Surface Charges for Gravity and Electromagnetismin the First Order Formalism

Surface Charges for Gravity and Electromagnetism in the First Order Formalism

Ernesto Frodden Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia, ChileDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile    and Diego Hidalgo Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia, ChileDepartamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Abstract

A new derivation of surface charges for 3+1 gravity coupled to Electromagnetism is obtained. Gravity theory is written in the tetrad-connection variables. The general derivation starts from the Lagrangian and uses the covariant symplectic formalism in the language of forms. For gauge theories surface charges disentangle physical from gauge symmetries through the use of Noether identities and the exactness symmetry condition. The surface charges are quasilocal, explicitly coordinate independent, gauge invariant, and background independent. For a black hole family solution the surface charge conservation implies the first law of black hole mechanics. As a check we show the first law for black hole electrically charged, rotating, and with an asymptotically constant curvature (the Kerr-Newman (anti-)de Sitter family). The charges, including the would-be mass term appearing in the first law, are quasilocal. It is not required a reference to the asymptotic structure of the spacetime nor boundary conditions, and therefore topological terms do not play a rôle. Finally, surface charges formulae for Lovelock gravity coupled to Electromagnetism are exhibited. It generalizes the one derived in a recent work by G. Barnich, P. Mao, and R. Ruzziconi. The two different symplectic methods to define surface charges are compared and shown equivalent.

\preprint

CECS-PHY-17/05

1 Introduction

To find the quantum degrees of freedom responsible for the black hole entropy remains one of the main questions that fuels the research of a quantum theory of gravity. The semiclassical analysis, the study of quantum field theory on fixed black hole spacetimes, ensures that the entropy is proportional to one-fourth of the horizon area (in units where ).

Because the entropy corresponds to the area of the horizon, the expectation is that the microscopic degrees of freedom responsible for the entropy are localized around the horizon itself. At least two prevailing approaches construct quantum horizons models dwelling on this later view.

A first one uses symplectic methods to build a Chern-Simons description of the horizon Ashtekar:2000eq (); Engle:2009vc (); Engle:2010kt (); Perez:2010pq () in the context of loop quantum gravity. A second one uses tools Brown:1986nw (); Barnich:2001jy () developed in the context of holography, specifically asymptotic symmetries and associated charges, to describe the black hole horizon (some examples are Strominger:1997eq (); Guica:2008mu (); Compere:2015mza (); Afshar:2016uax (); Hawking:2016msc ()). Both approaches start from a classical description of the black hole horizon through boundary conditions that left some freedom for the variables at the boundary. In both cases one may wonder if in that freedom there are true degrees of freedom. Surprisingly, in certain context they have been called would-be gauge degrees of freedom Carlip:1995zj (); Geiller:2017xad () as some of them would be degrees of freedom that come from the partial freezing of the gauge symmetries at the boundary. That seems artificial as they are strongly dependent on the particular choice of boundary conditions.

With this context in mind, the present work is a first step to explore, from a different perspective, the common basis where both approaches above are standing on. That is, the study of physical and gauge symmetries of the symplectic structure when boundary conditions are imposed. To do so, we focus on surface charges, motivated in subsection 1.1, as a main quantity that encodes the physical information related with symmetries, in the context of gauge theories.

As a result we rederive explicitly the general expression of surface charges from the covariant symplectic method in the language of forms, section 2. Then, to describe gravity, we choose to depart from the usual metric approach by using the tetradic-connection formalism. We also deal directly with the more general case of a gravity theory (with cosmological constant) coupled to Electromagnetism in four dimensions. That allows us to obtain, as a second and main result, compact formulae for the surface charges that are gauge invariant and do not require reference to a coordinate system, equation (69). A valuable conclusion of this work is an explicit exhibition of surface charges formula and the tight requirement for them to satisfy a conservation law. This paves the way to disentangle physical from gauge symmetries in the context of asymptotic symmetries and boundary conditions previuosly discussed.

The content of these notes goes as follow: In section 2 we rederive surface charges for a general gauge theory. In sections 3-5 we progressively establish the explicit formulae of the surface charges for the theory of gravity coupled to Electromagnetism. In subsection 3.1 we show that boundary terms in the Lagrangian have not effects on surface charges. In subsection 5.1 we perform a test of the reliability of the formalism by recovering the standard first law of black hole mechanics in a quasilocal way. In section 6 we study the generalization to an arbitrary dimension. We compute the surface charges formula for Lovelock gravity coupled to Electromagnetism. Finally, in the appendix A we further present a general comparison of the canonical covariant symplectic approach with other covariant techniques inherited from the BRST formalism Barnich:2016rwk ().

1.1 Why surface charges?

It is not enough well-known that the first Noether theorem does not apply for gauge symmetries. The reason is that the would-be Noether current is trivially conserved, i.e., it is conserved without requiring the equations of motion (for more discussion see Compere:2007az ()). As we will show, this is in fact a consequence of the second Noether theorem. Then, the would-be Noether charge is ambiguous as it has the arbitrary gauge parameter on it. For example, for the electromagnetic field the ‘gauge current’ for the gauge symmetry, , is and the ‘gauge charge’ depends explicitly on the arbitrary gauge parameter.

A usual way to cure this lack of meaning for the would-be Noether current is to assume extra structure for the fields and gauge parameters at an asymptotic spacetime region. With this, the charge computed out of the current may acquire a physical meaning. However, still ambiguities related with the action boundary term may affect the value of the current and extra input as differentiability of the action may be needed to have a well defined charge.

In the case of gravity, the asymptotic structure of flat, de Sitter, or anti-de Sitter spacetimes are drastically different. The boundary term in the action to guarantee differentiability of the action changes for each case. This fact makes the definition of asymptotic charges problematic. A more general approach not resting on the particular asymptotic structure of the spacetime is certainly desirable.

The quantities known as surface charges provide the necessary generalization Barnich:2001jy (); Barnich:2007bf (). They are conserved quantities for a physical symmetry in the context of gauge theories.

In the next section we show how to compute surface charges in general, from canonical symplectic methods. The computation relies on the Lagrangian but is not dependent of the ambiguities of the boundary terms that one may add to it.

An interesting property of surface charges is that they are quasilocal. It is not needed to use an asymptotic spacetime structure to define them. On the other hand, one may perfectly compute them on asymptotic regions too.

It is worth to notice that surface charges are a particular case of a generalization of conserved currents. In Barnich:2000zw () it is explained that the Noether’s first theorem can be rephrased as a cohomology in the context of the BRST symmetry. Then, it is proven that higher order conserved currents are in correspondence with a generalization of ‘global symmetries’. This is a generalization of the Noether’s first theorem. The usual Noether current is the first one of these currents. The surface charges are built out of the second of these currents. To understand the rest of the currents from a canonical symplectic perspective is an interesting problem left for a future work.

2 Surface Charges for Gauge Theories

In gauge theories field transformations due to gauge and rigid symmetries are entangled. This can puzzle the definition of physical quantities like charges. On the frame of covariant symplectic methods we can start studying both gauge and rigid symmetries on the same foot, and then to make the difference at a crucial step. We start by considering the Noether procedure for infinitesimal symmetry transformations in the language of forms. We specify this, first to the case of gauge symmetries, and then to the case where diffeomorphism is one of the gauge symmetries. At the end, we will define and assume the existence of exact transformations to produce physically sensible results. We follow in general lines the canonical covariant symplectic approach Wald:1999wa (); Iyer:1994ys () but having in mind the invariant symplectic approach used in Barnich:2001jy (); Barnich:2007bf () where surface charges are defined. Another useful reference is section 3 in Compere:2009dp (), or Prabhu:2017 () where a close general treatment is performed.

Consider a Lagrangian form for a collection of fields . The arbitrary variation is

 δL=E(Φ)δΦ+dΘ(δΦ), (1)

with the equations of motion, and a boundary term. The Lagrangian has a symmetry if for certain infinitesimal variations over the configuration space it becomes at most an exact form

 δϵL=dMϵ, (2)

we call the collection of parameters that generate the infinitesimal symmetry, and denotes the infinitesimal transformation generated over any quantity. In the case , the usual notion of symmetry for the action is recovered by choosing a vanishing of the symmetry parameters at a neighborhood of the boundary of the manifold. This can be done always only for gauge symmetries.

The fields transform under a symmetry as , therefore

 dMϵ=E(Φ)δϵΦ+dΘ(δϵΦ). (3)

Now, let us assume that the transformation is linear in the symmetry parameters . This assumption allows us to make a crucial step. We can remove the derivatives over all symmetry parameters and formally decompose

 E(Φ)δϵΦ=dSϵ−Nϵ, (4)

such that in the symmetry parameters appears only as factors. We will use a hat to remember that the equations hold on-shell, for instance or . Using the new expression for we obtain

 d[Θ(δϵΦ)−Mϵ+Sϵ]=Nϵ. (5)

Now, we restrict ourselves to gauge symmetries. For them, the very structure of the last equation implies111 can be factorized by the arbitrary parameters and at the same time it is equal to an exact form, this implies that vanishes. Proof: Integrate (5) and choose the parameters to vanish at a neighborhood of the boundary.

 Nϵ=0, (6)

these are called Noether identities and there is one of them for each independent gauge parameter. These are the usual constraints of the theory due to the redundancy of using gauge variables and are reason why the First Noether theorem does not apply for gauge symmetries.

Then, it is natural to define the form

 Jϵ≡Θ(δϵΦ)−Mϵ+Sϵ, (7)

which by virtue of the Noether identities satisfies

 dJϵ=0. (8)

Note that the statement is off-shell and therefore is not a current. However this quantity reduces on-shell to what is usually called the Noether current . As far as generates a gauge symmetry this current is trivial as its off-shell conservation law suggests. However, with two more ingredients this current generates non-trivial and finite charges. These extra assumptions are that is an exact symmetry of the fields, i.e., (further discussed in the paragraph before (21)), and that the boundary term in the Lagrangian is consistent with the boundary conditions Aros:1999id (). This is the standard Noether procedure which for gauge theories needs to be suplemented with extra information.

However, there is an alternative. We can follow a quasilocal approach that does not make use of the asymptotic structure. The cost is the relying on a linearized theory. This is shown in the following.

The Poincaré lemma ensures that a closed form is locally exact, that is, there exists such that222Equations (8) and (9) suggest that non-trivial on-shell currents are those which satisfy a conservation law in the whole spacetime but for which the Poincaré lemma can not extended to the whole spacetime, i.e., they correspond to the equivalence classes of closed forms which are not exact, i.e., the de Rham cohomology. We refer to Barnich:2000zw () for a rephrasing of Noether theorems using the cohomology of the BRST symmetry.

 Jϵ=d˜Qϵ. (9)

Now, consider an off-shell variation

 δΘ(δϵΦ)−δMϵ+δSϵ = dδ˜Qϵ. (10)

We assume that . The double hat will be used to remember that, besides the equations of motion, the linearized equations of motion hold too. For instance does not vanishes on-shell. We need the extra assumption that satisfies the linearized equations of motion, .

The presymplectic structure density is an antisymmetrized double variation of the fields on the phase space, defined by

 Ω(δ1,δ2) ˆˆ= δ1Θ(δ2Φ)−δ2Θ(δ1Φ)−Θ([δ1,δ2]Φ), (11)

where the boundary term in the action is also referred to as the presymplectic potential density. The variations, , are assumed to satisfy the linearized equation of motion. Note that is a double variation in the phase space and a -form in spacetime. The double variation can also be understood as a two-form in the phase space. The last term, , should be considered because variations of fields on the phase space do not commute in general Green:2013ica (). The prefix in presymplectic stands for the fact that variations on the fields can also be gauge symmetry transformations. We need it because by using gauge variables there is not a systematic way to disentangle the gauge redundancy from the phase space. In this sense the phase space is degenerated. It contains gauge orbits, i.e., family of points identified through gauge transformations. In other words, if is the manifold where is defined, then, on-shell, has degenerated directions Wald:1999wa (). Precisely those ones associated to infinitesimal gauge transformations.

Considering the presymplectic structure density evaluated in a gauge variation, , we rewrite (10) as

 Ω(δ,δϵ)=−δϵΘ(δΦ)−Θ([δ,δϵ]Φ)+δMϵ−δSϵ+dδ˜Qϵ, (12)

To go further let us assume that contains diffeomorphisms. More precisely, suppose the collection of gauge parameters can be split as , i.e., . Such that is a vector field generating infinitesimal diffeomorphisms and denotes the rest of infinitesimal gauge symmetry transformations. For a form that is invariant under , the infinitesimal diffeomorphism transformations are generated through a Lie derivative333If the form is a gauge variable an ambiguity arises for the Lie derivative and the Cartan formula (13) has to be corrected. We address this point in the examples.

 δξω=\mathfsLξω=d(ξ┘ω)+ξ┘dω, (13)

we use to denote the interior product over forms. For a vector field and a one-form , both expressed in coordinate components, the interior product is . The interior product distributes over the wedge product of forms exactly as the exterior derivative does. The exterior derivative, , and the interior product, , act only on the immediate term at the right of the symbol unless explicit parenthesis are drawn.

We assume the Lagrangian and the presymplectic potential density are left invariant under the transformation generated by ,444Note that if is a gauge transformation of a Chern-Simons theory the Lagrangian is not invariant. The genetalization is straightforward but we refer to Tachikawa:2006sz () for a discussion of this case. As a top form in the manifold the Lagrangian satisfies , then, . Here and in the following we assume . Then, we have

 δMϵ = ξ┘(EδΦ)+ξ┘dΘ(δΦ). (14)

On the other hand

 δϵΘ(δΦ)=δξΘ(δΦ)=dξ┘Θ(δΦ)+ξ┘dΘ(δΦ). (15)

And,

 Ω(δ,δϵ) = −dξ┘Θ(δΦ)−Θ([δ,δϵ]Φ)+ξ┘(EδΦ)−δSϵ+dδ˜Qϵ, (16)

after explicitly using the equations of motion and the linearized equations of motion, we obtain a simple expression for the presymplectic structure density

 Ω(δ,δϵ) ˆˆ= d(δ˜Qϵ−ξ┘Θ(δΦ))−Θ([δ,δϵ]Φ). (17)

In the case the parameters are extended non-trivially on the phase space, the last term does not vanish (we still assume but ). In the examples, it is going to be the case when gauge parameters get fixed to encode exact symmetries. Analogous to the decomposition , the term can be decomposed as

 Θ([δ,δϵ]Φ)=dBδϵ+Cδϵ, (18)

such that in , the varied parameters appear as factors. A similar argument that the one used to prove the off-shell Noether identity, , proves that on-shell .555In (17) the gauge parameters and are disentangled, then fixing and integrating over any -surface the arbitrariness of implies .

We define a -form in spacetime and first variation in phase space by

 kϵ ≡ δ˜Qϵ−ξ┘Θ(δΦ)−Bδϵ. (19)

As an abuse of name we may refer to this quantity as the would be surface charge integrand.

In the case represents a gauge symmetry we can choose such that . Then, equation (17) tells us the standard result: The presymplectic structure density for a gauge transformation is trivial, i.e., it is an exact form in spacetime666The presymplectic structure for an action is obtained by therefore, it is defined up to an exact form in spacetime.

 Ω(δ,δϵ) ˆˆ= dkϵ. (20)

A gauge symmetry is a degenerate direction in the presymplectic structure. Once integrated, last expression becomes an arbitrary boundary term that in particular can be chosen to vanish.

Now, in the particular case that generates an exact symmetry the presymplectic structure density vanishes. We call exact symmetry the condition where particular parameters solve the equation (Killing fields for the metric for instance). Then, because the presymplectic structure density is linear in the infinitesimal transformations, we have . Therefore, for exact symmetries

 dk¯ϵ ˆˆ= 0 (21)

The establishment of this equation is the main goal of this section. This equation is a second conservation law of one degree less than . It has a true physical meaning because it requires the use of the equations of motion besides the property . As commented in subsection 1.1, is a current in the context of a generalized Noether theorem Barnich:2000zw ().

Therefore, we define the surface charge by the integral

 ⧸δQ¯ϵ≡∮k¯ϵ. (22)

Note that it is called surface because it is naturally defined on a manifold which is a surface in four dimensions. And more important, it is called a charge because it is conserved. This happens only because the exactness of the symmetry guarantee the conservation law (21). That makes the integral independent of the closed surface where the integration is performed. On the other hand, one could compute non-vanishing quantities for gauge symmetries but these quantities are not charges. Following the notation proposed in Barnich:2007bf () we use to denote quantities that are not necessarily integrable on the phase space. In other words, the function such that its variation on the phase space satisfies may not exist. A sufficient condition for its existence is this is the condition of integrability for the surface charge to become a finite charge.

As explained before, a gauge symmetry produces a trivial Noether current in the sense that it is conserved even off-shell (8). Then, to use a physical symmetry or the equations of motion or nothing, in order to prove the conservation of does not make any difference. However, if for certain choice of the gauge parameters the gauge symmetry can be made exact, it will produce a second necessarily on-shell conservation law for . Let us remark that choosing the gauge parameters means that we are not dealing with a gauge symmetry anymore. However, in the derivation of we make intensive use of the presence of a gauge symmetry.777This is exacly what happens in the Hamiltonian approach for asymptotic charges: The parameters generating gauge symmetry are boiled down to exact symmetries in the asymptotic region.

In the four dimensional examples worked out in subsection 5.1, is a closed two-form in spacetime that can be used to relate quantities defined on two arbitrary disconnected closed two-surfaces which are the boundaries of a given three-volume. The integration of on a closed two-surfaces is trivial if the surface is contractible to a point. In the black hole example, will be integrated over spheres enclosing the singularity. Note that, as we are strongly using differentiability of fields, this should be guaranteed in the three-volume as well as in its boundary. Bulk singularities and spikes in boundaries have to be treated carefully.

Two remarks regarding possible ambiguities are in order. First, note that there is an ambiguity in the definition of , which percolates to an arbitrary exact form in the presymplectic structure density. However, for exact symmetries it simply vanishes and does not have any effect in the definition of . Second, another ambiguity could arise because does not change the equation . This ambiguity is harmless as far as is used only integrated over closed surfaces.

3 General Relativity

In this section we consider the action for gravity in four dimensions in the first order formalism. This formalism is fundamental in the sense that it is suitable to include fermionic fields. At the same time the metricity and parallelism properties of spacetime can be easily disentangled Zanelli:provisory ().

The language of forms allows us to write variables without doing explicit references to coordinates. We consider as independent variables the tetrad and the Lorentz connection, , both are one-forms. The curvature two-form read . Besides the standard Einstein-Hilbert term we consider a cosmological constant and a topological Euler term, all them arranged in the well-known McDowell-Mansouri action MM (); Aros:1999id (); Wise:2006sm (). In the following we will suppress the indexes and the wedge product to make the notation compact when possible.

Therefore, the action for gravity simple reads

 S[e,ω]=κ∫M¯R⋆¯R, (23)

with the barred curvature given by

 ¯RIJ≡RIJ±1ℓ2eIeJ, (24)

note that as before the wedge product between forms is understood. The stands for the dual of the internal group, in this case the Lorentz group, for instance . The stands for the both possible signs of the cosmological constant. The treatment is the same then we consider both at once. The overall constant has not effect in the following but we fix it to to make contact with standard approaches. We also choose the units to set the Newton constant .

The dependence on the cosmological constant can be consistently removed at the end of the calculation by considering the limit . Note that the Euler term is multiplied by , thus the limit can not be taken at this stage. In fact the Euler term can be thought as providing a regulator for the Einstein-Hilbert plus cosmological constant action and for the finite Noether charges derived from it Aros:1999id (). However, as far as we consider exact symmetries the present quasilocal approach is insensitive to it. This is made explicit at the end of this section.

The variation of the Lagrangian is

 δL=Eeδe+Eωδω+dΘ, (25)

if we get rid of the term by imposing boundary condition, as we will discuss in a moment, the variational principle implies the equations of motion (putting back the indexes, and )

 EI = ∓2κℓ2εIJKLeJ¯RKL=0, (26) EIJ = −κεIJKLdω¯RKL=∓2κℓ2εIJKLdωeKeL=0, (27)

where we use to denote the covariant exterior derivative, for instance the Bianchi identity reads , and the torsion . The second equation is equivalent to setting the torsion equal to zero, . Because this is an algebraic equation for the Lorentz connection, , it can be solved in terms of the tetrad, . The replacement of in the first equation produces the usual Einstein equation with cosmological constant written in forms.

As suggested before, to have a well-posed variational principle the term

 Θ=2κδω⋆¯R, (28)

must vanish at the boundary of the spacetime . If we allow for an arbitrary , the following boundary condition is required888Note that the imposed boundary condition has a symmetry larger than the local Lorentz group. It is invariant under the (anti-)de Sitter group [which contains the Lorentz group . Note also that for asymptotically flat spacetimes a well-defined action principle needs a boundary term different than the Euler one. In Ashtekar:2008jw () it is shown that corresponds to the Hawking-Gibbons term in the first order formalism, and therefore it allows a well-defined variational principle as well as asymptotic Hamiltonians. However, as explained in Subsection 3.1 boundary terms in the action do not contribute in surface charges approach that we follow.

 (29)

Note that this condition requires an a priori knowledge of the boundary of the spacetime. In other words, we are reducing the space of solution such that the previous equation can be satisfied. The family of spacetimes with this property are named locally asymptotic (anti-)de Sitter spacetimes. On the other hand, the standard assumption is more relaxed because it can be applied in principle to any patch of the spacetime. However, it is a strong condition because is a connection and we would need to fix the gauge in the boundary too.

The approach we follow to define the surface charges is quasilocal. It is insensitive to the chosen prescription for the boundary term. The only requirement is that a well-posed variational principle exists in order to obtain the equations of motion.

The gauge symmetries of the action are general diffeomorphisms and local Lorentz transformations. The infinitesimal transformations of the fields by the local Lorentz group is

 δλeI = λ\indicesIJeJ (30) δλω\indicesIJ = −(dωλ)\indicesIJ=−dλ\indicesIJ−ω\indicesIKλ\indicesKJ+ω\indicesKJλ\indicesIK, (31)

where are the parameters of the infinitesimal Lorentz transformation . Remember that it is a gauge symmetry, that is, the group elements take different values at different points of the manifold.

The infinitesimal transformations of the fields due to diffeomorphisms are normally assumed to be generated by an arbitrary vector field through a Lie derivative

 ~δξe = \mathfsLξe=d(ξ┘e)+ξ┘(de) (32) ~δξω = \mathfsLξω=d(ξ┘ω)+ξ┘(dω), (33)

where in the second equality we use the Cartan formula. However, note that due to the presence of exterior derivatives they are not homogeneous under local Lorentz transformation. The intuitive interpretation of and as infinitesimal variation require them to be homogeneous under the action of the local Lorentz group. More precisely, if we attach ourselves to the intuitive idea of variations as comparison of fields in a neighbourhood, , we expect them to have a covariant transformation under the local Lorentz group. This criteria is not satisfied by the infinitesimal diffeomorphism transformation presented before, and therefore we correct (32)-(33) by eliminating the non-homogeneous part. This can be done by adding an infinitesimal Lorentz transformation with a parameter . For a recent discussion see Jacobson:2015uqa (). This corrects the non-homogeneous part of both transformations at once, and we get

 δξe = \mathfsLξe+δξ┘ωe=dω(ξ┘e)+ξ┘(dωe) (34) δξω = \mathfsLξω+δξ┘ωω=ξ┘R. (35)

Another way to think about this, is that in the transformation of the tetrad, the exterior derivative is promoted to a covariant exterior derivative , while in the transformation of the Lorentz connection, because of the identity , the ill-transforming part, , is subtracted.

Therefore, the general infinitesimal gauge transformations, involving diffeomorphisms and local Lorentz transformations, with parameters , which are themselves homogeneous, are999Still other prescriptions for the infinitesimal transformations are possible. For instance the ones recently introduced in Montesinos:2017epa () differ from (36) and (37) by a term depending on the equation of motion. These kind of terms are known as trivial symmetries because they are present in any theory (section 3.1.5 in Henneaux:1992ig ()).

 δϵe = \mathfsLξe+δ(ξ┘ω+λ)e=dω(ξ┘e)+ξ┘(dωe)+λe (36) δϵω = \mathfsLξω+δ(ξ┘ω+λ)ω=ξ┘R−dωλ. (37)

Now, we follow the procedure detailed in section 2 to obtain the surface charges for General Relativity. The Lagrangian transforms as a total derivative, . Under a symmetry transformation

 d(ξ┘L−Θ(δϵω))=Eeδϵe+Eωδϵω, (38)

using explicitly the symmetry transformation on the variables we can mimic (4)

 Eeδϵe+Eωδϵω = d[Eωλ−Eeξ┘e]+(Eee−dωEω)λ+dωEeξ┘e+Eeξ┘dωe+Eωξ┘R (39) = d[Eωλ−Eeξ┘e]. (40)

In the first line we used exact forms to have the gauge symmetry parameters either inside an exact form or as a factor of a term. In the second line we used the Noether identities: and , which are a rewriting of the standard Bianchi identity. Then, we define

 Jϵ≡Θ(δϵω)−ξ┘L−Eeξ┘e+Eωλ, (41)

that trivially satisfies . Explicit computation results in an exact three-form that depends just on the gauge parameter of the Lorentz symmetry

 Jϵ=Jλ=−κ d(2λ⋆¯R) (42)

In spite its trivial conservation one may try to use this current to define global charges associated to exact symmetries at the asymptotic boundary. The result is non-trivial as shown in Aros:1999id (). It is in this context that the Euler density becomes crucial, to accomplish the boundary condition . This regularize the symplectic structure such that there is no leaking on the boundary for locally asymptotic (anti-)de Sitter spacetimes. Then, global finite charges can be asymptotically computed through that method.101010Note that in Aros:1999id () the integrand to define the charge is , which is gauge dependent, then, an explicit gauge fixing at the boundary is required such that it also respects the gauge dependent asymptotic symmetry condition . That is equivalent to our expression where we can use the integrand and can fix by the exactness condition expressed below in (50). However, these expressions are explicitly Lorentz invariant.

But here we intend for a quasilocal definition of charges. Following the prescription of section 2, we make a step further and perform an arbitrary variation of and compute each term of (16). To compute the contribution note that

 Θ([δ,δϵ]) = 2κ[δ,δϵ]ω⋆¯R (43) = −2κ dω(δλ+ξ┘δω)⋆¯R = −2κ d[(δλ+ξ┘δω)⋆¯R]+2κ(δλ+ξ┘δω)⋆dω¯R,

where we used . As it was shown in general, (18), the last term in the third line, correponding to , vanishes on-shell.

Then, from (28), (42), and (43) we obtain the surface charge integrand for General Relativity

 kϵ = −2κ(λ⋆δ¯R−δω⋆ξ┘¯R). (44)

Now, if the exact symmetry condition is satisfied, we have , and we can define surface charges . In the example we will be able to integrate the varied quantities on the phase space to find . The charges are varied on the phase space, through a family of solutions. The study of the phase space for a family of solutions can be done explicitly, for instance, when considering the variation of the integration constants that appear in a solution.

For completeness we write down the presymplectic structure density for pure gravity

 Ω(δ1,δ2) ˆˆ= 2κ(δ2ω⋆δ1¯R−δ1ω⋆δ2¯R) (45) ˆˆ= −18πδ[1ωIJ δ2]ΣIJ+2κd(δ1ωIJ⋆δ2ωIJ), (46)

where we used that , the value of , and the definition . The first term is the conjugate pair of gravity variables . The second term, consequence of the Euler term in the action, is an exact form and disappears when the density is integrated on a smooth boundary of a manifold, . Similarly, the Euler contribution to the surface charge will not have any effect because for exact symmetries it becomes and exact form. Explicitly, the contribution of the Euler term to (44) is

 kEulerϵ=−2κ[λ⋆δR−δω⋆ξ┘R]=−2κ[d(λ⋆δω)+δϵω⋆δω]. (47)

This confirms that the procedure is not affected by the action boundary terms.

Remember, to guarantee that is closed we need an exact symmetry such that . Therefore, we have to solve the parameters such that

 δϵe = 0, (48) δϵω = 0. (49)

In the following are solutions of the previous equation. The condition imposes a general relation between and . Exact symmetries are on-shell, thus we use , and solve from (48)

 λIJ=eI┘dω(ξ┘eJ)=eIμeJν∇[μξν], (50)

where is the interior product such that , in coordinates components it is . In the second equality we exhibit the solution in components with the spacetime covariant derivatve. This relation is a sufficient condition that gauge parameters should accomplish to encode an exact symmetry. Note that the Killing equation, with the metric , is a direct consequence of . The Killing equation in coordinate components, , can also be seen directly in the rightest expression for : It is encoded in the fact that is antisymmetric. On the other hand, because , the condition holds trivially.

Therefore, we have obtained the expresion of for the surface charges in the tetradic first order formalism (integration of (44)). We have also shown that the exact symmetries condition for the tetrad is the Killing equation in this language.

As a final remark, note that there is a second and straight way to obtain the same result. Consider the following expression for the contracting homotopy operator discussed in the appendix A

 Iδe,δω≡δeI∧∂   ∂dωeI+δωIJ∧∂   ∂RIJ. (51)

Acting with this operator on we obtain the following surface charge integrand

 k′ϵ≡Iδe,δωSϵ=kϵ−kEulerϵ. (52)

It has the advantage of giving directly the term that does not have a contribution from the action boundary term. Note that in the general framework is the boundary term independent part of . However, the difference is harmless because for exact symmetries we have , and therefore both prescriptions are equivalent. This fact has a straightforward generalization for Lovelock theories in dimensions, section 6.

3.1 Topological and boundary terms effect on surface charges

Surface charges are insensitive to action boundary terms. This is a remarkable property that is in high contrast with usual Noether procedures to compute charges (see for instance the recent review Corichi:2016zac ()). To see how this happens let us add a boundary term to the Lagrangian: , with the assumption that is gauge invariant, . Now we can repeat the procedure of section 2 by keeping track of this boundary term. We have , and the surface charge potential

 kϵ→kϵ+δ(ξ┘α)−ξ┘δα=kϵ, (53)

where we used . Thus, there is not change at all for the surface charges.

The previous case is quite general. For example in the Nieh-Yan topological term, with density , is in this category. However, there are examples where is not gauge invariant. This is the case for the Euler or the Pontryagin topological terms, with densities and , respectively. Both are exterior derivatives of Chern-Simons Lagrangian which are gauge quasi-invariant forms Zanelli:provisory (). For the Euler term the expression (47) shows that it does not affect the surface charge. A similar computation for the Pontryagin yields

 kPontryaginϵ=−2κ[λδR−δωξ┘R]=−2κ[d(λδω)+δϵωδω], (54)

then, surface charges are blind to the Pontryagin topological term too.

Finally, in we may also be interested in using the Holst term density, , inside the gravity action. This is not a topological term by itself but a part of the Nieh-Yan, and it does not affects the equations of motion either. To deal with it note that . The second term was already studied, then, it is enough to keep track of in the computation of surface charges potential. This term also does not produce any changes because already at the level of the presymplectic structure density, , the contributions are all proportional to the torsion and therefore vanish on-shell.

Then, boundary terms, and in particular topological terms, do not affect the surface charges. Note that this is already explicit for surface charges computed through the contracting homotopy operator (52) because it depends only of and not of the Lagrangian. In this sense here we have stressed what is already indirectly known due to the fact that surface charges obtained through both methods are equivalent (appendix A).

4 Electromagnetism

Before dealing with the more general case of General Relativity coupled to Electromagnetism we briefly review the pure electromagnetic theory. Because diffeomorphisms are not a gauge symmetry here, the procedure is simpler. The variable is the connection one-form . The field strength two-form is , and it posses the gauge symmetry, .

The action is

 S[A]=α∫MF∗F, (55)

with , and where the Hodge dual acting on the field strength in coordinate components or in tetrad components is respectively , with and .

The variation of the Lagrangian is

 δL(A) = EAδA+dΘ(δA) (56) = −2α(d∗F)δA+d(2αδA∗F), (57)

where because is not a dynamical field for this theory. As before, the boundary term should vanish. An option is to fix the connection at the boundary and consequently the gauge symmetry. Another option that does not restrict the connection is to assume a vanishing field strength at the boundary, . The infinitesimal symmetry gauge transformation is , and applying it to the variation of the Lagrangian, we get the Noether identity, which is the trivial equation . The Lagrangian is invariant under the gauge transformation, therefore, we have

 Jλ=EAλ+Θ(δA)=−2αd(λ∗F). (58)

Note the similarity with the corresponding equation for General Relativity (42).

The presymplectic structure density is . And for a gauge symmetry, , we get the surface charge integrand

 kλ=−2αλ∗δF. (59)

The exact symmetry condition is solved for , such that the gauge symmetry turns into a rigid symmetry. Note that here the exact condition is independent of the fields and admits a general solution. Then, we have which can be integrated in a three-surface enclosed by a two-surface , to define

 ⧸δQλ0=λ04π∮S∗δF, (60)

where we have restored the value of . For simplicity the parameter is chosen to be a constant in the phase space. Then, the variation can be trivially removed by an integration on phase space. We set the integration constant to zero. Then, we obtain the definition of the electric charge, enclosed by the surface

 Qλ0=λ04π∮∗F, (61)

the conservation ensures that for any other surface, obtained by a continuous deformation of , the electric charge is the same. If there are not sources can be contracted to a point and all charges are zero. This finishes the analysis of surface charges for Electromagnetism.

For completeness we derive the Noether current for a background spacetimes with rigid symmetries. Note that the spacetime may not be a solution of the Einstein equation. The rigid symmetries are controlled by a Killing field such that , where and is given by (50). The connection suffers from the same ambiguity of any gauge variable, then a Lie derivative on it, may be changed by an arbitrary gauge transformation while keeping the same information. Repeating the argument that led us to (35), the symmetry infinitesimal transformation which is also gauge invariant is

 δξA=ξ┘F, (62)

this transformation sometimes is called ‘improved.’ Note that while being assumed as a physical transformation it is not an exact symmetry transformation. Applied to the Lagrangian . Because is not an arbitrary parameter, but a fixed Killing field, there is not a Noether identity associated to it. The current , is a true Noether current because it is conserved just on-shell

 d(Θ(δξA)−ξ┘L) = −EAδξA ˆ= 0, (63)

explicitly

 Jξ=α(ξ┘F∗F−Fξ┘∗F), (64)

which is the dual of the standard four-current of Electromagnetism written in forms, , with the electromagnetic stress-energy tensor. Note that again . Instead, we used an infinitesimal transformation that is covariant (actually invariant) under the gauge transformation. This subtelty has produced debate in the literature and the transformation (62) has been settled as the right one because it produces a gauge invariant current (see for instance section 2.3 in the recent review Banados:2016zim ()). However, the Noether current belongs to an equivalence class of currents related by exact forms (and possibly terms vanishing on-shell). The gauge ambiguity in the chosen transformation for the symmetry (62) just change the representative current of the class. To define a charge the equation, , should be integrated on a four-dimensional manifold, which is splittable in two pieces. This procedure is insensitive to the ambiguity, and therefore the Noether charge.

5 General Relativity and Electromagnetism

Using the results of the previous sections the extension to the coupled theory is easy. Here we use for the Hodge dual and for the group dual. We also make explicit some indexes to differentiate the infinitesimal gauge parameter , from the infinitesimal gauge parameter .

The Lagrangian and its variation are given by

 L = κ¯RIJ⋆¯RIJ+αF∗F (65) δL = E′eδe+Eωδω+EAδA+dΘ(δω,δA), (66)

where . We use the notation . is the sum of the equation of motion of pure gravity (26) plus the contribution due to the electromagnetic stress tensor written as a form . The boundary term reads

 Θ(δω,δA)=2κδωIJ⋆¯RIJ+2αδA∗F. (67)

The infinitesimal gauge symmetry transformations are controlled by the parameters corresponding to diffeomorphisms, Lorentz local symmetry, and local symmetry, respectively. The transformations are the same for the gravity fields, (36) and (37). For the electromagnetic field we need to consider that it also transforms by diffeomorphisms. As discussed in the previous section the improved version is .

Following the procedure of section 2 we obtain

 Jϵ=−2 d(κλIJ⋆¯RIJ+αλ∗F), (68)

that is simply the sum of gravity and electromagnetic off-shell constributions found previously, (42) and (58). While expected this is non-trivial because there is a non obvious off-shell cancellation among the terms proportional to appearing in , , and .

The full surface charge integrand is

 kϵ=−2κ(λIJ⋆δ¯RIJ−δωIJ⋆ξ┘¯RIJ)−2α(λ δ∗F−δA ξ┘∗F). (69)

This is the sum of equations (44) and (59) plus a contribution due to diffeomorphism transformation of the electomagnetic field. Now, to ensure that we need the exactness of the symmetries , which is already solved by (50), but we also need

 δϵA=ξ┘F−dλ=0, (70)

from this equation can be solved in general. In coordinates components it is equivalent to solve from . It is in fact the equation for the electric potential for an electromagnetic field projected with . We note that is solution of the homogeneous equation, therefore, if is solution of the inhomogeneous one, we have . The plays exactly the same rôle that in pure Electromagnetism and therefore implies the conservation of the electric charge.

Note that in the surface charges formalism the definition of the electric charge and the charges due to spacetime Killing symmetries are on the same foot.

Before discussing an example let us remark the linearity property of the surface charge integrands. In the general derivation of section 2 we used the assumption . However, the obtained formula (69) is explicitly linear in the vector field generating diffeomorphism and in all the gauge parameters, i.e.

 α1kϵ1+α2kϵ2=kα1ϵ1+α2ϵ2, (71)

where can be arbitrary functions on the phase space. Thus, if and are closed forms for exact symmetries generated by and , then is also a closed form for the exact symmetry generated by with the precise identification . This fact is exploited in what follows.

5.1 Charged and rotating black hole

As an example we apply the result to a particular black holes familiy. We show that surface charges are compatible with the ones obtained through the standard asymptotic analysis. Then, we show how the quasilocal nature of surface charge allow to have a first law of black hole mechanics without relying on the asymptotic structure of spacetime Barnich:2003xg (). Note that this quasilocal perspective is the best that can be done when the black hole is embedded in and asymptotically de Sitter spacetime.

We consider a black hole solution family which is electrically charged, rotating, and satisfies the asymptotically constant curvature boundary conditions (29). It is known as the (anti-)de Sitter Kerr-Newman family. A possible tetrad and electromagnetic potential describing the solution are

 e0 = √Δrρ(dt−asin2θΞdϕ),e1=ρ√Δrdr , e2 = −ρ√Δθdθ ,e3=√Δθsinθρ(adt−a2+r2Ξdϕ), (72) A = −qrρ2(dt−asin2θΞdϕ), (73)

with , , , and . The upper sign is reserved for the anti-de Sitter family and the lower one for the de Sitter one. We stress that it is possible to use another set of variables related by a gauge transformation, but as the procedure is explicitly gauge invariant it will not have any impact on the results. In particular to rotate by an arbitrary Lorentz transformation or to add a term of the form to has not effect. From the equation we solve the connection and compute: , , , and . At this level we have reduced the phase space to the particular family solution spanned by the parameters , thus, the variation acts only on functions of those parameters.

In the metric formalisms and are two independent Killing fields. Through the solution of the exactness conditions for , (50), we get and respectively. Similarly through the exactness conditions on , (70), we obtain the corresponding and . Now we have the ingredients to compute surface charges. Plugging all the quantities in (69) we get the associated integrands and , one for each symmetry. The spacetime described by has non-contractible spheres due to the singularity. The integration can be performed over any two-surface enclosing the singularity. The surface charges associated to the exact symmetries generated by and are

 ⧸δQt = ∮kt=δmΞ±3amδaℓ2Ξ2 (74) ⧸δQϕ = ∮kϕ=−aδmΞ2+(3Ξ2−4Ξ3)mδa. (75)

The exactness condition has a further independent solution for a constant such that . The corresponding exact symmetry parameter is and the surface charge is

 ⧸δQλ0=∮kλ0=λ04π∮δ∗F=−λ0(δqΞ±2aqδaℓ2Ξ2). (76)

To proceed now we have two strategies: To fit the scheme in the results from the asymptotic picture or to insist with a quasilocal approach. We sketch both.

Asymptotic strategy: In order to fit with the asymptotic picture we can exploit the linearity of each surface charge, (71), and to adjust the freedom of the gauge parameters in the phase space to obtain the standard integrated charges (see for instance Caldarelli:1999xj ())

 M ≡ Qξ=∂t∓(a/ℓ2)∂ϕ=mΞ2 (77)