Claudio Bunster, Andrés Gomberoff and Alfredo Pérez
Abstract

We present a new application of the Regge-Teitelboim method for treating symmetries which are defined asymptotically. It may be regarded as complementary to the one in their original 1974 paper. The formulation is based on replacing the asymptotic planes by two patches of hyperboloids (advanced and retarded) of fixed radius and varying center. The motivation is to study radiation, and these hyperboloids are well suited to the task because they are asymptotically null, and thus are able to register the details of the process. The treatment produces naturally a Hamiltonian formulation of the symmetry of Bondi, van der Burg, Metzner and Sachs (BMS); it sheds light on the role of the Bondi “news” from the Hamiltonian viewpoint, on the role of magnetic flux and of the Taub-NUT space as a gravitational magnetic pole, and brings out the interrelationship between spin and charge. If a cosmological constant of either sign is brought in, the asymptotic symmetry of the gravitational field is that of de Sitter or anti de-Sitter, and there is no room for an analog of the BMS symmetry.

Chapter 2 Regge-Teitelboim analysis of the symmetries of electromagnetic and gravitational fields on asymptotically null spacelike surfaces 111To appear in the forthcoming volume “Tullio Regge: an eclectic genius, from quantum gravity to computer play,” Eds. L. Castellani, A. Ceresola, R. D’Auria and P. Fré (World Scientific).


To Tullio, for old times’ sake

Centro de Estudios Científicos (CECs), Avenida Arturo Prat 514, Valdivia, Chile
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Avda. Diagonal las Torres 2640, Peñalolén, Santiago, Chile

1 Introduction

The 1974 paper by Regge and Teitelboim [1] contained two main results: (i) A completion of Dirac’s analysis [2] of the role of constraints in field dynamics that was necessary in order to account for the different character of the gauge transformations which do not change the physical state (“proper gauge transformations”), from those which do (“improper gauge transformations”). It was found that, in the latter case, Dirac’s “weakly vanishing” generators have to be improved by the addition of a surface integral and do not vanish weakly. The surface integral gives the value of the charge associated to the improper transformations. (ii) An application of (i) to obtain a Poincaré invariant formulation of the theory of gravitation on spacelike surfaces which are asymptotically planes.

We develop below a new application of the method, which produces naturally a Hamiltonian formulation of the symmetry of Bondi, van der Burg, Metzner and Sachs (BMS)[3, 4, 5]; it sheds light on the role of the Bondi “news” from the Hamiltonian viewpoint, on the role of magnetic flux and of the Taub-NUT space as a gravitational magnetic pole, and brings out the interrelationship between spin and charge. If a cosmological constant of either sign is brought in, the asymptotic symmetry is that of de Sitter or anti de-Sitter, and there is no room for an analog of the BMS symmetry.

The treatment is based on replacing the planes by two patches of hyperboloids (advanced and retarded) of fixed radius an varying center. The motivation is to study radiation, and these hyperboloids are well suited to the task because they are asymptotically null, and thus are able to register the details of the process; whereas the asymptotic planes are not, because if one goes far enough along a spatial direction, the radiation emitted by a confined source has not enough time to reach there.

On the other hand, if compared with retarded and advanced light cones, the hyperboloids have the advantage of being spacelike and therefore permitting direct step by step use of the Dirac’s procedure with the Regge-Teitelboim complement, which has been battle-tested, and in which all the structures that appear (action, Hamiltonian, Poisson and Dirac brackets, surface deformations, most general permissible motion) are well, and tightly, defined from the start.

Furthermore, the two patches of hyperboloids of fixed radius and varying center have the essential property of covering smoothly the whole of spacetime, in contradistinction with the foliations by hyperboloids of fixed center and varying radius, used previously by many authors, which only cover part of it.

The structure of the paper is the following. In order to make the treatment self-contained and set the notation and terminology, section 2 begins by reviewing the general procedure. Section 3 discusses the foliation by hyperboloids of the same radius and different center; and gives a simple geometrical discussion of the antipodal identification which is necessary to match smoothly the retarded and advanced patches. Next, section 4 contains an analysis of the asymptotic properties of the free electromagnetic field on the hyperbolic foliation. The discussion is given in detail, because practically all the results derived for electromagnetism can be translated literally to the gravitational case, whose treatment becomes then considerably lighter. Section 5 is then devoted to the gravitational case without a cosmological constant (), while the case is discussed in section 6.

Four appendices are included: appendix A gives explicit expressions for the Poincaré generators on the hyperbolic foliation. Appendix B discusses details of the asymptotic conditions that are not explicitly used in the main text, but are necessary for consistency. Appendix C provides a “dictionary” for translating in the gravitational case the variables which appear in the present Hamiltonian treatment with those employed in the original BMS light cone analysis. Finally, appendix D exhibits the Penrose diagrams for our hyperbolic foliations, in the Minkowski, de Sitter and Anti-de Sitter spaces; and illustrates the limiting cases in which the radius of the hyperboloids tends to zero – approaching lightcones, or to infinity – approaching planes.

2 Hamiltonian field dynamics, surface deformations, gauge transformations, surface integrals.

In the formulation of field dynamics in which the state is defined on a general spacelike surface developed by Dirac [2], and completed by Regge and Teitelboim [1] to incorporate symmetries which are defined asymptotically, the generator–through Poisson brackets–of the “most general permissible motion” has the form

(0)

where is an integral over the spacelike surface on which the state is defined, of the form

(0)

and is a surface integral over the asymptotic boundary of that spacelike surface.

The surface integral is included to make well defined the functional derivatives of , so that one has,

(0)

without any surface terms. This means that the contribution of the asymptotic part of the field is already included in (2). Here we have abbreviated as all the canonical field variables of the theory.

In (2) the are the generators of deformations of the spacelike surface in which the state is defined, while the are the generators of internal gauge transformations. They are both constrained to vanish, that is, they are weakly equal to zero:

(0)

If the parameters are such that the surface integral vanishes, the motion generated by is called a “proper” gauge transformation [6], and it it is not a symmetry, but rather an expression of the fact that the system is described by variables which are redundant. This normally happens when they vanish at infinity, but there are important cases, in both electromagnetism and gravity [1] where the surface integrals vanish even though the parameters do not vanish at infinity but obey parity conditions there. In that case the transformation is still proper. For an internal symmetry one feels on safe grounds stating that a proper gauge transformation does not change the physical state on a given spacelike surface. For the case of a surface deformation this point of view may be kept for purely tangential deformations (changes of spatial coordinates), but if the deformation has a normal component the hypersurface is geometrically deformed, and therefore, if one can “perform local observations inside”, one would expect the physical state to change. On the other hand if one only performs observations at infinity, then one may safely take the point of view that proper normal deformations are also gauge transformations, and do not change the physical state either. If are such that the surface integral does not vanish the motion is called an “improper” gauge transformation, and one expects it to change the physical state. As a consequence of (2), the functional form of the transformation generated by (2) is the same if the transformation is proper or improper, the difference is only introduced by the asymptotic behavior of the transformation parameters.

The difference between proper and improper transformations manifest itself at the level of the action principle in which (2) is the Hamiltonian, in the fact that the constraints (2) are obtained from it by extremizing the action with respect to and keeping fixed the part that contributes to the surface integral . Thus, (2) states that the generator of proper gauge transformations vanishes weakly. On the other hand, for improper gauge transformations this does not happen:

If one is interested in the action of an asymptotic motion, one gives the asymptotic part of the transformation and continues it inside in an arbitrary manner. The way in which one chooses to continue inside is irrelevant because any two continuations differ by a proper gauge transformation. It is however necessary to continue, because it is only the sum of the volume part of the generator and the surface term which has well defined functional derivatives and is therefore capable of acting through a Poisson bracket. Neither the volume part alone, nor the surface integral alone has that capability. Alternatively, one may choose a particular continuation inside by fixing the gauge. If this is done, the combined set of the original gauge constraints and the gauge conditions become second class, and one can pass from the original Poisson bracket to the associated Dirac bracket, in terms of which the second class constrains vanish strongly and have zero bracket with everything. Then the surface term stands alone and is capable to act as itself as a generator through the Dirac bracket.

The constraint–generators , are first class. If we denote collectively by , , the parameters of any two motions one has, for proper gauge transformations,

(0)

where the commutator of the two original infinitesimal transformations is a bilinear expression of , and their derivatives (in practice, first derivatives, with coefficients which in general depend on the fields). For improper transformations it may happen that Eq. (2) is relaxed by the appearance of a central extension on its right hand side, that is, by the addition to of a term that has zero Poisson brackets with all the dynamical variables. This is not allowed for proper transformations because it would spoil the first class character of the constraints; i.e. the vanishing of the constraints would not be preserved by the transformation. This obstruction to the presence of a central extension does not happen for improper transformations because the charge is not constrained to be zero.

Given any theory one may calculate by working out directly the Poisson bracket on the left side of (2). However, if one has geometrical insight on the nature of the motions at hand one may write down the result without doing that calculation. For example, for a Yang-Mills theory, with structure constants , one has

and for two surface deformations within an arbitrary Riemannian spacetime with Lorentzian signature one has [7, 8],

(0)
(0)

where is the metric of the spacelike surface.

Lastly, we elaborate on the sentence “…The surface integral is included in (2) to make well defined the functional derivatives of …” written above. To achieve this it is necessary to find an appropriate set of boundary conditions. There is no foolproof, inductive method for that. One rather works by trial and error and there is no guarantee of success. The procedure in practice is as follows: (i) A tentative set of asymptotic conditions is obtained by applying the asymptotic transformations that one wants to have present, to a simple field configuration that one also wants to have present. For example, in the case of gravitation, one would boost a Schwarzschild field; or, in electromagnetism, a Coulomb field. (ii) One extracts properties of the result obtained that can be formulated independently of the specific original configuration, and uses them as a starting ansatz. For example, one may retain a decay rate in inverse powers of the radial distance and a parity condition for the coefficients.(iii) One finds the most general parameters which preserve the ansatz. It may happen then that that set of parameters does not contain all the symmetries that one was interested in (for example, the complete Poincaré group). If that happens one relaxes the ansatz to make room. If success is achieved – meaning, in the example just given, that one has the Poincaré group or more – one checks whether the surface integral that appears in the variation of the is the variation of a finite surface integral, or, as one says colloquially, “if the can be taken out”. If this happens one is done. If it does not, one modifies the boundary conditions in the light of the nature of the failure. With luck and dedication the process converges and one finally succeeds.

3 Foliation of Minkowski space by hyperboloids of the same radius and different centers

3.1 Retarded and advanced hyperboloids

In Minkowski space it is natural to define the state on a three-dimensional spacelike hyperboloid, because that surface is mapped onto itself by a Lorentz transformation. In this sense, hyperboloids are more adequate to the special principle of relativity than the planes corresponding to inertial frames for which the boosts are interchanged with spatial translations. This possibility was considered by Dirac in 1940 [9] and he called it the “point form of field dynamics” with the term “point” referring to the center of the hyperboloid.

A spacelike hyperboloid with center at and radius obeys the equation

(0)

Actually, Eq. (3.1) describes two disjoint hyperboloids, one with (“retarded hyperboloid”) and another with (“advanced hyperboloid”), as shown in Fig. 1. Although Dirac did not discuss foliations of spacetime by means of a family of hyperboloids, this has been done by many authors but to our knowledge in all cases treated so far the foliation has been defined by keeping fixed and letting vary as one passes from one hyperboloid to the next.

In other words, the foliations used previously have consisted of a sequence of hyperboloids with fixed center and varying radius222See, for example, [10], [11], [12], and also [13]\textcolorred and references therein. In some of these discussions timelike hyperboloids are employed (in which case in (3.1) is replaced by ).. These foliations have the advantage that the four are treated on the same footing so Lorentz invariance is manifest; but the price payed is extremely high, because only a small part of Minkowski space is covered, and moreover, a spurious explicit dependence on the varying , which is taken as the time , is introduced. Here we take the other natural option, we keep the radius fixed and we allow the position of the center to vary. This is a direct extension of what is done with null foliations, which may be regarded as being the limit . The actual value of will turn out to be irrelevant, since all the quantities of physical interest will incorporate naturally in their units.

For simplicity we take the center to move along the line . There are then two such foliations, one by means of retarded hyperboloids and the other by means of advanced ones. The retarded foliation covers all of spacetime except past timelike infinity, while, conversely, the advanced one covers all of spacetime except future timelike infinity. Thus, both the advanced and retarded patches are needed to cover all of spacetime; much in the same way as for a magnetic pole one needs two coordinate patches on the two-sphere, with one of them unable to cover the north pole and the other unable to cover the south pole.

In the two-patch formalism, a retarded hyperboloid together with an advanced one, neither of which is by itself a Cauchy surface, fulfill the role that a Cauchy surface plays in the single patch formalism. As illustrated and explained in figure 2, one may think of the retarded and advanced hyperboloids as being the upper and lower halves of a thickened Cauchy surface. Although in principle the two-patch and the single patch Hamiltonian formalisms are equivalent, when analyzing radiation the former is better suited to the task than the latter. This is because each member of the pair of hyperboloids on which the state is defined is asymptotically null, which makes the outgoing and incoming radiation emerge naturally. The replacement of a Cauchy surface by a pair of surfaces which intertwine past and future, so that the present becomes just an interpolation between the two, is of course the viewpoint that led to Feynman diagrams and other key developments in quantum field theory [14]. As the analysis below shows, this viewpoint is not incompatible with a Hamiltonian formulation but, on the contrary, it admits a natural one.

It is useful to introduce coordinates which are adapted to the geometry of the hyperboloid and to the foliation with fixed radius and varying center. Following standard practice for advanced and retarded light cones, we write

(0)
(0)

for the retarded patch, and similarly

(0)
(0)

For the retarded patch the metric reads

(0)

while for the advanced case one has

(0)

3.2 Overlap of retarded and advanced patches. Antipodal mapping.

To discuss the overlap it is necessary to specify the range of the variables. To this effect, we observe that equations (3.1)-(3.1) may be reduced to a formulation in two spacetime dimensions , where . They then become

(0)
(0)

and,

(0)
(0)

which imply,

(0)

and,

(0)

whereas,

(0)

It follows from (3.2) that the retarded and advanced times increase as one proceeds from to , for fixed , . The retarded time has the range while . In both cases is a unit normal to the two-sphere. It will be specified below whether that normal is taken to be the outward normal or the inward one.

In what regards the ranges of , there are two simple equivalent choices. One may take: (i) , covering the whole sphere; (ii) , covering half of the sphere. The first choice (i) is employed in Fig. 1, which illustrated the retarded and advanced patches (the corresponding Penrose diagram is given in appendix D), whereas the viewpoint (ii), with is taken in Fig. 2 where it is shown that one must have

(0)

for a given . If one reverts to the more usual practice of taking on both patches, the identification (3.2) is expressed as

(0)

Eq. (3.2) means that at the intersection of an advanced and a retarded hyperboloid, the corresponding advanced and retarded two-spheres are antipodically mapped into each other. Therefore, a continuous field will appear as discontinuous if the identification is ignored, simply because if one were to compare the values of a field with the same and at the joining, one would be attempting, because of Eq. (3.2), to relate the values of the field at points which are antipodal to each other, and therefore wildly distant in space.

The need for joining past and future null infinity, and for doing so through the antipodal identification, was discovered by Strominger when studying gravitational scattering [15] (see also [16] and references therein). We have included the discussion above, which is illustrated in Fig. 2, to emphasize its simple geometrical origin, and independence of any particular theory in consideration.

Ultimately, the opposite orientation of the Lienardretarded and advanced hyperbolas stems from having chosen to increase in the same direction as , while geometrically (and physically) the opposite choice would be more natural (an incoming wave converging to an absorber is really a time reversed advanced wave emitted by it). This is also the origin of the plethora of different signs that appear throughout the paper in corresponding formulae for the retarded and advanced cases.

Fig. 1.: Advanced and retarded hyperboloids The figure shows a set of retarded hyperboloids of constant as well as a set of advanced ones at constant . For a given (), the corresponding hyperbolas tend asymptotically to the retarded (advanced) lightcone that intersects the time axis at that value of (), respectively. A complete two-sphere is attached at every point of each hyperbola in the diagram. The corresponding two-spheres at the intersection of at advanced and a retarded hyperbola are antipodically mapped.
Fig. 2.: Overlap of retarded and advanced patches. The figure shows, in the two-dimensional formulation of Eqs. (3.2)–(3.2), the intersection of the retarded hyperbola and the advanced one , which intersect each other at two points (on each of which half of a two-sphere is attached), at , . The arrows shown on each hyperbola, which specify its orientation, are determined so that they coincide asymptotically with the direction of propagation of an outgoing null wave (retarded hyperboloid) or an incoming one (advanced hyperboloid). It is apparent from the figure that at each of the two intersections the components of the two hyperbolas have opposite directions. This means that the retarded and advanced patches have opposite orientations. Hence, as stated in Eq. (3.2), for a coordinate that runs smoothly over the complete hyperbola one must take for a given . The figure also illustrates how the matched hyperbolas fulfill the role of a Cauchy surface. To that effect one may imagine deforming the hyperbolas between and so that the gray region becomes very thin in the vertical direction; and taking any value of as large as one desires, but finite. Next consider for simplicity a massless scalar field ; one could take as a boundary condition for the action principle, and therefore, for the equations of motion, giving the value of on both hyperboloids, which would include giving its value on the upper and lower faces of the thickened segment . In the limit of the segment of vanishing thickness this would be equivalent to giving the initial conditions , at , which is appropriate because the equation of motion for is of second order in . However, asymptotically, the upper and lower faces of the thickened segment cannot be brought together because they become null towards the future and past respectively, and the segment splits. Therefore one cannot replace giving on the two faces by giving, say, and on one face. But this is just right, because in that region the equation of motion is of first order in the variables , , which become null coordinates. So it is appropriate to give just on the upper face ( fixed) and on the lower one ( fixed). Thus the formulation in terms of two patches of hyperbolas achieves automatically the matching of a surface in the interior, with two (past and future) null surfaces at infinity.

4 Electromagnetic field in Minkowski space

We will analyze in this section the case of the electromagnetic field on a fixed Minkowskian background. Practically all the features that will be encountered in the gravitational case already appear in this technically simpler context.

The main difference which does not hinder the analogy is that, since the background is fixed, its Poincaré symmetry appears as a global symmetry rather than an asymptotic gauge symmetry. There are no constraints associated with the surface deformation , which are not varied in the action principle. The in (2) are replaced by the energy and momentum densities of the electromagnetic field

(0)
(0)

The only gauge symmetry present in the problem is the electromagnetic one, whose generator is

(0)

Here is the vector potential, its conjugate momentum, and is the metric on the hyperboloid, and denotes its determinant.

If instead of having a fixed background we were considering dynamically coupled electromagnetic and gravitational fields, then expressions (4), (4) would be added to their gravitational counterparts discussed in section 5, and the sum would be constrained to vanish. The asymptotic analysis given below would still hold because at large distances the spacetime would be flat. Then the asymptotic symmetry transformations of the coupled Einstein-Maxwell system would be those discussed here (internal electromagnetic, and Poincaré transformations) and the additional gravitational supertranslations333If one does not want to bring in gravitation but still desires to have arbitrary surface deformations in the interior in Minkowski space, one may appeal to Dirac’s Hamiltonian form of field dynamics [17], introduce the constraint , and fix asymptotically the deformation freedom through (3.1)-(3.1). Here the ’s are the normal and tangential projections of the conjugates to the Minkowskian ’s. .

We will now discuss the Poincaré and proper and improper gauge transformations for the electromagnetic field on our hyperbolic slicing. In this case the time equal constant surface is left invariant under the Lorentz group, whereas it is mapped onto a different hyperboloid by spacetime translations. Thus if one compares the situation with constant planes, one sees that the roles of spatial translations and boosts are interchanged.

4.1 Asymptotic conditions

By applying the procedure described at the end of Sec. 2, starting from the Coulomb field written in hyperbolic coordinates, one is led to the boundary conditions,

(0)
(0)
(0)
(0)

One may verify using the equations of motion

(0)
(0)

that the conditions (4.1)-(4.1) are preserved if is one of the Killing vector of the Poincaré group given in appendix A, and has the form

(0)

4.2 Improved Hamiltonian

According to the Regge-Teitelboim procedure one endeavours to add to the provisional Hamiltonian444When the boundary conditions (4.1)-(4.1) hold, and is any Killing vector of the Poincaré group, this provisional Hamiltonian is finite. The symplectic term is also finite.

(0)

a surface integral , which ensures that the functional derivatives of the improved Hamiltonian

(0)

are well defined. It turns out that this cannot quite be achieved, and one can only go part of the way. The best one can do is to replace in (4.2) by with

and add a surface term

(0)

With this improvement it is direct to show that the variation of the Hamiltonian (4.2) is given by

(0)

where is the amount of spacetime translation involved in the deformation, the vector is given by

(0)

in the retarded patch, and

in the advanced one. Also, one has the expressions

(0)
(0)

where

(0)

is the order coefficient in the expansion of , which is the dual of the tangential magnetic field. The tensor is the metric of the unit sphere. The indices will be lowered with it and raised with its inverse. The determinant of is denoted by .

The vector corresponds in Cartesian coordinates to an electromagnetic field that decays as , that is to a wave emitted by a confined source, or converging towards an absorber. Outgoing waves are “painted on” retarded hyperboloids, so appears in eq. (4.2) in that case555For an electric charge with acceleration in its retarded rest frame, one has from the retarded Lienard-Wiechert field: . See for example, [18, 19]. . Similarly, incoming waves are painted on advanced hyperboloids, and is used then in the evaluation of (4.2). To have a complete record of the physical situation, keeping track in particular of the independent and , one must combine the data from both patches, because as explained in the previous section, only then one has the amount of data which are registered on a Cauchy surface. The vector is the electromagnetic analog of the “Bondi news” that will be encountered in gravitation below.

When the variation (4.2) does not vanish, the equations of motion do not follow from demanding that the improved action should be stationary. As a consequence, there is no Hamilton principle and the Hamiltonian has no well-defined Poisson brackets. The symmetries remain symmetries of the equations of motion but since there is no action principle they do not yield conserved charges.

Now, the variation (4.2) vanishes when even if . This means that the Lorentz and proper gauge symmetry charges have well-defined brackets in that case, and they obey a closed algebra which will be given below. However if , the variation vanishes only when . This means that one can realize canonically the action of the Lorentz and improper gauge transformations in that case. But not that of the translations . This is understandable because in the hyperbolic foliation the whole Lorentz group corresponds to tangential deformations of the surface, and one should not expect difficulties there. However, the translations move one away from the surface and if one wants them to be realized canonically, one must have .

Thus one expects that when the response of the charges under spacetime translations should depart from that dictated by the symmetry algebra.

This lack of conservation indeed occurs. One finds for example,

(0)

where

(0)

are the improper gauge charges, one per each point of the two sphere at infinity [16] , and,

(0)
(0)

On the right hand side of these equations, one takes the negative sign for the retarded patch and the positive sign for the advanced one. Equation (4.2) is illustrated in Fig. 3.

Fig. 3.: News at work. The charge was moving at (low) uniform velocity and it suddenly reversed its motion. Far away from it, the density of electric flux lines is that of the Coulomb field of a charge in uniform motion to the right. However, closer up the field is the Coulomb field of a point charge moving to the left. In both cases the strength of the field varies as . The change from “before” to “after” is produced by a thin shell of news carried by a wave emitted at the turning point, whose field is tangent to the wave front sphere and varies as . This diagram, based on J.J. Thomson way of looking at radiation, first appeared in [20].

The above relations may be verified directly from the equations of motion (4.1)-(4.1) and the boundary conditions (4.1)-(4.1), but it is more illuminating to realize that they may be obtained directly from the “lack of extremality” of t he action when the equations of motion hold due to the non-vanishing of the variation (4.2). Indeed, if one repeats the steps leading to Noether’s theorem, but taking into account (4.2), one finds that the lack of conservation of the charge under a spacetime translation is given by

(0)

where is the amount of infinitesimal translation, and is the change of given by the equations of motion (4.1)-(4.1) under the most general motion with parameter . Thus equations (4.2), (4.2), (4.2) follow from taking for gauge transformations, for time translations, and for rotations, as it follows from (4.1); with in all three cases. In the above equations , stand either for the retarded or advanced pairs , .

There is a message that comes out loud and clear from the relationship of (4.2) with the lack of conservation (4.2)-(4.2). It is this: one should not expect to be able to “take the delta outside” when there are news. The reason is that, had one succeeded in doing so, one would have obtained a conserved quantity, energy for example. But this cannot happen because, on an asymptotically null surface, the emitted the energy carried away by the news reaches infinity “instantly”, which is precisely why the formulation enables one the study of the details of the radiation process666On an asymptotic plane, in contradistinction, one can always go far enough so that the energy emitted by a source which has been radiating during a finite time has not yet reached there, so that there are no news. One should then be able to take the delta outside, but at the price of loosing all the details about of the radiation process. The BMS symmetry for electromagnetism, and previously also for gravity, has been recently tamed to fit a foliation by surfaces that are asymptotically planes [21, 22, 23]. This has required dexterity, since the symmetry is intimately related to radiation and its natural habitat is an asymptotically null surface, rather than a plane. .

So the “failure” is really a success, and a useful one.

4.2.1 Spin from charge

Before living this section we would like to bring out an interesting phenomenon which arises as a consequence of the improvement described in section 4.2 in the charges. As a consequence of it the Lorentz charge becomes

(0)

Therefore the internal symmetry charge contributes to the angular momentum and boosts charges. This phenomenon is similar to the modification of the angular momentum which appears in the presence of a magnetic pole in abelian and non-abelian gauge theories. The novelty here is that it occurs already without a magnetic pole. It should be pointed out that the zero mode of , that is the total electric charge, does not contribute to (4.2.1) but the higher ones, do. Further consequences of this are discussed in the next subsection.

The spin from charge phenomenon does not happen for energy and momentum because no surface term analogous to the one appearing in (4.2.1) is included in the translation charge.

4.3 Canonical realization of the symmetry algebra in the absence of news

We now consider the case of no news

(0)

in which there exists a Hamilton principle even when spacetime translations are included, so that the whole symmetry algebra can be realized canonically 777If is imposed one must demand that it be preserved under gauge and Poincaré transformations. The condition is gauge invariant, so there is no problem with gauge transformations proper or improper. Furthermore, under Lorentz transformations changes with the Lie derivative on the sphere under , so it is also preserved. However, under spacetime translations the situation is different because the preservation of leads to another condition involving higher order terms, and the preservation of the new condition gives yet another one, so one has an infinite sequence. It is the infinite sequence that should be regarded as the condition of no news, although only the first term in it, , appears in the variation of the action. As explained in App. B, the infinite sequence can be solved recursively and therefore poses no problems either..

The Poincaré generators obey, of course, the Poincaré algebra, and the electromagnetic charges are abelian. The charges transform as

(0)

under Lorentz transformations, and they obey

(0)

so that they are invariant under spacetime translations888If one brings in the gravitational field the translations are generalized to the supertranslations , which include as their spherical harmonics modes. Equation (4.3) is then generalized to ..

If the left hand side of (4.3) is read as giving the action of a Lorentz transformation on the electromagnetic charges is says that those charges are not Lorentz invariant, but rather, transform under themselves under Lorentz transformations. On the other hand, if the bracket in (4.3) is read as giving the action of an improper gauge transformation on a Lorentz generator, it expresses that the Lorentz generator is not invariant. This is of course the origin of spin from charge phenomenon which is revealed to be an expression of an important consequence of the electromagnetic BMS algebra, namely, that the Lorentz group can be “moved around” within the algebra by an improper gauge transformations. This property has an analog in the gravitational case, as it will be seen later on.

The fact that from the point of view of the algebra there is no natural choice of the Lorentz group as a subgroup has sometimes been considered as an unwelcome feature (“angular momentum problem” [4]). However in the present context, if that were a difficulty, it would be easy to remediate because one can always go to the BMS rest frame by demanding the stronger boundary conditions

(0)

The condition (4.3) can always be reached (when !) by an improper gauge transformation, and after it is reached it only allows for an improper gauge transformation which possesses just a zero mode. In other words, once (4.3) is imposed the only transformations that leave the boundary conditions invariant are Poincaré transformations and a single global group. The Lorentz group cannot any longer be moved around (except by the standard translations that shift the origin). In that sense these stronger boundary conditions are in the electromagnetic case the analog for the hyperboloids of the original Regge-Teitelboim conditions for surfaces that are asymptotically planes999In the case of planes gauge transformations appear whose parameter tends at infinity to a function which is odd under inversions, and yet are proper because they do not have an associated surface integral. They have no analog in the case of hyperboloids.. It should be stressed that in the BMS rest frame the charges are still present and have the same value than in any other BMS frame since they commute with each other, but their conservation appears as “accidental” since the associated symmetries are frozen.

4.4 Magnetic flux and charge

The boundary conditions (4.1)-(4.1) allow for the presence of a magnetic flux

(0)

at infinity, which is the analog of the electric flux . In order to display more distinctly the symmetry between electricity and magnetism. it is useful to recall that a vector on the sphere may be decomposed as the sum of a gradient and a curl

(0)

where the operator is defined as

(0)

so the magnetic flux (4.4) reads

(0)

Under an improper gauge transformation with parameter the function changes as

(0)

and remains invariant

(0)

If the function is regular the integral of on the sphere vanishes, so in order to include in (4.1)-(4.1) a magnetic pole of charge one must allow for a with a Dirac string singularity. For example, for the choice

(0)

the corresponding is given by

(0)

On the other hand, for an electric pole, one simply takes

(0)

without having to introduce singular functions. This asymmetry is due to the fact that equations (4.1)-(4.1) are written in the “electric representation”, in which a vector potential is only introduced for the magnetic field.

4.4.1 Two potential formulation

One may go to the a two potential representation by introducing an electric vector potential satisfying

Then, equations (4.1)-(4.1) are replaced by

(0)
(0)

The electric flux then reads

(0)

which is the counterpart of (4.4). There are now also magnetic gauge transformations with an associated parameter which is independent of the “electric” . Under a magnetic BMS transformation and transform according to

(0)
(0)

Also the surface term in the Lorentz generator (4.2.1) is now replaced by

where and similarly for , so that the magnetic charge density also contributes to the angular momentum if one is away from the “magnetic BMS rest frame” .

If one demands that and be regular on the sphere, Eqs. (4.4), (4.4.1) do not allow for a zero mode in the electric and magnetic charges. They must both be introduced through Dirac string singularities, which then requires to admit similar singularities in and in order, for example, to implement rotations.

The magnetic BMS charge density obeys (4.3) as well. The electric and magnetic charges and are independent and so are the symmetries that are associated to them.

In this two-potential formulation it becomes evident that are electric and magnetic BMS charges on equal footing, and also electric and magnetic BMS transformations, which are independent and obey identical algebras. In the purely electric formulation, the magnetic symmetries are frozen, und the associated conservation laws appear as “accidental”. One may think that in the electric formulation one is in the “magnetic BMS rest frame”.

The retarded and advances news vectors defined in (4.2) may be written as

where the hodge dual “” on the sphere is defined as in (4.4). It then follows that there are no separate “magnetic news”, they are the dual of the electric news:

(0)

Lastly we remark that the fields and may be regarded as canonically conjugate pairs on the sphere. This status is formally implemented by including a surface symplectic term ,

in the action. Then, extremization with respect to yields the conservation laws ); whereas extremization with respect to yields ), where are the electric and magnetic gauge transformation parameters.

5 Gravitational field

5.1 Correspondence with electromagnetism

In this section we analyze the gravitational field along the same lines that we analyzed above the electromagnetic field. The parallel between both cases is so close that it permits to make the following discussion succinct. The correspondence is as follows: The mode of the improper gauge symmetry generated by the total electric charge is the analog of the , modes of the Bondi-van der Burg-Metzner-Sachs supertranslation, which are the ordinary translations generated by . The modes with of the improper gauge symmetry correspond to the modes of the supertranslations. Therefore, altogether, one has the correspondence:

On the other hand, the Lorentz transformations play along side:

There is, as emphasized before, the difference that in the gravitational case all the generators are given by surface integrals, whereas in the electromagnetic one since the background was fixed, the spacetime translations and the Lorentz transformations were not. But this is just a technical point which is easily accounted for and does not hinder at all the close correspondence between both cases.

The important concept of “news” is also present here, of course since it is the context in which it was originally introduced by Bondi [24]. The only difference is that now it is a symmetric traceless tensor , appropriate to describe a gravitational wave, rather than the vector appropriate for an electromagnetic one. Thus, one has the correspondence:

Keeping this in mind, we will essentially write the corresponding equation without much discussion, because one may translate to gravitation word by word in each case the corresponding comments from electromagnetism.

5.2 Asymptotic conditions

For the gravitational field without a cosmological constant, the generators of surface deformation are given by,

Here we have set . The cosmological constant will be brought in below, in section 6. It makes a substantial difference.

Since our spacelike surfaces are asymptotically null, we must take as a starting point a coordinate system for the Schwarzschild metric which incorporates this property manifestly. This is provided by the Eddington-Finkelstein coordinates in terms of which the line element reads,

(0)

where the relative minus sign in the last term corresponds to the retarded case and the plus sign to the advanced one. Therefore the coordinates , respectively, are null not only asymptotically but everywhere.

The next step is to pass to hyperbolic coordinates, through the change of variables (3.1)-(3.1), and extract the asymptotic form of the resulting expression. In doing so, and throughout the rest of this section, we will use the radial variable

(0)

in place of . This turns out to be more convenient in keeping track of the different orders in the asymptotic form. The change of coordinates then reads

(0)

for the retarded patch, and

Unless explicitly noted we will work below in the retarded patch. The corresponding expressions for the advanced patch can always be obtained by the simple rule , as indicated before.

One finds from the asymptotic form of (5.2) the following expressions for the metric and its conjugate momentum,

(0)
(0)
(0)
(0)
(0)
(0)

whereas the lapse and the shift are given by

From the above expressions one sees that the deviations in the metric from the Minkowskian background begin at order , and in the momentum at order ; thus one must be prepared to go at least these orders in the final expression one is looking for. Keeping this in mind, one performs a boost in (5.2) and passes again to hyperbolic coordinates. After trial and error one arrives at the following analogue of the electromagnetic boundary conditions (4.1)-(4.1):

(0)
(0)
(0)
(0)
(0)