A Note on the Newman-Unti Group and the BMS Charge Algebra in Terms of Newman-Penrose Coefficients

Abstract

The symmetry algebra of asymptotically flat spacetimes at null infinity in four dimensions in the sense of Newman and Unti is revisited. As in the Bondi-Metzner-Sachs gauge, it is shown to be isomorphic to the direct sum of the abelian algebra of infinitesimal conformal rescalings with 𝔟𝔪𝔰₄. The latter algebra is the semidirect sum of infinitesimal supertranslations with the conformal Killing vectors of the Riemann sphere. Infinitesimal local conformal transformations can then consistently be included. We work out the local conformal properties of the relevant Newman-Penrose coefficients, construct the surface charges, and derive their algebra.

1. Introduction

The definitions of asymptotically flat four dimensional spacetimes at null infinity by Bondi et al. [1, 2] (BMS) and Newman-Unti (NU) [3] in 1962 merely differ by the choice of the radial coordinate. Such a change of gauge should not affect the asymptotic symmetry algebra if, as we contend, this concept is to have a major physical significance.

The problem of comparing the symmetry algebra in both cases is that, besides the difference in gauge, the very definitions of these algebras are not the same. Indeed, NU allow the leading part of the metric induced on Scri to undergo a conformal rescaling. When this generalization is considered in the BMS setting, it turns out that the symmetry algebra is the direct sum of the BMS algebra 𝔟𝔪𝔰₄ [4] with the abelian algebra of infinitesimal conformal rescalings [5, 6]. There are two novel and independent aspects in this computation.

(i) The first concerns the fact that the BMS algebra in 4 dimension involves the conformal Killing vectors of the unit, or equivalently, the Riemann sphere and can consistently accommodate infinitesimal local conformal transformations. The symmetry algebra 𝔟𝔪𝔰₄ then involves two commuting copies of the noncentrally extended Virasoro algebra, called superrotations in [7], and simultaneously the supertranslations generators are expanded in Laurent series. The standard, globally well-defined symmetry algebra 𝔟𝔪𝔰^glob₄ consists in restricting to the globally well-defined conformal Killing vectors of the sphere which correspond to infinitesimal Lorentz transformation, while the supertranslation generators are expanded into spherical harmonics.

This local versus global versions of the symmetry algebra are of course not related to the BMS gauge choice, but will also occur in alternative characterizations of the asymptotic symmetry algebra where the conformal Killing vectors of the sphere play a role. Examples of this are the geometrical approach of Geroch [8] based on Penrose's definition of null infinity [9] and also, as we will explicitly discuss in this paper, the asymptotic symmetries in the NU framework.

(ii) The second aspect is related to the modified Lie bracket that should be used when the vector fields parametrizing infinitesimal diffeomorphisms depend explicitly on the metric. Indeed, when using the modified Lie bracket, the spacetime vectors realize the asymptotic symmetry algebra everywhere in the bulk and furthermore, even on Scri, this bracket is needed to disentangle the algebra when conformal rescalings of the induced metric on Scri are allowed. Similarly, in the context of the AdS/CFT correspondence, this bracket allows one to realize the asymptotic symmetry algebra in the bulk and to disentangle the symmetry algebra at infinity when considering transformations that leave the Fefferman-Graham ansatz invariant only up to conformal rescaling of the boundary metric [10]. From a mathematical point of view, the modified Lie bracket is the natural bracket of the Lie algebroid that is associated to any theory with gauge invariance [11].

What we will do in this paper is to rederive from scratch the asymptotic symmetry algebra in the NU framework by focusing on metric aspects and on the two novel features discussed above. As expected, the symmetry algebra is again the direct sum of 𝔟𝔪𝔰₄ with the abelian algebra of infinitesimal conformal rescalings of the metric on Scri and thus coincides, as it should, with the generalized symmetry algebra in the BMS approach. A related analysis of asymptotic symmetries in the NU context from the point of view of Scri and emphasizing global issues instead can be found in [12, 13].

Even though the results presented here are not really surprising in view of those in the BMS framework and the close relation between the NU and BMS approaches, the exercise of working out the details is justified because the NU framework is embedded in the context of the widely used Newman-Penrose formalism [14] so that explicit formulae in this context are directly relevant in many applications, see for example, the review article [15].

As a first application, we study the transformation properties of the Newman-Penrose coefficients parametrizing solution space in the NU approach. Our main focus is on the inhomogeneous terms in the transformation laws that contain the information on the central extensions of the theory. We then discuss the associated surface charges by following the analysis in the BMS gauge [16] and briefly compare with standard expressions that can be found in the literature. The algebra of these charges is derived and shown to involve field dependent central charges in the case of 𝔟𝔪𝔰₄ which vanish for 𝔟𝔪𝔰^glob₄.

2. NU Metric Ansatz for Asymptotically Flat Spacetimes

The metric ansatz of NU is based on a family of null hypersurfaces labelled by the first coordinate, 𝑥⁰≡𝑢=const. The second coordinate 𝑥¹≡𝑟 is chosen as an affine parameter for the null geodesic generators 𝑙^𝜇 of these hypersurfaces, so that 𝑙^𝜇=−𝛿^𝜇_𝑟. Up to a change of signature from (+,−,−,−) to (−,+,+,+), a renumbering of the indices, and the tetrad transformation that makes the conformal factor real, the line element considered in section 4 of NU [3] can be written as 𝑑𝑠²=𝑊𝑑𝑢²−2𝑑𝑟𝑑𝑢+𝑔_𝐴𝐵𝑑𝑥^𝐴−𝑉^𝐴𝑑𝑢𝑑𝑥^𝐵−𝑉^𝐵𝑑𝑢, (2.1) with associated inverse metric 𝑔^𝜇𝜈=0−10−1−𝑊−𝑉^𝐵0−𝑉^𝐴𝑔^𝐴𝐵, (2.2) where 𝑔_𝐴𝐵𝑑𝑥^𝐴𝑑𝑥^𝐵=𝑟²𝛾_𝐴𝐵𝑑𝑥^𝐴𝑑𝑥^𝐵+𝑟𝐶_𝐴𝐵𝑑𝑥^𝐴𝑑𝑥^𝐵+𝑜(𝑟), (2.3) with 𝛾_𝐴𝐵 conformally flat. Below, we will use standard stereographic coordinates 𝜁=cot(𝜃/2)𝑒^𝑖𝜙,𝜁, 𝛾_𝐴𝐵𝑑𝑥^𝐴𝑑𝑥^𝐵=𝑒²𝜑𝑑𝜁𝑑𝜁, 𝜑=𝜑(𝑢,𝑥). In particular, we use the notation 𝑒²𝜑 for the conformal factor. In Section 4, we will give the explicit dictionary that allows one to translate to the quantities originally used by NU.

In addition, the choice of origin for the affine parameter of the null geodesics is fixed through the requirement that the term proportional to 𝑟⁻² in the expansion of the spin coefficient −𝜌=𝐷_𝜌𝑙_𝜈𝑚^𝜌𝑚^𝜈 is absent.

When expressed in terms of the metric, one finds 1𝜌=−4𝑔^𝐴𝐵𝑔_{𝐴𝐵,𝑟}1=−4𝜕_𝑟ln𝑔=−𝑟⁻¹+14𝐶^𝐴_𝐴𝑟⁻²+𝑜𝑟⁻², (2.4) where 𝑔=det𝑔_𝜌𝜈 and the index has been raised with the inverse of 𝛾_𝐴𝐵. The requirement is thus equivalent to the condition 𝐶^𝐴_𝐴=0. (2.5) In the following we denote by 𝐷_𝐴 the covariant derivative with respect to 𝛾_𝐴𝐵 and by Δ the associated Laplacian and by 𝑅 the scalar curvature. In complex coordinates 𝜁,𝜁, 𝐶_𝜁𝜁=0 and we define for later convenience 𝐶_𝜁𝜁=𝑒²𝜑𝑐,𝐶_𝜁𝜁=𝑒²𝜑𝑐. Finally, 𝑉^𝐴=𝑂𝑟⁻²,𝑊=−2𝑟𝜕_𝑢𝜑+Δ𝜑+𝑂𝑟⁻¹, (2.6) where Δ𝜑=4𝑒⁻²𝜑𝜕𝜕𝜑 with 𝜕=𝜕_𝜁,𝜕=𝜕_𝜁.

The more restrictive fall-off conditions in [3] are relevant for integrating the field equations but play no role in the discussion of the asymptotic symmetry algebra.

3. Asymptotic Symmetries in the NU Approach

The infinitesimal NU transformations can be defined as those infinitesimal transformations that leave the form (2.2) and the fall-off conditions (2.3)–(2.6) invariant, up to a rescaling of the conformal factor 𝛿𝜑(𝑢,𝑥^𝐴)=𝜔(𝑢,𝑥^𝐴). In other words, they satisfy ℒ_𝜉𝑔^𝑢𝑢=0,ℒ_𝜉𝑔^𝑢𝐴=0,ℒ_𝜉𝑔^𝑢𝑟𝜕=0,_𝑟1𝑔𝜕_𝜌𝑔𝜉^𝜌=𝑜𝑟⁻²ℒ_𝜉,𝑔^𝑟𝐴=𝑂𝑟⁻²,ℒ_𝜉𝑔^𝐴𝐵=−2𝜔𝑔^𝐴𝐵+𝑂𝑟⁻³,ℒ_𝜉𝑔^𝑟𝑟=2𝑟𝜕_𝑢𝜔+2𝜔Δ𝜑−Δ𝜔+𝑂𝑟⁻¹. (3.1)(3.2)(3.3) Equations (3.1) are equivalent to 𝜕_𝑟𝜉^𝜈=𝑔^𝜈𝜌𝜕_𝜌𝜉^𝑢⟺𝜕_𝑟𝜉^𝑢𝜕=0,_𝑟𝜉^𝐴=𝜕_𝐵𝜉^𝑢𝑔^𝐵𝐴,𝜕_𝑟𝜉^𝑟=−𝜕_𝑢𝜉^𝑢−𝜕_𝐴𝜉^𝑢𝑉^𝐴, (3.4) and are explicitly solved by 𝜉^𝑢𝜉=𝑓,_𝐴=𝑌^𝐴+𝐼^𝐴,𝐼^𝐴=−𝜕_𝐵𝑓^∞_𝑟𝑑𝑟^′𝑔^𝐴𝐵,𝜉^𝑟=−𝑟𝜕_𝑢𝑓+𝑍+𝐽,𝐽=𝜕_𝐴𝑓^∞_𝑟𝑑𝑟^′𝑉^𝐴, (3.5) with 𝜕_𝑟𝑓=0=𝜕_𝑟𝑌^𝐴=𝜕_𝑟𝑍. Equation (3.2) then implies 1𝑍=2Δ𝑓. (3.6) The first equation of (3.3) requires 𝜕_𝑢𝑌^𝐴=0, the second that 𝑌^𝐴 is a conformal Killing vector of 𝛾_𝐴𝐵, which amounts to 𝑌^𝜁≡𝑌=𝑌(𝜁),𝑌^𝜁≡𝑌=𝑌𝜁, (3.7) in the coordinates 𝜁,𝜁, and also that 𝜕_𝑢𝑓=𝑓𝜕_𝑢1𝜑+2𝜓, (3.8) with 𝜓=𝐷_𝐴𝑌^𝐴, or more explicitly in 𝜁,𝜁 coordinates, 𝜓=𝜕𝑌+𝜕𝑌+2𝑌𝜕𝜑+2𝑌𝜕𝜑, and 𝜓=𝜓−2𝜔. Finally, the last equation of (3.3) implies 2𝜕_𝑢𝑍+𝑍𝜕_𝑢𝜑=𝑌^𝐴𝜕_𝐴Δ𝜑+𝜓Δ𝜑+2𝜕_𝐴𝑓𝛾^𝐴𝐵𝜕_𝐵𝜕_𝑢𝜑+𝑓Δ𝜕_𝑢𝜑−Δ𝜔, (3.9) which is identically satisfied when taking the previous relations into account.

One approach is to consider that (3.8) fixes 𝜔 in terms of 𝑓 and 𝑌, 𝜔=(1/2)𝜓+𝑓𝜕_𝑢𝜑−𝜕_𝑢𝑓. Consider Scri, the space ℐ with coordinates 𝑢,𝜁,𝜁 and metric 𝑑𝑠²_ℐ=0𝑑𝑢²+𝑒²𝜑𝑑𝜁𝑑𝜁. (3.10) The NU algebra is then defined as the commutator algebra of the vector fields 𝜕𝜉=𝑓𝜕𝑢+𝑌^𝐴𝜕𝜕𝑥^𝐴, (3.11) with 𝑓=𝑓(𝑢,𝑥^𝐴) arbitrary and 𝑌^𝐴(𝑥) conformal Killing vectors of a conformally flat metric in 2 dimensions, or equivalently, the algebra of conformal vector fields of the degenerate metric (3.10).

This is not the symmetry algebra of asymptotically flat spacetimes in the sense of NU however. Indeed, 𝜑 is arbitrary, it can for instance be considered as the finite ambiguity related to Penrose's conformal approach [9, 17, 18] to null infinity. One can then interpret 𝜑 as part of the background structure, or in other words, of the gauge fixing [8], and compute the asymptotic symmetries for a fixed choice of 𝜑, that is, 𝜔=0 in the formulae above, or ask the more general question of how the asymptotic symmetries depend on changes in 𝜑 by an arbitrary infinitesimal amount 𝜔. In both cases, one has to consider (3.8) as a differential equation for 𝑓. As we now show, the symmetry algebra will then be isomorphic to the trivially extended 𝔟𝔪𝔰₄ algebra by the abelian algebra of infinitesimal conformal rescalings, as it should, and as a consequence, the Poincaré algebra is embedded therein in a natural way. Furthermore, there is a natural realization of the asymptotic symmetry algebra on an asymptotically flat 4 dimensional bulk spacetime. Note also that, for 𝜔=0, (3.8) has been interpreted from the point of view of Penrose's conformal approach to null infinity in [12] following [19] and related to the preservation of null angles, which is the standard way [9, 17, 20, 21] to recover the BMS algebra from geometrical data on Scri.

The general solution for (3.8) reads 𝑓=𝑒𝜑𝑇+12^𝑢₀𝑑𝑢^′𝑒⁻𝜑𝜓,𝑇=𝑇𝜁,𝜁, (3.12) and the general solution to (3.1)–(3.3) defining the asymptotic symmetries is given by 𝜉^𝜌 as in (3.5) where 𝑍, 𝑌^𝐴, 𝑓 satisfy (3.6), (3.7), (3.12) with 𝜔 arbitrary. Asymptotic Killing vectors thus depend on 𝑌^𝐴,𝑇,𝜔 and the metric, 𝜉=𝜉[𝑌,𝑇,𝜔;𝑔].

For such metric-dependent vector fields, consider on the one hand the suitably modified Lie bracket taking the metric dependence of the spacetime vectors into account, 𝜉₁,𝜉_2𝑀=𝜉₁,𝜉₂−𝛿^𝑔_𝜉1𝜉₂+𝛿^𝑔_𝜉2𝜉₁, (3.13) where 𝛿^𝑔_𝜉1𝜉₂ denotes the variation in 𝜉₂ under the variation of the metric induced by 𝜉₁, 𝛿^𝑔_𝜉1𝑔_𝜇𝜈=ℒ_𝜉1𝑔_𝜇𝜈.

Consider on the other hand the extended 𝔟𝔪𝔰₄ algebra, that is, the semidirect sum of the algebra of conformal Killing vectors of the Riemann sphere with the abelian ideal of infinitesimal supertranslations, trivially extended by infinitesimal conformal rescalings of the conformally flat degenerate metric on Scri. More explicitly, the commutation relations are given by [(𝑌₁,𝑇₁,𝜔₁),(𝑌₂,𝑇₂,𝜔₂)]=(𝑌,𝑇,𝜔) where 𝑌^𝐴=𝑌^𝐵₁𝜕_𝐵𝑌^𝐴₂−𝑌^𝐵₂𝜕_𝐵𝑌^𝐴₁,𝑇=𝑌^𝐴₁𝜕_𝐴𝑇₂−𝑌^𝐴₂𝜕_𝐴𝑇₁+12𝑇₁𝜕_𝐴𝑌^𝐴₂−𝑇₂𝜕_𝐴𝑌^𝐴₁,𝜔=0. (3.14) It thus follows the following.

Theorem 3.1. The spacetime vectors 𝜉[𝑌,𝑇,𝜔;𝑔] realize the extended 𝔟𝔪𝔰₄ algebra in the modified Lie bracket, 𝜉𝑌₁,𝑇₁,𝜔₁;𝑔,𝜉𝑌₂,𝑇₂,𝜔₂;𝑔_𝑀=𝜉𝑌,𝑇,𝜔;𝑔, (3.15) in the bulk of an asymptotically flat spacetime in the sense of Newman and Unti.

Note in particular that for two different choices of the conformal factor 𝜑 which is held fixed, 𝜔=0, the asymptotic symmetry algebras are isomorphic to 𝔟𝔪𝔰₄, which is thus a gauge invariant statement.

Proof. The proof follows closely the one in [6] for the BMS gauge. In order to be self-contained we recall the different steps here. In a first stage, one shows that on ℐ, the vectors fields 𝜉[𝑌,𝑇,𝜔;𝛾] given in (3.11) with 𝑓 as in (3.12) realize the extended 𝔟𝔪𝔰₄ algebra in terms of the modified Lie bracket. Indeed, this is obvious for the 𝐴 components which do not depend on the metric so that the modified bracket reduces to the standard Lie bracket for these components. For the 𝑢 component, taking into account that 𝛿^𝑔_𝜉1𝑓₂=𝜔₁𝑓₂+12𝑒𝜑^𝑢₀𝑑𝑢^′𝑒⁻𝜑−𝜔₁𝜓₂−2𝜔₂+2𝑌^𝐴₂𝜕_𝐴𝜔₁, (3.16) we have [𝜉₁,𝜉₂]^𝑢_𝑀|_𝑢=0=𝑒𝜑|_𝑢=0𝑇. Direct computation then shows that 𝜕_𝑢([𝜉₁,𝜉₂]^𝑢_𝑀)=𝑓𝜕_𝑢𝜑+(1/2)𝐷_𝐴𝑌^𝐴 with 𝑓 given by (3.12) with 𝑇,𝑌,𝜔 replaced by their hatted counterparts, implying the result for the 𝑢 component.
For the spacetime vectors, direct computation gives [𝜉₁,𝜉₂]^𝑢_𝑀=[𝜉₁,𝜉₂]^𝑢_𝑀=𝑓. Using the defining property (3.4), one then finds that 𝜕_𝑟([𝜉₁,𝜉₂]^𝜌_𝑀)=𝑔^𝜌𝜈𝜕_𝜈𝑓. For the 𝐴 components the result then follows from the one on ℐ, lim_𝑟→∞[𝜉₁,𝜉₂]^𝐴_𝑀=𝑌^𝐴. This is due to the fact that 𝐼^𝐴 goes to zero at infinity, that the nonvanishing term at infinity does not involve the metric, and that the correction term in the bracket does not change the asymptotic behaviour. Finally, for the 𝑟 component, we still need to check that the 𝑟 independent component of [𝜉₁,𝜉₂]^𝑟_𝑀 is given by (1/2)Δ𝑓, which follows by direct computation.

For completeness, let us also stress here that, if one focuses on local properties and expands the conformal Killing vectors 𝑌^𝐴𝜕_𝐴 and the infinitesimal supertranslations 𝑇 in Laurent series, 𝑙_𝑛=−𝜁^𝑛+1𝜕,𝜕𝜁𝑙_𝑛=−𝜁^𝑛+1𝜕𝜕𝜁𝑇,𝑛∈ℤ,_𝑚,𝑛=𝜁^𝑚𝜁^𝑛,𝑚,𝑛∈ℤ, (3.17) the commutation relations for the complexified 𝔟𝔪𝔰₄ algebra read 𝑙_𝑚,𝑙_𝑛=(𝑚−𝑛)𝑙_𝑚+𝑛,𝑙_𝑚,𝑙_𝑛=(𝑚−𝑛)𝑙_𝑚+𝑛,𝑙_𝑚,𝑙_𝑛𝑙=0,_𝑙,𝑇_𝑚,𝑛=𝑙+12−𝑚𝑇_{𝑚+𝑙,𝑛},𝑙_𝑙,𝑇_𝑚,𝑛=𝑙+12−𝑛𝑇_{𝑚,𝑛+𝑙}. (3.18) The 𝔟𝔪𝔰₄ algebra contains as subalgebra the Poincaré algebra, which we identify with the algebra of exact Killing vectors of the Minkowski metric equipped with the standard Lie bracket. It is spanned by the generators 𝑙₋₁,𝑙₀,𝑙₁,𝑙₋₁,𝑙₀,𝑙₁,𝑇_0,0,𝑇_1,0,𝑇_0,1,𝑇_1,1. (3.19)

Nontrivial central extensions of the algebra (3.18) have been studied in [7]: the computation of 𝐻²(𝔟𝔪𝔰₄) reveals that there is only the standard ones for the Virasoro algebra extending the first two commutation relations.

4. Explicit Relation between the NU and the BMS Gauges

The definition of asymptotically flat spacetimes in the BMS approach [1, 2, 4] as reviewed in [5, 6] amounts to replacing 𝑔_𝑢𝑢=1/𝑔^𝑢𝑢=−1 by 𝑔_𝑢𝑢=1/𝑔^𝑢𝑢=−𝑒^2𝛽,𝛽=𝑂𝑟⁻² (4.1) in (2.1) and (2.2) while imposing the additional requirement that det𝑔_𝐴𝐵=𝑟⁴det𝛾_𝐴𝐵. (4.2)

Both definitions then differ just by a choice of radial coordinate. Indeed, replacing the radial coordinate by a function of the 4 coordinates preserves the zeros in (2.1) and (2.2) (see e.g., the discussion in [22]). Furthermore, for first nontrivial order in 𝑟, the determinant condition leads to the same restriction (2.5) as the choice of the origin of the affine parameter. It follows that the relation between the two radial coordinates does not involve constant terms and is of the form 𝑟^′=𝑟+𝑂𝑟⁻¹. (4.3) More explicitly, starting from the NU approach, BMS coordinates are obtained by defining the new radial coordinates as [23] 𝑟_BMS=det𝑔_𝐴𝐵det𝛾_𝐴𝐵^1/4. (4.4) Conversely, starting from the BMS approach with radial coordinate 𝑟, NU coordinates are obtained by changing the radial coordinate to 𝑟_𝑁=𝑟−^∞_𝑟𝑑𝑟^′𝑒^2𝛽−1. (4.5) These changes of coordinates only affect lower-order terms in the asymptotic expansion of the metric that plays no role in the definition of asymptotic symmetries and explains a posteriori why the asymptotic symmetry algebras in both approaches are isomorphic.

At this stage, the dynamics of the theory comes into play. The Einstein equations are solved order by order in 𝑟. In the first orders, there are integrations “constants” that appear as free data characterizing asymptotically flat solutions. We will now work out the explicit relation between these data in both approaches. The inverse metric in the BMS gauge (as discussed in [6]) is given by 𝑔^𝜇𝜈_BMS=0−𝑒^−2𝛽0−𝑒^−2𝛽−𝑒^−2𝛽𝑉𝑟−𝑒^−2𝛽𝑈^𝐵0−𝑒^−2𝛽𝑈^𝐴𝑔^𝐴𝐵,𝑔_𝐴𝐵=𝑟²𝛾_𝐴𝐵+𝑟𝐶_𝐴𝐵+14𝛾_𝐴𝐵𝐶^𝐶_𝐷𝐶^𝐷_𝐶+𝑂𝑟⁻¹. (4.6) For simplicity, we assume here that there is no trace-free part 𝐷_𝐴𝐵 at order 0 and that the conformal factor is time-independent, 𝜕_𝑢𝜑=0, in which case the news tensor is simply 𝑁_𝐴𝐵=𝜕_𝑢𝐶_𝐴𝐵 and 𝑓=𝑇+(1/2)𝑢𝜓 with 𝑇=𝑒𝜑𝑇. Writing 𝐶_𝜁𝜁=𝑒²𝜑𝑐,𝐶_𝜁𝜁=𝑒²𝜑𝑐,𝐶_𝜁𝜁=0, (4.7) we have 1𝛽=−4𝑟⁻²𝑐𝑐+𝑂𝑟⁻⁴,𝑈^𝜁2=−𝑟²𝑒⁻⁴𝜑𝜕𝑒²𝜑𝑐−23𝑟³𝑁^𝜁−4𝑒⁻⁴𝜑𝑐𝜕𝑒²𝜑𝑐+𝑂𝑟⁻⁴,𝑉𝑟=4𝑒⁻²𝜑𝜕𝜕𝜑+𝑟⁻¹2𝑀+𝑂𝑟⁻², (4.8) which implies in particular that 𝑟_𝑁𝑐=𝑟+𝑐𝑟2𝑟+𝑂⁻³. (4.9) The only consequence of Einstein's equations on the angular momentum and mass aspects 𝑁^𝜁=𝑁^𝜁(𝑢,𝜁,𝜁),𝑀=𝑀(𝑢,𝜁,𝜁) is the evolution equations 𝜕_𝑢1𝑀=−8𝑁^𝐴_𝐵𝑁^𝐵_𝐴+18Δ1𝑅+4𝐷_𝐴𝐷_𝐶𝑁^𝐶𝐴,𝜕_𝑢𝑁_𝐴=𝜕_𝐴1𝑀+4𝐶^𝐵_𝐴𝜕_𝐵1𝑅+𝜕16^𝐴𝑁^𝐵_𝐶𝐶^𝐶_𝐵−14𝐷_𝐴𝐶^𝐶_𝐵𝑁^𝐵_𝐶−14𝐷_𝐵𝐶^𝐵_𝐶𝑁^𝐶_𝐴−𝑁^𝐵_𝐶𝐶^𝐶_𝐴−14𝐷_𝐵𝐷^𝐵𝐷_𝐶𝐶^𝐶_𝐴−𝐷_𝐴𝐷_𝐶𝐶^𝐵𝐶. (4.10)(4.11)

Consider now the “eth” operators [24] defined here for a field 𝜂^𝑠 of spin weight 𝑠 according to the conventions of [25] through ð𝜂^𝑠=𝑃^1−𝑠𝜕𝑃^𝑠𝜂^𝑠,ð𝜂^𝑠=𝑃^1+𝑠𝜕𝑃^−𝑠𝜂^𝑠,𝑃=√2𝑒⁻𝜑, (4.12) where ð,ð raise, respectively, lower the spin weight by one unit and satisfy ð,ð𝜂^𝑠=𝑠2𝑅𝜂^𝑠. (4.13) The spin weights of the various quantities are summarized in Table 1. Note that the 𝑃 used here differs from the one used in [3], which we will denote by 𝑃_𝑁 below. It also no longer denotes the particular function (1/2)(1+𝜁𝜁), contrary to the notation used in [6, 16].

Table 1 

Spin and conformal weights.

In order to compare with the notation used in [3], we use 𝜁=𝑥³+𝑖𝑥⁴. With 𝑥^′𝛼=𝑢,𝑟_N,𝑥³,𝑥⁴ and 𝑥^𝜇=𝑢,𝑟,𝜁,𝜁, computing 𝑔^𝛼𝛽_N(𝑥^′)=−((𝜕𝑥^′𝛼/𝜕𝑥^𝜇)𝑔^𝜇𝜈_BMS(𝜕𝑥^′𝛽/𝜕𝑥^𝜈))(𝑥(𝑥^′)), where the overall minus sign takes the change of signature into account, then gives the following dictionary by comparing with [3]: 𝑃_𝑁=1√2𝑒⁻𝜑=12𝑃,∇=2𝜕,𝜇⁰=−𝑃²𝜕1𝜕ln𝑃=2Δ1𝜑=−4Ψ𝑅,₀₂+Ψ⁰₂=−2𝑀−𝜕_𝑢(𝑐𝑐),𝜎⁰=𝑐,𝜔⁰=ð𝜎⁰,Ψ⁰₁=−𝑃𝑁_𝜁−𝜎⁰ð𝜎⁰−34ð𝜎⁰𝜎⁰. (4.14) For convenience, let us also use Ψ⁰₃=−ð𝜎̇⁰−14ð𝑅,Ψ⁰₄=−𝜎̈⁰. (4.15) In these terms, Ψ̇⁰₃=ðΨ⁰₄,Ψ̇⁰₂=ðΨ⁰₃+𝜎⁰Ψ⁰₄,Ψ̇⁰₁=ðΨ⁰₂+2𝜎⁰Ψ⁰₃. (4.16) Indeed, the first equation holds by definition and the assumed time-independence of 𝑃. The evolution equation (4.10) is equivalent to the real part of the second equation. Taking into account the on-shell relation of the NU framework, Ψ⁰₂−Ψ⁰₂=ð²𝜎⁰−ð²𝜎⁰+𝜎⁰𝜎̇⁰−𝜎⁰𝜎̇‌⁰, (4.17) we find 𝑀=−Ψ⁰₂−𝜎⁰𝜎̇‌⁰+12ð²𝜎⁰−12ð²𝜎⁰, (4.18) in terms of which (4.10) is fully equivalent to the second equation of (4.16) and (4.11) is equivalent to the last equation of (4.16), in agreement with [3].

5. Transformation Laws of the NU Coefficients Characterizing Asymptotic Solutions

Let 𝒴=𝑃⁻¹𝑌 and 𝒴=𝑃⁻¹𝑌. The conformal Killing equations and the conformal factor then become ð𝒴=0=ð𝒴,𝜓=ð𝒴+ð𝒴. (5.1) It follows for instance that ðð𝒴=−𝑅2𝒴,ð²𝜓=ð³1𝒴−2𝒴ð𝑅,1ðð𝜓=−2ð𝑅𝒴+ð𝑅𝒴. (5.2) Using the notation 𝑆=(𝑌,𝑇,𝜔), we have −𝛿_𝑆𝛾_𝐴𝐵=2𝜔𝛾_𝐴𝐵 for the background metric and −𝛿_𝑆,ð𝜂^𝑠=−𝜔ð𝜂^𝑠+𝑠ð𝜔𝜂^𝑠,−𝛿_𝑆,ð𝜂^𝑠=−𝜔ð𝜂^𝑠−𝑠ð𝜔𝜂^𝑠. (5.3)

To work out the transformation properties of the NU coefficients characterizing asymptotic solution space, one needs to evaluate the subleading terms in ℒ_𝜉𝑔_𝑁^𝛼𝛽 on-shell. This can also be done by translating the results from the BMS gauge, which yields −𝛿_𝑆𝜎⁰=𝑓𝜕_𝑢+𝒴ð+𝒴3ð+21ð𝒴−2ð𝒴−𝜔𝜎⁰−ð²𝑓,−𝛿_𝑆𝜎̇⁰=𝑓𝜕_𝑢+𝒴ð+𝒴ð+2ð𝒴−2𝜔𝜎̇⁰−12ð²𝜓,−𝛿_𝑆Ψ⁰₄=𝑓𝜕_𝑢+𝒴ð+𝒴1ð+25ð𝒴+2ð𝒴−3𝜔Ψ⁰₄,−𝛿_𝑆Ψ⁰₃=𝑓𝜕_𝑢+𝒴ð+𝒴ð+ð𝒴+2ð𝒴−3𝜔Ψ⁰₃+ð𝑓Ψ⁰₄,−𝛿_𝑆Ψ⁰₂=𝑓𝜕_𝑢+𝒴ð+𝒴3ð+23ð𝒴+2ð𝒴−3𝜔Ψ⁰₂+2ð𝑓Ψ⁰₃,−𝛿_𝑆Ψ⁰₁=𝑓𝜕_𝑢+𝒴ð+𝒴ð+2ð𝒴+ð𝒴−3𝜔Ψ⁰₁+3ð𝑓Ψ⁰₂. (5.4)

Following for instance the terminology in [26] Section 3, but now for general infinitesimal transformations 𝜁^′=𝜁+𝜖𝑌(𝜁), 𝜁^′=𝜁+𝜖𝑌(𝜁) instead of those associated to linear fractional transformations on the sphere and also considering 𝜁 as the holomorphic coordinate instead of 𝜁, a field 𝜂 has spin weight 𝑠 and conformal weight 𝑤 if it transforms as −𝛿_𝑌,𝑌𝜂=𝑌^𝐴𝜕_𝐴+𝑠2𝜕−𝑤𝑌−𝜕𝑌2𝜓𝜂. (5.5) A tensor density of rank 𝑠⩾0 and weight 𝑛 transforms as −𝛿_𝑌,𝑌𝐴_𝜁⋯𝜁=𝑌^𝐴𝜕_𝐴+𝑠𝜕𝑌+𝑛𝜕𝑌+𝜕𝑌𝐴_𝜁⋯𝜁, (5.6) while for rank 𝑠⩽0 and weight 𝑛, we have −𝛿_𝑌,𝑌𝐴_𝜁⋯𝜁=𝑌^𝐴𝜕_𝐴−𝑠𝜕𝑌+𝑛𝜕𝑌+𝜕𝑌𝐴_𝜁⋯𝜁. (5.7) It then follows that a tensor density of weights (𝑠,𝑛) defines a field of weights (𝑠,−(2𝑛+|𝑠|)) and conversely, a field of weights (𝑠,𝑤) defines a tensor density of weights (𝑠,−(1/2)(𝑤+|𝑠|)). For 𝑠⩾0, this is done through 𝜂=𝐴_𝜁⋯𝜁𝑃^2𝑛+𝑠 and 𝐴_𝜁⋯𝜁=𝑃^𝑤𝜂. For 𝑠⩽0, we have 𝜂=𝐴_𝜁⋯𝜁𝑃^2𝑛−𝑠 and 𝐴_𝜁⋯𝜁=𝑃^𝑤𝜂. Note that complex conjugation gives rise to opposite spin weight and rank but leaves the conformal and density weights unchanged. Alternatively, (5.5) can be written as −𝛿_𝒴,𝒴𝜂=𝒴ð+𝒴ð+𝑠−𝑤2ð𝒴−𝑠+𝑤2ð𝒴𝜂. (5.8)

When focusing on 𝑇=0=𝜔 at the surface 𝑢=0 and on the homogeneous part of the transformations, this gives the weights summarized in Tables 1 and 2. These tables are extended to the Lie algebra elements, which are passive in all our computations, by writing [𝑌,𝑇]=−𝛿_𝑌,𝑌𝑇 and [𝑌,𝑌^′]^𝐴=−𝛿_𝑌,𝑌𝑌^′𝐴.

Table 2 

Rank and density weights.

6. Surface Charge Algebra

In this section, 𝜔=0 so that 𝑓=𝑇+(1/2)𝑢𝜓 and we use the notation 𝑠=(𝒴,𝒴,𝑇) for elements of the symmetry algebra, which is given in these terms by [𝑠₁,𝑠₂]=𝑠̂ where 𝒴=𝒴₁ð𝒴₂−(1⟷2),𝒴=𝒴₁ð𝒴₂−(),𝒴1⟷2𝑇=₁ð+𝒴₁ð𝑇₂−12𝜓₁𝑇₂−(1⟷2). (6.1) The translation of the charges, the nonintegrable piece due to the news, and the central charges computed in [16] is given here 𝑄_𝑠[𝒳]1=−𝑑8𝜋𝐺²Ω^𝜑𝑓Ψ⁰₂+𝜎⁰𝜎̇‌⁰+𝒴Ψ⁰₁+𝜎⁰ð𝜎⁰+12ð𝜎⁰𝜎⁰,Θ+c.c._𝑠[]=1𝛿𝒳,𝒳𝑑8𝜋𝐺²Ω^𝜑𝑓𝜎̇⁰𝛿𝜎⁰,𝐾+c.c._𝑠1,𝑠2[𝒳]=1𝑑8𝜋𝐺²Ω^𝜑14𝑓₁ð𝑓₂ð1𝑅+2𝜎⁰𝑓₁ð²𝜓₂−(1⟷2)+c.c.. (6.2) Note that one could also write the charges 𝑄_𝑠[𝒳] by allowing for the additional terms ((1/2)ð²𝜎⁰−(1/2)ð²𝜎⁰) in the first parenthesis since these terms were cancelled with the corresponding terms in the complex conjugate expression. Note also that not Ψ⁰₂ but only Ψ⁰₂+Ψ⁰₂ is free data on-shell because of the relation (4.17).

We recognize all the ingredients of the surface charges described in [27], which in turn have been related there to previous expressions in the literature and, in particular, to the twistorial approach of Penrose [28]. More precisely, up to conventions, 𝑄_0,0,𝑇 agrees with Geroch's linear supermomentum [8] 𝑄_𝑔𝑛+𝑄_𝑔𝑛, as given in (A1.12) of [27]. The angular (super-)momentum that we get is 𝑄_𝒴,0,01=−𝑑8𝜋𝐺²Ω^𝜑𝒴Ψ⁰₁+𝜎⁰ð𝜎⁰+12ð𝜎⁰𝜎⁰−𝑢2ðΨ⁰₂+Ψ⁰₂+𝜕_𝑢𝜎⁰𝜎⁰. (6.3) It differs from 𝑄_𝜂𝑐 given in (4) of [27] by the explicitly 𝑢-dependent term of the second line. It thus has a similar structure to Penrose's angular momentum as described in (11), (12), and (17a) of [27] in the sense that it also differs by a specific amount of linear supermomentum, but the amount is different and explicitly 𝑢-dependent, 𝑄_𝒴,0,0=𝑄^𝑢=0_𝒴,0,0+12𝑢𝑄_0,0,ð𝒴. (6.4)

The main result derived in [16] states that (i)if one is allowed to integrate by parts, 𝑑²Ω^𝜑ð𝜂⁻¹=0=𝑑²Ω^𝜑ð𝜂¹, (6.5) where 𝑑²Ω^𝜑=(2𝑑𝜁∧𝑑𝜁)/𝑖𝑃²,(ii)if one defines the “Dirac bracket” through 𝑄_𝑠1,𝑄_𝑠2^∗[𝒳]=−𝛿_𝑠2𝑄_𝑠1[𝒳]+Θ_𝑠2−𝛿_𝑠1𝒳,𝒳, (6.6) then the charges define a representation of the 𝔟𝔪𝔰₄ algebra, up to a field-dependent central extension, 𝑄_𝑠1,𝑄_𝑠2^∗=𝑄_{[𝑠1,𝑠2]}+𝐾_𝑠1,𝑠2, (6.7) where 𝐾_𝑠1,𝑠2 satisfies the generalized cocycle condition 𝐾_{[𝑠1,𝑠2],𝑠3}−𝛿_𝑠3𝐾_𝑠1,𝑠2+cyclic(1,2,3)=0. (6.8)The representation theorem contained in (6.7) and (6.8) can be verified directly in the present context by starting from (6.2), (4.17) and using the properties (4.13), (6.5) of ð, the evolution equations (4.16), the conformal Killing equations (5.1), the 𝔟𝔪𝔰₄ algebra (6.1), and the transformation laws (5.4).

Several remarks are in order as follows.(i)Integrations by parts are justified for regular functions on the sphere and thus for 𝔟𝔪𝔰^glob₄ and regular solutions. In the case of Laurent series more care is needed, see for example, [29]. We will address this question elsewhere.(ii)For the globally well-defined 𝔟𝔪𝔰^glob₄ algebra on the sphere, the central charge 𝐾_𝑠1,𝑠2 vanishes.(iii)The nonconservation of the charges follows by taking 𝑠₂=(0,0,1) and 𝑠₁=𝑠. Indeed, since (𝑑/𝑑𝑢)𝑄_𝑠=(𝜕/𝜕𝑢)𝑄_𝑠−𝛿_(0,0,1)𝑄_𝑠, the equality of the right hand sides of (6.6) and (6.7) gives 𝑑𝑄𝑑𝑢^𝑠1=−𝑑8𝜋𝐺²Ω^𝜑𝜎̇⁰−𝛿_𝑠𝜎⁰+14ð𝑓ð1𝑅+2𝜎⁰ð²𝜓+c.c.. (6.9) For 𝑠=(0,0,1), this gives the standard Bondi-Sachs mass loss formula, 𝑑𝑄𝑑𝑢^0,0,11=−𝑑8𝜋𝐺²Ω^𝜑𝜎̇⁰𝜎̇⁰+c.c.. (6.10) It also follows that the standard 𝔟𝔪𝔰^glob₄ charges are all conserved on the sphere in the absence of news.

To the best of our knowledge, except for the previous analysis in the BMS gauge, the above representation result does not exist elsewhere in the literature. A more detailed discussion of its implications, a detailed comparison with results in the literature as well as a self-contained derivation of the 𝔟𝔪𝔰₄ transformation laws in the context of the Newman-Penrose formalism will be given elsewhere.

Acknowledgments

The authors thank Cédric Troessaert for useful discussions. This work is supported in part by the Fund for Scientific Research-FNRS (Belgium), by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P6/11, by IISN-Belgium, by “Communauté française de Belgique—Actions de Recherche Concertées”, and by Fondecyt Projects no. 1085322 and no. 1090753.