinates by
the integrals
M = 1
dz dsi
−∂iγjj − ∂iγzz + ∂jγij
dz dsi ∂
iγtz (7)
where indices are raised and lowered with the asymptotic metric ηab and the
area element dsi is that of a sphere S
D−3 at infinity in a slice of constant t
and z.
The ADM tension is similarly given by the integral [13][6][18]
T = −
∂Σ∞/S1
−∂iγjj − ∂iγtt + ∂jγij
. (8)
Note that in contrast with the ADM mass and momentum, the definition
of the tension does not include an integral in the z-direction. The ADM
mass is an integral over the boundary of a slice of constant t, which includes
the direction around the S1. The tension, on the other hand, is defined
[13][6][18] by an integral over the boundary of a slice of constant z. This
includes, in principle, an integration over time. However, if one expands the
integrand around spatial infinity, one finds that terms that make non-zero
contributions to the integral are always time independent. Time dependent
terms fall-off too rapidly to contribute. Hence, one can omit the integration
over the time direction and work with the quantity T defined above, which
is strictly speaking a ‘tension per unit time’.
We can evaluate these formulas for M, P and T in terms of the asymp-
totic parameters of our spacetimes. The spacetimes we consider have topol-
ogy RD−1 × S1, the coordinate z in the compact direction being identified
with period L. We can write the metric explicitly as
ds2 = gttdt
2 + 2gtzdtdz + gzzdz
2 + 2(gtidtdx
i + gzidzdx
i) + gijdx
idxj (9)
where xi with i = 1, . . .D − 2 are the non-compact spatial coordinates. We
assume the following falloff conditions at spatial infinity
gtt ≃ −1 + ct/rD−4, gzz ≃ 1 + cz/rD−4, gtz ≃ ctz/rD−4, (10)
and further that the coefficients gti and gzi falloff sufficiently fast that they
do not contribute to any ADM integrals at infinity. The mass, tension [7]
and momentum can then be shown, using the field equations, to be given in
terms of the asymptotic parameters ct, cz and ctz by
M = ΩD−3  L
((D − 3)ct − cz), T =
(ct − (D − 3)cz), (11)
P = −(D − 4)ΩD−3  L
ctz . (12)
4 The boosted black string
The boosted black string serves as a simple analytic vacuum spacetime in
which to check the results we present below for the first law, Smarr and
Gibbs-Duhem relations. The boosted black string metric may be obtained
starting from the uniform black string, performing a boost transformation
with parameter β and identifying the new, boosted z coordinate with period
L. This gives
ds2 = −(1 −
cosh2 β)dt2 + (1 +
sinh2 β)dz2 (13)
sinh β cosh βdzdt + (1 − c
)−1dr2 + r2dΩ2D−3
The horizon, which has topology SD−3×S1 is located at rH = c1/(D−4). From
the asymptotic form of the metric, one finds using the expressions (11) and
(12) that the ADM mass, tension and momentum are given as in [3] by
ΩD−3L
rD−4H ((D − 4) cosh2 β + 1) (14)
rD−4H (1 − (D − 4) sinh2 β) (15)
P = −
ΩD−3L
rD−4H (D − 4) sinh β cosh β (16)
Note that, as mentioned in the introduction, the tension becomes negative
for sinh2 β > 1/(D − 4). We can further compute, as in reference [3], that
the horizon area, surface gravity, and horizon velocity of the boosted black
string are given by
A = ΩD−3LrD−3H cosh β, κ =
D − 4
2rH cosh β
, vH = −
sinh β
cosh β
. (17)
5 Gauss’ Laws for Perturbations
Following the work of [9], we use the Hamiltonian formalism of general rela-
tivity to derive the first law for stationary, non-rotating Kaluza-Klein black
holes. Another of our goals is to derive a ‘first law’ for variations in the
tension as in reference [13], for this class of spacetimes. This requires a slight
generalization of the Hamiltonian formalism to accomodate evolution of data
on timelike surfaces in a spacelike direction. Although, as we discuss below
in section (8), we have not yet succeeded in providing a Hamiltonian deriva-
tion of the ‘tension first law’ in the stationary case, our presentation of the
Hamiltonian formalism will be general enough to provide the necessary tools.
The essence of the method is as follows. In vacuum gravity, suppose one
has a black hole solution with a Killing field. Now consider solutions that
are perturbatively close to this background solution, but are not required
to have the original Killing symmetry. The linearized Einstein constraint
equations on a hypersurface can be expressed in the form of a Gauss’ law
(see [10]), relating a boundary integral at infinity to a boundary integral at
the horizon. The physical meaning of this Gauss’ law relation depends on
the choice of Killing field, as well as on the choice of hypersurface. Taking
the generator la of a Killing horizon, together with an appropriate choice of a
spacelike hypersurface, yields the usual first law for variation of the mass [9].
In the case of solutions that are z translation invariant, choosing the spatial
translation Killing vector Za, again with an appropriate choice of a timelike
hypersurface, gives a ‘first law’ for variations in the tension [13].
The formalism then proceeds in the following way. Assume we have a
foliation of a spacetime by a family of hypersurfaces of constant coordinate
w. We denote these hypersurfaces, both collectively and individually, by V
and the unit normal to the hypersurfaces by wa. With the application to
tension in mind, we consider both timelike and spacelike normals by setting
a = ǫ with ǫ = ±1. This slight generalization introduces factors of ǫ
into a number of otherwise standard formulas. The spacetime metric can be
written as
gab = ǫwawb + sab (18)
where sab, satisfying sa
bwb = 0, is the metric on the hypersurfaces V . As
usual, the dynamical variables in the Hamiltonian formalism are the met-
ric sab and its canonically conjugate momentum π
ab = ǫ
|s|(Ksab − Kab).
Here Kab = sa
c∇cwb is the extrinsic curvature of a hypersurface V . We con-
sider Hamiltonian evolution along the vector field W a = (∂/∂w)a, which can
be decomposed into its components normal and tangential to V , according
to W a = Nwa + Na where Nawa = 0. As usual, we refer to N and N
respectively as the lapse function and the shift vector. The gravitational
Hamiltonian density which evolves the system along W a is then given by
H = NH + NaHa with
H = −R(D−1) + ǫ
D − 2
− πabπab) (19)
Hb = −2Da(|s|−
2πab). (20)
where R(D−1) is the scalar curvature for the metric sab and the derivative
operator Da on the hypersurface V . One further finds that the quantities H
and Ha are simply related to the normal components of the Einstein tensor,
H = 2ǫ Gabw
awb, Hb = 2ǫ Gacw
ascb (21)
These components of the field equations contain only first derivatives with
respect to the coordinate w, and hence represent constraints on the dynamical
fields, sab and π
ab, on V . This property is independent of whether the normal
direction is timelike, as in the usual ADM formalism, or spacelike. In vacuum,
the equations H = 0 and Hb = 0 are enforced in the Hamiltonian formalism as
the equations of motion of the nondynamical lapse and shift variables. These
are referred to as the Hamiltonian and momentum constraints, a terminology
we continue to use in the case that the normal wa is spacelike.
Let us now assume that the spacetime metric ḡab is a solution to the
vacuum Einstein equations2 with a Killing vector ξa. We decompose ξa into
components normal and tangent to the hypersurfaces V introduced above,
according to ξa = Fwa + βa. Now, let us further assume that the metric
gab = ḡab+δgab is the linear approximation to another solution to the vacuum
Einstein equations. Denote the Hamiltonian data for the background metric
by s̄ab, π̄
ab, the corresponding perturbations to the data by hab = δsab and
pab = δπab, and the linearized Hamiltonian and momentum constraints by
δH and δHa. As shown in [10, 9, 13], the following statement then holds as
a consequence of Killing’s equation in the background metric,
FδH + βaδHa = −D̄aBa (22)
where D̄a is the background covariant derivative operator on the hypersurface
and the vector Ba is given by
Ba = F (D̄ah−D̄bhab)−hD̄aF +habD̄bF +
βb(π̄cdhcds̄
b−2π̄achbc−2pab).
Here indices are raised and lowered with the background metric s̄ab. Since the
metric gab solves the vacuum Einstein equations by assumption, we know that
δH = δHa = 0. Therefore, we have the Gauss’ law type statement D̄aB
a = 0.
Note that the detailed form of this relation for the perturbations hab and p
depends on the the Killing vector ξa and the normal wa to the hypersurface,
2In this paper we will focus on the case when the background spacetime is vacuum.
It is straightforward to add stress-energy which is described by a Hamiltonian [13]. If
the matter Hamiltonian contains constraints–such as for Maxwell theory–then additional
charges appear in the first law. This was worked out for Einstein-Yang-Mills in [9]. The
general case when the background spacetime has stress-energy, such as a cosmology, was
studied earlier in [10]. In this situation, the criterion for a Gauss’ law on perturbations is
that the background have an Integral Constraint Vector.
as well as on the background spacetime metric. Making different choices for
the Killing vector and normal can lead to different relations of this form. We
can now integrate the relation D̄aB
a = 0 over the hypersurface V and use
Stokes theorem to obtain
c = 0, (24)
where for black hole spacetimes the boundary ∂V of the hypersurface V
typically has two components, one on the horizon and one at infinity3.
6 The first law for stationary, Kaluza-Klein
black holes
Following references [9, 8], we now use the Hamiltonian formalism presented
in the last section to derive the first law for stationary, non-rotating Kaluza-
Klein black holes. The first law relates the difference δA in the horizon
area between nearby solutions to the variations δM, δP and δ L in the mass,
momentum and length of the compact direction. As in reference [8], we carry
out the calculation first holding the length at infinity,  L, fixed, and then use
this result in order to do the calculation with δ L 6= 0.
We assume as in section (2) that we have a stationary, non-rotating
Kaluza-Klein black hole solution with metric ḡab and horizon generating
Killing field la, which is given at infinity by la = T a + vHZ
a. We further
assume as in section (5) that the metric gab = ḡab + δgab is a linear approx-
imation to a solution of the field equations. At this stage, we assume that
δgab is such that δ L = 0. Further on, we will relax this assumption.
The derivation of the mass first law is then quite similar to that for
rotating black holes [9]. Consider a spacelike hypersurface V , which intersects
the horizon at the bifurcation surface and has a unit normal approaching the
vector T a at infinity. Choose the Killing vector in the Gauss’ law construction
to be the horizon generator la. Let ∂V∞ and ∂VH denote the boundaries of the
hypersurface V at infinity and at the horizon bifurcation surface. Equation
(24) then implies that
IH + I∞ = 0 (25)
3Kaluza-Klein bubble spacetimes, which we do not consider here, provide an interesting
contrast . There is no interior horizon, but the rotational Killing field has an axis. Hence
to use Stokes theorem, one must exclude the axis, which introduces an inner boundary.
where
c, I∞ =
c. (26)
Let us first consider the calculation of IH . On the horizon bifurcation
surface, the quantities F and βa vanish, and the boundary integral on the
horizon reduces to
IH = −
daρ̂c(−h D̄cF + hcb D̄bF ) (27)
where ρ̂c is the unit outward pointing normal to the bifurcation surface within
V . One can show that the surface gravity is given by κ = ρ̂c∂cF , and it then
follows as in reference [9] that
IH = 2κ δA (28)
Now consider the boundary term at infinity. Many of the terms in (23) fall
off too rapidly to make non-zero contributions. In particular, it is straight-
forward to check that the DaF terms, as well as those including products of
π̄ab with the metric perturbation, fall off too rapidly as r → ∞ to contribute.
Furthermore, it is sufficient to take F ≃ 1 and βz = vH in this limit. We
then arrive at the expression
dz dsi(∂
ih− ∂jhij − 2vHpiz) (29)
At this point, we need to note that the formulas (6) and (8) for the ADM
mass, momentum and tension are written in terms of the variable γab defined
by gab = ηab + γab. In order to interpret the boundary integral (29) in
terms of variations in M, P and T , we need to relate the perturbations
γab and p
ab in the Hamiltonian formalism to a covariant perturbation δγab.
It is straightforward to show that to the required order of accuracy h =
δklδγkl + δγzz and h
ij ≃ δikδjlδγkl, while a further calculation reveals that
piz ≃ −(1/2) ∂iδhzt. We then find that
∂ih− ∂jhij + vH∂ihzt
= −16πG(δM− vHδP) (31)
Inserting these results into equation (25) then yields the mass first law for
boosted black strings (with the length  L at infinity held fixed)4
δM = κ
δA + vHδP (33)
We see that the momentum appears as an extensive parameter in the first law,
while vH , which for the boosted black string is the horizon velocity, appears
as an intensive parameter. This parallels the way angular momentum enters
the first law for rotating black holes. Equation (32) is easily verified for the
case of the boosted black string using the formulas of section (4).
We now generalize the first law (33) to include perturbations with δL 6= 0.
Our analysis of the boundary term is based on that in [8] for the static
case. The boundary integral at the horizon in this case remains unchanged
and is still given by equation (28). Additional terms, however, occur in
the boundary term at infinity. Given the results above, we can write the
boundary term at infinity as
I∞ = 16πG(−δM|δL=0 + vHδP|δL=0 + λδL), (34)
where λ remains to be determined. On the other hand, we know the L
dependence of M and P explicitly from the expressions (11) and (12). This
allows us to write
δM = δM|δL=0 +
δL (35)
δP = δP|δL=0 +
δL (36)
Combining these with equations (34), (25) and (28) then gives
I∞ = 16πG
−δM + vHδP + (λ +
. (37)
We can now further appeal to the results of [8] for the case P = 0 and write
λ = λ|P=0 + λ′. We know from [8] that λ|P=0 + M/L = T . Putting this
together, allows us to rewrite (37) as
I∞ = 16πG
−δM + vHδP + (λ′ + T − vH
. (38)
4We can also include perturbative stress energy in this relation, in which case the mass
first law becomes
δM = κ
δA+ vHδP +
(−δT a
b). (32)
We still need to calculate the quantity Î∞ = 16πGλ
′δL which includes
only the terms in I∞ that are proportional to both P and δL. It is noted in [8]
that in order for the perturbative Gauss’s law (24) to apply with δL 6= 0, one
need to make a coordinate transformation so that δL appears in the metric
perturbation, rather than in a change in the range of coordinates. Following
this procedure yields the metric perturbations
hzz ≃ 2
), hzt ≃
There are two terms in equation (23) that potentially contribute to Î∞ and
we accordingly write Î∞ = Î
+ Î(2)
. The first of these terms is given by
Î(1)
dzdac
−2βbπ̄achab
dzdai(−2vH π̄izhzz) (41)
dzdai vH∂iḡtz
= 16πG vH P
The second term, which requires some care in evaluating, is given by
Î(2)
−2βbδπcb
dai(−2vHpiz) (45)
daivH ∂iḡtz
(1 − 2 + 1) (46)
= 0. (47)
where the factor (1 − 2 + 1) in line (46) comes about in the following way.
We have piz ≃ δ(
sszzKiz) with Kiz ≃ −(1/2)∂igtz. The first 1 comes from
the variation of the volume element
s, the −2 comes from the variation
of inverse metric component szz following from equation (39), and the final
1 comes from the variation of gtz, also as in equation (39). Putting these
results together gives λ′ = 2vHP/L and hence
I∞ = 16πG
−δM + vHδP + (T +
. (48)
Finally, combining this with IH gives the first law
δA + vHδP + (T +
)δL (49)
From the δL term, we see that the coefficient of δL is an effective tension given
by T̂ = T + vHP/L. As mentioned in the introduction, the tension of the
boosted black string becomes negative for sufficiently large boost parameter.
It is straightforward to check that the first law (49) is satisfied for variations
within the family of boosted black string solutions in section (4), and also
that the effective tension is given by
rD−4H (50)
which is equal to the tension of the unboosted black string having the same
horizon radius.
7 Smarr formulas, scaling relations and Ko-
mar integrals
Smarr formulas are relations between the thermodynamic parameters that
hold for black hole solutions that have exact symmetries. In this section we
will derive the Smarr formula for stationary, but non-rotating Kaluza-Klein
black holes. We will also derive a second Smarr-type formula that holds in the
case of exact translation invariance in the z-direction, e.g. for the boosted
black string. We present two approaches to deriving these formulas. The
first is based on general scaling relations, which are familiar from classical
thermodynamics, and the second is based on Komar integral relations.
Given the statement of the first law (49) for stationary, non-rotating
Kaluza-Klein black holes, the Smarr formula can be derived by making use
of a simple scaling argument (see e,g, [11]). Given any stationary vacuum
solution to Einstein’s equations, we may obtain a one parameter family of
solutions by rescaling all the dimensionful parameters in the given solution in
an appropriate way. If a parameter µ has dimensions (length)n, we replace it
with λnµ. The parameters ct, cz and ctz that specify the asymptotics of the
stationary solution all scale as (length)D−4. If we rescale these accordingly,
and also replace L with λL, then the mass and momentum rescale as
M = λD−3M̄, P = λD−3P̄ (51)
where M̄ and P̄ are the mass and momentum of the original solution. Sim-
ilarly, the area of the event horizon of the family of spacetimes will be
A = λD−2Ā. Now consider how these quantities change under a small change
in λ. We have
dM = (D−3)Mdλ
, dP = (D−3)P dλ
, dA = (D−2)Adλ
, dL = Ldλ
The first law (49) must hold in particular for this variation in λ. This will
implies that
(D − 3)M = (D − 2) 1
κA + T̂ L + (D − 3)vHP (53)
which is the Smarr formula for stationary, non-rotating Kaluza-Klein black
holes. Note that via the scaling argument, the effective tension T̂ naturally
enters the Smarr formula as well as the first law.
We now derive a second Smarr formula that holds only for solutions, such
as the boosted black string, that have exact translation invariance in the z-
direction5. Note that the mass, momentum and horizon area are all extensive
quantities in the compactification length L and that different values of L give
another one parameter family of solutions. Within this family we have
dM = MdL
, dP = P dL
, dA = AdL
under a small variation in L. For the first law to be satisfied under such
variations, we must have
M = 1
κA + T̂ L + vHP (55)
Because of the simple extensivity of M, P, A and L itself in the length L
of the compact direction, this second Smarr formula takes the form of the
usual Euler relation for a thermodynamic system, without any additional
dimension dependent prefactors. Note that by taking a linear combination
of the two Smarr formulas (53) and (55), the horizon area term may be
eliminated, giving
M = (D − 3)T̂ L + vHP (56)
= (D − 3)T L + (D − 2)vHP (57)
5It appears likely that the sboosted black strings can be shown to be the only stationary,
non-rotating, z translational vacuum solutions with non-singular horizons (see reference
[19]).
For P = 0 this is the well known relation between the mass and tension for
a uniform black string.
The Smarr formulas may also be derived by geometrical means using
Komar integral relations. This is done in reference [12] for the first Smarr
formula in the case P = 0. For a vacuum spacetime with a Killing vector ka,
and a hypersurface Σ with boundaries ∂Σ∞ at infinity and ∂ΣH at the black
hole horizon, the Komar integral relation implies the equality I∂Σ∞ = I∂ΣH
where
IS = −
dSab∇akb. (58)
The first Smarr formula results from taking ka to be the horizon generator
la and Σ to be a spacelike hypersurface with normal dt at infinity. The com-
putation of the horizon boundary term in this case is by now quite standard
(see [20]). The horizon generator la is null on the horizon and consequently
normal to the boundary ∂ΣH . Let q
a be the second null vector orthogo-
nal to ∂ΣH , normalized so that l
aqa = −1. One then has on the boundary
dSab = 2l[aqb]dA, where dA is the surface area element. It then follows that
I∂ΣH =
κA (59)
where we have made use of the definition (3) of the surface gravity. The
boundary term at infinity may be straightforwardly computed using the
asymptotic form of the metric in (10) and the expressions (11) and (12)
for the ADM mass, tension and momentum. One finds
I∂Σ∞ =
ΩD−3L
(D − 4)ct (60)
D − 2
((D − 3)M−T L) − vHP. (61)
Equating the two boundary integrals correctly reproduces the first Smarr
formula (53).
The scaling argument that led to the second Smarr formula assumed
translation invariance in the z-direction, i.e. that Za as well as the horizon
generator la = T a + vHZ
a is a Killing vector. To give a geometric derivation,
we additionally assume, as in section (2.1), that the Killing vectors T a and Za
commute. Let us now take the Killing vector in the Komar construction to
be V a = vHT
a +Za, which is orthogonal to the horizon generator la both at
infinity and on the horizon. We take the hypersurface Σ to be timelike, with
normal equal to dz at infinity and proportional to Va at the horizon. The
normal to the horizon within Σ is then proportional to the horizon generator
la, and hence the boundary term at the horizon includes the factor
laVb∇aV b = VaVb∇alb = 0. (62)
In the first equality the commutivity of the Killing vectors is used and the
second equality follows from Killing’s equation. Hence, the boundary term
at the horizon I∂ΣH vanishes for this choice of Killing field and hypersurface.
The boundary term at infinity is again straightforward to compute using the
expressions in (10),(11) and (12). We find
I∂Σ∞ = −
(D − 4)(cz + vHctz) (63)
= − 1
D − 2
(M/L− (D − 3)T ) + vHP/L. (64)
Equating this with zero then gives the second Smarr formula (55).
8 Tension first law and Gibbs-Duhem rela-
A second kind of variational formula for static Kaluza-Klein black holes was
derived in reference [13]. This ‘tension first law’ states that
LdT = − 1
Adκ (65)
and holds for perturbations that take a static, translation invariant solution
into a nearby solution that is stationary, but not necessarily translation in-
variant. It applies, for example, to the perturbation between the marginally
stable uniform black string and the static non-uniform black string of ref-
erence [21]. In this section, we discuss the thermodynamic context of this
formula and conjecture its extension to include P 6= 0.
We regard the quantities M, A, L and P as extensive parameters, while κ,
T and vH are regarded as intensive parameters. For thermodynamic systems,
the first law relates variations in the extensive parameters, as does equation
(49). In classical thermodynamics a formula relating the variations of the
intensive parameters is known as a Gibbs-Duhem relation. A Gibbs-Duhem
relation can be derived from the first law, together with the variation of an
Euler formula, such as equation (55). In the present case, variation of the
Euler formula gives
dM = 1
(κdA + Adκ) + T̂ dL + LdT̂ + vHdP + PdvH (66)
Combining this with the first law then gives the Gibbs-Duhem relation
Adκ + LdT̂ + PdvH (67)
which reduces to (65) for P = 0.
Note, however, that the Euler formula (55) holds only for z-translationally
invariant solutions, and hence the result above holds only for perturbations
that respect this symmetry, i.e. within the boosted black string family of
solutions. Equation (65) was derived in [13] via the Hamiltonian perturba-
tion methods of section (5), and does not require that the perturbations are
invariant under z translations. We would like to extend the derivation of [13]
to the stationary non-rotating case, but have not yet accomplished this.
9 Conclusions
We have derived various thermodynamic relations for stationary, non-rotating
Kaluza-Klein black holes. As in reference [8], the derivation of the first law
required a careful application of Hamiltonian perturbation theory techniques.
Perhaps the most interesting aspect of the first law (49) is the appearance
of the effective tension T̂ which generally differs from the ADM tension. For
the boosted black string, the ADM tension becomes negative for large boost
parameter, while the effective tension remains positive. We note that the
gravitational contribution to the ADM tension was shown to be positive for
static spacetimes in reference [22] using spinorial techniques. It should be
interesting to see what these techniques yield in the stationary case, e.g. do
they prove positivity of the effective tension.
Our results concerning the Smarr formulas in section (7) are also of in-
terest. In particular, the parallels between the scaling argument and Komar
integral relation derivations are intriguing and can most likely be understood
in a more general setting. Finally, we would like to be able to give a Hamilto-
nian derivation of the Gibbs-Duhem, or ‘tension first law’, result in equation
(67).
Acknowledgments
The authors would like to thank Roberto Emparan and Henriette Elvang for
helpful discussions. This work was supported in part by NSF grant PHY-
0555304.
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	Introduction
	Stationary, non-rotating Kaluza-Klein black holes
	Two commuting Killing fields
	ADM mass, tension and momentum
	The boosted black string
	Gauss' Laws for Perturbations
	The first law for stationary, Kaluza-Klein black holes
	Smarr formulas, scaling relations and Komar integrals
	Tension first law and Gibbs-Duhem relation
	Conclusions

  We study the thermodynamics of Kaluza-Klein black holes with momentum along
the compact dimension, but vanishing angular momentum. These black holes are
stationary, but non-rotating. We derive the first law for these spacetimes and
find that the parameter conjugate to variations in the length of the compact
direction is an effective tension, which generally differs from the ADM
tension. For the boosted black string, this effective tension is always
positive, while the ADM tension is negative for large boost parameter. We also
derive two Smarr formulas, one that follows from time translation invariance,
and a second one that holds only in the case of exact translation symmetry in
the compact dimension. Finally, we show that the `tension first law' derived by
Traschen and Fox in the static case has the form of a thermodynamic Gibbs-Duhem
relation and give its extension in the stationary, non-rotating case.

Introduction
The physics of Kaluza-Klein black holes, i.e. black hole spacetimes asymp-
totic at infinity to M × S1, has proved to be a surprisingly rich subject, in-
cluding such phenomena as the Gregory-Laflamme instability, non-uniform
static black strings and the black hole/black string phase transition (see e.g.
the reviews [1, 2]). Research to date has focused primarilly on the static
case. However, it is also of interest to explore the properties of stationary
solutions. Accordingly, in this paper we will study the thermodynamics of
stationary Kaluza-Klein black holes1.
Static Kaluza-Klein black holes are characterized at infinity by the mass
M, tension T and the length L of the compact direction. The physical
meaning of the tension follows from its role in the first law for static S1
Kaluza-Klein black holes [6][7][8]
dM = κ
dA + T dL. (1)
We see that the tension determines the variation of the mass with varying
length of the compact direction, under the constraint that the horizon area is
held fixed. Within the thermodynamic analogy, it appears to be an intensive
parameter of the system, like temperature or pressure.
Stationary Kaluza-Klein black holes can carry linear momentum in the
compact direction, as well as angular momentum. In this paper, we will be
interested in this linear momentum, which we denote by P, and will assume
that the angular momentum vanishes. The simplest solutions with P 6= 0
are boosted black strings. These are obtained by starting from the infinite
uniform black string, boosting in the z direction and then identifying the new
z coordinate with period L. The boosted black string is then locally, but not
globally, the same as the static uniform black string. Further stationary, but
not z-translationally invariant, solutions may be obtained by giving localized
black holes or non-uniform black strings velocity in the compact direction.
In subsequent sections, we present the following results. We use the
Hamiltonian methods of [9][10][8] to establish the first law for stationary, non-
rotating Kaluza-Klein black holes. We also derive two Smarr formulas for
these spacetimes. These are exact relations between the geometric quantities
1Aspects of stationary Kaluza-Klein black holes have been studied in [3][4]. The ther-
modynamics of asymptotically AdS, boosted domain walls have been investigated in ref-
erence [5].
M, κA, vHP and T L, where the quantity VH is defined below. The first of
these formulas holds for the entire class of spacetimes under consideration.
The second Smarr formula holds under the additional assumption of exact
translation invariance in the compact direction. A linear combination of these
two formulas gives the relation between mass and tension for the boosted
black string. We derive each of these Smarr formulas in two ways, first using
scaling arguments (as in e.g. reference [11]) and second using Komar integral
relations (as in reference [12]). Finally, we present a Gibbs-Duhem formula
that relates variations in the tension to variations in the other intensive
parameters. This result generalizes the ‘tension first law’ of reference [13].
Our result for the first law resolves a small puzzle related to the boosted
black string, which formed part of the motivation for this work. It was found
in reference [3] that the tension of the boosted black string becomes negative
for values of the boost parameter in excess of a certain critical value, which
depends only on the spacetime dimension. If the physical interpretation of
the tension based on equation (1) were to continue to hold in the stationary
case, then the energy of the system would decrease with increasing L, which
seems counter-intuitive. This puzzle is resolved by showing that the coeffi-
cient of the dL term in the first law for black holes is an effective tension T̂ .
The effective tension T̂ is equal to the ADM tension in the static case, but
includes a contribution from the momentum in the stationary case. For the
boosted black string T̂ is always positive, and is in fact given by the tension
of the unboosted black string with the same horizon radius.
2 Stationary, non-rotating Kaluza-Klein black
holes
We consider stationary D-dimensional vacuum black hole spacetimes that
are asymptotic to MD−1 × S1, and assume that the black hole horizon is a
bifurcate Killing horizon. In accordance with our focus on linear momentum
around the S1, we take the ADM angular momentum to vanish. We denote
the horizon generating Killing field by la, and assume that at infinity it has
the form
la = T a + vHZ
a (2)
where T a = (∂/∂t)a and Za = (∂/∂z)a, with z being the coordinate around
the S1. The surface gravity κ of the black hole horizon is defined, as usual,
via the relation on the horizon
∇a(lblb) = −2κla. (3)
The form (2) of the horizon generating Killing field at infinity resem-
bles the decomposition of the horizon generating Killing field for a rotating,
asymptotically flat black hole. In that case, i.e. upon replacing vHZ
a, with φ an azimuthal coordinate, one can show that T a and φa are
themselves Killing vectors [14, 15, 16]. The quantity ΩH can then be in-
terpreted as the angular velocity of the horizon, and further shown to be
constant on the horizon [14].
Returning to the case of Kaluza-Klein black holes, the situation is quite
different. Already in the static case, solutions exist which are non-uniform in
the z direction. In the stationary case then, it will not generally be the case
that T a and Za are Killing vectors. For localized black holes or non-uniform
black strings with velocity around the S1, only the linear combination la is
a Killing vector.
2.1 Two commuting Killing fields
It is nonetheless useful to separately consider the case in which T a = (∂/∂t)a
and Za = (∂/∂z)a are two commuting Killing fields, and that the relation
(2) holds throughout the spacetime. The boosted black string falls into this
class of spacetimes. If both Za and T a are Killing fields, then the quantity
vH in equation (2) may be considered to be the velocity of the black hole
horizon. This identification follows in a similar way to that of ΩH as the
angular velocity in the rotating case (see e.g. the article by Carter in [17]).
It follows from equation (3), together with our assumption that T a and Za
are commuting Killing vectors, that in addition to lala = 0 on the horizon,
one also has there the orthogonality relations
laTa = 0, l
aZa = 0. (4)
Given these, one can then show that the metric components on the horizon
satisfy the two relations
(T aZa)
2 = (T aTa) Z
bZb, vH = −
T aZa
(ZbZb)
The second of these leads to the interpretation of vH as the velocity of the
horizon in the following manner. For rotating black holes, one considers ‘zero
angular momentum observers’ or ZAMO’s. The angular velocity ΩH of the
horizon is the limit of a ZAMO’s angular velocity as it approaches the horizon
radius. For a boosted black string, we may analogously consider observers
with zero linear momentum along the string, which we might be justified
in calling ZELMO’s. Let pa = m dxa/dτ be the momentum of a particle
following a geodesic. It’s energy E = −T apa and the z−component of its
momentum P = Zapa are both constants of motion. The condition P = 0
of vanishing linear momentum is then dz/dt = −gtz/gzz, and we see that on
the horizon the coordinate velocity of a ZELMO is equal to vH .
3 ADM mass, tension and momentum
We review the formulas for the ADM mass, tension and momentum. Let us
write the spacetime metric near infinity as gab = ηab + γab, where ηab is the
D-dimensional Minkowski metric. The components of γab are assumed to fall-
off sufficiently rapidly that the integral expressions for the mass, tension and
momentum are well-defined. In the asymptotic region, write the spacetime
coordinates as xa = (t, z, xi), where i = 1, . . . , D − 2 and the coordinate z
running around the S1 is identified with period L. Let Σ be a spatial slice
and ∂Σ∞ its boundary at spatial infinity. The ADM mass and momentum
in the z direction are then given in asymptotically Cartesian coordinates by
the integrals
M = 1
dz dsi
−∂iγjj − ∂iγzz + ∂jγij
dz dsi ∂
iγtz (7)
where indices are raised and lowered with the asymptotic metric ηab and the
area element dsi is that of a sphere S
D−3 at infinity in a slice of constant t
and z.
The ADM tension is similarly given by the integral [13][6][18]
T = −
∂Σ∞/S1
−∂iγjj − ∂iγtt + ∂jγij
. (8)
Note that in contrast with the ADM mass and momentum, the definition
of the tension does not include an integral in the z-direction. The ADM
mass is an integral over the boundary of a slice of constant t, which includes
the direction around the S1. The tension, on the other hand, is defined
[13][6][18] by an integral over the boundary of a slice of constant z. This
includes, in principle, an integration over time. However, if one expands the
integrand around spatial infinity, one finds that terms that make non-zero
contributions to the integral are always time independent. Time dependent
terms fall-off too rapidly to contribute. Hence, one can omit the integration
over the time direction and work with the quantity T defined above, which
is strictly speaking a ‘tension per unit time’.
We can evaluate these formulas for M, P and T in terms of the asymp-
totic parameters of our spacetimes. The spacetimes we consider have topol-
ogy RD−1 × S1, the coordinate z in the compact direction being identified
with period L. We can write the metric explicitly as
ds2 = gttdt
2 + 2gtzdtdz + gzzdz
2 + 2(gtidtdx
i + gzidzdx
i) + gijdx
idxj (9)
where xi with i = 1, . . .D − 2 are the non-compact spatial coordinates. We
assume the following falloff conditions at spatial infinity
gtt ≃ −1 + ct/rD−4, gzz ≃ 1 + cz/rD−4, gtz ≃ ctz/rD−4, (10)
and further that the coefficients gti and gzi falloff sufficiently fast that they
do not contribute to any ADM integrals at infinity. The mass, tension [7]
and momentum can then be shown, using the field equations, to be given in
terms of the asymptotic parameters ct, cz and ctz by
M = ΩD−3  L
((D − 3)ct − cz), T =
(ct − (D − 3)cz), (11)
P = −(D − 4)ΩD−3  L
ctz . (12)
4 The boosted black string
The boosted black string serves as a simple analytic vacuum spacetime in
which to check the results we present below for the first law, Smarr and
Gibbs-Duhem relations. The boosted black string metric may be obtained
starting from the uniform black string, performing a boost transformation
with parameter β and identifying the new, boosted z coordinate with period
L. This gives
ds2 = −(1 −
cosh2 β)dt2 + (1 +
sinh2 β)dz2 (13)
sinh β cosh βdzdt + (1 − c
)−1dr2 + r2dΩ2D−3
The horizon, which has topology SD−3×S1 is located at rH = c1/(D−4). From
the asymptotic form of the metric, one finds using the expressions (11) and
(12) that the ADM mass, tension and momentum are given as in [3] by
ΩD−3L
rD−4H ((D − 4) cosh2 β + 1) (14)
rD−4H (1 − (D − 4) sinh2 β) (15)
P = −
ΩD−3L
rD−4H (D − 4) sinh β cosh β (16)
Note that, as mentioned in the introduction, the tension becomes negative
for sinh2 β > 1/(D − 4). We can further compute, as in reference [3], that
the horizon area, surface gravity, and horizon velocity of the boosted black
string are given by
A = ΩD−3LrD−3H cosh β, κ =
D − 4
2rH cosh β
, vH = −
sinh β
cosh β
. (17)
5 Gauss’ Laws for Perturbations
Following the work of [9], we use the Hamiltonian formalism of general rela-
tivity to derive the first law for stationary, non-rotating Kaluza-Klein black
holes. Another of our goals is to derive a ‘first law’ for variations in the
tension as in reference [13], for this class of spacetimes. This requires a slight
generalization of the Hamiltonian formalism to accomodate evolution of data
on timelike surfaces in a spacelike direction. Although, as we discuss below
in section (8), we have not yet succeeded in providing a Hamiltonian deriva-
tion of the ‘tension first law’ in the stationary case, our presentation of the
Hamiltonian formalism will be general enough to provide the necessary tools.
The essence of the method is as follows. In vacuum gravity, suppose one
has a black hole solution with a Killing field. Now consider solutions that
are perturbatively close to this background solution, but are not required
to have the original Killing symmetry. The linearized Einstein constraint
equations on a hypersurface can be expressed in the form of a Gauss’ law
(see [10]), relating a boundary integral at infinity to a boundary integral at
the horizon. The physical meaning of this Gauss’ law relation depends on
the choice of Killing field, as well as on the choice of hypersurface. Taking
the generator la of a Killing horizon, together with an appropriate choice of a
spacelike hypersurface, yields the usual first law for variation of the mass [9].
In the case of solutions that are z translation invariant, choosing the spatial
translation Killing vector Za, again with an appropriate choice of a timelike
hypersurface, gives a ‘first law’ for variations in the tension [13].
The formalism then proceeds in the following way. Assume we have a
foliation of a spacetime by a family of hypersurfaces of constant coordinate
w. We denote these hypersurfaces, both collectively and individually, by V
and the unit normal to the hypersurfaces by wa. With the application to
tension in mind, we consider both timelike and spacelike normals by setting
a = ǫ with ǫ = ±1. This slight generalization introduces factors of ǫ
into a number of otherwise standard formulas. The spacetime metric can be
written as
gab = ǫwawb + sab (18)
where sab, satisfying sa
bwb = 0, is the metric on the hypersurfaces V . As
usual, the dynamical variables in the Hamiltonian formalism are the met-
ric sab and its canonically conjugate momentum π
ab = ǫ
|s|(Ksab − Kab).
Here Kab = sa
c∇cwb is the extrinsic curvature of a hypersurface V . We con-
sider Hamiltonian evolution along the vector field W a = (∂/∂w)a, which can
be decomposed into its components normal and tangential to V , according
to W a = Nwa + Na where Nawa = 0. As usual, we refer to N and N
respectively as the lapse function and the shift vector. The gravitational
Hamiltonian density which evolves the system along W a is then given by
H = NH + NaHa with
H = −R(D−1) + ǫ
D − 2
− πabπab) (19)
Hb = −2Da(|s|−
2πab). (20)
where R(D−1) is the scalar curvature for the metric sab and the derivative
operator Da on the hypersurface V . One further finds that the quantities H
and Ha are simply related to the normal components of the Einstein tensor,
H = 2ǫ Gabw
awb, Hb = 2ǫ Gacw
ascb (21)
These components of the field equations contain only first derivatives with
respect to the coordinate w, and hence represent constraints on the dynamical
fields, sab and π
ab, on V . This property is independent of whether the normal
direction is timelike, as in the usual ADM formalism, or spacelike. In vacuum,
the equations H = 0 and Hb = 0 are enforced in the Hamiltonian formalism as
the equations of motion of the nondynamical lapse and shift variables. These
are referred to as the Hamiltonian and momentum constraints, a terminology
we continue to use in the case that the normal wa is spacelike.
Let us now assume that the spacetime metric ḡab is a solution to the
vacuum Einstein equations2 with a Killing vector ξa. We decompose ξa into
components normal and tangent to the hypersurfaces V introduced above,
according to ξa = Fwa + βa. Now, let us further assume that the metric
gab = ḡab+δgab is the linear approximation to another solution to the vacuum
Einstein equations. Denote the Hamiltonian data for the background metric
by s̄ab, π̄
ab, the corresponding perturbations to the data by hab = δsab and
pab = δπab, and the linearized Hamiltonian and momentum constraints by
δH and δHa. As shown in [10, 9, 13], the following statement then holds as
a consequence of Killing’s equation in the background metric,
FδH + βaδHa = −D̄aBa (22)
where D̄a is the background covariant derivative operator on the hypersurface
and the vector Ba is given by
Ba = F (D̄ah−D̄bhab)−hD̄aF +habD̄bF +
βb(π̄cdhcds̄
b−2π̄achbc−2pab).
Here indices are raised and lowered with the background metric s̄ab. Since the
metric gab solves the vacuum Einstein equations by assumption, we know that
δH = δHa = 0. Therefore, we have the Gauss’ law type statement D̄aB
a = 0.
Note that the detailed form of this relation for the perturbations hab and p
depends on the the Killing vector ξa and the normal wa to the hypersurface,
2In this paper we will focus on the case when the background spacetime is vacuum.
It is straightforward to add stress-energy which is described by a Hamiltonian [13]. If
the matter Hamiltonian contains constraints–such as for Maxwell theory–then additional
charges appear in the first law. This was worked out for Einstein-Yang-Mills in [9]. The
general case when the background spacetime has stress-energy, such as a cosmology, was
studied earlier in [10]. In this situation, the criterion for a Gauss’ law on perturbations is
that the background have an Integral Constraint Vector.
as well as on the background spacetime metric. Making different choices for
the Killing vector and normal can lead to different relations of this form. We
can now integrate the relation D̄aB
a = 0 over the hypersurface V and use
Stokes theorem to obtain
c = 0, (24)
where for black hole spacetimes the boundary ∂V of the hypersurface V
typically has two components, one on the horizon and one at infinity3.
6 The first law for stationary, Kaluza-Klein
black holes
Following references [9, 8], we now use the Hamiltonian formalism presented
in the last section to derive the first law for stationary, non-rotating Kaluza-
Klein black holes. The first law relates the difference δA in the horizon
area between nearby solutions to the variations δM, δP and δ L in the mass,
momentum and length of the compact direction. As in reference [8], we carry
out the calculation first holding the length at infinity,  L, fixed, and then use
this result in order to do the calculation with δ L 6= 0.
We assume as in section (2) that we have a stationary, non-rotating
Kaluza-Klein black hole solution with metric ḡab and horizon generating
Killing field la, which is given at infinity by la = T a + vHZ
a. We further
assume as in section (5) that the metric gab = ḡab + δgab is a linear approx-
imation to a solution of the field equations. At this stage, we assume that
δgab is such that δ L = 0. Further on, we will relax this assumption.
The derivation of the mass first law is then quite similar to that for
rotating black holes [9]. Consider a spacelike hypersurface V , which intersects
the horizon at the bifurcation surface and has a unit normal approaching the
vector T a at infinity. Choose the Killing vector in the Gauss’ law construction
to be the horizon generator la. Let ∂V∞ and ∂VH denote the boundaries of the
hypersurface V at infinity and at the horizon bifurcation surface. Equation
(24) then implies that
IH + I∞ = 0 (25)
3Kaluza-Klein bubble spacetimes, which we do not consider here, provide an interesting
contrast . There is no interior horizon, but the rotational Killing field has an axis. Hence
to use Stokes theorem, one must exclude the axis, which introduces an inner boundary.
where
c, I∞ =
c. (26)
Let us first consider the calculation of IH . On the horizon bifurcation
surface, the quantities F and βa vanish, and the boundary integral on the
horizon reduces to
IH = −
daρ̂c(−h D̄cF + hcb D̄bF ) (27)
where ρ̂c is the unit outward pointing normal to the bifurcation surface within
V . One can show that the surface gravity is given by κ = ρ̂c∂cF , and it then
follows as in reference [9] that
IH = 2κ δA (28)
Now consider the boundary term at infinity. Many of the terms in (23) fall
off too rapidly to make non-zero contributions. In particular, it is straight-
forward to check that the DaF terms, as well as those including products of
π̄ab with the metric perturbation, fall off too rapidly as r → ∞ to contribute.
Furthermore, it is sufficient to take F ≃ 1 and βz = vH in this limit. We
then arrive at the expression
dz dsi(∂
ih− ∂jhij − 2vHpiz) (29)
At this point, we need to note that the formulas (6) and (8) for the ADM
mass, momentum and tension are written in terms of the variable γab defined
by gab = ηab + γab. In order to interpret the boundary integral (29) in
terms of variations in M, P and T , we need to relate the perturbations
γab and p
ab in the Hamiltonian formalism to a covariant perturbation δγab.
It is straightforward to show that to the required order of accuracy h =
δklδγkl + δγzz and h
ij ≃ δikδjlδγkl, while a further calculation reveals that
piz ≃ −(1/2) ∂iδhzt. We then find that
∂ih− ∂jhij + vH∂ihzt
= −16πG(δM− vHδP) (31)
Inserting these results into equation (25) then yields the mass first law for
boosted black strings (with the length  L at infinity held fixed)4
δM = κ
δA + vHδP (33)
We see that the momentum appears as an extensive parameter in the first law,
while vH , which for the boosted black string is the horizon velocity, appears
as an intensive parameter. This parallels the way angular momentum enters
the first law for rotating black holes. Equation (32) is easily verified for the
case of the boosted black string using the formulas of section (4).
We now generalize the first law (33) to include perturbations with δL 6= 0.
Our analysis of the boundary term is based on that in [8] for the static
case. The boundary integral at the horizon in this case remains unchanged
and is still given by equation (28). Additional terms, however, occur in
the boundary term at infinity. Given the results above, we can write the
boundary term at infinity as
I∞ = 16πG(−δM|δL=0 + vHδP|δL=0 + λδL), (34)
where λ remains to be determined. On the other hand, we know the L
dependence of M and P explicitly from the expressions (11) and (12). This
allows us to write
δM = δM|δL=0 +
δL (35)
δP = δP|δL=0 +
δL (36)
Combining these with equations (34), (25) and (28) then gives
I∞ = 16πG
−δM + vHδP + (λ +
. (37)
We can now further appeal to the results of [8] for the case P = 0 and write
λ = λ|P=0 + λ′. We know from [8] that λ|P=0 + M/L = T . Putting this
together, allows us to rewrite (37) as
I∞ = 16πG
−δM + vHδP + (λ′ + T − vH
. (38)
4We can also include perturbative stress energy in this relation, in which case the mass
first law becomes
δM = κ
δA+ vHδP +
(−δT a
b). (32)
We still need to calculate the quantity Î∞ = 16πGλ
′δL which includes
only the terms in I∞ that are proportional to both P and δL. It is noted in [8]
that in order for the perturbative Gauss’s law (24) to apply with δL 6= 0, one
need to make a coordinate transformation so that δL appears in the metric
perturbation, rather than in a change in the range of coordinates. Following
this procedure yields the metric perturbations
hzz ≃ 2
), hzt ≃
There are two terms in equation (23) that potentially contribute to Î∞ and
we accordingly write Î∞ = Î
+ Î(2)
. The first of these terms is given by
Î(1)
dzdac
−2βbπ̄achab
dzdai(−2vH π̄izhzz) (41)
dzdai vH∂iḡtz
= 16πG vH P
The second term, which requires some care in evaluating, is given by
Î(2)
−2βbδπcb
dai(−2vHpiz) (45)
daivH ∂iḡtz
(1 − 2 + 1) (46)
= 0. (47)
where the factor (1 − 2 + 1) in line (46) comes about in the following way.
We have piz ≃ δ(
sszzKiz) with Kiz ≃ −(1/2)∂igtz. The first 1 comes from
the variation of the volume element
s, the −2 comes from the variation
of inverse metric component szz following from equation (39), and the final
1 comes from the variation of gtz, also as in equation (39). Putting these
results together gives λ′ = 2vHP/L and hence
I∞ = 16πG
−δM + vHδP + (T +
. (48)
Finally, combining this with IH gives the first law
δA + vHδP + (T +
)δL (49)
From the δL term, we see that the coefficient of δL is an effective tension given
by T̂ = T + vHP/L. As mentioned in the introduction, the tension of the
boosted black string becomes negative for sufficiently large boost parameter.
It is straightforward to check that the first law (49) is satisfied for variations
within the family of boosted black string solutions in section (4), and also
that the effective tension is given by
rD−4H (50)
which is equal to the tension of the unboosted black string having the same
horizon radius.
7 Smarr formulas, scaling relations and Ko-
mar integrals
Smarr formulas are relations between the thermodynamic parameters that
hold for black hole solutions that have exact symmetries. In this section we
will derive the Smarr formula for stationary, but non-rotating Kaluza-Klein
black holes. We will also derive a second Smarr-type formula that holds in the
case of exact translation invariance in the z-direction, e.g. for the boosted
black string. We present two approaches to deriving these formulas. The
first is based on general scaling relations, which are familiar from classical
thermodynamics, and the second is based on Komar integral relations.
Given the statement of the first law (49) for stationary, non-rotating
Kaluza-Klein black holes, the Smarr formula can be derived by making use
of a simple scaling argument (see e,g, [11]). Given any stationary vacuum
solution to Einstein’s equations, we may obtain a one parameter family of
solutions by rescaling all the dimensionful parameters in the given solution in
an appropriate way. If a parameter µ has dimensions (length)n, we replace it
with λnµ. The parameters ct, cz and ctz that specify the asymptotics of the
stationary solution all scale as (length)D−4. If we rescale these accordingly,
and also replace L with λL, then the mass and momentum rescale as
M = λD−3M̄, P = λD−3P̄ (51)
where M̄ and P̄ are the mass and momentum of the original solution. Sim-
ilarly, the area of the event horizon of the family of spacetimes will be
A = λD−2Ā. Now consider how these quantities change under a small change
in λ. We have
dM = (D−3)Mdλ
, dP = (D−3)P dλ
, dA = (D−2)Adλ
, dL = Ldλ
The first law (49) must hold in particular for this variation in λ. This will
implies that
(D − 3)M = (D − 2) 1
κA + T̂ L + (D − 3)vHP (53)
which is the Smarr formula for stationary, non-rotating Kaluza-Klein black
holes. Note that via the scaling argument, the effective tension T̂ naturally
enters the Smarr formula as well as the first law.
We now derive a second Smarr formula that holds only for solutions, such
as the boosted black string, that have exact translation invariance in the z-
direction5. Note that the mass, momentum and horizon area are all extensive
quantities in the compactification length L and that different values of L give
another one parameter family of solutions. Within this family we have
dM = MdL
, dP = P dL
, dA = AdL
under a small variation in L. For the first law to be satisfied under such
variations, we must have
M = 1
κA + T̂ L + vHP (55)
Because of the simple extensivity of M, P, A and L itself in the length L
of the compact direction, this second Smarr formula takes the form of the
usual Euler relation for a thermodynamic system, without any additional
dimension dependent prefactors. Note that by taking a linear combination
of the two Smarr formulas (53) and (55), the horizon area term may be
eliminated, giving
M = (D − 3)T̂ L + vHP (56)
= (D − 3)T L + (D − 2)vHP (57)
5It appears likely that the sboosted black strings can be shown to be the only stationary,
non-rotating, z translational vacuum solutions with non-singular horizons (see reference
[19]).
For P = 0 this is the well known relation between the mass and tension for
a uniform black string.
The Smarr formulas may also be derived by geometrical means using
Komar integral relations. This is done in reference [12] for the first Smarr
formula in the case P = 0. For a vacuum spacetime with a Killing vector ka,
and a hypersurface Σ with boundaries ∂Σ∞ at infinity and ∂ΣH at the black
hole horizon, the Komar integral relation implies the equality I∂Σ∞ = I∂ΣH
where
IS = −
dSab∇akb. (58)
The first Smarr formula results from taking ka to be the horizon generator
la and Σ to be a spacelike hypersurface with normal dt at infinity. The com-
putation of the horizon boundary term in this case is by now quite standard
(see [20]). The horizon generator la is null on the horizon and consequently
normal to the boundary ∂ΣH . Let q
a be the second null vector orthogo-
nal to ∂ΣH , normalized so that l
aqa = −1. One then has on the boundary
dSab = 2l[aqb]dA, where dA is the surface area element. It then follows that
I∂ΣH =
κA (59)
where we have made use of the definition (3) of the surface gravity. The
boundary term at infinity may be straightforwardly computed using the
asymptotic form of the metric in (10) and the expressions (11) and (12)
for the ADM mass, tension and momentum. One finds
I∂Σ∞ =
ΩD−3L
(D − 4)ct (60)
D − 2
((D − 3)M−T L) − vHP. (61)
Equating the two boundary integrals correctly reproduces the first Smarr
formula (53).
The scaling argument that led to the second Smarr formula assumed
translation invariance in the z-direction, i.e. that Za as well as the horizon
generator la = T a + vHZ
a is a Killing vector. To give a geometric derivation,
we additionally assume, as in section (2.1), that the Killing vectors T a and Za
commute. Let us now take the Killing vector in the Komar construction to
be V a = vHT
a +Za, which is orthogonal to the horizon generator la both at
infinity and on the horizon. We take the hypersurface Σ to be timelike, with
normal equal to dz at infinity and proportional to Va at the horizon. The
normal to the horizon within Σ is then proportional to the horizon generator
la, and hence the boundary term at the horizon includes the factor
laVb∇aV b = VaVb∇alb = 0. (62)
In the first equality the commutivity of the Killing vectors is used and the
second equality follows from Killing’s equation. Hence, the boundary term
at the horizon I∂ΣH vanishes for this choice of Killing field and hypersurface.
The boundary term at infinity is again straightforward to compute using the
expressions in (10),(11) and (12). We find
I∂Σ∞ = −
(D − 4)(cz + vHctz) (63)
= − 1
D − 2
(M/L− (D − 3)T ) + vHP/L. (64)
Equating this with zero then gives the second Smarr formula (55).
8 Tension first law and Gibbs-Duhem rela-
A second kind of variational formula for static Kaluza-Klein black holes was
derived in reference [13]. This ‘tension first law’ states that
LdT = − 1
Adκ (65)
and holds for perturbations that take a static, translation invariant solution
into a nearby solution that is stationary, but not necessarily translation in-
variant. It applies, for example, to the perturbation between the marginally
stable uniform black string and the static non-uniform black string of ref-
erence [21]. In this section, we discuss the thermodynamic context of this
formula and conjecture its extension to include P 6= 0.
We regard the quantities M, A, L and P as extensive parameters, while κ,
T and vH are regarded as intensive parameters. For thermodynamic systems,
the first law relates variations in the extensive parameters, as does equation
(49). In classical thermodynamics a formula relating the variations of the
intensive parameters is known as a Gibbs-Duhem relation. A Gibbs-Duhem
relation can be derived from the first law, together with the variation of an
Euler formula, such as equation (55). In the present case, variation of the
Euler formula gives
dM = 1
(κdA + Adκ) + T̂ dL + LdT̂ + vHdP + PdvH (66)
Combining this with the first law then gives the Gibbs-Duhem relation
Adκ + LdT̂ + PdvH (67)
which reduces to (65) for P = 0.
Note, however, that the Euler formula (55) holds only for z-translationally
invariant solutions, and hence the result above holds only for perturbations
that respect this symmetry, i.e. within the boosted black string family of
solutions. Equation (65) was derived in [13] via the Hamiltonian perturba-
tion methods of section (5), and does not require that the perturbations are
invariant under z translations. We would like to extend the derivation of [13]
to the stationary non-rotating case, but have not yet accomplished this.
9 Conclusions
We have derived various thermodynamic relations for stationary, non-rotating
Kaluza-Klein black holes. As in reference [8], the derivation of the first law
required a careful application of Hamiltonian perturbation theory techniques.
Perhaps the most interesting aspect of the first law (49) is the appearance
of the effective tension T̂ which generally differs from the ADM tension. For
the boosted black string, the ADM tension becomes negative for large boost
parameter, while the effective tension remains positive. We note that the
gravitational contribution to the ADM tension was shown to be positive for
static spacetimes in reference [22] using spinorial techniques. It should be
interesting to see what these techniques yield in the stationary case, e.g. do
they prove positivity of the effective tension.
Our results concerning the Smarr formulas in section (7) are also of in-
terest. In particular, the parallels between the scaling argument and Komar
integral relation derivations are intriguing and can most likely be understood
in a more general setting. Finally, we would like to be able to give a Hamilto-
nian derivation of the Gibbs-Duhem, or ‘tension first law’, result in equation
(67).
Acknowledgments
The authors would like to thank Roberto Emparan and Henriette Elvang for
helpful discussions. This work was supported in part by NSF grant PHY-
0555304.
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	Introduction
	Stationary, non-rotating Kaluza-Klein black holes
	Two commuting Killing fields
	ADM mass, tension and momentum
	The boosted black string
	Gauss' Laws for Perturbations
	The first law for stationary, Kaluza-Klein black holes
	Smarr formulas, scaling relations and Komar integrals
	Tension first law and Gibbs-Duhem relation
	Conclusions
