Magnetic charges and Wald entropy

: Using Wald’s formalism, we study the thermodynamics (ﬁrst laws and Smarr formulae) of asymptotically-ﬂat black holes, rings etc. in a higher-dimensional higher-rank generalization of the Einstein-Maxwell theory. We show how to deal with the electric and magnetic charges of the objects and how the electric-magnetic duality properties of the theory are realized in their thermodynamics


Introduction
Over most of the past century, electric-magnetic duality has been one of the main research topics in Theoretical Physics.While it arises as a symmetry of the sourcefree equations of motion of electromagnetism in 4 dimensions, it has been extended and generalized in many directions.Of particular interest for us is the generalization to higher-rank fields in higher dimensions and the inclusion of localized sources.The former is quite natural in the context of brane physics and the second, which goes back to Dirac's discovery of the magnetic monopole [1], arises naturally when one considers classical point-like or brane-like sources: the rotations between electric and magnetic fields must be in one-to-one correspondence with rotations of electric and magnetic sources and their charges. 1n d dimensions, black holes and black branes can carry some of the electric and magnetic charges associated to (p + 1)-form fields: as a general rule, black p-branes (where p = 0 corresponds to black holes) can carry the electric charges of (p + 1)-form potentials and the magnetic charges of ( p + 1)-form potentials, with p = d − p − 4 and p = p.When p = p, they can carry electric and magnetic charges of the same (p + 1)-form field.
The dynamics of black p-branes should exhibit the same electric-magnetic duality properties as the theory whose equations of motion they solve.In particular, the laws of thermodynamics [4] should exhibit those properties.Thus, in d = 4, where black holes can carry electric and magnetic charges of the same electromagnetic field, the first law of black-hole mechanics must include work terms for the variations of both kinds of charges.The conjugate thermodynamical potentials are the electrostatic and magnetostatic potentials evaluated on the black-hole horizon.If the theory has electricmagnetic duality, then we expect the first law of black-hole mechanics in 4 dimensions and the Smarr formulae [5] to be invariant under the simultaneous rotations of the (variations of the) electric and magnetic charges and of the conjugate thermodynamical potentials.The invariance of the Smarr formula for axion-dilaton black holes was recently proven in Ref. [6] using Wald's formalism [7][8][9] and the generalized Komar formula [10] constructed in Refs.[11,12].The proof can be easily generalized to other theories with scalars and vectors minimally coupled to gravity with electric-magnetic dualities [13] like most ungauged supergravity theories.However, a similar proof for the first law using Wald's formalism is not yet available.
Actually, the presence of magnetic work terms and scalar work terms in the first law of 4-dimensional black hole mechanics was found by other methods and is well known [14].In the notation of that reference, it takes this form: where the q Λ s and p Λ s are, respectively, electric and magnetic charges with respect to the vector field A Λ and ψ Λ and χ Λ are the electrostatic and magnetostatic potentials evaluated on the horizon, G ab (φ ∞ ) is the metric of the scalar manifold evaluated at infinity, Σ a the scalar charges and φ a ∞ the vales of the scalars at infinity (moduli).The invariance of this formula under electric-magnetic duality transformations presents several problems because, on general grounds [13], the terms involving electric and magnetic charges should be combined in a manifestly symplectic-invariant expression, which is not the case.The term involving the scalar charges is manifestly invariant.However, there is no good definition of the scalar charge as a conserved charge and, while this does not invalidate the result, it obscures its meaning.
In this paper we want to study the electric-magnetic duality properties of the first law of black hole mechanics for asymptotically-flat black p-branes coupled to higherrank form potentials in d dimensions 2 using Wald's formalism, leaving the problem of understanding the term with scalar charges for later work.In previous work [16][17][18] we showed how to prove the first law in presence of matter fields by correctly taking into account the interplay between diffeomorphisms and gauge transformations.However, since there is only one gauge transformation per gauge field, we were unable to recover the work terms proportional to the variations of the magnetic charges or the moduli, even though the same methods correctly give the terms proportional to magnetic charges in the Smarr formula Refs.[11,12,6].In this paper we are going to show how those terms arise in a more careful calculation and how they do it with the appropriate sign to have electric-magnetic duality invariance.
The theory we are going to consider is the straightforward generalization of Einstein-Maxwell to higher dimensions and higher-rank form potentials. Thus, it differs from the one considered in Ref. [19] 3 by the absence of scalar fields, which we plan to study in future work.In Ref. [19] only electric charges and electrically-charged black branes were considered and, in this work, we are going to show how to include the magnetic ones.We will not constrain the dimension of the horizon and, therefore, we consider, simultaneously, black p-branes which are electrically charged with respect to the (p + 1)-form field and black p-branes which are magnetically charged with respect to it or electrically with respect to the dual ( p + 1)-form field, although we are always going to use the formulation in which the (p + 1)-form potential appears. 4 This paper is organized as follows: in Section 1 we introduce the theories we are going to consider and we are going to explain how electric-magnetic duality is realized in them.In Section 2 we are going to define the conserved charges of the theory: those associated to the gauge symmetries and the magnetic ones, whose nature is topological.In Section 3 we are going to prove the restricted form of the generalized zeroth laws which we will use in Section 4 to find the Smarr formulae and in Section 5 to prove the first law.We end by discussing the results obtained and proposing new directions of research in Section 6.
1 Electric-magnetic duality and (p + 1)-forms We are going to consider the generalization of the d-dimensional Einstein-Maxwell theory in which the Maxwell 1-form field is replaced by a (p + 1)-form whose (p + 2)-form field strength F6 is invariant under gauge transformations where χ is an arbitrary p-form.
The components of the dual ( p + 2)-form ⋆F ( p ≡ d − p − 4) are given by and We will call the dual form G G ≡ ⋆F . (1.8) G and F are forms of the same rank when p = p, which happens when d = 2(p + 2).Real (anti-) self-duality requires σ 2 = +1 and only (p + 1)-form fields with p odd can have this property.We choose the Vielbein e a = e a µ dx µ , (1.9) as the gravitational field.The Levi-Civita spin connection ω ab = −ω ba is defined through the first Cartan structure equation and the curvature 2-form is .11)We will also use the total covariant derivative ∇.It satisfies the Vielbein postulate that relates the components of the spin connection ω µ ab to those of the affine connection Γ µν ρ which are given by the Christoffel symbols In terms of these variables and objects, the action we want to consider is Since we are mainly interested in the electric-magnetic duality properties of the first law and Smarr formulae, we are not including the dilaton field that usually couples to these forms in supergravity/superstring theories.A more general study including general couplings to scalars will be made elsewhere.Under a general variation of the fields where the equations of motion E a (Einstein) and E (Maxwell) and the symplectic potential (d − 1)-form Θ(e a , A, δe a , δA) are given by7 where ı c stands for ı e c , i.e. the interior product with the vector field e c = e c µ ∂ µ .This set of equations can be enlarged with the Bianchi identity In order to explore the invariance of the enlarged set of equations of motion under electric-magnetic duality transformations, it is convenient to define the vector of field strengths in terms of which the equations take the form ) where we have defined the 2 × 2 matrix For σ 2 = +1, this matrix is the non-diagonal metric of O(1, 1) and for σ 2 = −1 it is the "metric" of Sp(2, R).It is, then, evident, that the above system of equations is invariant under linear transformations of F and that the groups of invariance are O(1, 1) and Sp(2, R).Of course, these linear transformations only make sense for p = p.However, when this is not the case, the equations are still formally invariant under the discrete subgroups of O(1, 1) and Sp(2, R) ∼SL(2, R) that simply interchange F and G (up to signs).However, these are not the duality groups of the theory because the transformations must respect the self-duality constraint ⋆ F = ΩF . (1.21) For σ 2 = +1, only the interchange of F and G (up to a global sign) survives, while, for Observe that the symplectic potential Θ(e a , A, δe a , δA) is not invariant under any electric-magnetic duality transformations.This is not surprising because the action is not invariant, either.

Lorentz charge
The action Eq.(1.14) is exactly invariant under local Lorentz transformations of the Vielbein which induce the following transformation of the spin connection and curvature (2.2) For these particular transformations and upon use of the Noether identity (the symmetry of the Einstein equations) we find The off-shell invariance of the action for arbitrary parameters σ ab and arbitrary integration region imply the closedness of J[σ] and its local exactness Using this (d − 2)-form one can construct for on-shell field configurations a conserved charge for each independent parameter σ ab that leaves that field configuration invariant [22].However, there are no non-trivial parameters σ ab that leave invariant a regular Vielbein σ a b e b = 0 and, therefore, there seems to be no conserved charges associated to this symmetry.In spite of this, this (d − 2)-form plays an important role in black-hole thermodynamics for a particular σ ab , as we are going to see and we have already pointed out in Refs.[17,18].

Electric charge
The action Eq.(1.14) is exactly invariant under the gauge transformations of the (p + 1)form field A Eq. (1.5).For those particular transformations, and using the Noether identity the general variation of the action Eq.(1.15) can be written in the form The off-shell invariance of the action for arbitrary p-forms χ and integration region imply the closedness of J[χ] and its local exactness The (d − 2)-form Q[χ] can be used to define a charge which is conserved on-shell (dG = 0) for each independent gauge parameter χ leaving invariant the field configuration [22,23].The p-form gauge parameters that leave invariant the potential A are the closed ones dχ = 0.In a compact manifold with no boundary, these can be decomposed in a linear combination of harmonic p-forms h i plus an exact p-form de.Only the harmonic ones give non-trivial conserved charges when integrated over closed codimension-2 surfaces and the addition of exact p-forms to h i does not change their values [20,19].Observe that the sign we have chosen in this definition is purely conventional.

Magnetic charge
Even though there are no more gauge symmetries in our theory, we can define magnetic charges which are conserved in exactly the same sense as the electric ones: where hm is a harmonic p-form.We are using a different set of indices for the magnetic charges since, in general, the number of harmonic pand p-forms need not be the same.It is unclear how duality rotations or Dirac-like quantization conditions for these charges can be defined, except for the special case in which p = p and hi = h i .In this case, the charges can also be arranged in vectors transforming in the same way as F .Observe that the definition of magnetic charge Eq. (2.10) becomes trivial and gives zero whenever F = dA globally.Thus, as expected, non-vanishing magnetic charges are an exclusive property of certain non-trivial gauge field configurations.

Noether-Wald charge
The action Eq.(1.14) is exactly invariant under infinitesimal diffeomorphisms (2.12) Then, if we consider only the infinitesimal transformations of the fields δϕ ≡ ϕ ′ (x) − ϕ(x) the action is invariant up to a total derivative The associated (d − 2)-form Q[ξ] is called the Noether-Wald charge [8].
As discussed in Refs.[16-18, 6, 21], 8 the transformation of fields with some gauge freedom under infinitesimal diffeomorphisms is, generically, of the form where £ ξ is the standard Lie derivative with respect to the vector field ξ and δ Λ ξ is a ("compensating" or "induced") gauge transformation whose parameter Λ ξ depends on ξ and on the fields on which the transformation acts. 9We should write, then, Λ(ξ, e a , A), but we will use Λ ξ for simplicity, keeping in mind the dependence on the fields.In general the value of this parameter is only fully determined when the diffeomorphism is a symmetry of the whole field configuration.In that case we will denote the vector field that generates it by k since, in particular, it must be a Killing vector and one can define where L k transforms covariantly under gauge transformations, hence the name covariant Lie derivative.This property (which is not shared by the standard Lie derivative) has to be checked case by case.It ensures that the annihilation of all the fields by the transformation δ k (or by the operator L k ) is a gauge-independent condition.In the case of the (p + 1)-form field A, the compensating p-form gauge parameter is given by [16][17][18] where the momentum map p-form P ξ satisfies, for ξ = k, the momentum map equation (2.17) Then which is guaranteed to vanish when ξ = k by virtue of the momentum map equation (2.17).
The (p + 2)-form field strength F is gauge invariant and, upon use of the Bianchi identity which, yet again, vanishes identically by virtue of the momentum map equation (2.17).
In the case of the Vielbein field, the compensating gauge (Lorentz) parameter is given by [31,32,[16][17][18] 10 where P ξ ab is the Lorentz momentum map which, for ξ = k, is defined to satisfy the Lorentz momentum map equation This equation is solved by the Killing bivector For the spin connection we have that vanishes when ξ = k by virtue of the Lorentz momentum map equation (2.21).Finally, as a simple exercise, we can consider the transformation of the curvature.According to the general rule and using the explicit form of the compensating Lorentz parameter, we get When ξ = k we can use the Lorentz momentum map equation and which vanishes identically (Ricci identity).
In order to find the Noether-Wald charge we just have to plug the above transformations into the general variation of the action Eq.(1.15) The term involving P ξ ab in Eq. (2.27) vanishes by virtue of the Noether identity associated to local Lorentz invariance Eq. (2.3).Integrating by parts the two terms of Eq. (2.27) that involve derivatives, we get (2.30) Using the Noether identities associated to gauge transformations Eq. (2.6) and diffeomorphisms11 we arrive at which, combined with Eq. (2.13), leads to As usual, this implies the local existence of the (d (2.34) which is a straightforward generalization of the Noether-Wald (d − 2)-form obtained in the Einstein-Maxwell case in Ref. [16].It is manifestly not invariant under any electric-magnetic duality transformations.

Restricted, generalized, zeroth laws
Before we derive the Smarr formula and the first law of black hole thermodynamics we must derive the generalized zeroth laws: the constancy of the potentials associated to the charges over the event horizon.Our techniques only allow us to prove them restricted to the bifurcation surface (hence the name restricted, generalized zeroth laws), but this is sufficient for our purposes.The statements may, in some cases, be extended to the rest of the horizon using the ideas proposed in Ref. [37] These laws apply to the bifurcation surfaces (BH) of Killing horizons (H) associated to the Killing vector k, which is also assumed to generate a diffeomorphism that leaves invariant all the fields of the theory.Thus, k 2 H = 0, k BH = 0.In stationary blackhole spacetimes, the Killing vector whose Killing horizon coincides with the black-hole event horizon, k is an asymptotically timelike linear combination of the one generating time translations t = t µ ∂ µ and those generating rotations in orthogonal planes where the constants Ω n are the associated angular velocities of the horizon.If the (p + 2)-form field F is invariant under the diffeomorphism generated by k, then we can define the momentum map p-form P k satisfying the momentum map equation (2.17) and, assuming that F is regular on the horizon, Then, using the Hodge decomposition theorem where the h i are harmonic p-forms on the bifurcation surface and the constants Φ i are going to play the role of potentials associated to the charges Q i defined in Eq. (2.9) now computed by integration over the bifurcation surface. 12e can also define potentials associated to the magnetic charges.The invariance of the metric and gauge field under the diffeomorphism generated by k, plus the equations of motion dG = 0, lead to the existence of a magnetic momentum map Pk In an analogous fashion, the regularity of G over the horizon leads to where the hm are harmonic p-forms on the bifurcation surface and the constants Φ m are going to play the role of potentials associated to the magnetic charges P m defined in Eq. (2.10), now computed by integration over the bifurcation surface.
Observe that the same reasoning can be applied to the Lorentz momentum map equation, obtaining which implies that P k ab can be expanded as a linear combination with constant coefficients of covariantly constant antisymmetric Lorentz tensors.It is a well-known result that where κ is the surface gravity (constant over the whole event horizon, according to the standard zeroth law) and n ab is the binormal to the horizon with the normalization n ab n ab = −2.Clearly, n ab is covariantly constant over the bifurcation surface and κ can be interpreted as the "potential" associated to the Lorentz charge, which is, essentially, the area of the (spatial sections of the) horizon.
The main observation is that, on-shell and for a Killing vector k that generates a symmetry of the whole field configuration, the only non-vanishing contribution to Furthermore, under the same conditions, which implies the local existence of the (d Since we have proven that J[ξ] = dQ[ξ], Eq. (4.1) leads to the identity for the Komar charge Smarr formulae for stationary black holes are obtained by integrating Eq. (4.4) on hypersurfaces Σ with boundaries at a spatial section of the event horizon ∂Σ h (usually, the bifurcation surface BH) and at spatial infinity ∂Σ ∞ .Applying Stokes' theorem to that integral one gets and performing the integrals one arrives at the Smarr formula.
In order to apply this algorithm we must first construct the Komar charge K[k] finding ω k .This can be done for general configurations using the techniques of Ref. [6].The trace of the Einstein equation (1.16a) can be written in terms of the Lagrangian as follows: which implies that the on-shell Lagrangian takes the value Next, using the momentum map equations (2.17) and (3.4) and integrating by parts and using the equation of motion and Bianchi identity, we arrive at The Komar charge (d − 2)-form is, then, given by When p = p (so d is even), defining the vector of momentum maps which transforms as F under electric-magnetic duality because it satisfies the equation K[k] can be rewritten in the manifestly duality-symmetric form We now plug the Komar charge Eq. (4.11) in the integrals of Eq. (4.6).For asymptoticallyflat black holes, only the gravitational term in the first line contributes to the integral over spatial infinity since the products of potentials and gauge fields fall off too fast approaching infinity if we impose adequate boundary conditions.Using also the restricted generalized zeroth laws for the momentum maps Eqs.(3.3) and (3.5), we get 1 16πG (4.15)For the Killing vector Eq.(3.1), the integral in the left-hand side of this equation gives where M is the mass and J n are the components of the angular momentum.Furthermore, using Eq.Thus, we get the Smarr equation For p = p this formula takes the manifestly electric-magnetic duality invariant form where Q i is the charge vector defined in Eq. (2.11) and Φi is the vector of potentials Φi ≡ Φi Φ i , (4.20) so that 5

First law
We are going to review the derivation of the first law in full detail, improving the derivations made in Refs.[16-18, 6, 21] and showing where and how the variation of the magnetic charges, missed in those works, arise.Following Refs.[7][8][9], and denoting by ϕ all the fields of the theory, we define the symplectic (d − 1)-form and we choose δ 1 ϕ = δϕ, variations which satisfy the linearized equations of motion but which are, otherwise, arbitrary, and δ 2 ϕ = δ ξ ϕ, the transformations under diffeomorphisms that we have defined in Section 2.4.On-shell Θ = Θ ′ and using the definitions of J[ξ] Eq. (2.33) and δ ξ Eq. (2. 14) 2) This result differs from the standard one by the last term, which does not look like a total derivative.Let us study it in more detail in the theory at hand: since G is gauge invariant.Now, let us consider the last term.By definition, and taking into account that the parameter of the compensating gauge transformation depends on the field on which the transformation acts By the same token while the first term transforms in the standard fashion Combining these results, integrating by parts and using the equation of motion dG = 0 we arrive at another total derivative which allows us to rewrite the complete symplectic (d − 1)-form as the total derivative of a (d − 2)-form that we will denote by Ω(ϕ, δϕ, δ ξ ϕ), which is defined up to total derivatives ω(ϕ, δϕ, δ ξ ϕ) .= −dΩ(ϕ, δϕ, δ ξ ϕ) , (5.8a) Plugging into Ω(ϕ, δϕ, δ ξ ϕ) the expressions we have obtained for Q[ξ] and Θ ′ (ϕ, δϕ) and operating, we can put Ω(ϕ, δϕ, δ ξ ϕ) in this form: (5.9) again, up to total derivatives.We are going to profit from this freedom to rewrite this charge as follows: Pξ ∧ δF . (5.10) Now, when ξ = k, since δ k ϕ = 0 implies ω(ϕ, δϕ, δ k ϕ) = 0, we have the identity where, upon use of the definition of the dual momentum map Eq. (3.4) Ω(ϕ, δϕ, δ k ϕ) takes the final form (5.12) To proceed, we integrate the identity Eq. (5.11) over the same hypersurface over which we integrated the analogous identity involving the Komar charge K[k] in the previous section.Using Stokes' theorem (5.13) For the Killing vector Eq.(3.1) the integral at spatial infinity can be shown to give [9,41] 14 (5.14) When evaluating the integral over the bifurcation surface, we can use the reasoning in Ref. [9] to show that the second term in Eq. (5.12) does not contribute and that the first gives, simply κδA/(8πG + Ω n δJ n + Φ i ΩδQ i . (5.17)

Discussion
In this paper we have studied how to deal with magnetic charges in a d-dimensional generalization of the Einstein-Maxwell theory with (p + 1)-form potentials.Our main results are 1.The Komar charge Eqs.(4.11), which, for p = p, takes the manifestly electricmagnetic duality-invariant form Eq. (4.14).

3.
The first law Eq.(5.16) and the manifestly electric-magnetic duality-invariant form Eq. (5.17) that it takes when p = p.
We have assumed in the derivation of these results the asymptotic flatness of the solutions.Thus, they are valid for black holes, black rings and their generalizations, but, in order to apply them to infinite, planar, p-branes, a few, simple, modifications would be necessary to replace mass by tension and charges by charge densities removing the infinite volume factors.Wrapping these branes on compact dimensions would introduce additional effects (KK and winding modes) that need to be studied separately. 15urthermore, observe that the Smarr formulae and first laws obtained are generic: a particular solution may not be able to carry the electric, the magnetic or either charge.For instance, a black hole in 6 dimensions in a theory with a 2-form will not be able to carry electric nor magnetic charge with respect to the 2-form.In 5 dimensions, a black hole can carry the electric charge of a 1-form potential but not the magnetic charge (electric with respect to a 2-form potential), while a black ring can, in principle, carry the opposite.
In the p = p cases, black p-branes can carry electric and magnetic charges of the same (p + 1) potential and, as it is well known, since electric-magnetic duality leaves invariant the metric, all their geometric properties including their surface gravity and area are also duality invariant.Thus, the first law of their dynamics should also be invariant.Our results show that this is, indeed, the case.
As mentioned in the Introduction, the first law also has a term proportional to the variation of the moduli where the proportionality constants are the scalar charges, for which no good definition as conserved charges has ever been given [14].Here we have avoided this problem by studying a theory with no scalar fields, but this is a problem that has to be confronted and understood and we plan to do so in future work.

. 22 )
As a matter of fact, for this value of the momentum map, the Lorentz momentum map equation (2.21) becomes the integrability condition of the Killing vector equation.Then, on the Vielbeinδ ξ e a = −(dı ξ + ı ξ d)e a + σ ξ a b e b = Dξ a + P ξ a b e b = − 1 2 ∇ µ ξ a + ∇ a ξ µ dx µ ,(2.23)which vanishes when ξ = k by virtue of the Killing vector equation.

1 16πG
(3.7), the integral in the right-hand side gives 1 16πG (d) N BH (−1) d−1 ⋆ (e a ∧ e b )P k ab = − where T is the Hawking temperature and S is the Bekenstein-Hawking entropy.