Wald entropy in Kaluza-Klein black holes

: We study the thermodynamics of the 4-dimensional electrically charged black-hole solutions of the simplest 5-dimensional Kaluza-Klein theory using Wald’s formalism. We show how the electric work term present in the 4-dimensional first law of black-hole thermodynamics arises in the purely gravitational 5-dimensional framework. In particular, we find an interesting geometric interpretation of the 4-dimensional electrostatic potential similar to the angular velocity in rotating black holes. Furthermore, we show how the momentum map equation arises from demanding compatibility between the timelike Killing vector of the black-hole solution and the spatial Killing vector of the 5-dimensional background.


Introduction
The possible existence of additional dimensions of different kinds has been a recurring theme in theoretical physics for most of the past century since the first proposal by Kaluza and Klein [1,2]. 1 In Kaluza-Klein (KK) theories, gravity with additional spatial dimensions gives rise to gauge and scalar fields, providing a framework for the unification of gravity with the rest of the fundamental interactions, which are described by Abelian and non-Abelian Yang-Mills-(YM) type fields.Actually, according to Ref. [4] it seems that the latter were discovered by Pauli precisely in this framework in which they arise quite naturally because, after all, YM theories are nothing but geometric theories with extra dimensions corresponding to the fibers of the corresponding principal bundle. 2his aspect of YM theories as theories of extra dimensions is often neglected because, when they are not the result of a KK theory, those extra dimensions are not standard spacetime dimensions and cannot be "seen" by the metric field that describes gravity as it happens in KK theories.It is this separation between spacetime dimensions that can be seen by the metric and extra, "gauge", dimensions, that cannot, that leads to the mathematical structure of principal bundle to describe the geometry of YM theories.In the context of KK theories, all dimensions can be treated in the same fashion and the simpler structure of a simple (pseudo-) Riemannian differential manifold suffices to describe geometries which include YM fields but can also be simultaneously described in terms of principal bundles.
Using this dual description of KK geometries as (pseudo-) Riemannian manifolds or as principal bundles one should be able to prove results using the former formulation, which is simpler and more widely understood.One of the main goals of this paper is to use this mechanism to study the spacetime symmetries of systems with YM fields in terms of the symmetries of (pseudo-) Riemannian manifolds which, as is well known, are isometries generated by Killing vectors.We will consider the simplest setting, 5dimensional general relativity in a manifolds with a single compact dimension which gives rise to a U(1) gauge field, but our results should apply to more general settings.
The study of the spacetime symmetries of gauge field configurations arises naturally in the construction of spacetime conserved quantities [6,7] and, via Wald's discovery of the connection between the Noether charge associated to diffeomorphisms and the black hole entropy [8,9], in the context of black-hole thermodynamics in presence of matter fields with gauge freedoms [10][11][12][13][14].The main observation is that one has to search for symmetries in the complete bundle and that those symmetries, when seen from (or projected to) the base space, are combinations of a diffeomorphism and a "induced" or "compensating" gauge transformation.This gauge transformation depends on the diffeomorphism and cannot be ignored or separated from it.As a consequence, most fields cannot be treated as simple tensors under diffeomorphisms as in [15].This is a fundamental fact that we are going to prove using the KK framework, 3 but one can arrive at this conclusion by considering spinors in curved spacetime.
Spinors (and Lorentz tensors) are defined in appropriate bundles connected to the tangent space on which local Lorentz transformations act.Usually, they are treated as scalars under diffeomorphisms but it is not difficult to see that this description is incorrect: let us consider spinor fields in Minkowski spacetime and let us consider the effect of an infinitesimal global Lorentz spacetime (i.e.not tangent space) transformation on the spinors with parameter σ ab .If they are treated as scalars they will transform as such, that is4 Thus, they will not transform as spinors under that spacetime transformation as they certainly should under a Lorentz transformation.The solution to th eabove problem comes from the following observation: the spacetime diffeomorphism generated by the Killing vector k (σ) induces a tangent space Lorentz transformation with a parameter that is minus the (automatically antisymmetric ) derivative of the Killing vector, also known as Killing bivector or Lorentz momentum map.In this case as expected.In more general settings the parameter of the induced local Lorentz transformation includes a term proportional to the spin connection [16] and is, indeed, local.
The combination of the Lie derivative and the compensating Lorentz transformation for the infinitesimal diffeomorphism generated by an arbitrary Killing vector field k is known as the spinorial Lie derivative L k and was first introduced by Lichnerowicz and Kosmann in Refs.[17][18][19] and later studied and extended in Refs.[20][21][22]16] also as the Lorentz-covariant Lie derivative or as the Lie-Lorentz derivative.One of its main properties is that it transforms covariantly under further diffeomorphisms and local Lorentz transformations.Thus, the invariance of the spinor field ψ under the infinitesimal diffeomorphism generated by k reads and is invariant statement. 5imilar considerations lead to the definition of more general Lie covariant derivatives [23,[12][13][14] with analogous properties.As stated above, one of our main goals is to test the simplest of these constructions (the Lie-Maxwell derivative for U(1) gauge fields) using the KK framework.
Another of our main goals in this paper is to study the thermodynamics of 4dimensional KK black holes directly from the 5-dimensional point of view using Wald's formalism.It is clear that we can only do this if we know the relation between the 4-and 5-dimensional spacetime symmetries and this is one of our motivations for its study.Furthermore, we know that the 4-dimensional event horizon of stationary solutions is the Killing horizon of a certain Killing vector but, does the existence of a 4-dimensional event horizon imply the existence of a 5-dimensional one?Will it also be a Killing horizon?With respect to which Killing vector?Will the 5-dimensional surface gravity be equal to the 4-dimensional one?To the best of our knowledge, there are no complete answers to these questions in the literature and we will try to find them for static 4-dimensional black holes.
Finally, we will apply all these results to the calculation of the Smarr formula and first law in 5 dimensions, aiming to obtain those of the 4-dimensional black holes in a sort of "dimensional reduction" of those formulae.We will succeed with the first but not with the second, which needs further research.
This work is organized as follows: in Section 1 we review the most basic Kaluza-Klein theory, its dimensional reduction and symmetries.In Section 2 we review the general electrically charged, static solution of the 4-dimensional theory obtained by dimensional reduction in the previous section.In Section 3 we study the thermodynamics of the black holes of the 4-dimensional theory using Wald's formalism and the extensions necessary to account for the gauge symmetries.In Section 4 we study 4dimensional black holes from a 5-dimensional perspective and use the results to try rederive Smarr formula and the first law Section 5. Finally, in Section 6 we discuss our results and future directions of work.

Basic Kaluza-Klein theory
Consider pure Einstein gravity in 5 dimensions parametrized by the coordinates x μ. 6 The only dynamical field is the 5-dimensional metric ĝ μ ν and the 5-dimensional line element is In this theory, the dynamics of the metric field is dictated by the Einstein-Hilbert action where G N is the 5-dimensional analog of the Newton constant.If the 5 th coordinate is periodic where ℓ is some length scale, all the components in the metric can be expanded in Fourier series. 7Since the higher modes correspond to fields which appear as massive from the non-compact 4-dimensional world perspective and since their masses can be made arbitrarily high by choosing the size of the 5 th direction small enough, we can safely ignore them at low energies and work with the zero modes only, which are the components of a z-independent metric Thus, in this scenario the metric admits an isometry generated by a spacelike Killing vector 6 We write hats over all 5-dimensional objects to distinguish them from the 4-dimensional ones.The 5 th coordinate will be denoted as x 4 = z and the corresponding (world) index will be z to distinguish it from the corresponding, not underlined, 5 th tangent space direction.Thus, ( μ) = (µ, z), ( â) = (a, z), etc.We use a mostly minus signature and the rest of the conventions are those used in Ref. [23]. 7A common choice in the literature is ℓ = R, the asymptotic radius of the compact dimension which will be defined below in Eq. (1.7).However, since that radius is invariant, as we are going to show, that choice (as well as the choice in which z is an angle and z ∼ z + 2π) prevents any redefinition of the coordinate z.We are, however, specially interested in studying the rescalings of z which induce a global rescaling symmetry of the 4-dimensional theory and, therefore, we have to distinguish very carefully between ℓ and R. k = k μ∂ μ = ∂ z , (1.5) and the coordinates we are using (x μ) = (x µ , x 4 ≡ z) are coordinates adapted to the isometry.
It is important to distinguish between the length scale ℓ and the radius of the compact direction, which is a local quantity R(x) since the geometry we are describing is that of a S 1 fibered over a 4-dimensional base space.In these adapted coordinates, we can define it in a coordinate-independent way as the integral of the 1-form along the S 1 at any point of the 4-dimensional base space: The constant asymptotic value of R(x) will be denoted by R and will be assumed to be finite.
The 5-dimensional metric can be decomposed in terms of fields which transform as 4-dimensional fields are expected to.There is a scalar (the Kaluza-Klein (KK) scalar field), k, defined by and constrained to take strictly positive values; the KK vector field and the KK metric (1.10) The 5-dimensional line element can be rewritten in terms of the 4-dimensional KK fields we have just defined as (1.11) The particular definitions of the 4-dimensional fields g µν , A µ , k are justified by their behaviour under 5-dimensional reparametrizations that respect the gauge choice in Eq. (1.4) (coordinates adapted to the isometry).
Most of these reparametrizations are generated by z-independent 5-dimensional vectors ξ μ which act on the 5-dimensional metric according to (1.12) It follows that their action on the 4-dimensional fields is These transformations can be interpreted as 4-dimensional general coordinate transformations generated by the 4-dimensional vector plus standard gauge transformations generated by the gauge parameter acting on A only.Therefore, A plays the role of a 1-form connection with gauge transformations and gauge-invariant field strength (1.17) There is only one z-dependent 5-dimensional general coordinate transformation that preserves the gauge Eq. (1.4).It is generated by the vector field η ≡ z∂ z , (1.18) and it only acts on the z coordinate as a rescaling: if α is an infinitesimal parameter, then Let us stress that this transformation does not change R.This transformation only acts on components of the metric with a z index: and its effect on the 4-dimensional fields is a rescaling ) Observe that the KK scalar will in general reach some constant value k ∞ = 1 at spatial infinity.According to Eq. (1.7), that value is related to the asymptotic value of the radius R and the length scale ℓ by Under the above rescaling Since the 5-dimensional theory we are starting with is diff-invariant, the 4-dimensional one will also be invariant under these global rescalings.Observe that the vector field that generates these rescalings does not commute with the Killing vector that generates translations in the internal dimension ( Thus it does not commute with the vectors that generate 4-dimensional gauge transformations (1.25) Following Scherk and Schwarz [24], in order to find the equations of motion that govern the dynamics of the 4-dimensional fields it is convenient to use the Vielbein formalism, making a particular choice for the decomposition of the 5-dimensional one ê â μ in terms of the 4-dimensional fields e a µ , A µ , k that breaks the 5-dimensional Lorentz group down to the 4-dimensional one:

.26)
Here A a = e a µ A µ and we will assume that all 4-dimensional fields with Lorentz indices a, b, c, . . .have been contracted with the 4-dimensional Vielbein.The above expressions can also be written in the form êa = e a , êa = e a − ı a A∂ z , (1.27a) where ı ξ indicates the inner product with the 4-dimensional vector ξ and ı a with e a , that is, ı a A = e a µ A µ .With this decomposition, the non-vanishing components of the spin connection8 are or, written as 1-forms (1.29) The components of the curvature 2-form are9 ( The Ricci 1-form Rĉ = R μ ĉdx μ is defined by where îˆb is the inner product with the vector 5-dimensional vector êˆb = êˆb μ∂ μ.We have where ı a = η ab ı b .Thus, the 5-dimensional Einstein equations Rĉ = 0 are equivalent to the following 3 equations involving 4-dimensional fields: The action from which these 4-dimensional equations can be derived can be obtained from the 5-dimensional Einstein-Hilbert action Eq.(1.2), which, in the Vielbein formalism, takes the form Substituting the above decompositions of the 5-dimensional Vielbein and curvature in terms of the 4-dimensional fields we get Integrating over the internal coordinate z ∈ [0, 2πℓ] and using the z-dependence of the 4-form, we get We have kept the total derivative because total derivatives can modify Noether currents and charges.
It is not too difficult to see that the equations that one gets from this action are combinations of Eqs.(1.34) and, therefore, equivalent to them.
The factor of k in front of the Einstein-Hilbert term in Eq. (1.36) indicates that the 4dimensional metric g µν is not in the (conformal) Einstein frame, in which, by definition, the Einstein-Hilbert term has no additional scalar factors.The Einstein-frame metric is clearly related to g µν by a Weyl rescaling with some power of the KK scalar k.If we want the rescaling to preserve the normalization of the metric in the non-compact directions, we must use a power of k/k ∞ and not just of k to rescale it.Thus, we define the 4-dimensional Einstein-frame metric g E µν and Vielbein e E a µ and the Einstein-frame KK vector field A E µ by Under this rescaling, and (1.40) We arrive to the Einstein-frame action with the 4-dimensional Newton constant given by Observe that the 4-dimensional Newton constant depends on the invariant radius R and not on ℓ or k ∞ , both of which transform under rescalings.Finally, we redefine k in terms of an unconstrained scalar field φ which can take any real value and we arrive to the final form of our action This is a particular Einstein-Maxwell-dilaton (EMD) model with a = − √ 3 in the parametrization used in Ref. [25], whose results we can use as long as we take into account the additional total derivative term. 10 After all these redefinitions, the relation between the 5-dimensional line element and the 4-dimensional Einstein-frame line element and other Einstein-frame fields is where Given a solution of the 4-dimensional theory Eq. (1.44) with fields g E µν , A E µ , φ, with φ → φ ∞ at infinity, the above relation allows us to rewrite it as a solution of pure 5-dimensional gravity.In the next section we are going to review the static, electrically charged, spherically symmetric 4-dimensional black hole solutions of this theory.
Let us now derive the equations of motion of the 4-dimensional theory.Under a general variation of the fields, the action Eq.(1.44) behaves as follows: where ϕ denotes collectively all the fields of the theory.Suppressing the overall factors of (16πG N ) −1 , the equations of motion are given by ) ) while 10 See also Ref. [26] for the definitionof the scalar charge.

Θ(ϕ, δϕ
(1.50) The action and equations of motion are evidently invariant under U(1) gauge transformations as well as under the global scale transformations which originate in the 5-dimensional general coordinate transformation z ′ = e α z.
The set of equations of motion of the 4-dimensional KK theory, enhanced with the Bianchi identity of the KK vector field strength are left invariant by the electric-magnetic duality transformation of the vector and scalar fields This transformation can be used to generate new from already known solutions and, in particular, magnetic from electric solutions and vice versa.In contrast with the other symmetries of the equations of motion of the KK theory (diffeomorphisms, U(1) gauge transformations and global rescalings) it is not clear whether this symmetry has a 5-dimensional, purely geometrical origin.To start with, it is unclear what the 5-dimensional origin of the Bianchi identity B E = 0 is.On the other hand, this is a symmetry of the equations of motion only: the transformations Eqs.(1.54) do not leave invariant the action.
Finally, notice that the scalar equation can be rewritten in the alternative form The term in square brackets is the on-shell conserved Noether (d − 1)-form current associated to the invariance under the global transformations Eq. (1.52).

Motion in a Kaluza-Klein spacetime
It is interesting to study the geodesic motion of test particles in the 5-dimensional space, which is controlled by the equations where α = 0 for massless particles and α = m 2 for massive particles.Rewriting these equations in terms of the 4-dimensional fields, we find plus the equation for z(ξ).This equation is complicated but it is entirely equivalent to the conservation of the momentum conjugate to z Using this relation to eliminate ż in Eqs.(1.57), they take the form which are the equations of motion of a 4-dimensional particle with electric charge P z and a spacetime-dependent effective mass squared α + k −2 P 2 z .The interaction with the scalar induces another force term proportional to P 2 z .Eqs. (1.56) can be derived from the Polyakov-type action (1.60) When m = 0, e can be eliminated replacing the (algebraic) solution to its equation of motion in the above action.One obtains the Nambu-Goto-type action (1.61) Rewriting the action Eq.(1.60) in terms of the 4-dimensional fields and performing a Legendre transformation to eliminate z(ξ), one arrives to and eliminating e and using the fact that P z is constant, we get The physical interpretation of this action is exactly the same as that of Eqs.(1.59), which, as expected, can be derived from Eq. (1.62).
In the Einstein frame, the above action takes the form and it describes a particle of electric charge and a position-dependent inertial mass that depends on the 5-dimensional mass and the charge and their couplings to the KK scalar.

The electric Kaluza-Klein black hole
The basic static, spherically symmetric, asymptotically flat, electric, solution of the 4-dimensional KK theory Eq. (1.44) has the form11 , where the functions H and W are given by and the integration constants h, w, α satisfy the relation In this solution we can always take h ≥ 0 and w ≤ 0. Therefore, for w = 0, there is always a regular horizon at r = r 0 = −w.When w = 0 (r 0 = 0), the horizon becomes singular and the solution is not a black hole.Multicenter solutions are possible in this case and we will refer to it as the extremal case and to r 0 as the non-extremality parameter.
Defining the mass M through the asymptotic expansion and the electric charge through the integral where Σ 2 is a 2-dimensional closed surface that encloses the black hole 13 we find that the integration constants w, h, α can be expressed in terms of M and q as follows: Then, extremality is reached when One can also define a charge for the scalar field, Σ, through its asymptotic expansion at spatial infinity (2.9) or equivalently, through the definition proposed in Ref. [26,25]. 12Notice that, with this definition, the electric charge q rescales as q → e −α/2 q , (2.5) aunder the transformations Eqs.(1.21a) and (1.21b). 13Usually, it is taken to be the 2-sphere at spatial infinity, but one gets the same result integrating on any other surface that can be obtained by deforming it, such as sections of the horizon, as long as we do not cross any singularities, the equations of motion are satisfied and the integrands remain closed 2-forms.
The scalar charge Σ is not expected to be an independent quantity ("charge") characterizing a black hole (i.e. an actual black hole with regular event horizon).As a matter of fact, the solutions of this theory in which Σ is independent do not have an event horizon (they are singular) and in the solutions at hand do have a regular event horizon Σ is given by the following function of the conserved charges [25]: (2.10) Let us now consider the thermodynamical properties of this black hole.This black hole, and the rest of the black holes that we are going to consider, being static, have a timelike Killing vector, l ≡ ∂ t and their event horizons, located at r = r 0 when r 0 = 0, coincide with the associated Killing horizon (2.11) The Hawking temperature and the Bekenstein-Hawking entropy are given by and they satisfy the relation

.13)
Observe that, in the extremal limit r 0 → 0, the temperature goes to infinity while the entropy goes to zero.
Eq. (2.13) can be used to derive a Smarr-like formula multiplying and dividing the right-hand side by r 0 + h.In the first case, we get and, comparing with we can identify with the difference between the value of electrostatic potential at the horizon and at spatial infinity.Indeed, one can check that, in the gauge we are using, in which the potential vanishes at spatial infinity Varying the entropy Eq. (2.12b) we get δS = 1 where T, Φ, Σ are given, respectively, by Eqs.(2.12a), (2.16) and (2.10), confirming our identification of the thermodynamical potentials conjugated to the charges.

4-dimensional thermodynamics à la Wald
As we have mentioned before, the 4-dimensional action Eq.(1.44) is a particular example of the Einstein-Maxwell-dilaton (EMD) model with a = − √ 3, up to a total derivative.The thermodynamics of the black-hole solutions of all these models has been studied in detail using Wald's formalism in Ref. [25].It can easily be shown that the total derivative has no effect on the first law, which is given by Eq. (2.18) when the magnetic terms are set to zero.
For later use we quote the expression of the Noether-Wald charge With this charge we can construct the generalized Komar charge [30] by the procedure explained in Refs.[31][32][33].First, we compute ω l , defined by which in this case is given by Then, the Komar charge is given by and, for purely electric black holes, it leads to the Smarr formula Eq. (2.15).Our goal now is to recover these results form the purely 5-dimensional point of view.First, we are going to study some general aspects of the 5-dimensional geometry of 4-dimensional Kaluza-Klein black holes.

Kaluza-Klein black-hole solutions from the 5-dimensional point of view
An important previous question is whether the presence of event horizons in the 4dimensional metric implies their presence in the 5-dimensional one.We need to study the behaviour of null geodesics in the 5-dimensional spacetime which is determined by Eqs.(1.56) which we have shown to be equivalent to the 4-dimensional Eqs.(1.59) plus the equation of conservation of P z .The second of Eqs.(1.59) is particularly interesting because it tells us that the lightcones of the 5-dimensional metric are equal to those of the 4-dimensional one, times a circle. 145-dimensional, massless, P z = 0 particles which move over the 5-dimensional lightcone are seen to move inside the 4-dimensional one.
In particular, this means that, if the 4-dimensional metric has event horizons, so does the 5-dimensional one at the same place in the 4-dimensional coordinates.The 5dimensional horizon simply has one more dimension, parametrized by z, fibered over the 4-dimensional one and we will denote both the 4-and 5-dimensional event horizons by H.
The main feature of the 5-dimensional geometries that correspond to the 4-dimensional static black holes we are considering is the fact that they all admit the 5-dimensional spacelike Killing vector k = ∂ z and a timelike Killing vector associated to the staticity of the 4-dimensional metric, l = ∂ t .The event horizon of the 4-dimensional black hole is also the Killing horizon of this vector and can be characterized by this property As we have just discussed, there is a 5-dimensional event horizon which is a S 1 fibration over the 4-dimensional event horizon H and the 4-dimensional timelike Killing vector l = ∂ t is also a Killing vector of the 5-dimensional metric.However, if we use the 5-dimensional metric Thus, from the 5-dimensional point of view, the event horizon is not the Killing horizon of l.
It is natural to search for a 5-dimensional extension of l, that we will denote by l, whose Killing horizon coincides with the event horizon, that is, Assuming that l has the form we have 14 The conformal rescaling that brings us to the Einstein metric leaves the lightcones invariant.
which means that we have to demand that and we are going to assume that While this is not the most general possibility, it will be good enough for us.
On the other hand, we want l to be a Killing vector of the 5-dimensional metric.If which is satisfied in the solutions considered.Furthermore, taking into account the previous result, and we conclude that the Lie derivative of the 1-form must vanish up to a gauge transformation with parameter f .Using Eq. (4.7) and differential-form language, and rescaling the equation with k −1/2 ∞ , this condition takes the form This is nothing but the Maxwell momentum map equation introduced in Ref. [12] with k −1/2 ∞ g playing the role of momentum map P E l , and, taking into account that γ| H = 0 and that the momentum map is defined only up to an additive constant, we conclude that and we arrive at Summarizing, the condition of invariance of the gauge field A E under the isometry generated by l is ) where the "compensating gauge transformation" parameter χ l is

14)
L l A E is the gauge-covariant Lie (or Lie-Maxwell) derivative of A E with respect to l mentioned in the introduction [23,12]. 15The emergence of this formula in the KK framework is one of our main results.
The last set of components of the 5-dimensional Killing vector equation for l are automatically satisfied upon use of all the previous results.Thus, we have constructed a 5-dimensional extension of l (the uplift of l), namely which is a Killing vector of the 5-dimensional metric and whose Killing horizon a S 1 fibration over the Killing horizon of l.
On the Killing horizon itself we can write where the constant Ω is given by In the solutions we are considering, Ω can be identified with the electrostatic potential evaluated over the horizon which is a gauge-dependent quantity.Since the gauge transformations of the 4-dimensional KK vector field are 5-dimensional diffeomorphisms which are not 5-dimensional isometries, this result is not surprising.However, the ambiguity in the value of Ω can be eliminated by demanding the 5-dimensional metric to be asymptotically flat with the following normalization16 or, equivalently, that the KK vector field vanishes at spatial infinity. 17Then, where Φ is the (gauge-invariant) difference of electrostatic potential between the horizon and spatial infinity.Without this condition, the coordinates t and z are entangled at infinity in electrically-charged black holes, for instance.There is another interpretation for the constant Ω: the linear momentum of freefalling observers in the direction z, given by is a conserved quantity.When the KK vector is electric, ı l A = A t = 0, observers with P z = 0, however, are moving in the z direction with velocity which may vanish depending on the chosen gauge or, equivalently, on the chose coordinate z.This fact can be interpreted as the dragging of inertial frames by the spacetime, which has momentum in the direction z.A particle that starts at infinity with zero velocity in the compact direction and falls radially towards the horizon will acquire a non-vanishing velocity in the internal direction that will equal k 1/2 ∞ Ω at the horizon.This is very similar to what happens in the Kerr spacetime to zero angular momentum observers (ZAMOs) and, geometrically, it has to do with the fact that the 5-dimensional vector ∂ t is not hypersurface-orthogonal.A difference, however, is that in these spacetimes there may not be a static limit where ĝtt = 0.
A similar phenomenon happens in the magnetic case in which ı ∂ ϕ A = 0.For vanishing P z , either ż = φ = 0 or To end this section, we can prove that the surface gravity of the 5-dimensional Killing horizon coincides with that of the 4-dimensional one.First, observe that the standard definition of the 4-dimensional surface gravity is invariant under Weyl rescalings of the metric when we write it in the form and, therefore, we can use this definition in the Einstein or KK frames.The 1-form dual to the Killing vector l if given by and so the pullbacks of the 1-forms l μdx μ and l µ dx µ are identical over the horizon even if the dual vectors are not.Then, on H only, using the vanishing of P E l and g tt there, we find ) thus showing that the 4-and 5-dimensional surface gravities are the same.

The electric KK black hole in 5d
Using Eq. (1.46) for the 4-dimensional electric solution Eqs. (2.1) we get the 5-dimensional Ricci-flat and asymptotically-flat metric Defining the coordinates and, using the relation between the integration constants h, ω, α Eq. (2.3), the metric can we rewritten in the form In this form we can clearly see that, in the extremal case w = 0 (so W = α = 1), the spacetime is a pp-wave with a flat 3-dimensional wavefront These coordinates are not so useful in the non-extremal case, which is the one we are most interested in.
The 5-dimensional metric Eq.(4.32) is singular at r = −w = r 0 , but this is just a coordinate singularity as can be seen by using 4-dimensional Eddington-Finkelstein coordinates together with in which the metric takes the form in which all the components are regular and the determinant is different from zero at r = r 0 .Actually, according to the general arguments we have given, there is an event horizon at r = r 0 .
The 5-dimensional Killing vector field that becomes null on the event horizon r = r 0 is just where Φ is the electrostatic potential evaluated on the horizon (normalized to vanish at infinity), given in Eq. (2.16).

5-dimensional thermodynamics à la Wald
In this section we want to see to which extent we can recover the 4-dimensional first law Eq.(2.18) and the 4-dimensional Smarr formula Eq. (2.15) working with the purely gravitational 5-dimensional action and solution using Wald's formalism.First, we have to find the 5-dimensional Noether-Wald charge for the 5-dimensional Killing vector l, defined in Eq. (4.16), whose Killing horizon coincides with the black hole's event horizon.
The 5-dimensional Noether-Wald charge [9] is nothing but the standard Komar charge 3-form [30], that, for a generic Killing vector ξ, can be written in the form where is the Lorentz momentum map associated to ξ [12], also known as the Killing bivector.
Let us compare this charge with the 4-dimensional one.The different components of the momentum map of l can be written as follows: Pˆl za = − 1 2 k ı l F E a + P E l e E a k , (5.3b) − ke E a P E l . (5.3c) Since l is, by hypothesis, a 5-dimensional Killing vector, we should get Pl za = − Pl az .Indeed, this property follows immediately from the momentum map equation.
The only term that will contribute to the integral when pulled back over 3-dimensional spacelike hypersurfaces is Using now this equation and the first of Eqs.(5.3) in the Komar charge Eq. ( 5.1), we find that plus components that will vanish when pulled back over the spacelike 3-surfaces we consider.Thus, where Q[l] is the Noether-Wald charge computed directly form the 4-dimensional action Eq.(3.1).Integrating over Let us now compare the 5-and 4-dimensional Komar charges.They are given by ) where ω l is given in Eq. (3.3).Taking into account Eq. (5.6) it is obvious that we are missing the terms that give rise to the 4-dimensional 2-form ω l .The apparent reason why there is a 4-dimensional ω l and not a 5-dimensional ωl is that which L .= 0 while L .= 0 (see Eq. (3.2)).This is a bit strange, because we have derived the 4-dimensional Lagrangian and equations of motion directly from the 5-dimensional one.On closer inspection, we see that, and (5.10) Since, by hypothesis, £ l φ = 0, we have that £ l ⋆ dφ = 0 and using the definitions of the electric and magnetic momentum maps we find, as in Refs.[26,25]), that, on-shell, ı l E φ is the total derivative of the closed 2-form that whose integrals give the black-hole scalar charge Σ, Q l .We have We can use this freedom directly in Q[l] as follows: let us consider the exterior derivative of the last term in Eq. (3.1) 1 √ 3 ı ξ ⋆ E dφ for ξ = l and let us use again the fact that, by assumption, £ l ⋆ E dφ = 0. We have, then, (5.12) after using the definitions of the momentum maps, integrating by parts and using the Maxwell equation of motion.Thus, we find that and, setting h = 0 and substituting this term back into Q[l] in Eq. (3.1), we recover the Komar charge Eq. (3.4).This trick can be played in the 5-dimensional Noether-Wald charge since we have seen that, after integration over the compact direction, it is equal to the 4-dimensional one.In this way, we can recover the 4-dimensional Komar charge from the 5-dimensional one and we are bound to obtain exactly the same Smarr formula.Nevertheless, it is interesting to carry out the analysis directly in 5 dimensions.

5-dimensional derivation of the 4-dimensional Smarr formula
The Smarr formula [34] for 5-dimensional static, asymptotically-flat black holes, can be obtained by integration of the exterior derivative of the Komar charge associated and equating this result to that of the integral over the bifurcation surface, Eq. (5.19) we recover the 4-dimensional Smarr formula Eq. (2.15).Observe that, in this framework, it does not seem possible to obtain the magnetic terms unless we introduce in a more or less arbitrary form the total derivative we discussed above.We can also try to derive the 4-dimensional version corresponding to the KK theory of the first law of black hole mechanics using the 5-dimensional Noether-Wald charge, since we have shown that it can be dimensionally reduced to the 4-dimensional one.However, as we are going to show, this is an entirely different calculation in which there are many subtleties.
(5.25) Furthermore, it is assumed that the variations of the Vielbeins δe satisfy the linearized equations of motion in the black-hole's background.In Ref. [36] it was found that, once all these details have been taken into account, Ŵ[ l] takes the form19 This identity is to be integrated over the same spacelike hypersurface we used to integrate the derivative of the Komar charge, leading to (5.27) In this case, since we are dealing with the Einstein-Hilbert action, following Ref.[15], almost by definition, the first integral simply gives the variation of the conserved charges associated to the Killing vector l, that is, a linear combination of the variation of the mass and the variation of the momentum in the z direction, which is, essentially, the 4-dimensional electric charge.The second integral, on the other hand, gives the surface gravity times the variation of the 5-dimensional area, which is related to the 4-dimensional one as we have shown above.In principle, there is no work term associated to the variation of the modulus φ ∞ but, since some of the 5-and 4-dimensional variables (charge, electrostatic potential, horizon area) are related by factors that involve the modulus φ ∞ and, therefore, their variations lead to an extra term involving the variation of the modulus. 20ll this is qualitatively correct, but we have not been able to recover the 4-dimensional first law with the correct numerical coefficients.We believe that a more detailed study of the definition of mass with KK asymptotics, is necessary and work in this direction is already underway [37].

Discussion
In this paper we have investigated the 5-dimensional geometry, spacetime symmetries and thermodynamics of 4-dimensional, static, KK black holes.In particular, we have determined the existence of a 5-dimensional stationary event horizon associated to the 4-dimensional static one (more precisely, a U(1) fibration over it) and we have shown how the 4-dimensional Killing vectors can be uplifted to 5-dimensional Killing vectors including a piece proportional to the generator of translations in the compact dimension.In other words: as stated in the introduction 4-dimensional spacetime isometries induce gauge transformations.The natural emergence of the covariant Lie derivative (here Lie-Maxwell derivative) in this context is quite remarkable.
Applying the relation between 4-and 5-dimensional Killing vectors to the generator of 4-dimensional time translations we have obtained a Killing vector which is null on the 5-dimensional event horizon, proving that it is a Killing horizon as well.This allows us to use all the geometrical properties enjoyed by Killing horizons.In particular, we have used the standard mathematical definition of surface gravity to show that those of the 4-and 5-dimensional horizons must be equal.On the other hand, the interpretation of the 4-dimensional electrostatic potential on the horizon as the velocity in the internal direction with which a particle with zero momentum in that direction (hence, zero electric charge in 4 dimensions) crosses the horizon provides a very interesting alternative physical interpretation of this quantity.
Our success in relating the 4-and 5-dimensional thermodynamics has not been complete, however.A more thorough study of how the definitions of gravitational conserved charges in [15] applies to spacetimes with KK asymptotics is necessary.
Furthermore, the answer to the question of how terms proportional to the variation of the 4-dimensional magnetic charge can be obtained in the first law directly from 5dimensional expressions is unclear.The 5-dimensional backgrounds that give rise to 4dimensional ones with magnetic charges do not asymptote to Ricci-flat metrics [38,39] and the formalism has to be revised in detail [37].
It is also unclear how 4-dimensional scalar charges may arise in the 5-dimensional framework.Recently, we have found a coordinate-independent definition for these (non-conserved!)charges that satisfies a Gauss law [25] and which is related to the global scaling symmetry of the 4-dimensional theory.The 5-dimensional generator of this global symmetry is the vector z∂ z which does not leave invariant the 5-dimensional metrics under consideration and, therefore, there is no 5-dimensional gravitational conserved charge associated to it, but we do not know how to construct another charge that satisfies a Gauss law even if it is not conserved.Given the ubiquity of scalar fields in KK theories, a deeper understanding of this problem and of the meaning of conserved charges is desirable.Work in this direction is in progress Ref. [40].

. 11 )
Thus, ω l can be understood as originating in the freedom that we have to add onshell closed 2-forms to the Noether-Wald charge Q[ξ], since we only compute directly dQ[ξ].